Flow states of two dimensional active gels driven by external shear.

Using a minimal hydrodynamic model, we theoretically and computationally study the Couette flow of active gels in straight and annular two-dimensional channels subject to an externally imposed shear. The gels are isotropic in the absence of externally- or activity-driven shear, but have nematic order that increases with shear rate. Using the finite element method, we determine the possible flow states for a range of activities and shear rates. Linear stability analysis of an unconfined gel in a straight channel shows that an externally imposed shear flow can stabilize an extensile fluid that would be unstable to spontaneous flow in the absence of the shear flow, and destabilize a contractile fluid that would be stable against spontaneous flow in the absence of shear flow. These results are in rough agreement with the stability boundaries between the base shear flow state and the nonlinear flow states that we find numerically for a confined active gel. For extensile fluids, we find three kinds of nonlinear flow states in the range of parameters we study: unidirectional flows, oscillatory flows, and dancing flows. To highlight the activity-driven spontaneous component of the nonlinear flows, we characterize these states by the average volumetric flow rate and the wall stress. For contractile fluids, we only find the linear shear flow and a nonlinear unidirectional flow in the range of parameters that we studied. For large magnitudes of the activity, the unidirectional contractile flow develops a boundary layer. Our analysis of annular channels shows how curvature of the streamlines in the base flow affects the transitions among flow states.

Chiral fluid membranes with orientational order and multiple edges.

We carry out Monte Carlo simulations on fluid membranes with orientational order and multiple edges in the presence and absence of external forces. The membrane resists bending and has an edge tension, the orientational order couples with the membrane surface normal through a cost for tilting, and there is a chiral liquid crystalline interaction. In the absence of external forces, a membrane initialized as a vesicle will form a disk at low chirality, with the directors forming a smectic-A phase with alignment perpendicular to the membrane surface except near the edge. At large chirality a catenoid-like shape or a trinoid-like shape is formed, depending on the number of edges in the initial vesicle. This shape change is accompanied by cholesteric ordering of the directors and multiple π walls connecting the membrane edges and wrapping around the membrane neck. If the membrane is initialized instead in a cylindrical shape and stretched by an external force, it maintains a nearly cylindrical shape but additional liquid crystalline phases appear. For large tilt coupling and low chirality, a smectic-A phase forms where the directors are normal to the surface of the membrane. For lower values of the tilt coupling, a nematic phase appears at zero chirality with the average director oriented perpendicular to the long axis of the membrane, while for nonzero chirality a cholesteric phase appears. The π walls are tilt walls at low chirality and transition to twist walls as chirality is increased. We construct a continuum model of the director field to explain this behavior.

Controlling the shape and topology of two-component colloidal membranes.

Changes in the geometry and topology of self-assembled membranes underlie diverse processes across cellular biology and engineering. Similar to lipid bilayers, monolayer colloidal membranes have in-plane fluid-like dynamics and out-of-plane bending elasticity. Their open edges and micrometer-length scale provide a tractable system to study the equilibrium energetics and dynamic pathways of membrane assembly and reconfiguration. Here, we find that doping colloidal membranes with short miscible rods transforms disk-shaped membranes into saddle-shaped surfaces with complex edge structures. The saddle-shaped membranes are well approximated by Enneper's minimal surfaces. Theoretical modeling demonstrates that their formation is driven by increasing the positive Gaussian modulus, which in turn, is controlled by the fraction of short rods. Further coalescence of saddle-shaped surfaces leads to diverse topologically distinct structures, including shapes similar to catenoids, trinoids, four-noids, and higher-order structures. At long timescales, we observe the formation of a system-spanning, sponge-like phase. The unique features of colloidal membranes reveal the topological transformations that accompany coalescence pathways in real time. We enhance the functionality of these membranes by making their shape responsive to external stimuli. Our results demonstrate a pathway toward control of thin elastic sheets' shape and topology-a pathway driven by the emergent elasticity induced by compositional heterogeneity.

Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers.

