There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

■ Abstract The phase-field method has recently emerged as a powerful computational approach to modeling and predicting mesoscale morphological and microstructure evolution in materials. It describes a microstructure using a set of conserved and nonconserved field variables that are continuous across the interfacial regions. The temporal and spatial evolution of the field variables is governed by the Cahn-Hilliard nonlinear diffusion equation and the Allen-Cahn relaxation equation. With the fundamental thermodynamic and kinetic information as the input, the phase-field method is able to predict the evolution of arbitrary morphologies and complex microstructures without explicitly tracking the positions of interfaces. This paper briefly reviews the recent advances in developing phase-field models for various materials processes including solidification, solid-state structural phase transformations, grain growth and coarsening, domain evolution in thin films, pattern formation on surfaces, dislocation microstructures, crack propagation, and electromigration.

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Phase-Field Models for Microstructure Evolution

Welding Metallurgy of

The phase-field method has become an important and extremely versatile technique for simulating microstructure evolution at the mesoscale. Thanks to the diffuse-interface approach, it allows us to study the evolution of arbitrary complex grain morphologies without any presumption on their shape or mutual distribution. It is also straightforward to account for different thermodynamic driving forces for microstructure evolution, such as bulk and interfacial energy, elastic energy and electric or magnetic energy, and the effect of different transport processes, such as mass diffusion, heat conduction and convection. The purpose of the paper is to give an introduction to the phase-field modeling technique. The concept of diffuse interfaces, the phase-field variables, the thermodynamic driving force for microstructure evolution and the kinetic phase-field equations are introduced. Furthermore, common techniques for parameter determination and numerical solution of the equations are discussed. To show the variety in phase-field models, different model formulations are exploited, depending on which is most common or most illustrative.

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An introduction to phase-field modeling of microstructure evolution

The physical basis behind the Ostwald ripening process for two-phase mixture has been reviewed in detail, using the various theories developed to describe this process. The Ostwald ripening, also termed second phase coarsening, is generally thought to be slow, diffusion-controlled process which occurs subsequent to phase separation under extremely small under-saturation levels. The major advance for the description of this process was made when Lifshitz, Slyozov and Wagner (also known as the LSW theory) published their papers more than fourty years ago. This classical LSW theory predicts that the ripening kinetics and the particle size distribution function are applicable to dilute systems only [i.e. when the volume fraction (Q) of second phase approaces zero: Q → 0], in which particle-particle interactions are not important. After the publication of the LSW theory, many experimentalists tested the veracity of the theory. Experimentalists have confirmed the prediction of self-similar ripening behavior at long times. However, virtually none of the reported distributions are of the form predicted by the LSW theory. The reported distributions are generally broader and more symmetric than the LSW predictions. It was later realized that a major problem with the LSW approach was the mean field nature of the kinetic equation. In order to remove the zero volume fraction assumption of the LSW theory, the many theories have been developed based on the statistically averaged diffusion interaction of a particle of given size class with its surroundings, using both analytic and numerical methods. Many attempts to determine the statistically averaged growth rate of a particle either do not account for the long-range nature of the diffusional field surrounding the particle, and/or employed ad hoc assumptions in an attempt to account for the diffusional interactions between particles. The strength of the diffusional interactions between particles stems from the long range Coulombic nature of the diffusion field surrounding a particle. As a result, particle interactions occur at distances of many particle diameters and restrict the validity of LSW theory to the unrealistic limit of zero volume fraction of coarsening phase. More realistic models of the ripening process at finite-volume fractions (Q) of coarsening phase have been proposed by various workers such as Brailsford-Wynblatt (1979), Voorhees-Glicksman (1983), Marqusee-Rose (1984), Tokuyama-Kawasaki (1984), Enomoto-Tokuyama-Kawasaki (ETK) (1986), and Yao-Elder-Guo-Grant (YEGG) (1993) models. Although a great deal of progress has been made in understanding Ostwald ripening, a fully satisfactory approach has not yet been found, and it has remained a vexing problem in the field. At present, it is very difficult to determine which of these theories best describes coarsening at finite volume fraction. The statistical mechanical theories, developed to describe systems in which Q ≪ 1, employed the same microscopic equation to describe the coarsening rates of individual particles, but different techniques to perform the statistical averaging. In addition, these theories can be distinguished on yet a finer scale. All of the theories predict that the rate constant will vary as Q1/2 in this low volume fraction limit. These theories predict that the scaled time-independent particle radius distributions become broader and more symmetric than those predicted by LSW as the volume fraction increases. Clearly more experimental and theoretical work is necessary in order to settle the subtle disagreement now existing between the various Ostwald ripening theories.

