| Hi-index | 0.00 |
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 morphology in solid state phase transformations. A diffuse interface model combined with the minimax technique is implemented to predict the morphology of critical nucleus during solid to solid phase transformations in both two and three dimensions. We use a particular example of cubic to cubic transformation within the homogeneous modulus approximation and study the effect of elastic energy contribution on the morphology of a critical nucleus. The results show that strong elastic energy interactions may lead to critical nuclei with a wide variety of shapes, including plates, needles and cuboids with non-convex interfaces. It is found that strong elastic energy contributions may lead to critical nuclei whose point group symmetry is below the crystalline symmetries of both the new and the parent phases. In Chapter 4, we develop a constrained string method to solve the saddle-point problem with general constraints. Based on the description of the string method, a smooth curve is evolved with intrinsic parametrization whose dynamics takes the string to the most probable transition path between two metastable regions in configuration space. Then Lagrange multiplier is applied for the extra constraint and numerical algorithm of the constrained string method is implemented to find the constrained MEP and saddle points. Numerical analysis includes the conservation of the constraint and the energy law. We also propose a simplified approach to implement the constraint by the Augmented Lagrangian method. By using the constrained string method, we investigate the morphological evolution during precipitation of a second-phase particle in a solid along the entire transformation path from nucleation to equilibrium in Chapter 5. We show that a combination of diffuse-interface description and a constrained string method is able to predict both the critical nucleus and equilibrium precipitate morphologies simultaneously without a priori assumptions. Using the cubic to cubic transformation as an example, it is demonstrated that the maximum composition within a critical nucleus can be either higher or lower than that of equilibrium precipitate while morphology of an equilibrium precipitate may exhibit lower symmetry than the critical nucleus resulted from elastic interactions. In Chapter 6, we present more applications for the nucleation in solids, including cubic to tetragonal transformation, and nucleation for two order parameters in solids. Our works on the mathematical modeling and computational algorithms open some new research directions and provide useful tools for the analysis of the nucleation phenomenon in general, for instance, we will consider the dynamic simulation of the nucleation process, inhomogeneous nucleation, and heterogeneous nucleation in the near future.