Controlling the shape and topology of two-component colloidal membranes.

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 have in-plane fluid-like dynamics and out-of-plane bending elasticity. Their open edges and micrometer-length scale provide a tractable system to study the equilibrium energetics and dynamic pathways of membrane assembly and reconfiguration. Here, we find that doping colloidal membranes with short miscible rods transforms disk-shaped membranes into saddle-shaped surfaces with complex edge structures. The saddle-shaped membranes are well approximated by Enneper's minimal surfaces. Theoretical modeling demonstrates that their formation is driven by increasing the positive Gaussian modulus, which in turn, is controlled by the fraction of short rods. Further coalescence of saddle-shaped surfaces leads to diverse topologically distinct structures, including shapes similar to catenoids, trinoids, four-noids, and higher-order structures. At long timescales, we observe the formation of a system-spanning, sponge-like phase. The unique features of colloidal membranes reveal the topological transformations that accompany coalescence pathways in real time. We enhance the functionality of these membranes by making their shape responsive to external stimuli. Our results demonstrate a pathway toward control of thin elastic sheets' shape and topology-a pathway driven by the emergent elasticity induced by compositional heterogeneity.

Axisymmetric membranes with edges under external force: buckling, minimal surfaces, and tethers.

We use theory and numerical computation to determine the shape of an axisymmetric fluid membrane with a resistance to bending and constant area. The membrane connects two rings in the classic geometry that produces a catenoidal shape in a soap film. In our problem, we find infinitely many branches of solutions for the shape and external force as functions of the separation of the rings, analogous to the infinite family of eigenmodes for the Euler buckling of a slender rod. Special attention is paid to the catenoid, which emerges as the shape of maximal allowable separation when the area is less than a critical area equal to the planar area enclosed by the two rings. A perturbation theory argument directly relates the tension of catenoidal membranes to the stability of catenoidal soap films in this regime. When the membrane area is larger than the critical area, we find additional cylindrical tether solutions to the shape equations at large ring separation, and that arbitrarily large ring separations are possible. These results apply for the case of vanishing Gaussian curvature modulus; when the Gaussian curvature modulus is nonzero and the area is below the critical area, the force and the membrane tension diverge as the ring separation approaches its maximum value. We also examine the stability of our shapes and analytically show that catenoidal membranes have markedly different stability properties than their soap film counterparts.

Force-Induced Formation of Twisted Chiral Ribbons.

We demonstrate that an achiral stretching force transforms disk-shaped colloidal membranes composed of chiral rods into twisted ribbons with handedness opposite the preferred twist of the rods. Using an experimental technique that enforces torque-free boundary conditions we simultaneously measure the force-extension curve and the ribbon shape. An effective theory that accounts for the membrane bending energy and uses geometric properties of the edge to model the internal liquid crystalline degrees of freedom explains both the measured force-extension curve and the force-induced twisted shape.

Chiral edge fluctuations of colloidal membranes.

We study edge fluctuations of a flat colloidal membrane comprised of a monolayer of aligned filamentous viruses. Experiments reveal that a peak in the spectrum of the in-plane edge fluctuations arises for sufficiently strong virus chirality. Accounting for internal liquid crystalline degrees of freedom by the length, curvature, and geodesic torsion of the edge, we calculate the spectrum of the edge fluctuations. The theory quantitatively describes the experimental data, demonstrating that chirality couples in-plane and out-of-plane edge fluctuations to produce the peak.