We use theory and numerical computation to determine the shape of an axisymmetric fluid membrane with a resistance to bending and constant area. The membrane connects two rings in the classic geometry that produces a catenoidal shape in a soap film. In our problem, we find infinitely many branches of solutions for the shape and external force as functions of the separation of the rings, analogous to the infinite family of eigenmodes for the Euler buckling of a slender rod. Special attention is paid to the catenoid, which emerges as the shape of maximal allowable separation when the area is less than a critical area equal to the planar area enclosed by the two rings. A perturbation theory argument directly relates the tension of catenoidal membranes to the stability of catenoidal soap films in this regime. When the membrane area is larger than the critical area, we find additional cylindrical tether solutions to the shape equations at large ring separation, and that arbitrarily large ring separations are possible. These results apply for the case of vanishing Gaussian curvature modulus; when the Gaussian curvature modulus is nonzero and the area is below the critical area, the force and the membrane tension diverge as the ring separation approaches its maximum value. We also examine the stability of our shapes and analytically show that catenoidal membranes have markedly different stability properties than their soap film counterparts.

Deformation and orientational order of chiral membranes with free edges.

Motivated by experiments on colloidal membranes composed of chiral rod-like viruses, we use Monte Carlo methods to simulate these systems and determine the phase diagram for the liquid crystalline order of the rods and the membrane shape. We generalize the Lebwohl-Lasher model for a nematic with a chiral coupling to a curved surface with edge tension and a resistance to bending, and include an energy cost for tilting of the rods relative to the local membrane normal. The membrane is represented by a triangular mesh of hard beads joined by bonds, where each bead is decorated by a director. The beads can move, the bonds can reconnect and the directors can rotate at each Monte Carlo step. When the cost of tilt is small, the membrane tends to be flat, with the rods only twisting near the edge for low chiral coupling, and remaining parallel to the normal in the interior of the membrane. At high chiral coupling, the rods twist everywhere, forming a cholesteric state. When the cost of tilt is large, the emergence of the cholesteric state at high values of the chiral coupling is accompanied by the bending of the membrane into a saddle shape. Increasing the edge tension tends to flatten the membrane. These results illustrate the geometric frustration arising from the inability of a surface normal to have twist.

Shapes of fluid membranes with chiral edges.

We carry out Monte Carlo simulations of a colloidal fluid membrane with a free edge and composed of chiral rodlike viruses. The membrane is modeled by a triangular mesh of beads connected by bonds in which the bonds and beads are free to move at each Monte Carlo step. Since the constituent viruses are experimentally observed to twist only near the membrane edge, we use an effective energy that favors a particular sign of the geodesic torsion of the edge. The effective energy also includes the membrane bending stiffness, edge bending stiffness, and edge tension. We find three classes of membrane shapes resulting from the competition of the various terms in the free energy: branched shapes, chiral disks, and vesicles. Increasing the edge bending stiffness smooths the membrane edge, leading to correlations among the membrane normals at different points along the edge. The normalized power spectrum for edge displacements shows a peak with increasing preferred geodesic torsion. We also consider membrane shapes under an external force by fixing the distance between two ends of the membrane and finding the shape for increasing values of the distance between the two ends. As the distance increases, the membrane twists into a ribbon, with the force eventually reaching a plateau.

Force-Induced Formation of Twisted Chiral Ribbons.

We demonstrate that an achiral stretching force transforms disk-shaped colloidal membranes composed of chiral rods into twisted ribbons with handedness opposite the preferred twist of the rods. Using an experimental technique that enforces torque-free boundary conditions we simultaneously measure the force-extension curve and the ribbon shape. An effective theory that accounts for the membrane bending energy and uses geometric properties of the edge to model the internal liquid crystalline degrees of freedom explains both the measured force-extension curve and the force-induced twisted shape.

Topological structure and dynamics of three-dimensional active nematics.

