Capillarity - Mediated Grain Growth in 3 - D

Space-filling in kinetically active 3-d network structures, such as polycrystalline solids at high temperatures, is treated using topological methods. The theory developed represents individual network elements—the polyhedral cells or grains—as a set of objects called average N -hedra, where N , the topological class, equals the number of contacting neighbors in the network. Average N -hedra satisfy network topological averages for the dihedral angles and quadrajunction vertex angles, and, most importantly, act as “proxies” for real irregular polyhedral grains with equivalent topology. The analysis provided in this paper describes the energetics and kinetics of grains represented as average N -hedra as a function of their topological class. The new approach provides a quantitative basis for constructing more accurate models of three-dimensional grain growth. As shown, the availability of rigorous mathematical relations for the curvatures, areas, volumes, free energies, and rates of volume change provides precise predictions to test simulations of the behavior 3-d networks, and to guide quantitative experiments on microstructure evolution in three-dimensional polycrystals. Background and Introduction The energetics and growth kinetics of polycrystals at high temperatures are important topics regarding microstructure evolution in the solid state. The kinetic foundation for understanding grain growth in two dimensions was established about 50 years ago by C.S. Smith [1], J. von Neumann [2], and W.W. Mullins [3]. J. von Neumann proved that a network of two-dimensional cells with uniform mobility, M , and boundary energy, γgb, obeys the kinetic law da dt = πγgbM 3 (n− 6), n ≥ 2, (1) where a is the area of a grain in two dimensions (R) with n sides or vertices. The von NeumannMullins relationship, Eq. 1, or so-called “n − 6 rule”, is the growth law for isotropic twodimensional polycrystals. It provides an excellent approximation for the behavior of grains forming networks in thin films, or, in general, the situation where one spatial dimension of a polycrystal is suppressed relative to the other two [4-7]. The “n−6” rule was known empirically to experimentalists [8, 9] before its mathematical basis was shown by von Neumann. The excess free energy (relative to a single crystal of equivalent volume) of an isotropic two-dimensional assembly of grains of unit thickness, ∆E, is given by the grain boundary free energies summed over the perimeters, Pi of the two-dimensional polygonal cells, ∆E2 = γgb 2 ∑ i Pi, (2) Materials Science Forum Vols. 467-470 (2004) pp 1025-1032 online at http://www.scientific.net © (2004) Trans Tech Publications, Switzerland Online available since 2004/Oct/15 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 130.203.133.33-17/04/08,16:04:01)