Multiscale Modeling and Simulation of a Cahn--Larché System with Phase Separation on the Microscale
Lisa Reischmann, Malte A. Peter · SIAM Journal on Applied Mathematics · 2020
We consider the process of phase separation of a binary system under the influence of mechanical stress and we derive a mathematical multiscale model, which describes an evolving microstructure taking into account the elastic properties of the involved materials. Motivated by phase-separation processes observed in lipid monolayers in film-balance experiments, the starting point of the model is the Cahn--Hilliard equation coupled with the equations of linear elasticity, the so-called Cahn--Larché system. Owing to the fact that the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level, a multiscale approach is imperative. We assume the pattern of the evolving microstructure to have an intrinsic length scale associated with it, which, after nondimensionalization, leads to a scaled model involving a small parameter $\epsilon>0$, which is suitable for periodic-homogenization techniques. The problem is formally homogenized using the method of two-scale asymptotic expansions, which leads to a model of distributed-microstructure type in the limit. Finally, numerical simulations based on finite elements showcase the model behavior of the distributed-microstructure model.