Topological structures are effective descriptors of the nonequilibrium dynamics of diverse many-body systems. For example, motile, point-like topological defects capture the salient features of two-dimensional active liquid crystals composed of energy-consuming anisotropic units. We dispersed force-generating microtubule bundles in a passive colloidal liquid crystal to form a three-dimensional active nematic. Light-sheet microscopy revealed the temporal evolution of the millimeter-scale structure of these active nematics with single-bundle resolution. The primary topological excitations are extended, charge-neutral disclination loops that undergo complex dynamics and recombination events. Our work suggests a framework for analyzing the nonequilibrium dynamics of bulk anisotropic systems as diverse as driven complex fluids, active metamaterials, biological tissues, and collections of robots or organisms.

Stability of the interface of an isotropic active fluid.

We study the linear stability of an isotropic active fluid in three different geometries: a film of active fluid on a rigid substrate, a cylindrical thread of fluid, and a spherical fluid droplet. The active fluid is modeled by the hydrodynamic theory of an active nematic liquid crystal in the isotropic phase. In each geometry, we calculate the growth rate of sinusoidal modes of deformation of the interface. There are two distinct branches of growth rates; at long wavelength, one corresponds to the deformation of the interface, and one corresponds to the evolution of the liquid crystalline degrees of freedom. The passive cases of the film and the spherical droplet are always stable. For these geometries, a sufficiently large activity leads to instability. Activity also leads to propagating damped or growing modes. The passive cylindrical thread is unstable for perturbations with wavelength longer than the circumference. A sufficiently large activity can make any wavelength unstable, and again leads to propagating damped or growing modes. Our calculations are carried out for the case of zero Frank elasticity. While Frank elasticity is a stabilizing mechanism as it penalizes distortions of the order parameter tensor, we show that it has a small effect on the instabilities considered here.

Enhancement of Microorganism Swimming Speed in Active Matter.

We study a swimming undulating sheet in the isotropic phase of an active nematic liquid crystal. Activity changes the effective shear viscosity, reducing it to zero at a critical value of activity. Expanding in the sheet amplitude, we find that the correction to the swimming speed due to activity is inversely proportional to the effective shear viscosity. Our perturbative calculation becomes invalid near the critical value of activity; using numerical methods to probe this regime, we find that activity enhances the swimming speed by an order of magnitude compared to the passive case.

Chiral edge fluctuations of colloidal membranes.

We study edge fluctuations of a flat colloidal membrane comprised of a monolayer of aligned filamentous viruses. Experiments reveal that a peak in the spectrum of the in-plane edge fluctuations arises for sufficiently strong virus chirality. Accounting for internal liquid crystalline degrees of freedom by the length, curvature, and geodesic torsion of the edge, we calculate the spectrum of the edge fluctuations. The theory quantitatively describes the experimental data, demonstrating that chirality couples in-plane and out-of-plane edge fluctuations to produce the peak.

Wrinkling of a thin film on a nematic liquid-crystal elastomer.

Wrinkles commonly develop in a thin film deposited on a soft elastomer substrate when the film is subject to compression. Motivated by recent experiments [Agrawal et al., Soft Matter 8, 7138 (2012)]1744-683X10.1039/c2sm25734c that show how wrinkle morphology can be controlled by using a nematic elastomer substrate, we develop the theory of small-amplitude wrinkles of an isotropic film atop a nematic elastomer. The directors of the nematic elastomer are initially uniform. For uniaxial compression of the film along the direction perpendicular to the elastomer directors, the system behaves as a compressed film on an isotropic substrate. When the uniaxial compression is along the direction of nematic order, we find that the soft elasticity characteristic of liquid-crystal elastomers leads to a critical stress for wrinkling which is very small compared to the case of an isotropic substrate. We also determine the wavelength of the wrinkles at the critical stress and show how the critical stress and wavelength depend on substrate depth and the anisotropy of the polymer chains in the nematic elastomer.

Probing a self-assembled fd virus membrane with a microtubule.

