The crucial role of adhesion in the transmigration of active droplets through interstitial orifices.

Active fluid droplets are a class of soft materials exhibiting autonomous motion sustained by an energy supply. Such systems have been shown to capture motility regimes typical of biological cells and are ideal candidates as building-block for the fabrication of soft biomimetic materials of interest in pharmacology, tissue engineering and lab on chip devices. While their behavior is well established in unconstrained environments, much less is known about their dynamics under strong confinement. Here, we numerically study the physics of a droplet of active polar fluid migrating within a microchannel hosting a constriction with adhesive properties, and report evidence of a striking variety of dynamic regimes and morphological features, whose properties crucially depend upon droplet speed and elasticity, degree of confinement within the constriction and adhesiveness to the pore. Our results suggest that non-uniform adhesion forces are instrumental in enabling the crossing through narrow orifices, in contrast to larger gaps where a careful balance between speed and elasticity is sufficient to guarantee the transition. These observations may be useful for improving the design of artificial micro-swimmers, of interest in material science and pharmaceutics, and potentially for cell sorting in microfluidic devices.

Microscale modelling of dielectrophoresis assembly processes.

This work presents a microscale approach for simulating the dielectrophoresis assembly of polarizable particles under an external electric field. The model is shown to capture interesting dynamical and topological features, such as the formation of chains of particles and their incipient aggregation into hierarchical structures. A quantitative characterization in terms of the number and size of these structures is also discussed. This computational model could represent a viable numerical tool to study the mechanical properties of particle-based hierarchical materials and suggest new strategies for enhancing their design and manufacture. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

The vortex-driven dynamics of droplets within droplets.

Understanding the fluid-structure interaction is crucial for an optimal design and manufacturing of soft mesoscale materials. Multi-core emulsions are a class of soft fluids assembled from cluster configurations of deformable oil-water double droplets (cores), often employed as building-blocks for the realisation of devices of interest in bio-technology, such as drug-delivery, tissue engineering and regenerative medicine. Here, we study the physics of multi-core emulsions flowing in microfluidic channels and report numerical evidence of a surprisingly rich variety of driven non-equilibrium states (NES), whose formation is caused by a dipolar fluid vortex triggered by the sheared structure of the flow carrier within the microchannel. The observed dynamic regimes range from long-lived NES at low core-area fraction, characterised by a planetary-like motion of the internal drops, to short-lived ones at high core-area fraction, in which a pre-chaotic motion results from multi-body collisions of inner drops, as combined with self-consistent hydrodynamic interactions. The onset of pre-chaotic behavior is marked by transitions of the cores from one vortex to another, a process that we interpret as manifestations of the system to maximize its entropy by filling voids, as they arise dynamically within the capsule.

Dynamics of polydisperse multiple emulsions in microfluidic channels.

Multiple emulsions are a class of soft fluid in which small drops are immersed within a larger one and stabilized over long periods of time by a surfactant. We recently showed that, if a monodisperse multiple emulsion is subject to a pressure-driven flow, a wide variety of nonequilibrium steady states emerges at late times, whose dynamics relies on a complex interplay between hydrodynamic interactions and multibody collisions among internal drops. In this work, we use lattice Boltzmann simulations to study the dynamics of polydisperse double emulsions driven by a Poiseuille flow within a microfluidic channel. Our results show that their behavior is critically affected by multiple factors, such as initial position, polydispersity index, and area fraction occupied within the emulsion. While at low area fraction inner drops may exhibit either a periodic rotational motion (at low polydispersity) or arrange into nonmotile configurations (at high polydispersity) located far from each other, at larger values of area fraction they remain in tight contact and move unidirectionally. This decisively conditions their close-range dynamics, quantitatively assessed through a time-efficiency-like factor. Simulations also unveil the key role played by the capsule, whose shape changes can favor the formation of a selected number of nonequilibrium states in which both motile and nonmotile configurations are found.

Soft channel formation and symmetry breaking in exotic active emulsions.

We use computer simulations to study the morphology and rheological properties of a bidimensional emulsion resulting from a mixture of a passive isotropic fluid and an active contractile polar gel, in the presence of a surfactant that favours the emulsification of the two phases. By varying the intensity of the contractile activity and of an externally imposed shear flow, we find three possible morphologies. For low shear rates, a simple lamellar state is obtained. For intermediate activity and shear rate, an asymmetric state emerges, which is characterized by shear and concentration banding at the polar/isotropic interface. A further increment in the active forcing leads to the self-assembly of a soft channel where an isotropic fluid flows between two layers of active material. We characterize the stability of this state by performing a dynamical test varying the intensity of the active forcing and shear rate. Finally, we address the rheological properties of the system by measuring the effective shear viscosity, finding that this increases as active forcing is increased-so that the fluid thickens with activity.

