Hydrodynamic crystals: collective dynamics of regular arrays of spherical particles in a parallel-wall channel

Capillary imbibition of confined monodisperse emulsions in microfluidic channels

We study imbibition of a monodisperse emulsion into high-aspect ratio microfluidic channels with the height h comparable to the droplet diameter d. Two distinct regimes are identified in the imbibition dynamics. In a strongly confined system (the confinement ratio d/h = 1.2 in our experiments), the droplets are flattened between the channel walls and move more slowly compared to the average suspension velocity. As a result, a droplet-free region forms behind the meniscus (separated from the suspension region by a sharp concentration front) and the suspension exhibits strong droplet-density and velocity fluctuations. In a weaker confinement, d/h = 0.65, approximately spherical droplets move faster than the average suspension flow, causing development of a dynamically unstable high-concentration region near the meniscus. This instability results in the formation of dense droplet clusters, which migrate downstream relative to the average suspension flow, thus affecting the entire suspension dynamics. We explain the observed phenomena using linear transport equations for the particle-phase and suspension fluxes driven by the local pressure gradient. We also use a dipolar particle interaction model to numerically simulate the imbibition dynamics. The observed large velocity fluctuations in strongly confined systems are elucidated in terms of migration of self-assembled particle chains with highly anisotropic mobility.

Annealing and melting of active two-dimensional soliton lattices in chiral nematic films

In this work, thousands of electrically driven dissipative solitons, called directrons, are generated in a chiral nematic liquid crystal. The directrons start with random motions but soon synchronize their motions and self-organize into a two-dimensional hexagonal lattice. The directron lattice moves collectively and forms a hexatic phase. By increasing the applied voltage, the lattice exhibits a first-order hexatic-to-liquid phase transition.

Hydrodynamic synchronization and clustering in ratcheting colloidal matter

Ratchet transport systems are widespread in physics and biology; however, the effect of the dispersing medium in the collective dynamics of these out-of-equilibrium systems has been often overlooked. We show that, in a traveling wave magnetic ratchet, long-range hydrodynamic interactions (HIs) produce a series of remarkable phenomena on the transport and assembly of interacting Brownian particles. We demonstrate that HIs induce the resynchronization with the traveling wave that emerges as a "speed-up" effect, characterized by a net raise of the translational speed, which doubles that of single particles. When competing with dipolar forces and the underlying substrate symmetry, HIs promote the formation of clusters that grow perpendicular to the driving direction. We support our findings both with Langevin dynamics and with a theoretical model that accounts for the fluid-mediated interactions. Our work illustrates the role of the dispersing medium on the dynamics of driven colloidal matter and unveils the growing process and cluster morphologies above a periodic substrate.

Exceptional topology in ordinary soft matter

Hydrodynamics is shown to induce non-Hermitian topological phenomena in ordinary, passive soft matter. This is demonstrated by subjecting a two-dimensional elastic lattice to a low-Reynolds viscous flow. The interplay of hydrodynamics and elasticity splits Dirac cones into bulk Fermi arcs, pairing exceptional points with opposite half-integer topological charges. The bulk Fermi arc is a generic hallmark of the system exhibited in all lattice and flow symmetries. An analytic model and simulations explain how the emergent singularities shape the spectral bands and give rise to a web of van Hove singularity lines in the density of states. The present findings suggest that non-Hermitian physics can be explored in a broad class of ordinary soft matter, living and artificial alike, opening avenues for topology-based technology in this regime.

Microfluidic formation of crystal-like structures

Crystal-like structures find application in several fields ranging from biomedical engineering to material science. For instance, droplet crystals are critical for high throughput assays and material synthesis, while particle crystals are important for particles and cell encapsulation, Drop-seq technologies, and single-cell analysis. Formation of crystal-like structures relies entirely on the possibility of manipulating with great accuracy the micrometer-size objects forming the crystal. In this context, microfluidic devices offer versatile tools for the precise manipulation of droplets and particles, thus enabling fabrication of crystal-like structures that form due to hydrodynamic interactions among droplets or particles. In this review, we aim at providing an holistic representation of crystal-like structure formation mediated by hydrodynamic interactions in microfluidic devices. We also discuss the physical origin of these hydrodynamic interactions and their relation to parameters such as device geometry, fluid properties, and flow conditions.