The self-assembly of highly anisotropic colloidal particles leads to a rich variety of morphologies whose properties are just beginning to be understood. This article uses computer simulations to probe a particle-scale perturbation of a commonly studied colloidal assembly, a monolayer membrane composed of rodlike fd viruses in the presence of a polymer depletant. Motivated by experiments currently in progress, we simulate the interaction between a microtubule and a monolayer membrane as the microtubule "pokes" and penetrates the membrane face-on. Both the viruses and the microtubule are modeled as hard spherocylinders of the same diameter, while the depletant is modeled using ghost spheres. We find that the force exerted on the microtubule by the membrane is zero either when the microtubule is completely outside the membrane or when it has fully penetrated the membrane. The microtubule is initially repelled by the membrane as it begins to penetrate but experiences an attractive force as it penetrates further. We assess the roles played by translational and rotational fluctuations of the viruses and the osmotic pressure of the polymer depletant. We find that rotational fluctuations play a more important role than the translational ones. The dependence on the osmotic pressure of the depletant of the width and height of the repulsive barrier and the depth of the attractive potential well is consistent with the assumed depletion-induced attractive interaction between the microtubule and viruses. We discuss the relevance of these studies to the experimental investigations.

Interaction of chiral rafts in self-assembled colloidal membranes.

Colloidal membranes are monolayer assemblies of rodlike particles that capture the long-wavelength properties of lipid bilayer membranes on the colloidal scale. Recent experiments on colloidal membranes formed by chiral rodlike viruses showed that introducing a second species of virus with different length and opposite chirality leads to the formation of rafts--micron-sized domains of one virus species floating in a background of the other viruses [Sharma et al., Nature (London) 513, 77 (2014)]. In this article we study the interaction of such rafts using liquid crystal elasticity theory. By numerically minimizing the director elastic free energy, we predict the tilt angle profile for both a single raft and two rafts in a background membrane, and the interaction between two rafts as a function of their separation. We find that the chiral penetration depth in the background membrane sets the scale for the range of the interaction. We compare our results with the experimental data and find good agreement for the strength and range of the interaction. Unlike the experiments, however, we do not observe a complete collapse of the data when rescaled by the tilt angle at the raft edge.

Creating arbitrary arrays of two-dimensional topological defects.

An atomic force microscope was used to scribe a polyimide-coated substrate with complex patterns that serve as an alignment template for a nematic liquid crystal. By employing a sufficiently large density of scribe lines, two-dimensional topological defect arrays of arbitrary defect strength were patterned on the substrate. When used as the master surface of a liquid crystal cell, in which the opposing slave surface is treated for planar degenerate alignment, the liquid crystal adopts the pattern's alignment with a disclination line emanating at the defect core on one surface and terminating at the other surface.

Instability of flat disks with respect to the formation of twisted ribbons in smectic-A* monolayers.

Smectic-A* monolayers self-assembled from aqueous solutions of chiral fd viruses and a polymer depletant can assume a variety of shapes such as flat disks and twisted ribbons. A first order phase transition from a flat disk to a ribbon occurs upon lowering the concentration of polymer depletant or the temperature. A theoretical model based on the de Gennes model for the smectic-A phase, the Helfrich model of membrane elasticity, and a simple edge energy has been previously used to calculate the disk-ribbon phase diagram. In this paper we apply this model to the nucleation process of ribbons. First, we study the "rippled disks" that have been observed as precursors of ribbons. Using a model shape proposed by Meyer which includes rippling in both the in-plane and out-of-plane directions, we study the energetics of the disks as functions of the edge energy modulus (a measure of the polymer concentration) and the mean curvature modulus k. We find that as the edge energy modulus is reduced the radial size of the ripples grows rapidly in agreement with experimental observations. For small enough k we find that the out-of-plane size of the ripples grows but its value saturates at a fraction of the twist penetration depth, too small to be experimentally observable. For large k the membrane remains flat though rippled in the radial direction. Such membranes do not have negative Gaussian curvature and thus will not likely spawn twisted ribbons. We also study the creation of twisted ribbons produced by stretching the edge of a flat membrane in a localized region. In experiments using a pair of optical traps it has been observed that once the membrane has been sufficiently stretched a ribbon forms on the stretched edge. We study this process theoretically using a free energy consisting of the Helfrich and edge energies alone. We add a small ribbonlike perturbation to the protrusion produced by stretching and determine whether it is energetically favorable as a function of the size of the protrusion. In qualitative agreement with experiment we find a nonzero value for the critical size of the protrusion needed to make a ribbon energetically favorable, though the value we find is an order of magnitude lower than the experimental value possibly due to our neglect of the director field. As in the case of the rippled disks we find that the mean curvature energy acts as a barrier between the disk and twisted ribbon structures.

