Self-organizing microfluidic crystals The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Uspal, William E., and Patrick S. Doyle. “Self-Organizing Microfluidic Crystals.” Soft Matter 10, no. 28 (2014): 5177–5191. As Published http://dx.doi.org/10.1039/c4sm00664j Publisher Royal Society of Chemistry (RSC) Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/107145 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Soft Matter View Article Online PAPER View Journal Self-organizing microfluidic crystals† William E. Uspala and Patrick S. Doyle*b Cite this: DOI: 10.1039/c4sm00664j We consider how to design a microfluidic system in which suspended particles spontaneously order into flowing crystals when driven by external pressure. Via theory and numerics, we find that particle–particle hydrodynamic interactions drive self-organization under suitable conditions of particle morphology and geometric confinement. Small clusters of asymmetric “tadpole” particles, strongly confined in one Received 26th March 2014 direction and weakly confined in another, spontaneously order in a direction perpendicular to the Accepted 1st May 2014 external flow, forming one dimensional lattices. Large suspensions of tadpoles exhibit strong density DOI: 10.1039/c4sm00664j heterogeneities and form aggregates. By rationally tailoring particle shape, we tame this aggregation and www.rsc.org/softmatter achieve formation of large two-dimensional crystals. 1 Introduction hydrodynamic interactions (HI) are central to most examples of ow-driven self-organization at the microscale.11,12 The tensorial The shape of interacting particles can control their spontaneous form and spatial decay of hydrodynamic interactions can organization into larger ordered structures. Recently, innova- change dramatically in the presence of conning boundaries. In tions in colloidal synthesis providing precise control over particular, we consider hydrodynamic interactions when the particle morphology have driven fresh efforts to relate shape to typical size of a particle is comparable to the height of a self-assembled equilibrium structure.1,2 Particle “designers” conning slit, such that the particles are constrained to “quasi- have exploited depletion interactions,3 steric effects,4 and Janus two-dimensional” (q2D) motion [Fig. 1(a)]. The tightly conned patterning5 for self-assembly of novel complex materials. On the particles experience strong friction from the conning plates, other hand, comparatively few studies have sought to relate and will therefore lag a pressure-driven external ow. Due to shape to self-organization out of equilibrium, despite the rich this lag, the particles create ow disturbances with a charac- set of static and dynamic structures that can be sustained teristic dipolar structure: moving upstream relative to the uid, Published on 08 May 2014. Downloaded 18/06/2014 20:14:29. through continuous dissipation of energy.6 In particular, particles push uid mass away from their upstream edges and suspensions that self-organize under ow into owing crystals draw uid mass into their downstream edges.13,14 The strength are of great interest from both theoretical and practical of this leading order ow disturbance is proportional to the perspectives. They provide a natural arena for extension and particle area, and decays as the inverse square distance from the revision of theoretical ideas developed in the context of equi- particle center. That mass conservation determines the leading librium crystallization. Moreover, they can be harnessed for order disturbance is ultimately due to the presence of the two microuidic and lab-on-a-chip applications. Orderly ow eases conning plates. The plates exert friction on the uid, removing recognition and interrogation of suspended objects in cytom- momentum from the system and screening long-range etry7 and bioassays.8 Flowing crystals could be used as dynam- momentum transport. In contrast, in three dimensions, ically programmable metamaterials, assembled with high conservation of momentum determines the leading order far- throughput in continuously operating microdevices, or as eld ow disturbance, the “Stokeslet.” tunable diffraction gratings.9 Researchers have achieved self- Quasi-two-dimensional microchannels have proven to be a organizing owing crystals with acoustically excited bubbles10 rich setting for collective phenomena involving owing droplets and weakly inertial spheres.11 However, the possibility that self- or solid particles, including owing crystals.15 One-dimensional organization of owing crystals can be encoded via design of owing crystals of “pancake shaped” droplets, ordered in the particle shape has remained largely