Membrane mechanics meet minimal manifolds
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 studied by the Sharma (IISc Bangalore) and Dogic (UCSB) Labs have in-plane fluid-like dynamics and out-of-plane bending elasticity, but their open edges and micron length scale provide a tractable system to study the equilibrium energetics and dynamic pathways of membrane assembly and reconfiguration. First, we discuss how doping colloidal membranes with short miscible rods transforms disk-shaped membranes into saddle-shaped minimal surfaces with complex edge structures. Theoretical modeling demonstrates that their formation is driven by increasing positive Gaussian modulus, which in turn is controlled by the fraction of short rods. Further coalescence of saddle-shaped surfaces leads to exotic topologically distinct structures, including shapes similar to catenoids, tri-noids, four-noids, and higher order structures. We then mathematically explore the mechanics of these catenoid-like structures subject to an external axial force and elucidate their intimate connection to two problems whose solutions date back to Euler: the shape of an area-minimizing soap film and the buckling of a slender rod under compression. A perturbation theory argument directly relates the tensions of membranes to the stability properties of minimal surfaces. We also investigate the effects of including a Gaussian curvature modulus, which, for small enough membranes, causes the axial force to diverge as the ring separation approaches its maximal value.
Internal structure of honey bee swarms for mechanical stability and division of labor
The western honey bee (Apis mellifera) is a domesticated pollinator famous for living in highly social colonies. In the spring, thousands of worker bees and a queen fly from their hive in search of a new home. They self-assemble into a swarm that hangs from a tree branch for several days. We reconstruct the non-isotropic arrangement of worker bees inside swarms made up of 3000 - 8000 bees using x-ray computed tomography. Some bees are stationary and hang from the attachment board or link their bodies into hanging chains to support the swarm structure. The remaining bees use the chains as pathways to walk around the swarm, potentially to feed the queen or communicate with one another. The top layers of bees bear more weight per bee than the remainder of the swarm, suggesting that bees are optimizing for additional factors besides weight distribution. Despite not having a clear leader, honey bees are able to organize into a swarm that protects the queen and remains stable until scout bees locate a new hive.
Frustrated Self-Assembly of Non-Euclidean Crystals of Nanoparticles
Self-organized complex structures in nature, e.g., viral capsids, hierarchical biopolymers, and bacterial flagella, offer efficiency, adaptability, robustness, and multi-functionality. Can we program the self-assembly of three-dimensional (3D) complex structures using simple building blocks, and reach similar or higher level of sophistication in engineered materials? Here we present an analytic theory for the self-assembly of polyhedral nanoparticles (NPs) based on their crystal structures in non-Euclidean space. We show that the unavoidable geometrical frustration of these particle shapes, combined with competing attractive and repulsive interparticle interactions, lead to controllable self-assembly of structures of complex order. Applying this theory to tetrahedral NPs, we find high-yield and enantiopure self-assembly of helicoidal ribbons, exhibiting qualitative agreement with experimental observations. We expect that this theory will offer a general framework for the self-assembly of simple polyhedral building blocks into rich complex morphologies with new material capabilities such as tunable optical activity, essential for multiple emerging technologies.