Active Liquid Crystals in Confinement

Jérôme Hardoüin

Aquesta tesi doctoral està subjecta a la llicència Reconeixement 4.0. Espanya de Creative Commons.

Esta tesis doctoral está sujeta a la licencia Reconocimiento 4.0. España de Creative Commons.

This doctoral thesis is licensed under the Creative Commons Attribution 4.0. Spain License.

Active Liquid Crystals in Confinement

Programa de doctorat en Nanociències Autor/a: Jérôme Hardoüin Director/a: Francesc Sagués Tutor/a: Jordi Ignés

Active Liquid Crystals in Conﬁnement

Jérôme Hardoüin

October 28, 2019

Contents

Introduction1

Preface 7

1 Flow and Order: the physics of Active Liquid Crystals9 1.1 Active Matter...... 9 1.1.1 Zoology of Active Matter...... 9 1.1.2 Emergent Properties of far from equilibrium systems ...... 10 1.1.3 From the microscopic to the macroscopic: modelling active systems 11 1.2 Liquid Crystals ...... 17 1.2.1 Mesophases ...... 17 1.2.2 Phase transitions...... 18 1.2.3 Liquids with elasticity ...... 18 1.2.4 Director field and order parameter ...... 18 1.2.5 Elastic energy...... 19 1.2.6 Topological defects...... 19 1.2.7 Liquid crystals in confinement: defects and topology ...... 23 1.3 Flow and Order: the physics of Active Nematics...... 27 1.3.1 An out-of-equilibrium liquid crystal...... 28 1.4 Scientific Objectives...... 34 1.4.1 Influence of lateral confinement ...... 35 1.4.2 Influence of topology...... 36 1.4.3 Active nematics at a wall...... 37 1.4.4 Active nematics at curved interfaces...... 38

2 Materials and Methods 41 2.1 Active Nematics Synthesis...... 41 2.1.1 Active Gel Preparation...... 41 2.1.2 Basic sample design...... 44 2.2 Imaging and Analysis...... 46 2.2.1 Optical setups...... 46 2.2.2 Image processing ...... 48 2.2.3 Determining director and defects in active nematics...... 50 2.2.4 Characterization of the flow field ...... 52 2.3 Numerical methods (chapter3) ...... 55 2.4 Experimental techniques...... 57 2.4.1 Surface Microfluidics (Chapter3,4 and5)...... 57

i CONTENTS

2.4.2 Active emulsions: droplets, shells and ellipsoids ...... 60

3 Active nematics under lateral Confinement 69 3.1 State of the art ...... 69 3.2 Experimental setup: surface microfluidics...... 69 3.3 Computational Setup...... 70 3.4 Results and Discussion...... 70 3.4.1 A defect-free state: shear flow disrupted by instabilities ...... 72 3.4.2 The dancing state: a one-dimensional line of flow vortices ...... 74 3.5 Summary ...... 76

4 Eﬀect of Topology 83 4.1 Experimental Setup...... 83 4.2 Results...... 84 4.2.1 Transport and polarization in isolated annuli...... 84 4.2.2 Cross-talks between ﬂows and order in connected annuli ...... 87 4.2.3 Frustated topology with a genus 3 topology ...... 91 4.3 Discussion...... 92

5 Active Nematics at a Wall: Description and Control 99 5.1 Motivation: perturbations from the boundaries...... 99 5.2 Properties of an isolated negative defect at a wall ...... 100 5.2.1 Nucleation, Structure and Motility ...... 101 5.2.2 Forces ...... 105 5.2.3 Hydrodynamics of active nematics using BioFlow ...... 109 5.2.4 Conclusion...... 117 5.3 Collective dynamics of negative defects at a wall ...... 118 5.3.1 On the road to spatio-temporal chaos ...... 118 5.3.2 Describing Chaos: the Kuramoto-Sivashinski equation ...... 120 5.3.3 KSE and 1D active nematics...... 122 5.3.4 Statistical properties of localized structures ...... 124 5.3.5 Energy spectrum in Active Nematics ...... 130 5.3.6 Conclusion...... 139 5.4 Controlling Wall Defects...... 140 5.4.1 Effect of an indentation ...... 140 5.4.2 Effect of a ratchet pattern...... 142 5.4.3 Effect of wall curvature...... 148 5.4.4 Conclusion...... 149

6 Active Nematics and Curvature 155 6.1 Introduction...... 155 6.1.1 Geometrical frustration and topological charge ...... 155 6.1.2 Defects and local curvature ...... 159 6.1.3 Experimental active nematics on curved interfaces ...... 160 6.1.4 Motivation...... 160 6.2 Synthesis of smectic ellipsoids ...... 163 6.3 Active nematic emulsions...... 165 ii CONTENTS

6.4 Results...... 167 6.4.1 Solid body dynamics...... 168 6.4.2 Active nematic deformations...... 170 6.5 Summary ...... 174 6.6 Simulation Results ...... 176 6.7 Role the the viscous anisotropy ...... 176 6.8 Conclusion and perspectives...... 177

Closing remarks 183

Conclusion 191

Acknowledgments 195

Bibliography 197

iii

List of videos

3.1 Shear flow of active nematics confined in a 50 µm channel. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 0.15 fps. scale:0.73 µm/px...... 80 3.2 Shear state from the simulations. Defects are highlighted by the green circle (+1/2) and the blue triangle (-1/2). Channel width is 32 lattice sites. 80 3.3 Defect tracking in the shear state. Active nematics confined in a 80 µm channel. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 1 fps. scale:0.69 µm/px. scale bar 50µm...... 80 3.4 Dancing disclination state in a 120 µm channel. Positive (+1/2) defects are overlaid with green disks. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 0.6 fps. scale:0.56 µm/px...... 80 3.5 Dancing state from the simulations. Channel width is 60 lattice sites. 81 3.6 Switching state in active nematics confined in a 90 µm channel. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 0.3 fps. scale:0.79 µm/px...... 81 3.7 Switching state from the simulations. Defects are highlighted by the green circle (+1/2) and the blue triangle (-1/2). Channel width is 40 lattice sites...... 81 3.8 Self-collapsing state from the simulations. The bend instabilities grow and then collapse in on themselves. Channel width is 30 lattice sites. 81

4.1 Symmetry Breaking Confocal fluorescence video of active nematics confined in a 60 µm wide annulus. scale bar: 100 µm...... 96 4.2 Switching state Confocal fluorescence video of active nematics confined in a 110 µm wide annulus. scale bar: 100 µm...... 96 4.3 Turbulent state Confocal fluorescence video of active nematics confined in a 200 µm wide annulus. scale bar: 100 µm...... 97 4.4 Synchronisation in a Genus 2 Confocal fluorescence video of active nematics confined in a genus 2 handle-body, with an overlapping distance

of D/2R0 = 0.94. The width of each annulus is w = 80 µm. scale bar: 100 µm...... 97

v LIST OF VIDEOS

4.5 Frustration in a Genus 3 Confocal fluorescence video of active nematics confined in a genus 3 handle-body. The width of each annulus is w = 80 µm and the overlapping distance is D/2R0 = 0.83. scale bar: 100 µm...... 98 5.1 Steady regime of an isolated wall-defect in a disk. Fluorescence micrograph of active nematics confined in a disk of 170 µm radius. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the flow regime described in Fig. 5.2 (c)...... 152 5.2 Nucleation and merging regime of an isolated wall-defect in a disk. Fluorescence micrograph of active nematics confined in a disk of 170 µm radius. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the flow regime described in Fig. 5.2 (d)...... 152 5.3 Drifting regime of an isolated wall-defect in a disk. Fluorescence micrograph of active nematics confined in a disk of 170 µm radius. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the flow regime described in Fig. 5.2 (e)...... 152 5.4 Collective dynamics at a wall. Fluorescent micrograph of active nematics in the vicinity of a flat wall. The wall is located at the bottom of the image. Frame rate: 2 fps. Scale bar: 100 µm. This video corresponds to the analysis proposed in Fig. 5.24...... 153 5.5 Periodic dynamics in a small disk Time-lapse of the periodic oscillation of active nematics confined in a 130 µm corrugated disk. The indentation is located to the left side of the disk. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the analysis performed in Fig. 5.29...... 153 5.6 Effect of a ratchet pattern Fluorescence micrograph of active nematics facing a wall patterned with ratchets. The wavelength of the pattern, λ =200 µm is fixed for all the experiments. In this example, the height is h =100 µm. Frame rate: 2 fps. Scale bar: 100 µm...... 154 6.1 Bursting of smectic shells into ellipsoids. The shells are immersed in a solution of 2% Pluronic dissolved in water. The image is acquired by polarized-light microscopy. Frame rate: 5 fps. Scale bar: 100 µm. The time-lapse of Fig. 6.8 (a) is taken from this video...... 181 6.2 Active nematic emulsion. (left) Fluorescence micrograph showing the dynamics of smectic ellipsoids immersed in an active solution. Within a few hours, a dense ative nematic layer forms onto their surface. Frame rate: 2 fps. Scale bar: 100 µm (right) The same active nematic emulsion observed by polarized-light microscopy. The motion is slower, and more erratic, because the video was taken soon after mixing the ellipsoids with the active gel, and so we believe the active nematic layer was not entirely formed yet. Frame rate: 2 fps. Scale bar: 100 µm ...... 181 6.3 Solid-body dynamics and active deformations of an ellipsoid (left) Fluorescence micrograph showing the dynamics of an active ellipsoid. Frame rate: 2 fps. Scale bar: 100 µm. (right) The same video in the reference frame of the ellipsoid, showing the deformations of the active nematic layer. 182

vi Abstract

Living systems flow. What appears obvious from our daily observation of people, birds or insects remains surprisingly true at the smallest scale of life. Even at the earliest stages of embryonic development, the most elementary units of living systems, cell tissues, exhibit sustained currents. This perpetual movement is a signature of one of the fundamental properties of living systems - their ability to consume energy and transform it into directed motion. Living systems also cooperate. In the same way as fish swimming collectively form large scale structures to fool their predators, cells self-organize in tissues of increasingly complex shapes. Pattern formation in biology involves many processes from chemical signalling to hydrodynamics. Yet, the striking similarity between the flows and shapes adopted by collective systems at all scales of life motivated the development of a unifying theory, containing the minimal physical processes involved. This framework is called active soft matter. It refers to any system composed of self-driven units that consume and convert energy into directed motion. In some cases, the particles are so densely packed that they can be described as a continuous phase with long-range orientational order. This particular class has been termed active liquid crystals, of which cell tissues are the flagship illustration. These systems are characterized by a peculiar interplay between order and flows. The constant energy consumption drives them out of thermodynamic equilibrium. As a consequence, they are constantly deforming by sustained - and typically chaotic - flows. Reciprocally, the flow pattern directly depends on the local ordering of the particles.

Beyond the apparent chaos, this interplay between activity and order also confers to active liquid crystals a fascinating ability to adapt to the environments where they reside. In this work, we investigate the interplay between the geometry, the order and the flows of an active liquid crystal. Using novel micro-printing techniques, we develop versatile experimental setups that allow us to study how geometrical confinement tames the active flows and defect properties. We specifically investigate the effect of lateral confinement, topology, boundary roughness and Gaussian curvature.

Our experimental system is an in vitro mixture of cytoskeletal proteins, created in the laboratory of Z. Dogic from Brandeis University (MA, USA) in 2012. In brief, ATP- fuelled kinesin motor clusters crosslink and drive bundled microtubules, giving rise to an active network of bioﬁlaments that develops far from thermodynamic equilibrium. The

1 Introduction active gel can also self-assemble at soft interfaces, forming a quasi-2d active nematic liquid crystal, which features spontaneous turbulent-like ﬂows.

We report dramatic transformations of the spatio-temporal dynamics of active nematics. The so-called active turbulence reorganizes into vortex lattices, directed, or defect-free unidirectional flows. Topological defects, which determine the active flow behavior, are created and annihilated on the boundaries rather than in the bulk, and acquire a strong orientational order in narrow channels. Their nucleation is governed by an instability whose wavelength is effectively screened by the lateral confinement. Their density, spatial distribution, orientation, and velocities evade most of the laws derived for unconfined active nematics. The careful description of the co-evolving order and flow patterns away from active turbulence enables us, to some extent, to disentangle the way they interact. In addition, we relate the transition to ordered regimes to generic descriptions of spatio- temporal chaos in out-of-equilibrium fluids, in an effort to understand the physics of these complex systems through universal laws.

Dramatic transitions also occur in the case of closed interfaces i.e surfaces with no boundaries. By condensing the active nematic material at the surface of spherical vesicles, previous experimental achievements had demonstrated how the interplay between activity, topology and vesicle deformability could produce a myriad of ordered dynamical states including a tunable periodic state that oscillates between two defect configurations, and shape-changing vesicles with streaming filopodia-like protrusions. In the last part of the manuscript, we report an original example of spinning active nematic droplets. We condense an active nematic layer on the outer surface of oil droplets with an ellipsoidal shape. In this configuration, topology and Gaussian curvature contribute to the emergence of a chiral symmetry breaking in the active deformations. This chirality is transferred to the solid-body dynamics of the ellipsoids, which rotate with a surprisingly constant pulsation. These results demonstrate how the non-equilibrium dynamics of active materials could be converted into macroscopic engines.

Our result not only improve the theoretical understanding of active liquid crystals. We also demonstrate promising strategies to control the spatial organization and the active flows through geometrical confinement, which could contribute to the design of autonomous microfluidic systems performing complex tasks without any external input.

2 Resum

Fluxen els sistemes de vida. El que sembla evident a partir de la nostra observació diària de persones, aus o insectes segueix sent sorprenentment cert a la menor escala de la vida. Fins i tot a les primeres etapes del desenvolupament embrionari, les unitats més elementals dels sistemes vius, els teixits cel·lulars, presenten corrents sostinguts. Aquest moviment perpetu és una signatura d’una de les propietats fonamentals dels sistemes vius: la seva capacitat de consumir energia i transformar-la en moviment dirigit. Els sistemes de vida també cooperen. De la mateixa manera que els peixos que neden formen col·lectivament estructures a gran escala per enganyar els seus depredadors, les cèl·lules s’autoorganitzen en teixits de formes cada cop més complexes. La formació de patrons en biologia implica molts processos des de la senyalització química fins a la hidrodinàmica. No obstant això, la sorprenent similitud entre els fluxos i les formes adop- tades pels sistemes col·lectius a totes les escales de la vida va motivar el desenvolupament d’una teoria unificadora, que contenia els processos físics mínims implicats. Aquest marc s’anomena textit matèria suau activa. Es refereix a qualsevol sistema compost per unitats impulsades per si mateixes que consumeixen i converteixen l’energia en moviment dirigit. En alguns casos, les partícules estan tan densament envasades que es poden descriure com una fase continua amb ordre orientatiu de llarg abast. Aquesta classe particular s’ha anomenat textit cristalls líquids actius, dels quals els teixits cel·lulars són la il·lustració insígnia. Aquests sistemes es caracteritzen per tenir una peculiar interacció entre ordre i fluxos. El consum d’energia constant els allunya d’un equilibri termodinàmic. Com a conseqüència, aquests sistemes es modifiquen constantment per fluxos sostinguts i típica- ment caòtics. Recíprocament, el patró de flux depèn directament de les propietats locals de la distribució espacial. Més enllà del caos aparent, aquesta interacció entre l’activitat i l’ordre també confereix als cristalls líquids actius una capacitat fascinant d’adaptar-se als entorns on resideixen.

En aquest treball s’investiga la interacció entre la geometria, l’ordre i els fluxos d’un cristall líquid actiu. Amb noves tècniques de microimpressió, desenvolupem configuracions experimentals versàtils que ens permeten estudiar els noms de confinament geomètrics dels fluxos actius i les propietats defectuoses del sistema nemàtic actiu microtúbul / kinesina. Investiguem específicament l’efecte del confinament lateral, la topologia, la rugositat del límit i la curvatura gaussiana. El nostre sistema experimental és una barreja in vitro de proteïnes citoesquelètiques, creada al laboratori de Z. Dogic de la Universitat de Brandeis (MA, EUA) el 2012. En resum, els clústers de motors de kinesina alimentats per ATP es

3 Introduction reticulen i condueixen microtúbuls agrupats, generant un xarxa activa de bioﬁlaments que es desenvolupa lluny de l’equilibri termodinàmic. El gel actiu també es pot autoensamblar en interfícies suaus, formant un cristall líquid nemàtic actiu quasi-2d, que presenta ﬂuxos espontànis de tipus turbulent.

Els experiments es basen en noves estratègies microfluídiques per modelar les interfí- cies oli-aigua amb formes controlades, desenvolupades durant el doctorat. El primer s’anomena microfluídics de superfície. Permet dividir una interfície plana oli- aigua en diversos patrons de qualsevol mida i forma amb una resolució micromètrica. El principal avantatge és que les constriccions microfluídiques estan incrustades en un dis- positiu portàtil que es pot utilitzar en qualsevol sistema obert. A més, la graella confina específicament la interfície, deixant la majoria de les dues fases sense definir. És particu- larment útil en el cas de la nàmatica activa on la solució d’aigua actua com a amortidor, garantint que les concentracions d’elements actius siguin iguals en tots els patrons com- parats. El segon protocol, anomenat smectic ellipsoids, permet sintetitzar les emulsions de petroli en aigua de gotetes allargades sub-milimètriques. A aquestes escales, les heterogeneïtats de curvatura són molt desfavorables a causa de la tensió superficial. Aquí, les gotetes estan compostes per un cristall líquid escènic, alineat de manera que les forces elàstiques internes conserven la forma el·lipsoïdal durant hores. La tècnica consisteix en dispositius microfluídics de vidre utilitzats convencionalment per preparar dobles emulsions, adaptats a la literatura. L’últim mètode, anomenat active emulsions, és una prova del concepte que demostra la possibilitat d’incorporar gotes de petroli en aigua a una solució activa i condensar les nàu- tiques actives a la seva superfície exterior. Fins ara, els treballs experimentals sobre nèmics actius en interfícies esfèriques es preparaven amb el material actiu encapsulat dins de les gotetes. Aquesta configuració requereix adaptar la configuració microfluídica a les restric- cions del material actiu (biocompatibilitat, durada de l’activitat, etc.). Aquí, les emulsions es poden preparar amb tècniques microfluídiques convencionals i emmagatzemar-les abans del seu ús. A més, es pot renovar el material actiu al voltant de les gotetes, fet que aug- menta considerablement la vida útil de les gotetes actives.

Informem de transformacions dramàtiques de la dinàmica espaciotemporal de la nà- matica activa. Els resultats es classifiquen en dues categories: els efectes sobre l’ordre de la fase nemàtica activa i l’impacte en els patrons de flux. La varietat de la dinàmica espaciotemporal observada només dóna el sabor de la fascinant adaptabilitat dels cristalls líquids actius a limitacions geomètriques. La població de defectes topològics experimenta transformacions dramàtiques. La seva densitat, distribució espacial, orientació i velocitats eviten la majoria de les lleis derivades per a una nèmica activa no definida. En les condicions més extremes de confinament lateral, els defectes desapareixen temporalment. De la mateixa manera, els fluxos transiten cap a patrons extremadament ordenats com ara xarxes de vòrtex, cisallament i fluxos dirigits. Sens dubte, aquestes transformacions espacials i dinàmiques estan relacionades. No obstant, descriure-les de manera independent no és suficient per entendre com sorgeixen els patrons espaciotemporals.

L’anomenada turbulència activa es reorganitza en ﬂuxos de vòrtex, ﬂuxos unidirec-

4 Introduction cionals dirigits o sense defectes. Els defectes topològics, que determinen el comportament del flux actiu, es creen i aniquilen als límits més que no a la massa, i adquireixen un fort ordre orientatiu en canals estrets. La seva nucleació es regeix per una inestabilitat la longitud d’ona de la qual és controlada efectivament pel confinament lateral. La seva densitat, distribució espacial, orientació i velocitats eviten la majoria de les lleis derivades per a una nèmica activa sense definir. Una acurada descripció de l’ordre i dels patrons de flux que evolucionen lluny de les turbulències actives ens permet, fins a cert punt, desvincular la forma en què interactuen. A més, relacionem la transició a règims ordenats a descripcions genèriques del caos espaciotemporal en fluids fora d’equilibri, en un esforç per comprendre la física d’aquests sistemes complexos a la llum de les lleis universals.

Les transicions dramàtiques també es produeixen en el cas d’interfícies tancades textit i.e superfícies sense límits. En condensar el material nemàtic actiu a la superfície de les vesícules esfèriques, els èxits experimentals anteriors havien demostrat com la inter- acció entre activitat, topologia i deformació de les vesícules podria produir una infinitat d’estats dinàmics ordenats incloent un estat periòdic ajustable que oscil·la entre dues configuracions de defectes i la forma. -cambic vesícules amb protuberàncies similars a la filopodia. A la darrera part del manuscrit, es presenta un exemple original de gota de gotes nemà- tiques actives. Condensem una capa nemàtica activa a la superfície exterior de gotes d’oli amb forma el·lipsoïdal. En aquesta configuració, la topologia i la curvatura gaussiana contribueixen a l’aparició d’una simetria quiral que trenca les deformacions actives. Aque- sta quiralitat es transfereix a la dinàmica del cos sòlid dels el·lipsoides, que giren amb una pulsació sorprenentment constant. Aquests resultats demostren com la dinàmica de no equilibri dels materials actius es pot convertir en motors macroscòpics.

El nostre resultat no només millora la comprensió teòrica dels cristalls líquids actius. També demostrem estratègies prometedores per controlar l’organització espacial i els fluxos actius mitjançant confinament geomètric, que podrien contribuir al disseny de sistemes microfluídics autònoms que realitzen tasques complexes sense cap entrada externa.

Preface

This dissertation is the result the three years I spent as a Ph.D. student in the Self- Organized Complexity and Self-Assembled Materials group (SOC & SAM) in the depart- ment of Materials Science and Physical Chemistry and in the Institute of Nanotechnology (IN2UB) of the University of Barcelona, from October 2016 to October 2019. The research has been conducted under the supervision of Jordi Ignés-Mullol and Francesc Sagués. The experiments were performed in collaboration with the team of Teresa Lopez-Leon from the group Eﬀets Collectifs et Matière Molle (EC2M) in the Gulliver Institute, at the Ecole Supérieure de Physique et Chimie Industrielle (ESPCI) in Paris, where I spent four months from September 2017 to January 2018, and four additional months between from September 2018 and January 2019. The present introduction gives the outline of the manuscript, and acknowledges the collaborators involved in each project.

In the first chapter, we provide the theoretical background and state of the art necessary to introduce the scientific context of this dissertation. We first present the paradigms of Active matter and Liquid crystals. Both are used as building blocks to understand the physics of Active liquid crystals. We also provide a short state of the art specific to our experimental system. Finally, we expose the scientific objectives of this dissertation.

The second chapter is devoted to the Materials and Methods. We detail the basic pro- tocols to prepare the active solution, and the analysis tools - adapted from the literature, or custom - used in the diﬀerent projects. The expression of kinesin was conducted at the BioNMR group at the University of Barcelona with the help of Irrem-Laareb Mohammad Jabeen and Berta Martinez. The image processing tools mostly rely on techniques developed by Perry Ellis and Alberto Fernandez Nieves from Georgia Tech University. Custom methods were developed with occasional help from Pau Guillamat and Berta Martinez Prat, both graduate students from SOC & SAM. We also present theoretical and computational frameworks to simulate active nematics. Finally, we present the experimental techniques developed during the PhD. The surface microﬂuidics, used in chapters3 to 5, would not have existed without the help of Justine Laurent, a research engineer at Laboratoire de Physique et Mécanique des Milieux hétérogènes (PMMH), CNRS, ESPCI Paris. The synthesis of smectic ellipsoids was developed in close collaboration with Dae Seok Kim, a postdoc at the Gulliver Institue.

7 Introduction

In the third chapter, we present the experimental results on the behaviour of active nematics under lateral conﬁnement. The experiments were performed at SOC & SAM and the samples were analyzed with a laser-scanning confocal microscope in the Unitat de Microscòpia Òptica Avançada (UMOA) in the Centres Cientíﬁcs i Tecnològics of the University of Barcelona (CCiTUB), with the technical support of Manel Bosch. The results are compared with simulations performed by Rian Hugues, a graduate student at the Rudolf Peierls Centre for Theoretical Physics in the Clarenton laboratory at the Uni- versity of Oxford, under the supervision of Amin Doostmohammadi and Julia M Yeomans.

In chapter four, we study the eﬀect of the topology on the conﬁning geometry. The experiments were performed both at SOC & SAM and EC2M.

In chapter five we present the work on the dynamics of active nematics close to a boundary. The first part focuses on the qualitative description of the observed phenoma, and the second part proposes strategies to control the flows. The experiments were exclusively conducted at EC2M. The qualitative description of the spatio-temporal dynamics was inspired by helpful discussions with Hugues Chaté and Olivier Dauchot. Some of the experiments on control were performed by Claire Doré, a master student at EC2M.

In chapter six, we explore the role of Gaussian curvature, and describe the physics of an active nematic emulsion composed of ellipsoidal droplets. The experiments presented in this dissertation were performed in collaboration with Dae Seok Kim. The project is currently continued by Martina Clairand, a graduate student at EC2M.

This work was funded by the European Union’s Horizon 2020 research and innova- tion program under grant agreement no. 674979-NANOTRANS. We are indebted to the Brandeis University Materials Research Science and Engineering Centers (MRSEC) Biosynthesis facility for providing the tubulin.

8 Chapter 1 Flow and Order: the physics of Active Liquid Crystals

The experimental work presented in this dissertation aims at understanding and controlling the spatio-temporal dynamics of a soft condensed system referred to as active nematics. These materials can be defined by two distinctive features. First, they are composed of a suspension of motile units that are able to consume energy locally and convert into directed motion. As such, they belong to the broader class of active matter, that we will introduce in the first place. Second, active liquid crystals are ordered phases. The spatial distribution and orientation of the elementary units obey the laws of passive liquid crystals, a class of soft matter systems whose characteristics will be described in details. Even so, despite these straightforward analogies, active liquid crystals display a wide range of spatio-temporal dynamics whose theoretical understanding and experimental control must not only include the framework of liquid crystals and active systems, but also account for the extreme sensitivity to external constraints that can be of many types: nature of the solvent, frictional forces, geometrical confinement, etc. The last part of this introduction will therefore be dedicated to the state of the art of active nematic experiments, and present the objectives of the project.

1.1 Active Matter

1.1.1 Zoology of Active Matter

As said in the preamble, the unifying characteristic of active matter systems is that they are composed of self-driven units, each capable of converting stored or ambient free energy into directed motion [1,2]. This deﬁnition encompasses a whole zoology of experimental realizations, starting from living systems.

In Nature Living systems flow. This property is ubiquitous at all scales of life (Fig. 1.1), from human crowds [3–5], terrestrial , aerial [6,7] and aquatic [8–11] flocks of animals, to insect colonies [12,13], cell layers and bacterial suspensions [14]. Even the flows induced by the cystoskeletal elements within the cells may be referred to as active matter [15–18].

9 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.1 – Active matter in Nature.

Despite the huge disparity in size, all these systems do have common features. They are all composed of propelled units that are similar in shape and uniform in size. Usually, they are elongated objects, and move along the direction given by their shape anisotropy. They have rather uniform absolute velocities, although they are allowed to change direction. Finally, they interact with their neighbours, a property that tends to align their direction of motion. However, living systems are also infinitely more complex than this. They are intelligent, differentiable, hungry, emotional... All these uncontrollable parameters are annoying for an active matter physicist. Therefore, several artificial active systems have emerged in the lab over the years.

In the lab Notable realizations include suspensions of vibrated rods [19,20] or disks [21], colloidal or nanoscale particles propelled through a ﬂuid by catalytic activity at their surface [22–25], and collections of robots [26]. For a closer connection with biology, in vitro active systems have also been developed, by growing bacterial colonies [27] or extracting the basic constituents of the cytoskeleton [28,29]. The common goal is to design systems containing the minimum features that lead to collective motion, and having access to control parameters to understand, and control, their emerging properties. The diversity of experimental systems is the measure of the variety of intriguing behaviours observed in active matter. Some of them are presented in the next section.

1.1.2 Emergent Properties of far from equilibrium systems Pattern Formation Patterns refer here to the spontaneous formation of hierarchical structures in space, that can be associated to coherent flows. The collection of motile units of an active matter system, in some cases, self-organizes into highly ordered structures at scales that are much larger than their individual size. Some examples are displayed in Fig. 1.2. Polar patterns are associated to directed transport. The particles align with each other and move in the same direction. They include spirals and travelling bands. Spirals in a fish flock and in a solution of cytoskeletal filaments are displayed in Fig. 1.2 (a) and (b). Nematic patterns are composed of large scale structures inside which particles are aligned but propel indifferently back and forth of the alignment directions. Typical realizations include a trail of working ants (Fig. 1.2 (c)) or active mixtures of cytoskeletal filaments (Fig. 1.2 (d)) Clustering effects may also occur [30], such as motility-induced phase separation. Starting from an isotropic distribution, particles aggregate to form amorphous clusters or crystals (Fig. 1.2 (e) and (f)). Liquid crystal phases are also observed. These patterns have the structure of a passive liquid crystal, but are continuously moving. Such

10 1.1. Active Matter conﬁgurations have been oberved in cell and bacterial colonies (Fig. 1.2 (g)) of dense cytoskeletal suspensions (Fig. 1.2 (h)).

Negative Viscosities The rheological properties of an active suspension under shear depends significantly on the particles concentration. Lopez et al. [36] have shown that a suspension of particularly active bacteria displays a superfluidlike transition where the viscous resistance to shear vanishes, thus showing that, macroscopically, the activity of the bacteria organized by shear is able to fully overcome the hydrodynamic dissipative effects.

Active Stiﬀening Unlike simple polymer gels, many biological materials including blood vessels stiﬀen as they are strained, thereby preventing large deformations that could threaten tissue integrity. The molecular structures and design principles responsible for this non-linear elasticity are still poorly understood [37–43].

Active Solids Fire ants link their bodies to form aggregations, forming variable structures that can ﬂow, drip, or withstand applied load. Tennenbaum et al. have shown that ant aggregates form a viscoelastic material whose properties strongly depend on ant density and on the applied loads [44,45].

All these intriguing eﬀects bear witness to the faculty of active matter systems to operate collectively, forming macroscopic materials. Although it appears tempting to infer the features of these new materials from their microscopic description, it is also very challenging. In the next section, we will brieﬂy describe how simple interactions between elementary active particles can explain collective behaviours. We will also present the limits of this approach, by giving examples of how sensitive these systems are to microscopic details and external constraints.

1.1.3 From the microscopic to the macroscopic: modelling active systems A. Microscopic interactions Active matter systems are composed of elementary self-driven units that can be of diﬀerent types. They can be segregated in two classes: self-propelled particles, and active gels.

Self Propelled particles encompasses most of the natural active matter systems one can think of: a fish, an ant, a cell, a bacteria are self-propelled particles. The elementary source of motion comes from an internal propulsion mechanism. They are characterized by three main features, summarized in Fig. 1.3. Their shape: isotropic (sphere or disk), anisotropic (elongated), or deformable (cells). The type of propulsion, polar or nematic, tells whether the particles move in directed motion, or randomly back and forth with respect to their polarity. A fish is a polar particle. On the contrary, some bacteria perform a run-and-tumble motion: every now and then, their flagella stop and reorient in a random direction. As such, this kind of particle cannot perform a sustained directed motion. Finally, the interactions between two particles can be of different type. If, when

11 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.2 – Pattern formation in active matter Polar patterns (a) Spiral in a school of fish. From [31] (b) Spiral in an in vitro actin-myosin cytoskeletal construct. The picture is taken from the website of A. Bausch lab, and related to article [32]. Nematic patterns (c) Trail of ants [31] (d) Nematic filaments in a microtubule active gel. scale bar: 80 µm. Adapted from [33]. Clustering (e) Crystallization of self-propelled hard discs [34]. (f) Motility-induced phase separation of colloids scale bar:100 µm. Adapted from [35]. Active liquid crystals (g) Liquid-crystalline order in a myxobacterial flock [2] (h) Microtubule- based active nematics. scale bar: 100 µm

12 1.1. Active Matter

Figure 1.3 – Microscopic properties of active matter systems (left) The class of self-propelled particles. Active particles are defined by their shape, their propulsion type, and the behaviour in pair collisions. (right) Microscopic features of active gels. This class of active system is composed of passive filaments driven by molecular motors. The ± signs indicate the polarity of the filaments with respect to the motor processivity.

they collide, their respective velocity vectors tend to align, they are said to have polar interactions. On the other hand, if their velocities indiscriminately align and anti-align, they are said to have nematic interactions.

Active Gels are a class of active matter systems aimed at describing the dynamics of cytoskeletal networks [46]. They are composed of a network of cross-linked filaments subjected to the action of energy-transducing molecular motors, as shown in Fig. 1.3. Note that the filaments have a polarity in terms of the displacement of molecular motors. Motors moving towards the positive end of the filaments are termed anterograde motors. Motors moving towards the negative end are termed retrograde motors. They are represented in red in Fig. 1.3. The elementary source of motion is a sliding between two filaments induced by the dimeric motors. The interaction is termed extensile if the motors tends to make the filaments slide away from each other, and contractile if they bring them closer. Generally, anterograde motors (ex: kinesins) are susceptible to generate extensile stresses, while retrograde motors (ex: myosin) generate contractile stresses. In practice, the distinction is very subtle and will be detailed later. Another important property is the alignment between filaments. Filaments whose ends tend to aggregate as a function of their polarity are said to form a polar pattern. On the contrary, filaments who align between each other regardless of their polarity are said to form a nematic pattern. Note that this property is not intrinsic to the filament interactions, because these objects are passive. As such, the pattern is already an emerging property that depends on external

13 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.4 – Density-dependent non-linear elasticity in a biological gel Elastic modulus vs applied strain in a network of fibrin protofilaments as a function of the filament concentration c. A: c =0.5 mg · mL−1. B: c =1 mg · mL−1. C: c =2 mg · mL−1. D: c =4.5 mg · mL−1. Adapted from [38]. The non-linear behaviour is only observed at high concentrations.

constraints (type of motor, solvent composition). Our work is focused of the study of a microtubule-based active gel referred to as active nematics [33]. Thus, we will focus the rest of the introduction on active matter systems to the class of active gels. Simple theoretical models based on this microscopic description have successfully reproduced some of the emerging phenomena in active matter. The most classical one is the Vicsek model [47–49], which reproduced the transition to net transport through a spontaneous symmetry breaking of the rotational symmetry in the case of directed particles. In active gels, unifying theories have emerged [46], but they do not rely on a microscopic description only. As a matter of fact, many other parameters may strongly inﬂuence the macroscopic behaviour of an active gel, some of which are detailed in the next section.

B. Coarse-graining aspects Starting from the knowledge we have of the elementary source of motion, the emerging behaviour is inﬂuenced by a large variety of parameters, ranging from molecular details to macroscopic eﬀects.

Particle density is the obvious parameter to start with. The non-linear elastic response reported in the strain stiﬀening of biopolymer networks only triggers at high ﬁlament density, as shown in Fig. 1.4.

Depletion interactions Actually, the absolute density may, in some cases, not be the relevant parameter. Density can be naturally heterogeneous due to attractive forces between ﬁlaments. Two ﬁlaments may aggregate through a soft interaction called depletion [50–55]. The mechanism can be understood as follows. Let us consider a suspension

14 1.1. Active Matter

Figure 1.5 – Depletion induced coexisting ordered states in active matter systems. Adapted from [57]. (a) Schematic of the actomyosin motility assay. PEG acts as a depletion agent. (b) Illustration of different filament collision geometries with an incoming angle θin (c) Corresponding binary collision curves. Whereas strong polar or nematic collision rules lead to full alignment or anti-alignment, weak collisions cause a gradual alignment and may exhibit both polar or nematic features. (d) Polar filament clusters formed in the absence of PEG. (e) A large network of lanes formed at a concentration of PEG of 3%. The image is composed of an overlay covering a period of 100s to show that the pattern is frozen and stable. Filaments move along the lane contours in opposite directions.

of rod-like particles (filaments) surrounded by smaller non-absorbing polymer coils, that 1 we represent as spheres of radius Rg. Rg is called the radius of gyration . To maximize the entropy, depletant molecules occupy all the free volume of the solution. Even so, there exists a cylindrical volume surrounding each filament, called excluded volume, that the center of mass of the depletants cannot access by steric repulsions. If the filaments are far apart, the total excluded volume is the sum of those surrounding each rod. On the other hand, if the filaments aggregate, the excluded volumes overlap, which maximizes the free volume available for the depletants. As there are much more depletant molecules than filament, the total entropy is maximized when filaments are packed. To understand how entropic forces may influence the collective behaviour of an active gel, let us consider the case of an actin-myosin motility assay [57]. Actin filaments (ubiquitous in the cell cortex) are placed onto a glass substrate where myosin motors have been attached, as shown in Fig. 1.5 (a). A depletant polymer coil called poly-ethylene glycol (PEG) is also added to the solution. Huber et al. have shown that the statistics of collisions between actin filaments are significantly modified by the presence of depletants. If depletion force is low, filaments tends to orient in a polar fashion, resulting in global polar patterns such as traveling bands displayed in Fig. 1.5 (d). On the other hand, if the PEG concentration is increased, filaments will be more likely to orient nematically after a collision, leading to nematic patterns such as the trails of ants pictured in Fig. 1.5 (e). Although the patterns are dramatically different, we can see that the statistical distribution of filament collisions is only slightly modified, as shown in Fig. 1.5 (c).

1 The square radius of gyration (Rg) is the average squared distance of the elements of a polymer coil from its center of gravity [56]

15 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Many-body microscopic interactions At high densities, the microscopic dynamics can no longer be described as a collection of pair collisions between active particles. As a matter of fact, Suzuki et al. [32] have shown that the polar patterns observed in the actomyosin motility assays presented above required the occurrence of non-binary collisions. To go further, the emerging patterns in active gels are extremely sensitive to microscopic details of its elementary units, that are not easily accountable for in theoretical models.

Dissipation must be considered as well. Active gels are not isolated and the energy constantly injected in the system may or may not be dissipated by various physical phenomena. Systems where momentum is not conserved are called dry systems. This class includes bacteria gliding on a surface, animal herds on land or vibrated grains on a plate, where momentum is damped by friction interactions with the substrate. In this case, the only conserved quantity is the number of particles and the associated hydrodynamic field is the local density of active units. The other class is called wet systems, and refers to cases where momentum is conserved. By extension, it refers to active systems embedded in a solvent where hydrodynamic interactions are not negligible. Strictly speaking, momentum is not conserved because it is damped by the viscosity of the bulk medium. Active gels are considered to be wet systems. For the sake of simplicity, we will not detail the theoretical models developed for these two classes of active systems. The essential point is that dissipation, either with a substrate or with the surrounding solvent, has a strong influence on the emerging collective behaviour. Previous work in our laboratory has demonstrated that the spatio-temporal patterns developed by a microtubule-based active liquid crystal could be drastically modified (and controlled) by tuning the properties of the solvent [58–60].

Boundaries. One of the most significant, and often unappreciated, features of active systems is their capability to adapt to the environment where they reside. For example, human cancer cells switch between distinct invasion modes when they encounter constrictions in the crowded environment of stroma [61], and the growth of bacterial biofilms can be directed by their surroundings [62]. Moreover, geometrical confinement tends to control active flows, replacing the bulk chaotic flow state often termed active turbulence, by more regular flow configurations. Understanding the subtleties of how this occurs will have relevance to possible future applications of active materials in microfluidics and self assembly, and in assessing the relevance of the concepts of active matter in the description of biological systems.

The scales involved in the emerging properties of dense active matter systems are so large compared to the elementary units of motion that they almost seem decorrelated, as much as describing the mechanical response of a material to an applied load from the interactions between its constitutive atoms is tedious. In addition, all the effects described above, many of which involve non-linear couplings, may strongly affect the resulting material properties. Overall, a bottom-up construction of the properties of active matter systems seems far from reach. Conversely, some of the active materials, either by their mechanical response, structure, or flow patterns, bear striking resemblance to other systems that are well described by long lasting theoretical models based on equilibrium physics. Hence, growing efforts have been made to describe active systems within the

16 1.2. Liquid Crystals framework of material science, perturbed by additional "active" terms. This is the case of active nematics, whose structure, depicted in Fig. 1.2 (h) is strikingly reminiscent of a passive liquid crystal phase. In the next section, we will provide a brief introduction to passive liquid crystals, before extending the theory to their active analogue.

1.2 Liquid Crystals

Self-assembly is not an exclusive property of active matter systems. The ﬁeld of soft matter physics encompasses many types of "loosely bound" systems, where the mechanical energies at play (deformations, translations) are comparable to kBT , the thermal energy. In such systems, small mechanical stresses are enough to trigger signiﬁcant mesoscopic deformations, and small temperature changes are able to trigger phases transitions [63]. Most of these systems have the ability to self-organize. Among them, liquid crystals (LCs) [64] are composed of anisotropic elementary units, referred to as mesogens.

1.2.1 Mesophases

Depending on the specificities of the mesogen interactions and on external constraints, these liquids self-organize in a variety of ordered phases, called mesophases. The most trivial one is called the isotropic phase, represented in Fig. 1.6 (a). In this configuration, both the orientation and the location of the mesogens are free to explore all possible configurations, as for the case of molecules in a liquid. If the interactions between mesogens become more prominent, the orientation of a particle may be dependent on the orientation of its neighbours. In the nematic phase, represented in Fig. 1.6 (b), the mesogens locally align along the same direction characterized by a unit vector n referred to as the director. Despite this alignment anisotropy, the center of mass of each particle is however not constrained - particles are still able to freely move within the sample, as in conventional liquids. In other words, the mesogens have an orientational order but no long-range positional order. If the interaction between mesogens become even more prominent, partial positional order may be observed. This is the case for the smectic phase. In this configuration, particles have a strong tendency to arrange side-by-side. As a consequence, they spontaneously assemble in layers, as shown in Fig. 1.6 (c). The positional order is said to be partial, because unlike in a crystal where the location of atoms is fixed, the mesogens are still allowed to flow like a liquid within the layer. However, displacements perpendicular to the layers are highly unfavourable. Smectics are further classified into a wide variety of sub-phases. The simplest ones are termed smectic A and smectic C, depending whether the nematogens are respectively oriented perpendicular or tilted with respect to the plane of the layers. Nematics and smectics are the two main mesophases2. The molecular structure of a mesogen prescribes the nature of the accessible phases. Some molecules can only form an isotropic or a nematic phase, while others will adopt either the nematic or the smectic phase depending on the conditions.

2Even so, there exists a variety of additional phases that are not relevant to the purpose of this dissertation.

17 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.6 – Canonical mesophases in a liquid crystal (a) Isotropic phase. The mesogens are randomly oriented and uniformly distributed as in a classical liquid. (b) Nematic phase. The mesogens are uniformly distributed, but their orientations are, on average, aligned along a common director n. (c) Smectic phase. The mesogens are aligned and arranged side-by-side in layers.

1.2.2 Phase transitions Usually, liquid crystals are classified in two categories: Lyotropic liquid crystals (LLCs) are controlled with temperature and with the concentration of mesogens. The higher the concentration, the more ordered the mesophases. They consist, for instance, of amphiphilic molecules dissolved in an aqueous solvent. Usually, the mesogen is not the molecule itself, but rather self-assembled molecular aggregates. Many biopolymers such as DNA or cellulose display a lyotropic behaviour at high concentrations. Thermotropic liquid crystals (TLCs) are controlled with temperature only. The higher the temperature, the more disordered the phase. The mesogens are usually composed of an organic molecule featuring internal frustration, combining polar and rigid with apolar and flexible parts, often giving them an elongated shape. with an elongated shape. The most studied type of TLCs belongs to the family of alkyl-cyanobiphenyls. These molecules are composed of an aromatic core terminated by a cyano group, and prolonged by an aliphatic chain. For this family of compounds, the type of mesophase accessible to a given mesogen depends on the length of the aliphatic chain. In the lab, they are mostly used in the form of two LCs, namely 40-pentyl-4-cyanobiphenyl (5CB) and 40-octyl-4-cyanobiphenyl (8CB). Although these two molecules only differ by the length of the aliphatic group, their phase diagrams are significantly different. 5CB displays a nematic phase between 18◦C and 35◦C, before turning isotropic at higher temperatures. Conversely, 8CB shows both nematic (33.4 – 40.4◦C) and smectic A phases (21.4 – 33.4◦C). A more detailed description of these compounds is available in the methods section (2.1.1).

1.2.3 Liquids with elasticity 1.2.4 Director field and order parameter In Fig. 1.6 (b), we can notice that the mesogens are not perfectly aligned with the main orientation: n, called the director field, only defines the average orientation, obtained on a mesoscopic volume. It has a unit norm, regardless of how well the mesogens are aligned.

