Designing pattern formation through anisotropy

Abstract

When governed by appropriate potentials, systems of particles interacting pairwise in three dimensions self assemble into diverse patterns near the surface of a sphere. The resulting structure of these minimal energy states can be altered through anisotropic effects. This leads to the inverse problem of finding anisotropic potentials that produce specific targeted equilibrium structures. To study this problem, continuous versions of the discrete particle interaction equations are employed so that a leading order approximation can be obtained. Linear stability is then determined through a Fourier type analysis in terms of spherical harmonics. This allows us to solve the linearized inverse problem: for a targeted equilibrium structure, where the particles congregate along a finite set of spherical harmonics, construct an anisotropic potential that induces the same finite set of linear instabilities. Several examples of anisotropic potentials that cause known linear instabilities are presented. The resulting minimal energy configurations are approximated through a gradient descent of the discrete particle energy. These numerical experiments corroborate that the linear instabilities can be used to predict the minimal energy structure in the full nonlinear dynamics. Solving the linearized inverse problem yields a clear path to designing pattern formation through anisotropic effects.