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Hydrodynamics of Soft Active Matter

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Hydrodynamics of Soft Active Matter REVIEWS OF MODERN PHYSICS, VOLUME 85, JULY–SEPTEMBER 2013 Hydrodynamics of soft active matter M. C. Marchetti* Physics Department and Syracuse Biomaterials Institute, Syracuse University, Syracuse, New York 13244, USA J. F. Joanny Physicochimie Curie (CNRS-UMR168 and Universite´ Pierre et Marie Curie), Institut Curie Section de Recherche, 26 rue d’Ulm, 75248 Paris Cedex, 05 France S. Ramaswamy Department of Physics, Indian Institute of Science, Bangalore, 560 12 India and TIFR Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, 21 Brundavan Colony Narsingi, Hyderabad 500 075 India T.B. Liverpool School of Mathematics, University of Bristol, Bristol, BS8 1TW United Kingdom J. Prost Physicochimie Curie (CNRS-UMR168 and Universite´ Pierre et Marie Curie), Institut Curie Section de Recherche 26 rue d’Ulm, 75248 Paris Cedex, 05 France and E.S.P.C.I, 10 rue Vauquelin, 75231 Paris Cedex 05 France Madan Rao Raman Research Institute, Bangalore, 560 080 India and National Centre for Biological Sciences (TIFR), Bangalore, 560065 India R. Aditi Simha Department of Physics, Indian Institute of Technology Madras, Chennai 600 036 India (published 19 July 2013) This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments. This approach offers a unified framework for the mechanical and statistical properties of living matter: biofilaments and molecular motors in vitro or in vivo, collections of motile microorganisms, animal flocks, and chemical or mechanical imitations. A major goal of this review is to integrate several approaches proposed in the literature, from semimicroscopic to phenomenologi- cal. In particular, first considered are ‘‘dry’’ systems, defined as those where momentum is not conserved due to friction with a substrate or an embedding porous medium. The differences and similarities between two types of orientationally ordered states, the nematic and the polar, are clarified. Next, the active hydrodynamics of suspensions or ‘‘wet’’ systems is discussed and the relation with and difference from the dry case, as well as various large-scale instabilities of these nonequilibrium states of matter, are highlighted. Further highlighted are various large-scale insta- bilities of these nonequilibrium states of matter. Various semimicroscopic derivations of the continuum theory are discussed and connected, highlighting the unifying and generic nature of the continuum model. Throughout the review, the experimental relevance of these theories for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material is discussed. Promising extensions toward greater realism in specific contexts from cell biology to animal behavior are suggested, and remarks are given on some exotic active-matter analogs. Last, the outlook for a quantitative understanding of active matter, through the interplay of detailed theory with controlled experiments on simplified systems, with living or artificial constituents, is summarized. DOI: 10.1103/RevModPhys.85.1143 PACS numbers: 05.65.+b, 87.18.Hf, 82.70. y, 87.16.Ln À CONTENTS 1. Homogeneous steady states 1150 2. Properties of the isotropic state 1151 I. Introduction 1144 II. Dry Active Matter 1148 3. Properties of the ordered state 1151 A. Polar active systems: Toner and Tu continuum B. Systems with nematic interactions on a substrate 1154 model of flocking 1148 1. Active nematic 1155 2. Self-propelled hard rods: A system of ‘‘mixed’’ symmetry? 1157 *[email protected] 0034-6861= 2013=85(3)=1143(47) 1143 Ó 2013 American Physical Society 1144 M. C. Marchetti et al.: Hydrodynamics of soft active matter C. Current status of dry active matter 1160 III. Active Gels: Self-driven Polar and Apolar Filaments in a Fluid 1161 A. Hydrodynamic equations of active gels 1161 1. Entropy production 1161 2. Conservation laws 1161 3. Thermodynamics of polar systems 1162 4. Fluxes, forces, and time reversal 1162 B. Linear theory of active polar and nematic gels 1163 1. Constitutive equations 1163 2. Microscopic interpretation of the transport coefficients 1163 3. Viscoelastic active gel 1164 C. Active polar gels 1164 FIG. 1 (color). Liquid-crystalline order in a myxobacterial flock. 1. Polarity effects 1164 Figure from Gregory Velicer (Indiana University Bloomington) and 2. Noise in active gels 1164 Juergen Bergen (Max-Planck Institute for Developmental Biology). 