18 1.2. Liquid Crystals

Therefore, we need another parameter to quantify the alignment. We will first derive this quantity in a 2D system. Consider a collection of particles indexed by α = 1, 2...N, laying on a plane. Let u(α) be the orientation of each particle. Because the mesogens (α) head and tail are not differentiable, we necessarily have hu iα = 0, where h·iα defines an ensemble average over all the particles. To quantify the order we define a symmetric traceless rank-2 tensor in the following form:

1 Q = u(α) u(α) 1 (1.1) h ⊗ − 2 i where ⊗ is the outer product. Q is called the nematic order parameter. In order to ﬁnd the director ﬁeld n, we need to diagonalise Q. Doing so, one can show that Q is given by the following expression [64]:

1 Q = (n n 1) (1.2) S ⊗ − 2

2 (α) where S = h3 cos θ − 1iα, and θ = arccos(u · n). S is called the scalar order parameter. We easily verify that S is bounded between 0 and 1. In the ideal case of a perfect nematic alignment (u(α) = n), we get θ = arccos(u(α) · n) = 0 and S = 1. Conversely, if the nematogens are in isotropic phase, we obtain S = 0. S is therefore a good measure of the alignment.

1.2.5 Elastic energy The tendency for the nematogens to a align has so far been described with a microscopic approach. The macroscopic consequence is that the systems acquires elasticity, that can be understood as the response of the material to distortions from the perfectly aligned state. Any distortion can be decomposed in the base of three elementary distortions 3: splay (Fig. 1.7 (a)), twist (Fig. 1.7 (b)), and bend (Fig. 1.7 (c)). Each of them has a diﬀerent cost depending of the molecular interactions between the mesogens. Generally, the energy associated to any distortion of the director ﬁeld is called the Frank-Oseen free energy density, and is expressed as follows: 1 −→ 1 −→ 1 −→ = ( n)2 + (n n)2 + (n ( n))2 (1.3) f 2K1 ∇ · 2K2 · ∇ × 2K3 × ∇ ×

where K1, K2, K3 are respectively the elastic constants associated to splay, twist and bend. In the Frank-Oseen description, orientationnal elasticity is therefore described by three parameters. The elastic constants Ki are quite weak (typically, tens of picoNewtons [64]), and of similar orders of magnitude. Therefore, theoreticians often consider them equal to make computations easier. In the so-called one-constant approximation, K = K1 = K2 = K3.

1.2.6 Topological defects

External constraints applied to a liquid crystal phase may impose the presence of topological defects. Formally, defects in an ordered medium are deﬁned as regions where the

3Traditionally, it omits two higher order modes, namely saddle-splay and splay-bend

19 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.7 – Long-range distortions of nematic liquid crystals (a) Splay (b) Twist (c) Bend. characteristic order of the phase is not satisﬁed. In a liquid crystal phase, a defect corresponds to a region where n is not deﬁned. In a 2D nematic, these regions can be 0-dimensional (point defects) or 1-dimensional (line defects).

Point defects In order to understand how point defects arise, let us consider the case of a 2D liquid crystal. The director ﬁeld n can be described with a single angle ψ as follows:

−→ −→ n = cos ψ e x + sin ψ e y (1.4) Written in the one-constant approximation, the Frank-Oseen free energy can be written as follows:

1 −→ −→ = [( n)2 + ( n)2] (1.5) f 2K ∇ · ∇ × which becomes:

1 −→ = ( )2 (1.6) f 2K ∇ψ The free energy is then minimized through the Euler-Lagrange equation:

∂f −→ ∂f = ∇ · −→ (1.7) ∂ψ ∂ ∇ψ which yields the Laplace equation ∆ψ = 0, whose solution corresponding to a point defect are of the following form:

ψ(r, θ) = mθ + ψ0 (1.8) The parameter m is called the strength, or topological charge, of the defect. It can only take integer or half integer values. To summarize, the localized structures that minimize the distortions have the following form:

n = cos(mθ + ψ0)ex + sin(mθ + ψ0)ey 1 3 (1.9) = 1 2 m ±2, ± , ±2, ± ...

20 1.2. Liquid Crystals

Figure 1.8 – Field lines for various point defects The ﬁeld lines indicate the local orientation of the mesogens. The defect is point-like and located at the center of the disk.

These structures are usually visualized by means of their field lines, as shown in Fig. 1.8. Note that the strength of the defect can be inferred from these representations. Going back to equation 1.9, we understand that m represents the number of times the director n rotates around the point defect. In order to compute the charge from the field lines, we visualize the evolution of ψ around a counter-clockwise loop Γ encircling the defect (represented in blue is Fig. 1.9). Formally, the rotation of the director field around a given point is quantified by the winding number:

1 I s = n · ∇ψ (1.10) 2π Γ To compute it, we sum up the tilt angles between consecutive vectors n while moving along Γ, as shown in Fig. 1.9. s is defined at any point of the direction field, and is equal to the topological charge of a defect when computed at the defect site. As a matter of fact, this is the way defects are identified in a director field. s is computed everywhere and the points where it acquires integer of half-integer values are attributed to defect sites. Finally, note that although all defects are obtained by minimization of the free energy, they are not energetically equivalent. In polar coordinates, the local elastic energy f can be written as follows:

1 m2 f(r, θ) = K (1.11) 2 r2 We can integrate f on a disk of radius R around the defect core to compare the energies for various m. However, as f diverges around r = 0 we prescribe the integration on an interval ]rc,R] (rc > 0) and attribute an energy Fc to the defect core. The integration yields:

2 R F = πKm ln + Fc (1.12) rc Therefore, we can see that F increases quadratically with the defect strength. As a consequence, defects of small charge are energetically more favourable. Typically, disclinations with |m| > 1 are not stable in conventional nematic liquid crystals. For a two- dimensional nematic liquid crystals two types of topological defects predominate [65]: comet-like s = +1/2 defects, and trefoil-like s = −1/2 defects.

21 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.9 – Winding number and topological charge (a) Case of a m = +1/2 defect. When moving along the loop Γ surrounding the defect, the mesogens perform a rotation of π, which yields s = +1/2 according to Eq. 1.10. (b) Case of a m = +1 defect. The mesogens perform a rotation of 2π. (c) Case of a m = −1/2 defect. The mesogens rotate by −π.

Figure 1.10 – Line defects are topologically unstable in 2D (a) A line defect in a 2D nematic director ﬁeld. The defect can be annihilated by a continuous transformation. Examples of pure splay (b) and bend (c) deformations restoring the continuity of the director ﬁeld.

Line defects

A sketch of a line defect is depicted in Fig. 1.10 (a). The director n is undefined within the defect line, pictured in red. These structures can be temporarily observed in nematic phases, but they are unstable. As a matter of fact, they can be continuously transformed into structures that no longer contain singularities, as shown in Fig. 1.10. In other words, lines defects are topologically equivalent to an undistorted state. They can however be observed in particularly constrained configurations, such as in strong spatial confinement or in the presence of an external field [64, 66]. As a matter of fact, they will play an important role in chapter5.

22 1.2. Liquid Crystals

Figure 1.11 – Orientation of nematic liquid crystal constituents at an interface (a) The orientation of the local director n is giving by azimuthal (φ) and polar (θ) angles. Green ﬁlaments represent the polymer coating generally used to favour a given anchoring condition (b) Case of a planar anchoring (θ = π/2)(c) Case of a homeotropic anchoring (θ = 0) (c) Case of a tilted (or conical) anchoring (0 < θ < π/2).

1.2.7 Liquid crystals in confinement: defects and topology The presence of topological defects in a liquid crystal phase implies that a significant energy input has been transferred to the mesogens in the form of distortions, which where concentrated into the cores of topological defects. However, passive liquid crystals obey the laws of equilibrium thermodynamics. Therefore, in the absence of external constraints, the energy stored in these defects should eventually be released and the defects should disappear. Even so, experimental liquid crystals are always finite systems, and boundary effects can induce permanent constraints to the director field requiring the presence of topological defects.

Anchoring The alignment of the mesogens may be significantly perturbed in the vicinity of an interface - solid (a glass plate), or liquid (an immiscible fluid). By tuning the properties of the interface, it is possible to control the orientation of the mesogens. We call this effect anchoring. The orientation of the mesogens at the interface is represented by two angles, as shown in Fig. 1.11 (a): θ measures the tilt with respect to the normal of the surface (z-axis), and φ the in-plane orientation of the mesogen. There exist three elementary cases. A uniform alignment with θ = π/2 is called a planar anchoring, shown in Fig. 1.11 (b). All the mesogens are lying flat onto the substrate. In a pure planar configuration, φ is also uniform, such that the mesogen have a nematic order. If instead φ is a random variable the anchoring is called planar degenerate. The second limit case corresponds to θ = 0, and is called homeotropic anchoring. This configuration is represented in Fig. 1.11 (c). All the mesogens are perpendicular to the surface. An intermediate case, called conical anchoring (Fig. 1.11 (d)) corresponds to a tilted homeotropic anchoring: θ 6= 0 is uniform, as well as φ. The description made above pictures cases where the nematogens perfectly follow the alignment imposed by the interface. In practice, the constraint imposed by the boundary propagates, creating distortions that can be energetically costly in the bulk. The equilibrium director field will result in a competition between the anchoring constraint

23 Chapter 1. Flow and Order: the physics of Active Liquid Crystals and the elasticity of the bulk, which tends to reorient the mesogens along the far ﬁeld director, which may be diﬀerent from the direction imposed by the anchoring. The bulk elasticity is characterized by the Frank-Oseen free energy with elastic constant K (in the one-constant approximation). Conversely, the anchoring strength can be modelled through the Rapini-Papoular anchoring potential [67]:

2 fs = W sin(φ − φ0) (1.13)

where φ0 is the tilt constraint imposed at the boundary (we notice here that the in- plane orientation at the surface, θ, does not enter in the energy balance). The anchoring strength is quantiﬁed by the Kleman-de Gennes length K/W [68]. We can now distinguish two cases. A weak anchoring corresponds to the case where K dominates over W . In this case, bulk elastic distortions are costlier that distortions of the anchoring. Therefore, the bulk orientation will propagate at the boundary and the anchoring will not be well-deﬁned. On the other hand, a strong anchoring corresponds to the case where W dominates over K. In this case, φ = φc at the interface and the anchoring will propagate in the bulk over a distance at least of the order of K/W . In experiments, the anchoring can be controlled by various techniques. Generally, they consist in adding polymer chains or surfactants at the interface, which align the mesogens by physico-chemical interactions. The choice of polymer depends on the desired anchoring, on the type of liquid crystal and on the type of interface. More details will be provided in section 2.1.1.

Confinement properties The presence of boundary conditions, and specially in the case of strong anchoring, will generate distortions in the bulk that may result in the permanent formation of topological defects. Several concepts need to be introduced to understand such a coupling. From now on, we will focus on the case of 2D liquid crystals, both for simplicity and because our experimental system is 2D. Let us consider a liquid crystal sheet formed by nematogens in a nematic phase, as pictured in Fig. 1.12 (a). In the absence of boundaries, the nematic field is uniformly aligned along n. Let us now confine the nematic sheet within a disk. We distinguish two extreme cases. In the first one, represented in Fig. 1.12 (b), we impose an infinitely weak anchoring (W K). As a consequence, the director field remains uniform, unperturbed by the presence of the walls. Conversely, in Fig. 1.12 (c) we impose an infinitely strong anchoring (W L). The alignment propagates into the bulk. Upon reaching the center of the disk, the director field can no longer satisfy the alignment, and a s = +1 defect forms. This example is just an illustrative sketch of the relation between anchoring effects and defect nucleation. The formal relation is obtained by topological arguments, as described in the next section.

Topology of a surface: the Gauss-Bonnet theorem

In topology, surfaces are characterized by their Euler characteristic, χ. Two surfaces with the same Euler characteristic are said to be topologically equivalent, which means that there exists a continuous transformation from one to the other. For a compact surface without boundaries, χ satisﬁes the Gauss-Bonnet theorem:

24 1.2. Liquid Crystals

Figure 1.12 – Anchoring and topological defects (a) In the absence of boundaries, a nematic phase at equilibrium is uniformly aligned along a common director n. In confinement, the equilibrium state depends on the anchoring strength (b) In the extreme case of weak anchoring, bulk elastic forces dominate and the nematic field is not affected by the presence of the walls. (c) In the extreme case of strong anchoring (planar in this case), the alignment with the wall propagates to the bulk. At the center of the disk, the alignment is frustrated by topology, generating a s = +1 defect.

χ = 2(1 − g), (1.14) where the genus number g counts the number of holes in the surface. A sphere has no hole, which yields χ = 2. Conversely, a doughnut has one hole, giving χ = 0. Similarly, a coﬀee mug has only one hole (also called handle), which suggested the famous phrase according to which "a mathematician cannot distinguish his or her doughnut from his or her coﬀee mug". A similar formula can be generalized to the case of a compact surface with a boundary:

χ = 2(1 − g) − h (1.15) where h counts the number of boundaries. For instance, the disk drawn in Fig 1.12 (b) has no hole (g = 0), and one boundary (h = 1), which yields χ = 1. Note that this formula is particularly restrictive: a compact surface with a boundary is by deﬁnition topologically equivalent to a disk. This formula just tells that any 2D shape without hole and with a single boundary has χ = 1.

Conﬁnement and topological charge: the Poincaré-Hopf theorem The sum of the defect charges in a 2D nematic phase is linked to the Euler characteristic of the conﬁning surface through the Poincaré-Hopf theorem.

X si = χ, (1.16) i

where si is the topological charge of the i-th defect. This equation enables a direct interpretation of the sketch of Fig 1.12 (c) corresponding to the strong anchoring. There

25 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.13 – Edge charge computation (a) The edge charge is computed in the Frenet- Serret frame represented in blue. In this frame, the director orientation is given by θ. (b) An example of a uniform director ﬁeld in a disk. The corresponding orientation of the director at the boundary in the Frenet-Serret frame is displayed below. We can see that the director rotates by 2π, leading to an edge charge of sedge = 1 according to Eq. 1.18. (b) Case with a perfect planar anchoring at the wall. The orientation of the director is

ﬁxed in the Frenet-Serret frame, leading to sedge = 0. (c) Case with a perfect homeotropic anchoring at the wall, also giving sedge = 0.

is a single defect of charge s = +1, which satisﬁes the Euler characteristic of the disk. It is however less clear in the case of weak anchoring, Fig. 1.12 (b), for no defects are present in the bulk. The reason is that the computation of the total charge in the nematic phase must account for the charge at the boundary, also called the edge charge [69]:

X si = χ i bulk (1.17) X X si = sj + sedge i j

The bulk charge simply corresponds to the sum of the bulk defects present in the nematic phase. The edge charge measures how much the tilt angle changes around the boundary. This angle, called θ in Fig. 1.13 (a) has to be computed in the reference frame of the boundary, also called the Frenet-Serret frame, represented by the base (et, en). Then, the computation of the edge charge is performed in a similar fashion to the winding number formula (1.10).

1 I sedge = n · ∇θ (1.18) 2π Γ

In the weak anchoring conﬁguration, reproduced in Fig. 1.13 (b), we can see that the director rotates by 2π along the boundary, which yields sedge = 1 = χ. We therefore recover equation 1.17. Similarly, in the strong anchoring cases pictured in Fig. 1.13 (c) and (d), the orientation of the director is ﬁxed with respect to the boundary, giving sedge = 0. On the other hand, the bulk hosts a s = +1 charge, which complies again with the Poincaré-Hopf theorem.

26 1.3. Flow and Order: the physics of Active Nematics

Figure 1.14 – Examples of active liquid crystals (a) Growing colony of E.Coli bacteria [70]. (b) Epithelial tissue of Madine–Darby canine kidney (MDCK) cells. Scale bar is 10 µm [71]. (c) d Dense monolayer of mouse ﬁbroblast cells [72] showing −1/2 and +1/2 defects marked with blue and orange circles. (d) Monolayer of neural progenitor stem cells [73]. Cells are depleted from −1/2 defects (blue, trefoil symbols) and accumulate at +1/2 ones (red, comet-like symbols). (e) Granular nematic of vibrated rods [20]. Positive defects are marked with red circles and negative defects are marked wit green circles. Scale bar: 1 mm.

1.3 Flow and Order: the physics of Active Nematics

In section 1.1.2, we discussed the various emergent properties of active matter systems. One of them concerned pattern formation. The elementary units composing the system interact with each other forming large scale structures. At the highest concentrations of mesogens, some active matter systems have the ability to spontaneously self-organize into a liquid crystal phase. Examples of such systems are proposed in Fig. 1.14. They are all quasi 2D, composed of a sheet of active particles. The mesogens appear to develop a long range alignment, interrupted by the presence of topological defects. Furthermore, we notice that the charge of the defects is exclusively s = ±1/2, in agreement with the observations on 2D passive nematic LC phases (1.2.6). However, unlike the passive analogue, a constant energy input on each mesogen maintains the system far from equilibrium, giving rise to a variety of phenomena.

27 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

1.3.1 An out-of-equilibrium liquid crystal Defect nucleation: the bend instability In passive liquid crystals, we have seen that the presence of topological defects was associated to the minimization of elastic distortions imposed by external constraints (see section 1.2.6). In active nematics, the director field is entirely covered with half-integer defects, as shown in Fig. 1.14 (d). The distortions are a consequence of the active stresses acting on the individual filaments. Even so, the mechanism of defect nucleation is non trivial. As presented in section 1.1.3 and sketched in Fig. 1.15 (a), the elementary source of motion is a sliding between two filaments induced by the molecular motors. In the case of kinesin-tubulin active nematics, the active stresses are extensile, which means that they tend to elongate a bundle of parallel filaments in the direction of alignment n. As such, they do not a priori induce distortions of the director field, because the filaments remain aligned along n. However, as we suggested in section 1.1.2, the emergent properties of active matter systems have to be interpreted beyond the microscopic description. On a mesoscopic scale, an extensile bundle of aligned filaments is unstable to bend deformations. A local perturbation perpendicular to n will tend to be enhanced (Fig. 1.15 (c)) leading to the nucleation of a defect pair. This mechanism, called the bend instability, involves a hydrodynamic coupling with the solvent [2,74–79]. The whole system, composed of the active nematic layer embedded in its solvent, is not only extensile but dipolar extensile. The active stresses push the surrounding solvent along the long direction, but also aspirate it along the short direction [80], as shown in Fig. 1.15 (b). This situation is unstable to bend distortions, as depicted in panel (c). An experimental time-lapse of this process is displayed in Fig. 1.15 (d). In practice, not all perturbations are enhanced. Any distortion of the director field can be decomposed on the Fourier basis of sinusoidal perturbations. The growth rate Ω of a given mode will depend on its wave- number k. Theoretical and experimental studies have shown that the system selects the mode k∗ corresponding to the highest growth rate [81]. This means that the process of defect nucleation is characterized by a unique length scale, referred to as the active length scale. The experiments by Martinez et al. [81] give a particularly compelling evidence of this selection mechanism. The authors were able to temporarily suppress all the defects in an active nematic layer by aligning the filaments in a radial arrangement (Fig. 1.15 (e)). When the alignment constraint is released, the radial pattern rapidly destabilizes into a well-defined pattern characterized by a single wave number, as shown in Fig. 1.15 (f). Furthermore, they showed that the selected wave number and growth rates could be related to the material parameters as follows:

α k∗ ∝ ( )1/2 K (1.19) K γ Ω∗ ∝ [1 + (1 − ν)2]k∗2 γ 4η where K is the elastic constant associated to bend distortions, α is the activity parameter 4, and γ, η, ν are respectively the rotational and shear viscosities, and the flow alignment coefficient, all three related to a specific type of hydrodynamic coupling with the solvent. The expression of k∗ shows how the interplay between activity and filament

4Details on the modelling are provided in section 2.3

28 1.3. Flow and Order: the physics of Active Nematics

Figure 1.15 – Bend instability in active nematics (a) Sketch of the defect nucleation process. (b) The microscopic source of motion is a sliding between parallel filaments, which is unstable to bend deformation due to a hydrodynamic coupling with the solvent. The distortion is further enhanced until a defect pair is created. (d) A sequence of images demonstrates buckling, folding and internal fracture of a nematic domain leading to the nucleation of a defect pair. Time lapse, 15 s; scale bar, 20 µm. Adapted from [33]. (e) Fluorescence micrograph of an active nematic interface aligned radially (left). The bend instability creates concentric circular cracks separated by a well-defined spacing (right). Scale bar: 100 µm. (f) Power spectrum of the intensity profile of (e) showing the predominance of a peak, which defines k∗. Adapted from [81].

ﬂexibility determines the emergent properties of active nematics [82,83]. The constant defect nucleation is balanced by a competing process of pair-annihilation. As a matter of fact, defects with opposite charge have a tendency to attract and cancel out, restoring the nematic order. Overall, the density of defects remains constant over time. Finally, coming back to the topological considerations made in section 1.2.7, the Euler characteristic of a compact 2D interface without boundaries is χ = 0. Although the number of defect exceeds by far the topological requirements, the Poincaré-Hopf theorem (Eq. 1.17) is still respected: defects nucleate and annihilate by pair of oppositely-charged defects, which has no impact on the total charge of the system: P = = 0. i si χ

29 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Forces and ﬂows around topological defects

The microscopic interaction between two ﬁlaments is said to be nematic, for the action of motors is a shear stress, and not a propulsion force. As a consequence, a nematic bundle cannot, in principle, exhibit directed motion. This argument is however only reasonable provided the bundle remains uniformly aligned along the director n. As soon as curvature heterogeneities arise, the symmetry of the active stresses may be broken.

Formally, one can model this eﬀect by the introduction of body force fa driving the active ﬂows [79,84,85]:

fa = −α∇ · Q, (1.20) where α is an activity parameter, and Q defines the tensor order parameter associated to the nematic phase (section 1.2.4). In short, α represents the isotropic energy input associated to activity, and ∇ · Q infers how this energy is redirected by the local disorder of the material. In a well-aligned region, ∇ · Q = 0, so there is no body force. As a matter of fact, the only regions with significant distortions of the order parameter are the defect sites. As a consequence, topological defects may be propelled by a bulk force whose direction depends on their symmetry. In Fig. 1.15 (d), we can observe that the core of the positive defect is moving away from the negative defect. This motion is part of the defect nucleation mechanism, and is referred to as defect unbinding: the positive defect moves away, while the negative defect stays still. This distinctive feature of topological defects can be explained by symmetry. The theoretical flow fields generated by active nematic disclinations were computed by Giomi and al. [84]. They are displayed in Fig. 1.16. In these representations, the flow pattern corresponds to the back-flow generated by a defect, artificially pinned at the center of a disk of radius R. The disk represents the defect-free region surrounding a given defect, and is also called the defect range.

dr+ ζ = v0x dt (1.21) R v = α , 0 4η

where r+ corresponds to the location of the defect core, and ζ is an eﬀective drag coeﬃcient proportional to the rotational viscosity γ. The most important point to note here is the proportionality between the v0 and the activity parameter α. The sign of α is given by the nature of the microscopic stress. It the system is extensile (resp. contractile), as for our case, α < 0 (resp α > 0). With the conventions of Fig. 1.16 (a), it means that the positive defect will propel toward negative x, as represented with the white arrow. Conversely, negative defects will not propel. Experimentally, negative defects do move, but their motion is only attributed to the elastic distortions induced by neighbouring positive defects.

Active turbulence

Although positive defects self-propel with a relatively constant velocity v0, and despite the symmetry of the back-ﬂow they generate, the collective ﬂow pattern is extremely chaotic. Positive defects constantly move past each other in the form of vortices with various sizes

30 1.3. Flow and Order: the physics of Active Nematics

Figure 1.16 – Flow ﬁelds for active defects Flows generated by a s = +1/2 (a) and a s = −1/2 (b) disclination. Due to their polar structure, positive disclinations propel along their polarity, represented by the white arrow. On the contrary, negative defects are not propelled due to their three-fold symmetry.

[84, 86–89]. On top of the pair-interactions of existing defects, the permanent nucleation and annihilation cycle further perturbs the flow pattern [90]. This flow regime, termed active turbulence [91–93], has been reported in various biological systems [14, 94–101]. The suggestive analogy with hydrodynamic turbulence only reflects the apparent lack of order of the flows, illustrated in Fig. 1.17 (a). However, the comparison is abusive over many respects. In classical turbulence, the control parameter is the Reynolds number:

LV R = (1.22) e ν where L, V, ν are respectively the typical size and flow velocity of the system, and the kinematic viscosity of the solvent. Turbulence is known to occur for very large Reynolds numbers (typically, Re > 1000). In our case, a quick estimation of the Reynolds number based on the active length scale λ ∼ 100 µm, the propulsion velocity of the positive −1 −6 2 −1 defects v0 ∼ 10 µm · s and the kinematic viscosity of water ν = 10 m · s leads −3 to Re ∼ 10 1. As a consequence, the underlying physics is inherently different. Furthermore, the distribution of vortex sizes follows an exponential law, as shown in Fig. 1.17 (b), which is a direct evidence that the system is characterized by a unique length scale [91]. On the contrary, the very definition of turbulence implies a cascade process involving a large band of wave numbers. Even so, the bridge between active nematic flows and classical turbulence can be made within the theory of spatio-temporal chaos as we will see later in Chapter5. Spatio-temporal chaos (STC) is a particular class of dynamical pattern arising in systems lacking long-time, large-distance coherence in spite of an organized behaviour at the local scale. According to Manneville [102], it is located in the middle of a triangle, the corners of which are temporal chaos, prevalent for a few spatially frozen degrees of freedom, spatial chaos, in time-independent patterns, and turbulence, with cascading processes over a wide range of space and time scales. Spatio-temporal chaos is characterized at the local scale by an instability mechanism that generates dissipative structures, which fits the description of active nematic dynamics. In other words, active turbulence is a type of STC that has a lower degree of chaos than

31 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.17 – Active turbulence (a) An example of active turbulence characterised by chaotic ﬂows with vortices of diﬀerent sizes. Black lines illustrate streamlines and the colourmap shows vorticity normalised with its maximum value, varying from +1 (red) for anticlockwise to −1 (blue) for clockwise vorticity. Adapted from [103]. (b) Experimental distribution of vortex sizes in an active nematic in the regime of active turbulence, adapted from [60].

classical turbulence.

Experimental system The experimental system we used in this project is an vitro biological active liquid crystal. It consists of a 2D liquid crystalline phase composed of densely packed biological ﬁlaments called microtubules, which are constantly sliding between one another though molecular motors called kinesins. The 2D phase is stabilized at an oil-water interface by surfactants. The motion is powered by adenosine triphosphate (ATP). The dense packing of the microtubules is favoured by the introduction of polyethylene glycol (PEG), a small polymer that induces ﬁlaments aggregation through depletion interactions (1.1.3). Additional constituents are added to the water phase to stabilize the biological proteins. Precise details on the synthesis and sample preparation are available in section 6.2. An illustrative sketch of the system is displayed in Fig. 1.18.

From now on, the denomination active nematic will exclusively refer to this system. The qualitative properties of active liquid crystals described so far are rather general to all the experimental realizations introduced at the beginning of the section (Fig. 1.14). However, it is important to bear in mind the variety of speciﬁc properties that may alter the generality of the results obtained for a given system: interactions with a substrate, strength and nature of the hydrodynamic coupling, elastic properties, etc. The results obtained in this project will occasionally be compared to analogue experiments with cell tissues, or bacterial baths, in order to illustrate common phenomena and interpret potential discrepancies.

32 1.3. Flow and Order: the physics of Active Nematics

Figure 1.18 – A microtubule-based active nematic (a) Illustration of the active nematics basic constituents. The process starts by synthesizing microtubules with a well defined length of 1 µm. Then, we prepare motor clusters by binding two biotinylated kinesins to a streptavidin molecule. Through the consumption of ATP, these motor clusters slide along the microtubules, creating a shear stress. Microtubules are packed together by a depleting agent called poly-ethylene glycol (PEG). The close packing combined with shear stresses creates large scale filaments which continuously elongate, break and merge with one another. When one of such filaments touches the oil interface, surfactant molecules bind to the microtubules preventing them from going back to the bulk. Eventually, all microtubules condense onto the interface forming a liquid crystal phase. (b) Fluorescence micrograph of an active nematic phase captured with confocal microscopy. The image is acquired at the oil-water interface. Scale bar: 100 µm

33 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

1.4 Scientiﬁc Objectives

The properties of active liquid crystals described earlier are somehow ambiguous. On the one hand, they spontaneously self-organize into a nematic phase. On the other hand, the order is continuously destabilized by the active stresses, generating chaotic flows at all achievable activity ranges. These results were however limited to the simplest case of a unconfined active liquid crystal layer. Conversely, in section 1.1.3, we pointed out the fascinating adaptability of these systems to external constraints. For example, human cancer cells switch between distinct invasion modes when they encounter constrictions in the crowded environment of stroma [61], and the growth of bacterial biofilms can be directed by their surroundings [62]. Therefore, active flows seem to acquire much more order in confined geometries. Similarly, recent studies have shown that topological defects in epithelial tissues govern cell death and extrusion [104, 105] suggesting both that the presence of topological defects is essential to its development, and that their location is crucial to the structuring of the tissue. Our research extends recent efforts devoted to studying confined biological fluids, like bacterial baths [106–109], cell tissues [72, 110–112] or cytoplasmatic preparations [113]. In particular, microtubule-based systems have already demonstrated to be versatile enough to ordering effects by responding to frictional/viscous gradients [58, 114] or to geometric and topological constraints [60, 115–120]. Recent contributions dealing with confinement in bacterial suspensions and cell layers have demonstrated a rich range of behaviour. Competition between wall orientation, hydrodynamic interactions, topology and activity lead to a wide variety of flow patterns: spiral vortices [107, 108], synchronised vortex lattices [109], unidirectional flows [111, 121, 122], shear flows [112] and freezing [72, 123]. Work on confining active mixtures of microtubules and motor proteins to circular domains [60, 120, 124, 125], in vesicles [115] or droplets [118, 126] and to more complex geometries such as tori [116, 127], have already probed the specific effects of interfacial viscosity, curvature and 3D confinement. These experimental results have prompted parallel simulations [128–142].

In this work, we aim at exploring the interplay between geometrical confinement, topological landscape, and active flows, as sketched in Fig. 1.19 (a). Experimentally, the approach we took is to study how both the flow patterns and the defect landscape respond to geometrical constraints. Starting from a 2D flat space without boundaries, a geometrical reshaping can be decomposed into four elementary transformation: a lateral constriction (Fig. 1.19 (b)) a change of topology (Fig. 1.19 (c)), a change of the boundary roughness (Fig. 1.19 (d)), and the addition of curvature (Fig. 1.19 (e)). These four transformations have been studied independently in chapters3,4,5 and6 respectively. The objectives are two-fold. First, the information we obtain from the emergence of order in controlled geometries may help disentangle the roles of hydrodynamics and elasticity in the structuring of soft active matter systems. On a more technological side, controlling the self-assembly of topological disclinations and dynamically structured flow fields in engi- neered geometries could pave the road to the design of new active topological microfluidic devices [143].

34 1.4. Scientiﬁc Objectives

Figure 1.19 – Scientific objectives Draft of the dissertation structure (a) schematic representation of the interplay between geometry, order and active flows in active nematics. We confine a layer of active nematics in various geometries and measure the impact on the spatio-temporal organization. (b) Effect of lateral confinement (Chapter3). (c) Effect of topology (Chapter4)(c) Effect of a wall roughness (Chapter5). (d) Effect of curvature (Chapter6).

1.4.1 Inﬂuence of lateral conﬁnement

We aim at characterizing the response of an active nematic layer when confined in channels of variable width. Theoretical models predict dramatic transitions in the flow states of confined active nematics, such as a transition to a spontaneously flowing state in strong confinement [76]. Recent simulations on active nematics have predicted a significant structuring of the flow pattern, associated to a well-defined arrangement of topological defects [139]. Different flow regimes were observed: for the narrowest channels, they recovered the emergence of a net transport predicted by theory. At intermediate width, the flow pattern was composed of a steady array of co-rotating vortices, associated to a regular "dancing motion" of positive defects as shown in Fig. 1.20 (a). In addition, simulations predicted a heterogeneous distribution of topological defects, with negative defects confined at the walls, and positive defects laying at the center of the channel. On the experimental side, the works on cellular nematics unveiled a large variety of behaviours, none of which could be directly related to the simulations of active nematics. For instance, cultured neural progenitor cells (NPCs) form a nematic phase under strong confinement [73], with most of the cells aligning with each other, and a few topological defects moving along the walls (Fig. 1.20 (b)). In addition, rapid cell accumulation towards the +1/2 topological defects was reported. Conversely, spindle-shaped cells plated in narrow stripes spontaneously develop a defect free shear flow regime [112] as shown inf Fig. 1.20 (c). Finally, epithaelial cells confined in thin stripes collectively migrate [144], at a velocity than increases for decreasing stripe width as shown in Fig. 1.20 (d). The lack of generality of these results on active liquid crystals in confinement motivated the design of a specific experiment for the case of microtubule-based active nematic, to verify the theoretical predictions. However, conventional microfluidic techniques used in the cell projects were not suitable for our system. Contrary to these systems, which grow onto a solid substrate, active nematics form at an oil/water interface. We have developed an

35 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

Figure 1.20 – Effect of lateral confinement (a) Dancing state in confined active nematics, predicted by simulations and characterized by a lattice of co-rotating vortices. Adapted from [139]. (b) Cultured neural progenitor cells (NPCs) form a nematic phase under strong confinement [73], with most of the cells aligning with each other, and a few topological defects moving along the walls. (c) Spontaneous shear-flow in a population of spindle-shaped cells confined in a narrow channel. Cells develop a nematic alignment with a tilt angle with respect to the wall direction that depends on the constriction width. Adapted from [112]. (d) Collective migration of epithelial cells in narrow constrictions. White arrows indicate the higher migration speed for narrower stripes. Adapted from [144].

experimental setup to laterally conﬁne an active nematic layer at an oil/water interface by fully conﬁgurable biocompatible polymer grids. The protocol is describe in section 2.4.1 and the results presented in chapter3.

1.4.2 Influence of topology In passive liquid crystals, the topological landscape is very sensitive to geometrical confinement as shown in part 1.2. The energy associated to elastic distortions imposed by geometrical constraints is minimized by the nucleation of topological defects. The case of active liquid crystals is a priori more complex: the constant energy injected to the system through the active stresses may or may not destabilize the equilibrium structures predicted by topology. Previous studies on spindle-shaped cells confined in a disk [72] with planar anchoring proved the dominance of elastic forces over activity in the shaping of the tissue. Such a geometry has an Euler characteristic of χ = 1 (cf. 1.2.7). Under strong confinement the cell tissue develops a nematic order disrupted by two half integer defects, thus fulfilling the topological Poincaré-Hopf theorem (Eq. 1.17). Moreover, elastic repulsions between defects force them to align along a disk diameter. This state remains frozen in time. On the contrary, a similar experiment on active nematics gives rise to much more dynamic patterns [120], as shown in Fig. 1.21 (b). Geometry still governs the structure of the director field to some extent, with the minimal number of defects satisfying the topological requirements. However, the order is disrupted by the emergence of a persistent spiralling motion, accompanied by occasional nucleations of additional defect pairs from the walls. To go further, theoretical predictions suggest the insensitivity of active nematics dynamics to topological constraints [141]. By prescribing the edge charge sedge (1.2.7) of the director field onto the disk boundary, the authors control the bulk charge sbulk required in the nematic field through the relation sedge + sbulk = χ. They

36 1.4. Scientiﬁc Objectives

Figure 1.21 – Effect of topology (a) (left) Nematic ordering of spindle-shaped cells confined in a disk. (right) Corresponding director field, showing the presence of two +1/2 defects whose positions are well described by the elastic properties of a passive nematic liquid crystal. Adapted from [72]. (b) Spontaneous spiralling motion of an active nematic layer confined in a disk. Scale bar: 20 µm. The equilibrium defect landscape is periodically disrupted by activity-driven defect nucleations from the walls. Adapted from [120]. (c) Insensitivity of active nematics to topological constraints. The spiralling motion emerges regardless of the bulk charge imposed within the active nematic layer. Adapted from [141].

show that the spiralling motion persists regardless of sbulk, as shown in Fig. 1.21 (c). Nevertheless, this discussion on topological constraints is somehow ambiguous. The real measure of the topology of a surface is the Euler characteristic, which is not affected by modifications of the edge charge. Even so, their results suggest that the spiralling flow pattern is inherent to the disk geometry, and probably more general to any surface with an Euler characteristic of χ = 1. On the other hand, the defect dynamics are somehow "slaved" to this emerging flow pattern. Then, a natural question is to wonder how this flow pattern emerges, and how it is transformed by a change of topology. To investigate this question motivated by recent simulations on active nematic annuli [145], we confined active nematics in surfaces with varying Euler characteristic, by changing their genus number g (section 1.2.7), as sketched in Fig. 1.19.

1.4.3 Active nematics at a wall

The experiments presented so far remain very phenomenological. The decomposition of a geometrical transformation into four elementary transformations tends to suggest that they can be treated independently. However, lateral conﬁnement remains a critical parameter in the dynamics of active nematics at higher topologies. In addition, the geometries with high genus number are signiﬁcantly curved, and discontinuous: for instance, the connection between two annuli in the g = 2 surface forms cusps, as sketched in Fig. 1.19 (b). The interpretation of the resulting dynamics requires to account for all these geometrical features that are not easily separable.

37 Chapter 1. Flow and Order: the physics of Active Liquid Crystals

The very essence of confinement effect is the presence of a wall. In this part, we come back to a more elementary case of active nematics in contact with a single boundary. The problem will be approached from various perspectives. First, we will focus on the structure of the director field of nematic layer in the vicinity of the boundary. One of the most striking effects of the wall is to align the nematic field in a planar configuration, suggesting a strong anchoring (section 1.2.7). On the other hand, active nematics are unstable to bend deformations, which necessarily leads to enhanced defect nucleations at the wall. Even more striking is the structure of the wall defects, which is studied in detail.

The second part of the project is devoted to the collective dynamics of wall defects within the framework of spatio-temporal chaos. The work presented so far aimed at dis- entangling the elastic and active contributions to the emergence of ordered phases. In this part, we propose a more general approach and put the results we obtained in the perspective of the transition to chaos in generic out-of-equilibrium systems. For instance, the emergence of the ordered regime composed of vortex lattices predicted in simulations of conﬁned active nematics [139] might be reminiscent of the convection rolls observed in the canonical Rayleigh-Bénard instability [146, 147]. As a matter of fact, both systems emerge from an underlying instability mechanism generating dissipative structures [148].

The last part is devoted to specific effects relative to the interaction of an active matter system with an asymmetric boundary. We propose model experiments to study the effects of a discontinuity at the wall, of a ratchet pattern, and finally of wall curvature. All these features are known to organize the flows of an active matter system. The orientation of this project is more applicative, as it proves the possibility to tame the chaotic flows by engineering the confining boundaries.

1.4.4 Active nematics at curved interfaces

The relation between order and curvature is well known from everyday physics. Funda- mental topological laws (see section 6.1.1) prevent the wrapping of a curved surface with a perfectly ordered phase without topological defects. It explains, for example, why it is not possible to pave a football ball with hexagons only. This coupling has direct reper- cussions at the smallest scales of life. So far, we have exclusively studied active nematics on flat interfaces. Yet, the geometry of a substrate has proved influential on the growth of a cell tissue, and particularly its curvature [149–151] and roughness [152, 153]. Recent experiments and simulations on active liquid crystals confined in curved geometries have proposed three different consequences. In the case of a fixed geometry, the distribution of topological defects depends on the local curvature of the interface [117]. More precisely, Ellis et al. have shown that the topological defects of an active nematic layer confined at the surface of a torus had a tendency to migrate towards regions of like-sign gaussian curvature. The opposite case involves deformable interfaces. Keber et al. encapsulated active nematics within lipid vesicles, which are highly deformable objects. They showed that the topological constraints imposed by the spherical geometry caused important distortions to the nematic field, which in return generated extreme deformations to the vesicle surface, as shown in Fig. 1.22 (b). Finally, an intermediate - and more intricate - coupling

38 1.4. Scientiﬁc Objectives

Figure 1.22 – Eﬀect of curvature. (a) Curvature-induced defect unbinding in active nematics. Topological defects migrate towards regions of like-sign gaussian curvature when conﬁned at the surface of a torus. Adapted from [117]. (b) Extreme deformations of an active nematic vesicle. Adapted from [154]. (c) Spontaneous division of an active nematic droplet, resulting from a coupling between the droplet curvature and the dynamics of the active liquid crystal phase. Adapted from [131].

was demonstrated in simulations of active nematic droplets. Starting from a spherical droplet filled with an extensile active nematic material, Giomi et al. [131] showed that the motion of topological defects induced curvature heterogeneities. In return, the director field was affected by the curvature gradients, resulting in the migration of the topological defect towards the regions with highest curvature. Finally, the propulsion of the defects within the highly curved regions further enhanced the elongation of the droplet. In extreme cases, this positive feedback lead to the division of the active droplet, as shown in Fig. 1.22. As such, the minimal model proposed by Giomi et al. mimicks two essential properties of living cells: their spontaneous motility and division. Experimentally, this situation has never been observed. In this part, we study the dynamics of active nematic condensed around ellipsoidal droplets. In the perspective of the projects presented above, this experiment represents a bridge between the results on the torus and the simulations by Giomi et al. The geometry of the ellipsoids is fixed, as in the torus experiments. However, their dimensions are small enough that the nematic phase contains the minimum number of defects required by topology, increasing the chances of coherent flows. As such, the experimental setup we propose may prove the possibility to organize active flows through curvature effects.

Chapter 2 Materials and Methods

In this part, we provide details on the experimental protocol. The first part is dedicated to the composition and preparation of the active gel. These techniques have had been mostly established previously in the research group, based in the literature and in a collaboration with the group of Z. Dogic, from Brandeis University. The second part is dedicated to the imaging techniques. We present the different microscopes that have been used, and we describe the image processing software to analyse active nematic experiments. Finally, we present the three experimental methods developed during the PhD. The first one, surface microfluidics, is the core technique exploited in chapters3,4,and5. The second one, active ellipsoids, is at the source of the results presented in chapter6.

2.1 Active Nematics Synthesis

2.1.1 Active Gel Preparation In this section, we provide a step-by-step recipe to prepare the active material. A sketch presenting the main ingredients of the system is provided in Fig. 1.18. An active nematic is a 2D liquid crystalline phase composed of densely packed biological ﬁlaments called microtubules, which are constantly sliding between one another though the action of molecular motors. The 2D phase is stabilized at an oil-water interface by surfactants. The motion is powered by adenosine triphosphate (ATP). The dense packing of the microtubules is favoured by the introduction of polyethylene glycol (PEG), a small polymer that induces ﬁlament aggregation through depletion interactions. Additional constituents are added to the water phase to stabilize the biological proteins.

Microtubules polymerization

Microtubules (MTs) were polymerized from heterodimeric (ζ, β)-tubulin from bovine brain [a gift from Z. Dogic’s group at Brandeis University (Waltham, MA)], incubated at 37◦C for 30 min in aqueous M2B buﬀer (80 mM Pipes, 1 mM EGTA, 2 mM MgCl2) prepared with Milli-Q water. The mixture was supplemented with the reducing agent dithio- threthiol (DTT) (Sigma; 43815) and with guanosine-5-[(ζ, β)-methyleno]triphosphate (GM- PCPP) (Jena Biosciences; NU-405), a slowly hydrolysable analog of the biological nu- cleotide guanosine-5´-triphosphate (GTP) that completely suppresses the dynamic insta-

41 Chapter 2. Materials and Methods bility of the polymerized tubulin. GMPCPP enhances spontaneous nucleation of MTs, obtaining high-density suspensions of short MTs (1–2 µm). For ﬂuorescence microscopy, 3% of the tubulin was labeled with Alexa 647.

Kinesin synthesis Drosophila melanogaster heavy-chain kinesin-1 K401-BCCP-6His (truncated at residue 401, fused to biotin carboxyl carrier protein (BCCP), and labeled with six histidine tags) was expressed in Escherichia coli using the plasmid WC2 from the Gelles Laboratory (Brandeis University) and puriﬁed with a nickel column. After dialysis against 500 mM imidazole aqueous buﬀer, kinesin concentration was estimated by means of absorption spectroscopy. The protein was stored in a 60% (wt/vol) aqueous sucrose solution at −80◦C for future use. This process was carried out in the labs of the BioNMR group at the insitute of bioengineering of Catalunya (IBEC), with assistance from their researchers.

Protocol Biotinylated kinesin motor protein and tetrameric streptavidin (Invitrogen; 43-4301) aqueous suspensions were incubated on ice for 30 min at the specific stoichiometric ratio 2:1 to obtain kinesin–streptavidin motor clusters. MTs were mixed with the motor clusters that acted as cross-linkers, and with ATP (Sigma; A2383) that drove the activity of the gel. The aqueous dispersion contained a nonadsorbing polymeric agent (PEG, 20 kDa; Sigma; 95172) that promoted the formation of filament bundles through depletion. To maintain a constant concentration of ATP during the experiments, an enzymatic ATP-regenerator system was used, consisting on phosphoenolpyruvate (PEP) (Sigma; P7127) that fueled pyruvate kinase/lactate dehydrogenase (PK/LDH) (Invitrogen; 434301) to convert ADP back into ATP. Several antioxidant components were also included in the solution to avoid protein denaturation, and to minimize photobleaching during characterization by means of fluorescence microscopy. Anti-oxydant solution 1 (AO1) contained 15 mg/mL glucose and 2.5M DTT. Anti-oxidant solution 2 contained 10 mg/mL glucose oxydase (Sigma G2133) and 1.75 mg/mL catalase (Sigma, C40). Trolox (Sigma, 238813) was used as an additional anti-oxidant. A high-salt M2B solution was used to raise the MgCl2 concentration. The PEG-based triblock copolymer surfactant Pluronic F-127 (Sigma; P-2443) was added to procure a biocompatible water/oil interface in subsequent steps. Buffer for stock solutions of PEP, DTT, ATP, PEG and Streptavidin was M2B, and we added

20 mM of K2HPO4 to the buﬀer of Catalase, Glucose, Glucose Oxydase and Trolox. A typical recipe is summarized in Table 2.1.

