Achiral symmetry breaking and positive Gaussian modulus lead to scalloped colloidal membranes.

In the presence of a nonadsorbing polymer, monodisperse rod-like particles assemble into colloidal membranes, which are one-rod-length-thick liquid-like monolayers of aligned rods. Unlike 3D edgeless bilayer vesicles, colloidal monolayer membranes form open structures with an exposed edge, thus presenting an opportunity to study elasticity of fluid sheets. Membranes assembled from single-component chiral rods form flat disks with uniform edge twist. In comparison, membranes composed of a mixture of rods with opposite chiralities can have the edge twist of either handedness. In this limit, disk-shaped membranes become unstable, instead forming structures with scalloped edges, where two adjacent lobes with opposite handedness are separated by a cusp-shaped point defect. Such membranes adopt a 3D configuration, with cusp defects alternatively located above and below the membrane plane. In the achiral regime, the cusp defects have repulsive interactions, but away from this limit we measure effective long-ranged attractive binding. A phenomenological model shows that the increase in the edge energy of scalloped membranes is compensated by concomitant decrease in the deformation energy due to Gaussian curvature associated with scalloped edges, demonstrating that colloidal membranes have positive Gaussian modulus. A simple excluded volume argument predicts the sign and magnitude of the Gaussian curvature modulus that is in agreement with experimental measurements. Our results provide insight into how the interplay between membrane elasticity, geometrical frustration, and achiral symmetry breaking can be used to fold colloidal membranes into 3D shapes.

Colloidal membranes of hard rods: unified theory of free edge structure and twist walls.

Monodisperse suspensions of rod like chiral fd viruses are condensed into a rod-length thick colloidal monolayers of aligned rods by depletion forces. Twist deformations of the molecules are expelled to the monolayer edge as in a chiral smectic A liquid crystal, and a cholesteric band forms at the edge. Coalescence of two such isolated membranes results in a twist wall sandwiched between two regions of aligned rods, dubbed π-walls. By modeling the membrane as a binary fluid of coexisting cholesteric and chiral smectic A liquid-crystalline regions, we develop a unified theory of the π-walls and the monolayer edge. The mean-field analysis of our model yields the molecular tilt profiles, the local thickness change, and the crossover from smectic to cholesteric behavior at the monolayer edge and across the π-wall. Furthermore, we calculate the line tension associated with the formation of these interfaces. Our model offers insights regarding the stability and the detailed structure of the π-wall and the monolayer edge.

Imprintable membranes from incomplete chiral coalescence.

Coalescence is an essential phenomenon that governs the equilibrium behaviour in a variety of systems from intercellular transport to planetary formation. In this report, we study coalescence pathways of circularly shaped two-dimensional colloidal membranes, which are one rod-length-thick liquid-like monolayers of aligned rods. The chirality of the constituent rods leads to three atypical coalescence pathways that are not found in other simple or complex fluids. In particular, we characterize two pathways that do not proceed to completion but instead produce partially joined membranes connected by line defects-π-wall defects or alternating arrays of twisted bridges and pores. We elucidate the structure and energetics of these defects and ascribe their stability to a geometrical frustration inherently present in chiral colloidal membranes. Furthermore, we induce the coalescence process with optical forces, leading to a robust on-demand method for imprinting networks of channels and pores into colloidal membranes.

Reconfigurable self-assembly through chiral control of interfacial tension.

From determining the optical properties of simple molecular crystals to establishing the preferred handedness in highly complex vertebrates, molecular chirality profoundly influences the structural, mechanical and optical properties of both synthetic and biological matter on macroscopic length scales. In soft materials such as amphiphilic lipids and liquid crystals, the competition between local chiral interactions and global constraints imposed by the geometry of the self-assembled structures leads to frustration and the assembly of unique materials. An example of particular interest is smectic liquid crystals, where the two-dimensional layered geometry cannot support twist and chirality is consequently expelled to the edges in a manner analogous to the expulsion of a magnetic field from superconductors. Here we demonstrate a consequence of this geometric frustration that leads to a new design principle for the assembly of chiral molecules. Using a model system of colloidal membranes, we show that molecular chirality can control the interfacial tension, an important property of multi-component mixtures. This suggests an analogy between chiral twist, which is expelled to the edges of two-dimensional membranes, and amphiphilic surfactants, which are expelled to oil-water interfaces. As with surfactants, chiral control of interfacial tension drives the formation of many polymorphic assemblages such as twisted ribbons with linear and circular topologies, starfish membranes, and double and triple helices. Tuning molecular chirality in situ allows dynamical control of line tension, which powers polymorphic transitions between various chiral structures. These findings outline a general strategy for the assembly of reconfigurable chiral materials that can easily be moved, stretched, attached to one another and transformed between multiple conformational states, thus allowing precise assembly and nanosculpting of highly dynamical and designable materials with complex topologies.

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.

Self-limited self-assembly of chiral filaments.

