Large-scale simulations of Ostwald ripening in elastically stressed solids: I. Development of microstructure

Abstract

We present the results from large-scale simulations of Ostwald ripening of misfitting second-phase particles in an elastically anisotropic system. We employ the sharp interface description and perform simulations in two dimensions using boundary integral methods combined with state-of-the-art numerical methods such as the fast multipole method. We find that particle shapes are perturbed by elastic interactions at sufficiently large area fractions and particle sizes, and thus the shapes are not given by the equilibrium morphology of an isolated particle. Unlike isolated particles, the morphology transitions from fourfold to twofold shapes are not sharp, but are smeared due to interparticle interactions. However, for a very low area fraction system, the particles remain fourfold symmetric well beyond the bifurcation point, indicating that elastic interactions are essential in inducing a particle shape bifurcation. We find that the evolution of the microstructure is not scale invariant. However, the microstructure is unique, in a statistically averaged sense, for a given ratio of the elastic and interfacial energies.

Introduction

Ostwald ripening or coarsening is a process in which larger second-phase particles grow at the expense of smaller particles. It occurs in a vast range of two-phase mixtures, from liquid–liquid to solid–solid mixtures. The coarsening process in the absence of elastic stress has been examined extensively (see for example [1], [2] and references therein). However, recent theoretical and experimental work has indicated that the coarsening process in solid systems can be qualitatively and quantitatively different from more classical interfacial-energy-driven coarsening processes due to the presence of elastic stress. The elastic stress can come from many sources, such as the difference between the lattice parameters of the particle and the matrix, or an applied stress. As virtually all precipitates have a misfit, or a different lattice parameter from that of the matrix, elastic stress is generic to the Ostwald ripening in two-phase coherent solids. Our work focuses on such systems.

In two-phase solids, the dynamics of coarsening are governed by the interfacial and the elastic energies. The elastic stress can have a large effect on the coarsening process in coherent solids because the total elastic energy of the system can easily be of the same order as the total interfacial energy. This can be shown by examining the magnitude of a dimensionless parameter, which is a measure of the relative importance of elastic and interfacial energies in the system [3]. This parameter is given by

$$\text{L=ϵ}{}^2\text{lC}{}_{44}\text{/σ,}$$

where ϵ is the particle–matrix misfit, l is a characteristic length of a particle, C₄₄ is an elastic constant, which is chosen for non-dimensionalization of other elastic constants, and σ is the interfacial energy. For a system with multiple particles, 〈L〉=ϵ²〈l〉C₄₄/σ, where 〈L〉 is the average L and 〈l〉 is an average characteristic length associated with the particles in the system. L is a ratio of a characteristic elastic energy, ϵ²C₄₄l³, to a characteristic interfacial energy, σl² (in three dimensions). We take l to be the circularly equivalent dimensional radius, $\text{R=}\text{A/}\text{π}$ in two dimensions, where A is the dimensional area of the particle. In many systems, even those with small misfits, it is likely that particles attain an average size during coarsening to yield 〈L〉 of order 1 or larger. For example, in a Ni–Al alloy, L=5 corresponds approximately to a particle size of 0.09 μm, which is well within the range of technological interest. Since the elastic and interfacial energies are then of the same order of magnitude, the coarsening process must proceed by a decrease in the sum of the elastic and interfacial energies in the system. Thus, the dynamics of the coarsening process may not be similar to those given by an interfacial-energy-driven coarsening process, since the microstructures that minimize the interfacial energy are frequently unlike those that minimize the elastic energy.

This qualitatively different behavior of coarsening in the presence of elastic stress compared to that in the absence of stress is illustrated by many theoretical and experimental investigations. The classic work of Ardell and Nicholson [4] showed clearly that particles change their morphology from spheres to cuboids to plate- or rod-like shapes, and align along the elastically soft directions of the crystal as they increase in size, or equivalently, 〈L〉. Theoretical investigations have shown, for example, that elastic stress can give rise to large-scale particle migration through the matrix [5], [6], [7] and inverse coarsening where small particles grow at the expense of large particles [6], [8], [9], [10], [11].

In simulating Ostwald ripening in elastically anisotropic solids, it is absolutely necessary that the particle morphology be unconstrained, even if the particle shapes are not of direct concern. For example, two elastically and diffusionally interacting particles were found to be unstable with respect to coarsening; a small change in the area of one particle will always lead to the smaller particle shrinking [10]. In contrast, when the morphology of the particles was fixed to be a circle, the two particles could be stable with respect to coarsening. When the morphology of the particles is not constrained, the extra degrees of freedom associated with changes in morphology allow the particle to access evolutionary pathways that decrease the energy of the system and that are not open when the particle is constrained to a simple shape. These results suggest that it is necessary to solve a challenging multibody free-boundary problem in order to understand the statistically averaged properties of ensembles of coarsening particles in elastically stressed solids. There are several approaches that can be used to determine microstructural evolution in elastically stressed solids. Each approach has its own unique advantages. One is to employ a diffuse interface, or phase field, theory. There are several advantages to this approach: (1) one equation holds at all points in the system, (2) it provides an elegant description of the evolutionary process, (3) it allows topological changes such as particle coalescence, and (4) it is simple to solve numerically [6], [12], [13], [14], [15], [16]. The Ising model and discrete atom methods offer similar advantages [17], [18], [19]. The disadvantage of these methods, however, is that it is difficult to include large numbers of particles in the calculations with realistically chosen thermophysical parameters, although there has been some recent progress in this direction [16]. To avoid these difficulties, we will use a sharp interface description of the coarsening process, which provides excellent resolution of the interfaces [11], [20]. The challenge in using this approach, however, is that the evolution of the interfaces must be followed explicitly in time, and the resulting boundary integral equations can be difficult to solve numerically. In addition, it is necessary to develop an efficient algorithm for determining the elastic stress in the system. Fortunately, major advances in recent years in the algorithms used to solve these boundary integral equations now make this approach very efficient [21]. A fast multipole method for anisotropic elasticity has also been developed to obtain a similar increase in computational efficiency [22].

In this paper, we will study the microstructural evolution of an elastically stressed solid, focusing on the particle morphology and the particle spatial correlations. The coarsening kinetics and particle size distribution evolution is examined in [23] (hereafter, Paper II). The results are analyzed to answer questions such as:

• How do interparticle elastic interactions modify the particle shapes compared to the equilibrium shapes of isolated particles?

• How do particle correlations develop and how strong are they?

• Is the evolution self-similar in some way, and if not, is there an evolving attractor state that depends only on the characteristic strength of the elastic stress?

Using our large-scale simulations, we are now equipped to answer these questions. In this paper, we provide detailed analyses of the microstructural evolution obtained from the simulations. First, we give a qualitative description of the phenomena that occur during coarsening in both physical space and reciprocal (Fourier) space. Next, we present quantitative measures for the evolution of the particle morphology, and then for the spatial distribution of particles. Finally, we compare the results from various area fractions, focusing on the changes in the particle morphology due to interparticle elastic and diffusional interactions.