Phase-ﬁeld modeling of equilibrium shapes of precipitate and growth instabilities in the presence of coherency stresses
Precipitation-hardened alloys are one of the most technologically signiﬁcant materials that are used for structural applications, where an important mode of strengthening is due to the impediment to the movement of dislocations. The alloys, particularly possessing coherent precipitate-matrix interface, give rise to the coherency strain ﬁelds producing the coherency stresses in the matrix, which further interact with the dislocations to provide necessary strengthening. In this context, the control of the shape and distribution of the precipitates as a function of material and process parameters is important. In this thesis, we propose a diﬀuse-interface approach in order to compute the equilibrium shape of precipitates, which also allows us to minimize the grid-anisotropy related issues that occur in the classical sharp interface methods. The method is based on the minimization of the functional consisting of the elastic free energy and the interfacial energy, while the volume of the precipitate is conserved. Using this technique we reproduce the shape bifurcation diagram from 2D simulations for isotropic, inhomogeneous elastic energy with dilatational misﬁt, which is compared against the analytical solution provided by Johnson-Cahn and an existing sharp interface FEM technique. Thereafter the model has been utilized for the investigation of equilibrium shapes for diﬀerent combinations of elastic misﬁt matrices and cubic anisotropy. Additionally, we incorporate the anisotropy in the interfacial energy, which has not been studied by previous sharp interface techniques and investigate its inﬂuence on shape bifurcation. Finally, we extend the model to calculate the equilibrium shapes of the precipitate in 3D which resembles the precipitate structures observed in real microstructures. We notice that the nature of shape bifurcation diagram in 3D is diﬀerent than that observed in 2D, which is principally because there exist multiple variants of precipitate shapes for the same precipitate size e.g. prolate-like or oblate-like structures that are not equivalent. We also compute a range of equilibrium shapes in 3D, that can form as a result of symmetry breaking for diﬀerent forms of anisotropy in the elastic energy. In the next part of the thesis, we extend the phase-ﬁeld model for computing the equilibrium shape of single precipitates for consideration of multiple variants that allows us to investigate the equilibrium conﬁgurations of precipitates. Here, given the properties of the material such as the magnitude of misﬁt strain and its signs, the magnitude of the shear moduli and the size of the precipitate, one can determine the equilibrium conﬁguration that can form during solid-state precipitation reactions. Using this method, we investigate three solid-state precipitation reactions, i.e. formation of a core-shell type of precipitates in the microstructure and two symmetry-breaking transitions namely, cubic to tetragonal (typically observed in the superalloys with γ' − γ'' microstructure) and hexagonal to orthorhombic (formation of multi-variant precipitate pattern in Ti-based alloys). We evaluate the criteria for the formation of core-shell type microstructures and show that while the formation of such structures is purely due to interfacial conditions (satisfying the wetting condition), the reaction pathway leading to their formation is assisted by elastic interactions between the precipitates. In the symmetry-breaking transitions, we investigate the formation of equilibrium conﬁgurations of the multi-variant precipitates using energetic calculations. Here, we observe that the formation of such conﬁgurations involving multiple variants is favored over the nucleation of a single precipitate of the same equivalent volume beyond a certain precipitate size. In the last part of the thesis, we relax the condition of volume preservation of the precipitates and study precipitate growth in a supersaturated matrix in the presence of anisotropy in the elastic energy. To achieve this, we utilize a phase-ﬁeld model based on the grand-potential formulation for coupling the chemical driving forces with coherency stresses. Using this model we investigate the speciﬁc problem of solid-state dendrite formation occurring during certain precipitation reactions. Here, we ﬁnd that the anisotropy in the elastic energy gives rise to the formation of the dendrite-like structures that are typically observed in solidiﬁcation microstructures as an eﬀect of the Mullins-Sekerka in stabilities. We determine the dendrite tip shape and tip velocity, as the precipitate grows in size as a function of diﬀerent materials parameters such as the magnitude of misﬁt strain, supersaturation in the matrix, and the anisotropy strength in both the energies. We notice that in all the cases, the dendrite tip shape and tip velocity does not achieve steady-state which is in contrast with the dendrites observed during solidiﬁcation. This modiﬁcation is due to the presence of coherency stresses.