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.

We have studied precipitate growth and coarsening in ternary alloys using two different phase held models. The first one is a ternary extension of the classical Cahn-Hilliard (C-H) model in which both the phases are characterized using conserved held variables i.e. composition (cB; cC ); mobility matrix and gradient energy efficient are the other input parameters in this model. In the second model, each phase is treated as separate, and phase identify cation is through a (non-conserved) phase held variable ; we have used a grand potential-based (GP) formulation, due to Plapp [1], Choudhury and Nestler [2], where interfacial energy and interface width, as well as free energy and diffusivity matrix for the relevant phases are the input parameters. The first model i.e. the Cahn-Hilliard (C-H) type model is conceptually simple. The model for ternary is a straight forward extension of the binary. The grand potential (GP) formulation has the advantage of being able to incorporate thermodynamic database like Thermocalc in it. We present below a summary of the findings of our research on (a) precipitate growth, precipitate coarsening, and (c) a critical comparison between results from phase held simulations and those from experiments on an Ni-Al-Mo alloy Precipitate growth In our study of precipitate growth in ternary alloys, we end that when both the solute elements have the same diffusivity, precipitate growth behaviour in ternary alloys is identical to that binary alloys; specifically, we recover the temporal power law r2 = kgt relating the particle radius to time, and the growth kg depends only on supersaturation (i.e., equilibrium volume fraction of the precipitate phase), and is independent of the slope of the tie line. However, when one solute element, (say, C) di uses slower than the other (i.e. (DCC =DBB) < 1,(where DBB, DCC are intertie suavities’ in the lab frame of reference), the ux of C at the interface is smaller than that of species B, causing the precipitate to become depleted in C and enriched in B; this process continues until the growth phase enters a scaling regime where we recover the temporal law for growth: r2 = kgt. In this regime, the tie line selected by the precipitate and matrix interfacial compositions is different from the thermodynamic tie line containing the alloy, a result first reported by Coates [3]. After validating our phase held model quantitatively through a critical comparison with Coates' theory of tie line selection, we have characterized the growth behaviour: specifically, we end that growth kg drops with decreasing value of DCC ; the magnitude of this drop is stronger for alloys which (a) are on higher-C tie lines (i.e., the slope of the tie line is higher), and (b) have smaller precipitate volume fractions. Precipitate coarsening In our simulations, we end that precipitate coarsening does indeed enter a scaling regime where the temporal power law r3 = kt (which relates the average precipitate radius r to (b) time t) is valid; the coarsening rate k depends, as expected, not only on precipitate volume fraction, but also on the slope of the tie line and diffusivity ratio (DCC =DBB). (c) (d) When the solutes have equal diffusivity (i.e., (DCC =DBB) = 1), the coarsening behaviour is essentially the same as that in a binary alloy. However, when solute C (say) is the slower di using species, the coarsening rate k drops, with a deeper drop in alloys on higher-C tie lines. Both these conclusions are similar to those from our study of precipitate growth. (e) (f) However, there is a crucial difference between precipitate growth and coarsening in ternary alloys: The suppression in coarsening rate (for DCC < DBB) in ternary alloys is accompanied by another e ect: larger (and growing) precipitates are richer in the faster di using species B, while the smaller and shrinking precipitates are richer in the slower di using C. In other words, during coarsening in ternary alloys, the tie line selected by precipitate and matrix interfacial components depends on precipitate size; during growth, however, the scaling regime is characterized by the same tie line, independent of precipitate size. (g) (h) (i) Critical comparison between theory and experiment (j) (k) (l) We have used the grand potential based phase held model [1] [2] to study coarsening in Ni-Al-Mo alloys. This model has the advantage of ease with which we can incorporate the thermodynamic and kinetic data on real alloys. (m) (n) A comparison of coarsening rate from our 3D simulations with the experimentally observed rate reveals that diffusivity of the faster di using species (which, in Ni-Al-Mo alloys, is aluminium) from our simulations is within an order of magnitude from the experimental value. However the dominant term in the (@ =@c) matrix is underestimated by 2 to 3 orders of magnitude (compared to its value computed from CALPHAD-based thermodynamic data).