Circular 2D solid domains in fluid-solid composite vesicles: Axisymmetric and non-axisymmetric elastic ground states*

While the elasticity and morphology of fluid vesicles from homogeneous vesicles to fluid-fluid phase separated vesicles are well studied, those of solid-fluid composite vesicles are much less understood. For very thin solids, such as 2D solid domains in giant unilamellar vesicles, changes of Gaussian curvature induce high stretching energy due to the costs of in-plane shear, which grow prohibitive for large domains. Here, we consider a model of simplest case, a composite vesicle possessing a single circular solid domain, which is planar its stress-free state. The simplifying assumptions of strain free solids, we consider the axisymmetric shape equilibria including two branches of oblate (lower elastic energy) and prolate (higher elastic energy) shapes. Because the strictly planar solid domain shape limits the inflation of the axisymmetric composite, the bending energy at the solid-fluid domain edge diverges as the inflation approaches the maximal value. Given these diverging elastic energies, we address the question about whether high inflation triggers isometric (i.e. strain free) deformations of the solid that break axisymmetry. To explore the shape-equilibria outside of the assumption of axisymmetry, we conduct Surface Evolver simulations which vary the solid-fluid composition, vesicle inflation and ratio of bending to in-plane strain moduli (i.e. the effective elastic thickness of the solid). We first study the evolution from stretchable regime to nearly isometric folding regime for decreasing the thickness, showing the transition for spherical to isometric solid domain shapes at high inflation. Next, we study the symmetries and emergent shapes of elastic ground states in distinct regimes, such as small/large thickness, or low/high inflation, and attempt to classify the distinct ranges of nearly-isometric solid domain configurations in terms of localized vs. distributed patterns of (mean curvature) bending. Last, we connect the symmetry breaking of circular solid domains to the case of elasticity of composite domains including non-convex and flower-like domains shapes, recently observed in experiments.