Lattice Boltzmann simulation of deformable fluid-filled bodies: progress and perspectives.

With the rapid development of studies involving droplet microfluidics, drug delivery, cell detection, and microparticle synthesis, among others, many scientists have invested significant efforts to model the flow of these fluid-filled bodies. Motivated by the intricate coupling between hydrodynamics and the interactions of fluid-filled bodies, several methods have been developed. The objective of this review is to present a compact foundation of the methods used in the literature in the context of lattice Boltzmann methods. For hydrodynamics, we focus on the lattice Boltzmann method due to its specific ability to treat time- and spatial-dependent boundary conditions and to incorporate new physical models in a computationally efficient way. We split the existing methods into two groups with regard to the interfacial boundary: fluid-structure and fluid-fluid methods. The fluid-structure methods are characterised by the coupling between fluid dynamics and mechanics of the flowing body, often used in applications involving membranes and similar flexible solid boundaries. We further divide fluid-structure-based methods into two subcategories, those which treat the fluid-structure boundary as a continuum medium and those that treat it as a discrete collection of individual springs and particles. Next, we discuss the fluid-fluid methods, particularly useful for the simulations of fluid-fluid interfaces. We focus on models for immiscible droplets and their interaction in a suspending fluid and describe benchmark tests to validate the models for fluid-filled bodies.

Dispersion of activity at an active-passive nematic interface.

Efficient nutrient mixing is crucial for the survival of bacterial colonies and other living systems known as active nematics. However, the dynamics of this mixing is non-trivial as there is a coupling between nutrients concentration and velocity field. To address this question, we solve the hydrodynamic equation for active nematics to model the bacterial swarms coupled to an advection-diffusion equation for the activity field, which is proportional to the concentration of nutrients. At the interface between active and passive nematics the activity field is transported by the interfacial flows and in turn it modifies them through the generation of active stresses. We find that the dispersion of this conserved activity field is subdiffusive due to the emergence of a barrier of negative defects at the active-passive interface, which hinders the propagation of the motile positive defects.

The role of disorder in the motion of chiral active particles in the presence of obstacles.

The presence of obstacles is intuitively expected to hinder the diffusive transport of active particles. However, for chiral active particles, a low density of obstacles near a surface can enhance their diffusive behavior. Here, we study numerically the role that disorder plays in determining the transport dynamics of chiral active particles on surfaces with obstacles. We consider different densities of regularly spaced obstacles and distinct types of disorder: noise in the dynamics of the particle, quenched noise in the positions of the obstacles as well as obstacle size polydispersity. We show that, depending on the type and strength of the disorder, the presence of obstacles can either enhance or hinder transport, and discuss implications for the control of active transport in disordered media.

Model for active particles confined in a two-state micropattern.

We propose a model, based on active Brownian particles, for the dynamics of cells confined in a two-state micropattern, composed of two rectangular boxes connected by a bridge, and investigate the transition statistics. A transition between boxes occurs when the active particle crosses the center of the bridge, and the time between subsequent transitions is the dwell time. By assuming that the rotational diffusion time τ is a function of the position, some experimental observations are qualitatively recovered as, for example, the shape of the survival function. τ controls the transition from a ballistic regime at short time scales to a diffusive regime at long time scales, with an effective diffusion coefficient proportional to τ. For small values of τ, the dwell time is determined by the characteristic diffusion timescale which is constant for very low values of τ, when the rotational diffusion is much faster than the translational one and decays with τ for intermediate values of τ. For large values of τ, the interaction with the walls dominates and the particle stays mostly at the corners of the boxes increasing the dwell time. We find that there is an optimal τ for which the dwell time is minimal and its value can be tuned by changing the geometry of the pattern.

Dynamics of flowing 2D skyrmions.

We investigate, numerically, the effects of externally imposed material flows on the structure and temporal evolution of liquid crystal (LC) skyrmions. The dynamics of a 2D system of skyrmions is modeled using the Ericksen-Leslie theory, which is based on two coupled equations, one for material flow and the other for the director field. As the time scales of the velocity and director fields differ by several orders of magnitude for realistic values of the system parameters, we have simplified the calculations by assuming that the velocity relaxes instantaneously when compared to the relaxation of the director field. Thus, we have used a finite-differences method known as artificial compressibility with adaptive time step to solve the velocity field and a fourth-order Runge-Kutta method for the director field. We characterized the skyrmion shape or configuration as a function of the time and the average velocity of the flow field. We found that for velocities above a certain threshold, the skyrmions stretch in the direction perpendicular to the flow, by contrast to the regime of weak flows where the skyrmions stretch along the streamlines of the flow field. These two regimes are separated by an abrupt (first-order) dynamical transition, which is robust with respect to e.g., the LC elastic anisotropy. Additionally, we have found how the presence of a second skyrmion affects the evolution of the shape of the skyrmions, by comparing the evolution of pairs of skyrmions to the evolution of a single-skyrmion.