Liquid Crystals In part of the experiments reported in this thesis we have combined active and passive thermotropic liquid crystals (Chapters6). The passive analogues are always thermotropic, and feature nematic and/or smectic-A phases. We have mainly used two homologous alkyl-cyanobiphenyl compounds, which are chemically very stable:

• 4-pentyl-40-cyanobiphenyl (5CB, Synthon Chemicals, ST00683, Fig. 2.1(a)) features a nematic phase between 18 and 35◦C.

42 2.1. Active Nematics Synthesis

Compound Stock solution v/Vtotal PEG 12 % w/vol 0.139 PEP 200 mM 0.139 High-salt M2B 69 mM MgCl2 0.05 Trolox 20 mM 0.104 ATP 50 mM 0.03 Catalase 3.5 mg.ml−1 0.012 Glucose 300 mg.ml−1 0.012 Glucose Oxydase 20 mg.ml−1 0.012 PK/LDH 600 − 1000 units.ml−1 0.03 DTT 0.5 M 0.012 Streptavidin 0.352 mg.ml−1 0.023 Kinesin 0.07 mg.ml−1 0.234 Microtubules 6 mg.ml−1 0.167 Pluronic 17 % 0.027 Table 2.1 – Composition of all stock solutions, and their volume fraction in the ﬁnal mixture.

Figure 2.1 – Structures of 5CB (a) and 8CB (b).

• 4-octyl-40-cyanobiphenyl (8CB, Synthon, ST01422, Fig. 2.1 (b)) features LC behaviour between 21.4 and 40.4◦C. Diﬀerently from 5CB, 8CB shows both nematic (33.4 – 40.4◦C) and Smectic-A phase (21.4 – 33.4◦C).

Oils and surfactants

The active material used in the experiments is always in contact with an oil layer. The use of surfactants is required. Their primary function is to prevent direct contact between the protein constituents of the active gel and the oil molecules. An additional requirement in the particular case of active nematics is the presence of a poly-ethylene glycol (PEG) group in the surfactant composition. In a process that is not fully understood, PEG chains enable the microtubules to condense at the interface (see Fig. 1.18), which is necessary for the formation of the active nematic layer. During the PhD, we used various PEG-lated surfactants (see Fig. 2.2). Their choice was driven by two distinctive properties. First, their compatibility with the type of oil employed in a given experiment. Second, in the cases where the oil was a passive liquid crystal (5CB or 8CB), the type of surfactant controls the anchoring of the mesogen molecules. All surfactants have been used at concentrations around 1 − 3%wt/vol in the water phase.

43 Chapter 2. Materials and Methods

Silicon oils and Pluronic ® For an interface with silicon oils, we used a triblock copolymer consisting of a central hydrophobic block of polypropylene glycol (PPG) ﬂanked by two hydrophilic blocks of PEG (Fig. 2.2 (a)). In our case, we used Pluronic F-127 (Sigma,P 2443), with approximately 101 repeat units of ethylene glycol per PEG block and approximately 56 units of propylene glycol. This surfactant was used in all the projects of surface microﬂuidics (chapters3,4,5), as well as during the active ellipsoid synthesis(chapter6). The silicon oil (BlueStar Silicones) has a viscosity of 100cS.

Anchoring control in passive liquid crystals In this thesis, we have sometimes used 5CB and 8CB (chapter6) as the oil phase. The choice of surfactant determines the disposition of the LC molecules at the interface with the active aqueous suspension (see Fig. 1.11). For the synthesis of smectic ellipsoids, that required a planar anchoring at the interface, we used Pluronic F-127 (chapter6). In the project on active emulsions, where, droplets of active gel were immersed in a continuous phase of 5CB, we used Tween 80 (5 to 10% (w/v); P4780, Sigma) to obtain a conical anchoring (Fig. 2.2 (b)), and a modified version of a phospholipid with a 2000-Da PEG moiety in the polar head (Fig. 2.2 (c)), namely, 1, 2-distearoyl-sn-glycero-3-phosphoethanolamine-N- [methoxy(polyethylene glycol)-2000] ammonium salt (DSPE-PEG1; 880120,Avanti Polar Lipids), to obtain homeotropic anchoring. This surfactant was first dissolved in chloroform/methanol (2:1 v/v) with a final concentration of 25 mg/ml. The lipid solution was then dried for 20 min under a gentle N2 stream, followed by 1 hour under vacuum. The dry lipid cake was then hydrated for 1 hour to a final concentration of 12.5 mg/ml in a phosphate buffer [20mM K2PO4 + 100 mM KCl (pH 7.4)]. Vesicles spontaneously form after a few cycles of sonication and mixing, giving the solution a milky aspect. The suspensions were stored at 4 ◦C and sonicated for a few seconds before their use. Lipid vesicles were dispersed in the aqueous active solution to achieve a final lipid concentration of 0.2% (w/v).

Fluorinated oil and surfactant Preparing oil in water emulsions with microfluidic techniques is more convenient with low viscosity oils, such as fluorinated oils. For sta- bilization of interfaces between water and fluorinated oil, we used the triblock PFTE- PEG600Da-PFTE surfactant (RAN Biotechnologies) dispersed in the oily phase at 2%w/w concentrations.

2.1.2 Basic sample design Once the active gel has been prepared, we need to prepare an observation chamber. Because it is a biological fluid, the gel needs to be protected from air, which could denature the proteins through oxidation. It also needs to be protected from glass, onto which proteins have a tendency to stick. The minimal setup to from an active nematic interface is sketched in Fig. 2.3. It is composed of a block of poly-dimethylsiloxane (PDMS) with a cylindrical well of diameter 5mm and 5mm height. The block is prepared by filling a 3D-printed Polyactic acid (PLA) positive mold composed of cylindrical pillars (Fig. 2.3 (a)), and baking at 70◦C for 3 overnight. The PDMS block is bound onto a bioinert and superhydrophilic polyacrylamide (PAA)-coated glass, using a UV-curing adhesive (Norland, NOA81) (Fig. 2.3 (c)). The pool is filled with 2 µL of active gel, that directly

44 2.1. Active Nematics Synthesis

Figure 2.2 – Most commonly used PEG-lated surfactants. The PEG chains are pointed with red circles. (a) Pluronic F-127. x = 101, y = 56. (b) TWEEN-80. (c) DSPE-PEG-NH2.

Figure 2.3 – Basic sample design (a) A 3D-printed mould made of 5 mm heigh cylindrical pillars of 5 mm diameter is filled with liquid PDMS. (b) PDMS is cooked at 70 ◦C overnight and removed from the mould. (c) The PDMS pool is bound with UV curable glue to a glass slide previously coated with acrylamide brushes. (d)The pool is filled 2 µL of active gel and covered it with 100 µL of 100 cS silicon oil. spreads onto the hydrophilic surface (Fig. 2.3 (d)). As soon as possible, typically a few seconds later, the sessile droplet of active gel is covered with 200 µL of 100 cS. After several minutes at room temperature, the active nematic is spontaneously formed at the flat water/oil interface.

Glass coating

The glass coating is prepared as follows. Clean and activated glass (activation with an alkaline solution or with O2 plasma treatment) is ﬁrst silanized with an acidiﬁed ethano- lic solution of 3-(trimethoxysilyl) propylmethacrylate (Sigma, 440159), which will act as a polymerization seed. The silanized substrates are rinsed with ethanol and deionized water and subsequently immersed in a degassed solution of acrylamide monomers (for at least 2h) in the presence of the initiator ammonium persulfate (APS, Sigma, A3678) and N,N,N0,N0-tetramethylethylenediamine (TEMED, Sigma, T7024), which catalyses both initiation and polymerization of acrylamide. Glass substrates are stored in the polymerization solutions and they are used for up to three weeks.

45 Chapter 2. Materials and Methods

2.2 Imaging and Analysis

2.2.1 Optical setups In this section, we brieﬂy introduce the three main optical techniques employed for the characterization of the experimental samples.

Fluorescence and confocal fluorescence microscopes During the polymerization of microtubules (section 2.1.1), 3% of tubulin is labelled with Alexa 647 fluorescent dye, a bright, far-red fluorescent molecule. The principle of fluorescence microscopy is to irradiate the sample over a specific band of wavelength (excitation wavelength), and simultaneously record the light emitted by the fluorescent molecules. The intensity of the emitted light is orders of magnitude smaller than the excitation beam. Therefore, it is necessary to filter out the excitation light from the observation path of the microscope. To do so, the excitation wavelength is chosen sufficiently far from the emission peak, such that both signals can be separated through a dichroic mirror. A dichroic mirror is a semi-reflective mirror that selectively transmits light of a small range of wavelengths while reflecting others, as shown in Fig. 2.4 (a). The image is recomposed by an objective onto the detector of a highly sensitive camera. Fluorescence microscopy is relatively simple to implement. In the lab, we used a custom-built inverted microscope with a white led light source (Thorlabs MWWHLP1) and a Cy5 filter set (Edmund Optics). Image acquisition was performed with an ANDOR Zyla 4.2 sCMOS camera operated with ImageJ µ-Manager open-source software. The image formed at the camera detector corresponds to the portion of space located at the focal plane of the objective. A point-like fluorescent source located at the focal plane will recombine into a point onto the camera detector. Ideally, a good microscopy image is composed of a black background, and neat fluorescent textures coming from the fluorophores located at the focal plane. However, fluorescent sources located slightly out of the focal plane still contribute to the image. Because they are out of focus, point like sources will be significantly distorted, leading to blurry textures in the final image, as shown in Fig. 2.4 (b). This drawback is inherent to conventional fluorescent microscopes. Thick samples appear blurry because of the out-of-focus fluorescent sources. For sharper imaging of fluorescent samples, one can use a confocal fluorescent microscope. The principle is very similar to a conventional fluorescence microscope, but additional elements are placed along the light path to improve the selectivity of both the illumination and light collection modules. In the light path of the light source, a pinhole is added to reduce the extension of the illumination beam in the sample, prescribing it (ideally) to a point-like source in the focal plane. The image forming in the camera detector corresponds to the emission of the fluorescent molecules located in the excited point-like region. In practice, the source is never perfectly localized, and out-of-focus emissions may still degrade the image. To filter them out, a pinhole is added in the observation path, a shown in Fig. 2.4 (c). Ideally, the image formed at the detector is uniformly black, except in the point-like region illuminated by the point-like source at a given time. The intensity of the point directly depends on the local concentration of fluorophores. A complete confocal image is formed by the recombination of several of such point-like images, as the light source scans the field of view of the camera. Two main strategies

46 2.2. Imaging and Analysis

Figure 2.4 – Fluorescence and confocal fluorescence microscopy (a) Principle of a standard fluorescence microscope. (b) Intermediate thickness sample images of a 20 mm- thick mouse kidney section, labeled with DAPI, Alexa 488-wheat germ agglutinin (membrane stain), and Alexa 568-phalloidin (actin filament staining) imaged with a fluorescence microscope. Adapted from [155]. (c) Principle of a standard confocal fluorescence microscope. Pinholes are added to remove fluorescence signals coming from fluorophores located out of the focal plane. (d) Same sample as (b) imaged with a confocal microscope, for comparison. Adapted from [155].

have been developed. The classical technique, described above, is called laser scanning. In the lab, we used a Leica TCS SP2 laser-scanning confocal microscope equipped with a photomultiplier as detector and a HeNe–633nm laser as light source. A 10x oil immersion objective was employed. The main advantage of this setup is that it can be used with two modes simultaneously: confocal fluorescence, and confocal reflection. The confocal fluorescence mode captures the emission light coming from the fluorophores, as shown in Fig. 2.5 (a). Conversely, the reflection mode captures the light coming from the sample at the excitation wavelength. In our experiments, this technique is remarkably complementary to the fluorescence microscopy because the active nematics appear scattered, as shown in Fig. 2.5 (b). The scattering gives more contrast to the image, which enables to perform reliable Particle Image Velocimetry (PIV) measurements (see section 2.2.4). However, the laser scanning technique is relatively slow. Typically, the frame rate in our experiments was limited to 0.5Hz. This frame rate is not fast enough to image active nematics at high activity. A more recent and faster confocal microscopy technique has been developed. Instead of exciting the sample with a single point-like source, a pattern of several points is drawn onto the focal plane by means of a disk patterned with small holes. The disk spins at high velocity in front of the light source, such that the pattern of holes changes over time. The patterns are efficiently arranged such that the whole sample is illuminated in the shortest possible delay. This technique is called spinning disk confocal microscopy. It has the advantage of being faster (we could image the active nematics at a frame rate up to 5Hz), but the confocality, that is, the quality of the images, is not as good as the laser scanning. The choice of either of the two techniques depended on the type of experiment we wanted to perform.

47 Chapter 2. Materials and Methods

Figure 2.5 – Confocal fluorescence and reflection modes Simultaneous snapshots in fluorescence (a) and reflection (b) modes of active nematics confined in a 140 µm wide channel, acquired with a laser scanning confocal microscope. The fluorescence mode captures the light emitted by the fluorophores embedded in the sample, while the reflection captures the light absorbed by the sample. The reflection mode is more textured, which enables a better tracking of the flows. Scale bar: 50 µm.

Polarized-light microscopy Passive liquid crystals are usually characterized by means of a polarized-light microscope, as it is the most suitable technique to image optically anisotropic (birefringent) materials. Such devices are based on a conventional microscope, supplemented by two polarizers in the light path. A polarizer is an anisotropic optical device that has the property to select a specific polarization of a light beam in the direction of its easy axis. In the microscope, the polarizers are oriented at right angles with each other, as shown in Fig. 2.6. This means that the first polarizer selects the polarization that is perpendicular to the second easy axis. Thus, in the absence of a sample, all the incoming light is filtered out by the crossed/polarizers, and the field of view is completely dark. When introducing a birefringent sample such as a LC cell in between the two polarizers, plane-polarized light splits into two individual wave components polarized in mutually perpendicular planes. The velocities of these components, which are called ordinary and extraordinary wave fronts, are different, and vary with the direction of propagation through the sample. As a consequence, they do not cross the sample at the same speed and become out of phase. When passing through the second polarizers, ordinary and extraordinary wave fronts are recombined by interference. The latter may be constructive of destructive, leading to bright or dark signals, depending on the value of the phase shift. The intensity variations provide information on the local average orientation of the sample constituents. In the lab, we used a Nikon Eclipse 50iPol upright microscope. Images were acquired with an ANDOR Zyla 4.2 sCMOS camera operated with ImageJ µ-Manager open-source software.

2.2.2 Image processing In the following section, we briefly describe the methods we used to characterize the experiments. Most of the image processing was performed with matlab routines, either custom made or taken from the literature and ImageJ plugins. The characterization of active nematics (director field, defects) was entirely developed and kindly shared by Perris Ellis and Alberto Fernandez-Nieves from Georgia Tech. We developed additional simple routines to detect the orientation of defects as well as the spatial distribution of active stresses based on the director field. These programs were extensively used in chapters3,4, and5. Measurements on the flow field were performed with ImageJ plugins. Conventional velocimetry techniques were ready to use with open source plugins. We also developed

48 2.2. Imaging and Analysis

Figure 2.6 – Polarized-light microscopy Adapted from [156].

49 Chapter 2. Materials and Methods simple scripts for speciﬁc applications.

2.2.3 Determining director and defects in active nematics Here we briefly present the program developed by Ellis et al. [117] to detect the director field and the associated topological defects from a fluorescence image of active nematics.

The director field n is extracted using coherence-enhanced diffusion filtering (CEDF) [157], a technique that has been extensively used in biology for the segmentation of cells [158], as well as in computer vision. The purpose of the technique is to infer, from the intensity fluctuations of the fluoresccence image, the local alignment of the microtubules with a pixel resolution. Starting from the raw image, an example of which is given in Fig. 2.7 (a), the first step consists in applying a Gaussian blur, characterized by its standard deviation σ to remove the random noise of the signal. Then, the local orientation u is obtained by finding the direction along which the fluorescence intensity fluctuates the least. u, also called the "molecular director", represents the local orientation of the microtubules. However, it is quite noisy as shown in Fig. 2.7 (c). Thus, the output is filtered with an additional Gaussian blur of standard deviation ρ. The result of this computation is displayed in Fig. 2.7 (d). From u we obtain the nematic order parameter Q = u u 1 1 where is an ensemble average over all the "molecules", or pixels, in a h ⊗ − 2 iβ h·iβ disk of radius β around each point. Q is then diagonalized to recover both the scalar order parameter, S (Fig. 2.7 (f)), and the director field n (Fig. 2.7 (g)) through the following relation: Q = (n n 1 1). In practice, all these operations are performed within S ⊗ − 2 the program and the user has access to three control parameters σ, ρ and β. All three parameters are adjusted manually for each experiment, because the noise intensity and image features may depend a lot on the experimental conditions (objective, illumination, fluorescence efficiency etc). In our experiments, the best results are obtained by keeping σ = 0.5px and varying ρ = 10 − 20px and β = 5 − 6px. The optimization is always controlled by eye. Typically, for each video, we fine-tune these parameters by checking the fidelity of the director field and the defect detection (described below) on a random set of 5 frames. σ, ρ and β are then fixed for all the frames of the experiment.

The topological defects are singularities in the director field. They are characterized by two features. First, the scalar nematic order S is small (S 1). Therefore the program scans the image in the regions of smallest nematic order, the threshold being fixed to S < 0.1 (black spots in Fig. 2.7 (g)). Topological defects are also characterized by their winding number w, which measures the structure of the director field around a defect. The program numerically evaluates = 1 H around a counter-clockwise w 2π ∂φ/∂udu square loop of pixels around the point of interest, where u is the position in the loop and φ the orientation of n. At a defect location, w = s = ±1/2 where s is the topological charge of a defect in active nematics. In brief, the program evaluates w at each pixel that has S < 0.1, and defines a defect if w ∈ ±[0.49, 0.51].

The orientation of a defect is related to the divergence of Q. The body force driving the active ﬂow is fa ∼ ∇ · Q. Explicitly, we can deﬁne the polarity of the positive defect

50 2.2. Imaging and Analysis

Figure 2.7 – Finding the director field and defects in active nematics Step-by- step output to find the director and defects from an active nematic image. (a) Confocal fluorescence image of active nematics confined at the surface of a torus. Inset: Close up of a defect pair. (b) Image from (a) after applying a Gaussian blur with σ = 5px (c) Coherence directions of the tensors formed from the gradient of the image in (b). Black represents 0° and white represents 180° measured CW from the horizontal. (d) Coherence directions of the structure tensors formed by component-wise averaging the gradient tensors formed from image. The Gaussian filter has ρ = 29px. (e) The scalar order parameter S obtained by diagonalizing the Q formed from the directions in image (d). Q is formed for each point by considering the directions of all points in a β = 5px radius. (f) The director obtained by diagonalizing the Q formed from the directions in image. The defects are calculated by considering points of low S and calculating the n-rotation along a path encircling the point. Positive (resp. negative) defects are marked with red (resp. blue) triangles. Adapted from [159].

51 Chapter 2. Materials and Methods by p = ∇ · Q/|∇ · Q|. p is a vector and its orientation ψ denotes the orientation of the positive defect. We compute ψ for a s = +1/2 defect through the following expression:

h∂ Q − ∂ Q i ψ = arctan( x xy y xx ), (2.1) h∂xQxx + ∂yQxyi where h·i denotes an average over the shortest counter-clockwise loop of pixels surrounding the defect. This formula can be extended to the case of a s = −1/2 defect but we do not need it here. In parallel, we compute the norm of the divergence of Q by the following expression:

2 2 1/2 |∇ · Q| = (h∂xQxy − ∂yQxxi + h∂xQxx + ∂yQxyi ) (2.2)

as an estimate of the intensity of the local active stress fa. In Fig. 2.8, we provide a step-by-step example of the computation. From the fluorescence image (2.8 (a)), we extract the director field and detect the positive defect locations (2.8(b)) using the method presented above. Then, we compute ψ and |∇ · Q| not only around the defect cores, but everywhere in the image. The result is displayed in Fig. (2.8(c)). The vector field has the orientation ψ and the arrows are scaled with the magnitude of |∇ · Q|. An evaluation of ψ at the defect sites gives their orientation, displayed by black lines in Fig. (2.8(d)).

The distribution of active stresses is evaluated by the map of ∇ · Q, displayed in Fig. 2.8(c). The relation between the body force driving the active flow and ∇ · Q has only been defined around the core of a defect. However, this result can be extended to any point in the director field. The topological defects are the regions where |∇ · Q| should be maximal, because it is where the distortions are maximal. In Fig. 2.8(c) and particularly in the inset, the map of |∇ · Q| shows bright spots at the positive defects locations. However, we also notice bright lines that are fairly extended in the image. These lines correspond to regions of the director field where distortions are important, but not enough to form a proper defect. We call them "cracks". These cracks will be studied in detail in chapter5 (see Fig. 5.5).

2.2.4 Characterization of the flow field PIV measurements PIV measurements were obtained using confocal images in reflection mode. In this mode, the active nematic layer exhibits textures that can quite efficiently act as tracers for PIV softwares. The images were treated with ImageJ PIV plugin [160]. The data was then processed with custom Matlab codes.

FFT Velocimetry (Chapter3) In chapter3, flow profiles were computed using a custom ImageJ plugin. PIV measurements failed to give reliable measurements close to the walls for narrow channels because of a substantial drop of resolution that we attribute to artefacts in the confocal reflection mode. In particular, the light intensity decreases significantly close to the walls. Because of this, for a system in the shear flow state (active filaments aligned parallel to the walls),

52 2.2. Imaging and Analysis

Figure 2.8 – Defect and Force directions Step-by-step process to infer the directon of local active stress as well as the polarity of positive defects (a) Confocal fluorescence image of an unconfined active nematic layer. Inset: Close-up of a pair of defects. (b) Overlay of the director field. Positive defects are marked with black disks. (c) Map of the active stresses. White arrows represent the vector field ∇·Q. The pink color map overlays the magnitude of |∇ · Q| for a smoother representation. (d) Overlay of positive defects with their orientation taken from the direction of ∇ · Q at the defect site. The black bar is placed in such a way that it forms an arrow where the black disk is the pointer.

the longitudinal velocity Vx(y) was determined as follows. For each y position, a kymograph is built by assembling the image profile along the channel (x coordinate) at different times. The inhomogeneous reflected-light intensity results in traces in the x − t plane of the kymograph, whose slope gives Vx(y). The average value of this slope is obtained by computing the FFT of the kymograph image and finding at which angle the FFT has the highest intensity. We repeat the same process for each y coordinate, resulting in the desired flow profile.

Flow Order Parameter (Chap.4)

In chapter4, the flow order parameter Sˆ is defined and computed as follows. For each frame, we select the pixels located at the center of the annulus (green line) in Fig. 2.9 (a). This pixel ring is then unwrapped to a 360-pixel line, where the intensity of pixel θ ∈ [0, 360] is equal to the average intensity of the pixels located between the angles θ and θ + 1 around the ring. This process is repeated at each time t, and the results are stacked on a kymograph as shown in Fig. 2.9 (b). As the active nematics moves, the intensity gradients of the fluorescence signal follow the positions of the microtubules. If the active nematics is in a transport state at time t, the intensity gradients in the kymograph will exhibit stripes, whose slope (positive of negative) give information on the handedness of the rotation (clockwise or anti-clockwise). The symmetry breaking is visualised by computing the FFT of the time series image around times [t−∆t, t+∆t] with ∆t = 20s. ∆t needs to be large enough to actually capture spatio-temporal dynamics, and small enough to get a suitable resolution in time. A typical FFT image for a clockwise transport is shown in Fig. 2.9 (c). The stripes with a positive slope in the time series result in a cloud of points with a negative slope in the FFT. A typical FFT image for a turbulent phase is shown in Fig. 2.9 (d). The intensity is uniform in all directions. A typical FFT image for a counter-clockwise transport is shown in Fig. 2.9 (e). The stripes

53 Chapter 2. Materials and Methods

Figure 2.9 – Flow order in the switching state. This figure presents the computation of the flow order parameter based on the time series of a pixel ring at the center of the annulus. (a) Fluorescence micrograph of an active nematic confined in a 160µm annulus. The pixel line used to construct the kymographs of the annulus dynamics is overlaid in green (b) Space-time plot of the annulus center dynamics. (c) Typical FFT image for a clockwise transport. The stripes with a positive slope in the time series result in a cloud of points with a negative slope in the FFT. (d) Typical FFT image for a turbulent phase. The intensity is uniform in all directions. (e) Typical FFT image for a counter-clockwise transport. The stripes with a negative slope in the time series result in a cloud of points with a positive slope in the FFT. (f) Symmetry Breaking parameter Sˆ as a function of time compared with the PIV measurement of the azimuthal velocity for a 160µm wide annulus in the switching state. Sˆ compares the integrated intensity of the FFT image between the top-right and bottom right corners. Sˆ = +1 (resp. Sˆ = −1) corresponds to clockwise CW (resp. counter-clockwise CCW) flows.

with a negative slope in the time series result in a cloud of points with a positive slope in the FFT. We take advantage of the fact that the FFT image is symmetric with respect to its center, and divide the FFT image in 4 panels as shown in the inset of Fig. 2.9 (f). The top-left panel is the symmetric of the bottom-right, an the bottom-left panel is the symmetric of the top-right panel. As a consequence, information regarding the transport state is available in the top-right and bottom-right panels. Looking at the FFT image for a clockwise transport Fig. 2.9 (c) one notices that the intensity of the bottom-right panel, ˆ I− is higher than the intensity in the top-right panel, I+. We define S as the normalized 2 2 ˆ I−−I+ intensity difference between the 2 panels: S = 2 2 . As such, positive (resp. negative) I++I− values of Sˆ corresponds to clockwise CW (resp. counter-clockwise CCW) flows. The closer to | Sˆ |= 1, the stronger the symmetry breaking is. Typically, | Sˆ |= 0.5 already corresponds to highly directional flows, while | Sˆ |= 0.2 the flow is considered chaotic.

Theoretical ﬂow ﬁelds (Chapter4)

Positive and negative defects have been proven to generate ﬂow ﬁelds of the following form [161].

54 2.3. Numerical methods (chapter3)

α va = {[3(R + r) + r cos 2φ]ˆx +,x 12η α va = [r sin 2φ]ˆy +,y 12η αr 3 R va = {[( R − r) cos 2φ − cos 4φ]ˆx −,x 12ηR 4 5 αr 3 R va = − [( R − r) sin 2φ + sin 4φ]ˆy −,y 12ηR 4 5 We have computed these flow patterns in Matlab setting the ratio α = 1, = 0 8. 12η R . The half-width of the constructed images corresponds to r = 1. We have then placed each source at a distance d = 0.4 from the image center. The flow fields have been rotated in order to account for the observed orientation of the defects. The corresponding plots are displayed in Fig. 4.4.

2.3 Numerical methods (chapter3)

This section summarizes the numerical model used by Rian Hughes, from Oxford Univer- sity, to perform simulations that match our confined active nematics experiments. The model of the microtubule-motor system is based on a continuum description of a two- dimensional, active gel [2, 162–164]. The fields that describe the system are the total density ρ, the velocity u, and the nematic tensor Q = 2q(n ⊗ n − I/2), that describes both the orientation (n) and the magnitude (q) of alignment of the nematogens (in this model, q is an equivalent representation of the scalar order parameter defined in Eq. 1.2, with 2q = S). The nematic tensor is evolved according to the Beris-Edwards equation [165]

(∂t + u · ∇) Q − SA = ΓQH, (2.3) where SA = ξE − (Ω · Q − Q · Ω) is a generalised advection term, characterising the response of the nematic tensor to velocity gradients. Here, E = (∇u + ∇uT )/2 is the strain rate tensor, Ω = (∇uT − ∇u)/2 the vorticity tensor, and ξ is the alignment parameter representing the collective response of the microtubules to velocity gradients. Γ is a rotational diffusivity and the molecular field H = δF + I Tr δF , models the Q − δQ 2 δQ relaxation of the orientational order to minimise a free energy F. The free energy includes two terms. The first is an elastic free energy density, 1 ( Q)2, 2 K ∇ which penalises any deformations in the orientation field of the nematogens and where we assume a single elastic constant K. We note that the free energy functional does not include any Landau-de Gennes bulk free energy terms: all the ordering in the simulations arises from the activity [166]. This is motivated by the fact that there is no equilibrium nematic order in the experimental system without ATP (i.e. in the absence of active driving). The second contribution to the free energy is a surface anchoring, 1 Tr(Q Q )2. 2 W − D To correspond to the experiments QD is chosen so that the director prefers to align parallel to the boundary walls. The strength of anchoring at the boundaries, W, is set to values corresponding to weak anchoring so that the nematogens can re-orientate at the walls to allow defects to form there.

55 Chapter 2. Materials and Methods

The total density ρ satisﬁes the continuity equation and the velocity u evolves according to

ρ(∂t + u · ∇)u = ∇ · Π, (2.4) where Π is the stress tensor. The stress contributions comprise the active stress Πactive = −αQ where α is the activity coefficient, viscous stress Πviscous = 2ηE, where η is the viscosity, and the elastic stresses Πelastic = I 2 H + Q H H Q Q δF , where −P − ξq · − · − ∇ δ∇Q = K ( Q)2 is the modified pressure. Eqs. (2.3)–(2.4) were solved numerically using P p − 2 ∇ a hybrid lattice-Boltzmann method [89, 167]. In the experiments the microtubules slide over the walls and therefore no-stick boundary condition were imposed on the velocity field. The equations (2.3-2.4) of the manuscript are solved using a hybrid lattice-Boltzmann (LB) method. A finite difference scheme was used to solve equation (2.4) on a five-point stencil to discretize the derivatives on a square grid. These are coupled to Navier-Stokes equations, that are solved using lattice-Boltzmann method with the Bhatnagar-Gross- Krook (BGK) approximation and a single relaxation time for the collision operator [168]. The time integration is performed using PECE predictor-corrector method. The discrete space and time steps are chosen as unity for the LB method. The algorithm is implemented using the C++ programming language. All simulations were performed on a 1800 × w rectangular grid, where w varied from 30 to 160 lattice sites. The boundary conditions for the velocity field along all walls and corners are free-slip. The boundary conditions for the director field are weak planar along the channel walls, which is implemented via the free energy term 1 Tr(Q Q )2. The 2 W − D corners have QD such that the director field aligns at an angle of 45 degrees to the length of the channel, on the front-right and back-left corners, and at an angle of 135 degrees on the front-left and back-right corners. The left and right walls have no free energy term applied, and have Neumann boundary conditions. The parameters used in the simulations are given in Table 2.2. Initially, we chose parameters in a range that has previously been successful in reproducing the dynamics of experiments in microtubule bundles [169,170], and then further refined the parameters via a search in phase space. All simulations were run for up to 300,000 simulation time steps. Some simulations were performed up to 1,000,000 time steps to further test the stability of the simulations. The initial velocity field was set to zero everywhere in the domain. The orientation of the director field was initialised at an angle of either 0, 35 or 90 degrees to the length of the channel, up to some noise. The noise was implemented using the uniform_real_distribution() function in the standard C++ library. In all simulations, the parameters were set to lattice units. The figures of the simulations in Fig. 3.1 and Fig. 3.2 (a-c) were created using the programme Paraview. In Fig. 3.1, the director field was overlaid on the order parameter field q. In Fig. 3.2 (a-c), the director field was overlaid on the vorticity field. The kymograph of the velocity components Vx and Vy in Fig. 3.2 (d-e) was created using Matlab, by calculating the mean value of the velocity components in a subsection of the channel at any given time. The distributions of the defect positions across the channel width in simulations shown in Fig. 3.5 and 3.9 were created using Matlab. Each bar represents the normalised number of defects along a strip of lattice sites parallel to the

56 2.4. Experimental techniques channel length, of width one lattice site. The supplementary videos of the simulations were created using Paraview, and any defect tracking was created by overlaying the Paraview images with markers from textscMatlab.

In the simulations, λi was calculated as follows. At any given channel width and timestep, the distance between each +1/2 defect and its nearest neighbour was calculated. The mean of this quantity was then taken, to give the mean defect-defect distance at a

fixed time. Then an average over time was taken to give λi for a single simulation. This process was repeated for three different simulations with different initial conditions. The initial conditions were distinguished by the initialisation angle of the director field, which was selected to be 0, 45 and 90 degrees to the length of the channel. The mean value of these λi’s was then taken to give the value plotted in Fig. 3.4. The standard deviation of these values were used to get the error bars.

Parameter Symbol Value Alignment parameter ξ 0.9 Anchoring strength W 0.002 Total density ρ 40 Rotational diﬀusivity ΓQ 0.4 Elastic constant K 0.015 Viscosity η 1/6 Channel length LX 1800 Channel width LY 30 - 160 Activity α 0.0072 - 0.01

Table 2.2 – Parameters used in the simulations. The values for α are 0.0072, 0.0075, 0.0082, 0.0092 and 0.01. All values are in LB units.

2.4 Experimental techniques

In this section, we describe the experimental setups developed during the PhD.

2.4.1 Surface Microﬂuidics (Chapter3,4 and5) In this section, we present the solution developed in the lab to laterally conﬁne an active nematic interface.

Grid manufacturing The grids are printed using a two-photon polymerization printer, a Nanoscribe GT Pho- tonic Professional device, with a negative-tone photoresist IP-S (Nanoscribe GmbH, Ger- many) and a 25× objective. The grids were directly printed on silicon substrates without any preparation to avoid adhesion of the resist to the substrate (plasma cleaner of the substrate, for example, would increase the adhesion). After developping 30 minutes in Propylene Glycol Monomethyl Ether Acetate (PGMEA 99,5%, Sigma Aldrich) and 5

57 Chapter 2. Materials and Methods minutes in isopropanol (Technical, VWR), a batch polymerization is performed with UV- exposure (5 min at 80% of light power). After printing onto a silicon wafer, the grids are bound to a vertical glass capillary with a UV-curable glue. The capillary is then delicately manipulated to detach the grids from the printing support. The grids are washed in three steps (iso-propanol, DI water, ethanol) and dried with a nitrogen stream before each experiment. The thickness of the grids is 100µm, to ensure good mechanical resistance. In chapter3, we have used grids with rectangular openings 1.5 mm long and widths ranging from 30 to 300µm. Each grid contains different channel widths so that simultaneous experiments can be performed with the same active nematic preparation, thus ensuring that material parameters remain unchanged when comparing different confinement conditions. In chapter4, we have printed annular geometries with varying breadth, as shown in Fig. 2.10 (a). The geometries tested belong to the class of 2D handle-bodies. They are composed of connected elementary annuli. We will refer to them by the use of their genus number, g, which counts the number of holes of a given handle-body: an annulus will have g = 1, two connected annuli will have g = 2 and three connected annuli will have g = 3, as shown in Fig. 2.10(b). The manufacturing of handle-bodies adds technical constraints. It requires to build additional overhangs, bridging the central disk of the annuli to the main body of the grids, as sketched in Fig. 2.10 (c). In this work, we have studied the dynamics of active nematics confined in handle-bodies up to g = 3.A scanning electron microscope (SEM) image of such a grid with g = 2 handle-bodies is displayed in Fig. 2.10(d). The micropatterning has a resolution of around 50 nm, much higher than the needs of the experiment. The boundary conditions also appear to be very well defined. We never observe microtubules sticking to the wall. As shown by the flow profiles measured in the shear state, the shear profile extends up to the wall, attesting for a free slip boundary condition. However, the wetting properties of the grid are still under scrutiny. In our experience, the surface of the grids deteriorates after the first use, therefore the samples are considered single-use.

Experimental protocol

The active nematic layer is first prepared using an open-cell design (2.1.2), in which 2µL of the active aqueous microtubule-mixture was placed inside a custom-made pool of 5mm diameter and was covered with 60µL of 100cS silicon oil. Within 30 minutes, an active nematic layer extends over the whole surface of the pool. The layer of active nematic is confined in rectangular enclosures by means of micro- printed polymer grids of 100µm thickness that are placed in contact with the oil/aqueous interface. The active nematic is in contact with the active bulk solution underneath, thus ensuring activity and material parameters are equal in all channels. A detailed sketch of the experimental protocol is available in Fig. 2.11. The grids are 100 µm thick, while the thickness of the active nematic layer is of the order of a few microns. When the grid comes to the interface, we observe that the bound interface tends to lie somewhere in between the top and bottom of the grid, as shown in Fig. 2.11(c). When the grid impacts the interface, uncontrolled flows may destabilize the active nematic layer. These flows mostly depend on the parallelism upon impact, but also on the wetting properties of the grid. The parallelism is relatively easy to control with micro-screw adjustment.

58 2.4. Experimental techniques

Figure 2.10 – Polymer grid design and fabrication (a) Sketch of a grid design in the case of annuli experiments. 6 Annuli with equal inner radius Ri and varying outer radius Ro are printed onto the same grid. In order to attach the inner disks to the grid, 4 overhangs per annulus are added on top of the grid. These overhangs have not been displayed here for more clarity (b) Sketch of the detailed design of an annulus, showing the overhangs in top and side views. (c) Scanning Electron Microscope (SEM) image of a grid containing 6 g = 2 handle-bodies (ﬁgure-of-eight). As the picture is taken from the top with a small tilt allowing to clearly visualize the overhangs. In order to reduce the printing time and lighten the grid, additional circular holes have also been included around the main channels. Scale bar: 500 µm.

y (a) Micro-screw (b) h

z L x Nematic interface y PDMS pool w

micro-screw (c) Silicon oil z y Glass plate Channels

Fluorescence Active solution imaging Glass plate Figure 2.11 – Experimental setup (a) A polymer grid with rectangular openings is placed, by means of a micropositioner, in the custom-made elastomer pool that contains the aqueous active ﬂuid and the passive oil. (b) Top view with a sketch of the grid including the relevant spatial dimensions. (c) The grid is placed in contact with the oil/water interface. In contact with the grid, the active nematic layer forms only inside the grid openings.

59 Chapter 2. Materials and Methods

Advantages The active gel is prepared by mixing more than ten species at very small volumes, down to 0.2 microliters. With conventional lab equipment, these operations may lack repro- ducibility. For instance, even with the simplest experimental setup, the one used to study unconfined interfacial active nematics, a measure of the defect density may vary by up to 100% when studied with active material prepared in different batches. Therefore, for a quantitative analysis, we need to make all the measurement with the same batch. The problem is that each batch has a lifetime of just a few days, limiting the number of measurements that can be done with the same batch of active material. The measurements for laterally confined active nematics are especially long, since we need to ensure that the grid is correctly positioned at the interface and that the active nematic phase is well established within the channel. Overall, a complete measurement with different channel widths and four different activities took a full week. Taking into account the lifetime of the prepared protein suspensions, this sets a practical limit to the number of experiments for each system configuration (channel width and [ATP]). In order to comply with these constraints, a choice has been made to design an experiment that gives the most reliable results with a single batch of active material. We would like to stress some advantages of the solution proposed in this manuscript. First, channels of different widths are printed onto the same grid, so that all channels are in contact with the same bulk active material underneath. This is crucial in these experiments, where having a different bulk reservoir for each channel would change the surface to volume ratio from one channel to another and consequently introduce a variability in motor and ATP concentrations. On the contrary, in our configuration the bulk is a good buffer as all species are well mixed and uniform under the channels. In addition, the grid is bound to the interface after a condensation period of 45 minutes, leaving enough time for a thick active nematic layer to form at the interface. As a consequence, the initial condition is an active nematic layer of the same thickness and composition in all channels.

2.4.2 Active emulsions: droplets, shells and ellipsoids Introduction to emulsions Emulsions are composed of a dispersion of small liquid droplets into another liquid phase, immiscible with the ﬁrst one. The thermodynamical equilibrium corresponds to a state where interfacial tension, proportional to the contact area between the two phases, is minimized. As a consequence, an emulsion will always tend to destabilize, small droplets merging with bigger ones, up to the complete phase separation of the two liquids. This process is usually relatively fast. When preparing a vinaigrette, everyone has experienced that after a few minutes oil and vinegar fully separate. However, phase separation can be drastically slowed down by the addition of surfactants. Surfactants are composed of amphiphilic molecules that preferentially sit at the interface, thereby reducing the surface energy. Emulsions of water and oil stabilized by sophisticated surfactants may remain stable for several months: instead of a thermodynamical equilibrium, the emulsion is at equilibrium kinetically. The decrease of surface tension depends monotonically on the surfactant concentration, up to a critical point called the critical micellar concentration (CMC). The concentration of surfactants at the interface results from a equilibrium

60 2.4. Experimental techniques with free surfactant molecules in the bulk. Above CMC, exceeding surfactant molecules self-assemble into micelles within either of the two continuous phases, leaving the concentration of free surfactants (and the surface concentration) unchanged. An emulsion of oil droplets in water is usually referred to as a direct emulsion, and the opposite conﬁg- uration is called inverse emulsion [171]. In the next section, we will brieﬂy introduce the conventional techniques to produce controlled emulsions in the lab.

Microfluidic techniques There are two main microfluidic techniques to produce emulsions of controllable size. The first one, and most common, relies of the fabrication of small circuits out of a polymer called polydimethylsiloxane (PDMS) [172]. PDMS chips have several advantages. First, the design is very versatile and a large variety of tasks can be achieved within a single circuit. Second, once a design has been optimized, it is relatively cheap and extremely fast to reproduce it in large quantities with highest fidelity. The other branch of microfluidics relies on glass circuits [173]. A typical device is composed of glass capillaries fitted into one another, each of them being connected to fluid circuits controlled by syringe pumps. The main advantage of this technique is its chemical versatility and the possibility to withstand highier pressures (thus more viscous fluids) than PDMS. Each capillary is prepared separately before the assembly of the device, allowing a good control over its geometry and surface treatment. In the next part, we will describe the canonical setups to produce simple emulsions and doube emulsions (shells).

Principle of glass microﬂuidics

The simplest setup to produce single emulsions, described in Fig. 2.12 consists in a cylindrical capillary embedded within a large square capillary. The inner capillary, also called injection capillary, is filled with oil and the outer is filled with water (and surfactant). The flow rates of both phases is controlled by syringe pumps. In Fig. 2.12 (b) and (c), we can see that the exit of the inner capillary is tapered. The two immiscible liquids meet at the end of the tapered region in an unstable parallel stream. This configuration is called co-flowing. Usually, droplets are created with a diameter that is larger than the tip of the cylindrical capillary. The size is controlled by the relative flow rates. In the lab, we have used this technique to produce direct emulsion of 100 µm droplets with various oil compositions, from fluorinated oils to passive liquid crystals. As a matter of fact, one advantage of glass microfluidics is its versatility. Once assembled, a device is reusable for months over a wide range of solvents. In order to produce double emulsions, we need to add another capillary downstream, as shown in Fig. 2.13. The additional capillary is also tapered, with a larger opening than the first one. From now on, we will refer to it as the collection capillary. It is introduced at the opposite end of the square capillary, such that the two conical openings face one another at a short distance. The device is connected to three fluids circuits. The first one, referred to as the Inner, contains the liquid phase that will end up at the core of the double emulsion. It is connected to the injection capillary. The second one, referred to as Middle, contains the phase that will form a shell in the double emulsion (in our case, the oil phase). It is connected to the left end of the square capillary. In production regime,

61 Chapter 2. Materials and Methods

Figure 2.12 – Single emulsion setup (a) Sketch of the setup. The device is composed of a square glass capillary bound to a glass slide with epoxy glue. A cylindrical capillary is inserted within the square one. The cylinder is tapered to its end. Both capillaries are connected to independent fluid circuits containing the immiscible constituents of the emulsions. Each fluid is injected through the capillaries by means of syringe pumps. The phases meet at the tip of the cylinder forming the emulsion. (b) Close up of the junction showing the emulsification process. (c) Bright-field image of the tapered extremity of a cylindrical capillary. More details on its manufacturing are available in 2.4.2. Scale bar: 50 µm.

these two phase will co-flow producing a jet in between the glass tips. Finally, the Outer phase is connected to the other end of the square capillary. The role of this phase is to flow-focus the incoming jet of Inner and Middle phases into the collection capillary. When doing so, the Middle phase is isolated from the glass walls by the immiscible Outer phase. Eventually, the co-flow destabilizes leading to the formation of the double emulsion. In practice, many details must be finely tuned. The dimensions of the cylindrical and square capillaries are critical, as well as the shape of the conical tapers, their relative distance and their co-axial alignment. A specific surface treatment must be performed separately onto the injection capillary to ensure a good adhesion of the middle phase. In the next section, we provide specific details to the synthesis of smectic shells.

