Large Scale Simulation of Spinodal Decomposition

de Xiang Zheng

Spinodal decomposition is a process in which a system of binary mixture eventually evolves to the separation of two macroscopic phases. Such phase separation occurs in a thermodynamically unstable state. A number of binary mixture experiments have demonstrated the phenomenon of spinodal decomposition. Many models have been proposed to describe the evolution of the spinodal decomposition. The Cahn-Hilliard (CH) partial differential equation, which includes an order parameter and a free energy, and evolves to minimize the energy, has frequently been used as a phase field model. Due to random thermal fluctuations that are inevitable in physical systems, the CH equation might be unrealistic for the overall decomposition process. Experimental results demonstrate the existence of Brownian motion in the spinodal decomposition, which suggests that diffusion (deterministic contribution) and the noise (stochastic contribution) both have an essential influence on the rate of spinodal decomposition. Therefore, a stochastic process should be part of a realistic mathematical model of the overall decomposition process. In order to overcome the disadvantage that the CH equation ignores physically significant thermal fluctuation, the CH equation with a thermal fluctuation term has been proposed, where the thermal fluctuation is modeled by a time-space Brownian motion. The CH equation with the thermal fluctuation was first considered by Cook, so the extended CH equation is also known as the Cahn-Hilliard-Cook (CHC) equation. For studying the CHC equation, we are primarily interested in the properties of steady state, such as the energies, statistical moments, and morphology. This motivates our choices for the numerical frameworks for analyzing the CHC equation. The CHC equation is a stochastic partial differential equation involving a biharmonic form and a noise forcing term. When the potential term is a polynomial, the CHC equation is split into a lower order PDE system of two harmonic equations. T he space is discretized by the standard finite element method. The evolution of the spinodal decomposition and the effect of the thermal fluctuation are studied in 2D. For obtaining numerical results of the CHC equation with a more realistic logarithmic potential efficiently, especially in 3D, a fully implicit, cell-centered, finite difference scheme in the original biharmonic form, and an adaptive time-stepping strategy are combined to discretize the space and time. The numerical scheme is verified by a comparison with an explicit scheme. At each time step, the parallel NKS algorithm is used to solve a nonlinear spatially discretized system. We discuss various numerical and computational challenges associated with the cell-centered finite difference-based, massively parallel implementation of this framework. We present steady state solutions of the CHC equation in 2D and, for the first time, in 3D. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We demonstrate that the thermal fluctuation is able to accelerate the spinodal decomposition process, and change the final steady morphology. We study the evolution of energies and statistical moments, from the initial stage to the steady state. Next, we study the CHC equation from the statistical perspective. A parallel domain decomposition method, based on the Wiener chaos expansion (WCE) and the Karhunen-Loeve expansion (KLE), is presented. Applying the two expansions to time-space white noise, we transform the CHC equation into a deterministic form. The main advantage of the Wiener chaos approach is that it separates deterministic and random effects, and factors the latter out of the primary stochastic partial differential equation effectively and rigorously. Therefore, the stochastic partial differential equation can be reduced to its propagator: a system of deterministic equations for the coefficients of the Wiener chaos expansion. Formulae for the expansion of high order nonlinear terms are presented, which involve the solutions of the propagator. Compared to the Monte Carlo (MC) method, the Wiener chaos approach does not require the generation of random numbers. The Karhunen-Loeve expansion is able to capture the principal component of the random field. A domain decomposition method is used to solve the equation system, which is discretized by a stabilized implicit cell-centered finite difference scheme. An NKS algorithm is applied to solve the nonlinear system of equations at each time step. The evolution of the spinodal decomposition and respective variances are demonstrated. Numerical results demonstrate that the parallel domain decomposition method scales well to a thousand processor cores. For short time, the Wiener chaos Karhunen-Loeve expansion (WCKLE) method is more efficient than the Monte Carlo simulation. We simulate the whole spinodal decomposition process by the Wiener chaos Karhunen-Loeve expansion Monte Carlo (WKCLE-MC) h