3. Multicomponent active gels 1164 D. Active defects 1165 E. Current status on active gels 1166 composed of self-driven units, active particles, each capable IV. Hydrodynamic Consequences of Activity 1167 of converting stored or ambient free energy into systematic A. Instabilities of thin liquid active films 1167 movement (Schweitzer, 2003). The interaction of active par- 1. Spontaneous flow of active liquid films 1167 ticles with each other, and with the medium they live in, gives 2. Instabilities of thin films 1168 rise to highly correlated collective motion and mechanical B. Polar active suspensions with inertia 1171 stress. Active particles are generally elongated and their C. Rheology 1171 direction of self-propulsion is set by their own anisotropy, 1. Linear rheology of active isotropic matter 1172 rather than fixed by an external field. Orientational order is 2. Linear rheology of active oriented matter 1174 thus a theme that runs through much of the active-matter 3. Nonlinear rheology of active nematics 1175 narrative as can be seen, for instance, in the image of a swarm D. Applying the hydrodynamic theory to of myxobacteria, shown in Fig. 1. The biological systems of phenomena interest to us include in vitro mixtures of cell extracts of in living cells 1176 biofilaments and associated motor proteins (see Fig. 2), the V. Derivation of Hydrodynamics from whole cytoskeleton of living cells, bacterial suspensions (see Microscopic Models of Active Matter 1177 Fig. 3), cell layers (see Fig. 4), and terrestrial, aquatic (see A. Microscopic models 1177 Fig. 5), and aerial flocks. Nonliving active matter arises in 1. Self-propelled particles 1178 layers of vibrated granular rods, colloidal or nanoscale par- 2. Motors and filaments 1179 ticles propelled through a fluid by catalytic activity at their B. From stochastic dynamics to macroscopic equations 1179 surface (see Fig. 6), and collections of robots. A distinctive, 1. Smoluchowski dynamics 1180 indeed, defining feature of active systems compared to more 2. From Smoluchowski to hydrodynamics 1181 familiar nonequilibrium systems is the fact that the energy 3. An example: Derivation of continuum equations input that drives the system out of equilibrium is local, for for aligning Vicsek-type particles 1181 example, at the level of each particle, rather than at the 4. Hydrodynamic interactions 1182 C. Current status of microscopic theories of active matter 1183 VI. Conclusions, Outlook, and Future Directions 1184 Acknowledgments 1185 References 1185 I. INTRODUCTION The goal of this article is to introduce the reader to a FIG. 2. Patterns organized in vitro by the action of multimeric general framework and viewpoint for the study of the me- kinesin complexes on microtubules, imaged by dark-field micros- chanical and statistical properties of living matter and of copy. The concentration of motor proteins increases from left to some remarkable nonliving imitations on length scales from right. (a) A disordered array of microtubules. The other two images subcellular to oceanic. The ubiquitous nonequilibrium con- display motor-induced organization in (b) spiral and (c) aster densed systems that this review is concerned with (Toner, Tu, patterns. The bright spots in the images correspond to the minus and Ramaswamy, 2005; Ju¨licher et al., 2007; Joanny and end of microtubules. These remarkable experiments from Surrey Prost, 2009a; Ramaswamy, 2010) have come to be known et al. (2001) led the way to the study of pattern formation in active as active matter. Their unifying characteristic is that they are systems. Adapted from Surrey et al., 2001. Rev. Mod. Phys., Vol. 85, No. 3, July–September 2013 M. C. Marchetti et al.: Hydrodynamics of soft active matter 1145 FIG. 3. Bacterial ‘‘turbulence’’ in a sessile drop of Bacillus sub- FIG. 5 (color). A remarkable demonstration of polar order in a tilis viewed from below through the bottom of a petri dish. Gravity sardine school. Figure from Jon Bertsch, from underwater images is perpendicular to the plane of the picture, and the horizontal white from the Sea of Cortez: http://www.thalassagraphics.com/blog/? line near the top is the air-water-plastic contact line. The central p=167. fuzziness is due to collective motion, not quite captured at the frame rate of 1=30 s. The scale bar is 35 m. Adapted from Dombrowski et al., 2004. signals and stimuli, rendering the accessible parameter space immense. Perhaps, therefore, global principles such as conservation laws and symmetries constrain the possible system’s boundaries as in a shear flow. Each active particle dynamical behaviors of cells or, indeed, of organisms and consumes and dissipates energy going through a cycle that populations, such as collections of bacteria (see Fig. 3), fish fuels internal changes, generally leading to motion. Active schools (see Fig. 5), and bird flocks. Quantifying the sponta- systems exhibit a wealth of intriguing nonequilibrium prop- neous dynamical organization and motion of living systems is erties, including emergent structures with collective behavior the first step toward understanding in a generic
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