Director alignment at the nematic-isotropic interface: elastic anisotropy and active anchoring.

Activity in nematics drives interfacial flows that lead to preferential alignment that is tangential or planar for extensile systems (pushers) and perpendicular or homeotropic for contractile ones (pullers). This alignment is known as active anchoring and has been reported for a number of systems and described using active nematic hydrodynamic theories. The latter are based on the one-elastic constant approximation, i.e. they assume elastic isotropy of the underlying passive nematic. Real nematics, however, have different elastic constants, which lead to interfacial anchoring. In this paper, we consider elastic anisotropy in multiphase and multicomponent hydrodynamic models of active nematics and investigate the competition between the interfacial alignment driven by the elastic anisotropy of the passive nematic and the active anchoring. We start by considering systems with translational invariance to analyse the alignment at flat interfaces and, then, consider two-dimensional systems and active nematic droplets. We investigate the competition of the two types of anchoring over a wide range of the other parameters that characterize the system. The results of the simulations reveal that the active anchoring dominates except at very low activities, when the interfaces are static. In addition, we found that the elastic anisotropy does not affect the dynamics but changes the active length that becomes anisotropic. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Phase behavior of a binary mixture of patchy colloids: Effect of particle size and gravity.

We study theoretically the effect of size difference and that of gravity in the phase behavior of a binary mixture of patchy particles. The species, 2A and 3B, have two A and three B patches, respectively, and only bonds between patches A and B (AB bonds) are allowed. This model describes colloidal systems where the aggregation of particles (3B) is mediated and controlled by a second species, the linkers (2A) to which they bind strongly. Thermodynamic calculations are performed using Wertheim's perturbation theory with a hard sphere reference term that accounts for the difference in the size of the two species. Percolation lines are determined employing a generalized Flory-Stockmayer theory, and the effects of gravity are included through a local density approximation. The bulk phase diagrams are calculated, and all the stacking sequences generated in the presence of gravity are determined and classified in a stacking diagram. The relative size of the particles can be used to control the phase behavior of the mixture. An increase in the size of particles 3B, relative to the size of the linkers 2A, is found to promote mixing while keeping the percolating structures and, in certain cases, leads to changes in the stacking sequence under gravity.

Propagation of active nematic-isotropic interfaces on substrates.

Motivated by results for the propagation of active-passive interfaces of bacterial Serratia marcescens swarms [Nat. Commun., 2018, 9, 5373], we used a hydrodynamic multiphase model to investigate the propagation of interfaces of active nematics on substrates. We characterized the active nematic phase of the model through the calculation of the spatial and temporal auto correlation functions and the energy spectrum and discussed its description of the statistical dynamics of the swarms reported in the experiment. We then studied the propagation of circular and flat active-passive interfaces. We found that the closing time of the circular passive domain decays quadratically with the activity and that the structure factor of the flat interface is similar to that reported for the swarms, with an activity dependent exponent. Finally, the effect of the substrate friction was investigated. We found an activity dependent threshold, above which the turbulent active nematic forms isolated islands that shrink until the system becomes isotropic and below which the active nematic expands, with a well defined propagating interface. We also found that the interface becomes static in the presence of a friction gradient.

Ordering of binary colloidal crystals by random potentials.

Structural defects are ubiquitous in condensed matter, and not always a nuisance. For example, they underlie phenomena such as Anderson localization and hyperuniformity, and they are now being exploited to engineer novel materials. Here, we show experimentally that the density of structural defects in a 2D binary colloidal crystal can be engineered with a random potential. We generate the random potential using an optical speckle pattern, whose induced forces act strongly on one species of particles (strong particles) and weakly on the other (weak particles). Thus, the strong particles are more attracted to the randomly distributed local minima of the optical potential, leaving a trail of defects in the crystalline structure of the colloidal crystal. While, as expected, the crystalline ordering initially decreases with an increasing fraction of strong particles, the crystalline order is surprisingly recovered for sufficiently large fractions. We confirm our experimental results with particle-based simulations, which permit us to elucidate how this non-monotonic behavior results from the competition between the particle-potential and particle-particle interactions.

Active nematic-isotropic interfaces in channels.

