Modeling of grain growth characteristics in three-dimensional domains and two-dimensional cross sections
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I. INTRODUCTION
THE two-dimensional (2-D) grain growth of ideal single-phase materials has been studied extensively both by theoretical deductions[1] and by computer simulations.[2,3,4] However, the grain growth process represents in fact a threedimensional (3-D) phenomenon. While the generalization of von Neumann’s law from 2-D to 3-D was attempted recently,[5,6] it is still not widely accepted, and further development and validation of 3-D models is necessary.[5,7,8] When considering technical difficulties associated with direct experimental studies of 3-D grain growth, computer simulation represents one effective tool when exploring this 3-D microstructural event. Some Monte Carlo (MC) Potts simulations of the 3-D process have been reported.[9,10,11] In these simulations, the grain growth exponent was found to either asymptotically approach the theoretical value in the long time limit[10,12] or to be significantly lower than the theoretically expected value over the entire time domain.[9,13] It has not yet been confirmed that the grain growth kinetics obtained from cross sections is in agreement with that for the entire 3-D domain.[9,10] Furthermore, the normalized grain size distribution was reported to vary with time in 3-D MC Potts simulations,[14] a result that is re-examined here. In addition, the dimensions of the 3-D lattices employed in these earlier simulations were usually limited to 100 100 100, and thus, the simulations had to be terminated at relatively small grain sizes in order to preserve adequate numbers of grains for statistically significant analysis. Therefore, the previous 3-D MC Potts simulation results could not fully support the theoretical work. QIANG YU, Application Engineer, is with New Dimension Technolgies, East Windsor, CT 06088. YUJIE WU, Research Assistant, and SVEN K. ESCHE, Associate Professor, are with the Department of Mechanical Engineering, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030. Contact e-mail: [email protected] This article is based on a presentation made in the symposium entitled “Three Dimensional Materials Science” during the 2003 MS&T ’03: Materials Science & Technology Conference 2003 in Chicago, Illinois, on November 11 & 12, 2003, under the auspices of the ASM/MSCTS: Materials Science Critical Technology Sector Committee and the TMS/SMD: Structural Materials Division Committee. METALLURGICAL AND MATERIALS TRANSACTIONS A
Recently, a modified MC Potts algorithm was developed.[4] It produced the theoretically expected grain growth exponent over the entire time domain in 2-D MC Potts simulations. This MC Potts algorithm was expanded to model 3-D grain growth in this article.
II. MC POTTS ALGORITHM The 3-D MC Potts algorithm for isotropic, single-phase grain growth simulation was implemented as follows.[4] A continuum microstructure was mapped onto a 200 200 200 cubic lattice with 26 nearest neighbors (i.e., including the third nearest neighbors). A Monte Carlo Step (MCS) was defined as N reorientation attempts, where N 8
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