Phase-field models of microstructure evolution and new numerical strategies

A basic interest in materials science is the determination of microstructures of materials in order to understand why materials have the properties they exhibit and how these properties can be controlled to serve technological uses. In this dissertation, the phase-field approach is combined with other models and algorithms to study the effect of elastic energy, anisotropic mobility and structural defects on phase separation kinetics and morphological evolution in bulk systems.;A multi-dimensional general phase-field model has been developed by combining the iteration method for calculating the elastic energy and a semi-implicit spectral method for solving the Cahn-Hilliard equation. The model can efficiently calculate the microstructure evolution for systems with rather general elastic anisotropies, arbitrarily large elastic inhomogeneity, general misfit strain, and large anisotropic mobility. The microstructure and structure functions have been studied to show the effect of large anisotropic mobility together with the elastic energy.;To address the challenge of large scale simulations of microstructures which are still computationally expensive to date, an adaptive semi-implicit Fourier spectral (AFSIM) method is developed to solve Allen-Cahn equation by making grid points spatially adaptive in the physical domain via a moving mesh strategy, while maintaining a uniform grid in the computational domain for the spectral implementation. The moving mesh method is adopted since it is particularly efficient among recent efforts in developing advanced numerical algorithms. The newly developed approach not only provides more accurate treatment at the interfaces requiring higher resolution, but also retains the numerical efficiency of the semi-implicit Fourier spectral method. Numerical examples using the adaptive moving mesh semi-implicit Fourier spectral method are presented for both two and three space dimensional microstructure simulations, and they are compared with those obtained by other methods. By maintaining similar accuracy, the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial width to the domain size.;The AFSIM is further implemented to solve Cahn-Hilliard equation with inhomogeneous, anisotropic elasticity. Numerical implementations and test examples in both two and three dimensions are considered with a particular illustration using the well-studied example of misfitting particles in a solid as they approach to their equilibrium shapes. It is shown that significant savings in memory and computational time is achieved while accurate solutions are preserved.;The dynamics of the structure defects such as dislocations generally control the material properties during non-equilibrium processing. In phase-fields description, the dislocations exist only along the dislocations lines parallel to the slip planes. The total dislocation line length to volume ratio in dislocation motion is expected to be significantly smaller than the total area to volume ratio for interface motion. Therefore, AFSIM is extended to study the dislocation motion. By comparing the results with analytical solutions for accuracy and with results from the uniform Fourier-spectral semi-implicit method (UFSIM) for efficiency, it is shown that AFSIM yields much more accurate results for the dislocation stress field and is an order magnitude more efficient than the UFSIM for the same accuracy.