Three-dimensional monte carlo simulation of grain growth in zone-refined iron
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I. INTRODUCTION
GRAIN-SIZE changes during processing of metals and alloys may affect their strength, toughness, ductility, and corrosion resistance.[1] As a result, the prediction and control of grain-structure evolution have received considerable attention. Analytical models have been developed to understand the effects of time and temperature on the grain-size changes. These models often describe grain growth by the following parabolic equation, derived by considering growth due to the change in grain-boundary energy.[2–6] Ln ⫽ Ct
[1]
where L is the average grain size; n is the grain-growth exponent, which was derived[2–6] as 2.0; C is a temperaturedependent parameter; and t is time. However, measurements[7] on various metallic systems have shown that the grain-growth exponent may be more than 2.0 in many cases. The discrepancy between the experimental data[7,8] and the results from analytical models[2–6] can be attributed to the simplifying assumptions inherent in the models. The most common assumption of a spherical grain shape leads to an unrealistic estimation of the grain surface area and violates the fundamental topological mandate to share interfacial planes between neighboring grains in a dense crystalline solid. Furthermore, the previous analytical models considered growth of isolated grains which were unaffected by the neighboring grains in the matrix. A rigorous analytical treatment of grain growth, considering all its component physical processes and topological constraints, is not yet available. With the development of computational techniques in the last two decades that take into account both the grain-growth kinetics and the topological features, some of the common assumptions in the analytical models have been relaxed. As S. SISTA, formerly Graduate Student, Department of Materials Science and Engineering, The Pennsylvania State University, is Computer Aided Design Engineer, Intel Corporation, Hillsboro, OR 97124. T. DEBROY, Professor, is with the Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802. Manuscript submitted November 14, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS B
a result, more-realistic calculations of grain growth have become possible. Among the computational models, Monte Carlo (MC) simulation[7,8,10–22] has been widely used to simulate grain growth under both isothermal and nonisothermal conditions. Most of the calculations reported in the literature have been performed in two dimensions.[7,8,10–12,16,17] A few notable exceptions include some recent works, where calculations were done in three dimensions.[8,14,15,20–22] There are considerable computational challenges in conducting meaningful MC calculations. The main difficulty is in the processing of a large volume of data in a realistic time frame. For example, the simulation in a 100 ⫻ 100 two-dimensional grid, considering 200 MC simulation steps, involves 2.0 million (100 ⫻ 100 ⫻ 200) data points per variable. For a more realistic simulation, a three dimension
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