Collective vibrations of a hydrodynamic active lattice

Recent experiments show that quasi-one-dimensional lattices of self-propelled droplets exhibit collective instabilities in the form of out-of-phase oscillations and solitary-like waves. This hydrodynamic lattice is driven by the external forcing of a vertically vibrating fluid bath, which invokes a field of subcritical Faraday waves on the bath surface, mediating the spatio-temporal droplet coupling. By modelling the droplet lattice as a memory-endowed system with spatially non-local coupling, we herein rationalize the form and onset of instability in this new class of dynamical oscillator. We identify the memory-driven instability of the lattice as a function of the number of droplets, and determine equispaced lattice configurations precluded by geometrical constraints. Each memory-driven instability is then classified as either a super- or subcritical Hopf bifurcation via a systematic weakly nonlinear analysis, rationalizing experimental observations. We further discover a previously unreported symmetry-breaking instability, manifest as an oscillatory-rotary motion of the lattice. Numerical simulations support our findings and prompt further investigations of this nonlinear dynamical system.

Self-Arranged Levitating Droplet Clusters: A Reversible Transition from Hexagonal to Chain Structure

Water microdroplets condense over locally heated water-vapor interfaces and levitate in an ascending vapor-air flow forming self-assembled ordered monolayer clusters. The droplets do not coalesce due to complex aerodynamic interactions between them. The droplet cluster formation is governed by the condensation/evaporation balance and by coupling of heat flux and vapor flow with aerodynamic forces. Here, we report the observations of a reversible structural transition from the ordered hexagonal-structure cluster to the chain-like structure and provide an explanation of its mechanism and conditions under which the transition occurs. The phenomenon provides new insights on the fundamental physical and chemical processes with microdroplets including their role in reaction catalysis in nature and their potential for aerosol and microfluidic applications.

Mechanisms of spontaneous chain formation and subsequent microstructural evolution in shear-driven strongly confined drop monolayers

It was experimentally demonstrated by Migler and his collaborators [Phys. Rev. Lett., 2001, 86, 1023; Langmuir, 2003, 19, 8667] that a strongly confined drop monolayer sheared between two parallel plates can spontaneously develop a flow-oriented drop-chain morphology. Here we show that the formation of the chain-like microstructure is driven by far-field Hele-Shaw quadrupolar interactions between drops, and that drop spacing within chains is controlled by the effective drop repulsion associated with the existence of confinement-induced reversing streamlines, i.e., the swapping trajectory effect. Using direct numerical simulations and an accurate quasi-2D model that incorporates quadrupolar and swapping-trajectory contributions, we analyze microstructural evolution in a monodisperse drop monolayer. Consistent with experimental observations, we find that drop spacing within individual chains is usually uniform. Further analysis shows that at low area fractions all chains have the same spacing, but at higher area fractions there is a large spacing variation from chain to chain. These findings are explained in terms of uncompressed and compressed chains. At low area fractions most chains are uncompressed (spacing equals lst, which is the stable separation of an isolated pair). At higher area fractions compressed chains (with tighter spacing) are formed in a process of chain zipping along y-shaped structural defects. We also discuss the relevance of our findings to other shear-driven systems, such as suspensions of spheres in non-Newtonian fluids.

Blood Crystal: Emergent Order of Red Blood Cells Under Wall-Confined Shear Flow

Driven or active suspensions can display fascinating collective behavior, where coherent motions or structures arise on a scale much larger than that of the constituent particles. Here, we report numerical simulations and an analytical model revealing that deformable particles and, in particular, red blood cells (RBCs) assemble into regular patterns in a confined shear flow. The pattern wavelength concurs well with our experimental observations. The order is of a pure hydrodynamic and inertialess origin, and it emerges from a subtle interplay between (i) hydrodynamic repulsion by the bounding walls that drives deformable cells towards the channel midplane and (ii) intercellular hydrodynamic interactions that can be attractive or repulsive depending on cell-cell separation. Various crystal-like structures arise depending on the RBC concentration and confinement. Hardened RBCs in experiments and rigid particles in simulations remain disordered under the same conditions where deformable RBCs form regular patterns, highlighting the intimate link between particle deformability and the emergence of order.

Activity-induced instability of phonons in 1D microfluidic crystals

One-dimensional crystals of passively-driven particles in microfluidic channels exhibit collective vibrational modes reminiscent of acoustic 'phonons'. These phonons are induced by the long-range hydrodynamic interactions among the particles and are neutrally stable at the linear level. Here, we analyze the effect of particle activity - self-propulsion - on the emergence and stability of these phonons. We show that the direction of wave propagation in active crystals is sensitive to the intensity of the background flow. We also show that activity couples, at the linear level, transverse waves to the particles' rotational motion, inducing a new mode of instability that persists in the limit of large background flow, or, equivalently, vanishingly small activity. We then report a new phenomenon of phonons switching back and forth between two adjacent crystals in both passively-driven and active systems, similar in nature to the wave switching observed in quantum mechanics, optical communication, and density stratified fluids. These findings could have implications for the design of commercial microfluidic systems and the self-assembly of passive and active micro-particles into one-dimensional structures.

