# Uli Sack:

## Numerical Simulation of Phase Separation in Binary and Multicomponent Systems

### Abstract

The ban of lead in electronics solder by EU directives results in the technological challenge to develop lead-free alternatives with comparable life span and processing properties. Numerical simulations of the microstructure evolution may contribute to identify promising candidates and thus focus the immense experimental effort. Aim of this work is on the one hand to develop a numerical framework for the efficient and robust simulation of the microstructure evolution in binary alloys combining adaptive finite element methods with fast solvers for the Cahn-Hilliard model. On the other hand we will extend the existing fast solvers for the discrete scalar Cahn-Hilliard equation to the vector-valued case. After some preliminary remarks on phase diagrams, phase separation, and phasefield models in Chapter 1 we will firstly discuss anisotropic Allen-Cahn equations in Chapter 2. Alle-Cahn-like problems arise as subproblems in the Nonsmooth Schur-Newton (NSNMG) method for Cahn- Hilliard equations in Chapters 3 and 4. Here we prove existence and uniqueness of solutions to the anisotropic Allen-Cahn equation with logarithmic potential using the theory of maximal monotone operators. For the numerical solution we introduce an adaptive spatial mesh refinement cycle for evolution problems and several variants of implicit Euler time discretization. We prove stability for the latter and numerical experiments conclude the chapter. Chapter 3 combines existing and newly developed numerical tools to a simulation software for microstructure evolution in binary alloys. Key ingredients are the adaptive mesh refinement cycle of Chapter 2, the NSNMG solver, a quantification algorithm for measuring "coarseness" of microstructures and a quotient space multigrid method for indefinite problems. An application of this software to simulate the microstructure evolution in a eutectic AgCu alloy shows only marginal impact of elastic stresses on coarsening in the setting considered; while the use of a smooth interpolant of the logarithmic potential affects the coarsening dynamics considerably. In the final chapter we consider the multicomponent Cahn-Hilliard equation and derive a unified formulation for the discrete problems which allows a direct application of the NSNMG method. Existence and uniqueness of discrete solution are proved and numerical examples illustrate the robustness of the scheme with respect to temperature, mesh size, and number of components.