The interplay between geometry, topology and order can lead to geometric frustration that profoundly affects the shape and structure of a curved surface. In this commentary we show how frustration in this context can result in the faceting of elastic
vesicles. We show that, under the right conditions, an assortment of regular and irregular polyhedral structures may be the low energy states of elastic membranes with spherical topology. In particular, we show how topological defects, necessarily present in any crystalline lattice confined to spherical topology, naturally lead to the formation of icosahedra in a homogeneous elastic vesicle. Furthermore, we show that introducing heterogeneities in the elastic properties, or allowing for non-linear bending response of a homogeneous system, opens non-trivial pathways to the formation of faceted, yet non-icosahedral, structures.
We use an elastic model to explore faceting of solid-wall vesicles with elastic heterogeneities. We show that faceting occurs in regions where the vesicle wall is softer, such as areas of reduced wall thicknesses or concentrated in crystalline defect
s. The elastic heterogeneities are modeled as a second component with reduced elastic parameters. Using simulated annealing Monte Carlo simulations we obtain the vesicle shape by optimizing the distributions of facets and boundaries. Our model allows us to reduce the effects of the residual stress generated by crystalline defects, and reveals a robust faceting mechanism into polyhedra other than the icosahedron.
We study closed liquid membranes that segregate into three phases due to differences in the chemical and physical properties of its components. The shape and in-plane membrane arrangement of the phases are coupled through phase-specific bending energ
ies and line tensions. We use simulated annealing Monte Carlo simulations to find low-energy structures, allowing both phase arrangement and membrane shape to relax. The three-phase system is the simplest one in which there are multiple interface pairs, allowing us to analyze interfacial preferences and pairwise distinct line tensions. We observe the systems preference for interface pairs that maximize differences in spontaneous curvature. From a pattern selection perspective, this acts as an effective attraction between phases of most disparate spontaneous curvature. We show that this effective attraction is robust enough to persist even when the interface between these phases is the most penalized by line tension. This effect is driven by geometry and not by any explicit component-component interaction.
