Diffuse-interface approach to predicting morphologies of critical nucleus and equilibrium structure for cubic to tetragonal transformations

Abstract

A cubic to tetragonal transformation leads to simultaneous nucleation and growth of several orientation variants. The size and morphology of a critical nucleus, comprising of one or more variants, are determined by the competition among bulk thermodynamic driving force, elastic strain energy, and interfacial energy. In this work, a diffuse-interface model is developed to predict both the critical nucleus morphology of a single variant tetragonal precipitate described by a conserved field and the critical morphology of a two-variant nucleus and equilibrium multivariant twinning microstructure resulted from a structural transformation described by non-conserved fields. A constrained string method was employed to compute the probable minimum energy paths and the microstructures along each path. Numerical experiments indicate that our approach works effectively for predicting the critical nucleus morphology both the precipitation processes and structural transformations.

Introduction

Nucleation in solid-state phase transformations is generally believed to be one of the most difficult processes to model. It takes place when a material becomes metastable with respect to its transformation to a new crystal structure. During a natural precipitation process in a supersaturated solid or liquid solution, e.g. during isothermal annealing of a quenched homogeneous alloy within a two-phase-field of a phase diagram, the majority of precipitation reactions in solids occur through a nucleation-and-growth mechanism followed by particle coarsening. There are two thermodynamically well-defined morphologies: the morphology of a critical nucleus and the equilibrium morphology of a precipitate particle. Predicting precipitate particle morphology in solids becomes a main challenge because of the presence of both interfacial energy anisotropy and anisotropic elastic interactions. Moreover, since the actual nucleation process is a rare event and the critical nucleus only exists transiently, it is of great interest for developing effective methods that can be used to study the morphological relation between the critical nucleus and the equilibrium precipitation.

The diffuse-interface description, or the nonclassical nucleation theory, is based on the gradient thermodynamics of nonuniform systems and provides a general framework of treating nucleation. Rather than assuming a particular geometry for a critical nucleus which is determined by a competition between a bulk free energy decrease which is proportional to volume and an interfacial energy increase which is proportional to interfacial area, a critical nucleus is defined as the composition or order parameter fluctuation having the minimum free energy increase among all fluctuations which lead to nucleation, i.e., the saddle point configuration along the minimum energy path (MEP) between the metastable initial phase represented by a local minimum in the free energy landscape and the equilibrium phase represented by the global minimum. Therefore, nucleation of new phase particle requires overcoming a minimum thermodynamic barrier. The magnitude of the nucleation barrier, and thus the nucleation rate, or the resulted new phase particle density, is strongly dependent on the morphology of critical nucleus. On the other hand, following nucleation-and-growth, the morphology and volume fraction of new phase particle during coarsening are generally close to equilibrium. The particle morphology and volume fraction during coarsening together with the particle density predicted from nucleation provide the essential information that is needed for predicting the strength of a solid in mechanistic models.

Early studies of equilibrium shapes in solids often use both sharp- and diffuse-interface approaches [1], [2], [3], [4], [14], [15]. Recent attempts have also been made to predict the morphology of a critical nucleus in solids by taking into account both interfacial energy anisotropy and anisotropic elastic interactions [16], [17], [18], [20], [21], [22], [23]. For example, we showed that one can predict the morphology of a critical nucleus in a system going through a phase transition [20], [21] using a combination of the diffuse-interface (phase-field) description and the minimax algorithm based on the mountain pass theorem [12], [22]. Diffuse-interface critical nuclei have also been incorporated in the dynamic phase-field simulation through the explicit nucleation algorithm in [28]. Recently, in [23], a new approach is developed to simultaneously predict the morphology of a critical nucleus as well as the equilibrium morphology of a precipitate within the same diffuse-interface model described by a single phase-field function. In this paper, we consider the more complex nucleation in cubic to tetragonal transformations which leads to the various crystallographic orientation variants (structure domains). Computationally, our approach is based on the calculation of MEPs with the unconstrained and constrained string methods which we describe in Section 2. We also compare such an approach with the corresponding modification to the Nudged Elastic Band method. The main results of the paper are given in Section 3. There, we first discuss the diffuse-interface model corresponding to the cubic to tetragonal transformations, with either a single order parameter or a pair of phase-field functions. We then present the algorithm for computing both the critical nucleus and the equilibrium microstructure simultaneously by combining the diffuse-interface model with the constrained string method. Results of several numerical examples in the single order parameter case are provided. An example on the extension to the two-variant nucleation case is also shown. These examples demonstrate the great potential in enlarging the applicability of our method to the complex cubic to tetragonal nucleation. Final conclusions are given in Section 4.