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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.

SUMMARY: Plant cells and organs grow into a remarkable diversity of shapes, as directed by cell walls composed primarily of polysaccharides such as cellulose and multiple structurally distinct pectins. The properties of the cell wall that allow for precise control of morphogenesis are distinct from those of the individual polysaccharide components. For example, cellulose, the primary determinant of cell morphology, is a chiral macromolecule that can self‐assemble in vitro into larger‐scale structures of consistent chirality, and yet most plant cells do not display consistent chirality in their growth. One interesting exception is the Arabidopsis thaliana rhm1 mutant, which has decreased levels of the pectin rhamnogalacturonan‐I and causes conical petal epidermal cells to grow with a left‐handed helical twist. Here, we show that in rhm1 the cellulose is bundled into large macrofibrils, unlike the evenly distributed microfibrils of the wild type. This cellulose bundling becomes increasingly severe over time, consistent with cellulose being synthesized normally and then self‐associating into macrofibrils. We also show that in the wild type, cellulose is oriented transversely, whereas in rhm1 mutants, the cellulose forms right‐handed helices that can account for the helical morphology of the petal cells. Our results indicate that when the composition of pectin is altered, cellulose can form cellular‐scale chiral structures in vivo, analogous to the helicoids formed in vitro by cellulose nano‐crystals. We propose that an important emergent property of the interplay between rhamnogalacturonan‐I and cellulose is to permit the assembly of nonbundled cellulose structures, providing plants flexibility to orient cellulose and direct morphogenesis. Significance Statement: Some plants exhibit a striking pattern of growth with helically twisted cells and organs, but most plants do not display such chirality, despite the presence in their cell walls of chiral macromolecules. We show that the composition of pectin, a mix of multiple cell wall polysaccharides, determines whether or not cellulose assembles into helical structures that subsequently impart chirality to cells and organs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]