Phase Field Model for the Nucleation in Solid State Phase Transformations : Theories , Algorithms and Applications

Nucleation takes place when a material becomes thermodynamically meta-stable with respect to its transformation to a new state or new crystal structure. Very often, the nucleation process dictates the microstructure of a material. Predicting the shape of a critical nucleus in solids has been a long-standing problem in solid state phase transformations. It is generally believed that nucleation in solid is the most difficult process to model and predict. The main focus of this dissertation is the development of mathematical models and numerical algorithms for various nucleation phenomena in solid state phase transformation. Motivated by a general phase field framework with a diffuse interface description of the phase transformation, we develop a new computational approach to predict the morphology of a critical nucleus in solids under the influence of both interfacial energy anisotropy and long-range elastic interactions. The approach can help us uncover the wealth of fascinating topics in the solid states. The dissertation is organized as follows: In Chapter 1, we give an overview of the nucleation in the solid states and existing nucleation theories, including the classical nucleation theory and the diffuse interface theory. Then we introduce some numerical methods to compute the saddle point and the Minimum Energy Path (MEP). In Chapter 2, we investigate a phase-field model for finding the critical nucleus morphology in the homogeneous nucleation of solids. We analyze the mathematical properties of a free energy functional that includes the long-range, anisotropic elastic interactions. Based on a minimax technique and the Fourier spectral implementation, the numerical algorithms is developed to search for the saddle points. We demonstrate that the phase-field model is mathematically well defined and is able to efficiently predict the critical nucleus morphology in elastically anisotropic solids without making a priori assumptions. In Chapter 3, we present numerical simulations of the critical nucleus morphol-