We use numerical simulations to investigate the hydrodynamic behavior of the interface between nematic (N) and isotropic (I) phases of a confined active liquid crystal. At low activities, a stable interface with constant shape and velocity is observed separating the two phases. For nematics in homeotropic channels, the velocity of the interface at the NI transition increases from zero (i) linearly with the activity for contractile systems and (ii) quadratically for extensile ones. Interestingly, the nematic phase expands for contractile systems while it contracts for extensile ones, as a result of the active forces at the interface. Since both activity and temperature affect the stability of the nematic, for active nematics in the stable regime the temperature can be tuned to observe static interfaces, providing an operational definition for the coexistence of active nematic and isotropic phases. At higher activities, beyond the stable regime, an interfacial instability is observed for extensile nematics. In this regime defects are nucleated at the interface and move away from it. The dynamics of these defects is regular and persists asymptotically for a finite range of activities. We used an improved hybrid model of finite differences and the lattice Boltzmann method with a multi-relaxation-time collision operator, the accuracy of which allowed us to characterize the dynamics of the distinct interfacial regimes.

Annealing cycles and the self-organization of functionalized colloids.

The self-assembly of functionalized (patchy) particles with directional interactions into target structures is still a challenge, despite the significant experimental advances in their synthesis. Self-assembly pathways are typically characterized by high energy barriers that hinder access to stable (equilibrium) structures. A possible strategy to tackle this challenge is to perform annealing cycles. By periodically switching on and off the inter-particle bonds, one expects to smooth-out the kinetic pathways and favor the assembly of targeted structures. Preliminary results have shown that the efficiency of annealing cycles depends strongly on their frequency. Here, we study numerically how this frequency-dependence scales with the strength of the directional interactions (size of the patch σ). We use analytical arguments to show that the scaling results from the statistics of a random walk in configurational space.

Demixing of active particles in the presence of external fields.

Self-propelled active particles are inherently out of equilibrium as they collect energy from their surroundings and transform it into directed motion. A recent theoretical study suggests that binary mixtures of active particles with distinct effective diffusion coefficients exhibit dynamical demixing when their diffusion coefficients differ by more than one order of magnitude. Here, we show that this difference may be reduced drastically in the presence of external fields even when the response to the field is the same for both species. We investigate this demixing as a function of the ratio of the diffusion coefficients and discuss the implications of the results for active systems.

Effect of curvature on cholesteric liquid crystals in toroidal geometries.

The confinement of liquid crystals inside curved geometries leads to exotic structures, with applications ranging from biosensors to optical switches and privacy windows. Here we study how curvature affects the alignment of a cholesteric liquid crystal. We model the system on the mesoscale using the Landau-de Gennes model. Our study was performed in three stages, analyzing different curved geometries from cylindrical walls and pores, to toroidal domains, in order to isolate the curvature effects. Our results show that the stresses introduced by the curvature influence the orientation of the liquid crystal molecules, and cause distortions in the natural periodicity of the cholesteric that depend on the radius of curvature, on the pitch, and on the dimensions of the system. In particular, the cholesteric layers of toroidal droplets exhibit a symmetry breaking not seen in cylindrical pores and that is driven by the additional curvature.

Nematic liquid crystals on sinusoidal channels: the zigzag instability.

Substrates which are chemically or topographically patterned induce a variety of liquid crystal textures. The response of the liquid crystal to competing surface orientations, typical of patterned substrates, is determined by the anisotropy of the elastic constants and the interplay of the relevant lengths scales, such as the correlation length and the surface geometrical parameters. Transitions between different textures, usually with different symmetries, may occur under a wide range of conditions. We use the Landau-de Gennes free energy to investigate the texture of nematics in sinusoidal channels with parallel anchoring bounded by nematic-air interfaces that favour perpendicular (hometropic) anchoring. In micron size channels 5CB was observed to exhibit a non-trivial texture characterized by a disclination line, within the channel, which is broken into a zigzag pattern. Our calculations reveal that when the elastic anisotropy of the nematic does not favour twist distortions the defect is a straight disclination line that runs along the channel, which breaks into a zigzag pattern with a characteristic period, when the twist elastic constant becomes sufficiently small when compared to the splay and bend constants. The transition occurs through a twist instability that drives the defect line to rotate from its original position. The interplay between the energetically favourable twist distortions that induce the defect rotation and the liquid crystal anchoring at the surfaces leads to the zigzag pattern. We investigate in detail the dependence of the periodicity of the zigzag pattern on the geometrical parameters of the sinusoidal channels, which in line with the experimental results is found to be non-linear.

