![]() Line tension favors circular flat membrane that minimizes the exposed edge. The exact structure of the scalloped edge is determined by the competition between the line tension and the Gaussian curvature modulus. ![]() In this limit, edge-bound rods exhibit achiral symmetry breaking, forming domains of opposite twist that are separated by cusp-like point defects, where the membrane escapes into the third dimension. With decreasing chirality, which is accomplished by mixing rods of opposite handedness, flat 2D circular membranes become unstable, and instead develop complex scalloped edges. For membranes composed of single-component chiral rods, the handedness of the edge twist along the entire circumference is uniform and dictated by the microscopic chirality of the constituent rods. The semicircular edge profile requires twisting of the rods at the edge, and this twist penetrates into the membrane interior over a characteristic length scale ( 16, 18, 19). ![]() In comparison, significantly less is known about the Gaussian modulus, despite the significant role it plays in fundamental biological and technological processes such as pore formation as well as vesicle fusion and fission ( 7– 11). Consequently, experiments that interrogated mechanics or shape fluctuations of vesicles provided extensive information about the membrane curvature modulus and how it depends on the structure of the constituent particles ( 4– 6). Thus, the shape fluctuations of a closed vesicle only depend on the membrane-bending modulus. In particular, integrating Gaussian curvature over any simply closed surface yields a constant ( 3). However, lipid bilayers almost always appear as edgeless 3D vesicles, which further simplify theoretical modeling. Because an arbitrary deformation of a thin layer can have either mean and/or Gaussian curvature, the full theoretical description of membranes, in principle, requires two parameters, the bending and Gaussian curvature moduli. The possible configurations and shapes of 2D fluid membranes can be described by a continuum energy expression that accounts for the membrane’s out-of-plane deformations as well as the line tension associated with the membrane’s exposed edge ( 1, 2). 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. A simple excluded volume argument predicts the sign and magnitude of the Gaussian curvature modulus that is in agreement with experimental measurements. 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. In the achiral regime, the cusp defects have repulsive interactions, but away from this limit we measure effective long-ranged attractive binding. Such membranes adopt a 3D configuration, with cusp defects alternatively located above and below the membrane plane. 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. In comparison, membranes composed of a mixture of rods with opposite chiralities can have the edge twist of either handedness. Membranes assembled from single-component chiral rods form flat disks with uniform edge twist. 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. 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.
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