Smectic ellipsoids synthesis (chapter6)

Smectic ellipsoids are obtained by ﬁrst synthesizing smectic shells using the setup described in Fig. 2.13. We provide here a step-by-step protocol to build the glass microﬂu- idic device, and run the production of shells.

Matching a pair of capillaries is the ﬁrst step. The outer diameter of the cylindrical capillary (used to create the injection and collection capillaries) has to precisely match the inner dimension of the square capillary, minus 25 µm. If the gap between the two is too large, the connection will degrade the coaxial centering of the injection and collection capillaries. If it is too narrow, the inner capillary will stick to the square. In the lab, we match calibrated square capillaries of 1.02 ± 0.01mm inner opening with cylindrical capillaries of 1mm outer diameter. The matching is performed manually, by choosing the inner capillary that ﬁts the most tightly without sticking.

62 2.4. Experimental techniques

Figure 2.13 – Double emulsion setup Sketch of the setup. The device is similar to the single emulsion device, except that an additional cylindrical capillary, called collection capillary is inserted in the right end of the square capillary. The injection capillary is connected to the inner phase (yellow). The hermetic inlet is made of a syringe needle glued with epoxy. The left-end of the square capillary is filled with the middle phase (pink). The right-end of the square capillary is connected to the outer phase (green). The exit of the collection capillary is connected to a vial by a flexible tube (blue). Coloured dashed lines represent tubing connections that has been cut from the sketch for clarity. In the production of smectic shells, the inner phase is a 2 % in weight solution of Pluronic in water. The outer phase is a 60 % glycerol, 2 % Pluronic water solution and the middle phase is 8CB. Inset: (adapted from [174]) Close-up of the microfluidic junction. Inner and middle phase co-flow at the exit of the injection capillary, forming a jet. The outer phase flow-focuses the jet into the collection capillary. The jet eventually destabilizes forming the desired double emulsion.

63 Chapter 2. Materials and Methods

Glass tapering is performed using a pipet puller, commonly used in biology to prepare micro-pipets for cell handling or glass electrodes. The device is composed of a 5 × 5mm2 box-shaped heating filament that melts the capillary locally. The latter is held by two arms that simultaneously pull both ends apart with a controlled force. Upon melting of the central region, the two sides of the capillary move away from each other and the center acquires a hourglass shape. Eventually, the hourglass breaks leaving two conical tips. In the lab, we used a Sutter P-1000 pipet puller. We adjusted the parameters (filament temperature, pulling force, etc) to obtain rather blunt tips (typically, not longer than twice the initial outer diameter of the capillary). This shape guarantees a good stability of the flow.

Glass cutting is performed using a micro forge. When the capillaries come out of the pipet puller, the conical tips are clogged by residual glass. Moreover, the diameter of the opening has to be carefully controlled. One of the two halves of the capillary will become the injection capillary, with an opening of 60 µm diameter. The other half will be the collection capillary, with an opening of 120 µm. The micro forge enables to cut the tips at the desired dimension with good precision. In the lab, we used a Narishige MF-830. The device consists in a glass droplet heated by a filament and controlled by a three axes micro-screw. The process is quite simple. The droplet is placed right below the tip, along its symmetry axis. It is then heated and moved up. As it enters in contact with the glass tip, the droplet wets it forming a well-defined meniscus. The position of the meniscus will set the cutting line. We therefore raise the cutting line along the cone until it reaches the desired diameter. Then, the glass bead is rapidly cooled down. Once solidified, it is pulled away from the capillary, until the tip breaks along the cutting line.

Glass treatment is later performed on all the capillaries. The square and collection capillaries are simply carefully rinsed with acetone, ethanol and DI water. To prevent any deposition of dust, they are stored in a sealed box. In order to ensure the wetting of the (organic) middle phase on the tip of the injection capillary, the latter has to be treated with a robust hydrophobic coating. To do so, it is treated with a solution of Octadecyltrichlorosilane (OTS). It is a silane molecule with a very long aliphatic chain. The silane group chemically bonds to the glass, exposing the hydrocarbon (hydrophobic) tail to the surface. In practice, the glass tip is dipped for 5 min into a 2 mmol · L−1 OTS solution 4 : 1 in volume hexane-chloroform mixture. For example, we mix 8 mL of hexane with 2 mL of chloroform and 20 µL of OTS. The tip is then dried with nitrogen, rinsed with chloroform, heated at 120 ◦C for 5 min and then at 200 ◦C overnight.

Device assembly starts by gluing the square capillary onto a large glass slide. The injection and collection capillaries are then introduced at both ends and moved close to each other. Ideally, they should be separated by a distance equal to the diameter the opening in the collection capillary (120 µm in our case). The tricky step here is the alignment of the tips. The injection and collection openings should be both in focus, at the center of the glass capillary, and aligned coaxially. Capillaries are rotated and translated until the best compromise is reached. They are then glued onto the glass slide, carefully controlling the alignment at all times (pouring glue may move the capillaries). Once the microfluidic setup is fixed, we need to connect the inlets to the fluid circuits.

64 2.4. Experimental techniques

Figure 2.14 – Assembled device Picture of a double emulsion production device before the initialization step. The inlets are not connected to the ﬂuid circuits yet. Scale bar: 1 cm

The connections need to be perfectly hermetic as the pressures imposed by the pumps are quite large. The three inlets are built out of syringe needles and epoxy glue, as shown in Fig. 2.14. The collection capillary is extended by a ﬂexible collection tube.

Initialization . We connect all three inlets with water syringes. We flow water for 5 min to remove any air bubble and residual dust. Then, we prepare all the desired solutions. The inner phase is a 2 % in weight solution of Pluronic in water. The outer phase is a 60 % glycerol, 2 % Pluronic water solution. Both water phases go through a 0.45 µm pore size filter before injection, to avoid dust. The middle phase, in our case, is 8CB. All three syringes are connected to three independent pumps (Harvard Apparatus Elite 11). They are finally plugged to the inlets of the device by flexible tubing, making sure the connections do not introduce new bubbles.

Shell production . Once everything is connected, the syringe pumps are activated. The flow rates are 7500 µL · h−1 for the outer phase, 400 µL · h−1 for the middle phase, and 1000 µL · h−1 for the inner phase. We would start by switching on the outer phase, then the inner, and finally the middle phase. In practice, the flow rates have to be adjusted to each device, and may vary from one experiment to the other. They are fine-tuned by looking at the output in the collection capillary by means of a high speed camera. Two stable flow regimes are observed. The first one, displayed in Fig. 2.15 is called the dripping regime. The middle phase pinches of the tip of the injection capillary is a very reproducible manner, leading the a highly monodispersed double emulsion. The second regime, displayed in Fig. 2.15 (bottom) is called the jetting regime. The co-flow of inner and middle enters the collection capillary forming a parallel jet. Eventually, the jet destabilizes through the Rayleigh-Plateau instability forming shells with a more heterogeneous size. Although the dripping regime is preferred, both are acceptable in our experiments, for the monodispersity of the emulsion is not critical. During the production, the sample is continuously heated through a hair dryer to keep

65 Chapter 2. Materials and Methods

Figure 2.15 – Stable production regimes Typical dripping (a) and jetting (b) regimes observed during the shell production. Scale bar: 100 µm. Adapted from [63].

the temperature between 33.5 ◦C and 40.5 ◦C, to maintain 8CB in a nematic phase. The shells are collected in a vial ﬁlled with inner solution. The collection vial is also heated above 33.5 ◦C during collection.

From smectic shells to ellipsoids. Once the production is ﬁnished, the vials are kept at room temperature, which leads to the transition of the nematic shells to the smectic A phase. The transition is abrupt and most shells burst into ellipsoids. The latter are metastable and tend to become spherical after 24 to 48 hours approximately. Nevertheless, the elongated shape of the droplets remains long enough to be manipulated and observed.

Active ellipsoids synthesis. Ellipsoids are then dispersed in the active system. To do so, we sample 1 µL of the supernatant of the vials containing the shells (8CB is less dense than water, so ellipsoids tend to ﬂoat). The sample is mixed with 5 µL of active solution, within a square capillary of 0.6 mm opening, previously coated with acrylamide (section

66 2.4. Experimental techniques

2.1.2). The ends of the capillary are kept open so that active gel can be reintroduced over time to maintain the activity. Within few hours, a highly active two-dimensional nematic liquid crystal covers the entire surface of smectic droplets.

Chapter 3 Active nematics under lateral Conﬁnement

3.1 State of the art

As said in the introduction of the manuscript, active matter systems and particularly active nematics exhibit a fascinating adaptability to external constraints (see section 1.1.3). Here we concentrate on the microtubule-based active nematics (described in section 1.3.1) confined to two-dimensional channels. As said in the scientific objectives (section 1.4), early theoretical work predicted that laterally-confined active nematics undergo an instability to spontaneous laminar flow when the channel width reaches a typical length scale that depends on the strength of the activity [76]. This prediction has been recently confirmed in experiments with spindle-shaped cells [112]. On the other hand, simulations have predicted that, at higher activities or in wider channels, a structured ’dancing’ state can be stable in active nematics [139]. Our aim here is to assess, in a well-controlled and tunable experimental system, and with the support of numerical simulations, the role of confinement in the patterns and dynamics of an active nematic. In particular, we explore the emergence of a new length scale different from the active length that characterizes the unconfined systems. We find a rich dynamical behaviour, summarised in Fig. 3.1. More specifically, we uncover a defect-free regime of shear flow in narrow channels. This regime is unstable with respect to the nucleation of short-lived defects at the walls. By increasing the channel width, defect lifetime increases, developing a spatio-temporal organization that corresponds to the predicted state of dancing vortical flows [139], before full disorganization into the active turbulence regime for still wider channels, as is typical of the unconfined active nematic. We stress the close interplay between the velocity field and the defect dynamics, and highlight the emergence of a new length scale that, contrary to the classical active length scale, does not depend on the activity level but merely on geometrical parameters.

3.2 Experimental setup: surface microﬂuidics

The active nematic was prepared using an open-cell design (see section 2.1.2), in which 2µL of the active aqueous microtubule-mixture was placed inside a custom-made pool

Chapter 3. Active nematics under lateral Conﬁnement

(g) h

(a) (d) (e)

w L

(b) (e) (f) 1 q y (c) x 0 Figure 3.1 – Flow States. (a-c) Confocal fluorescence micrographs of an active nematic interface confined in channels of different widths. Scale bar: 100µm. (d) Top view of the experimental setup including the relevant spatial dimensions. A polymer plate with rectangular openings is placed, by means of a micropositioner, at the interface between the active fluid and silicon oil, thus constraining the existing active nematic. (e-g) Corre- sponding simulations of the experimental system. Streaking patterns follow the director field tangents. They are produced using a Line Integral Convolution of the director field [175] with Paraview software.The color map corresponds to the computed nematic order parameter, q. Panels a and e correspond to the active turbulence regime. Panels b and f illustrate the effects of moderate confinement, forcing a new dynamical regime of the defects. Panels c and g correspond to strong confinement, where the filaments are organized into a unstable alignment regime. of 5mm diameter and was covered with 60µL of 100cSt silicon oil. Within 30 minutes, an active nematic layer extends over the whole surface of the pool. The layer of active nematic is confined in rectangular enclosures by means of micro-printed polymer grids (see section 2.4.1) of 100µm thickness that are placed in contact with the oil/aqueous interface (see Fig. 3.1). The active nematic is in contact with the active bulk solution underneath, thus ensuring activity and material parameters are equal in all channels. A detailed sketch of the experimental protocol is available in Fig. 2.11.

3.3 Computational Setup

The simulations were performed by Rian Hugues, a graduate student in the laboratory of Julia Yeomans at Oxford university. The details of the computational setup are extensively described in section 2.3.

3.4 Results and Discussion

The main experimental control parameters of our system are the channel width, w, and the concentration of ATP, which determines the activity. In this section, we describe experiments and simulations showing how both the defect landscape and flow patterns evolve as w is increased. We identify two well defined regimes: a shear flow regime, observed for w < 80 µm, which is transiently defect-free (Fig. 3.2 (a-b) and 3.1, 3.2 and 3.3), and the dancing defects regime, for w > 90 µm (Fig. 3.2 (c) and 3.4 and 3.5). The transition between these two regimes is not sharp. For values of w in

70 3.4. Results and Discussion

Figure 3.2 – (a-d) Flow patterns. Each panel is composed of (top) a snapshot of a typical confocal fluorescence image of the active nematic, with defect locations overlaid, (center) experimental PIV measurement of the flow patterns, colored by the normalized vorticity, with velocity streamlines overlaid and (bottom) numerical simulations colored by the normalized vorticity. Lines in the simulations correspond to the director field. (a) Shear flow state, λ/w ∼ 1, in the shear phase. (b) Shear flow state, λ/w ∼ 1, in the instability phase. (c) Dancing state, λ/w ∼ 0.7. Scale bars: 50 µm. (d-e) Periodic instabilities that disrupt the shear state. Experimental kymographs for Vx averaged along the channel and Vy averaged across the channel are shown in (d) while the corresponding simulations appear in (e). The color encodes the normalized velocity components. One instance of stable shear flow regime and one instance of transversal instability are sketched.

71 Chapter 3. Active nematics under lateral Conﬁnement

λ/w 2 2

1.5 1.5 high activity medium activity low activity 1 1 la/w

0.5 0.5

Shear Dancing Turbulent w(μm) 0 0 // 00 100 200 200 400 300 600 400 800500 1000 Figure 3.3 – Defect Spacing. Experimental mean defect spacing, λ, rescaled by the channel width, w, as a function of w. Different colors correspond to different activities (high, medium and low correspond to and ATP concentration of c0 = 1.5 mM, c0/5 and c0/10, respectively). The error bars correspond to the standard deviation for measurements at 10 random times for each experiment. The non-scaled data is displayed in Fig. 3.6. The dotted line corresponds to a fit with w−1/2 .The continuous lines correspond to λ/w = la/w for each experiment, with la being the active length scale corresponding to the value of λ in the unconfined case.

the range 80 − 90 µm, the direction of the shear is not uniform along the channel, but rather composed of patchy domains where shear flow spontaneously arises with a random direction, and is then quickly disrupted by instabilities (see 3.6 and 3.7). We find that the relevant parameter setting the dynamic state in the system is λ/w, where λ is the mean defect separation. The latter is defined as λ = (Lw/N)1/2, where L is the length of a given channel, w its width, and N the number of defects averaged in q time. For unconfined active nematics, this length scale coincides with la = K/α, often referred to as the active length-scale, which determines the vortex size distribution and corresponds to the mean defect separation [60,89,176]. However, we find that, in confined active nematics, λ is no longer equal to la. Instead, it significantly decreases with the channel width (see Fig. 3.3 and Fig. 3.6).

3.4.1 A defect-free state: shear ﬂow disrupted by instabilities

The shear ﬂow regime is a defect free state that appears for λ/w > 1 (Fig. 3.3), which is experimentally realized for w < 80 µm. The active material is primarily aligned parallel to the walls over distances that can persist along the whole channel as shown in Fig. 3.2 (a). A global shear deformation is observed, with ﬂows along the channel (Fig. 3.7). The maximum velocities are measured at the walls, with opposite signs, and the velocity

72 3.4. Results and Discussion

Figure 3.4 – Instability wavelength.(a) Measurement of the instability wavelength λi vs w for different values of activity, controlled by the ATP concentration, whose values are displayed in the legend . (b) Fluorescent micrographs displaying the measurement of the instability wavelength λi at the onset of defect nucleation in the shear state for three different channels widths w. λi is taken as the distance between two neighbouring negative defects along a given wall, as indicated. Scale bar: 50µm. (c) λi vs w in simulations, for different values of activity as indicated. The unit of the axes is the number of lattice sites (l.s). In (a), error bars correspond to the standard deviation of the measurements for about 10 independent instability onsets per channel. In (c) error bars correspond to the standard deviation of the measurements.

perpendicular to the walls is negligible. The shear rate, characterised by the slope of the velocity profile, is approximately constant over a relatively large range of channel widths. This is as expected because the activity, and hence the energy input per unit area, which must be balanced by the viscous dissipation due to the shear, is the same for all channels. Extensile, aligned active nematics are intrinsically unstable to bend deformations [58, 76, 177, 178]. As a consequence, the sheared state eventually leads to local bend instabilities as shown in Fig. 3.2 (b). As a result, the velocity field repeatedly switches between two different states: longitudinal shear flow and a transversal instability regime (see 3.1 and 3.3). The instability takes the form of a sinusoidal deformation of the aligned nematic field, with a well defined length-scale along the channel. As the perturbation progresses, defects are rapidly nucleated from the walls, at regularly-spaced positions coinciding with the maxima of the sinusoidal perturbation. We measured the wavelength of the instability and found that it scales with the channel width (Fig. 3.4). This is strong evidence that the hydrodynamics is screened, and that the channel width is important in controlling the flows. Upon their nucleation at the boundaries, the orientation of the +1/2 defects is strongly anisotropic (see Fig. 3.5(a)). They preferentially align perpendicular to the walls and, due to their active self-propulsion [131,179,180], they move away from the walls into the bulk. On the contrary, because of their three-fold symmetric configuration, −1/2 defects have no self-propulsion and remain in the vicinity of the walls [139]. Eventually, the +1/2 defects reach the opposite wall and annihilate with negative defects residing close to it. In this way, the defect-free phase is periodically restored (see 3.3). Remarkably, even though no chirality is observed in the sheared state, as we repeat the experiments, the handedness of the shear flow initially selected is preserved through successive instability

73 Chapter 3. Active nematics under lateral Confinement cycles. This memory can be explained by observing that the instability is triggered locally and that it is entrained by the neighboring sheared regions. The dynamics of the switching behavior and the coexistence of the shear and the instability states can be best illustrated in a space-time diagram of the averaged velocity components in the channel, as shown in Fig. 3.2 (d-e). In the shear state, the velocity perpendicular to the boundaries and averaged across the channel width, hVy(x)iy, vanishes, while the velocity component parallel to the channel walls and averaged along the channel length, hVx(y)ix, is maximum at the walls. Once bend instabilities are triggered, defect pairs form and +1/2 defects propagate across the channel and dismantle the shear state, which leads to the emergence of non-zero values of hVy(x)iy with a well-defined length scale along the channel. The defects eventually reach the opposing wall and annihilate, such that the shear state is reestablished. As is apparent from Fig. 3.2 (d-e), over time the active system alternates between the two regimes. Simulations allowed to test channel widths well below the experimental capabilities, allowing to explore the λ w regime. For these conditions, we observed stable shear flow, as any pairs of defects that were generated at the channel walls immediately self-annihilated (see 3.8). In a recent paper Opathalage et al [120] have reported similar defect nucleation at the boundaries for a microtubule/kinesin mixture in circular confinement with no-slip boundary conditions and planar anchoring. They attribute the rate of defect formation to a combination of the build-up of microtubule density at the boundary increasing local active stresses and a change in the azimuthal force as circular flows wind the microtubules around the confining disk. In our experiments and simulations variation of microtubule density across the channel is small (except in defect cores) and the instability period is controlled by the time taken to realign the microtubules parallel to the walls after each instability. This realignement period not only decreases significantly for increasing channel width, but also for increasing activity as shown in Fig. 3.8.

3.4.2 The dancing state: a one-dimensional line of ﬂow vortices

Increasing the channel width to values between 90µm and 120µm, one-dimensional arrays of vortices are observed as shown in Fig. 3.2 (c). A close look at the defect distribution reveals that the transition to the flow state with organised vorticity arrays corresponds to the point when the channel can accommodate more than one defect in its cross-section, i.e., λ/w < 1 (Fig. 3.3). One could expect that this criterion is reached when the channel width becomes comparable to the active length-scale. However as shown in Fig. 3.3, la/w is still much larger than 1 in the range of the dancing state, i.e., λ la. Furthermore, contrary to the la/w curves, λ/w does not seem to depend on activity, supporting the idea that λ is indeed a pure geometrical feature. Dynamically, the system behaves as if two distinct populations of positive defects are travelling along the channel in opposite directions, passing around each other in a sinusoidal-like motion. A similar state had been predicted by simulations and was referred to as the dancing state [139]. However, contrary to the published simulations, the dancing state observed in our experiments is quite fragile, and vortex lattices are always transient and localised in space. Defects may annihilate with their negative counterparts, or even switch their direction of motion, thus perturbing the flow pattern (see 3.4 and 3.5). The difference between the spatial

3.4. Results and Discussion (b)

p p

0.1 0.1

y y 0 0 Experiments w 0 Simulations w

Figure 3.5 – Defect nucleation. (a) Statistical distribution of positive defect orientations. The angle ψ corresponds to the orientation of the defect with respect to the wall. ψ = π/2 (resp. ψ = 3π/2) corresponds to a defect perpendicularly colliding with (resp. moving away from) the wall. The red distribution is the result for defects close to the wall, while the yellow distribution refers to bulk defects.(b) Experimental (left) and simulated (right) probability distributions of defect position across a channel. Green (blue) distribution refers to positive (negative) defects.

75 Chapter 3. Active nematics under lateral Confinement organisation of oppositely charged defects in the confined active nematic is manifest in their arrangement across the width of the channel (Fig. 3.5 (b)). The distribution of the +1/2 defects has a single peak at the centre of the channel (Fig. 3.5 (b), green). On the other hand, the −1/2 defect distribution has two peaks, one at each of the boundary walls (Fig. 3.5(b), blue) and the profiles do not rescale with the channel width. Instead, the wall-peak distance is approximately constant at a separation ∼ 18µm from the wall, as shown in Fig. 3.9(d). This can be attributed to −1/2 defects having no self-propulsion and thus interacting elastically with the channel walls [181]. The distance of the −1/2 defect from the walls is therefore expected to be controlled by the intrinsic anchoring penetration length of the nematic ln = K/W, which is set by the competition between the orientational elastic constant K and the strength of the anchoring at the wall W, and is independent of the channel width and activity of the particles. As expected, for wider channels, w > 120µm, the difference between the +1/2 and −1/2 defect distributions diminishes as active turbulence is established (see Fig. 3.9(c)). We observed a similar behaviour in the simulations, but the -1/2 defects were more strongly localised near the walls, and the +1/2 defects consequently tended to lie towards the centre of the channel (Fig. 3.5(b)). Simulations have also allowed us to pinpoint the necessary boundary conditions at the channel walls to trigger defect nucleation. We find that such a localised defect formation at the walls is obtained only for free-slip boundary conditions for the velocity, and weak planar anchoring boundary conditions for the nematic director field. This is because the free-slip velocity, together with the parallel anchoring of the director, allows for strong tangential active flows, and hence strong tangential nematic order, to develop along the boundaries. This results in bend instabilities that grow perpendicular to the walls with the weak strength of the anchoring allowing the director to deviate from a planar configuration at the positions where the bend instability is developed. Our previous simulations that assumed no-slip velocity and strong alignment conditions on the confinement did not observe defect nucleation at the boundaries [139, 182], and showed insensitivity of the active nematic patterns to the boundary conditions [182]. This is because the strong anchoring used in these works prevented defects forming at the walls. It is, however, interesting to note that a recent computational study, based on a kinetic approach, has reported a special case of defect nucleation at the boundaries of an active nematic confined within a circular geometry with no-slip velocity boundary condition and free anchoring [136]. However, in that work the wall-bound defect nucleation was observed only for confining disks of small sizes and had a very different dynamics than that reported here: the +1 defect imposed by the circular geometry was first dissociated into two +1/2 defects and then, for sufficiently small confinement one of the +1/2 defects kept moving into and out of the boundary. This is in contrast to our results that show regularly-spaced defect pair nucleation sites at the boundaries for a range of channel widths. Moreover, we find that the corresponding defect spacing is governed by an instability wavelength that is no longer given by the conventional active length scale.

3.5 Summary

We have presented experimental results, supported by continuum simulations, investi- gating the flow and defect configurations of an active nematic confined to rectangular channels of varying width. Our experiments have identified a new dynamical state, where

76 3.5. Summary

Figure 3.6 – Defect Spacing. Experimental mean defect spacing, λ, as a function of channel width, w. Different colors correspond to different activities. The error bars correspond to the standard deviation for measurements at 10 random times for each experiment. The dotted line corresponds to a fit with a power law that yields λ ≈ w1/2. well-defined shear flow alternates in a regular way with bursts of instability characterised by +1/2 topological defects moving across the channel. We have also shown that, for wider channels, it is possible to identify the dancing state [139], although the particular boundary conditions considered in the present work make it less stable. Our work highlights the importance of topological defects in controlling the confined flows. Because the microtubules have weak planar anchoring and can freely slide along the channel walls, pairs of ±1/2 defects form at the walls of the channel. The +1/2 defects are self-propelled and move away from the walls whereas the −1/2 defects remain close to the boundaries. The distance to the boundaries is set by the anchoring penetration length. In bulk active nematics the defect spacing is set by the active length scale and, although there is some evidence of long-range ordering [183–185], defect motion is primarily chaotic. In confinement, however, the defect spacing and the wavelength of the instability are set by the channel width and the defect trajectories are more structured. Together, experiments and simulations demonstrate a surprisingly rich topological defect dynamics in active nematics under channel confinement, and a sensitive dependence on both channel width and boundary conditions. Therefore, confinement provides a way of controlling active turbulence and defect trajectories, a pre-requisite for using active systems in microfluidic devices.

77 Chapter 3. Active nematics under lateral Conﬁnement

4 Vx (μm/s) 3

1 0 y

-1 80 m -2 70 m 60 m -3 50 m y(μm) -4 -40 -20 0 20 40

Figure 3.7 – Shear flow profile. Experimental flow profile of Vx across the channel. Different colors refer to different channel widths. Data for all channels are plotted on the same graph with no rescaling. The origin of the y-axis is taken at the center of each channel. Velocities are computed with a custom ImageJ plugin (see Methods section 2.2.4 for details).

Figure 3.8 – Instability Cycles. Average time between two instability cycles as a function of the channel width. Diﬀerent colors correspond to diﬀerent ATP concentrations. Error bars correspond to the standard deviation of around 10 instability cycles for each channel width.

78 3.5. Summary

P (a) P 0.15 0.12 (b)

0.1

0.1 0.08

0.06

0.05 0.04

0.02 y (μm) y (μm) 0 0 0 10 20 30 40 50 60 70 0 20 40 60 80 100 P 0.1 (c) P 0.05 (d) 0.08 0.04 70 μm 0.06 80 μm 0.03 100 μm 140 μm 0.04 0.02

0.02 0.01 y (μm) y (μm) 0 0 0 20 40 60 80 100 120 140 0 10 20 30 40 Figure 3.9 – Defect Unbinding. Experimental probability distribution of topological defect positions across the channel width. (a), (b) and (c) correspond to the profiles for different channel widths. Green (resp blue) curves stand for +1/2 (resp -1/2) defects. (d) Distribution of −1/2 defects position close to a wall. The position of the wall is given by y = 0. Different colors stand for different channel widths. The maxima of all the distributions coincide at a distance around 18 µm from the wall.

79 Chapter 3. Active nematics under lateral Conﬁnement

3.1 – Shear ﬂow of active nematics conﬁned in a 50 µm channel. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 0.15 fps. scale:0.73 µm/px.

3.2 – Shear state from the simulations. Defects are highlighted by the green circle (+1/2) and the blue triangle (-1/2). Channel width is 32 lattice sites.

3.3 – Defect tracking in the shear state. Active nematics conﬁned in a 80 µm channel. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 1 fps. scale:0.69 µm/px. scale bar 50µm.

3.4 – Dancing disclination state in a 120 µm channel. Positive (+1/2) defects are overlaid with green disks. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 0.6 fps. scale:0.56 µm/px.

80 3.5. Summary

3.5 – Dancing state from the simulations. Channel width is 60 lattice sites.

3.6 – Switching state in active nematics conﬁned in a 90 µm channel. The images were acquired with a laser scanning confocal microscope (see Methods section 2.2.1). frame rate: 0.3 fps. scale:0.79 µm/px.

3.7 – Switching state from the simulations. Defects are highlighted by the green circle (+1/2) and the blue triangle (-1/2). Channel width is 40 lattice sites.

3.8 – Self-collapsing state from the simulations. The bend instabilities grow and then collapse in on themselves. Channel width is 30 lattice sites.

Chapter 4 Eﬀect of Topology

Flowing active matter provides a fascinating arena to investigate a variety of phenomena in non-equilibrium Physics [2, 162]. One of the present challenges in this field is to make active materials functional by engineering devices that could rectify and upscale the power endowed to molecular motors. In-vitro preparations, singularly those based on the coop- erative action of microtubules and protein-based motors [186], have been proved robust enough in laboratory tailored conditions to hold big promises in this exciting venture. In the light of the previously announced goal, the question to be investigated is whether these active flows admit being regularized into directed currents and be further dispensed into microfluidic designs for autonomous operation. Progress in this direction can be only achieved by first understanding the interplay between geometry, order and collective flows, paralleling what has been recently demonstrated in passive liquid crystals [187]. We demonstrate on what follows that two-dimensional microtubule-based active nematics can be ordered into coherent, defect-free, flows under annular confinement. Below a critical confinement width, a regime of directed transport with random handedness emerges inside isolated annuli. This flow symmetry-breaking is accompanied by a polarization of the nematic order in the direction of the flow. Adding asymmetric corrugations to the boundaries enables to control the transport handedness in an isolated annulus. In the case of two interconnected annuli we show the possibility to generate stagnation points in the active flow pattern, thereby localizing topological charges in space. Moreover, going to higher order platforms of connected annuli, examples of dynamic flow synchronization, anti-correlation and frustration are uncovered.

4.1 Experimental Setup

The experimental setup is rigorously the same as the one used in chapter3, except for the conﬁning patterns. The geometries tested belong to the class of 2D handle-bodies. They are composed of connected elementary annuli. We will refer to them by the use of their genus number, g, which counts the number of holes of a given handle-body: an annulus will have g = 1, two connected annuli will have g = 2 and three connected annuli will have g = 3. In this work, we have studied the dynamics of active nematics conﬁned in handle-bodies up to g = 3. A sketch of the grid design is available in Fig. 2.10.

83 Chapter 4. Eﬀect of Topology

Figure 4.1 – Active flows confined in annuli. Panels a-d are composed of a fluorescence micrograph of the confined active nematic on the left half, and the intensity map of the azimuthal velocity on the right half. The lower limit of the colormap is set to 0.2 µm.s−1 such that regions with negligible velocity remain colorless for a better contrast (specially in panel (c)). The annuli width are (a) 60 µm (b) 160 µm and (c) 200 µm. Scale bars:100 µm. (d) Time-averaged profiles of the velocity tangential and normal to the walls, ut(˜r) = hvθiθ and un = hvriθ, as a function of the radial coordinate r˜ = (r − Ri)/(Ro − Ri), where Ri and Ro denote the annulus inner and outer radius, respectively. (e) Mean global speed, V (t) = hv(r, θ, t)iΩ, and azimuthal velocity, V t(t) = hvθ(r, θ, t)iΩ averaged over time, as a function of annulus width. Error bars correspond to the standard deviation of the measurements with time. (f) Time evolution of the flow order parameter, Sˆ, for different channel widths. Sˆ is computed as described in the text and in Fig. 2.9.

4.2 Results

4.2.1 Transport and polarization in isolated annuli Under lateral confinement, active nematic flows reorganize into regular patterns [188], and the spatial distribution, size, and velocity of topological defects change dramatically. While, for straight channels, the control parameter is the channel width, annular confinement adds another control parameter: the mean curvature of the annulus, controlled by the inner radius. In order to decouple the effects of these control parameters, the inner radius is kept constant at Ri = 150 µm and the outer radius is chosen such that the annulus width spans a range of 40 < w < 300 µm. For narrow annuli, typically below 80 µm for the material parameters used here, we observe the emergence of a persistent directed flow, which is not possible in straight channels because of the end effects. Its handedness is random, but persistent over the

84 4.2. Results whole duration of the experiment, until activity dies out. In our experiments, the velocity field is obtained by applying Particle Image Velocimetry (PIV) algorithms to the confocal fluorescence images, using the evolving features of the active nematic as flow tracers (Fig. 4.6(a)). The velocity field is resolved in space and time and projected onto the radial and azimuthal directions: vr(r, θ, t) and vθ(r, θ, t) respectively. The time-averaged azimuthal velocity, vθ, is uniform along the annulus azimuth as shown in Fig. 4.1 (a), and only depends on the position across the channel width, with an apparent slow-down towards the walls. We call this regime the transport state (see 4.1). For wider annuli, typically 80 < w < 200 µm, the flow pattern is less stable in time [Fig. 4.1 (b)]. Transport may be observed for a while, before breaking down into a chaotic regime. Occasionally the symmetry breaks again, such that transport and chaos never cease to compete over time. The transition between both states occurs randomly, each of them persisting from a few seconds to minutes before breaking down. In this particular state, that we will call the switching state (see 4.2), the transport chirality may change randomly between two turbulent phases. Above 200µm, the flows are essentially chaotic rendering turbulent-like currents (see 4.3). As shown in Fig. 4.1 (c), vθ averages to zero everywhere around the annulus.

Given the observed rotational invariance of the velocity in the transport state, we eliminate the azimuthal dependence by computing the average tangential velocity profile ut(˜r) = hvθiθ (r˜ denotes the scaled radial coordinate). The resulting data yields to a symmetric concave profile, with maximum velocity at the center of the annulus cross- section (see Fig. 4.1 (d)). Previous simulations [142] have predicted the emergence of a Poiseuille profile. A parabolic fit to the experimental data is proposed in Fig. 4.6(b). The −1 equation of the fit has the form ut = u0 ·r˜(1−r˜) with u0 = 18 µm.s . The agreement with the experimental flow profile is satisfying around the center of the cross-section. However, the Poiseuille profile seriously underestimates the velocity at the walls (slip velocity). We may explain the discrepancy by noting that a no-slip boundary condition was imposed in simulations. On the contrary, in our case there seems to be low friction at the walls, as evidenced by the linear shear flow profile in narrow channels, shown in Fig. 3.7 in chapter 3.

We can perform a similar analysis for the radial component of the velocity, vr. We denote the velocity normal to the walls as un(˜r) = hvriθ. Although un is an order of magnitude smaller than ut, its profile is notably anti-symmetric. The mean radial velocity vanishes at the center of the annulus cross section (r˜ = 0.5), which implies that there is no global transport across the annulus. However, the magnitude increases as we move away from the center, with negative (resp. positive) values towards the inner (resp. outer) wall. This result reflects the anisotropy of positive defect interactions with a boundary. When positive defects move closer to a boundary, they tend to align perpendicularly to it, as shown in Fig. 3.5 in chapter3. The opposite sign of un comes from the fact that the surface normals are both taken pointing towards the center of the annulus, which means that defects colliding to the inner (resp. outer) wall will lead to un < 0 (resp. un > 0). The flow profiles obtained for the switching and turbulent regimes are much noisier, and little information can be extracted from the velocity distribution in r˜. To quantify how much of the system’s activity is devoted to transport, we compute a couple of space independent observables. In particular, V (t) = hv(r, θ, t)iΩ and Vt(t) =

85 Chapter 4. Eﬀect of Topology

hvθ(r, θ, t)iΩ are respectively the mean speed and the mean azimuthal velocity averaged in the whole space Ω. The contribution to transport is given by the ratio q(t) = Vt/V . For a series of experiments (Fig. 4.6(c)), this parameter fluctuates about 0.8 in the transport regime, about 0.3 with large fluctuations in the switching state, and becomes noisy with vanishing transport in the turbulent state. By plotting the time-averaged velocities V t and V , the decay of active transport with channel width becomes clear (Fig. 4.1(e)). The observed symmetry breaking can be more readily quantified in the form a scalar flow-order parameter, denoted Sˆ, whose values are bounded between -1 and +1. The way Sˆ is defined corresponds to an alternative representation of the transport around the annulus based on space-time images (see Methods section 2.2.4 and Fig. 2.9) of the dynamics at the center of the annulus. The technique is similar to the one we used in chapter3 to compute the flow profile in the shear state (Fig. 3.7). The symmetry breaking is directly interpreted from the left-right symmetry breaking in these images, as shown in Fig. 2.9. According to the chosen definition, Sˆ = +1 (resp. S = −1) denotes an ideal clockwise CW (resp. counter-clockwise CCW) transport currents, while values close to zero correspond to the turbulent state. Typical evolutions of Sˆ(t) are displayed in (Fig. 4.1(f)). In the transport state, Sˆ remains very close to 1 with little fluctuations. On the contrary, Sˆ fluctuates dramatically in the switching and turbulent states, with frequent drops towards Sˆ = 0, characteristic of an absence of transport. Typically, events of transport are associated to values of Sˆ > 0.5. Flows are largely correlated with the organization of positive defects [164] as illustrated in Fig. 4.2. For all but the widest annuli, positive defects acquire a remarkable polar order, denoted Pθ(t), measured along the azimuthal direction (Fig. 4.2 (a)). Pθ(t) fluctuates in time about a well-defined mean value, which is largest for the narrowest annuli (Fig. 4.2 (b)). The distribution of orientations, displayed in Fig. 4.7 (a) gives a clear signature of the polar order, with a high peak in the azimuthal direction (ϕ = 0). Interestingly, we also notice two minor peaks in the radial direction (ϕ = ±π/2), which could be interpreted as a slight signature of nematic order. However, plotting the average orientation of the defects as a function of the radial coordinate (Fig. 4.7 (b)) shows that the peaks come from the special ordering of defects in the vicinity of the walls commented earlier (see Fig. 3.5). As we move closer to the walls, positive defects tend to orient perpendicularly. The orientation of the defects is coherent with the anti-symmetric shape of the radial velocity plot (Fig. 4.6 (b)) giving signature of a perpendicular approach to the walls. This way, they appear to be protected from annihilation at the boundaries. We speculate that this is in agreement with the argument raised by Opathalage et al. [120], who proposed a mechanism for active +1/2 defect annihilation that is different with respect to conventional mechanisms in classical liquid crystals. More precisely, annihilation of positive active defects would not be driven by a simple local reorientation of the nematic field, but imply material-transport ahead of the positive defect. The interactions between the defect ordering and the flows at a boundary suggest the possibility to control active flows through "smart" geometrical confinement. In the transport state, a simple way to break the chiral symmetry without altering the topology of the channels consists in slightly redesigning the tracks from circular to squared and impose a symmetric arrangement of convex indentations at either side of each of the four corners (see Fig. 4.8). This minor intervention suffices to achieve currents that are either CW (indentation located right of the corner) or CCW (indentation located left of the

86 4.2. Results

Figure 4.2 – Defect polarization in the active nematic annular ﬂow. Positive defects positions and orientations are detected as described in the Methods. Defect orientations are then projected onto the local azimuthal direction (angle ϕ in (a)). (a) Fluorescence micrograph of an active nematic conﬁned in a 80 µm annulus. Positive defects positions and orientations are overlaid. (b) Azimuthal polar order of positive defects,

Pθ =< cos ϕ > as a function of time, for four diﬀerent annuli width. See Methods for details on the computation of Pθ. corner).

4.2.2 Cross-talks between ﬂows and order in connected annuli

In the previous section, we studied the evolution of active flows confined in annuli, i.e., g = 1 handlebodies. We want to explore the effect of more complex topologies, first by studying a transition into g = 2 handlebodies, realized by placing two annuli with increasing degrees of overlapping (see Fig. 4.3). The width of each annular channel is fixed at w = 80µm, at the transition between transport and switching states. This dynamically rich flow regime allows to observe that the normally random and uncorrelated switching events in each annulus become synchronized, and that this process is mediated by a dynamically self-assembled defect pattern in the overlapping region. For the smallest overlap (Fig. 4.3 (a)), flows and defect landscapes behave very similarly to isolated annuli. The flow order parameter (Fig. 4.3 (b)) reveals that annulus number 2 (red line) has a persistent CW transport, while annulus number 1 (blue line) is in the switching state. This is a clear evidence that active flows are not synchronised. The flow patterns (Fig. 4.3 (c)) do not give any sign of interpenetration between the two annuli. Finally, the defect distribution shows an accumulation of negative defects close to the walls and a higher concentration of positive defects at the center (Fig. 4.3 (d)), which is a sign of the confinement-induced defect unbinding reported in the case of channels (see Fig. 3.5 (b)). On the contrary, in the case of the largest overlap (Fig. 4.3 (e)), the flow order parameter of both annuli (Fig. 4.3 (f)) are perfectly anti-correlated. This tendency to rotate with opposite chirality should be expected in order to preserve the continuity of the velocity field in the overlapping region, and has also been reported in interacting bacterial vortices [109]. This anti-correlation is persistent within episodic switching bursts in which

87 Chapter 4. Effect of Topology the order parameter Sˆ changes in a symmetric and synchronized fashion in the two coupled annuli. The corresponding flow pattern (Fig. 4.3 (g)) features an unprecedented top- bottom symmetry breaking. The bright yellow spot located below the cusp connecting the annuli is the signature of a strong jet of active material from that cusp towards the center. On both sides of the cusp one can also notice two blue spots that are the signature of stagnation points in the flow field. These stagnation points are attributed to recirculation vortices on each side of the jet. The case of intermediate overlap (Fig. 4.3 (i)) suggests the existence of an intricate coupling between defects and flow patterns. The flow order parameter (Fig. 4.3 (j)) gives evidence of anti-correlations between the flow regimes in the two annuli. However, one can also observe that the average value of Sˆ, below 0.5, is significantly lower than either less overlapping or more overlapping annuli. As a matter of fact, the dynamics in this configuration is mostly turbulent, with no sign of transport. In spite of this, the flow pattern (Fig. 4.3 (k)) appears strikingly regular, with 5 equidistant stagnation points centered in the intersection region and 2 jets located at either cusp of the g = 2 handlebody. Similarly, defect distributions are also very well organised (Fig. 4.3 (l)). Negative defects concentrate at two locations along the main axis of the g = 2 handlebody (blue dotted circles). On the other hand, positive defect concentrate in the vicinity of the top and bottom cusps (green dotted circles). Remarkably, there is an apparent topological protection of the center of symmetry of the g = 2 handlebody, which is defect-free at all times. This region is characterized by a thick active bundle oriented parallel to the main axis of the genus two handlebody. Apparently, jets coming from the facing cusps align the active material in this area, rendering defect crossing rather unlikely. The corresponding dynamics are observable in 4.4. This scenario points to the existence of a cross-talk between singularities in co-evolving fields, velocity and nematic orientation in this case. This principle was recently raised by Giomi et al. in a study of passive liquid crystals forced through microfluidic channels [187], where the geometry of the channels led to the stabilisation of topological defects that were not energetically favoured. The scenario presented here is an active analogue, where the flows are not directly prescribed, but rather emerging from the coupling between activity and topology.

The above flow patterns can be reconstructed from the superposition of the ideal flow fields around individual defects, as derived by Giomi et al. [161]. From the outcome of defect tracking displayed in Fig. 4.3 (l), we have computed an average local topological charge, T , by counting the total number of positive (N+) and negative (N−) defects detected at a given location over the whole duration of the experiment: T = 1/2(N+ − N−)/Nf with Nf the number of frames. The resulting topological landscape (Fig. 4.4 (a)) confirms the clear charge localization seen in the experiments and corresponding velocity maps (Fig. 4.4 (b)) . In the model, the spots of positive (resp. negative) charges are chosen as flow "sources", that generate the theoretical flow pattern of a positive (resp. negative) defect (Fig. 4.4 (c)). Although this "toy model" neglects interaction between flow sources, it is useful as a means to understand the cross-talk . The acceleration areas in the cusp are efficiently recovered, as well as the stagnation points, although the latter appear more spread out. Note that in the representation of the computed flow fields (Fig. 4.4 (d)), the overlaying mask of the G = 2 handlebody is only illustrative, and no boundary conditions are imposed in the calculations.

88 4.2. Results

Figure 4.3 – Active ﬂows in annuli connected in a genus two handlebody design. (a, e, i) Fluorescence micrographs of 80µm wide annuli connected side by side, for various inter-center distance D. (a)D/2Ro = 0.97 (e)D/2Ro = 0.8 (i) D/2Ro = 0.94 (b, f, j) Flow order parameter as a function of time for each annulus in (a, e, i) connected annuli. scale bars: 100 µm. (c, g, k) Local speed averaged over time at the center of (a, e, i). (d, h, l) Overlay of positive (green) and negative (blue) defects positions at the center of (a, e, i) connected annuli. The overlay comprises detections over 600 frames.

89 Chapter 4. Eﬀect of Topology

Figure 4.4 – Cross-talk between stagnation points in active flow and defect orientation. (a) Overlay of the local average topological charge T at the center of two connected annuli with an inter-center distance D/2Ro=0.94. Regions with higher negative (resp. positive) topological charge are highlighted in blue (resp. red). (b) Corresponding velocity map. The speed has been normalized to 1 for simplicity. Dark blue spots are the signature of stagnation points, while bright yellow spots correspond to the regions with highest velocities. (c) Schematic reconstruction of the flow pattern. We superimpose the theoretical flow fields generated by two positive (top) and two negative (bottom) defects located at the positions defined in (a). See methods for the expression of the flow fields. (d) The resulting flow field both captures the stagnation points and the accelerated areas close to the positive defects. The mask overlay (black regions) does not correspond to any boundary condition imposed in the computation.

