Energy Driven Pattern Formation in Planar Dipole-Dipole Systems in the Presence of Weak Noise

Abstract

We study pattern formation in planar fluid systems driven by intermolecular cohesion (which manifests as a line tension) and dipole-dipole repulsion which are observed in physical systems including ferrofluids in Hele-Shaw cells and Langmuir layers. When the dipolar repulsion is sufficiently strong, domains undergo forked branching reminiscent of viscous fingering. A known difficulty with these models is that the energy associated with dipole-dipole interactions is singular at small distances. Following previous work, we demonstrate how to ameliorate this singularity and show that in the macroscopic limit, only the relative scale of the microscopic details of a system are relevant, and develop an expression for the system energy that depends only on a generalized line tension, {\Lambda}, that in turn depends logarithmically on that scale. We conduct numerical studies that use energy minimization to find equilibrium states. Following the subcritical bifurcations from the circle, we find a few highly symmetric stable shapes, but nothing that resembles the observed diversity of experimental and dynamically simulated domains. The application of a weak random background to the energy landscape stabilizes a sm\"org\r{a}sbord of domain morphologies recovering the diversity observed experimentally. With this technique, we generate a large sample of qualitatively realistic shapes and use them to create an empirical model for extracting {\Lambda} using only a shape's perimeter and morphology with high accuracy.