For experiments on chiral self-assembly, we used a two-component mixture consisting of 880 nm long rod-like fd viruses and the non-adsorbing polymer Dextran. In aqueous suspension, fd viruses alone exhibit purely repulsive interactions 13. Adding non-adsorbing polymer to a dilute isotropic suspension of fd rods induces attractive interactions via the depletion mechanism and leads to their condensation into colloidal membranes, equilibrium structures consisting of one-rod-length thick liquid-like monolayers of aligned rods (Fig. 1a) 11. Despite having different structures on molecular lengthscales, the longwavelength coarse-grained properties of colloidal membranes are identical to those of conventional lipid bilayers. However, unlike their amphiphilic counterparts, colloidal membranes do not form vesicles and are instead observed as freely suspended disks with exposed edges. Here, we investigate the behavior of these exposed edges in a manner analogous to previously studied liquid-liquid domains embedded in lipid bilayers [14][15][16] . For our experiments, it is essential that fd viruses are chiral, i.e. a pair of aligned viruses minimizes their interaction energy when they are slightly twisted in a preferred direction with respect to each other. The strength of chiral interactions can be continuously tuned to zero through either genetic or physical methods ( Supplementary Fig. 1) 13,17 .Before investigating chiral membranes, we determined the structure of a membrane's edge composed of simpler achiral rods using three complimentary imaging techniques, namely 2D and 3D polarization microscopy and electron microscopy. The local tilting of the rods within a membrane was determined using 2D LC-PolScope, which produces images in which the intensity of each pixel represents the local retardance of the membrane (Fig. 1d) 18. Such images can be quantitatively related to the tilting of the rods away from the layer normal, the z-axis 19. Rods in the bulk of a membrane are aligned along the zaxis, so that 2D LC-PolScope images appear black in that region (Fig. 1e). In contrast, the bright birefringent ring along the membrane's periphery reveals local tilting of the rods (Fig. 1e, Supplementary Fig. 2). For achiral rods, this indicates that a membrane has a hemi-toroidal curved edge (Fig. 1b, c). In comparison to an untilted edge, a curved edge structure lowers the area of the rod/polymer interface, thus reducing interfacial tension, at the cost of increasing the elastic energy due to twist distortion. This hypothesis is confirmed by visualizing the 3D membrane structure using electron tomography, whichshows that the viruses' long axis transitions from being parallel to the z-axis in the membrane bulk to perpendicular to the z-axis and tangent to the edge along the membrane periphery ( When viewed with optical microscopy, a membrane's edge exhibits significant thermal fluctuations, the analysis of which yields the line tension γ eff , the energetic cost required to move rods from the membrane interior to the periphe...

We combine simulations and experiments to elucidate the molecular forces leading to the assembly of two dimensional membrane-like structures composed of a one rod-length thick monolayer of aligned rods from an immiscible suspension of hard rods and depleting polymers. Computer simulations predict that monolayer membranes are thermodynamically stable above a critical rod aspect ratio and below a critical depletion interaction length scale. Outside of these conditions alternative structures such as stacked smectic columns or nematic droplets are thermodynamically stable. These predictions are confirmed by subsequent experiments using a model system of virus rod-like molecules and non-adsorbing polymer. Our work demonstrates that collective molecular protrusion fluctuations alone are sufficient to stabilize membranes composed of homogenous rods with simple excluded volume interactions.

The assembly of filamentous bundles with controlled diameters is common in biological systems and desirable for the development of nanomaterials. 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 as the equilibrium state, 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 surprising in their complexity.

Using simulations, we construct the effective dynamics in metabasin space for a Lennard-Jones glass former. Metabasins are identified via a scheme that measures transition rates between inherent structures and generates clusters of inherent structures by drawing in branches that have the largest transition rates. This construction is fundamentally different from the stochastic approach based on molecular-dynamics trajectories. The effective dynamics in this metabasin space is shown to be Markovian but to differ significantly from the simplest trap models. We specifically show that retaining information about the connectivity in this metabasin space is crucial for reproducing the slow dynamics observed in this system.

We formulate a density functional theory that describes the phase behavior of hard rods and depleting polymers, as realized in recent experiments on suspensions of fd virus and non-adsorbing polymer. The theory predicts the relative stability of nematic droplets, stacked smectic columns, and a recently discovered phase of isolated monolayers of rods, or colloidal membranes. We find that a minimum rod aspect ratio is required for stability of colloidal membranes and that collective protrusion undulations are the dominant effect that stabilizes this phase. The theoretical predictions are shown to be qualitatively consistent with experimental and computational results.

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