90 4.2. Results

Figure 4.5 – Active ﬂows in annuli connected in a genus three handlebody platform. (a) Three connected annuli of 80µm width with D/2Ro = 0.83. scale bar: 100 µm. (b) Flow order parameter as a function of time for each annulus in the assembled geometry.

4.2.3 Frustated topology with a genus 3 topology

In this section, we investigate how the active nematic flows respond to geometric frustration. In the case of the g = 2 handlebody, we have seen that the degrees of freedom in the flow are already more constrained than in the case of isolated annuli: the transport chiralities in both annuli are necessarily anti-correlated in order to minimise the friction forces within the connected area as shown for the largest overlap in Fig. 4.3 (e). In spite of this slight decrease in the degrees of freedom, transport was topologically allowed in all annuli. This is no longer true in the case of a g = 3 handlebody geometry shown in Fig. 4.5 (a). Considering that two connected annuli tend to circulate with opposite chiralities, such a flow pattern is not compatible with transport in neither of the two chiralities for the third one, leading to frustration. The transport dynamics of each annulus, presented in Fig. 4.5 (b) and observable in 4.5, shows strong evidence of frustration. At time t = 0, annuli number 2 (red line) and 3 (green line) annuli are anti-correlated, with the first one rotating CW and the second one CCW. On the other hand, annulus number 1 (blue line) is in a turbulent regime. This state is not stable in time as it enters in a CW motion around t = 100 s. Interestingly, at the same time the flow order in annulus 2 dramatically drops to 0. Therefore, at time t = 200 s annuli 1 and 3 are anti-correlated, and annulus 2 is turbulent. Eventually, around t = 300 s, the annulus 2 enters a CCW rotation, while the flow order in annulus 3 drops to 0. Such a process of alternating pair anti-correlations is not exclusive, and temporary events of transport in all annuli are sometimes observed (at times around t = 500 s in this particular experiment). However, these results suggest that the emerging flow pattern tends to favour two counter-rotating annuli facing a turbulent one. In other words, the system behaves as if frustration was accumulated in either of the three annuli, and exchanged over time. Such an intricate flow coupling requires further investigation. In particular, the time scale between the frustration exchanges could give information on what triggers the transitions.

91 Chapter 4. Eﬀect of Topology

4.3 Discussion

The symmetry breaking in annuli appears to have a twofold origin. On the one hand, the flow patterns spontaneously (and randomly) transit to directed transport. One the other hand, topological defects acquire a strong polar order, reminiscent of a polar flocking. However in the case of active nematics, the cross-talk between defects and flows in highly non trivial. More precisely, there is no way to tell a priori how both symmetry breakings are related. One possibility is to invoke a sort of polar flocking transition, which would suggest that elastic interactions between defects favour a polar alignment that necessarily leads to directed flows. As such, the symmetry breaking would originate away from the walls. However, one cannot rule out the idea of a spontaneous hydrodynamic symmetry breaking that would not be related to the nematic order, and which could transiently align the positive defect populations. For example, the transition to coherent flows in annuli has been observed in the microtubule-based active gel in its three-dimensional form i.e in the absence of liquid crystal order [116]. Another hydrodynamic source of symmetry breaking could be the emergence of edge currents, which are essential to understand bacterial vortices [109]. Although these currents are driven by the thinnest layer of bacteria directly in contact with the walls, they completely determine the handedness of the bulk vortex. As a matter of fact, our preliminary experiments on active nematics have proved the ability to control the handedness of the transport by small rectifications of the wall properties (see Fig. 4.8). This observation remarkably confirms the responsiveness of the AN to, even minimal, geometric details of the confining walls, and opens an avenue of future research that is presently investigated. Another interesting cross-talk between topological landscape and active flows is observed in the case of connected annuli. For moderate annulus overlap, both defects and flows exhibit stagnation points. At the center of the genus two handlebody a very stable topological structure emerges. Its net charge is zero, for it is made of two negative defects facing one another along the main axis, and two positive defects facing one another in the vicinity of the cusps. The flow field develops an array of five stagnation points, with two converging jets in the vicinity of the cusps. A very simple correspondence between both fields can be drawn by treating the topological defects as flow sources and adding up their theoretical flow fields. We believe that the agreement between the experimental and computed flow fields captures the main features of the cross-talk. To further understand why the particular genus two handlebody geometry stabilizes this pattern, we would like to stress the fundamental role of the cusps. As microtubules have a strong tendency to align at the boundaries, active nematics orient planar to either side of the cusps. Upon reaching the tip of the cusp, the nematic field cannot comply with the discontinuity at the boundary. The bending rigidity determines the maximal curvature above which defect nucleation occurs. As a result, the AN bends around the cusp, which acts as the core of +1/2 defects. This idea means that the +1/2 defects should form as long as the curvature of the cusp is "too high". This defect is oriented towards the center of the genus two handlebody, and as a consequence propels inwards. The same takes place at the opposite cusp, leading to the two jets displayed in Fig. 4.3 (k). This two defects eventually collide towards the center of symmetry, and the collision deforms their shape by aligning their fronts parallel to the main axis of the genus two handlebody. This is why the center of symmetry of the genus two handlebody is defect free, as evidenced in Fig. 4.3 (l). As we

92 4.3. Discussion

Figure 4.6 – Particle Image Velocimetry of active flows confined in annuli. (a) Fluorescence micrograph of an active nematic confined in a 60 µm annulus, with the corresponding velocity field overlaid in blue. The velocity field is averaged over 600 frames at 2 frames per second. (b) Tangential velocity ut fitted with a Poiseuille-like velocity profile. The quadratic fit has the form ut = u0 · r˜(1 − r˜). (c) Transport coefficient q(t) = Vt/V , for three different annulus widths. In the transport state (red line), the ratio remains close to 0.8. This is coherent with the average profiles described in Fig. 4.1 (d) showing that the radial component of the velocity is negligible. Furthermore, the time dependence is weak compared to the other signals: the ratio typically fluctuates about 0.82 ± 0.09. In the switching state (green line), this ratio drops significantly, and the fluctuations are larger (0.31 ± 0.22). Sometimes, the transport velocity changes sign, which attests for occasional flow reversals. Finally, in the turbulent state (blue line), the transport is almost negligible, with oscillations about 0.05 ± 0.2. Note that, in the particular experiment displayed here, a large fluctuation is observed at time t = 160s, with the ratio shooting down to −0.6. move outwards along the main axis from the center, we eventually reach the normal to the inner disk of each annulus. There, microtubules are oriented parallel to the wall and therefore perpendicular to the main axis. To comply with this director change along the main axis, negative defects spontaneously nucleate in between the center and the inner disks. This behaviour is reminiscent of the work performed by Giomi and al. [187], where controlled flows in microfluidic chips have proved to stabilize topological structures in passive liquid crystals. The fundamental difference in our case is that the flow pattern itself emerges flow the coupling between activity, order and geometrical constraints. Even richer flow synchronization events emerge when active nematics are confined in frustrated geometries such as a genus three handlebody. We hope that the presented results uncover the richness of topological active matter, and will motivate further investigations in the context of active microfluidics.

93 Chapter 4. Eﬀect of Topology

Figure 4.7 – Distribution of defect orientations Defect positions and orientations are computed with Matlab routines (see Methods). The orientations are then projected onto the azimutal direction θ. Statistics are obtained from a video of 600 frames at 2 frames/s. The average number of defects per frame was 18. The average lifetime of a given defect track was 40 frames. As a consequence, the number of independent defects in the statistics Ni is estimated around Ni ∼ 600/40 × 18 = 270 defects. (a) Distribution of defect orientations in a 80 µm wide annulus. (b) Average defect orientation as a function of the radial coordinate. Error bars correspond to the standard deviation of the measurements.

94 4.3. Discussion

Figure 4.8 – Wall-induced rectiﬁcation in asymetric annuli (a,e) Sketches of square annuli with asymetric outer boundaries. (b,f) Fluorescence micrographs of active nematics conﬁned in asymetric square annuli. (e,g) Time-series of a pixel ring at the center of the annuli. (d,h) FFT computation of the times series (e,g) showing clockwise (resp. counter- clockwise) transport.

95 Chapter 4. Eﬀect of Topology

4.1 – Symmetry Breaking Confocal ﬂuorescence video of active nematics conﬁned in a 60 µm wide annulus. scale bar: 100 µm.

4.2 – Switching state Confocal ﬂuorescence video of active nematics conﬁned in a 110 µm wide annulus. scale bar: 100 µm.

96 4.3. Discussion

4.3 – Turbulent state Confocal ﬂuorescence video of active nematics conﬁned in a 200 µm wide annulus. scale bar: 100 µm.

4.4 – Synchronisation in a Genus 2 Confocal ﬂuorescence video of active nematics conﬁned in a genus 2 handle-body, with an overlapping distance of D/2R0 = 0.94. The width of each annulus is w = 80 µm. scale bar: 100 µm.

97 Chapter 4. Eﬀect of Topology

4.5 – Frustration in a Genus 3 Confocal ﬂuorescence video of active nematics conﬁned in a genus 3 handle-body. The width of each annulus is w = 80 µm and the overlapping distance is D/2R0 = 0.83. scale bar: 100 µm.

98 Chapter 5 Active Nematics at a Wall: Description and Control

5.1 Motivation: perturbations from the boundaries

In the previous chapter, we discussed about the origin of the symmetry breaking in annuli, providing two possible interpretations. The transition observed both in the flow and order of positive defect points to a type of flocking transition as observed in polar fluids. However, this interpretation tends to oversimplify the modelling of positive defects as self propelled polar particles. In fact, the lifetime of a defect is short compared to the typical time-scale of the transport. In other words, a given topological defect annihilates well before closing a revolution around the annulus. The density of positive defects remains constant in time because these annihilation events are compensated by defect nucleations. In confinement, these nucleations almost exclusively occur at the boundaries, i.e the inner and outer circles. One has to remember that recently created defects orient perpendicular to the walls, and therefore, orthogonal to the polar order in the bulk as recalled in 4.7. As these defects move into the bulk, they will locally disrupt the polar order. In the case of narrow annuli, this effect is minor because as soon as defects are nucleated they orient with the azimuthal direction. This is no longer the case in the switching state where new born defects seem to have enough "power" to disrupt the polar order. A similar mechanism is proposed by Chen et al. [145] in their simulations of confined active nematics in an annulus, where a periodic switching in the flow chirality is attributed to synchronized defect nucleations from the boundaries. The goal of this chapter is to characterize the dynamics of active nematics close to a wall. In order to investigate the nucleation dynamics at the walls, we have built space-time plots of the pixels rings closest to the inner and outer walls, with the same technique as the one presented in Fig. 2.9 of the Methods section 2.2.4. The results, displayed for the inner ring in figure 5.1, are striking over many respects. First, the appearance of the kymographs is significantly different from the ones obtained in the bulk shown in Fig. 2.9. The latter only exhibited textures, corresponding to intensity gradients of microtubule bundles crossing the pixel line. The assymetry in the traces gave information of eventual transport along the annulus. These textures were spread uniformly on the image. Close to the wall, the kymograph appears much more

99 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.1 – Active nematic dynamics close to a wall (a) Fluorescence micrograph of a 200 µm annulus. Coloured dotted lines correspond to the pixel ring used for the space-time plot (b). Scale bar: 100 µm.In the space-time plot (b), time is represented in the vertical axis and increases downwards. The horizontal axis is 360 pixels wide corresponding to 360 degrees around the pixel ring. (b) Space-time plot of the pixel ring at r˜ = 0. (c) Inset of space-time plot (b) showing three blossoms (red,yellow and green circles) followed by merging branches (d) Larger inset of space-time plot (b) showing the large-scale attraction between the branches. structured, with high intensity lines branching up like trees separated by smoother regions of uniform intensity. Bearing in mind that these images are space-time plots, with time increasing downwards, one realises that the branching points, represented in Fig. 5.1 (c) as white circles, correspond to merging events between two branches. These merging events are not only observed locally between two neighbouring branches: the converging paths shown if Fig. 5.1(d) suggest that they attract each other over large distances. If we track back in time the paths of these lines, by moving up the image, we notice that they originate from dark "blossoms" (red, yellow and green circles in 5.1 (c)) that spontaneously pop up from the uniform regions. In this section, we will try to interpret the tree patterns reported. In the ﬁrst part, we will deﬁne the structure, birth and dynamics of a single branch as an intriguing variant topological defects bearing distinctive characteristics. In the light of this elementary description we will then relate the collective behaviour of wall defects to a classical model of spatio-temporal chaos.

5.2 Properties of an isolated negative defect at a wall

From the previous chapters on active nematics in conﬁnement we know that walls are preferential sites of defect nucleation. More importantly, we have shown that as soon as positive defects are created, they move away from the wall. It is also well-known

100 5.2. Properties of an isolated negative defect at a wall that topological defects always nucleate by pairs, which means that positive defects must leave negative image charges at the wall. This result is well illustrated with the wall- induced defect unbinding principle described in chapter3. Based on this scenario of defect nucleation, one is naturally drawn to conclude that the wall-defects are negative defects. However, the behaviour of wall defects is surprisingly different from the properties of bulk negative defects. As shown by the tree branches in the space-time plots obtained for a wide annulus (Fig 5.1 (b)), wall defects are very motile and prone to merging with their peers. In order to reduce the apparent complexity of the collective dynamics, we will first study the evolution of an isolated wall-defect in a disk. As a matter of fact, we have observed that when confined in a disk, in some cases active nematics exhibit quasi-steady flows and robust structural patterns.The dynamics are described in Fig. 5.2 and observable in 5.1. In panel (a), a snapshot of a typical experiment shows the structure of an isolated defect. It is composed of a triangular hole in contact with the boundary, prolonged by a more fluorescent filament that we will refer to as a plume. On both sides of the plume, we note the presence of highly bent regions, that will be called cracks. In the bulk, positive defects periodically form as the plume oscillates transversally. In Fig. 5.2 (b), a kymograph of the pixels line along the disk boundary shows the trace of the negative defect moving along the wall. The whole dynamics can be classified in three phases. The first one, referred to as the steady regime ( 5.1), corresponds to periods of time during which the position of the wall defect around the disk is essentially fixed (i.e) the defect is not drifting. Furthermore, there are no additional defects along the wall. In that case, the dynamics is characterised by quasi-periodic oscillations of the plume, leading to defect nucleations in the bulk. A time-lapse of this behaviour is displayed in Fig. 5.2 (c). The second phase corresponds to events of wall-defect nucleation, soon followed by a merging event 5.2. In the space- time plot of Fig. 5.2 (b), it is clearly visible through the characteristic branches described earlier. A time-lapse of this behaviour is reproduced in Fig. 5.2 (d). The third phase corresponds to periods of time where the isolated wall defect drifts along the boundary ( 5.3). A time-lapse of this behaviour is reproduced in Fig. 5.2 (e). These phases are very convenient case studies, each showing the elementary properties of wall defects that we aim at understanding here.

5.2.1 Nucleation, Structure and Motility Nucleation An event of defect nucleation is shown in Fig. 5.3 (a-b). Starting from a uniformly aligned state, a bend distortion grows from the wall. In the fluorescence image, the core of the distortion appears black, as shown in Fig. 5.3 (b), t=1 s. At first, the bend distortion is quite extended laterally, so the black hole is large. Like conventional liquid crystals, the system minimizes the energy by shrinking distortions into small size (ideally point-like) defects. As a consequence, the hole quickly heals (Fig. 5.3 (b), t=2 s). The initial extension followed by a quick healing of the distortion hole explains the shape of the "blossoms" observed in the space-time plots. After healing, the center of the core is no longer black, but on the contrary appears more fluorescent than its surroundings as shown in 5.3 (b), t=4 s. This fluorescence enhancement explains the brightness of the tree branches repro-

101 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.2 – Isolated wall-defect in a disk (a) Fluorescence micrograph of active nematics conﬁned in a disk of 170 µm radius. The blue dashed line corresponds to the pixel line used in the kymograph (b). Scale bar: 100 µm. Inset: close-up of the wall defect structure, showing a triangular hole, prolonged by a bright ﬁlament referred to as "plume", and two cracks on both sides. (b). (b) Kymograph showing the wall dynamics of active nematics in a disk. The black line is the trace of a stable wall defect. Branches indicate occasional nucleation of additional wall defects. (c) Time lapse showing quasi-periodic transversal oscillations of the plume extending from the wall-defect. The interval between two frames is 2 s. (d) Time lapse showing a wall defect nucleation event followed by its immediate merging. The newborn defects are pointed by coloured arrows. The interval between two frames is 2 s. (e) Time lapse showing the slow drift of the defect along the wall. The interval between two frames is 25 s.

duced in Fig. 5.3 (c). The position of the nucleation along the walls does not seem random. Going back to Fig. 5.1, we can distinguish two types of event. Typically, the origin of the tree branches - corresponding to a defect nucleation - is located around the middle of a uniform region between neighbouring boundary defects. The global structure of the kymograph captures this phenomenon, as tree branches seem to be spaced with a well-deﬁned length scale (see section 5.3.2 for a detailed analysis). This result is not surprising given the properties of the bend instability in bulk active nematics, where the typical distance between neighbouring defects is governed by the active length scale (see section 1.3.1). However, on top of these regularly spaced nucleations, we notice that the tree branches are unexpectedly rich in short-lived blossoms. A large number of defects seem to nucleate randomly at a short distance from a given branch, before quickly disappearing. This last category of nucleation is the only one observed for the disk experiment, as shown in Fig. 5.2 (b). The nucleation event is random in time, but new born defects are always created in the vicinity of the main defect. On the contrary, at the opposite side of the disk, microtubules remain strongly aligned to the wall at all times. In section 5.13, we will further discuss this peculiar localisation eﬀect.

Merging When two boundary defects are close enough, they attract and merge. An illustrative sketch and corresponding experimental time lapse are displayed in Fig. 5.3 (d) and (e)

102 5.2. Properties of an isolated negative defect at a wall

Figure 5.3 – Birth and death of a wall defect Birth (a) Sketch of the dynamics and (b) corresponding experimental time-lapse of a wall-defect nucleation. Eventually, the plume ﬁlls the enter of the hole. Kymograph (c) Kymograph extracted from Fig. 5.1 showing the blossoms characteristic of a defect birth, and the merging pattern characteristic of defect annihilation. Death(d) Sketch of the dynamics and (e) corresponding experimental time-lapse of the merging between two wall-defects. The conﬁning wall, although not depicted in these panels, is located close to the bottom line of the images.

respectively. In between the two objects, the nematic ﬁeld is shaped as a positive defect facing the wall (Fig. 5.3 (e), t=0s), whose lateral size constantly shrinks over time (Fig. 5.3 (e), t=2s). Eventually, the front of this positive defect detaches from the wall (Fig. 5.3 (e), t=4s). As it happens, the wall defect cores merge, and the intercalary defect is zipped out into the bulk (Fig. 5.3 (e), t=8s). Eventually, a unique wall defect remains. Initially, both wall defects have the same structure. As a consequence, they must have the same topological charge. Their merging is surprising because in the bulk, it is known that topological defects exclusively annihilate by pairs of opposite-sign charge. On the contrary, defects with like-sign charge repel each other. In the next section, we will look closely at the charge distribution around such objects in order to lift the apparent paradox.

Topological Charge At the defect core, the anchoring with the wall changes from planar to homeotropic, as sketched in Fig. 5.4 (a). When computing the edge charge from Eq. 1.18, we obtain sedge = −1/2. This result is coherent with the history of the wall-defect. The nucleation process described in Fig. 5.3 (a-b) implies the creation of a defect pair, with the positive defect moving away from the wall and the negative defect remaining at the nucleation site, which turns out to be the wall. However, the structure is notably diﬀerent from bulk negative defects. In particular, the usual three-fold symmetry (shown in Fig. 5.4 (b)) is replaced by a mirror symmetry perpendicular to the wall. One of the consequences is that on both sides of the symmetry axis, the nematic ﬁeld is bent at an angle of π/2, instead of 2π/3 in the case of the three-fold symmetry. These bend distorsions are too important for the deformation to be continuous, and we observe the formation of "cracks", as shown in Fig. 5.5 (a). The cracks

103 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.4 – Topological charge of a wall defect (a) Sketch of the orientation ﬁeld of active nematics close to a wall defect, with the corresponding edge charge computed from Eq. 1.18. (b) Structure of a bulk negative defect, with the corresponding topological charge computed from the winding number (Eq. 1.10).

Figure 5.5 – From a line defect to cracks (a) Reproduction of the inset of Fig. 5.2 (a) showing a wall defect. On the left side, the idealized director field is represented by white lines. On the right side, the presence of cracks (dark spots corresponding to holes in the nematic layer) attests for significant distortions of the director field. (b) Sketch of a line defect, reproduced from Fig. 1.10. (c) Continuous bend deformation of the line defect, removing the singularity. (d) Activity-induced bend instability leading to the formation of cracks (defect pairs) [189].

correspond to regions where the director field is not defined, because of the absence of active material. We interpret these cracks as a signature of the presence of a line defect. As said in the introduction (section 1.2.6), line defects may be observed in 2D nematic phases in case of strong confinement. The presence of cracks is interpreted as an energy balance between elastic and active stresses. The elasticity of the filaments would transform the configuration of Fig. 5.5(b), where distortion is sharp, to Fig. 5.5 (c), where gradients of deformation are smooth. Fig. 5.5 (c). Conversely, active nematics are unstable to bend deformations. As a consequence, activity tends to destabilize the lane through the nucleation of defect pairs [189]. In such strong confinement conditions, the defect pairs cannot unbind, and remain within the defect line in the form of cracks. A priori, the presence of a line defect does not impact the topological charge of the structure, because its winding number is 0 (see section 1.2.6 and Fig. 5.6 (a)). Never- theless, let us consider two wall-defects sitting side by side, as depicted in Fig. 5.6 (b). If they are far appart, the winding number associated to the line defects remains s = 0. However, is they are close enough, we can see that the defect lines of both defects combine,

104 5.2. Properties of an isolated negative defect at a wall

Figure 5.6 – Winding number around a wall defect (a) Sketch of the orientation field of a wall-defect. The winding number associated to the blue loop is s = 0. (b) Sketch of the orientation field in the presence of two wall-defects that are far apart. (c) Sketch of the orientation field in the presence of two wall-defects sitting close to each other. The overlap of the two defects materialises a s = +1/2 defect.

yielding to a winding number of s = +1/2, characteristic of a positive defect. In other words, the presence of a neighbouring wall-defect changes the energy balance in favour of a spontaneous positive defect nucleation. According to charge conservation, this means that we need to account for a positive charge de-localized around each wall-defect. An- other interpretation is that the positive defect pre-exists in the bulk, and is only attracted by the two wall-defects.

These subtleties only reflect a fundamental deviation from the theoretical framework: defects are not 0-dimensional in experimental active nematics. Their cores are spatially extended, and distortions are more spread out. According to Eq. 1.20, the body force driving the active flows fa will necessarily be more spread out as well. As a consequence, considerations of charge conservation are not sufficient to understand the dynamics of wall defects. We will need to account for the full distribution of active stresses, and in particular around the cracks, to understand how these objects move, attract, and merge.

5.2.2 Forces

The map of the active force fa can only be deﬁned to within a constant, because the activity parameter α is not experimentally accessible. Therefore, from now on we will refer to a reduced form of the active force, f deﬁned as:

f = ∇ · Q, (5.1)

such that fa = −αf. An example of the step-by-step protocol to obtain f from a fluorescence image is proposed in Fig. 5.7 (a). The tensor order parameter is computed using a method developed by Perry Ellis and Alberto Fernandez-Nieves from Georgia Tech. University [117, 159], and presented in section 2.2.3. It provides Q with a pixel- level resolution. The diagonalization of Q gives access to the director field n, overlaid in green in Fig. 5.7 (b). Then, we obtain f through Eq. 5.1, as shown in Fig. 5.7 (c) . The magnitude of f is overlaid in pink over the fluorescence image (green). Pink regions corresponds to regions where f is maximal. Far from the wall, we notice that the maxima of f are located around the defect sites, and relatively localized in ellipsoidal spots. In comparison, close to the wall defect, we observe a U-shaped force line, symmetrically

105 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.7 – Active Forces (a) Fluorescence micrograph of an active nematic confined in a disk. Green disks indicate the position of positive defects. The orange triangle indicates the location of a wall defect. (b) Corresponding director field overlaid on top of the fluorescence image. (c) Active force map overlaid (in pink) on top of the fluorescence image (in green). Arrows indicate the direction of the force, while the color intensity maps the magnitude of the force.

spread with respect to the defect core but quite extended on both sides. Looking more closely at the nematic ﬁeld below the force line, we notice that it corresponds to the cracked regions, previously described as line defects.

Looking at the direction of the force along the cracks, represented by the black arrows, we ﬁnd that both arms of the force lane point towards the defect core. In other words, the cracks are exerting a force that is pushing against the wall. This result is better understood using symmetry arguments. In Fig. 5.8 (a-c), we have sketched an overlay of the force maps obtained for wall-defect, negative and positive defects respectively. While the study on the topological charge proposed in the previous section suggested that wall- defect were more similar to negative defects, it becomes clear that their axial symmetry is much closer to that of positive defects. As such, the resultant of active forces is non-zero along the symmetry axis.

This result does not explain a priori either the apparent motility of wall defects along the wall, or their tendency to merge: horizontal forces should precisely cancel out by symmetry. However, this ideal view is blind to eventual perturbations of the axial symmetry. In order to gain more insights on defect pair interactions, we have studied the force map generated by two neighbouring wall defects. To do so, we have focused on the Nucleation and merging phase described in Fig. 5.2 (b). When defects are close enough, as is the case in the ﬂuorescence snapshot of Fig. 5.9 (a) the force lines are no longer symmetric with respect to the defect cores. Instead, a merged W-shaped force line links the two defects, as shown in Fig. 5.9 (b). The magnitude of active stresses does not inform us whether the interaction force is attractive or repulsive. To account for the directionality of the forces, we have looked at the component of f parallel to the boundary, by projecting it on the azimuthal direction, as shown in Fig. 5.9(c). The color code is green for clockwise (CW) force, and pink for counter-clockwise (CCW) force. We can still see that around each defect core, both arms

106 5.2. Properties of an isolated negative defect at a wall

Figure 5.8 – Symmetry and Active Forces Red dotted lines indicate the symmetry axes. Blue areas indicate the regions with higher active stress. Blue arrows indicate the direction of the resulting active force (a) Sketch of a wall defect. The horizontal components of the active forces cancel out at the defect core, while the vertical contributions add up. (b) Sketch of a negative defect. Due to the three-fold symmetry, all components of the active force cancel out at the core. (c) Sketch of a positive defect. The horizontal components of the active forces cancel out at the defect core, while the vertical contribution add up.

are pointing inwards, in coherence with picture Fig. 5.8 (a). However, because of the W-shaped connection, the arms no longer have the same length. More precisely, the force arms located in between the two defects are shorter. This imbalance creates an eﬀective attraction, as pictured in Fig. 5.9 (d). This attraction mechanism is reminiscent of the famous "Cheerios eﬀect", sketched in Fig. 5.9 (e), describing the interaction between solid particles dispersed at an air/water interface [190], or alternately, liquid droplets resting onto a deformable substrate [191]. Particles are bound to the interface by a contact line, whose shape depends on their wetting properties.The surface tension acts along the contact line, at a given angle (contact angle) from the interface. If a particle is isolated, the contact angle is uniform around it, which means that the net horizontal force is self-balanced. On the other hand, in the presence of a neighbouring particle, the contact angle may change because of the resulting deformation of the interface, causing a net horizontal force. This force is attractive if the particles have the same wetting properties.

Tangential forces and defect nucleation Tangential stresses could also explain wall defect nucleation. In a study recently published [120], Opathalage et al. suggest that a nucleation occurs at the wall as soon as the total tangential active stress reaches a minimum. Using the videos of their publication, we have computed the active force map and its projection in the azimuthal direction the same way as presented above. Then, the azimuthal projection has been integrated over the whole surface of the disk. The results are shown in Fig. 5.10 (a) and (b). We have then performed the same analysis with our experiments and the results are displayed in Fig. 5.10 (c) and (d). We want to warn the reader that the experimental conditions, as well as the observed flow patterns and defect landscapes are very different between the two experiments. The main difference is that, in the experiments of Opathalage, the wall

107 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.9 – "Cheerios Effect" between two wall defects (a) Fluorescence micrograph of active nematics in a disk of 350 µm diameter, in the Nucleation and Merging phase (b) Magnitude of the active force f overlaid (in pink) on top of the fluorescence micrograph (in green). (c) Projection of f in the azimuthal direction. The color code is green for counter-clockwise (CW) force, and pink for clockwise (CCW) force. (d) Sketch of two wall defects side by side. Force lines are shortened in between the defect cores, causing unbalanced net horizontal forces (e) "Cheerios effect" between liquid drops on a solid substrate, after [191]. (A) Surface tension of a liquid drop on a compliant substrate causes significant deformation. The local shape at the contact line (i.e., the angles at which the three interfaces meet) is determined by balance of surface stresses, which is Neumann’s triangle. (B) Side-view of an isolated liquid drop at the surface of a soft interface, showing that the net horizontal force is balanced. (C) When two drops approach each other, their surface shape changes, especially in the region between the two drops. (D) This shape change leads to an unbalanced in-plane force in response to which the drops move toward each other.

108 5.2. Properties of an isolated negative defect at a wall defects all disappear soon after nucleation. Most of the time, the disk is only filled with two positive defects moving towards the center in a spiralling motion. As they approach the center, their azimuthal velocity increases. Conversely, the azimuthal stress decreases, as shown in Fig. 5.10 (b). Eventually, as the azimuthal stress reaches a minimum, a defect pair nucleates from the boundary at a random location. Soon, a positive defect recombines with the negative charge at the wall to annihilate. In our case, the wall defect is never disappearing. From time to time, an additional wall defect emerges, before quickly merging with the main defect. We want to test whether or not these nucleations also happen at the minima of the azimuthal stress. Fig. 5.10 (d) seems to go against such a hypothesis. Nucleations do coincide with local extrema of the azimuthal stress. However, the absolute value of the azimuthal stress does not seem to be a reliable threshold to predict the nucleation events, that scatter randomly onto the peaks of the signal. Another significant difference between the two results is the location of the nucleations. As said previously, the nucleation sites in Opathalage et al. experiments are randomly distributed around the disk boundary. In our case, we never observe any nucleation at the opposite side of the wall defect. This result seems intuitive when we look at the flow pattern produced by the wall defect. Along the symmetry axis, a fast jetting expels microtubules from the wall defect towards the opposite wall. Along the way, microtubules re-align parallel to the boundary. It is as if the jet pushed them flat towards the opposite wall. A nucleation onto the opposite wall appears quite unlikely, as the jet constantly "pushes" the active material towards the wall. More precisely, we may have to introduce a pressure term to understand where defects nucleate. However, the definition of pressure needs to be clarified.

5.2.3 Hydrodynamics of active nematics using BioFlow

The notion of pressure in active systems is ambiguous. By definition, pressure is the mechanical force per unit area that a confined system exerts on its container. Consider a fluid confined in a box. At equilibrium, the statistical description of the collisions between the fluid molecules with the walls can be reduced to an equation of state, where the thermodynamic pressure P only depends on density and temperature. Let us now consider a model suspension of self-propelled particles confined in a box. First, we will suppose that these particles are spherical with isotropic repulsive interactions. As such, they belong to the general class of Active Brownian Particles (ABPs). Occasionally, the particles will collide with the walls and bounce back to the bulk. Similarly to the case of fluids at equilibrium, a mechanical pressure can be derived from these microscopic collisions. In the case of ABPs, Solon et al. have shown that this mechanical pressure can be related to bulk correlators that only depend on density, ensuring that pressure fulfils an equation of state independent of the details of the particle interactions with the walls [192]. However, in a later publication the same authors have shown that this result is not generalizable to all active systems [193], but on the contrary rather specific to the simplest case of ABPs, supposing torque-free interactions with the walls and torque-free pairwise interparticle forces. Unfortunately, in most experimental systems, active particles are not perfectly symmetric, readily introducing torques in pair collisions and wall-particle interactions. Conversely, the presence of a solvent is a source of hydrodynamic torques.

109 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.10 – Tangential Forces and Defect Nucleation (a) Fluorescence micrograph of active nematics confined in a disk of 80 µm diameter (green) with the corresponding map of f = ∇ · Q overlaid. The experimental video is taken from [120]. Scale bar: 20 µm. R (b) Spatially integrated total active force in the tangential direction (eθ · ∇ · Q) as a function of time. Maxima correspond to configurations where the two +1/2 defects are far from one another. Minima occur directly before nucleation when defects have effectively merged. Nucleation events are marked with orange disks. (c) Fluorescence micrograph of active nematics confined in a disk of 350 µm diameter (green) with the corresponding map of f = ∇ · Q overlaid. Scale bar: 100 µm. (d) Spatially integrated total active force in the tangential direction as a function of time. Nucleation events correspond to local extrema of the tangential force, but the absolute value of the tangential stress does not seem to be a good criterion to predict defect nucleation in this particular experiment.

110 5.2. Properties of an isolated negative defect at a wall

In the case of active nematics, particles are far from spherical with obvious alignment interactions. As a consequence, active pressure is not a useful parameter in our system. On the other hand, one needs to remember that microtubules are immersed in a solvent, water, for which the classical definition of hydrodynamic pressure rigorously applies. This pressure could in principle be computed from the Navier-Stokes equation, provided the velocity field and external forces are well-defined. As a matter of fact, it is useful to remind here that the microtubules are non motile in the local reference frame of the solvent. As such, the fluorescence images captured provide a reliable velocimetry for the water flows. The water phase obeys the incompressible Navier-Stokes equations [103]:

∇ · u = 0 (5.2) ρ(∂tu + u · ∇u) = ∇ · Π where u is the velocity ﬁeld, ρ the density that we assume uniform and constant 1, and Π is a generalized stress tensor including the pressure, elastic, active and viscous stresses [103,188]:

Π = −pI + Πviscous + Πelastic + Πactive (5.3) Πviscous = 2ηE (5.4) δF Πelastic = −2ξqH + Q · H − H · Q − ∇Q (5.5) δ∇Q Πactive = −αQ (5.6)

Where p is the hydrodynamic pressure. All the notations a modelling details are available in section 2.3. Note that the contribution of active nematics to the mechanical pressure at the walls has not been ruled out here: it is entirely included in the term of active stresses. Considering the typical size of the individual ﬁlaments (1 µm) and corresponding velocities (∼1 µm · s−1), the Reynolds number is small enough to neglect inertial terms. Moreover, is has been shown that the active stresses dominate the elastic stresses, given that the typical time scale of energy injection is orders of magnitude faster than the elastic relaxation of the nematic orientation [166]. Therefore, equations 5.2 reduce to the following form:

∇ · u = 0 (5.7) 0 = ∇ · (−pI + 2ηE − αQ) (5.8)

which we can rewrite in the form of a classical Stokes equation:

1This assumption is general in the modelling of active nematics. In the disk experiment, we observe significant fluorescence heterogeneities, specially within the symmetry axis of the wall defect where the jetting occurs. This fluorescent bundle is suspected to reflect a densification of active material towards the wall, which is then re-injected into the bulk through the jet. However this phenomenon is very localized, and, there is no clear evidence of continuous density gradients.

111 Chapter 5. Active Nematics at a Wall: Description and Control

∇ · u = 0 (5.9)

−η∆u + ∇p = fa (5.10)

where fa = −αQ is the active force. As shown in 5.2.2, the force map fa = −α∇ · Q can be well estimated to within the constant −α as Q is reliably extracted from the fluorescence images. Nevertheless, there is currently no satisfying model to relate the parameter α to experimental conditions. Additionally, the flow field u could in principle be estimated using conventional PIV techniques. However, these methods are not sufficient to retrieve a reliable velocity field for two reasons. First, PIV algorithms detect changes in light intensity. In the case of active nematics, the intensity is uniform along a bundle of aligned microtubules. As a consequence, the algorithm under-estimates the velocities that are tangent to uniformly aligned regions. More importantly, classical velocimetry techniques do not necessarily satisfy the incompressibility condition. Generally, the output flow patterns are smoothed by arbitrary regularisation methods that do not account for the underlying transport mechanism. Overall, the independent computation of p based on the force and velocity inputs is inherently inaccurate.

Facing similar issues in the characterization of cell streaming dynamics, Boquet et al. have recently developped an implicit method that solves the Stokes equations for the velocity, pressure and external force simultaneously. The principle, detailed in the original publication [194], is summarized below. Consider two consecutive ﬂuorescent images I1 and I2 separated by a time interval ∆t. The displacement ﬁeld dx = u∆t is the optimal transformation relating the pixel intensities of images I1 and I2 through the following expression:

∇I2 · dx + (I2 − I1) = 0 (5.11) In classical methods, u is obtained by minimizing an energy functional of the form:

 J(dx) = J (dx) + J (dx)  data reg  Z  2 Jdata(dx) = (∇I2 · dx + (I2 − I1)) dΩ (5.12) Ω  Z  2  Jreg(dx) = γ ||dx|| dΩ Ω

where Jreg is the regularisation term which - when weighted by a positive constant γ - guarantees an arbitrary smoothness to the data. This parameter is essential in order to overcome the noise of experimental data, but has no physical meaning. The minimisation method developed by Boquet et al. can be written as follows:

(c, dx) = argmin(Jdata(dx) + Jreg(c, dx)) subject to A(c, dx) = 0 (5.13)

where the function [xi] = argmin[y(xi)]) provides the values of variables xi for which y reaches its minimum. The variable c includes all the state variables of the problem, and A(c, dx) are the constitutive equations of the system. This method is generic. Applied to our system, the problem folds down onto the following system of equations:

112 5.2. Properties of an isolated negative defect at a wall

(u, p, f, g) = argmin(Jdata(u∆t) + Jreg(f, g)) subject to A(u, p, f, g) = 0 (5.14) with

 ∆u f in Ω ∇p − µ − ext  A(u, p, f, g) = ∇ · u in Ω (5.15)  u − g in Γ

where g is the velocity along the boundary Γ, Ω is the experimental volume and fext includes all the external forces. The regularisation term Jreg takes the following form: Z I 2 2 Jreg(f, g) = α ||fext|| dΩ + γ ||∇Γg|| dΓ (5.16) Ω Γ Note that there is no condition on the form of f. As a matter of fact, the active nematic layer is only treated as an external force applied to the water phase with no preconception on its underlying physics, except the smoothness guaranteed by equation 5.16.

Preliminary results The technique has been implemented into an open-source plugin called BioFlow, ready to use within the bio-image analysis environment called Icy [195]. In this section, we provide preliminary results to the case of active nematics confined in a disk. The analysis is based on the experimental video described in Fig. 5.2. More specifically, we restricted the study to the steady regime, where there is only one wall-defect whose position remains fixed along the disk boundary. The flow field is characterized by a strong jet induced by the plume, which oscillates in a quasi-periodic motion as shown in Fig. 5.2 (c). We also note an important recirculation current along the walls, such that on average, the flow pattern is composed of two counter-rotating vortices symmetric with respect to the axis of the wall-defect. The sequence is composed of 183 frames covering 7 oscillation periods. The program evaluates the velocity, external force and pressure gradients from consecutive pairs of images. We will either present instantaneous fields (single pair of images) or average fields over the whole video. Finally, note that according to Eq. 5.15, the program output on pressure and force fields is renormalized by the fluid viscosity, ˜ such that the observables are u, p˜ = p/µ and fext = fext/µ. Nevertheless, at this stage of the project we are only interested in the spatial distribution of these quantities, and not the absolute values.

Velocity measurements A comparison between the velocity fields obtained by conventional PIV methods and the output of BioFlow is diplayed in Fig. 5.11. The calculation was performed on a single pair of consecutive images. Panel (b) shows the result of the open source PIV imageJ plugin. The jetting behaviour induced by the plume seems well captured, with highest velocities measured at the center of the jet. The vector field attests for two recirculating vortices on both sides of the jet. However, the velocity dramatically drops to zero close to the disk boundary. Looking at the fluorescence micrograph of panel

113 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.11 – Comparing velocity measurements. (a) Confocal fluorescence micrograph of active nematics confined in a disk with 350 µm diameter. The snapshot corresponds to the timestep at which the velocity field is computed. (b)Velocity field using conventional PIV methods. The highest velocities are measured within the jet. The velocity at the walls is zero. The speed has been normalized (the absolute value is not of interest for now). (c) Velocity field obtained through the method of Boquet et al. The maximal velocity is measured along the disk boundary, on both sides of the wall defect. (d) Time-lapse of the active nematic dynamics during one oscillation period of the plume, with an overlay the velocity field computed from BioFlow. The delay between each frame is δt = 2s.

(a), we can see that the microtubule filaments are well aligned with the boundary. As said in section 2.2.4, in classical PIV the component parallel to the filaments is seriously under estimated owing to a lack of contrast. We believe that the velocity drop at the boundary is an artefact that directly comes from this technical limitation. On the contrary, the output of BioFlow displayed in panel (c) seems to successfully capture the recirculation at the boundary. The highest velocities are measured along the wall, on both sides of the defect. Overall, the whole flow pattern looks much more faithful to what the eye may catch from the experimental video. Moreover, in the time-lapse depicted in panel (d), we can see that the flow field correctly captures the oscillations of the central plume.

External Force In Eq. 5.15, the deﬁnition of fext includes all the external forces applied to the system. In our case, the system is the water phase, and the only external force is the action of the active nematic layer. In principle, the contribution of the active stress to the active ﬂows is modelled by fa (see section 1.3.1). As a consequence, fa and fext should be similar. In Fig. 5.12, we compare the maps of active force f inferred from the tensor order parameter (Eq. 5.1) with the external force computed from BioFlow.

114 5.2. Properties of an isolated negative defect at a wall

The map of f, depicted in In Fig. 5.12 (a), can be split into three zones. To the right side of the disk where the nematic phase is well-aligned, f is negligible. Around the center of the disk, f is mostly localized around the core of a positive defect recently formed through the oscillation of the plume. Its direction is aligned with the polarity of the defect. Finally, the region of the wall defect is characterized by a V-shaped force line, described in details in section 5.2.2. ˜ The map of fext bears similarities with f. It is also minimal in the regions where the nematic layer is aligned. Furthermore, at the center of the disk, the magnitude of the force is highest within the core of the defect. However, it is also more spread out than f. Its orientation is roughly aligned with the defect polarity, although the force line appears more bent than f. The most significant discrepancy between the two force maps concerns the region of the ˜ wall-defect where the characteristic V-shaped force line is absent in fext. To understand this point, we propose the following interpretation. By definition, f maps the divergence of the tensor order parameter. As such, it shows the distribution of "internal stresses" i.e the regions of space where the local symmetry is such that the active stresses combine into a directed force. For instance, the axial symmetry of a positive defect results in a force aligned with its symmetry axis. If the positive defect is isolated within a well- aligned region, the defect will be propelled by f and generate flows. Conversely, consider a hypothetical case of two positive defects facing one another. They are both propelled by a force f aligned with their polarity. However, these forces cancel out by symmetry and the defects stay still. ˜ In the map of fext, we only see the forces that contribute to flows in the water phase. Around the wall-defect, the forces induced by both arms of the V-shape line cancel out by symmetry: the component normal to the wall is balanced by the reaction of the wall, and the tangential components or both arms counter-balance. As a consequence, these internal stresses do not contribute to the external force.

Pressure field The pressure field has not been studied in detail yet. In Fig. 5.13 (a), we can see that the instantaneous pressure field shows only subtle variations. Yet, it seems to suggest that the pressure around the wall-defect is higher than anywhere else. A closer look suggests that the pressure is even higher within the core of the positive defects, as shown in the inset. The time-averaged pressure map depicted in In Fig. 5.13 (b) gives a clearer picture of the distribution, confirming that the wall-defect increases the local hydrodynamic pressure on a quite extended area. Such a distribution is coherent with the flow field because the plume creates a strong jet expelling the solvent from high to low pressure zones.

Perspectives: understanding wall defect nucleation In Fig. 5.11, we have seen that maximal velocities are observed at the center of the disk and even more so along the wall, in the vicinity of the defect. Interestingly, these maxima at the wall coincide with the regions where velocity gradients are highest, as shown in Fig. 5.14. Here, we have plotted ||∇u|| where u represents an average of the velocity ﬁeld over the 183 frames and || · || is the frobenius norm for matrices 2. In our

2 qP P 2 The frobenius norm of a matrix A is deﬁned [196] as ||A|| = i j Aij

115 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.12 – From internal stresses in active nematics to external force onto the solvent. (a) Map of f = ∇ · Q (pink) overlaid onto the corresponding ﬂuorescence image (green). The magnitude has been scaled. Inset Close-up around a s = +1/2 defect ˜ at the center of the disk. White arrows indicate the direction of f. (b) Colormap of fext computed by BioFlow. Inset Close-up around a s = +1/2 defect at the center of the disk. ˜ White arrows indicate the direction of fext.

