Elsevier

Acta Materialia

Volume 49, Issue 11, 22 June 2001, Pages 1879-1890
Acta Materialia

A phase-field model for evolving microstructures with strong elastic inhomogeneity

https://doi.org/10.1016/S1359-6454(01)00118-5Get rights and content

Abstract

An efficient phase-field model is proposed to study the coherent microstructure evolution in elastically anisotropic systems with significant elastic modulus inhomogeneity. It combines an iterative approach for obtaining the elastic displacement fields and a semi-implicit Fourier–spectral method for solving the time-dependent Cahn–Hilliard equation. Each iteration in our iterative numerical simulation has a one-to-one correspondence to a given order of approximation in Khachatuyran's perturbation method. A unique feature of this approach is its ability to control the accuracy by choosing the appropriate order of approximation. We examine shape dependence of isolated particles as well as the morphological dependence of a phase-separated multi-particle system on the degree of elastic inhomogeneity in elastically anisotropic systems. It is shown that although prior calculations using first-order approximations correctly predicted the qualitative dependence of a two-phase morphology on elastic inhomogeneity, the local stress distributions and thus the driving force for microstructure evolution such as coarsening were in serious error quantitatively for systems with strong elastic inhomogeneity.

Introduction

Essentially all solid–solid phase transformations produce coherent microstructures at their early stages. In a coherent microstructure, the lattice directions and planes are continuous across the interfaces separating the parent and product phases or separating different orientation domains of the product phase. In order to maintain this lattice continuity, the lattice mismatch between the product and the parent phases and among the orientation domains of the product phase is accommodated by elastic displacements of atoms from their equilibrium lattice positions. Therefore, formation of coherent microstructures generates elastic strain energy whose magnitude depends on the degree of lattice mismatch, the elastic properties of each phase, and the shape and spatial distributions of coherent domains [1], [2].

The effect of elastic strain energy on coherent precipitate morphology and its temporal evolution has been a subject of extensive experimental and theoretical studies, for reviews see [3], [4]. Most of the existing theoretical analysis and numerical modeling of elastic effect assumed that the elastic modulus is homogeneous [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], or that the modulus inhomogeneity is small so that first-order approximations may be employed [23], [24], [25], [26], [27], [28], [29], [30].

There have been a number of approaches proposed for modeling the elastic effect on precipitate morphology in coherent systems with a significant elastic inhomogeneity. For example, Lee proposed a discrete atom method (DAM) which allows rather arbitrary elastic inhomogeneity and precipitate morphology [31], [32]. Due to the discrete nature of the method, the spatial scale of the precipitates described by this method is atomic. Schmidt and Gross [33], [34] studied the equilibrium shapes of a coherent precipitate using a boundary integral method and a sharp-interface description. In [33], [34], the temporal evolution of precipitate morphology through diffusion transport of atoms was not considered. Jou et al. examined the temporal evolution of precipitate shapes in elastically inhomogeneous systems by simultaneously solving a diffusion equation and the elasticity equation using the boundary integral method [35]. Since the interfaces are considered to be sharp, it is difficult to handle certain topological changes which take place, e.g., during initial stage of spinodal phase separation, and during precipitate coalescence and splitting. Leo et al. developed a diffuse-interface model for modeling elastically inhomogeneous systems by coupling the Cahn–Hilliard diffusion equation with an elasticity equation. In this method, the elasticity equation is numerically solved by a conjugate gradient method (CGM) at any given moment during microstructure evolution [36]. A similar diffuse-interface model using CGM is proposed by Zhu et al. [37].

Recently, Khachaturyan et al. developed an analytical solution for the elastic field in an elastically inhomogeneous system using a perturbation method (PM) and sharp-interface description [38]. The strain energy is expressed as a sum of multiparticle interactions between finite elements of the constituent phases, pairwise, triplet, quadriuplet and so on, the n-particle interaction energy being related to the (n−2)th order term in the Taylor expansion of the Green function with respect to the elastic modulus misfit. The order of approximation required for a given system depends on the desired accuracy and the degree of elastic inhomogeneity. However, direct application of the analytical elastic energy expression to numerical simulation of coherent microstructure evolution in elastically inhomogeneous systems is difficult since the elastic strain energy involves multi-dimensional integrals in both real and Fourier spaces.

The main purpose of this paper is to present an efficient diffuse-interface phase-field model for elastically inhomogeneous systems. We used an iterative approach for numerically solving the elastic equilibrium equation and employed a diffuse-interface description. A unique feature of this method is the fact that each iteration in our numerical method corresponds to a given order of approximation in Khachaturyan's perturbation method. To be consistent with [38], we will simply call our numerical method the “perturbation method” or PM. We will apply PM to elastically anisotropic coherent systems with strong elastic inhomogeneity and compare our results with those predicted previously by others using first-order approximations. Application of the proposed PM to calculating effective elastic modulus of a two-phase mixture and its temporal evolution will be discussed in a future publication.

Section snippets

Mechanical equilibrium equation

We consider a simple binary solid solution with a compositional inhomogeneity described by X(r), representing the mole or atom fraction X at position r. We assume that the local elastic modulus tensor can be described in terms of the compositional inhomogeneity throughλijkl(r,t)=λmijklXpeq−X(r)Xpeq−XmeqpijklX(r)−XmeqXpeq−Xmeqwhere λmijkl and λpijkl are the elastic modulus tensors for the matrix with equilibrium composition Xmeq and for the precipitate with equilibrium composition Xpeq,

Diffusion equation in elastically inhomogeneous systems

For a binary substitutional solid solution, the diffusion flux (in unit of atoms per unit area per unit time) is given byJ=−NvM∇μwhere Nv is the number of atoms per unit volume, M is a mobility given byM=X(1−X)[XM1+(1−X)M2]where X is the composition of species 2 (mole or atom fraction), M1 and M2 are atomic mobilities of species 1 and 2, respectively. They are related to the diffusivity throughMi=DikBTwhere kB is the Boltzmann constant, T is the temperature, and Di is the diffusion coefficient

Convergence of the proposed method

In principle, the convergence and accuracy of the proposed perturbation method for a given order of approximation can be tested against analytical solutions which are available for certain special precipitate shapes [2]. However, the analytical solutions are available only for systems with a sharp-interface description whereas the interfaces in our numerical calculation are diffuse. Therefore, to examine the accuracy, we compare the results from the proposed method with those obtained from an

Summary

A diffuse-interface field model is proposed for predicting the morphological and microstructural evolution in elastically anisotropic systems with strong elastic inhomogeneities. Within this model, the elastic solutions are obtained using an iterative method with each iteration corresponding to a given order of approximation in Khachaturyan's perturbation method. Highly accurate results can be obtained using various orders of approximations for different degrees of elastic inhomogeneity. We

Acknowledgements

The authors are grateful for the financial support from the NSF under grant No. DMR-96-33719. The simulations were performed at the San Diego Supercomputer Center and the Pittsburgh Supercomputing Center.

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