Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Abstract

Coherency misfit stresses and their related anisotropies are known to influence the equilibrium shapes of precipitates. Additionally, mechanical properties of the alloys are also dependent on the shapes of the precipitates. Therefore, in order to investigate the mechanical response of a material which undergoes precipitation during heat treatment, it is important to derive the range of precipitate shapes that evolve. In this regard, several studies have been conducted in the past using sharp-interface approaches where the influence of elastic energy anisotropy on the precipitate shapes has been investigated. In this paper, we propose a diffuse interface approach which allows us to minimize grid-anisotropy-related issues applicable to sharp-interface methods. In this context, we introduce a novel phase-field method where we minimize the functional consisting of the elastic and surface energy contributions while preserving the precipitate volume. Using this technique, we reproduce the shape bifurcation diagrams for the cases of pure dilatational misfit that have been studied previously using sharp-interface methods and then extend them to include interfacial energy anisotropy with different anisotropy strengths which has not been a part of previous sharp-interface models. While we restrict ourselves to cubic anisotropies in both elastic and interfacial energies in this study, the model is generic enough to handle any combination of anisotropies in both the bulk and interfacial terms. Further, we have examined the influence of asymmetry in dilatational misfit strains along with interfacial energy anisotropy on precipitate morphologies.

Investigation of the γ′ Precipitates Dissolution in a Ni-Based Superalloy During Stress-Free Short-Term Annealing at High Homologous Temperatures

“…[35] The morphology of c¢ precipitates is determined by minimizing the sum of the precipitate elastic strain energy, the precipitate surface energy, and the elastic interaction energy between precipitates. [46][47][48] Balancing of internal energy occurs continously at each temperature. It should be emphasized that the equilibrium state at temperature no.…”

Equilibrium Multi-precipitate Configurations

“…Similarly, the interfacial energy, which is the other component of the energy in the system, varies with the interfacial area. In our earlier work, [44] we formulated a phase-field model for the computation of equilibrium shapes of precipitates, where we minimize the sum of the elastic and interfacial energy contributions while preserving the volume of the precipitate. In this paper, since we are interested in computing equilibrium configurations of two or more precipitates, we need to extend the formulation to incorporate multiple phases.…”

“…Here, we solve for multiple Allen-Cahn equations along with the condition of mechanical equilibrium, while conserving the volume of each of the precipitates during the minimization. This constrained minimization is achieved through the technique of volume preservation which is also described elsewhere [45] and also in our earlier work, [44] that is essentially the coupling of the Allen-Cahn type equations with correction terms using Lagrange parameters that ensure the conservation of the volume of the precipitates during evolution.…”

Quantitative Shape-Classification of Misfitting Precipitates during Cubic to Tetragonal Transformations: Phase-Field Simulations and Experiments

“…The phase-field method is frequently used to study the shape evolution of γ/γ′ microstructures [ 13 , 14 ] as well as systems with γ/γ′/γ′′ microstructure [ 15 ]. The variational formulation of the phase-field method makes it a useful tool to determine precipitate shapes of minimum energy [ 16 , 17 , 18 ].…”