Figure 5.13 – Hydrodynamic pressure. (a) Normalized pressure gradients ∇p˜ overlaid on top of the ﬂuorescence micrograph of the active nematic layer. The computation is based on a single pair of images. Inset Close-up around a s = +1/2 defect at the center of the disk (b) Normalized pressure gradients ∇p˜ averaged over the 183 frames of the video. The area to the left, where pressure is maximal, corresponds to the location of the wall-defect.

116 5.2. Properties of an isolated negative defect at a wall

Figure 5.14 – Velocity gradients and defect nucleation. (a) Normalized velocity gradient, averaged over the 183 frames. The red contour represents the typical extension around the wall-defect where additional defect occasionally appear. (b) Snapshot of the active nematic in the Nucleation a Merging phase, described in Fig. 5.2 (d), and showing two additional defects recently nucleated. Note that compared to Fig. 5.2 (d), the image has been rotated by 90° CCW such that the position of the central defect matches with that of (a). The comparison only aims at showing that the extension of the unstable region around the central defect is comparable with the region of large velocity gradient.

interpretation, ||∇u|| provides an estimation of the local heterogeneity of the flow pattern. From section 1.3.1, we know that the flow field may perturb the alignment of the nematic phase by enhancing bend distortions. As a consequence, regions with higher velocity gradients could be more prone to defect nucleation. As a matter of fact, in section 5.2.1 we showed that the sides of a wall-defect were preferential sites for defect nucleation. In the snapshot from the nucleation and merging phase depicted in Fig. 5.14 (b), we can see that the additional defects appear within a zone that roughly corresponds to the zone of high velocity gradients. However these results need to be complemented with theoretical arguments and deeper analysis.

5.2.4 Conclusion The study of an isolated defect at a wall has provided insightful information on its properties.

• The topological charge of a wall-defect is not well deﬁned. The edge charge is

sedge = −1/2. Even so, the cracks forming on both sides of the defect core appear quite unstable, and prone to form s = +1/2 defects. Overall, the description of these objects in terms of topological charge looks too simplistic.

• The motility of wall defects can be explained by symmetry arguments. These structures have the same mirror symmetry as positive defects, conferring them a net propulsion force. Most of this force is pointing towards the wall onto which they are bound, but tangential forces appear as soon as the mirror symmetry of the defect is distorted. In particular, when two defects are close enough, they attract.

• We propose two distinct mechanisms for defect nucleation. The ﬁrst one is responsible for the appearance of wall-defects separated by a well-deﬁned length scale,

117 Chapter 5. Active Nematics at a Wall: Description and Control

reminiscent of the nucleation process in the bulk described in section 1.3.1. In the vicinity of these main structures, additional and short-lived defects often appear before quickly merging. These nucleations are more frequent, and appear more random.

Several hypotheses have been proposed to understand the second nucleation mechanism.

• Tangential stress. In section 5.2.2, we argued that a nucleation at the wall might occur as soon as the tangential stresses decrease below a critical value. The idea was that tangential stresses tend to ﬂow-align the nematic ﬁeld parallel to the wall, damping the bend distortions. This argument has not permitted to reliably predict the nucleations in our experiments.

• Pressure. The second idea we have tested involves a pressure term. In simple words, we had the intuition that the bend distortions at the wall could be damped by a hydrodynamic pressure "pushing" the nematic phase onto the wall. The computation of the hydrodynamic pressure was performed using an image processing software called BioFlow. Contrary to our ﬁrst intuition, the regions where defects never nucleate correspond to the lowest hydrodynamic pressures.

• Velocity gradient BioFlow also provides a much better resolution in the velocity ﬁeld. We observe that the surrounding of a wall-defect also correspond to the regions with highest velocity gradients. This criterion could explain defect nucleation, in the sense that large gradients may create important distortions in the nematic ﬁeld.

• Finally, we believe that the computation of the external force applied to the solvent using BioFlow could provide useful information on how the active force coming from the distortions of the active nematic layer transfers to the solvent.

5.3 Collective dynamics of negative defects at a wall

In this part, we will describe the tree patterns observed in active nematics at a wall based on the formalism of spatio-temporal chaos. Most of the content presented here is adapted from the literature. The qualitative comparison with results on active nematics is still preliminary and most of the concepts introduced here are meant to open and motivate future work.

5.3.1 On the road to spatio-temporal chaos Under non-equilibrium conditions, a uniform and extended system will likely transit to a state with spatial variations, a "pattern" [197, 198]. Pattern formation has been extensively studied in very diverse fields, from chemistry [199, 200] and biology [201, 202] to plasma physics [203] and fluid dynamics [204, 205]. This wide interest is justified by the fact that patterns arise from nonlinear effects which can lead to qualitatively new phenomena that do not occur in linear systems. However, including nonlinear terms in the constitutive equations of a given system considerably increases the complexity of a

118 5.3. Collective dynamics of negative defects at a wall theoretical description, often preventing the prediction of analytical solutions. On the other hand, some features of pattern formation, and in particular the transition to chaos, are common to a wide variety of systems with very distinct constitutive equations (e.g. hydrodynamics, chemical systems, plasma physics). As a consequence, considerable eﬀort has been made to compare experiments with the most generic mathematical expressions yielding chaotic behaviour [206,207].

Spatio-temporal chaos (STC) [208] is a particular class of dynamical pattern arising in systems lacking long-time, large-distance coherence in spite of an organized behaviour at the local scale. According to Manneville, STC is located in the middle of a triangle, the corners of which are temporal chaos [209], prevalent for a few spatially frozen degrees of freedom, spatial chaos [210,211], in time-independent patterns, and turbulence [212,213], with cascading processes over a wide range of space and time scales. STC is characterized at the local scale by an instability mechanism that generates dissipative structures. However, the nature of the dissipative structure can be of very different type, and vary greatly if an additional specific frequency (ωc) and/or spatial periodicity (kc) are introduced in the system, for example through geometrical confinement. In that case, the most important parameter that controls the transition to chaos is the aspect ratio. This quantity corresponds to the ratio of the lateral extension of the confining geometry to the typical size of the dissipative structures. When this ratio is small, confinement effects are strong and the chaos is purely temporal. STC emerges when the confinement is partially relaxed, and it definitely relates to the process of transition to turbulence when confinement effects are weak. It appears to occupy a central position at the cross-roads of nonlinear dynamics, mathematical stability theory, and statistical physics of many-body systems and non-equilibrium processes, with a wide potential for applications [214]. Early work focused on temporal chaos because the lesser degrees of freedom enabled a close connection with the field of dynamical systems [215–217]. The treatment of spatial variations was permitted by the development of envelope equations [218–220], that enabled to illustrate the route to STC in many experimental systems such as the Rayleigh-Bénard instability. These equations describe the dynamics of spatio-temporal modes that have become unstable above a critical value of the control parameter. They are composed of two parts: a linear term that describes the instability mechanism, and non linear terms which control the behaviour of the amplitude of unstable modes, for instance, their saturation. Notably, in some cases the transition to chaos in laterally-confined systems can be well recovered by envelope equations that are only one-dimensional, which considerably simplifies the problem. The simplest envelope equation that yields spatio temporal chaos is called the Kuramoto- Sivashinski equation (KSE) [219–223]. In this section, we will show that the spatio- temporal "tree" patterns obtained for active nematics close to a wall are reminiscent of the patterns predicted by KSE. This result is of interest for two reasons. First, KSE is purely 1D, which implies that the influence of the active nematics far from the wall can be neglected. In other words, wall defects nucleate and merge in a way that does not depend on the bulk behaviour. In addition, one of the main issues with spatio-temporal chaos that prevents to treat the problem with the framework of dynamical systems is the great (infinite) number of degrees of freedom. However, studies of envelope equations prove that the energetically dominant modes of the dynamics are large scale and low

119 Chapter 5. Active Nematics at a Wall: Description and Control frequency [222]. In other words, the eﬀective number of degrees of freedom is "slaved" by the ratio between the system size and the wavelength of the dominant modes. In the case of KSE, the problem is even simpler because the large scale structures are characterized by a single length-scale. Theoretical eﬀorts have been made to reduce the complexity of modelling STC by constructing low-dimensional models based on the interactions between localized (soliton-like) structures [224–226]. In active nematics, the localized structures are the wall defects.

5.3.2 Describing Chaos: the Kuramoto-Sivashinski equation

As Manneville writes in his review on KSE [222], instabilities which develop in continuous media often lead to the formation of cellular structures periodic in space and/or time. The transition to turbulence is controlled by the aspect ratio, defined as the ratio of the lateral extension of the experimental enclosure to the typical size of the periodic structure. When this ratio is large, the most important features of the dynamics of these structures close to onset are related to long-wavelength low-frequency spatio-temporal modulations. These modulations can be accounted for by envelope equations, the envelope being generically a complex function slowly varying at the scale of the individual cellular structures. In some cases, the envelope modulus is slaved to an equilibrium value while the phase ψ remains the actual dynamical variable. The function ψ is a phase term, that is left very generic. It has been shown to describe the evolution of the perturbations to chemical waves propagating in a bistable system, the dynamics of flame front modulations, but also the evolution of a homogeneous medium unstable with respect to a spatially uniform oscillating chemical reaction. The KSE is the simplest model describing the transition to spatio-temporal chaos for a spatially extended systems subject to an instability at a well defined length scale. Consider a function ψ(x, t) defined in a domain of size L. The canonical Kuramoto- Sivashinski equation can be written as follows:

ψt + ψψx + ψxx + ψxxxx = 0 x ∈ [0,L] (5.17) + initial condition + boundary conditions,

n n where ψt = ∂ψ/∂t and ψnx = ∂ ψ/∂x . The boundary conditions usually considered are ψ = ψx = 0 at x = 0,L or periodicity ψ(L) = ψ(0). The choice of boundary conditions have very little influence on the bulk behaviour when L is large enough (typically L > 20). Similarly, the bulk behaviour does not depend on initial conditions. We note that the equation admits a trivial solution, = 0. Linear fluctuations around = 0 have the ψ ψ√ 2 4 growth rate σ = k − k , which gives two different wave numbers: km = 1/ 2 correspond to the fluctuation with highest growth rate, while k0 = 1 corresponds to the vanishing growth rate. These length scales play an important role in understanding KSE chaos [227]. Written in its non-dimensional form as 5.17, we see that the only control parameter is the system size, L. As L becomes large, typically L > 30, spatio-temporal chaos is observed.

120 5.3. Collective dynamics of negative defects at a wall

Figure 5.15 – Energy spectrum of KSE chaos. Adapted from Toh et al. [227]. It is −4 characterized by a high peak at k0 followed by a power law decrease as k . The behaviour at large k is characterized by an exponential decrease.

Energy Spectrum The energy spectrum is a concept introduced in hydrodynamics to understand how and why kinetic energy is transferred to structures over a broad spectrum of sizes. In the context of KSE, it is deﬁned as the Fourier transform of the two-point spatial correlation 2 function: E(k) = |ψk| , where k is the wave number. The typical shape of the energy spectrum is KSE is depicted in Fig. 5.15. It is characterized by a ﬂat part near = 0. √ k As k increases, we notice a high peak reaching a maximum near km = 1/ 2. This means that the energy input is maximal at the length scale corresponding to the most unstable

fluctuations of the instability. For k ≥ km the energy decreases first with a power law −4 as k and then exponentially as exp −k/k1 for large enough k. In KSE, the energy input is guaranteed by the diffusive term, ψxx, dissipation is ensured by the stabilizing term ψxxxx, and the energy transfer is enabled by the non-linear term ψψx. However, the mechanism of energy transfer is still poorly understood. One of the open questions that we will discuss later is the role of the regulation process, that is, the growth and death, of localized structures, on the energy mitigation.

A measure of chaos: Lyapunov Exponents Envelope equations are so-to-speak a bridge between Partial Differential Equations (PDEs) and dynamical systems. In the previous part on energy spectra, chaos was thought as an energy transfer across different length scales whose mechanism can be associated to specific terms of the PDE. The formalism of dynamical systems provides a different point of vue, where chaos is described in terms of degrees of freedom. If we consider a phase ψ defined in a discrete system of size N, the degrees of freedom are the values of ψ on the grid points. The Lyapunov exponents or Lyapunov Numbers (LN) measure the degree

121 Chapter 5. Active Nematics at a Wall: Description and Control of instability of trajectories in phase space [228]. More precisely, they correspond to the divergence rate between two infinitesimally close points in the phase space. If a LN is positive, the trajectories diverge exponentially and the system is non predictable. For a discrete system, there are as many LNs as there are degrees of freedom. If the maximum Lyapunov exponent (MLE) is positive, the system is chaotic. Furthermore, the number of positive LN measures the number of "effective" degrees of freedom of a given system. We would like to lay the emphasis on two results that are of interest in the context of our study. First, it was found that KSE chaos is extensive [225] (and even microextensive [226]): this means that the number of positive LNs and consequently the dynamical complexity, increase linearly with the system size L. Second, an in-depth study on the Lyapunov Numbers [225] has argued against the mechanism of energy cascade in the case of KSE chaos, suggesting on the contrary that the energy mitigation process is driven by the growth and death of localized structures. Both results stress the importance of localized structures and have motivated efforts to relate the "chaotic degrees of freedom" to these objects.

Localized Structures

A typical spatio-temporal pattern of the solution ψ(x, t) for L = 100 is displayed if Fig. 5.16 (a), taken from Wittenberg et al. [224]. Note that the origin of time in this picture does not correspond to the ﬁrst timestep of the simulation (t = 0 in the plot corresponds to t = 1 · 105 in the numerical simulation). In this representation, the base solution ψ(x, t) = 0 would appear uniformly grey. Instead, we observe very contrasted structures, in the form of white lines with a dark shade. These lines originate from uniform (grey) regions, oscillate rather erratically and merge with each other. Their connectivity is better appreciated in Fig. 5.16 (b) where we have manually drawn the main path lines. The spacing between adjacent lines seems to be controlled by a well-deﬁned length scale.

5.3.3 KSE and 1D active nematics The analogy between KSE and active nematics is far from obvious from the constitutive equations of active liquid crystals (see Eq. 5.2). As explained previously, the aim of this study is not to give an analytical description of active nematics, but to use well-established theoretical models of spatio-temporal chaos to understand wall defects dynamics. Still, we can provide arguments to support the analogy. First, KSE describes spatially-extended systems which spontaneously develop instabilities with well-defined length and time scales. Unconfined active nematics provide a good example of emergent STC. The bend instability generates topological defects at a frequency given by the nucleation rate ωa 6= 0, and the defect density is controlled by the active length-scale ka 6= 0 (section 1.3.1). The emerging pattern is consequently time-dependent and spatially heterogeneous. KSE is designed for a field supported by a finite domain of size L. In our experiments, the tree patterns are obtained by analysing the dynamics of active nematics in the vicinity of the wall, which is by definition a 1D problem. Conventionally, the boundary conditions are taken as periodic. As shown in Fig. 5.1, the wall considered is the perimeter of a disk, which provides periodic boundary conditions. In KSE, the field ψ is a phase term. It is not clear which physical field should be considered in our case. The space-time plots represent the fluorescence intensity variations along the wall. This fluorescence is directly

122 5.3. Collective dynamics of negative defects at a wall

Figure 5.16 – Spatio-temporal patterns of ψ as a solution of the KSE. (a) Spatio- temporal pattern ψ(x, t) of a solution of KSE chaos with L = 100. Adapted from Wit- tenberg et al. [224]. Contrary to the original ﬁgure, time goes downwards. (b) Binary overlay of the large-scale structures in panel (a). The overlay is drawn by hand with the help of a graphic table. (c) Spatio-temporal pattern of wall defects in active nematics. Time goes downwards. The data was obtained around the inner wall of a 200µm wide annulus with inner radius Ri = 150µm. (d) Binary overlay of the large-scale structures in panel (c). The overlay is drawn by hand with the help of a graphic table.

123 Chapter 5. Active Nematics at a Wall: Description and Control related to the microtubule density ρ. We also know that the regions of uniform intensity correspond to domains where microtubules are aligned with the walls, while contrasted regions correspond to the location of defects. As such, ψ could be related to the orientation of the microtubules at the wall. More importantly, KSE admits ψ = 0 as a solution. In our case, this solution would correspond to a perfect alignment with the wall. We know from Chapter4 that microtubules have a strong tendency to align with the walls, and temporary events of defect-free alignment are observed in narrow conﬁnement. As said in the previous part, much of the interest in KSE simulations has been driven by the behaviour of emerging localized, large-scale structures. In the next part, we propose a comparative study of KSE emerging structures with wall defect dynamics.

Localized structures in active nematics Figures 5.16 (c) and (d) allow a comparison of KSE structures with the typical tree patterns observed in the case of active nematics at a wall. The raw images are similar, for they both contain wavy lines with a black and white contrast merging with each other. Also, the structures observed in Fig. 5.16 (c) look very localized, while the structure obtained by KSE simulation appear more spread out. As a consequence, the uniform regions of grey intensity occupy more space in Fig. 5.16 (c). Another diﬀerence is the nucleation of the lines. In the previous section we described the branches to appear in the form of a dark spot (a "blossom"). On the contrary, in Fig. 5.16 (a) the branches seem to nucleate more "smoothly" from the uniform regions. The higher localization of the branches in active nematics is a sign that these structures are true singularities (defects) in the nematic ﬁeld, while the structures in KSE are localized (but continuous) phase extrema. Still, the nucleation and interactions between branches is very similar, as shown in Figs. (b) and (d). They both nucleate from uniform regions, oscillate erratically, and merge to form binary trees. Furthermore, despite the apparent randomness of merging events, the trees appear regularly spaced. As such, the KSE equation captures the essential features of wall defects that we were discussing in the previous section.

5.3.4 Statistical properties of localized structures Defect classiﬁcation

The localized structures in the chaotic phase of KSE are not defects per se, because ψ is a continuous function of x. An example of the profile of ψ in the chaotic state is displayed in Fig. 5.17 (a) reproduced from Toh [227]. The phase oscillates between −π < ψ < π with well defined peaks. These peaks can be followed in time, producing characteristic space-time tree patterns as shown in Fig. 5.17 (b) (time goes upwards). The magnitude of the peak is later chosen as a criterion to define localized structures: Figures. 5.17 (c), (d) and (e) correspond to the criterion ψ > 0, 1, 2, respectively. The higher peaks (ψ > 1) are called "pulses", and the smaller ones "humps". These two structures have different behaviours. According to Toh, "small humps are on and off within short time intervals" and "seem to be fluctuations on the shoulder of large humps [pulses]". These two populations are reminiscent of the discussion we made in section 5.2.1: on top of regularly spaced nucleations, we notice that the tree branches are unexpectedly rich in short-lived blossoms. Later in the paper, Toh computes statistical information on these

124 5.3. Collective dynamics of negative defects at a wall

Figure 5.17 – Localized structures in KSE chaos. Figure adapted from Toh et al. [227]. (a) Spatial evolution of solution ψ at t = 105000 for L = 512. (b) Temporal evolution of maxima. 100 time-steps separated by ∆t = 0.5 from t = 105000 are plotted. (c), (d) and (e) are for the maxima larger than 0, 1 and 2.

125 Chapter 5. Active Nematics at a Wall: Description and Control two populations, and in particular the distribution of distances.

We would like to compare his results with our experimental data. To do so, we need to find a good definition (and detection method) of humps and pulses in our images. We cannot use the fluorescence intensity as a criterion to distinguish humps from pulses, because the relation between the image intensity and ψ is not straightforward. However, bearing in mind that humps are short-lived, the defect lifetime can be a good criterion. The detection process is presented in Fig. 5.18. Starting from the raw intensity image, we first need to extract the defect tracks to isolate the tree pattern. Various thresholding methods have been tried. Unfortunately, they all produced "ghost" branches and artefacts that we did not manage to overcome. Instead, we found it straightforward and more reliable to manually trace the defect tracks. The result, shown in Fig. 5.18 (b) is a binary image containing the tree structure. Using custom-made processing tools, we can then easily detect all the nucleation events (Fig. 5.18 (c)), and follow the defects until they disappear. When two defects are merging, the remaining track is attributed to the oldest defect. The outcome of the tracking is pictured in Fig. 5.18 (d), where different colors stand for different defect tracks. We readily notice a large dispersion of track lengths. The lifetime threshold is arbitrarily set at tc = 50s, with tracks longer than tc attributed to pulses. The result is displayed in Fig. 5.18 (d). The red tracks correspond to pulses, and the blue tracks correspond to humps. In the particular experiment displayed in Fig. 5.18, the number of defect tracks detected (panel (c)) was 181. From these detections, 169 correspond to humps and only 12 correspond to pulses. We notice that some of the pulses percolate from the top to the bottom of the image, which means that their lifetime is longer than the measurement time (300s). The full distribution of defect lifetime, called δt, for the two populations is displayed in Fig. 5.19. In the case of humps (Fig. 5.19 (b)) , the distribution is relatively narrow and centered below 10s, with a peak at δt = 3.7s and a median at δt = 6s. On the contrary, the distribution of δt for pulses is much broader, with a peak at δt = 300s and a median at δt = 86s. Although statistical information on 12 pulses is not very relevant, we see that 4 pulses out of 12 last longer than the measurement time, which is 2 orders of magnitude longer than the typical lifetime of a hump. This result justifies to treat pulses and humps as distinct objects, and corroborates the classification made by Toh.

Distribution of distances Once the populations are clearly defined, Toh computes the statistical distribution of inter-defect distance. To do so, he records the distances between closest neighbours at each time, distinguishing two cases. First, he takes into account all the peaks (pulses and humps). The distribution, overlaid with grey disks in Fig. 5.20 (a), has two peaks. He attributes the higher one, localized around δH,1 = 7, to the hump-to-pulse distance. The second one, centered around δH,2 = 9, corresponds to the pulse-to-pulse distance. This means that the humps are sharply distributed with shorter distances than those of the pulses. In the second distribution, overlaid in grey triangles in Fig. 5.20 (a), the humps are filtered out, and the computation only accounts for the pulses. The profile is localized around a single peak at δP = 9 = δH,2. This means that pulses are line up nearly peri-

126 5.3. Collective dynamics of negative defects at a wall

Figure 5.18 – Humps and pulses in active nematics. (a) Spatio-temporal pattern of the fluorescence intensity at the inner wall of a 200µm wide active nematic annulus with inner radius Ri = 150 µm. Time goes downards. Scale bar: 100 µm. (b) Binary overlay of the localized structures in (a). The overlay is drawn by hand to prevent detection artefacts with the help of a graphic table. (c) Automatic detection of defect nucleations from the binary image. A detection is defined as the beginning of a white line in (b). (d) Output of the defect tracking algorithm. Different colors stand for different defects.

(d) Output of the defect classiﬁcation algorithm. Defect tracks shorter than tc = 50s are attributed to humps, and colored in blue. Defect tracks longer than tc = 50s are attributed to pulses, and colored in red.

Figure 5.19 – Lifetime distribution of humps and pulses. (a) Sample of Fig. 5.18 (e) showing pulses in red and humps in blue. The lifetime δt is deﬁned as the time between the nucleation and the merging of a given structure. (b) Distribution of lifetimes of humps. Statistics obtained over 169 humps. (c) Distribution of lifetimes of pulses. Statistics obtained over 12 pulses Note that the lifetime is underestimated because the measuring window is too short.

127 Chapter 5. Active Nematics at a Wall: Description and Control odically. On the contrary, humps are nucleating randomly because of fluctuations on the shoulder of the pulses. This qualitative description fits well with the space-time map of humps and pulses obtained for active nematics, recalled in Fig. 5.19 (a). The quantitative comparison with the distribution of distances is in good agreement, despite some discrepancies. The experimental distribution of all defects, pictured in blue in Fig. 5.20 (b), also has two peaks. The first one, the highest, is located around δH,1 = 42±10 µm. If the interpretation by Toh is valid, this value would correspond to the hump-to-pulse distance.

The second peak is less pronounced, and located around δH,2 = 115±10 µm. This value should correspond to the pulse-to-pulse distance. On the other hand the distribution of pulses pictured in red has a single peak, although relatively broad, centered around

δP = 192±10 µm. Consequently, in our case δH,2 < δP . We explain this result by the fact that humps are in large majority over pulses. Two pulses are almost never closest neighbours because of the presence of at least one intercalary hump. As a consequence, δH,2 does not correspond to the characteristic pulse-to-pulse distance δP . In our interpretation, it measures the characteristic distance between two "trees", deﬁned by a main pulse with its surrounding humps nucleating on both sides, which leads to δH,2 ∼ δP − 2δH,1.

Such details do not impede the generality of Toh’s attempt to reduce the complexity of KSE chaos to two main features: chaos consists of localized structures, and these structures are located according to a speciﬁc distribution in the statistical sense. In the next section, we will explain how these ideas are helpful to understand chaos, and in particular, the energy spectra, through statistical models.

Defects and soliton-like pulses

In this section we briefly summarize the statistical model developed by Toh. We consider a domain of length L with periodic boundary conditions. This domain is populated by localized structures with a fixed shape f(x), called soliton-like pulses (SLPs). The dynamics of these SLPs, characteristic of KSE chaos, are accounted for by a statistical distribution of locations constructed as follows. The distances between neighbours δi are controlled by the distribution of pulse-pulse distances p0(δ), previously computed and displayed in Fig. 5.20 (b). The number of SLPs, called N˜, fluctuates about a mean value given by L/δP . The distances between neighbours δi are supposed mutually independent. This is not entirely true because of the boundary conditions but L is supposed large P enough such that L− δi << L. A sketch of the model is displayed in Fig. 5.21 (a). The domain of length L is wrapped into a circle to represent the periodic boundary conditions. The white disks correspond to the location of the SLPs distributed around the circle. In Fig. 5.21 (b), the same sketch is overlaid onto the inner an outer walls of an active nematic annulus in the chaotic state, to show that this description applies to both walls independently. The SLPs would correspond to the wall defects. Let Ln be the location of the n-th SLP, such that δn = Ln+1 − Ln. The shape of the n-th SLP is given by f(x + Ln) and its Fourier transform F (k) exp iLnk. From this model, it is possible to compute an energy spectrum. According to Toh, the energy spectrum is given by the following equation:

128 5.3. Collective dynamics of negative defects at a wall

Figure 5.20 – Frequency distribution of the distances. (a) In the case of KSE chaos, adapted from Toh et al. [227]. The distribution including humps and pulses is overlaid with black circles. The distribution including pulses only is overlaid with black triangles.

(e) Probability density function p0(δ) for inter-pulse distance. (c) Inset of Fig. 5.17 (b) showing the spatio-temporal pattern of localized structures used in the computation. (d) In the case of active nematics. We used the spatio-temporal pattern presented in Fig. 5.18 (a). The distribution including humps and pulses is overlaid in blue. The distribution including pulses only is overlaid in red. (b) Probability density function p0(δ) for inter- pulse distance. (f) Inset of the tracking showing an example of the distance δ between two neighbouring pulses.

129 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.21 – Soliton-like pulses on a periodic domain. (a) Sketch of the statistical model proposed by Toh et al. [227], reducing the spatio-temporal patterns of ψ to a distribution of a random number N of soliton-like pulses distributed over a domain of size L, and where the distribution of defect distances δi satisﬁes p0(δ). (b) Example of application of the model to the inner and outer walls of a 200µm wide annulus with inner radius Ri =150 µm. Note that the green and yellow systems are 1D and supposed independent. The pulses correspond to the location of wall-defects.

1 ( ) = ( ) 2 ( ) E k 2|F k | g k N (5.18) X g(k) = exp [i(Ln − Lm)k] n,m=1 The term F (k) only depends on the shape of the soliton. On the contrary, g(k) depends on the spatial distribution of the localized structures along the domain. The author discriminates two cases. In the ﬁrst case, the distances between pulses are governed by p0(δ), as explained in the previous paragraph. The resulting spectrum, pictured in Fig. 5.22 (a), is strikingly similar to KSE spectrum. In particular, the characteristic peak at km is recovered. In the second case, the author gets rid of the regulation of distances between pulses by supposing uniform spatial distribution. The resulting energy spectrum, pictured in Fig. 5.22 (b), corresponds exactly to the spectrum of a single solitary wave, and the signature of KSE chaos is lost. The author concludes that the characteristic of the energy distribution of KSE chaos are directly driven by the growth and death of localized structures.

5.3.5 Energy spectrum in Active Nematics In order to push the analogy between KSE chaos and active nematics at a wall, it would be helpful to compute an energy spectrum. As previously said, it is deﬁned as the Fourier transform of the two-point correlation function of ψ. The process, described in Fig. 5.23, is in three steps. At each time t, we recover the values of ψ(x). An example of proﬁle

130 5.3. Collective dynamics of negative defects at a wall

Figure 5.22 – Energy spectrum of the SLP model. Figure adapted from Toh et al. [227]. (a) Energy spectrum multiplied by the factor L/2π for L = 512. The straight line corresponds to the SLP model while the dotted line corresponds to KSE simulations. (b) Energy spectrum of the solitary wave multiplied by the factor L/2π for L = 512. The outcome of SLP model has the same shape if we suppose a uniform distribution of SLPs along L.

131 Chapter 5. Active Nematics at a Wall: Description and Control is shown in Fig. 5.23 (b) From this array, we compute the autocorrelation function of position C(δ, t) = E[ψ(x, t)ψ∗(x + δ, t)] where E is the expected value operator. Then, we average the correlation function over time to get C(δ) = hC(t, δ)it. The result is displayed in Fig. 5.23 (c). Finally, we compute the Fourier transform of C to recover the energy spectrum. In order to test the computation method, we ﬁrst performed simulations of KSE chaos based on a matlab code taken from the literature.

The output energy spectrum for one of such simulations is displayed in Fig. 5.23 (d). The shape is coherent with the predictions of the literature: we can clearly see a flat profile close to = 0, followed by a high peak and a fast decrease with a characteristic k √ −4 power law close to k . However, the peak is not centred at km = 1/ 2 contrary to our expectations. We attribute this discrepancy to a different scale in the KSE simulation framework, that does not impact neither the shape not the values of the power-law exponents. The additional peaks observed in the tail of the signal (k > 10−1) are attributed to artefacts due to the pixelation of the image, and should not be considered.

Once the computation method has been calibrated, we may perform the same analysis for active nematics. Our best approximation of ψ is the fluorescence intensity space-time plot, I(x, t). For this example we used the kymograph obtained of the inner wall of a 200 µm-wide annulus with Ri =150 µm. The instantaneous intensity profile, displayed in Fig. 5.23 (f), is noisier than the simulation output, although we distinguish regularly spaced peaks. The correlation function, displayed in Fig. 5.23 (g), is quite different from the simulations. The minimum is attained for x = 30px. At larger distances, C increases to form a plateau up to x = 120px. On the contrary, the simulated C (panel (c)) showed periodic oscillations with a decaying envelope.

The discrepancy between KSE simulations and active nematics is also reﬂected in the energy spectrum. None of the characteristics of KSE chaos are convincingly recovered. The ﬂat plateau for small k is not clearly visible, although this could be due to the limited size of the images. The energy seems to reach a maximum close to k = 10−2, but the peak is weaker and broader compared to the one observed is Fig. 5.23 (d). Past the maximum, E(k) decreases as a power low k−α. This could be reminiscent of the k−4 power law of KSE, however in this particular experiment we found α = 1.71. Furthermore, the power-law regime extends over a large band of wave number (2 · 10−2 < k < 2 · 10−1) while the power law region of KSE is much narrower (5 · 10−2 < k < 3 · 10−2). For larger values of k, the exponential decrease characteristic of KSE is replaced by a power law as k−β. In this particular experiment, we found β = 3.4. However, these quantitative values of α and β need not be considered as consolidated results 3.

The significant difference between the energy spectra is surprising given the similarity of the spatio-temporal patterns, as depicted in Fig. 5.16. Certainly, our measurements are not energy spectra, strictly speaking. They are kymographs obtained from the fluorescence intensity contrasts close to the wall. As such, even if the spatial arrangement of

3The analysis on energy spectra proposed here is still preliminary, and the values of the exponents are not reliable, both because of a lack of experimental repetitions, and possible artefacts in the processing. On the other hand, we are conﬁdent on the shape of the spectra.

132 5.3. Collective dynamics of negative defects at a wall the localized structures, outlined in Fig. 5.16 (b) and (d), is very similar, the textures of the original images Fig. 5.16 (a) and (c)) are quite different. The size of the defect cores is smaller in the case of active nematics, and the defect-free regions appear quite uniform while the KSE patterns are almost exclusively composed of sinusoidal intensity modulations. Although these differences do not play any role in the statistical description presented before (section 5.3.4), they may have a strong impact on the spatial correlations computed in the energy spectrum. More fundamental differences may also affect the resulting spectra. For instance, the defect pattern in active nematics (Fig. 5.16 (b)) seems to contain more short-lived humps than the KSE counterpart (Fig. 5.16 (b)). A conclusion of the statistical analysis in section 5.3.4 was that the energy spectrum of KSE chaos could be recovered from the interactions between the main (soliton-like) pulses. An excess of humps may overshadow the regularity of the pulse arrangement, by degrading the spatial correlations.

In order to gain more insights on the shape of the spectrum obtained in Fig. 5.23 (h), we first need to assess its generality among various experimental conditions. First, we will investigate the role of curvature. So far, all the results presented have been obtained in annular geometries, on the grounds that most theoretical and simulation works suppose periodic boundary conditions. However, boundary conditions are known to play little influence on the chaotic dynamics (see section 5.3.2), and an annular configuration adds unnecessary curvature effects whose consequences on the dynamics are not known. As a consequence, we will perform the same analysis on a flat wall. Second, we need to distinguish which features of the spectrum of Fig. 5.23 (h) are intrinsic to the 1D correlations between defects, solely observed in the vicinity of the wall. To do so, we will compute similar spectra far away from the wall. Later, we will assess the influence of lateral confinement. As said in the introductory part of this chapter, KSE chaos describes "the transition to chaos when confinement is partially relaxed". The spectrum of Fig. 5.23 (h) was obtained in a 200 µm-wide annulus. At such a width, the flow patterns within the annulus are already well into the turbulent regime as described in chapter4. It would be interesting to study how the spectra evolve in more confined configurations. The last section aims at bridging the gap between the statistical analysis on defect interactions and the energy spectra. Namely, we will try to relate the relevant length scales of the spectra with the statistical distribution of distances between humps and pulses in a given experiment.

Energy spectrum on a flat wall In this experiment we study the spatio-temporal dynamics of active nematics close to a flat wall. As presented in the Methods section (2.4.1), annuli an channels are printed onto rectangular grids. Here, instead of imaging the dynamics within the confining annuli, we visualize the behaviour of the active layer on either of the four edges of the rectangular grid. A snapshot of the experiment is displayed in Fig. 5.24 (a). The grid is located at the bottom on the image, and the wall is represented by a black rectangle located at y = 0. The active layer is unconfined in the sense that it extends up to y ∼ 1mm. The video corresponding to this analysis is shown in 5.4. The space-time plot is obtained from the pixel intensity of a flat line parallel to the wall as a function of time. The

133 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.23 – Computing energy spectra from a spatio-temporal pattern (a) Spatio-temporal pattern of ψ from a simulation of KSE. (b) Instantaneous spatial profile ψ(x) at a given time. As ψ is a phase term it is bounded by [−π, π]. (c) Auto-correlation function of ψ in space, R, averaged over time. (d) Corresponding energy spectrum, computed as the Fourier transform of C. The computation method recovers all the features of KSE energy spectrum. The behaviour for k > 10−1 is attributed to artefacts due to the image pixelation. (e) Spatio-temporal pattern of the fluorescence intensity I(x, t) at the inner wall of a 200µm wide active nematic annulus with inner radius Ri = 150 µm. Time goes downards. (f) Instantaneous spatial profile I(x) at a given time. (g) Auto-correlation function of I(x, t) in space, C, averaged over time. (h) Corresponding energy spectrum, computed as the Fourier transform of C.

134 5.3. Collective dynamics of negative defects at a wall results are displayed in Fig. 5.24. We compare two cases. First, the flat line is taken in the close vicinity of the wall (see black dotted line in Fig. 5.24 (a). The corresponding space-time plot is plotted in Fig. 5.24 (b). We recognize the wall defect tree structures with a similar pattern as in the case of circular walls. The energy spectrum, displayed in Fig. 5.24 (c) in black, also has the same shape as Fig. 5.23 (h), although the values of the α and β exponents are different (α = 1.17 and β = 3.3 in this particular experiment). We repeat the same experiment, this time with a pixel line located 140µm away from the wall, where the dynamic is 2D. The space-time plot, pictured in Fig. 5.24 (d), shows smooth textures. The contrast is given by the motion of defects which constantly cross the pixel line. However the dynamics is not fully traceable because these defects evolve in a 2D space. The corresponding energy spectrum is displayed in red in Fig. 5.24 (c). The profiles are comparable for extremal values of k: they both exhibit a plateau at vanishing k and a k−β power law for large k (k > 10−1). However, the k−α power-law is not visible in the bulk spectrum. Note that this result does not necessarily mean that the bulk energy is not distributed the same way at the walls and in the bulk: the information extracted in the bulk is only partial, because the pixel line only accounts for one dimension of space. However, it means that the k−α power-law observed at the wall is a clear signature that wall defects are subject to a specific mechanism of energy transfer that is essentially unidimensional. The nature of this k−α power-law is, however, still poorly understood. This experiment on a flat wall suggests it is purely a wall effect. Moreover, the shape of the spectrum is not too different from the annular case, which suggests that the effect of curvature does not fundamentally impact the spatio-temporal dynamics of the wall-defects 4. In the next section, we explore the influence of lateral confinement on the shape of the energy spectra.

Effect of confinement We study the shape of the energy spectra on the inner and outer walls of active nematic annuli of decreasing width. The results are displayed in Fig. 5.25. They are still very preliminary, and at this stage we would like to lay the emphasis on two very general results. First, the kymographs (and their associated spectra) differ quite significantly between the inner and outer walls of a given annulus. Such a difference could be attributed to the difference of perimeter between the the inner and outer walls. In KSE chaos, the length of the domain L is the critical parameter that determines the dynamical regime. More importantly, the k−α power-law is only clearly distinguishable for the widest annuli of 200 and 110µm (panels (a-d)). Conversely, the profiles for smaller w (80 or 60µm) seem to acquire features of the KSE chaos. Namely, the behaviour at small k is essentially flat, although in some cases (panels (e-g)) we notice a first peak at the smallest value of k = 10−2.556. This specific data point is not considered relevant because it corresponds to δ = 1/k = 360px, which is the image size. Before the energy decreases for large k, we observe a significant peak at a specific wave number, reminiscent of the shape of KSE −α spectra about km. However, above the peak the k power-law range is less clearly defined and we do not measure any type of exponential decay for larger k.

4Even so, it may play an important role in the value of the exponents α and β

135 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.24 – Energy spectrum on a ﬂat wall (a) Fluorescent micrograph of active nematics in the vicinity of a ﬂat wall. The wall is located by a dark rectangle at the bottom. The black and red dotted lines correspond to the pixel arrays used to compute the space-time plots of the dynamics close (b) and far (d) from the wall, respectively. scale bar: 100µm. (c) Energy spectra extracted from the spatio-temporal patterns (b) and (d). the black line is obtained in the vicinity of the wall, while the red line is obtained 140µm away from the wall.

136 5.3. Collective dynamics of negative defects at a wall

Overall, it seems that the more confined the system is, the more the energy is localized about a well-defined length scale. This effect is particularly prominent for w <110 µm. Interestingly, in this particular set of experiments, w = 110µm corresponds to the transition from a turbulent flow pattern to the transport regime presented in chapter4. Even though a direct connection between the two events is non trivial, it seems intuitive that the higher order observed in the transport in active nematic annuli should be associated to the higher localization of the energy about a dominant wave number.

From localized structures to energy spectra In order to bridge the gap between the analyses on defect interactions and energy spectra, we can compare the length scales involved in the spectra with de statistical distribution of defect distances, extracted from the previous section (5.3.4). Such a comparison is presented in Fig. 5.26 in the case of a 110 µm-wide annulus. The distances are kept in pixels for simplicity. The energy spectrum, shown in panel (a), peaks at δ1 = 44.9px (with δ = 1/k being the wave length), which sets the upper limit of the power-law range (i.e the lower limit in the k space). The lower limit is given by δ2 = 6.3px (i.e upper limit is k space). On the other hand, the statistical distribution of distances, displayed in panel (b) yields δH,1 = 18.5px, δH,2 = 40.7px, and δP = 66px. Previously, we suggested that δH,2 measures the characteristic distance between trees (see section 5.3.4). Having δ1 ∼ δH,2 suggests that the k−α power-law decrease is related to the tree structures. More precisely, the power law decrease extends over the continuous range of wave numbers spanned by the inter-defect distances (δ2 < δH,1). This observation may provide information on a defect mediated energy mitigation process in active nematics. Further investigation would require theoretical developments to understand how the defect dynamics translate speciﬁcally to a power-law decrease in the energy spectra.

137 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.25 – From turbulence to KSE-like energy spectra under conﬁnement Energy spectra on the inner (left column) and outer (right column) walls of active nematic annuli of diﬀerent channel width (w decreases fron 200, 110, 80, 60 µm from top to bottom. As usual, the inner radius Ri = 150 µm is kept constant. Each panel is composed of (left) the spatio-temporal pattern and (right) the corresponding energy spectrum.

138 5.3. Collective dynamics of negative defects at a wall

Figure 5.26 – Energy and localized structures scalings (a) Energy spectrum computed from the inner wall of a 110µm wide annulus, whose spatio-temporal pattern is displayed as an inset. δ1 = 1/k1 and δ2 = 1/k2 correspond to the limits of the power-law range. (b) Corresponding frequency distribution of distances between localized structures, computed as Fig. 5.20.

5.3.6 Conclusion In this part, we have studied the collective dynamics of wall-defects and observed the following results.

• The characteristic spatio-temporal pattern of wall-defect interactions is strikingly similar to the dynamics of localized structures in unidimensional models of spatio- temporal chaos, and in particular the patterns produced by the Kuramoto-Sivanshinki Equation (KSE).

• This result suggests that wall-defect dynamics can be eﬃciently modelled as a 1D system, relatively independent on the neighbouring bulk chaotic ﬂows.

• Wall-defects are classiﬁed in two categories. Long-lived structures are termed pulses, and short-lived structures are termed humps.

• In KSE, the spatial probability distribution of pulses obeys a regulation mechanism maintaining a periodic arrangement with a well-deﬁned spacing. On the contrary, humps appear randomly in the vicinity of a pulse before quickly merging with it. Both types of defects are observed in active nematics, and each of them behaves in a similar fashion as for KSE.

• KSE chaos is recognizable by its energy spectrum, which depicts the spectral behaviour of the spatial correlations. It is mostly characterized by a noticeable peak at the wave-number corresponding to the regulation wavelength between pulses.

• Preliminary analyses on the equivalent energy spectra in active nematics give less compelling evidence of an analogy with KSE chaos. The characteristic peak is not clearly recovered, and replaced by a monotonous power-law decrease.

139 Chapter 5. Active Nematics at a Wall: Description and Control

• Some of the spectra obtained in narrow annuli, where the ﬂow patterns are more ordered, appear more similar to KSE spectra, which could suggest that unconﬁned active nematics are simply "too chaotic" to be described by simple KSE models.

5.4 Controlling Wall Defects

In the previous section, we have shown that the dynamics of active nematics at a wall were well characterized by unidimensional models of spatio-temporal chaos. This result is surprising because it means that the dynamics of the bulk, and in particular, the dynamics of bulk topological defects, do not have a great influence on the dynamics of wall defects. As such, we can say that the bulk does not send information to the walls. On the contrary, wall defects are responsible for the jetting of positive defects in the direction perpendicular to the walls. In other words, walls send positive defects to the bulk. The location of the jets is given by the distribution of wall defects. If we managed to control the location of the wall defects, we might be able to influence the bulk flow pattern which could help design autonomous out of equilibrium microfluidic devices. When the wall has a uniform curvature, we have seen that, although the distances between adjacent wall defects are regulated by a well-defined probability distribution, there is a uniform probability to find a wall-defect at a given location x along the wall. However, we will show in the preliminary results presented in this section that the distribution of defects can be significantly modified by slight modifications of the wall structure. The first part is dedicated to the effect of an indentation at the wall. The two others relate the effect of indentations with broken left-right symmetry, called ratchet patterns. The last part is devoted to the role of curvature. Ratchet and curvature experiments were performed by another student, Claire Doré, as part of a master project at Gulliver Institute, ESPCI, Paris, under my supervision. The results presented here are preliminary and will be pushed further in Claire’s PhD.