Modeling drug delivery from multiple emulsions.

We present a mechanistic model of drug release from a multiple emulsion into an external surrounding fluid. We consider a single multilayer droplet where the drug kinetics are described by a pure diffusive process through different liquid shells. The multilayer problem is described by a system of diffusion equations coupled via interlayer conditions imposing continuity of drug concentration and flux. Mass resistance is imposed at the outer boundary through the application of a surfactant at the external surface of the droplet. The two-dimensional problem is solved numerically by finite volume discretization. Concentration profiles and drug release curves are presented for three typical round-shaped (circle, ellipse, and bullet) droplets and the dependency of the solution on the mass transfer coefficient at the surface analyzed. The main result shows a reduced release time for an increased elongation of the droplets.

Lattice Boltzmann simulations capture the multiscale physics of soft flowing crystals.

The study of the underlying physics of soft flowing materials depends heavily on numerical simulations, due to the complex structure of the governing equations reflecting the competition of concurrent mechanisms acting at widely disparate scales in space and time. A full-scale computational modelling remains a formidable challenge since it amounts to simultaneously handling six or more spatial decades in space and twice as many in time. Coarse-grained methods often provide a viable strategy to significantly mitigate this issue, through the implementation of mesoscale supramolecular forces designed to capture the essential physics at a fraction of the computational cost of a full-detail description. Here, we review some recent advances in the design of a lattice Boltzmann mesoscale approach for soft flowing materials, inclusive of near-contact interactions (NCIs) between dynamic interfaces, as they occur in high packing-fraction soft flowing crystals. The method proves capable of capturing several aspects of the rheology of soft flowing crystals, namely, (i) a 3/2 power-law dependence of the dispersed phase flow rate on the applied pressure gradient, (ii) the structural transition between an ex-two and ex-one (bamboo) configurations with the associated drop of the flow rate, (iii) the onset of interfacial waves once NCI is sufficiently intense. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Microvorticity fluctuations affect the structure of thin fluid films.

The dynamic interaction of complex fluid interfaces is highly sensitive to near-contact interactions occurring at the scale of ten of nanometers. Such interactions are difficult to analyze because they couple self-consistently to the dynamic morphology of the evolving interface, as well as to the hydrodynamics of the interstitial fluid film. In this work, we show that, above a given magnitude threshold, near-contact interactions trigger nontrivial microvorticity patterns, which in turn affect the effective near-contact interactions, giving rise to persistent fluctuating ripples at the fluid interface. In such a regime, near-contact interactions may significantly affect the macroscopic arrangement of emulsion configurations, such as those arising in soft-flowing microfluidic crystals.

Rheology of active polar emulsions: from linear to unidirectional and inviscid flow, and intermittent viscosity.

The rheological behaviour of an emulsion made of an active polar component and an isotropic passive fluid is studied by lattice Boltzmann methods. Different flow regimes are found by varying the values of the shear rate and extensile activity (occurring, e.g., in microtubule-motor suspensions). By increasing the activity, a first transition occurs from the linear flow regime to spontaneous persistent unidirectional macro-scale flow, followed by another transition either to a (low shear) intermittent flow regime with the coexistence of states with positive, negative, and vanishing apparent viscosity, or to a (high shear) symmetric shear thinning regime. The different behaviours can be explained in terms of the dynamics of the polarization field close to the walls. A maximum entropy production principle selects the most likely states in the intermittent regime.

Curvature dynamics and long-range effects on fluid-fluid interfaces with colloids.

We investigate the dynamics of a phase-separating binary fluid, containing colloidal dumbbells anchored to the fluid-fluid interface. Extensive lattice Boltzmann-immersed boundary method simulations reveal that the presence of soft dumbbells can significantly affect the curvature dynamics of the interface between phase-separating fluids, even though the coarsening dynamics is left nearly unchanged. In addition, our results show that the curvature dynamics exhibits distinct non-local effects, which might be exploited for the design of new soft mesoscale materials. We point out that the inspection of the statistical dynamics of the curvature can disclose new insights into local inhomogeneities of the binary fluid configuration, as a function of the volume fraction and aspect ratio of the dumbbells.

Lamellar ordering, droplet formation and phase inversion in exotic active emulsions.

