Bifurcation Analysis of the 1D and 2D Cahn-Hilliard Equations

Abstract

The Cahn-Hilliard (C-H) equation is a phenomenological model for the physical process of phase separation in a binary mixture [1]. The model finds applications across a broad variety of fields, and has been used to model the separation of binary alloys, multiphase fluid flows, T-cell receptor clustering, the organization of the mitotic spindle, and the dynamics of stress granules [1,5]. This work examines the one-dimensional (1D) and two-dimensional (2D) C-H equations, focusing on analyzing the steady state attractors of the system. In 1D, the bifurcation diagram for the spinodal and metastable regimes is numerically generated as a function of the spatial length, and the free energies of the steady state curves are analyzed. A model is then developed to consider the probability of nucleation in the metastable regime. In 2D, the bifurcation curves are generated in the spinodal and metastable regimes with respect to different spatial domain height to width ratios and compared to 1D bifurcation curves. Conditions under which real 2D solutions can emerge are considered.