5.4.1 Eﬀect of an indentation

Although the previous section has shown that wall defects obey well defined regulation rules on the distances between new born defects, the precise location of a nucleation seems to be random. The idea of this project is to favour a specific location by introducing a spatial heterogeneity. In this part, we add an indentation at the surface of a disk of diameter D. The indentation has the form of a triangular cut somewhere along the perimeter of the disk. The cut is 10 µm wide and 10 µm deep, and these dimensions have been kept constant throughout the experiments. We consider this size to be small enough compared to the characteristic instability wavelength that it can be considered a fluctuation. An example of corrugated disk is displayed in Fig. 5.27 (a). The triangular indentation is located at the bottom of the disk. Because the size of the indentation is smaller than the persistence length of the system, most of the time the active nematic layer does not fill the hole, but instead follows the tangent to the disk as shown in Fig. 5.27 (a) and (b). However, sometimes, the filaments hanging above the indentation buckle and enter the indentation, as shown in Fig. 5.27 (c). This buckling is responsible for a secondary buckling event that propagates to the bulk (Fig. 5.27 (d)) eventually forming

140 5.4. Controlling Wall Defects

Figure 5.27 – Effect of an indentation on defect nucleation (a) Fluorescence micrograph of a corrugated disk of 350 µm diameter. The indentation is located at the bottom of the disk. (b-e) Time-lapse of a defect nucleation event induced by the indentation. The interval between each snapshot is 2 s. The scale bar is 50 µm. (b) The microtubule bundles are all aligned parallel to the wall, leaving an empty space in the hole of the indentation (c) The microtubules buckle at the location of the indentation and fill the hole (d) A secondary buckling event occurs in the vicinity of the indentation, to the right side (see red arrow) (e) The buckling amplifies to the bulk forming a wall defect. (f-i) sketches of each step described in snapshots (b-e).

a wall defect (Fig. 5.27 (e)). A simplified sketch of the mechanism is displayed in Fig. 5.27 (f-i). This process occurs randomly over time, but quite frequently. Notably, the nucleation comes with a jet perpendicular to the walls, since the positive defect is expelled as explained in the previous section 5.2.1. Therefore, adding a indentation is a way to induce localized jetting points. This idea has been fruitfully exploited in the annuli project, presented in chapter4. By adding indentations on the four corners of a square annulus, we can exploit the occasional jets to dictate the otherwise random handedness of the transport is active nematic annuli. The results are summarised in Fig. 5.28. Indentations can also contribute to flow regulation under confinement. In section 5.2, we have seen that the confinement of active nematics in a disk significantly tames the number of wall-defect nucleations. In the experiment presented in Fig. 5.29, we add an indentation to disks of decreasing diameter. A sketch of the geometry is given in panel (a). The diameter is changed from D = 400µm to D = 80 µm. In the largest disks, the dynamics are chaotic as expected. Looking at the binarised space-time plot of the wall defect dynamics, shown in Fig. 5.29 (f), we can see that the number of wall defects is relatively large. More importantly, we notice that although many defects randomly nucleate everywhere around the disk, they all merge onto a central tree. A red triangle locating the angular position of the indentation in the space-time plot shows that the central tree emerges from the indentation site. Therefore, the effect of the indentation is

141 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.28 – Controlling transport handedness with wall indentations (a,e) Sketches of square annuli with asymetric outer boundaries. (b,f) Fluorescence micrographs of active nematics conﬁned in asymetric square annuli. (e,g) Time-series of a pixel ring at the center of the annuli. (d,h) FFT computation of the times series (e,g) showing clockwise (resp. counter-clockwise) transport. The dotted lines are parallel to the brightest bands of the FFT signal.

already clearly visible. As D is decreased to 200 µm, the number of defect nucleations is expectedly smaller, as shown if Fig. 5.29 (g) showing less branches. The wall defects still seem attracted towards the indentation, but they are still largely spread around the disk. A peculiar configuration is observed when further decreasing D to 180 µm. The branches almost disappear, which means that defect nucleation is strongly prevented. The space- time plot shown in Fig. 5.29 (h) has two lines. One is located at the indentation site, and the other one at the opposite side of the disk. This means that the equilibrium state is not only composed of a wall defect at the indentation site, but also by a mirror defect at the opposite side of the disk, which is illustrated by the experimental snapshot in Fig. 5.29 (d). Finally, when D is decreased below 130 µm, a single wall defect is observed. Its position is fixed at the indentation site, a shown in Fig. 5.29 (i), although we distinguish small amplitude oscillations around it. A time-lapse of the dynamics in a 90 µm disk in Fig. 5.29 (j) shows the origin of these oscillations. This time, the indentation is located on the left side of the disk. We distinguish the wall defect, in the form of a very fluorescent jet, oscillating in a cilia-beating like motion [229]. The space-time plot of a pixel line located at a distance from the wall shows that the oscillations are remarkably periodic. The video corresponding to this analysis is shown in 5.5.

5.4.2 Effect of a ratchet pattern In the previous section, we have exploited spatial fluctuations to favour defect nucleations, and consequently, normal jets, at specific location along the walls. These jets were used

142 5.4. Controlling Wall Defects

Figure 5.29 – Periodic oscillations in a corrugated disk (a) Sketch of the corrugated disk geometry. The diameter varies between 80 µm and 400 µm. The indentation is a triangle with a fixed width and height of 10 µm. This size is chosen to be much smaller than the characteristic length scale of the system. (b), (c), (d), (e) are fluorescence micrographs of active nematics confined in corrugated disk with D = 400, 200, 180, 130µm respectively. The red triangle locates the indentation. The scale bar is common to all four panels and is equal to 100 µm (f), (g), (h), (i) are binary overlays of the space-time plots along the perimeter of each disk. As usual, the white lines correspond to the location of wall defects. The red triangles locate the position of the indentation. (j) Time-lapse of the periodic oscillation of active nematics confined in a 130 µm corrugated disk. This time, the indentation is located to the left side of the disk. The interval between two frames is 0.8 s. The blue dotted line on the first image shows the location of the pixel line used to compute the space-time plot in (k). scale bar: 100 µm (k) Space-time plot showing the oscillations of the bright filament pictured in (j). scale bar: 10 s.

143 Chapter 5. Active Nematics at a Wall: Description and Control to control the direction of transport in active nematic annuli. In this part, we explore the possibility to directly control tangential flows through a patterning of the wall. In confined situations, we have seen that tangential flows could emerge (shear flow in chapter 3 and transport in annuli in chapter4). On the contrary, in unconfined cases the left- right symmetry is always preserved due to the inherent chaotic flows. As the microtubules collide with the wall, they are as likely to flow towards either direction. However, the symmetry can be broken by exploiting the famous ratchet effect, sketched in Fig. 5.30.

This effect has proved to generate large scale horizontal flows in very diverse experimental systems from levitating droplets (Leiden-frost effect [230]) to bacterial colonies [231]. The principle is to break the symmetry of collisions by inducing a structural asymmetry at the wall. Let us draw a periodic repetition of asymmetric triangular protrusions. The triangles are defined by their height h and the wavelength λ, as sketched in Fig. 5.30 (a). The asymmetry is controlled by the ratio between the short and long sides, a and b. Let us consider two active particles (blue and red ellipses). The blue particle is ap- proaching the wall with a negative horizontal velocity, Vx < 0. During the collision, it is redirected towards the bulk, keeping Vx almost unchanged. The red particle has, on the contrary, Vx > 0. During the collision, it is redirected towards the neighbouring triangle, and may remain locked in the concave corner. As a result, the horizontal velocity decreases significantly. As the slow down only occurs for particles with a positive velocity, the collective flows develop into a horizontal drift with negative speed. The principle was explained in terms of active particles. However, the effect has been experimentally observed over a wide variety of systems and scales. In the case of the Leindefrost effect [230], it is the current of vapour below a millimetre-sized levitating droplet that is deflected by the indentation of the supporting plate, generating a propulsive force as shown in Fig. 5.30 (b). At smaller scales, the asymmetric collision of bacteria with the teeth of a micron-sized corrugated disks lead to its rotation [231], as shown in Fig. 5.30 (c). In all these cases, the dimensions of the pattern, h, λ and a, b have to be adjusted to the characteristic length scales of a given system. In our case, the active length scale is of the order of 100 µm. As a first experiment, we have kept λ = b = 200µm constant, and screened 4 values of h about the active length scale: h = 25, 50, 100, 200µm as sketched in Fig. 5.31 (a). These four patterns were printed on the four edges of a rectangular grid, such that all experiments could be performed within the same sample. The flow field was obtained through conventional PIV methods. A example of the resulting average vector field for a given experiment at h = 200µm is displayed in 5.31 (b). We can see that most vectors are aligned towards x. In order to quantify the alignment we decompose the velocity into its tangential and normal components, averaged in time, Vx and Vy, displayed in Fig. 5.31 (c) and (d) respectively. In the vicinity of a flat wall (h = 0), both components are rather homogeneous and close to 0, attesting for the symmetry of the chaotic flows. However, in all the other cases, the magnitude of the tangential flows is much higher than the normal component, indicating that the flow is directed along the wall. We also notice that Vx is essentially positive, which is coherent with the direction of the ratchet indentations. Furthermore, the magnitude of Vx appears uniform along x, except in the case h = 25µm where a slight slow down is observed at large x. The indentations seem to build up a uniform current of order 6 µm · s−1. In most cases, except for h = 100µm, this current does not extend very far

144 5.4. Controlling Wall Defects

Figure 5.30 – Ratchet effect with active particles (a) Sketch of a ratchet pattern drawn at a boundary. It is made of a repetition of asymmetric triangles, defined by their height h and the short and long sides a and b. The length of the basis is given by λ = a+b. The coloured ellipses represent active particles colliding with the boundary. The red particle colliding from left to right remains stuck on the indentation, while the blue one is reoriented towards the bulk. (b) A drop deposited on a hot ratchet (of temperature much higher than the boiling temperature of the liquid, so that a vapour film separates the solid from the liquid) self-propels in the direction indicated by the arrow. Here the drop has an equatorial radius R of 3 mm. The ratchet is made of brass and brought to a temperature T = 350 ◦C, much greater than 200 ◦C, the Leidenfrost temperature for ethanol, at which boiling would be observed. The teeth have a length of 1.5 mm and a height of 300 µm. After a short transient regime of acceleration, the drop moves at a constant velocity V =14 cm · s−1. Adapted from [230]. Scale bar: 1 mm (c) Bacterial driven micromotor. A nanofabricated asymmetric gear (48 µm external diameter, 10 µm thickness) rotates clockwise at 1 rpm when immersed in an active bath of motile E. coli cells, visible in the background. The gear is sedimented at a liquid–air interface to reduce friction. The yellow circle points to a black spot on the gear that can be used for visual angle tracking. Adapted from [231]. Scale bar: 50 µm.

145 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.31 – Ratchet effect in Active Nematics (a) Sketch of the ratchet pattern. The wave length of the pattern, λ = 200µm is fixed. The height of the triangles is varied from 25 µm to 200 µm . (b) Average velocity field of active nematics in the vicinity of a ratchet wall (h = 100µm). (c) Maps of the tangential velocity Vx for all the ratchet −1 patterns tested. Vx is given in µm · s . (d) Maps of the normal velocity Vy for all the −1 ratchet patterns tested. Vy is given in µm · s . (e) Profile of the tangential velocity averaged in x, as a function of the distance from the wall. (f) Time-lapse of the periodic dynamics observed within the ratchet indentations. The red dots correspond to positive defects. Arrows indicate the local direction of motion. Blue (resp. red) arrows indicate positive (resp. negative) tangential velocities. scale bars: 100 µm.

146 5.4. Controlling Wall Defects into the bulk, but remains localized to a band of about 200 µm thickness near the wall. In the case of h = 100µm, the current extends further than the field of view. A video corresponding to this configuration is shown in 5.6. In order to get more details on the current propagation, and considering the apparent invariance of Vx along the wall, we plot the velocity profile Vx(y), averaged in x. The results are displayed in Fig. 5.31 (e). The origin of y is taken at the tip of the indentations. In all cases, the profiles have a maximum reached at about y =100 µm. The value of the maximum depends non monotonically on h. The optimal height in terms of maximal velocity seems to be h = 50µm, which yields a maximal speed of 5.5 µm · s−1. However, although the maximal speed for h = 100µm is only 4.2 µm · s−1, the width of the profile is broader, with velocities higher than 3 µm · s−1 up to y =400 µm. All these results are still very preliminary. Still, they prove the robustness of the ratchet effect over a relatively broad range of indentation dimensions. The mechanism in the case of active nematics is yet to be understood. A time-lapse of the dynamics of the active nematic layer close to the indentations is displayed in Fig. 5.31 (f). It shows that microtubules are well aligned with the walls. At t =0 s, a positive defect seems trapped within the concave corner of the ratchet. Along the longer edge of the triangles, the active layer extends. To its right, the extension leads to a jet of active material directed towards the bulk, as represented by the blue arrow. To its left, the extension is redirected by the vertical edge of the triangle, as represented by the red arrow. Accordingly, the positive defect rises up into the bulk. This motion destabilizes the aligned region located upstream, creating an additional positive defect as shown at t = 17s. The two defects rotate around each other, such that the original defect is deflected into the bulk with a positive velocity. Eventually, the new born defect gets trapped within the concave corner and the process repeats itself periodically. Therefore, the positive drift is explained by the deflection of extensile forces towards x > 0, mediated by the nucleation and pair interactions of topological defects within the concave corners. The robustness of this mechanism comes from the remarkable adaptability of the size of topological defects to geometrical constrictions, a property underlined many times in chapters3 and4.

These results should be consolidated by more experimental repetitions and longer averaging times. It would be interesting to push the experiments to lower values of h, where defect nucleations would no longer be possible. We would also like to compute the horizontal velocity further into the bulk, in order to track the presence of a possible back-flow. Many other parameters could also be explored, such as the role of the length scale = + as well as the role of the asymmetry = a−b . λ a b s | λ | The objectives of this project would be two-fold. First, find the optimal parameters to redirect active nematic chaotic flows into a coherent wall streaming current. Secondly, we could use the patterning of the wall as a tool to study the compressibility of the active nematic layer. Consider for example two large square containers filled with active nematics, and connected through a ratchet patterned channel. The asymmetric transport would tend to flush either of the two containers, depending on the orientation of the ratchets. The densitification of an active system through a wall of funnels has already been proven in the case of swimming bacteria and cells [232, 233]. In some other cases, it has been shown that collective behaviours between bacteria may compete over the ratchet effect to maintain a uniform density [234]. In the case of active nematics, the active layer is much

147 Chapter 5. Active Nematics at a Wall: Description and Control denser. Furthermore, the contribution of elasticity should be increased compared to the case of bacteria, where friction forces with the substrate dominate. As a consequence, we should expect a balance to establish between the asymmetric transport favouring density gradients and the extensile elasticity of the gel conferring an incompressible behaviour.

5.4.3 Eﬀect of wall curvature

In the two previous sections, we have seen that the addition of wall indentations was associated to defect nucleations. The first section showed that even a small indentation favoured defect nucleation and could even attract wall defects. The ratchet patterns also generate defects, the motion of which mediates the deflection of the flow field. In chapter 4, we have seen that the cusps of a genus 2 handle-body were also a source of positive defects. This result is not surprising because in all these cases, the presence of a defect is imposed by the geometrical constraints. Cusps, indentations and ratchets share a common feature: they have sharp edges. As sketched in Fig. 5.32 (a), the nematic field cannot comply with planar anchoring at the boundary at the location of a geometrical discontinuity. The alignment would result in a "crack" at the location of the cusp, as shown by the grey dotted lines. Instead, the system escapes perpendicularly to form a defect at the wall, following the director field represented by the red lines. Consider now the case of a smooth corrugation, pictured in Fig. 5.32 (b) and (c). In that case, the planar anchoring is allowed everywhere and the topology does not require a defect to be formed. Yet, theoretical studies have shown that, even in the case of passive liquid crystals, the presence of a smooth bump ("Gaussian bump") may trigger the spontaneous nucleation of a pair of defects [235]. In the case of active nematics, the nucleation looks even more likely because of the inherent instability of the system to bend deformations. In this part, we aim at exploring the influence of local curvature on defect nucleation. To do so, we have printed undulations of controlled radius of curvature, R, with the usual polymer grids. An example of such a realization is displayed in Fig. 5.33 (a). In this particular case, the pattern is a succession of hemicycles with equal radius of curvature R but alternated sign. The surface is arbitrarily oriented towards the bulk, such that a positive radius of curvature (resp. negative) corresponds to configuration (b) (resp. (c)). From the snapshots of Fig. 5.33 (b), we can see that the active layer is perfectly aligned to the wall with R > 0. However, the bend distortions relax into a positive defect centred with the hemicycle, at a distance from the wall. Conversely, the nematic field does remain planar to the wall with R < 0 as shown in Fig. 5.33 (c). A wall defect forms somewhere around the center of symmetry of the hemicycle. In order to confirm the generality of this result, we compute the local average topological charge T everywhere in the sample. N+−N− T is defined the same way as in Fig. 4.4 from chapter4: T = 1/2 , where N+ Nf (resp. N−) is the number of positive (resp. negative) defects detected at a given point on the Nf frames of an experimental video. In the case of an unconfined active layer, and provided the experiment is long enough, we should obtain T = 0 everywhere. This is almost the case in the vicinity of a flat wall, as shown in Fig.5.33 (d). The colormap is rather uniformly green everywhere. As soon as curvature modulations are added, we observe a clear localization of wall defects nucleations, as shown in Fig. 5.33 (e). We can see that the location of the blue

148 5.4. Controlling Wall Defects

Figure 5.32 – Anchoring propagation in liquid crystals The grey dotted lines correspond to the vertical projection of the local tangent lines to the boundary. The red lines correspond to the nematic field adopted by the liquid crystal phase (a) Case with a geometrical discontinuity. The nematic field escapes in the vertical direction, forming a topological defect. Case of smooth negative (b) and positive (c) curvatures. There is no discontinuity in the director field.

spots, corresponding to a negative topological charge, corresponds to the maxima of the sinusoidal patterns where curvature is negative. Conversely, the localization of positive charge is much weaker. One may distinguish slight yellow shades in between the blue spots, that are not really signiﬁcant. When the undulations are more pronounced, the localization of charges is indubitable, as evidenced by the red and blue spots in Fig. 5.33 (f). As expected from the analysis of the ﬂuorescence image in panels (b) and (c), the positive topological charge is located somewhere at a distance from the wall. On the contrary, the negative charge is sitting close to the wall.

These results imply that defect nucleation is more favourable in regions of like-sign curvature. In a recent publication, Ellis et al. [117] have shown that topological defects were attracted towards regions of like-sign Gaussian curvature. This effect will be extensively discussed in chapter6. Let us note that the effect presented here is fundamentally different. Adding non-zero Gaussian curvature to an ordered phase imposes the nucleation of defects, as shown in section 6.1.1. In our case, the Gaussian curvature is zero everywhere, an the modulations at the wall do not impose the presence of a defect. They only induce bend distortions that are unstable in active nematics, leading to defect nucleations. Finally, we can observe that curvature modulations affect the motility of the defects. In the experiment shown Fig. 5.33 (a), the wall defects are free to move within the R < 0 bumps where they were nucleated. However, they never cross the surrounding R > 0 regions, where the anchoring is strongly planar at all times. As such, curvature heterogeneities can act as "topological traps" to reduce the motility of wall defects.

5.4.4 Conclusion The first two sections of this chapter were dedicated to the nucleation properties and dynamics of topological defects at a "smooth" boundary. We have suggested that their number and spatial distribution are defined by a undimensional regulation process involving an instability with a well-defined length scale as well as a peculiar annihilation process. Due to the horizontal invariance, the probability of presence of a defect at a given location was uniform, although the distances between defects at a given time were

149 Chapter 5. Active Nematics at a Wall: Description and Control

Figure 5.33 – Active nematics at a curved wall (a) Fluorescence micrograph of active nematics at the surface of a curved wall. The undulations are made of a succession of hemicycles of 100 µm. The surface is oriented such that a positive (resp. negative) radius of curvature R corresponds to case (b) (resp. case (c)). (b) Snapshot in a region with R > 0. The red disk corresponds to a positive defect. (c) Snapshot in a region with R < 0. The blue triangle corresponds to a negative defect. (d) Average local topological charge in the case of a ﬂat wall. (e) Average local topological charge in the case of a wall patterned with small sinusoidal undulations. The walength is 150 µm and the amplitude is 20 µm. (f) Average local topological charge in the case of a wall patterned with hemicyclic undulations of D = 100µm diameter.

150 5.4. Controlling Wall Defects controlled by the instability wavelength. Thanks to the high resolution and fidelity of the micro-printing techniques, we have explored the effect of controlled surface alterations of the wall. In the first part, we have seen that the defect nucleation is not always the result of a large scale regulation process. It can also be triggered by local fluctuations of the planar anchoring. Such fluctuations can be obtained by cutting a sharp indentation at a given location. The indentation not only favours defect nucleations, but also pins the new born defect at its nucleation site. Because wall defects are associated to vertical plumes, it is possible to create localized jets at a precise location. The controllability over active flows is particularly evident in confinement, where the constriction in a corrugated disk leads to the emergence of a remarkably periodic oscillating flow pattern. In the second part, we have used the ratchet effect to redirect active nematic chaotic flows into a coherent wall streaming current. We have shown that the deflection of active flows is mediated by a regular arrangement of topological defects around the sharp edges of the ratchet pattern. In all these cases, the nucleation of defects is dictated by the incom- patibility between the planar anchoring of active filaments and the presence geometrical discontinuities. However, the last part has shown that even smooth modulations of the wall curvature have drastic effects on the nucleation and spatial distributions of defects. Because active nematics are inherently unstable to bend deformations, the planar alignment is prone to instabilities in regions of negative Gaussian curvature. Conversely, wall defect nucleation is very unlikely is regions of positive Gaussian curvature. This effect must play an important role in the annular confinement, where the inner and outer walls have opposite curvature. Although preliminary, these projects all bear substantial evidence that the wall alignment is strong. Because of the strong alignment, the topology of the boundary has a direct influence on the topological landscape of the active layer. Discontinuities are transmitted to cracks and defects, curved boundaries create bend deformations which are enhanced by the bend instabilities. The effect of the wall is far from being restricted to the closest filament bundles in contact with the boundary, but extends over a large distance into the bulk. As such, understanding and controlling wall interactions constitute a promising step towards the control of active flows.

151 Chapter 5. Active Nematics at a Wall: Description and Control

5.1 – Steady regime of an isolated wall-defect in a disk. Fluorescence micrograph of active nematics conﬁned in a disk of 170 µm radius. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the ﬂow regime described in Fig. 5.2 (c).

5.2 – Nucleation and merging regime of an isolated wall-defect in a disk. Fluo- rescence micrograph of active nematics conﬁned in a disk of 170 µm radius. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the ﬂow regime described in Fig. 5.2 (d).

5.3 – Drifting regime of an isolated wall-defect in a disk. Fluorescence micrograph of active nematics conﬁned in a disk of 170 µm radius. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the ﬂow regime described in Fig. 5.2 (e).

152 5.4. Controlling Wall Defects

5.4 – Collective dynamics at a wall. Fluorescent micrograph of active nematics in the vicinity of a ﬂat wall. The wall is located at the bottom of the image. Frame rate: 2 fps. Scale bar: 100 µm. This video corresponds to the analysis proposed in Fig. 5.24.

5.5 – Periodic dynamics in a small disk Time-lapse of the periodic oscillation of active nematics conﬁned in a 130 µm corrugated disk. The indentation is located to the left side of the disk. Frame rate: 5 fps. Scale bar: 100 µm. This video corresponds to the analysis performed in Fig. 5.29.

153 Chapter 5. Active Nematics at a Wall: Description and Control

5.6 – Eﬀect of a ratchet pattern Fluorescence micrograph of active nematics facing a wall patterned with ratchets. The wavelength of the pattern, λ =200 µm is ﬁxed for all the experiments. In this example, the height is h =100 µm. Frame rate: 2 fps. Scale bar: 100 µm.

154 Chapter 6 Active Nematics and Curvature

6.1 Introduction

So far, we have exclusively studied active nematics on flat interfaces. In this part, we provide preliminary results showing the possibility to organize active flows through curvature effects. The discussion is organized in three parts. First, we will introduce the concept of geometrical frustration to explain the interplay between the topological landscape of a liquid crystal and the curvature of a surface. We will illustrate these effects through recent experiments of active nematics in a curved space. Then, we will present the experimental technique we developed to generate oil droplets of heterogeneous curvature dispersed in water. Finally, we will present the preliminary results showing regular flow patterns around ellipsoidal droplets.

6.1.1 Geometrical frustration and topological charge Let us first recall the definition of curvature in a flat space. Consider a differentiable curve r(s) defined in R2, with s the arclength parameter. An example of such a curve is sketched in Fig. 6.1 (a). The radius of curvature at a given point is given by the radius of the osculating circle, R(s) represented in red and blue in two different points in our example. The curve is oriented by its normal vector n, pointing upwards in the sketch. The curvature is defined as k(s) = (s).1/R(s) with (s) = −1 (resp. (s) = +1) if the osculating circle is below (resp. above) the curve.

Gaussian curvature

Let us now consider a surface defined in R3. We provide two such examples in Fig. 6.1 (b) and (c). At each point on the surface, one can define a normal vector n that is normal to the surface. Planes containing that vector are called normal planes. The intersection between one of such normal planes and the surface will be a curve similar to the one in Fig. 6.1 (a). The curvature of this curve at the point of interest is called the normal curvature. Different normal planes will give different normal curvatures, and the maximum and minimum curvatures, k1 and k2, are called principal curvatures. The local Gaussian curvature κ is defined as κ = k1k2. In the case of a flat plane, the radius of curvature is infinite everywhere, which yields κ = 0. Nevertheless, it is not the unique surface that

155 Chapter 6. Active Nematics and Curvature

Figure 6.1 – Gaussian Curvature (a) Definition of curvature of a differentiable curve. n, t are the local normal and tangent vectors to the curve. R is the radius of the osculating circle at a given location. (b) An example of a surface with positive Gaussian curvature. κ is the product of the two principal curvatures, which are here negative (blue). (b) An example of a surface with negative Gaussian curvature. has κ = 0. A cylinder is one example. Surface with κ = 0 are called developable surfaces. It means that they can be "developed", or simply flattened to a plane without distortions. To understand this point with hands, consider a sheet of paper, a pair of scissors and glue. One can easily roll the sheet to form a cylinder, because it does not change the Gaussian curvature. Conversely, if one tries to bend the sheet in two directions at the same time, the paper will crumple. Crumples correspond to points of compression and stretching of the paper fibres. To change the local Gaussian curvature of a surface, one needs to locally stretch or compress it (mathematically, change the metric of the surface). Now, let us replace the paper fibres by an ordered phase, for example a crystalline arrangement of particles as represented in Fig. 6.2 (a). Each particle is surrounded by 6 neighbours in this hexagonal lattice. Let us now change the Gaussian curvature of this surface to κ > 0, as in Fig. 6.2 (b). Because of the metric is changed, the distances between neighbours no longer comply with the hexagonal order. Instead, a pentagon forms, as overlaid in red. The pentagon is a defect in the crystalline phase. This is why football balls are patched with hexagons and pentagons. Instead, if we induce a negative Gaussian curvature, the defect will be a heptagon, as overlaid in blue in Fig. 6.2 (c). The example of a crystalline phase helps to understand how curvature affects the order on a surface. Consider now a flat and infinite surface paved with elongated rods, such as in a liquid crystal. This geometry allows a long range alignment without defects, as shown in Fig. 6.3 (a). Let us now fold it in one direction to form a cylinder. The distances between the layers of rods are preserved, and so does the alignment, as shown in Fig. 6.3 (b). On the contrary, if one folds the sheet on both directions to form a shell, the rod lanes will necessarily cross at the poles forming defects as shown in Fig. 6.3 (c). This sketches capture the qualitative coupling between curvature and defect nucleation in liquid crystals.

Topology and Curvature The drawing of Fig. 6.3 (c) represents a spherical shell. However, one could similarly fold a ﬂat plane onto an ellipsoidal shape, as shown in Fig. 6.4 (a). The resulting director ﬁeld

156 6.1. Introduction

Figure 6.2 – Geometrical Frustration in a crystal (a) Sketch of a plane covered with a crystalline phase. Each node is surrounded by 6 neighbours. The yellow edges represent the bound between edges. (b) Frustrated order in a surface of positive curvature. The red pentagon is a defect in the hexagonal lattice. (c) Similar example with a surface of negative Gaussian curvature. The blue heptagon is a defect in the hexagonal lattice.

Figure 6.3 – Geometrical Frustration in a liquid crystal (a) Left. Sketch of a plane covered with a nematic liquid crystal. The LC molecules are represented in grey ellipses. The director field, represented by black lines, is defect free. Right The same plane wrapped into a cylinder. The director field is still defect-free. (b) Left. Same sketch as (a), this time with folding lines in two directions to wrap the sheet into a sphere. Right. The plane is wrapped onto a sphere. The director field spontaneously develops two topological defects.

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Figure 6.4 – Euler characteristic (a) Sketch of ellipsoids covered by a LC phase. The black line corresponds to the director field. We can see that, although the shape of the ellipsoids is very different, the director field has the same structure. (b) Euler characteristic of a sphere. A sphere has no handles so χ = 2. (c) Euler characteristic of a torus. A torus has one handle so χ = 0. (c) Euler characteristic of a double torus. is not significantly different from the spherical case: the number and structure of defects are the same. Therefore, there seems to be a relation between curvature and topology that does not depend on local curvature. As explained in section 1.2.7, in topology surfaces are characterised by their Euler characteristic, χ given by Eq. 1.14. χ only depends on the genus number g, which counts the number of handles or "holes" in the surface. A sphere and an ellipsoid have no hole, which yields χ = 2. A torus has g = 1, giving χ = 0, as shown in Fig. 6.4 (c) etc. The relation between the curvature of a surface and its topology is given by the Gauss- Bonnet theorem. There are subtleties on its formulation, and we will only need a simplified version here. In the case of a differentiable closed surface, the total curvature, given by the integral of the Gaussian curvature over the whole surface, is related to the Euler characteristic through the following expression:

1 ZZ χ = κd2r = 2(1 − g) (6.1) 2π

Topology and defects: the Poincaré-Hopf theorem Let us now cover the closed surface with an ordered phase, as we did in Fig.6.3. The number of defects in the director ﬁeld N and their topological charge si are associated to the Euler characteristic through the Poincaré-Hopf theorem:

158 6.1. Introduction

N X si = χ (6.2) i=1 To illustrate this theorem, consider the case of a spherical ellipsoid, for which χ = 2. It means that the nematic phase will necessarily contain topological defects, whose total charge will sum up to +2. In the case of active nematics1, the charge of the topological defects is s = ±1/2 which means that the director ﬁeld will be disrupted by at least four half integer defects at all times. Additional defects may nucleate under the inﬂuence of activity. However the total charge around the shell will always sum up to χ = 2.

6.1.2 Defects and local curvature

In the previous section, we have seen that the Euler characteristic of a surface sets the total topological charge in the director field. This results implied that only the total curvature of a surface influenced the topological landscape of the director field. Yet, the location of the defects results from the minimization of the free energy associated to the distortions of the director field, Fd. If φ is the local orientation of the nematic field, Fd takes the following form [66,237,238]:

1 ZZ = (r) 2r (6.3) Fd 2K |∇φ |d

where K is the elastic constant of the liquid crystal. If we want to rewrite this expression in terms of the topological defects, we need to introduce a charge density function, ρ(r). We note ri the location of defects of charge si:

X ρ(r) = 2π δ(r − ri) (6.4) i

where δ(r − ri) is the Dirac operator. It has been shown that Fd can be developed as follows [237,238]:

1 ZZZZ = [ (r) (r)][ (r0) (r0)] (r r0)d2rd2r0 (6.5) Fd 2K ρ − K ρ − K GL ,

where GL is the Green’s function of the Laplace-Beltrami operator, the standard Laplace operator generalized to curved space [238]. This expression shows that the free energy associated to distortions of the director ﬁeld can be written as an integral involving defect pair interactions. The charge of each defect is however not directly ρ(r), but it seems to be "screened" by the local curvature K(r). In the literature, this expression has been compared to the Coulomb energy of a plasma with charge density ρ in a background of charge density −K [238]. This means that topological defects can be treated as charged particles that couple to the surface via the background charge provided by the Gaussian curvature of the surface.

1And in general, for a suﬃciently thin layer of passive nematic liquid crystal [236]

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6.1.3 Experimental active nematics on curved interfaces

The previous analogy to charged particles has two consequences. First, defects of like-sign topological charge will repel. In the particular case of a spherical active nematic shell, where there must be at least four positive defects, the latter will tend to maximize their relative distance. This result has been experimentally proven by Keber et al. [154]. When an active nematic film of microtubules and molecular motors is embedded in a lipid vesicle with small enough radius the nematic layer develops only four topological defects. Most of the time, these defects are arranged on the corners of a tetrahedron. This configuration maximizes the average angular distance between the defects to hαi = 109.5◦ as shown in Fig. 6.5 (b). Another local minimum of the energy is reached when the defect are arranged in a planar configuration, which yields hαi = 120◦. Although the distortion energy is higher, the high activity of the system allows the defects to transit, almost periodically, between the two states as shown in Fig. 6.5 (c). The second consequence of the previous analogy is that defects are interacting with a background charge provided by the Gaussian curvature, κ. More precisely, topological defects are attracted towards regions of like-sign Gaussian curvature. This result has recently been shown experimentally by Ellis et al. [117], who encapsulated active nematic in large toroidal droplets. A torus has a strong heterogeneity of Gaussian curvature. On the inner side, represented in blue in Fig. 6.6 (a), κ is negative. Conversely, κ is positive on the outer region represented in red. In the case of a torus, χ = 0, and therefore topology does not require the presence of any defect at all. However, the structures synthesized by Ellis et al. were very large compared to the active length scale, with guaranteed the presence of a large number of defects. A snapshot of a portion of an active nematic torus is displayed in Fig. 6.6 (b). From such fluorescence images, they automatically detected the defects and their position around the torus, as shown in Fig. 6.6 (c). From the topological landscape, they could compute the local average topological charge in various N +−N − areas of the torus , = A A , where + and − correspond to the average number A hsAi 2 NA NA of positive and negative defects in A over the experimental time. They finally correlated N +−N − = A A with the local curvature, 1 2 RR . The results, displayed in Fig. 6.6 hsAi 2 / π KdA (d), give clear evidence of the attraction of topological defects towards regions of like-sign Gaussian curvature. The apparent linear dependence further confirms the sensitivity of this interaction.

6.1.4 Motivation

The experimental results exposed in the previous section have proven two strong features of the interactions between the nematic field and the curvature of the surface. First, the Euler characteristic governs the minimal number of defects in the director field. Second, the location of the defects around the surface is very sensitive to the local curvature. As a consequence, surfaces with equal Euler characteristic are no longer equivalent. The minimal number of defects will be the same, but their spatial arrangement, and consequently, their dynamics, will highly depend on the curvature heterogeneities. In living systems, cells are influenced by surface curvature as demonstrated by cell movements in the developing corneal epithelium leading to vortex patterns [239]. Furthermore, recent studies have emphasized on the role of topological defects in the growth of cell tissues [70]. The

160 6.1. Introduction

Figure 6.5 – Active Nematic shells Adapted from [154]. Sketch of an active nematic vesicles. with four topological defects, pictures in yellow, red, purple and green. At the top and bottom of the sketch are added confocal fluorescence images showing the experimental visualization of the top and bottom hemispheres. (b) Energy landscape of the distorsions of the director field as a function of the average angular distance between defects. Two local extrema are observed for the planar and tetrahedral configurations. (c) Experimental measurement of the average angular distance, as a function of time. The characteristic angle for a tetrahedral (resp. planar) configuration is 109.5◦ (resp. 120◦, marked in blue (resp. red).

161 Chapter 6. Active Nematics and Curvature

Figure 6.6 – Curvature-induced defect unbinding in Active Nematics Adapted from [117]. (a) 3D Sketch of a torus. The blue (resp. red) region has negative (resp. positive) Gaussian curvature. (b) Confocal fluorescence micrograph of a cross section of an active nematic torus. scale bar:200 µm. (c) Director field corresponding to image (b). Positive (resp. negative) defects are marked with red (resp. blue) triangles. (c) Time-averaged topological charge in a region versus the integrated Gaussian curvature of that region for five experiments. The error bars on the data points are the standard error N +−N − of the mean. = A A , where + and − correspond to the average number of hsAi 2 NA NA positive and negative defects in area A over the experimental time.

study on active nematic vesicles has shown that restricting the number of defects to the minimum topological requirement lead to very organized spatial patterns and periodic dynamics. The role of local curvature in organizing flow patterns has not been tested yet. In spherical vesicles, the Gaussian curvature is constant. In tori, κ is highly heterogeneous, but the number of defects exceeds by far the topological requirements. Because of the constant nucleation and annihilation of defect pairs in the system, the statistical curvature-induced defect unbinding does not lead to a significant ordering of the flows. In order to explore the curvature-induced flow organization, one needs to generate surfaces that are small enough to prevent defect proliferation. Unfortunately, at microscopic scales, surface tension is highly unfavourable to curvature heterogeneities and most interfaces will relax into a spherical shape.

In this part, we present an experimental technique to generate small-sized long-lasting ellipsoidal droplets that are able to accommodate active nematics on their surface. The droplets are composed of the liquid crystal 8CB in the smectic-A. The elongated shape is a metastable state that remains stable for hours due to a competition between the viscous anisotropy of the passive liquid crystal and surface tension. The active nematic is formed at the outer surface of the spheroids and stabilized by surfactants. We uncover a very regular ﬂow organization, associated to a strong attraction of topological defect towards the regions of maximal Gaussian curvature. These results are compared to simulations works, both from the literature and from a recent collaboration with J. de Pablo’s group at the University of Chicago. The work presented here is still preliminary. It has been done in collaboration with ﬁrst-year PhD student Martina Clairand at Gulliver Institute, ESPCI, Paris.

162 6.2. Synthesis of smectic ellipsoids

6.2 Synthesis of smectic ellipsoids

A detailed protocol of the synthesis of smectic ellipsoids is available in the Methods section 2.4.2. For the sake of clarity, and to lay the emphasis on the preliminary results, we will only give a brief summary of the main steps of the fabrication and the characteristics of these objects. The first step consists in synthesizing smectic shells, using the protocol developed by Teresa Lopez-Leon et al. [240]. A smectic shell is composed of a droplet of water coated with a layer of the smectic-A liquid crystal, 40-Octyl-4-biphenylcarbonitrile (8CB), immersed in water. The inner and outer water phases contain 0.1wt% of polyvinyl alcohol (PVA), which stabilizes the double emulsion and enforces planar anchoring of the liquid crystal at the two interfaces. The system is prepared using glass micro-fluidic techniques, detailed in the Methods and in Fig. 2.13. In the smectic-A phase, the molecules of 8CB self-organize into parallel layers called smectic layers. In the case of shells, because of the spherical confinement, the structure is complex, and characterized by a set of curvature walls that divide the sphere into crescent domains, causing the undulation of the smectic layers. This structure is very well recognizable through crossed-polarizers, as shown in Fig. 6.7 (a) and (b). The smectic layers are tilted with respect to the domain boundaries as shown in Fig. 6.7 (c). The tilt angle depends on the thickness of the shell. The walls of the crescent domains cross at opposites poles of the sphere, forming topological defects. Two configurations are observed. In the first one, pictured in Fig. 6.7 (d), all crescent domains converge towards two s = +1 topological defects, diametrically opposed in the sphere. In the second one, sketched in in Fig. 6.7 (e), the sphere is divided into two hemispheres of converging crescents, tilted at a 90◦ angle between one another. The converging points correspond to four s = +1/2 defects. Notably, both configurations respect the Euler characteristic of the sphere. When looking at a cross-section of the sphere, one notices that the shell thickness is not homogeneous. Due to buoyancy, the inner water droplet is offset at a distance ∆ with respect to the center of the external shell, as shown in Fig. 6.7 (f). In some cases, the thinner side of the shell breaks open, connecting the two water phases. This event leads to the break up of the shell, whose shape rapidly changes. In the paper by Lopez et al. [240], broken shells are said to shrink into spherical droplets, some of which are visible in Fig. 6.7 (a). The equilibrium shape is controlled by surface tension, that minimizes interfacial energy. In the lab, we have repeated the same protocol, only changing the nature of the surfactant from PVA to Pluronic. The motivation of this change is that we wanted to condense active nematics around the smectic shells. In order to bind microtubule filaments, one needs to use a surfactant that contains a PEG chain, and PVA does not. However, changing surfactants has other consequences. The surface tension of an aqueous solution of 0.1wt% PVA is estimated [241] around 50 mN.m−1 , while the surface tension of a Pluronic solution above the critical micellar concentration (CMC) [242] is below 40 mN.m−1 2. The significant decrease of surface tension may affect the stability of the smectic shells. Second, the surfactant molecules control the anchoring of the 8CB molecules at the interface. Both PVA and Pluronic induce planar anchoring. However,

2These surface tensions correspond to the air/solution interface. Although we do not know the corresponding values at a water/8CB interface, the comparison only aims at stressing the potential diﬀerence between PVA and Pluronic.

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Figure 6.7 – Smectic shells Adapted from [240]. (a) Cross polarized image of a set of monodisperse smectic shells surrounded by smaller smectic droplets. The smectic single droplets result from the destabilization of some of the shells during manipulation of the samples. (b) Smectic texture of an experimental shell. The shell displays four s = +1/2 defects located at the equatorial plane (only two visible here, labelled with numbers), and a set of curvature walls that divide the shell into crescent domain (c) Birefringent textures within the crescent domains. The smectic layers are tilted by an angle of φ = +13◦ with respect to the domain boundaries. (d) The simplest smectic texture on a spherical surface has two s = +1 defects on the poles, in a configuration where the smectic layers (continuous lines) run along the parallels of the sphere and the director field (discontinuous lines) runs along the meridians of the sphere. (e) The s = +1 defects can split in two s = +1/2 defects by a simple transformation that costs no energy; the resulting configuration has four s = +1/2 defects organized along a great circle of the sphere. (f) The smectic phase is confined between two spheres of radii R and a that are not concentric. ∆ is the center-to-center distance between the spheres.

164 6.3. Active nematic emulsions the strength of this anchoring may be diﬀerent, which could dramatically change the structure of the smectic layers.

The stability of the smectic shells does not seem to be strongly affected. As a matter of fact, a very surprising phenomenon is observed when the smectic shells break. As the inner droplet breaks out, the smectic layer does not shrink into a spherical droplet. Instead, it stabilizes into and elongated shape, as shown in the time-lapse Fig. 6.8 (a) and 6.1. The bursting of the thin smectic layer happens very fast: the image sequence shows that at t = 0.2s, most of the upper hemisphere of the shell has disappeared. This phase is reminiscent of the burst of a soap film, and must be driven by surface tension. There remains a banana-shaped droplet, corresponding to the thicker hemisphere of the shell. The banana shape is not stable either, and it straightens into an ellipsoidal droplet. This process is a bit slower, and probably involves the elasticity of the smectic liquid crystal. In the shell geometry, the smectic layers are curved causing significant splay distortions that are energetically costly. When the shell bursts the curvature is not longer imposed which could explain the straightening. However the mechanism is still largely unknown and requires more scrutiny. Because the formation of smectic ellipsoids starts by a bursting event, the process is quite disordered and leads to a wide variety of shapes and aspect ratios. A typical sample is displayed in Fig. 6.8 (c). Some of the droplets are spherical, but most of them are sub- stantially elongated. Notably, one notices important light distortions within the droplets. These distortions reflect the orientation field of the smectic liquid crystal mocelules. In the sperical droplets, the textures appear particularly disordered. On the contrary, in the ellipsoidal droplets the smectic layers seem to be organized in a similar fashion to what we described in the case of smectic shells as shown in Fig. 6.8 (d). We readily recognize regularly spaced crescent domains composed of tilted smectic layers, as sketched in Fig. 6.8 (e). The orientation of the smectic planes, despite the tilt induced by crescent domains, is on average perpendicular to the long axis of the ellipsoid. This information is valuable to understand the stability of these objects. In addition to the orientational order inherent to liquid crystals, smectic liquid crystals have a partial positional order. This means that liquid crystal molecules are free to flow within the smectic layer in which they reside, but cannot move across different layers because the energy cost to elastic distortions is highly anisotropic. As a consequence, smectic molecules do not flow easily along the main axis of the ellipsoid. As illustrated in Fig. 6.8 (f), this effect competes with the surface tension by increasing the effective viscosity along the main axis by several orders of magnitude. Because the viscosity is not infinite, surface tension eventually takes over and the ellipsoids recover a spherical shape, as sketched in 6.8 (g). Nevertheless, this process takes approximately 20 hours, which leaves enough time to interface them with active nematics.

6.3 Active nematic emulsions

So far, the experimental studies on active nematics at curved interfaces have exclusively been performed by preparing water-in-oil emulsions, where the suspension of microtubules and motors is embedded within the droplets. This conﬁguration is not required a priori, because the active nematic layer is to be conﬁned at the interface. Therefore, the

165 Chapter 6. Active Nematics and Curvature

Figure 6.8 – Smectic ellipsoids (a) Time-lapse of the transformation from a smectic shell (t = 0 s) to a smectic ellipsoid. The dotted lines at t = 0 s correspond to the inner and outer shell boundaries. The breakup occurs where the shell is thinnest. Scale bar: 50 µm. (b) Sketch of the dynamics of ellipsoid formation. (c) Snapshot of a sample of smectic ellipoids 5mn after the shell synthesis. The shells have all collapsed into ellipsoidal droplets. scale bar: 100 µm. Coexisting spherical droplets are also visible. (d) Color image of a smectic ellipsoid. The deﬂection of light induced by the smectic layers reveals the textures of the director ﬁeld organized in crescent domains similar to the one observed in shells. (e) Comprehensive sketch of the structure of ellipsoid (d). The LC molecules are organized within parallel layers delimited by the red boundaries. (f) Qualitative interpretation of the energy balance preserving the elongated shape. The smectic layers (red) are oriented perpendicular to the main axis. The shrinking of the interface into a droplet - the shape that minimizes the surface energy Eγ - is slowed down because of the high energy cost of elastic distortions Ed perpendicular to the smectic planes. (g) Typical evolution of the ellipsoidal shape as a function of time. The aspect ratio decreases with in a scale of 10 hours.

