Numerical solution of the reaction-diffusion Cahn-Hilliard equations using an isogeometric analysis based method -

Abstract

Phase-field models are becoming more popular in approaching interfacial phenomena by describing a microstructure with a set of continuous field variables across the interfacial region. The field variables are assumed to be continuous across this region, which is opposite to the situation encountered in sharp interface models, where they are discontinuous. In the case of conserved field variables, their temporal evolution is governed by the Cahn-Hilliard equation [1,2]. The Cahn-Hilliard equation was devised from a phase-separation model called the spinodal decomposition in a binary alloy [3] by which a solid solution can separate into distinctly different phases that have different compositions and physical properties. This scenario is different from the classical nucleation-driven mechanism and takes place throughout the material, not just at discrete nucleation sites. Because there is no thermodynamic barrier for nucleation, decomposition is only driven by the diffusion process. Due to this simple description, this model is also used in other fields such multiphase fluid flow [4-7], phase separation [8], formation of quantum dots [9] , Taylor flow in micro-channels [10], pore migration in a temperature gradient [11], tumor growth [12,13] and precipitation reactions [14]. In particular, the reaction-diffusion Cahn-Hilliard equations (RDCH) arise in a class of chemical systems whereby precipitation reactions induced by the diffusion of initially separated species lead to the emergence of Liesegang bands. Moreover, these equations are capable of reproducing the transition from traditional Liesegang bands or stripes to unusual spotted patterns with a square-hexagonal symmetry. In this thesis, we propose to numerically solve the RDCH equations using the isogeometric analysis (IGA) and the finite element method (FEM), in circular geometries, and compare the two methods. We show that our numerical solution can capture the pattern formation and the aforementioned transition from stripes to spots. In that regard, the