We study numerically the behaviour of a two-dimensional mixture of a passive isotropic fluid and an active polar gel, in the presence of a surfactant favouring emulsification. Focussing on parameters for which the underlying free energy favours the lamellar phase in the passive limit, we show that the interplay between nonequilibrium and thermodynamic forces creates a range of multifarious exotic emulsions. When the active component is contractile (e.g., an actomyosin solution), moderate activity enhances the efficiency of lamellar ordering, whereas strong activity favours the creation of passive droplets within an active matrix. For extensile activity (occurring, e.g., in microtubule-motor suspensions), instead, we observe an emulsion of spontaneously rotating droplets of different size. By tuning the overall composition, we can create high internal phase emulsions, which undergo sudden phase inversion when activity is switched off. Therefore, we find that activity provides a single control parameter to design composite materials with a strikingly rich range of morphologies.

Lattice Boltzmann study of chemically-driven self-propelled droplets.

We numerically study the behavior of self-propelled liquid droplets whose motion is triggered by a Marangoni-like flow. This latter is generated by variations of surfactant concentration which affect the droplet surface tension promoting its motion. In the present paper a model for droplets with a third amphiphilic component is adopted. The dynamics is described by Navier-Stokes and convection-diffusion equations, solved by the lattice Boltzmann method coupled with finite-difference schemes. We focus on two cases. First, the study of self-propulsion of an isolated droplet is carried on and, then, the interaction of two self-propelled droplets is investigated. In both cases, when the surfactant migrates towards the interface, a quadrupolar vortex of the velocity field forms inside the droplet and causes the motion. A weaker dipolar field emerges instead when the surfactant is mainly diluted in the bulk. The dynamics of two interacting droplets is more complex and strongly depends on their reciprocal distance. If, in a head-on collision, droplets are close enough, the velocity field initially attracts them until a motionless steady state is achieved. If the droplets are vertically shifted, the hydrodynamic field leads to an initial reciprocal attraction followed by a scattering along opposite directions. This hydrodynamic interaction acts on a separation of some droplet radii otherwise it becomes negligible and droplets motion is only driven by the Marangoni effect. Finally, if one of the droplets is passive, this latter is generally advected by the fluid flow generated by the active one.

Switching dynamics in cholesteric liquid crystal emulsions.

In this work we numerically study the switching dynamics of a 2D cholesteric emulsion droplet immersed in an isotropic fluid under an electric field, which is either uniform or rotating with constant speed. The overall dynamics depend strongly on the magnitude and on the direction (with respect to the cholesteric axis) of the applied field, on the anchoring of the director at the droplet surface and on the elasticity. If the surface anchoring is homeotropic and a uniform field is parallel to the cholesteric axis, the director undergoes deep elastic deformations and the droplet typically gets stuck into metastable states which are rich in topological defects. When the surface anchoring is tangential, the effects due to the electric field are overall less dramatic, as a small number of topological defects form at equilibrium. The application of the field perpendicular to the cholesteric axis usually has negligible effects on the defect dynamics. The presence of a rotating electric field of varying frequency fosters the rotation of the defects and of the droplet as well, typically at a lower speed than that of the field, due to the inertia of the liquid crystal. If the surface anchoring is homeotropic, a periodic motion is found. Our results represent a first step to understand the dynamical response of a cholesteric droplet under an electric field and its possible application in designing novel liquid crystal-based devices.

Shear dynamics of an inverted nematic emulsion.

Here we study theoretically the dynamics of a 2D and a 3D isotropic droplet in a nematic liquid crystal under a shear flow. We find a large repertoire of possible nonequilibrium steady states as a function of the shear rate and of the anchoring of the nematic director field at the droplet surface. We first discuss homeotropic anchoring. For weak anchoring, we recover the typical behaviour of a sheared isotropic droplet in a binary fluid, which rotates, stretches and can be broken by the applied flow. For intermediate anchoring, new possibilities arise due to elastic effects in the nematic fluid. We find that in this regime the 2D droplet can tilt and move in the flow, or tumble incessantly at the centre of the channel. For sufficiently strong anchoring, finally, one or both of the topological defects which form close to the surface of the isotropic droplet in equilibrium detach from it and get dragged deep into the nematic state by the flow. In 3D, instead, the Saturn ring associated with the normal anchoring disclination line can be deformed and shifted downstream by the flow, but remains always localized in the proximity of the droplet, at least for the parameter range we explored. Tangential anchoring in 2D leads to a different dynamic response, as the boojum defects characteristic of this situation can unbind from the droplet under a weaker shear with respect to the normal anchoring case. Our results should stimulate further experiments with inverted liquid crystal emulsions under shear, as most of the predictions can be testable in principle by monitoring the evolution of liquid crystalline orientation patterns or by tracking the position and shape of the droplet over time.

