Multiplicative Noise in Microstructure Evolution
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Multiplicative Noise in Microstructure Evolution K. G. Wang, M. E. Glicksman and P. Crawford Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180 ABSTRACT Multiparticle diffusion equations were modeled to simulate the dynamics of phase coarsening. Local environmental information and particle interactions within the microstructure are included in our simulations. These studies reveal that the growth rates of particles with the same radii can differ, and that particles with the average radius can grow, shrink, or remain conditionally stable. These results are in contrast to mean-field predictions, where particle growth rates are strictly deterministic. Multiparticle simulations prove that fluctuations occur in the particle growth rates, even at extremely low microstructural densities. Multiplicative noise provides a good basis to describe microstructural fluctuations. INTRODUCTION Predicting microstructure evolution in alloys remains one of the cornerstones of materials science. Phase coarsening is a common relaxation process in microstructural evolution that leads to a decrease in the excess total interfacial energy of the system. The system typically consists of separate and identifiable units, or domains, such as second-phase particles distributed within a contiguous matrix. During phase coarsening, larger particles tend to grow by absorbing solute at the expense of small particles that tend to dissolve by losing solute. Over time, this “competitive diffusion” results in an increase in the average size of the particle population, and in a concomitant decrease in the number density of particles. Indeed, the physical properties of two-phase materials, such as strength, toughness, ductility, and electrical conductivity, all depend on the material’s average particle size and particle size distribution function (PSD). Understanding and controlling microstructures in two-phase systems are clearly of widespread technological importance and raise fundamental materials science issues. In 1946, Todes [1] published a seminal paper suggesting the influence of capillary coarsening on late-stage phase separation. It took another 15 years, however, to complete the initial, and now classical, theory of capillary-mediated phase coarsening published by Lifshitz and Slyozov [2], and, independently, by Wagner [3]. This theory is often referred to as LSW theory, and retains full validity only in the limit of zero volume fraction. The prediction of LSW theory that the cube of the average length scale of second-phase particles increases linearly with time—a fact borne out by numerous experiments performed on many different systems over the past forty years—is shown to be valid even in the case of finite volume fractions. The theoretical PSD predicted from LSW theory and the associated rate constants controlling the kinetics of the evolving microstructural length scales, however, do not agree with careful quantitative experiments [4]. The evolution of two-phase microstructures, in the case of finite volume f
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