166 6.4. Results

Figure 6.9 – Active ellipsoid emulsion (a) Color image of ellipsoids dispersed in active gel. Scale bar: 50 µm. (b) Comprehensive sketch of the active ellipsoid emulsion. The 8CB droplets are immersed in the conventional active gel solution containing, among other constitutents, microtubule filaments, molecular motors and ATP. Pluronic surfactant is added at a concentration above CMC. It migrates at the water-8CB interface and stabilizes the microtubules there. Within a few hours, a thick active nematic layer forms at the interface. (c) Confocal fluorescence micrograph of the active ellipsoid emulsion after 2 hours. Most of the fluorescence is signal detected at the ellipsoids interface, although residual microtubule filaments are visible is the bulk as well. Scale bar: 50 µm.

behaviour should be analogous for an oil droplet with the same dimensions immersed in a continuous phase of active material. In the methods section, we give examples of various experimental realizations of oil-in-water active nematic emulsions performed in the lab. The principle is to prepare oil droplets with conventional microfluidic techniques, before dispersing them in a solution of active gel. This technique has several advantages. Essentially, it enables to use any microfluidic device without substantial adaptation to the active material (surface treatment, volume). Second, the continuous phase acts as a buffer solution for all the components of the active solution (activity, motor, filaments). Therefore, all the droplets of a given sample are directly comparable, because they are in contact with the same concentration of active material. In addition, the continuous solution is accessible , and its composition can be changed or renewed easily, ensuring very long experimental lifetimes. In this work, we disperse smectic ellipsoids in a solution of active gel. Details of the experimental protocol are provided in the methods section. Within a few hours, a thick microtubule layer condenses onto the surface of the ellipsoids, as shown in Fig. 6.9 and 6.2. In the next section, we will describe the preliminary results on the dynamics of active nematics around ellipsoidal droplets.

6.4 Results

The results presented here are still preliminary. They aim at setting the framework of the study, by showing the parameters of interest and describing the phenomenology of the dynamics. The elongated droplets coated with active nematics will be referred to as active ellipsoids. These objects are active in two respects. In the ﬁrst part, we will

167 Chapter 6. Active Nematics and Curvature describe the solid body transformations. Active ellipsoids move around in the continuous phase. The translational displacements are attributed to the activity of the bulk active gel. In addition, persistent rotations are observed. In the second part, the description of the periodic deformations of the active layer gives insights on the rotation mechanism: the curvature heterogeneities contribute to transform the chaotic ﬂows of active nematics into a sustainable torque dipole.

6.4.1 Solid body dynamics

The analysis is based on fluorescence image sequences acquired with a spinning disk confocal microscope (see Methods section 2.2.1). A snapshot of a typical experiment is displayed in Fig. 6.10 (a). Using custom-made matlab routines, we track the position of the centroid, r(t), and the orientation of the long axis, φ(t), of the active ellipsoids as a function of time, as shown in Fig. 6.10 (b). In the second part, we will use r(t) and φ(t) to recompose the video within the reference frame of the active ellipsoid, by rotating (Fig. 6.10 (c)) and shifting (Fig. 6.10 (d)) the original frames. An example of such an operation is displayed in Fig. 6.10 (e) and shown in 6.3. It will enable to study the deformations of the active nematic layer. For the moment, we focus on the physics of the solid-body transformation. From the tracking of r(t) one can compute the mean square 2 displacement (MSD) of the centroid given by h(r(t) − r0) i with r0 = r(t = 0). The first analysis gives evidence of a clear super-diffusive behaviour. The MSD depends on 2 a time with a power law as h(r(t) − r0) i ∼ t as shown in Fig. 6.10 (f). In this particular experiment, we found a = 1.83 ∼ 2 which corresponds to a near ballistic motion. However the value of the exponent needs to be consolidated with more experiments. The super-diffusive behaviour is not surprising considering that the active ellipsoids are immersed in a bath of active material. In Fig. 6.10 (a), one can notice a weak fluorescence in the continuous phase. As a matter of fact, it still contains microtubule filaments, whose interactions with molecular motors and ATP generates large-scale chaotic flows. Tracer particles embedded in active nematics are known to exhibit near ballistic motion [33]. Similarly, inside cellular organisms, the motion of cytoskeletal elements contributes to the superdiffusive transport of endogeneous intracellular particles in a similar fashion [243]. More intriguing is the evolution of φ(t), displayed in Fig. 6.10 (g). In the plot, φ is normalized by 2π, such that an increment of 1 in the vertical axis corresponds to a complete revolution of the active ellipsoid. A positive slope (dφ/dt > 0) corresponds to a counter-clockwise (CCW) rotation. The profile is characterized by a V-shape, with a CW rotation from t = 0 to t = 500s, followed by a CCW rotation from t = 800s to t = 2400. The angular velocity ω = dφ/dt, is remarkably constant throughout these periods. Moreover, the magnitude of ω is equal in the CW and CCW phases. The striking regularity of the ellipsoid rotation over more than 1000 s rules out the influence of the chaotic bulk flows. On the other contrary, deformations of the active nematic layer around the ellipsoid look surprisingly. In the next section, we will characterize them and propose a mechanism explaining the persistent rotations3.

3Note that apart from this mode of rotation the ellipsoids may display a rotation about the main axis. However such a rotation is not detectable through the ﬂuorescence images and we will not consider it.

168 6.4. Results

Figure 6.10 – Solid-body dynamics of an active ellipsoid (a) Fluorescence micrograph of an active ellipsoid immersed in a bulk of active gel. scale bar: 50 µm. (b) Binarised version of image (a) showing the detection of the ellipsoid, represented by the position of its centroid, r, and the orientation of the main axis, φ. (c) We can realign the ellipsoid to the horizontal, by rotating the image by −φ.(d) An example of the centroid displacement over a 300 s period. (e) The informations on r and φ enable to reconstruct the dynamics of the active nematic layer within the reference frame of the ellipsoid, by subtracting the solid-body displacements. (f) Mean-Squared Displacement (MSD) of the ellipsoid centroid as a function of time. The grey shade corresponds to the error bar of the measurements, which are composed of four videos of duration 700 s taken during the lifetime of a single active ellipsoid. Inset: Log-Log plot of the MSD, showing a power law as ta with a = 1.83. (g) Evolution of the main axis orientation φ, normalized by 2π, as a function of time. The grey dotted lines have the same slope, but opposite sign.

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6.4.2 Active nematic deformations

A periodic twist

As previously recalled, the dynamics of active nematics confined to a spherical shell of small size (typically 50 to 100 µm diameter) is characterized by a periodic oscillation of four positive defects between a tetrahedral and a planar configuration. The ellipsoids have been synthesized within the same size range, which guarantees that the number of positive defects only rarely exceeds the topological requirement of four. However, the dynamics observed are strikingly different from the spherical case. To characterize the evolution of the director field, we simultaneously image the top and bottom of the ellipsoid by means of a spinning disk confocal microscope. The bottom face is coloured in purple, and the top face is coloured in orange (see Fig. 6.11). Both images are overlapped, such that we get an representation of the director field around the whole ellipsoid in one frame. A time-lapse of this kind is proposed in Fig. 6.11 (a). In order to ease the interpretation of these images, a corresponding sketch of the director field at each time step is displayed in Fig. 6.11 (b). At t = 0s, we do not distinguish any defect on neither of the two faces. The microtubules seem to wrap around the ellipsoid. The alignment is essentially perpendicular to the main axis of the ellipsoid, although we notice a small tilt, with opposite angle on the two sides. The apparent absence of defects only means that they are located at the poles of the ellipsoid, which are not visible in this representation. Because active nematics are unstable to bend deformations, the aligned state rapidly collapses, and we notice important distortions at t = 1s. These deformations destabilize into four cracks along the main axis of the ellipsoid, leading to the migration of the four topological defects from one pole to the other. The defects migrate along the cracks, in a process is illustrated in Fig. 6.12. Right after the migration (t = 3s), the director field is aligned along the main axis on both faces, perpendicularly to the initial state (t = 0s). So far, the displacement of the defects remains purely vertical. However, as they move close to the poles the defect pairs start winding around each other in a spiralling motion. Notably, the spirals have opposite handedness, which creates a torsion of the director along the ellipsoids, as evidenced at t = 5s. The torsion amplifies in time, which gradually realigns the director field perpendicular to the main axis, as in the initial state. The positive defects at the poles cannot annihilate. They continuously spiral, amplifying the shear stress until the bend instability triggers again. The process repeats itself periodically.

Chiral symmetry breaking

The opposite handedness of the spirals, responsible for the torsion of the nematic ﬁeld, also confers chirality to the deformations of the active ellipsoids. This chiral symmetry breaking is reminiscent of a process described by Naganathan et al. [244] in the early stages of morphogenesis. According to the authors, the Left-Right (LR) symmetry breaking emerges in the ﬁrst stage of the development of an embryo. The primary determinant of this process has been linked to the cytoskeleton. Bilateral symmetry breaking requires a chiral process. In their article, the authors argue that the acto-myosin cortex generates chiral torques that facilitate the symmetry break-

170 6.4. Results

Figure 6.11 – Active nematic periodic deformations around an ellipsoid (a) Hemi- sphere projection of a 3D confocal stack of an active ellipsoid. The top (resp. bottom) image is shaded in orange (resp. purple). (b) Time-lapse of the deformations of the active nematic layer. Top and bottom images of the confocal stack are overlaid. Scale bar: 50 µm. The overlay at t = 3s corresponds to the two images displayed in (a). (c) Comprehensive sketch of the periodic dynamics displayed in (b). Topological defects are marked with coloured disks.

Figure 6.12 – Migration of a positive defect along a crack Defect lines are migration pathways for s = +1/2 defects. A positive defect, pictured with a blue disk, is sitting below a bent lane. Due to activity the bent lane destabilizes to form a defect line, in a process similar to the one described in Fig. 5.5. The bundles open up, and the s = +1/2 defect zips along the defect line. Eventually, the defect line disappears. This process happens simultaneously for the four defects of the ellipsoid, which are exchanged from one pole to the other at each instability cycle.

171 Chapter 6. Active Nematics and Curvature ing. The authors propose an underlying microscopic mechanism, reproduced in Fig. 6.13. The contractile stress created by the action of a myosin between two actin ﬁlaments also generates a microscopic torque dipole, from which a macroscopic twist emerges. This eﬀect can be understood quantitatively based on the physics of active gels with chiral asymmetries [245, 246]. However, in the discussion of the article, the authors concede that the chiral interactions are not the only features determining the symmetry breaking, and suggest that the mechanism may involve non-trivial tension-torque couplings evoked in anterior publications [247,248].

An important distinction needs to be made on the nature of the chiral symmetry breaking. In our case, we have not seen evidence of a global symmetry breaking, in the sense that two independent ellipsoids do not necessarily have the same handedness 4. In the experiments by Naganathan et al., the authors precise that the chiral torques generated by the actin network are always clockwise. Nevertheless, the fundamental question behind these two experimental problems remains to identify the chiral process that triggers the LR symmetry breaking in an isolated system - may it be an embryo, or an active ellipsoid. In our experiments, the symmetry breaking mechanism in active ellipsoids is far from clear, but may not be too different from the one we reported in chapter3 in the shear flow regime. In this case, the nematic phase aligns with the confining walls, and a shear flow with random handedness emerges. As such the LR symmetry breaking reported in this chapter does not seem to be inherent to the ellipsoidal geometry.

To summarize, active gel models are able to reproduce global chiral symmetry breakings in active systems that explicitly include microscopic chiral interactions. In this project, we report examples of partial chiral symmetry breaking in active systems that do not, to our knowledge, contain microscopic chiral interactions. Our understanding of this particular point remains preliminary and requires further investigation.

Persistent rotations

The chiral symmetry breaking does not explain a priori why the ellipsoids continuously rotate. As a matter of fact, because the spirals are counter-rotating at the poles, there is no net torque acting on the main axis of the ellipsoids. Moreover, even if they were co-rotating, the torque would aligned with the main axis, which cannot explain a rotation perpendicular to it. In order to understand the observed dynamics, one needs to account for the presence of walls. The droplets are not isolated in an infinite bulk. They are confined in a glass cell. 8CB is less dense than water, which means that the droplets tend to float on top of the sample, protected from air by a glass slide as shown in Fig. 6.14 (c). As a consequence, the top side of the ellipsoids is sitting close to a boundary. The vortices at the poles are responsible for a slip velocity perpendicular to the main axis of the ellipsoid, as shown by the red and blue arrows in Fig. 6.14 (a). The friction with the walls creates a counteracting force that tends to move the ellipsoid in opposite direction. A naive analogy to understand this point is to picture the vortices at the poles like wheels rolling onto the glass surface.

4This result should however be consolidated with a statistical analysis

172 6.4. Results

Figure 6.13 – Active torque generation Adapted from [244]. The cortex actively generates torques. (A) Left, myosin heads consume ATP to pull and twist actin ﬁlaments, leading to the generation of a force dipole (top, magenta) and a torque dipole (bottom, beige). Right, these can generate an active tension and an active torque density at larger scales, causing an isolated piece of cortex to contract (top) and rotate (bottom). The gray arrow points from the outside to the inside of the cell and the rotation is clockwise when viewed from the outside. (B) Top, myosin intensity (blue markers) and velocity proﬁles

(magenta markers, AP ﬂow velocity vx; beige markers, y-velocity vy) along the AP axis for the non-RNAi condition (averaged over 25 embryos). Bottom, sketch of a C. elegans embryo with clockwise active torques in beige (as viewed from the outside of the embryo). A gradient in myosin concentration along the AP axis (see plot above) leads to a gradient in active torques (shown here with varying sizes of the clockwise torques), resulting in a chiral ﬂow (red and green arrows) orthogonal to the gradient

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Because the wheels are rotating with opposite handedness, the ellipsoids start spinning, as shown in Fig. 6.14 (d). To validate this interpretation, we come back to the evolution of the angular velocity with time proposed in Fig. 6.10 (g). The regime of steady CCW rotation between t = 800s and t = 2000s is indeed associated to the above-mentioned chiral symmetry breaking in the active nematic layer, reproduced in Fig. 6.14 (a). Conversely, between t = 500s and t = 800s, we notice that the ellipsoid does not rotate, and the deformations of the active nematic layer are quite diﬀerent. They are still periodic, but this time the vortices at the poles are co-rotating, as shown through the time-lapse of Fig. 6.14 (f). To our understanding, what happens in this case is that instead of migrating toward the poles, two of the four topological defects remain within the equator of the ellipsoid. The resulting ﬂow pattern is composed of two co-rotating vortices at the poles, and one counter-rotating vortex around the equator, as sketched in Fig. 6.14 (g). In this case, the interactions with the wall cancel out by symmetry, as shown in Fig. 6.14 (e). According to Fig. 6.10 (g), this regime seems less stable, as it only appears for 300s over the 2400s of the experimental time.

6.5 Summary

We have synthesized ellipsoidal droplets made of smectic liquid crystal 8CB. Their shape is metastable, and only maintained because of a competition between the surface energy of the droplets and the energy cost of elastic deformations between the smectic layers 5. When dispersed in an active solution of microtubules, a dense active nematic layer condenses at the interface. Topology requires the presence of defects, whose dynamics are notably different from the spherical case. Defects are essentially prescribed by pairs around the poles of the ellipsoids, where the Gaussian curvature is highest. Because of repulsive interactions, they continuously rotate around each other, creating vortices at the poles. Notably, most of the time the vortices have opposite handedness, attesting for a chiral symmetry breaking of the active flows. The active ellipsoids become torque dipoles. The interaction with a glass plate leads to a sustained and remarkably stable solid rotation of the ellipsoid. The interpretations provided so far are very qualitative, and we would like to stress several questions that need to be addressed. First and foremost, the influence of the smectic interface has been neglected. We suggested that the attraction of the topological defects towards the poles is purely a curvature effect. In parallel, we explained the persistence of the ellipsoidal shape by the alignment of the smectic layers perpendicular to the main axis. In a previous work, our lab has shown that the orientation of the smectic layers could, by itself, strongly orient the active flows [58], as explained in section 6.7. Second, the origin of the chiral symmetry breaking is not clear. While previous studies on cytoskeletal streaming provided a microscopic mechanism based on the active torque at the scale of individual motors, our results suggest a more macroscopic interpretation. Again, some of the simulations from the literature will support our discussion.

5The origin of the anisotropy is believed to be a combined eﬀect of elasticity and viscosity

174 6.5. Summary

Figure 6.14 – From active nematic deformations to solid rotation (a) Time-lapse of the dominant periodic regime observed in active ellipsoids. Positive defects are marked with coloured disks, and their direction of motion is given by coloured arrows. The red a blue horizontal arrows represent the direction of the flow field at the poles of the ellipsoid. (b) Sketch of regime (a). The active ellipsoid becomes a torque dipole. (c) Side-view of the experimental sample, showing the ellipsoid in contact with the upper glass plate. The vicinity of a boundary breaks the symmetry of the viscous forces around the ellipsoid. (d) Top-view of the sample. The interaction between the boundary and the torque dipole leads to a solid rotation of the ellipsoid. The angular velocity is called ω. (e) Sketch of an ellipsoid with symmetric active deformations. This situation does not generate solid rotations. (f) Time-lapse showing a periodic regime with symmetric deformations, corresponding to the configuration in (e). The vortices have the same handedness at the poles. This state is observed very temporarily (from t = 500s to t = 800s in the experiment of Fig. 6.10)and appears less stable than state (a). (g) Sketch of regime (f).

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6.6 Simulation Results

Recent simulation work focused on the case of active nematics confined at the surface of spheroids [249]. The results gave evidence of a substantial attraction of positive defects at the poles, but did not reproduce the chiral symmetry breaking. A spheroid is an ellipsoid that has a symmetry of revolution along the main axis. Therefore, such ellipsoids are characterized by two parameters, a and b who are respectively the short and long axis of the ellipse formed by the intersection of the ellipsoid with any plane containing the axis of symmetry. From now on, the symmetry axis will be the vertical axis z. The shape of a spheroid is governed by its aspect ratio, η = b/a. Spheroids with η > 1 are referred to as prolate, while spheroids with η < 1 are referred to as oblate. A sketch of a generic spheroid is displayed in Fig. 6.15 (a). The Gaussian curvature κ, is positive everywhere, and maximal at the poles of the main axis. From the previous considerations, we expect an attraction of the positive defects towards the poles. However, defect pair repulsions will tend to maximize their spacing, leading to a non trivial equilibrium. A snapshot of the director field obtained in the simulations of Alaimo et al. [249] is shown in Fig. 6.15 (b). It is composed of four defects. Two of them (green and purple) are pinned at the poles, while two others (blue and red) are located somewhere around z = 0. This configuration seems to be stable in time, as shown by the overlay of the defect positions throughout the whole simulation window. Notably, the defects at the poles are extremely localized and standing still. The blue and red defects are more spread out around the spheroid, although they never reach the poles. When resolving the relative positions in time, as shown in Fig. 6.15 (d), we can see that their positions oscillate around z = 0 in an anti-correlated fashion. The pinning of two positive defects at the poles clearly suggests a curvature-induced attraction. However, the defect pair-repulsions seem to prevail in the distribution of defect locations: only two defects occupy the poles, and the others move around the equator, which maximizes the distances between defects. This configuration is reminiscent of the regime described in Fig. 6.14 (f). However it has proved to be less stable than the chiral symmetry breaking regime, that is not observed in these simulations.

6.7 Role the the viscous anisotropy

The interpretations presented so far have focused on the predominant role of curvature, and overshadowed the demonstrated ability of smectic liquid crystals to tame the flows of an active nematic layer [58]. When the oil phase of an active nematic interface is replaced by a smectic liquid crystal in the smectic-A phase, the active flows are dramatically affected by the viscous anisotropy induced by the smectic planes. This effect is illustrated in Fig. 6.16 (a-c). Pau et al. showed that the active flows strongly align with the orientation given by the smectic planes. This is coherent with the argument proposed in section 6.2, according to which the viscosity perpendicular to the smectic planes is several order of magnitude higher than the viscosity parallel to the planes. In the ellipsoids, smectic planes are oriented perpendicular to the main axis 6, which means that most of the active flows should be oriented perpendicular to the main axis.

6on average (see section 6.2)

176 6.8. Conclusion and perspectives

Figure 6.15 – Simulated active nematic spheroids Adapted from [249]. (a) Sketch of a spheroid. A spheroid is an ellipsoid that has a symmetry of revolution around the main axis. It is characterized by its short and main axes a and b. The colormap corresponds to the local Gaussian curvature κ. κ is positive everywhere and maximal at the poles. (b) Snapshot showing the defect conﬁguration within a simulation of 1000 particles on a prolate spheroid with a/b = 0.25. The four 1/2 disclinations are highlighted, the director ﬁeld - black lines - is also shown. (c) Overlay of the defect trajectories over the whole simulation. The colours correspond to the four defects pictured in (b). (d) Height hi for each defect with respect to the waist a function of time. We can see that two defects remain pinned at the poles while the two others oscillate around the equator.

This result is coherent with the orientation of the vortices observed at the poles. Further- more, the motion of topological defects along the main axis comes from the fact that bend instabilities grow perpendicular to the main alignment direction. However, the viscous anisotropy does not explained a priori their attraction towards the poles.

Neither the curvature nor the viscous effects alone seem to be sufficient to recover the features and the robustness of the observed spatio-temporal patterns. A new collaboration with the lab of Juan de Pablo will rely on simulations of active nematics confined to spheroids, with and without viscous anisotropy, to help disentangle these two effects.

6.8 Conclusion and perspectives

When conﬁned at the surface of liquid crystal ellipsoids, active nematics develop a remarkably regular ﬂow pattern. This regime is characterized by a strong attraction of topological defects towards the poles, continuously spinning around the symmetry axis. The spirals are counter-rotating, conferring chirality to the active deformations 7. Peri- odically, bend instabilities disrupt the steady vortices, leading to a transfer of the defects along the main axis, from one pole to the other. This regime depends very little on the aspect ratio of the ellipsoids, but is never observed on spherical droplets. The attraction of positive defects towards regions of like-sign Gaussian curvature is pre-

7We can attribute a handedness to the torque dipoles

177 Chapter 6. Active Nematics and Curvature

Figure 6.16 – Control of active flows with a patterned smectic interface Adapted from [58]. (a) Polarizing optical micrograph, and configuration of the underlying molecular planes in the SmA phase of the passive liquid crystal. (b) Fluorescence confocal micrograph revealing the correlation between the aligned active nematic and the anisotropic SmA phase. (c) Time average of the dynamic pattern. The arrows depict the antipar- allel flow directions along the lanes of defect cores. The topological defects of the active nematic layer are preferentially flowing in the direction given by the smectic planes Scale bars: 100 µm. (d) Sketch of an ellipsoidal droplet with some smectic layers overlaid. The easy-flow direction (along the smectic layers) is perpendicular to the main axis.

178 6.8. Conclusion and perspectives dicted by theory and evidenced in experiments. Previous simulations from the literature give evidence of curvature induced defect attraction in active nematic ellipsoids, but predict very different flow patterns. This project will be continued by Martina Clairand, both at ESPCI and the University of Barcelona. A collaboration has been initiated with Juan de Pablo’s laboratory at the University of Chicago. In their simulations, they will explore the role of the viscous anisotropy induced by the smectic phase. In experiments, systematic observations are still missing to fully characterize the dynamics. In particular, a precise comparison of the dynamics between droplets of different shapes would provide insightful information on the interaction between positive defects and curvature.

In this work, the shape of the active droplets is fixed, because surface tension is too high for the active stress to deform the interface. Other experimental systems have shown that if surface tension is low enough, active nematic droplets may experience extreme deformations. To go further, simulations predict that an active nematic droplet may spontaneously elongate and divide under the influence of the active stresses [250]. The director field adapts to curvature heterogeneities, which in return change the distribution of active forces. In a feedback loop, the active forces further deform the interface, which may enhance, or damp, the curvature heterogeneities. In simulations performed by Giomi et al. [250], we can see that active droplets spontaneously deform into an ellipsoidal shape. The director field is modified accordingly, with positive defects located at the poles. It turns out that the resulting flow pattern further enhances the elongation, until the droplet divides in two as shown in Fig. 6.17.

179 Chapter 6. Active Nematics and Curvature

Figure 6.17 – Spontaneous division and motility in active droplets Adapted from [250]. Behavior of a 2D active nematic droplet immersed in a Newtonian fluid for a low surface tension and high activity. The colormap represents the velocity field around the droplet. The black lines correspond to the director field within the nematic droplet. If the activity is large enough, the capillary forces are no longer sufficient to balance the initial straining flow and the droplet divides.

180 6.8. Conclusion and perspectives

6.1 – Bursting of smectic shells into ellipsoids. The shells are immersed in a solution of 2% Pluronic dissolved in water. The image is acquired by polarized-light microscopy. Frame rate: 5 fps. Scale bar: 100 µm. The time-lapse of Fig. 6.8 (a) is taken from this video.

6.2 – Active nematic emulsion. (left) Fluorescence micrograph showing the dynamics of smectic ellipsoids immersed in an active solution. Within a few hours, a dense ative nematic layer forms onto their surface. Frame rate: 2 fps. Scale bar: 100 µm (right) The same active nematic emulsion observed by polarized-light microscopy. The motion is slower, and more erratic, because the video was taken soon after mixing the ellipsoids with the active gel, and so we believe the active nematic layer was not entirely formed yet. Frame rate: 2 fps. Scale bar: 100 µm

181 Chapter 6. Active Nematics and Curvature

6.3 – Solid-body dynamics and active deformations of an ellipsoid (left) Fluo- rescence micrograph showing the dynamics of an active ellipsoid. Frame rate: 2 fps. Scale bar: 100 µm. (right) The same video in the reference frame of the ellipsoid, showing the deformations of the active nematic layer.

182 Closing remarks

The main experimental methods and results obtained during this Ph.D. are graphically summarized in Fig. 6.18. The experiments rely on new microfluidic strategies to pattern oil-water interfaces with controlled shapes, developed during the PhD (chapter2). The first one is referred to as surface microfluidics (see 2.4.1). It enables to divide a flat oil-water interface into multiple patterns of any size and shape with a micrometric resolution. The main advantage is that the microfluidic constrictions are embedded in a portable device that can be used in any open system. Moreover, the grid specifically confines the interface, leaving the bulk of both phases unconfined. It is particularly useful in the case of active nematics where the water solution acts as a buffer, ensuring that the concentrations of active elements are equal in all the compared patterns. The second protocol, termed smectic ellipsoids (see 2.4.2), enables to synthesize oil-in- water emulsions of sub-millimetric elongated droplets. At such scales, curvature heterogeneities are highly unfavourable due to surface tension. Here, the droplets are composed of a smectic liquid crystal, aligned in such a way that the internal elastic forces preserve the ellipsoidal shape for hours. The technique involves glass microfluidic devices conventionally used to prepare double emulsions, adapted from the literature. The last method, termed active emulsions (see 2.4.2), is a proof of concept demonstrating the possibility to embed oil-in-water droplets into an active solution, and condense active nematics at their outer surface. Until now, the experimental works on active nematics at spherical interfaces were prepared with the active material encapsulated inside the droplets. This configuration requires to adapt the microfluidic setup to the constraints of the active material (biocompatibility, activity lifetime etc). Here, the emulsions can be prepared with conventional microfluidic techniques and stored before use. Moreover, the active material around the droplets can be renewed, which considerably increases the lifetime of the active droplets.

The results are classified in two categories: the effects on the order of the active nematic phase, and the impact on the flow patterns. The variety of the spatio-temporal dynamics observed only gives the flavour of the fascinating adaptability of active liquid crystals to geometrical constraints. The population of topological defects undergo dramatic transformations. Their density, spatial distribution, orientation, and velocities evade most of the laws derived for unconfined active nematics. In the most extreme conditions of lateral confinement, the defects temporarily disappear. Similarly, the flows

183 Chapter 6. Active Nematics and Curvature transition to extremely ordered patterns such as vortex lattices, shear and directed ﬂows. Undoubtedly, these spatial and dynamical transformations are related. However, describing them independently is not enough to understand how the spatio-temporal patterns emerge.

The fundamental specificity of the experiments on geometrical confinement presented in chapters3 and4 is the presence of a boundary. In chapter5 we investigate in more details the interaction between active nematics and a single wall. We uncover a peculiar competition between boundary conditions and activity, that could be summarized under the denomination active planar anchoring. On the one hand, the walls constrain the nematic field to a planar anchoring. The strength of the alignment is hard to assess, because anchoring and elasticity are not the only forces at play. To our mind, it can be considered as strong in the sense that the nematic field close to the walls "does not depend much" on the nematic phase in the bulk. This idea, expanded in chapter5, essentially means that the distortions of the bulk hardly propagate to the wall. On the other hand, the coupling between activity and hydrodynamic interactions destabilizes the planar alignment, leading to defect nucleations. As such, the anchoring is "weak enough" to allow defect pairs to form right at the wall surface. The bend distortions are by definition transversal to the alignment direction, which means that newborn positive defects are oriented perpendicularly to the walls, and quickly move away into the bulk. On the contrary, negative defects remain at the wall surface.

Thereby, the active planar anchoring is characterized by a parallel alignment of the microtubules, regularly interrupted by the appearance of negative defects. The latter are given the distinctive denomination wall-defects because their properties are diametrically opposed to that of the bulk negative defects. Their symmetry is that of a positive defect, which confers them a propulsive force. Furthermore, they attract and merge with their peers. As a matter of fact, the dynamics at the wall is dominated by the collective interactions of wall-defects that can be relatively well-described by unidimensional models. In that sense, in section 5.2.2 we proposed an analogy with the Cheerios effect which may help to understand the specificity of wall-defects. The presence of the wall imposes a different energy balance as the one for bulk defects, as much as particles at an interface are subject to an additional energy provided by surface tension. As such, wall-defects evolve in their own 1D interface.

More precisely, we found notable similarities between the behaviour of wall defects and the localized structures in the solutions of the Kuramoto-Sivashinski equation. These models were designed to capture the transition to chaos in generic out-of-equilibrium systems under conﬁnement. The analogy may give a new perspective in the interpretation of the patterns on conﬁned active nematics, in terms of generic transitions observed in other out-of-equilibrium hydrodynamic problems. For instance, the regime of vortex lattices is reminiscent of the emergence of convection rolls in the canonical Rayleigh-Bénard instability. In this particular case, the most suitable envelope equation describing the transition is not the Kuramoto-Sivashinski equation, but the principle is the same: the transition to chaos is the result of an underlying instability mechanism generating dissipative struc-

184 6.8. Conclusion and perspectives tures. The ratio between the size of the dissipative structure and the confinement width determines the emerging flow pattern, as for what we found in lateral confinement (see Fig. 3.3 in chapter3).

Although the analysis on wall-defects presented in chapter5 remains preliminary, the methodology employed, involving three distinct approaches, provides a complementary overview of the problem. The first one concerned individual wall-defects and pair interactions. It showed how their structure could explain their motility, as well as their surprising tendency to attract and merge. A detailed description of the hydrodynamics of the solvent finally gave insights on their nucleation mechanism. The second part considered collective dynamics of wall-defects with a statistical approach based on the literature. It showed that the tree patterns observed could be well described using a classical 1D envelope equation (KSE), thus implying that wall-defect were relatively isolated from the bulk. Finally, preliminary studies aimed at exploring the energetics of the spatio-temporal dynamics through energy spectra. The apparent discrepancy between our results and KSE predictions could help us go beyond this simple model. For instance, the statistical model proposed by Toh [227] and described in section 5.3.4recovered the energy spectrum of KSE chaos by considering pulses only, and ruling out humps. In active nematics, the nucleations of humps look more frequent. A more refined statistical model should include them because their predominance may strongly impact the energy distribution across length scales and minimize the role of the regulation between pulses. At a lower level, the statistical model supposed interactions between "soliton-like pulses", that is, structures with the theoretical shape of a soliton. The detailed description of the wall-defect structure could help design a more realistic modelling of the pulses, which could change their statistical interactions. Overall, the constant interaction between these three approaches could potentially improve our understanding on the emergence, structure and energetics of chaos in active nematics.

Aside from the theoretical description of confined active nematics, the results presented in this manuscript provide promising strategies to control the active flows. The experiments presented in chapter3 and4 belong to the class of topological microfluidics [251]. They give insights on how to obtain ordered shear flows, directed transport, stagnation points, and synchronized flows by engineering complex geometries. At the level of a single boundary, the preliminary experiments presented in chapter5 propose promising methods to directly take advantage of the active planar anchoring to reshape the active nematic dynamics. By altering the surface of the wall - through indentations, curvature, or asymmetric patterns - the competition between wall alignment and instability to bend can be turned into a real power engine driving the flows. The chaotic dynamics of wall-defects can be tamed, and their location precisely controlled. When properly controlled, these principles could contribute to the design of autonomous microfluidic systems, performing complex tasks without any external control.

The active planar anchoring with a wall is not the only effect through which geometrical confinement may tame the active flows. Dramatic transitions also occur in the case of closed interfaces i.e surface with no boundaries.

185 Chapter 6. Active Nematics and Curvature

In chapter5, we have seen that confining an active nematic layer at the surface of a disk imposes, according to the Poincaré-Hopf theorem, the presence of two positive defects. Similar topological concepts extend to closed surfaces, and impose the presence of four positive defects at the surface of a sphere [154]. In chapter3, we have seen that the presence of a wall leads to an unbinding of the topological defects as a function of their charge. Similarly, in chapter5 we have demonstrated the possibility to separate defects according to their charge, by playing with the local curvature of the wall. Recent experiments have shown a similar effect of defect unbinding at the surface of a torus [117]. In this case, the mechanism is different, and involves Gaussian curvature. Namely, topological defects are attracted towards regions of like-sign Gaussian curvature. The experiments on active ellipsoids presented in chapter6 demonstrate how topology and curvature effects cooperate to form an original example of chiral active particles. When immersed in an active solution, and as soon as an active nematic layer condensed onto their surface, the ellipsoids display intriguing solid-body dynamics characterized by a super-diffusive behaviour and sustained rotations. These dynamics are tightly linked to an emerging chirality in the deformations of the nematic layer.

Perspectives

In this section, we just provide a brief list of experimental ideas, with suggestions on what would be worth exploring, based on the experience I have acquired during these three years. They are sketched in Fig. 6.19.

Probing the extensivity of active nematics The ratchet experiments presented in section 5.4.2 demonstrated the possibility to direct the active flows close to the boundaries. We suspect that we could achieve material transport by patterning both walls of a channel, as shown in Fig. 6.19. A similar geometry has been tested with colonies of bacteria [232]. The authors connected two baths of swimming bacteria by ratchet funnels. They have shown that the asymmetric transport induced by the funnels could concentrate the bacteria in one bath. We suspect that the microtubule-based active nematic would behave differently. The extensile elasticity inherent to this system may be unfavourable to a compression of the nematic phase. In simple words, we believe that the system is "less compressible" than its bacterial analogue. Such an experimental would provide insights on the elasticity of active nematics. It would be interesting to observe whether or not a steady state is reached, with a different concentration in each bath. If so, we could investigate the effect of the container size and shape. We could also study how the topological landscape (defect density, distribution, velocity) compare in both containers.

Active nematic vehicles The experiments on the control of active ﬂows through the wall alteration presented in section 5.4 demonstrated the possibility to localize wall defects by playing with indentations or curvature. In addition, in section 5.2.3, we showed that the wall-defects expelled a permanent jet of active material perpendicular to the wall. Using hydrodynamic models, we demonstrated that the location of the wall-defects were

186 6.8. Conclusion and perspectives

Figure 6.18 – Main results presented in the manuscript The asterisks (*) point out the results that are less consolidated, and which would require further investigation.

187 Chapter 6. Active Nematics and Curvature associated to higher hydrodynamic pressures. As a consequence, by tuning the shape of the wall, we can create pressure gradients. This idea is reminiscent of the basic principles of aerodynamics, where the asymmetry of the flow field around a wing results in a lift force (although the physical phenomena are quite different). Hence the idea of designing asymmetric particles propelled by the active flows. For instance, in a moon-shaped particles, negative defects would only nucleate on the concave side, and "push" the object in opposite direction. A simpler geometry of a square patterned with an indentation on one side may also be propelled by a wall-defect pinned onto the indentation.

Active nematic vesicles The experiments on active emulsions demonstrated the possibility to condense the active nematic material on the outer surface of any fluid interface provided it is coated with a PEGylated surfactant (see section 2.4.2). As said in the introduction, previous work by Kleber et al. has shown a myriad of dynamical states when active nematics were embedded inside vesicles [154]. The very low surface tension lead the vesicle to deform dramatically under the active forces, with the formation of streaming filopodia-like protrusions. In parallel, simulation works by Giomi et al. on active nematic droplets with low surface tension reported cases where the coupling between curvature and active deformations could lead to even more drastic deformation, motility and even to the division of the droplets [131]. In this project, instead of embedded active solution inside the vesicles, we would like to immerse the vesicles inside active solution. Our intuition is that this configuration could lead to dynamical states closer to the ones reported by Giomi et al. When the active solution is inside the vesicles, the quantity of microtubules available to condense at the interface is limited by the volume of solution inside the vesicles. As a consequence, the active nematic layer is not very dense. Conversely, we expect that the opposite configuration could lead to a much denser active nematic shell at the surface of the vesicles, which could further enhance the deformations until vesicle division.

Active micro-machines In a similar spirit to the work of active ellipsoids and active vesicles, our team is trying to condense and active nematic layer onto a solid surface. To do so, we need to ﬂuidize the interface, in order to bridge the highly deformable active nematic layer with a no-slip boundary. The strategy tested so far involves the formation of lipid bilayers onto a glass surface. We hope that the lipid bilayer is ﬂuid enough to follow the active streams while maintaining adhesion with the nematic layer. However, we have not achieved a reliable coupling yet. In parallel, we have noticed that when crosslinking PDMS inside water, there remained a lubrication layer at the surface of the polymer. As such, although the cross-linked PDMS is solid, its surface could be compatible with active nematics. Our idea is to synthesize 3D particles of various shapes, and embed them in active solution. The works on active ellipsoids demonstrate how topology and curvature may cooperate to convert the active deformations into power engines. Using the same 3D-printer used to prepare the polymer grids (see section 2.4.1), we could synthesize particles of any shape and design active micro-machines powered by the deformations of an active nematic layer condensed at their surface.

188 6.8. Conclusion and perspectives

Temperature-controlled self-assembly During this thesis, we conducted preliminary studies on the temperature-controlled self-assembly of active nematics. By replacing the depletion agent PEG with a temperature-responsive polymer called pnipam, we were able to reversibly form and dissolve an active nematic layer. Below 23 ◦C, the polymer was hydrophilic and acted as a depletant with triggered the self-assembly of the nematic layer. Above that critical temperature, pnipam turned hydrophobic, and the depletion force vanished, causing the collapse of the nematic phase. There remain experimental issues which degrade the reversibility of this process. In particular, we found that, in the hydrophobic state, the pnipam polymers tend to cross-link, forming aggregates which no longer contribute to the depletion force in the hydrophilic state. However, when properly-controlled, this tool could have direct impact on the control of active flows. For instance, it would enable to switch on and off the net transport in three-dimensional active fluids confined in annular geometries, observed by Wu et al. [116]. Moreover, repeating the experiments on depletion-induced pattern formation in actin-myosin motility assays described in section 1.1.3, and replacing PEG with pnipam, we may be able to reversibly switch between polar and nematic patterns using temperature.

Measuring activity through hydrodynamics The analysis performed in section 5.2.3 on the hydrodynamics of active nematics oﬀers a new perspective on how to characterize the system. In this model, the active layer is treated as an external force, which drives the ﬂows of the solvent. The latter rigorously obeys the Navier-Stokes equation, as written in Eq. 5.15, which implies:

f = ∇p − µ∆u ext (6.6) ∇ · u = 0, In the paragraph called External Force in section 5.2.3, we provide arguments supporting the idea that fext essentially corresponds to the bulk active force fa, except that fa gives a more complete map of the internal stresses within the active layer, some of which do not contribute to the propulsion of the solvent. Yet, it seems possible to draw a correspondence between these two variables.

In parallel, we know that fa = −α∇ · Q from Eq. 1.20 (with α the activity parameter). The map of ∇·Q can be directly obtained from the image processing techniques developed by Ellis et al. (see section 2.2.3). The only unknown of the equation is α. As such, it may be possible to estimate the activity parameter quantitatively. Undoubtedly, there remain several assumptions to conﬁrm before doing such an analysis. First, the model supposes that the water phase behaves as a purely viscous ﬂuid, without any elastic response. To do so, the elastic component due to the active nematic layer has been neglected, which may not be valid in this case. The opinion of theoreticians would be highly valuable to push further the investigation.

189 Chapter 6. Active Nematics and Curvature

Figure 6.19 – Perspectives The asterisks (*) point out the results that are less consolidated, and which would require further investigation.

190 Conclusion

In this manuscript, we study the spatio-temporal dynamics of a microtubule-based two- dimensional active nematic system under confinement. We are particularly interested in the interplay between the geometry of the confining space, the spatial ordering of the nematic phase, and the dynamics of the active flows. We specifically investigate the effect of lateral confinement (chapter3), topology (chapter4), boundary roughness (5) and Gaussian curvature (chapter6). In each case, we describe how the active nematic phase adapts to the geometrical constraints and what are the emerging flow patterns.

In the third chapter, we disclose a defect-free regime of shear flow in narrow channels. This regime is unstable to bend deformations, which lead to the nucleation of short- lived defects at the walls. The wavelength of the bend deformations no longer corresponds to the classical active length scale. As a matter of fact, it does not depend on activity, but merely on geometrical parameters. By increasing the channel width, defect lifetime increases, developing a spatio-temporal organization that corresponds to the predicted state of dancing vortical flows [139], before full disorganization into the active turbulence regime for still wider channels, as is typical of the unconfined active nematic. We stress the close interplay between the velocity field and the defect dynamics by connecting the transitions in the flow patterns with the ratio between the mean defect spacing and the width of the confining channel.

In chapter four, we demonstrate that active nematics can be ordered into directed flows under annular confinement. Below a critical confinement width, a regime of transport with random handedness emerges inside isolated annuli. This flow symmetry- breaking is accompanied by a polarization of the nematic order in the direction of the flow. Adding asymmetric corrugations to the boundaries enables to control the transport handedness in an isolated annulus. In the case of two interconnected annuli we show the possibility to generate stagnation points in the active flow pattern, thereby localizing topological charges in space. Moreover, going to higher order platforms of connected annuli, examples of dynamic flow synchronization, anti-correlation and frustration are uncovered.

In chapter ﬁve, we elucidate the dynamics of active nematics in the vicinity of a boundary. We demonstrate how the interplay between anchoring and activity leads to the emergence of a special type of topological defect, whose properties are described in details. The collective dynamics of wall-defects are interpreted in the light of one-

191 Chapter 6. Active Nematics and Curvature dimensional models of spatio-temporal chaos, in an effort to understand the transitions observed in chapters 3 and 4 with generic principles of out-of-equilibrium fluids. Finally, we demonstrate strategies to control the dynamics of wall-defects, and thereby the active flows, by tuning the boundary roughness.

In chapter six, we investigate the dynamics of active nematics at the surface of ellipsoidal droplets. We demonstrate how topology and curvature eﬀects may cooperate to form an original example of chiral active particles. When immersed in an active solution, and as soon as an active nematic layer condensed onto their surface, the ellipsoids display intriguing solid-body dynamics characterized by a super-diﬀusive behaviour and sustained rotations. These dynamics are tightly linked to an emerging chirality in the deformations of the nematic layer.

192 Publications

1. P. Guillamat, Ž. Kos, J. Hardoüin, J. Ignés-Mullol, M. Ravnik, and F. Sagués, “Active nematic emulsions,” Science advances, vol. 4, no. 4, p. eaao1470, 2018 2. J. Hardoüin, R. Hughes, A. Doostmohammadi, J. Laurent, T. Lopez-Leon, J. M. Yeomans, J. Ignés-Mullol, and F. Sagués, “Reconfigurable flows and defect landscape of confined active nematics,” arXiv preprint arXiv:1903.01787, 2019 3. J. Hardouin, P. Guillamat, F. Sagues, and J. Ignes, “Dynamics of ring disclinations driven by active nematic shells,” Frontiers in Physics, vol. 7, p. 165, 2019 4. J. Hardouin, J. Laurent, T. Lopez-Leon, F. Sagues, and J. Ignes, “Flow and order in active nematics confined in annuli,” In prep.

193

Acknowledgments

195

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