Emergent hydrodynamic bound states between magnetically powered micropropellers

Hydrodynamic interactions (HIs), namely, solvent-mediated long-range interactions between dispersed particles, play a crucial role in the assembly and dynamics of many active systems, from swimming bacteria to swarms of propelling microrobots. We experimentally demonstrate the emergence of long-living hydrodynamic bound states between model microswimmers at low Reynolds numbers. A rotating magnetic field forces colloidal hematite microparticles to translate at a constant and frequency-tunable speed close to a bounding plane in a viscous fluid. At high driving frequency, HIs dominate over magnetic dipolar ones, and close propelling particles couple into bound states by adjusting their translational speed to optimize the transport of the pair. The physical system is described by considering the HIs with the boundary surface and the effect of gravity, providing an excellent agreement with the experimental data for all the range of parameters explored. Moreover, we show that in dense suspensions, these bound states can be extended to one-dimensional arrays of particles assembled by the sole HIs. Our results manifest the importance of the boundary surface in the interaction and dynamics of confined propelling microswimmers.

Two-dimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier

We discuss the basic physics of the flow of micron-scale droplets in 2D geometry.

Collective waves in dense and confined microfluidic droplet arrays

Excitation mechanisms for collective waves in confined dense one-dimensional microfluidic droplet arrays are investigated by experiments and computer simulations. We demonstrate that distinct modes can be excited by creating specific 'defect' patterns in flowing droplet trains. Excited longitudinal modes exhibit a short-lived cascade of pairs of laterally displacing droplets. Transversely excited modes obey the dispersion relation of microfluidic phonons and induce a coupling between longitudinal and transverse modes, whose origin is the hydrodynamic interaction of the droplets with the confining walls. Moreover, we investigate the long-time behaviour of the oscillations and discuss possible mechanisms for the onset of instabilities. Our findings demonstrate that the collective dynamics of microfluidic droplet ensembles can be studied particularly well in dense and confined systems. Experimentally, the ability to control microfluidic droplets may allow the modulation of the refractive index of optofluidic crystals, which is a promising approach for the production of dynamically programmable metamaterials.

Collective dynamics in a binary mixture of hydrodynamically coupled microrotors

We study, numerically, the collective dynamics of self-rotating nonaligning particles by considering a monolayer of spheres driven by constant clockwise or counterclockwise torques. We show that hydrodynamic interactions alter the emergence of large-scale dynamical patterns compared to those observed in dry systems. In dilute suspensions, the flow stirred by the rotors induces clustering of opposite-spin rotors, while at higher densities same-spin rotors phase separate. Above a critical rotor density, dynamic hexagonal crystals form. Our findings underscore the importance of inclusion of the many-body, long-range hydrodynamic interactions in predicting the phase behavior of active particles.

Waves and instability in a one-dimensional microfluidic array

Motion in a one-dimensional (1-D) microfluidic array is simulated. Water droplets, dragged by flowing oil, are arranged in a single row. Due to their hydrodynamic interactions, the spacing between these droplets oscillates with a wave-like motion that is longitudinal or transverse. The simulation yields wave spectra that agree well with experiment. The wave-like motion has an instability which is confirmed to arise from nonlinearities in the interaction potential. The instability's growth is spatially localized. By selecting an appropriate correlation function, the interaction between the longitudinal and transverse waves is described.

Active and driven hydrodynamic crystals

Motivated by the experimental ability to produce monodisperse particles in microfluidic devices, we study theoretically the hydrodynamic stability of driven and active crystals. We first recall the theoretical tools allowing to quantify the dynamics of elongated particles in a confined fluid. In this regime hydrodynamic interactions between particles arise from a superposition of potential dipolar singularities. We exploit this feature to derive the equations of motion for the particle positions and orientations. After showing that all five planar Bravais lattices are stationary solutions of the equations of motion, we consider separately the case where the particles are passively driven by an external force, and the situation where they are self-propelling. We first demonstrate that phonon modes propagate in driven crystals, which are always marginally stable. The spatial structures of the eigenmodes depend solely on the symmetries of the lattices, and on the orientation of the driving force. For active crystals, the stability of the particle positions and orientations depends not only on the symmetry of the crystals but also on the perturbation wavelengths and on the crystal density. Unlike unconfined fluids, the stability of active crystals is independent of the nature of the propulsion mechanism at the single-particle level. The square and rectangular lattices are found to be linearly unstable at short wavelengths provided the volume fraction of the crystals is high enough. Differently, hexagonal, oblique, and face-centered crystals are always unstable. Our work provides a theoretical basis for future experimental work on flowing microfluidic crystals.

