Shaping fluid membranes by controlling their gaussian modulus

Similar to lipid bilayers, one-rod-length thick colloidal membranes have in-plane fluid-like dynamics and out-of-plane bending elasticity. However, in contrast to the edgeless lipid vesicles, colloidal membranes assume a flat disk-like shape. Their open edges and micron length scale provide a tractable experimental system to study the interplay between the Gaussian curvature energy and the edge tension. By doping colloidal membrane with short miscible rods, we demonstrate that disk-shaped membranes transform into saddle-shaped surfaces with complex edge structures. Further coalescence of saddle-shaped surfaces leads to higher-order folded structures with exceedingly complex shapes and topologies, including catenoids, tri-noids, four-noids, and many others. At long time scales, we observe the formation of a system spanning sponge-like phases. The unique features of colloidal membranes allow us to visualize the topology changing coalescence pathways in real-time with molecular-level detail. We enhance the functionality of these membranes by making their shape responsive to external stimuli. A phenomenological model shows that the saddle-shaped membranes are Enneper-like minimal surfaces and that the miscible short rods increase the value of the positive Gaussian curvature modulus. Our results demonstrate an unconventional pathway towards shape control of thin sheets, one which is driven by the emergent elasticity induced by the compositional heterogeneity.