Theory of depletion-induced phase transition from chiral smectic- A twisted ribbons to semi-infinite flat membranes.

We consider a theoretical model for the chiral smectic A twisted ribbons observed in assemblies of fd viruses condensed by depletion forces. The depletion interaction is modeled by an edge energy assumed to be proportional to the depletant polymer in solution. Our model is based on the Helfrich energy for surface bending and the de Gennes model of chiral smectic A liquid crystals with twist penetration at the edge. We consider two variants of this model, one with the conventional Helfrich Gaussian curvature term, and a second with saddle-splay energy. A mean field analysis of both models yields a first-order phase transition between ribbons and semi-infinite flat membranes as the edge energy is varied. The phase transition line and tilt angle profile are found to be nearly identical for the two models; the pitch of the ribbon, however, does show some differences. Our model yields good qualitative agreement with experimental observations if the sign of the Gaussian curvature or saddle-splay modulus is chosen to favor negative Gaussian curvature.

Nematic cells with quasicrystalline-patterned alignment layers.

Nematic cells with surface anchoring fields exhibiting quasicrystalline symmetry can be fabricated using linear photopolymerizable polymers and suitable optics. Using the Lebwohl-Lasher model we carry out Monte Carlo simulations of a nematic cell where the top and bottom surfaces have an anchoring field with the symmetry of a periodic approximant to a Penrose lattice. We show that the anchoring field has point topological defects, nearly all with topological charge +/-1 . Our simulations, which assume infinitely strong anchoring of the nematic to the anchoring field, show that half-integer disclination lines emerge from the point defects and either traverse the thickness of the cell for small cell thicknesses, hug the patterned surfaces for large thicknesses, or combine these behaviors for intermediate thicknesses. We show that estimates of the values of the thicknesses separating these three behaviors can be obtained using the properties of Penrose tilings.

Direct measurement of the twist penetration length in a single smectic A layer of colloidal virus particles.

In the 1970s, deGennes discussed the fundamental geometry of smectic liquid crystals and established an analogy between the smectic A phase and superconductors. It follows that smectic layers expel twist deformations in the same way that superconductors expel magnetic field. We make a direct observation of the penetration of twist at the edge of a single isolated smectic A layer composed of chiral fd virus particles subjected to a depletion interaction. Using the LC-PolScope, we make quantitative measurements of the spatial dependence of the birefringence due to molecular tilt near the layer edges. We match data to theory for the molecular tilt penetration profile and determine the twist penetration length for this system.

Nematic cells with defect-patterned alignment layers.

Using Monte Carlo simulations of the Lebwohl-Lasher model we study the director ordering in a nematic cell where the top and bottom surfaces are patterned with a lattice of +/-1 point topological defects of lattice spacing a . As expected on general physical grounds we find that the nematic order depends on the ratio of the height of the cell H to a . For thick cells (Ha > or = 0.9) we find that the system is very well ordered and the frustration induced by the lattice of defects is relieved in a novel way by a network of half-integer defect lines which emerge from the point defects and hug the top and bottom surfaces of the cell. When Ha < or = 0.9 the system has zero nematic order parameter and the half-integer defect lines thread through the cell joining point defects on the top and bottom surfaces. We present a simple physical argument in terms of the length of the defect lines to explain these results. To facilitate eventual comparison with experimental systems we also simulate optical textures in the presence of crossed polarizers.