Developments in the theory of Ostwald ripening since the classic work of I. M. Lifshitz and V. V. Slyozov (LS) are reviewed and directions for future work are suggested. Recent theoretical work on the role of a finite volume fraction of coarsening phase on the ripening behavior of two-phase systems is reformulated in terms of a consistent set of notation through which each of the theories can be compared and contrasted. Although more theoretical work is necessary, these theories are in general agreement on the effects of a finite volume fraction of coarsening phase on the coarsening behavior of two-phase systems. New work on transient Ostwald ripening is presented which illustrates the broad range of behavior which is possible in this regime. The conditions responsible for the presence of the asymptotic state first discovered by LS, as well as the manner in which this state is approached, are also discussed. The role of elastic fields during Ostwald ripening in solid-solid mixtures is reviewed, and it is shown that these fields can play a dominant role in determining the coarsening behavior of a solid-solid system.

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The Theory of Ostwald Ripening

The critical shear stress τc to move a dislocation through a random array of obstacles in the glide plane is calculated using a statistical theory. The result is an expression for τc in terms of the obstacle concentration, the line tension of the dislocation, and of the interaction force between the dislocation and a single obstacle. Fleischer's solution of the same problem is not reproduced by the statistical theory. Quantitatively the two results are not very different, but our new result is supported by some recent experimental evidence. Furthermore the theory provides a definite prescription how to combine the concentrations and interaction forces of obstacles of different kinds in the expression for τc.

A Statistical Theory of Solid Solution Hardening

Bulk metallic glass (BMG) formers are multicomponent alloys that vitrify with remarkable ease during solidification. Technological interest in these materials has been generated by their unique properties, which often surpass those of conventional structural materials. The metastable nature of BMGs, however, has imposed a barrier to broad commercial adoption, particularly where the processing requirements of these alloys conflict with conventional metal processing methods. Research on the crystallization of BMG formers has uncovered novel thermoplastic forming (TPF)-based processing opportunities. Unique among metal processing methods, TPF utilizes the dramatic softening exhibited by a BMG as it approaches its glass-transition temperature and decouples the rapid cooling required to form a glass from the forming step. This article reviews crystallization processes in BMG former and summarizes and compares TPF-based processing methods. Finally, an assessment of scientific and technological advancements required for broader commercial utilization of BMGs will be made.

Processing of Bulk Metallic Glass

Effect of interstitial solutes on the strength and ductility of titanium

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.

P. Haasen

Papers

Nucleation and growth of γ′-Precipitates in Ni-14 at.% Al

Atom probe field ion microscopy of a FeNiB glass

Solid solution hardening of gold and other f.c.c. single crystals

Kriechen von Silizium-Einkristallen

Measurement of dislocation velocities in Cu-Al single crystals—II☆