We discuss dynamical simulations and free energy calculations on patchy spheres with chiral pair interactions that spontaneously assemble into filamentous bundles. The chirality frustrates long-range crystal order by introducing twist between interacting subunits. For some ranges of system parameters this constraint leads to bundles with a finite diameter, and in other cases frustration is relieved by the formation of defects. While some self-limited structures can be modeled as twisted filaments arranged with local hexagonal symmetry, other structures are surprisingly complex.

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.

Dynamics of the molecular orientation field coupled to ions in two-dimensional ferroelectric liquid crystals.

Molecular orientation fluctuations in ferroelectric smectic liquid crystals produce space charges, due to the divergence of the spontaneous polarization. These space charges interact with mobile ions, so that one must consider the coupled dynamics of the orientation and ionic degrees of freedom. Previous theory and light scattering experiments on thin free-standing films of ferroelectric liquid crystals have not included this coupling, possibly invalidating their quantitative conclusions. We consider the most important case of very slow ionic dynamics, compared to rapid orientational fluctuations, and focus on the use of a short electric field pulse to quench orientational fluctuations. We find that the resulting change in scattered light intensity must include a term due to the quasistatic ionic configuration, which has previously been ignored. In addition to developing the general theory, we present a simple model to demonstrate the role of this added term.

Role of electrostatics in the texture of islands in free-standing ferroelectric liquid crystal films.

Curved textures of ferroelectric smectic-C* liquid crystals produce space charge when they involve divergence of the spontaneous polarization field. Impurity ions can partially screen this space charge, reducing long-range interactions to local ones. Through studies of the textures of islands on very thin free-standing smectic films, we see evidence of this effect, in which materials with a large spontaneous polarization have static structures described by a large effective bend elastic constant. To address this issue, we calculated the electrostatic free energy of a free-standing film of ferroelectric liquid crystal, showing how the screened Coulomb interaction contributes a term to the effective bend elastic constant, in the static long-wavelength limit. We report experiments which support the main features of this model.

Unwinding of a strained cholesteric elastomer by disclination loop nucleation.

The application of a sufficiently strong strain perpendicular to the pitch axis of a monodomain cholesteric elastomer unwinds the cholesteric helix. Previous theoretical analyses of this transition ignored the effects of Frank elasticity which we include here. We find that the strain needed to unwind the helix is reduced because of the Frank penalty and the cholesteric state becomes metastable above the transition. We consider in detail a previously proposed mechanism by which the topologically stable helical texture is removed in the metastable state: namely, by the nucleation of twist disclination loops in the plane perpendicular to the pitch axis. We present an approximate calculation of the barrier energy for this nucleation process which neglects possible spatial variation of the strain fields in the elastomer, as well as a more accurate calculation based on a finite-element modeling of the elastomer.

Textural transformations in islands on free standing smectic-C* liquid crystal films.

We report on and analyze the textural transformations in islands, thicker circular domains, floating in very thin free standing chiral Smectic-C* liquid crystal films. As an island is growing, an initial pure bend texture of the director changes into a reversing spiral at a critical size. Another distinct spiral texture is induced by changing the boundary condition at the central point defect in the island. To understand these transformations from a pure bend island, a linear stability analysis of the c-director free energy is developed, which predicts a state diagram for the island. Our observations are consistent with the theoretical phase diagram.

Buckling instability in liquid crystalline physical gels.

In a nematic gel we observe a low-energy buckling deformation arising from soft and semisoft elastic modes. We prepare the self-assembled gel by dissolving a coil-side-group liquid-crystalline polymer-coil copolymer in a nematic liquid crystal. The gel has long network strands and a precisely tailored structure, making it ideal for studying nematic rubber elasticity. Under polarized optical microscopy we observe a striped texture that forms when gels uniformly aligned at 35 degrees C are cooled to room temperature. We model the instability using the molecular theory of nematic rubber elasticity, and the theory correctly captures the change in pitch length with sample thickness and polymer concentration. This buckling instability is a clear example of a low-energy deformation that arises in materials where polymer network strains are coupled to the director orientation.

Isotropic-cholesteric transition in liquid-crystalline gels.

In a nematic gel, the appearance of nematic order is accompanied by a spontaneous elongation of the gel parallel to the nematic director. If such a gel is made chiral, it has a tendency to form a cholesteric helical texture, in which local elongation of the gel parallel to the nematic director is suppressed due to the requirement of elastic compatibility. We show that a conical helix in which the director makes an oblique angle with respect to the helix axis serves as an energy minimizing compromise between the competing tendencies for elongation and twisting. We find the dependence of the helical cone angle and pitch on the strength of the chirality, and determine the change in sample shape at the isotropic to cholesteric phase transition.

Electroclinic effect and modulated phases in smectic liquid crystals.

We explore the possibility that the unusually large electroclinic effect observed in the smectic-A phase of a ferroelectric liquid crystal arises from the presence of an ordered array of disclination lines and walls in a smectic-C* phase. If the spacing of these defects is in the subvisible range, this modulated smectic-C* phase would be similar macroscopically to a smectic-A phase. The application of an electric field distorts the array, producing a large polarization, and hence a large electroclinic effect. We show that with suitable elastic parameters and sufficiently large chirality, the modulated phase is favored over the smectic-A and helically twisted smectic-C* phases. We propose various experimental tests of this scenario.