An Analysis of Phase Separation Processes for Stochastic and Nonlocal Extensions of the Classical Phase-Field Model

Abstract

Phase separation in composite materials, or materials that are made up of two or more different components, is a process in which the material separates over time into regions of predominantly one component or the other. Because these composite materials are often developed to be used under certain conditions, significant changes of this nature in their material composition can dramatically affect their performance, thus it is important to understand the dynamic behavior of these changes. One mathematical model of phase separation is the classical phase field model, which is a system of nonlinear evolution equations that describes nonisothermal phase separation. Instead of a composite material, the phase field model describes pure materials which can assume two different phases; for example, a solid and a liquid phase. In this dissertation, I analyze the behavior of solutions of the phase field model, along with two extensions to the classical model. One extension is the addition of stochastic terms to the model, thus incorporating random behavior into the equations to address fluctuations in temperature, structural imperfections of the material, or other unknown effects. The other extension is the addition of a nonlocal operator to account for long range interactions. The behavior of all three variations of the phase field model is studied through numerical simulations, and the results compared to describe the differences between the models. In addition, I present a proof showing that the numerical method converges to the true solution of the phase field model.