# Numerical modelling of phase separation on dynamic surfaces

## TL;DR

This paper develops a fully Eulerian numerical method for simulating phase separation on dynamic, deforming surfaces using a surface-independent mesh, enabling analysis of complex surface phenomena like splitting and pattern formation.

## Contribution

It introduces a hybrid finite difference and trace finite element method for Cahn-Hilliard equations on evolving surfaces, avoiding surface triangulation and handling topological changes.

## Key findings

- Method accurately models phase separation on moving surfaces.
- Capable of simulating topological transitions like surface splitting.
- Successfully applied to pattern formation and droplet splitting scenarios.

## Abstract

The paper presents a model of lateral phase separation in a two component material surface. The resulting fourth order nonlinear PDE can be seen as a Cahn-Hilliard equation posed on a time-dependent surface. Only elementary tangential calculus and the embedding of the surface in $\mathbb{R}^3$ are used to formulate the model, thereby facilitating the development of a fully Eulerian discretization method to solve the problem numerically. A hybrid method, finite difference in time and trace finite element in space, is introduced and stability of its semi-discrete version is proved. The method avoids any triangulation of the surface and uses a surface-independent background mesh to discretize the equation. Thus, the method is capable of solving the Cahn-Hilliard equation numerically on implicitly defined surfaces and surfaces undergoing strong deformations and topological transitions. We assess the approach on a set of test problems and apply it to model spinodal decomposition and pattern formation on colliding surfaces. Finally, we consider the phase separation on a sphere splitting into two droplets.