Starting with the Mean Field Method (MFM) and Boundary Element Method (BEM), we investigate a mathematical model based on these two methods for studying particle-coarsening process in alloys. With MFM, second-phase particles are considered to be merged into bulk matrix, which greatly simplifies computation. However, the Mean-Field model itself is limited to a system with extremely small volume fractions of second phase. By combining BEM with MFM, this mathematical model shows the influence of second phase in particle-coarsening process. Our primary work demonstrates the robustness and capability of this model. This model is however limited to particle coarsening that is far away from grain boundaries. In this dissertation, we successfully extend the model to particle coarsening near grain boundaries. A major improvement made to the previous mathematical model is based on solute atoms conservation and diffusion theory. The capability and validity of the novel model is demonstrated by a binary alloy system. The simulation results are shown to quantitatively reproduce the essential features of particle coarsening near grain boundaries in certain alloys: a) precipitation Free Zones (PFZs) form near grain boundaries, b) the width of PFZs is proportional to square root of time, c) particles at the edge of PFZs are larger than those inside the grain. This novel model is shown to be well suited in describing particle coarsening near grain boundaries. On the other hand, it proves the credibility of the theories built in our mathematical model, i.e., the formation of PFZs near grain boundaries is caused by diffusion of solute atoms. / Thesis / Master of Applied Science (MASc)

In this thesis, we have used a phase field model for studying the effect of atomic mobility inside the precipitate phase on coarsening behaviour in two dimensional (2D) systems. In all the available coarsening theories, the diffusivity inside the precipitate phase is not explicitly taken into account; this would imply that there is no chemical potential gradient inside the precipitate. This assumption is valid if (a) the atomic mobility inside the precipitate is much higher than that in the matrix, or (b) the precipitate volume fraction is small (i.e. the interparticle spacing is far higher than the average particle size). We undertook this study to evaluate the potential effect of diffusivity in the precipitate on coarsening in situations where conditions (a) and (b), above, do not hold, by studying systems with moderate volume fractions (20% and 30%) and with low atomic mobilities in the precipitate. In our study, we have fixed the atomic mobility in the matrix at a constant value. We have used the well known Cahn-Hilliard model in which the microstructure is described in terms of a composition field variable. The evolution of microstructure is studied by numerically solving a non-classical diffusion equation known as the Cahn-Hilliard equation. We have used a semi-implicit Fourier spectral technique for solving the CH equation using periodic boundary conditions. The coarsening behaviour is tracked and analyzed using number density of particles, their average size and their size distribution. The main conclusion from this study is that, contrary to expectations, the atomic mobility in the precipitate phase has only a small effect on coarsening behavior. Specifically, with decreasing atomic mobility in the precipitate phase, we report a small increase in the number density, a slightly wider size distribution and a slightly smaller coarsening rate. We also add that these effects are too small to allow experimental verification. These results indicate that the need for chemical potential equilibration within each precipitate is not an important factor during coarsening.