Percolation of Diffusionally Evolved Two-Phase Systems
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Percolation of Diffusionally Evolved Two-Phase Systems Victor E. Brunini, Christopher A. Schuh, and W. Craig Carter Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, 02139
ABSTRACT Although the phase fractions in dual phase systems are often compared with the percolation threshold for a randomly-assembled composite, most two-phase systems are nonrandom by virtue of correlations introduced during processing or as a consequence of microstructural evolution. This study examines the two dimensional percolation threshold in systems with soft impingement, i.e., when the phase distribution is affected by diffusional interactions between growing second phase particles. Phase field modeling is used to simulate the nucleation and growth process, with many simulations conducted at various system sizes and equilibrium phase fractions to obtain percolation probabilities. The value of the percolation threshold in the thermodynamic limit is estimated based on the finite size scaling behavior of the system. The value of the critical exponent ν (the size scaling exponent) is also estimated. INTRODUCTION The properties of a multi-phase system can depend strongly on whether or not the individual phases percolate. For example, in a two phase system where one phase is highly conductive and the other is not, the sample will be highly conductive if and only if the conductive phase has percolated. Since properties transition sharply over a small range of phase fractions, precise knowledge of the threshold location is important for microstructural engineers to tailor material properties. In general, the percolation threshold is regarded as an empirical parameter, with precise estimates of the threshold being available only for a few specific families of microstructures. For example, the threshold has been mapped for structures consisting of randomly placed discs (or spheres in three dimensions) with various distributions of radii [1-3]. While these models provide a useful analog to the microstructures of some materials, they do not explicitly consider the physical phase correlations that result from microstructural evolution. Our purpose in this paper is to explore the percolation transition in a model two-phase system that evolves through diffusion, starting from an initially unstable concentration. PROCEDURE The phase field method is a useful tool for simulating the kinetics of a variety of processes, and has previously been applied to the solidification of eutectic systems [4]. Here we simulate a eutectic system derived from the model used by Wheeler et al. [4] This model is chosen instead of a simple double-well free energy model because it allows the simulation of
nucleation and growth at concentrations that would lie in the spinodal region of the double-well model. To reduce computational complexity the system is assumed to be isothermal and entirely solid, and we limit our discussion to a two-dimensional system. This leaves two phase field va
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