Soft modes near the buckling transition of icosahedral shells

TL;DR

This paper investigates the soft-mode transition in icosahedral shells, revealing how their shape changes from smooth to faceted as elastic properties vary, with implications for biological and synthetic shell structures.

Contribution

It introduces a phonon spectrum analysis of a simplified shell model to interpret the buckling transition as a soft-mode phenomenon, highlighting the role of curvature and elastic susceptibilities.

Findings

Vertex susceptibility peaks at the transition

Edge and face susceptibilities dominate in faceted shells

Susceptibility behaviors relate to ridge-scaling in elastic sheets

Abstract

Icosahedral shells undergo a buckling transition as the ratio of Young's modulus to bending stiffness increases. Strong bending stiffness favors smooth, nearly spherical shapes, while weak bending stiffness leads to a sharply faceted icosahedral shape. Based on the phonon spectrum of a simplified mass-and-spring model of the shell, we interpret the transition from smooth to faceted as a soft-mode transition. In contrast to the case of a disclinated planar network where the transition is sharply defined, the mean curvature of the sphere smooths the transitition. We define elastic susceptibilities as the response to forces applied at vertices, edges and faces of an icosahedron. At the soft-mode transition the vertex susceptibility is the largest, but as the shell becomes more faceted the edge and face susceptibilities greatly exceed the vertex susceptibility. Limiting behaviors of the susceptibilities are analyzed and related to the ridge-scaling behavior of elastic sheets. Our results apply to virus capsids, liposomes with crystalline order and other shell-like structures with icosahedral symmetry.