Many technologically important alloys such as Ni base superalloys and Ti-Al base alloys benefit from the precipitation of an ordered β phase from a disordered α matrix. When the crystallographic symmetry of the β phase is a subgroup of that of the disordered α phase, the microstructure may contain multiple orientational variants of the β phase, each with its own (anisotropic, crystallographically equivalent) misfit (lattice parameter mismatch) with the matrix phase. Examples include orthorhombic precipitates in a hexagonal matrix in Ti-Al-Nb alloys, and tetragonal precipitates in a cubic matrix in ZrO2-Y2O3. We have studied two-phase microstructures containing multiple variants of the precipitate phase. In particular, we have used phase field simulations to study the effect of elastic stresses in a two dimensional system containing a disordered matrix and three different orientational variants of the precipitate phase, with a view to elucidate the effect of different levels of anisotropy in misfit. We consider a two dimensional, elastically homogeneous and isotropic model system in which the matrix (α) and precipitate (β) phases have hexagonal and rectangular symmetries, respectively, giving rise to three orientational variants of the β phase. Therefore, our phase field model has composition (c) and three order parameters (η1, η2, η3) as the field variables.Due to the difference in crystallographic symmetry, the precipitate-matrix misfit strain tensor, ε*, can be anisotropic. ε*maybe represented in its principal form as ε *= (ε xx 0 ) 0 εyy where ε xx and ε yy are the principal components of the misfit tensor. We define t= εyy/εxx as the parameter representing anisotropy in the misfit. In this thesis, we report the results of our systematic study of microstructural evolution in systems with different values of t, representing different levels of anisotropy in misfit: •Case A: t=1 (dilatational or isotropic misfit) • Case B: 0 <t<1 (principal misfit components are unequal but have the same sign) • Case C: t=0 (the principal misfit along the y direction is zero) • Case D: -1 <t<0 (principal misfit components have opposite signs and unequal magnitudes) • Case E: t= -1 (principal misfit components are equal, but with opposite signs; pure shear) In Cases D and E, there is an invariant line along which the normal misfit is zero. In Case D, this invariant line is at ±54.72◦, and in Case E, it is at ±45◦, with respect to the x-axis. Our simulations of microstructural evolution in this system are based on numerical integration of the Cahn-Hilliard and Cahn-Allen equations which govern the evolution of composition and order parameter fields, respectively. In each case, we have studied two different situations: isolated particle (single variant) and many interacting particles (multivariant). Dynamical growth shape of an isolated precipitate In systems with an isotropic misfit (Case A), the precipitate shape remains circular at all sizes. In Cases B and C, the precipitate shape is elongated along the y-axis, which is also the direction in which the magnitude of the misfit strain is lower. In all these cases, the symmetry of the particle shape remains unaltered at all sizes. In contrast, in Cases D and E, the particle shape exhibits a symmetry-breaking transition. In Case D, the precipitate elongates initially along the y direction (i.e. the direction of lower absolute misfit), before undergoing a transition in which the mirror symmetry normal to x and yaxes is lost. In Case E, the particle has an initial square-like shape (with its sides normal to the 11directions) before losing its four-fold rotation axis to become rectangle-like with its long axis along one of the the 11directions. The critical precipitate size at which the symmetry-breaking shape transition occurs is obtained using bifurcation diagrams. In both Cases D and E, the critical size for the dynamical growth shapes is larger than those for equilibrium shapes[1].This critical size is larger when the matrix supersaturation is higher or shear modulus is lower. Microstructural Evolution In all the five cases, the elastic stresses have a common effect: they lead to microstructures in which the precipitate volume fraction is lower than that in a system with no misfit. This observation is consistent with the results from the thermodynamics of stressed solids that show that a precipitate-matrix misfit increases the interfacial composition in both the matrix and the precipitate phase. In systems with isotropic misfit (Case A), the microstructure consists of isolated circular domains of the precipitate phase that retain their circular shape during growth and subsequent coarsening. In Cases B and C with anisotropic misfit with t≥0, the three orientational variants of the precipitate phase are elongated along the directions of lower misfit (y-axis and ±120◦to y-axis). At a given size, particles in Case C (in which one of the principal misfits is zero) are more elongated than those in Case B. Systems with a higher shear modulus enhance the effect of misfit stresses, and therefore, lead to thinner and longer precipitates. When the precipitate volume fraction is increased, these elongated precipitates interact with (and impinge against) one another to a greater extent, and acquire a more jagged appearance. For Cases D and E, each orientation domain is associated with an invariant line along which the normal misfit is zero. Thus, in Case D, early stage microstructures show particles elongated along directions of lower absolute misfit (y-axis and ±120°to y-axis). At the later stages, the domains of the precipitate phase tend to orient along the invariant lines; this leads some of the particles to acquire a ‘Z’ shape before they completely re-orient themselves along the invariant line. In Case E, each variant grows as a thin plate elongating along the invariant line. The growth and impingement of these thin plates leads to a microstructure exhibiting complex multi-domain patterns such as stars, wedges, triangles, and checkerboard. These patterns have been compared (and are in good agreement) with experimental observations in Ti-Al-Nb alloys containing the precipitate (O) and matrix (α2)phases[2]. Since in Case E the sum of misfit strains of the three variants is zero, elastic energy considerations point to the possibility of compact, self-accommodating clusters of the three variants, sharing antiphase boundaries (APBs). Thus, if the APB energy is sufficiently low, the microstructure may consist of such compact clusters. Our simulations with such low APB energy do show triangle shaped clusters with six separate particles (two of each variant)in a self-accommodating pattern. (Refer PDF file)