Instabilities in Colloidal Crystals on Fluid Membranes

TL;DR

This paper analyzes the stability of colloidal crystals on fluid membranes, identifying conditions under which regular arrangements become unstable and bifurcate into non-planar configurations, with implications for designing metamaterials.

Contribution

It introduces a parametric analysis of instabilities in colloidal crystals on fluid membranes, distinguishing between long- and short-wavenumber modes, and connects theoretical results to experimental observations.

Findings

Instabilities occur via bifurcations to non-planar configurations.

Two distinct instability modes identified: long-wavenumber and short-wavenumber.

Analysis provides boundary conditions for stable and unstable regimes.

Abstract

The complex physics of self-assembly in colloidal crystals on deformable interfaces and surfaces poses interesting possibilities for the designability and synthesis of next-generation metamaterials. The goal of this article is to characterize instabilities arising in colloidal crystals assembled on fluid membranes. The colloidal particles are modeled as pair-wise interacting point particles, constrained to lie on a fluid membrane and yet free to reorganize, and the membrane's elastic energy is modeled via the Helfrich energy. We find that when a collection of particles is arranged on a planar membrane in some regular fashion -- such as periodic lattice -- then the regular configuration admits bifurcations to non-planar configurations. Using the Bloch-wave anstaz for the mode of instabilities, we present a parameteric analysis of the boundary between the stable and unstable regimes. We find that instabilities can occur through two distinct kinds of modes, when the parameters belong in certain physically interesting regimes, referred to as long-wavenumber modes ($L$ modes) and short-wavenumber modes ($S$ modes) in the article. We discuss some connections between these results and recent experiments, as well as the open problem of budding in biomembranes.