Layering instability in a confined suspension flow

We show [J. Fluid Mech. 592, 447 (2007)] that swapping (reversing) trajectories in confined suspension flows prevent collisions between particles approaching each other in adjacent streamlines. Here we demonstrate that by inducing layering this hydrodynamic mechanism changes the microstructure of suspensions in a confined Couette flow. Layers occur either in the near-wall regions or span the whole channel width, depending on the strength of the swapping-trajectory effect. While our theory focuses on dilute suspensions, we postulate that this new hydrodynamic mechanism controls the formation of a layered microstructure in a wide range of densities.

Scattering and nonlinear bound states of hydrodynamically coupled particles in a narrow channel

We model a pair of hydrodynamically interacting particles confined in a channel with thin rectangular cross section. We find that the particles have a finite region of attraction, which arises from the screening of dipolar hydrodynamic interactions by the side walls. Outside this region, the two particles break apart and scatter; inside, they oscillate together as an effectively free quasiparticle. We demonstrate that modulation of channel geometry provides a means to irreversibly manipulate bound pairs.

Hydrodynamic coupling of spherical particles to a planar fluid-fluid interface: theoretical analysis

We have developed a new technique (based on our Cartesian-representation method) to describe hydrodynamic interactions of a spherical particle with an undeformable planar fluid-fluid interface under creeping-flow conditions. The interface can be either surfactant-free or covered with an incompressible surfactant monolayer. We consider the effect of surface incompressibility and surface viscosity on particle motion. The new algorithm allows to calculate particle mobility coefficients for hydrodynamically coupled particles, moving either on the same or on the opposite sides of the interface.

The intensity correlation function in evanescent wave scattering

As a first step toward the interpretation of dynamic light scattering with evanescent illumination from suspensions of interacting spheres, in order to probe their near wall dynamics, we develop a theory for the initial slope of the intensity autocorrelation function. An expression for the first cumulant is derived that is valid for arbitrary concentrations, which generalizes a well-known expression for the short-time, wave-vector dependent collective diffusion coefficient in bulk to the case where a wall is present. Explicit expressions and numerical results for the various contributions to the initial slope are obtained within a leading order virial expansion. The dependence of the initial slope on the components of the wave vector parallel and perpendicular to the wall, as well as the dependence on the evanescent-light penetration depth are discussed. For the hydrodynamic interactions between colloids and between the wall, which are essential for a correct description of the near-interface dynamics, we include both far-field and lubrication contributions. Lubrication contributions are essential to capture the dynamics as probed in experiments with small penetration depths. Simulations have been performed to verify the theory and to estimate the extent of the concentration range where the virial expansion is valid. The computer algorithm developed for this purpose will also be of future importance for the interpretation of experiments and to develop an understanding of near-interface dynamics, at high colloid concentrations.

Burgers shock waves and sound in a 2D microfluidic droplets ensemble

We investigate the collective motion of a two-dimensional disordered ensemble of droplets in a microfluidic channel far from equilibrium and at Reynolds number approximately 10(-4). The ensemble carries ultraslow shock waves and sound, propagating at approximately 100 microm s(-1) and superposed on diffusive droplets motion. These modes are induced by long-range hydrodynamic dipolar interactions between droplets, the result of the symmetry breaking flow. The modes obey the Burgers equation due to a local coupling between droplets velocity and number density. This stems from a singular effect of the channel sidewall boundaries upon summation of the hydrodynamic interaction in two dimensions.

Self-sustained motion of a train of haptotactic microcapsules

Using theory and simulation, we design a "train" of N microcapsules that undergoes self-sustained, directed motion along an adhesive surface in solution. The motion is initiated by the release of nanoparticles from a single "signaling" capsule at one end of the train. The released nanoparticles can bind to the underlying surface and thereby induce an adhesion gradient on the substrate. Through the combined effects of the self-imposed adhesion gradient and hydrodynamic interactions, the N microcapsules autonomously move in single file toward the region of greatest adhesion. At late times, this train reaches a steady-state velocity U, which decreases with train length as N(-1/2). We calculate the maximum length for which the train maintains this cooperative, autonomous motion.

Hydrodynamic interactions between two semiflexible inextensible filaments in Stokes flow

Hydrodynamic interactions between two semiflexible inextensible filaments are shown to have a significant impact on filament buckling and their subsequent motion in Stokesian fluids. In linear shear flow, hydrodynamic interactions lead to filament shear dispersion that depends on the filament aspect ratio and the initial filament separation. In linear extensional flow, hydrodynamic interactions lead to complex filament dynamics around the stagnation point. These results suggest that hydrodynamic interactions need to be taken into account to determine the self-diffusion of non-Brownian semiflexible filaments in a cellular flow [Y.-N. Young and M. J. Shelley, Phys. Rev. Lett. 99, 058303 (2007)].