Review Progress in Ostwald ripening theories and their applications to nickel-base superalloys Part I: Ostwald ripening theories

Phase separating systems present a unique opportunity for designing composites with hierarchical microstructure at different length scales. We report here our success in synthesizing phase separating metallic glasses exhibiting the entire spectrum of microstructural possibilities expected from a phase separating system. In particular, we report novel core shell and hierarchical structures of spherical glassy droplets, resulting from critical wetting behavior and limited diffusion. We also report synthesis of a bulk phase separating glass in a metallic glass system. The combination of unique core shell and hierarchical structures in metallic glass systems opens a new avenue for the microstructure design of metallic glasses.

Phase Separating Bulk Metallic Glass: A Hierarchical Composite

Phase field study of precipitate rafting under a uniaxial stress

We examine the influence of elastic stress on the Ostwald ripening kinetics of two elastically and diffusionally interacting, misfitting spherical particles in an anisotropic crystal. The coupled equations of elasticity and diffusion are solved analytically in series form in a bispherical coordinate system when the matrix supersaturation is small, local equilibrium obtains at the interface, particle and matrix possess the same cubic elastic constants, and the stress engenered by compositional inhomogeneity is negligible. Expressions are obtained for the matrix composition field, the local normal interfacial velocities of the particles, the isotropic particle growth rates, and the velocity of the particles' centers of mass. Inverse coarsening, or the growth of a smaller particle at the expense of a larger particle, is predicted for particle alignments along the elasticity soft 〈100〉 directions in nickel and for the 〈110〉 and 〈111〉 directions in molybdenum. Coarsening rates are often significantly enhanced for other particle orientations with respect to those of the stress-free case. The elastic stresses also change the functional dependence of the particle growth rate on particle size suggesting that the temporal exponents observed during classical ripening may not obtain in stressed systems. These predictions indicate that elastically-induced preferential coarsening strongly influences microstructural development in two-phase coherent alloys.

The coarsening kinetics of two misfitting particles in an anisotropic crystal

Development of solidification microstructure in a laser welded Ti/Ni dissimilar binary couple is presented. At the fusion interfaces in both Ti and Ni, growth of the base metal grains into the weld pool is inhibited by the presence of composition gradients in the melt. Ti2Ni dendrites grow toward the base metal at the Ti fusion interface. In the Ni side, appearance of a nickel solid solution phase is followed by layers of Ni3Ti, Ni3Ti+NiTi eutectic, and NiTi. NiTi dendrites and Ti2Ni constitute the microstructure in the middle of the weld. Isolated titanium dendrites are observed throughout the weld, but their preferential occurrence toward the top surface of the weld is more prominent. Results are rationalised on the basis of interplay of the transport processes in the weld with the thermodynamics of the Ti-Ni system.

T.A. Abinandanan

Papers

Phase Separating Bulk Metallic Glass: A Hierarchical Composite

Phase field study of precipitate rafting under a uniaxial stress

The coarsening kinetics of two misfitting particles in an anisotropic crystal

Microstructure development during dissimilar welding: case of laser welding of Ti with Ni involving intermetallic phase formation

Grain boundary effects on spinodal decomposition. II. Discontinuous microstructures