Nonequilibrium self-organization of colloidal particles on substrates: adsorption, relaxation, and annealing.

Colloidal particles are considered ideal building blocks to produce materials with enhanced physical properties. The state-of-the-art techniques for synthesizing these particles provide control over shape, size, and directionality of the interactions. In spite of these advances, there is still a huge gap between the synthesis of individual components and the management of their spontaneous organization towards the desired structures. The main challenge is the control over the dynamics of self-organization. In their kinetic route towards thermodynamically stable structures, colloidal particles self-organize into intermediate (mesoscopic) structures that are much larger than the individual particles and become the relevant units for the dynamics. To follow the dynamics and identify kinetically trapped structures, one needs to develop new theoretical and numerical tools. Here we discuss the self-organization of functionalized colloids (also known as patchy colloids) on attractive substrates. We review our recent results on the adsorption and relaxation and explore the use of annealing cycles to overcome kinetic barriers and drive the relaxation towards the targeted structures.

Pattern-induced anchoring transitions in nematic liquid crystals.

In this paper we revisit the problem of a nematic liquid crystal in contact with patterned substrates. The substrate is modelled as a periodic array of parallel infinite grooves of well-defined cross-section sculpted on a chemically homogeneous substrate which favours local homeotropic anchoring of the nematic. We consider three cases: a sawtooth, a crenellated and a sinusoidal substrate. We analyse this problem within the modified Frank-Oseen formalism. We argue that, for substrate periodicities much larger than the extrapolation length, the existence of different nematic textures with distinct far-field orientations, as well as the anchoring transitions between them, are associated with the presence of topological defects either on or close to the substrate. For the sawtooth and sinusoidal cases, we observe a homeotropic to planar anchoring transition as the substrate roughness increases. On the other hand, a homeotropic to oblique anchoring transition is observed for crenellated substrates. In this case, the anchoring phase diagram shows a complex dependence on the substrate roughness and substrate anchoring strength.

Percolation in binary and ternary mixtures of patchy colloids.

We investigate percolation in binary and ternary mixtures of patchy colloidal particles theoretically and using Monte Carlo simulations. Each particle has three identical patches, with distinct species having different types of patch. Theoretically we assume tree-like clusters and calculate the bonding probabilities using Wertheim's first-order perturbation theory for association. For ternary mixtures, we find up to eight fundamentally different percolated states. The states differ in terms of the species and pairs of species that have percolated. The strongest gel is a trigel or tricontinuous gel, in which each of the three species has percolated. The weakest gel is a mixed gel in which all of the particles have percolated, but none of the species percolates by itself. The competition between entropy of mixing and internal energy of bonding determines the stability of each state. Theoretical and simulation results are in very good agreement. The only significant difference is the temperature at the percolation threshold, which is overestimated by the theory due to the absence of correlations between bonds in the theoretical description.

Wetting of cholesteric liquid crystals.

We investigate theoretically the wetting properties of cholesteric liquid crystals at a planar substrate. If the properties of substrate and of the interface are such that the cholesteric layers are not distorted, the wetting properties are similar to those of a nematic liquid crystal. If, on the other hand, the anchoring conditions force the distortion of the liquid crystal layers the wetting properties are altered, the free cholesteric-isotropic interface is non-planar and there is a layer of topological defects close to the substrate. These deformations can either promote or hinder the wetting of the substrate by a cholesteric, depending on the properties of the cholesteric liquid crystal.

Self-assembly of colloidal bands driven by a periodic external field.

We study the formation of bands of colloidal particles driven by periodic external fields. Using Brownian dynamics, we determine the dependence of the band width on the strength of the particle interactions and on the intensity and periodicity of the field. We also investigate the switching (field-on) dynamics and the relaxation times as a function of the system parameters. The observed scaling relations were analyzed using a simple dynamic density-functional theory of fluids.

The effect of anchoring on the nematic flow in channels.

Understanding the flow of liquid crystals in microfluidic environments plays an important role in many fields, including device design and microbiology. We perform hybrid lattice-Boltzmann simulations of a nematic liquid crystal flowing under an applied pressure gradient in two-dimensional channels with various anchoring boundary conditions at the substrate walls. We investigate the relationship between the flow rate and the pressure gradient and the corresponding profile of the nematic director, and find significant departures from the linear Poiseuille relationship. We also identify a morphological transition in the director profile and explain this in terms of an instability in the dynamical equations. We examine the qualitative and quantitative effects of changing the type and strength of the anchoring. Understanding such effects may provide a useful means of quantifying the anchoring of a substrate by measuring its flow properties.