unexplored. streamwise direction, can sustain transverse and longitudinal When particles are driven by ow, they are coupled via the acoustic waves, or “microuidic phonons”.16,17 Small clusters of disturbances they create in the suspending uid. These discs, ordered perpendicular to the ow direction, can maintain relative positions as they are carried by the ow.18 Various two- dimensional crystal lattices are possible in unbounded q2D.19 a Department of Physics, Massachusetts Institute of Technology, USA However, in each example of a owing crystal, the crystal is only bDepartment of Chemical Engineering, Massachusetts Institute of Technology, marginally stable: the amplitude of a collective mode neither Cambridge, MA, 02139, USA. E-mail: [email protected] † Electronic supplementary information (ESI) available. See DOI: grows nor decays in time. Consequently, crystals have no “ ” 10.1039/c4sm00664j restoring force against perturbation by channel defects, and This journal is © The Royal Society of Chemistry 2014 Soft Matter View Article Online Soft Matter Paper effects, the aligned and focused conguration is an attractor for particle dynamics. This single particle picture provides a starting point for consideration of how owing crystals might self-organize in multiple particle systems. In Fig. 1(c), we consider a single aligned and focused particle. The particle translates in the ow direction without any rotation or lateral motion, and is part of an innite lattice of real and image particles. If one or more of the image particles is exchanged for a real particle, and the associated image channels are exchanged for uid, the resulting conguration should also steadily translate with no relative particle motion. Each real particle in the “triplet” at the right of Fig. 1(c) experiences the same ow elds as the “singlet” at le, and therefore must have the same motion as the singlet. The triplet conguration is a “xed point” in phase space for the dynamics of the system, and, in view of the innite lattice, can be regarded as a one-dimensional owing crystal. In general, for a group of n real particles, the n-let conguration is a xed point. For an n-let, the real particles are located at y ¼ W/2n + Fig. 1 (a) Illustration of quasi-2D hydrodynamics. A disc is tightly (i À 1)W/n, where i ˛ {1.n}, ˆy is the transverse direction, and y fi fl con ned between parallel plates and subject to an external ow (black ¼ W/2 is the channel centerline. For instance, particle i ¼ 1ina vectors). The particle is advected downstream (blue vector) by the ¼ flow. However, due to strong friction from the confining plates, the triplet is located at y W/6 and has two neighbors: a real particle lags the external flow, and moves upstream relative to it (green particle at y ¼ W/2, and its reected image particle at y ¼W/6. vector). The particle therefore creates a characteristic dipolar flow The transverse positions are determined by translational disturbance field; fluid mass is pushed away from its upstream edge symmetry: each particle, real or image, is separated from its and drawn into its downstream edge. (b) A single fore-aft asymmetric nearest neighbors by a distance W/n. dumbbell is stably attracted to the centerline through hydrodynamic self-interaction and interaction with its hydrodynamic images. (c) A Furthermore, symmetry considerations extend to two- single aligned and focus particle is part of an infinite lattice of real and dimensional crystals. Fig. 1(d) shows two “columns” of a image particles. When one or more image particles are exchanged for “doublet crystal” that has translational symmetry in the real particles, the resulting configuration should also steadily translate streamwise direction. This conguration is also a xed point. along the channel with no relative particle motion. Each of the real and For instance, for the column shown at le, the inuence of all image particles is separated by W/n, where n is the number of real particles. (d) An infinite two-dimensional lattice should likewise other columns vanishes by symmetry. In general, two-dimen- steadily translate. The lattice length a is determined by particle density. sional n-let crystals are dynamical xed points for suspension Published on 08 May 2014. Downloaded 18/06/2014 20:14:29. dynamics. While we have argued that one and two-dimensional owing do not self-organize from disorder. However, disordered droplet crystals are dynamical xed points – i.e. have no relative particle suspensions exhibit large, freely propagating uctuations in motion – we have not examined their stability. For instance, a density,20 as well as directionally dependent, long-range orien- lattice might be linearly unstable, subject