A minimal physical model captures the shapes of crawling cells.

Cell motility in higher organisms (eukaryotes) is crucial to biological functions ranging from wound healing to immune response, and also implicated in diseases such as cancer. For cells crawling on hard surfaces, significant insights into motility have been gained from experiments replicating such motion in vitro. Such experiments show that crawling uses a combination of actin treadmilling (polymerization), which pushes the front of a cell forward, and myosin-induced stress (contractility), which retracts the rear. Here we present a simplified physical model of a crawling cell, consisting of a droplet of active polar fluid with contractility throughout, but treadmilling connected to a thin layer near the supporting wall. The model shows a variety of shapes and/or motility regimes, some closely resembling cases seen experimentally. Our work strongly supports the view that cellular motility exploits autonomous physical mechanisms whose operation does not need continuous regulatory effort.

Spontaneous motility of passive emulsion droplets in polar active gels.

We study by computer simulations the dynamics of a droplet of passive, isotropic fluid, embedded in a polar active gel. The latter represents a fluid of active force dipoles, which exert either contractile or extensile stresses on their surroundings, modelling for instance a suspension of cytoskeletal filaments and molecular motors. When the polarisation of the active gel is anchored normal to the droplet at its surface, the nematic elasticity of the active gel drives the formation of a hedgehog defect; this defect then drives an active flow which propels the droplet forward. In an extensile gel, motility can occur even with tangential anchoring, which is compatible with a defect-free polarisation pattern. In this case, upon increasing activity the droplet first rotates uniformly, and then undergoes a discontinuous nonequilibrium transition into a translationally motile state, powered by bending deformations in the surrounding active medium.

Switching hydrodynamics in liquid crystal devices: a simulation perspective.

In liquid crystal devices it is important to understand the physics underlying their switching between different states, which is usually achieved by applying or removing an electric field. Flow is known to be a key determinant of the timescales and pathways of the switching kinetics. Incorporating hydrodynamic effects into theories for liquid crystal devices is therefore important; however this is also highly non-trivial, and typically requires the use of accurate numerical methods. Here, we review some recent advances in our theoretical understanding of the dynamics of switching in liquid crystal devices, mainly gained through computer simulations. These results, as we shall show, uncover interesting new physics, and may be important for future applications.

Bistable defect structures in blue phase devices.

Blue phases are liquid crystals made up by networks of defects, or disclination lines. While existing phase diagrams show a striking variety of competing metastable topologies for these networks, very little is known as to how to kinetically reach a target structure, or how to switch from one to the other, which is of paramount importance for devices. We theoretically identify two confined blue phase I systems in which by applying an appropriate series of electric field it is possible to select one of two bistable defect patterns. Our results may be used to realize new generation and fast switching energy-saving bistable devices in ultrathin surface treated blue phase I wafers.

Thermal and hydrodynamic effects in the ordering of lamellar fluids.

Phase separation in a complex fluid with lamellar order has been studied in the case of cold thermal fronts propagating diffusively from external walls. The velocity hydrodynamic modes are taken into account by coupling the convection-diffusion equation for the order parameter to a generalized Navier-Stokes equation. The dynamical equations are simulated by implementing a hybrid method based on a lattice Boltzmann algorithm coupled to finite difference schemes. Simulations show that the ordering process occurs with morphologies depending on the speed of the thermal fronts or, equivalently, on the value of the thermal conductivity ξ. At large values of ξ, as in instantaneous quenching, the system is frozen in entangled configurations at high viscosity while it consists of grains with well-ordered lamellae at low viscosity. By decreasing the value of ξ, a regime with very ordered lamellae parallel to the thermal fronts is found. At very low values of ξ the preferred orientation is perpendicular to the walls in d=2, while perpendicular order is lost moving far from the walls in d=3.

Phase separation of binary fluids with dynamic temperature.

Phase separation of binary fluids quenched by contact with cold external walls is considered. Navier-Stokes, convection-diffusion, and energy equations are solved by lattice Boltzmann method coupled with finite-difference schemes. At high viscosity, different morphologies are observed by varying the thermal diffusivity. In the range of thermal diffusivities with domains growing parallel to the walls, temperature and phase separation fronts propagate toward the inner of the system with power-law behavior. At low viscosity hydrodynamics favors rounded shapes, and complex patterns with different length scales appear. Off-symmetrical systems behave similarly but with more ordered configurations.