Energetics of polycrystals
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The energetics of polycrystalline solids at high temperatures is treated using topological methods. The theory developed represents individual irregular polyhedral grains as a set of symmetrical abstract geometric objects called average N-hedra (ANH’s), where N, the topological class, equals the number of contacting neighbor grains in the polycrystal. ANH’s satisfy network topological averages in three-dimensions for the dihedral angles and quadrajunction vertex angles, and, most importantly, can act as “proxies” for irregular grains of equivalent topology. The present analysis describes the energetics of grains represented as ANH’s as a function of their topological class. This approach provides a quantitative basis for constructing more accurate models of three-dimensional well-annealed polycrystals governed by capillarity. Rigorous mathematical relations, derived elsewhere, for the curvatures, areas, and volumes of ANH’s yields quantitative predictions for the excess free energy. Agreement is found between the analytic results and recently published computer simulations.
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- 1.C. S. SMITH, “Grain Shapes and other Metallurgical Applications of Topology,” Chapter in Metal Interfaces (American Society for Metals, Cleveland, OH, 1952) p. 65.Google Scholar
- 2.J. VON NEUMANN, in “Metal Interfaces, written discussion” (American Society for Metals, Cleveland, OH, 1952) p. 108.Google Scholar
- 9.Idem., in “Metal Interfaces, written discussion” (American Society for Metals, Cleveland, OH, 1952) p. 110.Google Scholar
- 10.F. GRANER, Y. JIANG, E. JANIAUD and C. FLAMENT, Phys. Rev. E 63 (2001) 402.Google Scholar
- 17.D. WU, Private Communication, 2003.Google Scholar
- 19.Idem. (http://geom.umn.edu/software/evolver)), (2002).
- 22.S. J. COX and M. A. FORTES, Phil. Mag. Lettr 83 (2003) 28.Google Scholar
- 23.S. J. COX, Private Communication, 2003.Google Scholar
- 25.J. A. GLAZIER and B. PRAUSE, in “Foams, Emulsions and Their Applications” edited by P. Zitha et al. (MIT-Verlag, Bremen, 2000) p. 120.Google Scholar
- 27.M. M. LIPSCHUTZ, “Differential Geometry, Schaums Outline Series” (McGraw-Hill, New York, 1981).Google Scholar
- 31.E. W. WEISSTEIN, “Reuleaux Tetrahedron,” MathWorld, 2004, http://mathworld.wolfram.com/ReuleauxTetrahedron.html..
- 33.E. E. UNDERWOOD, “Quantitative Metallography, in Metals Handbook,” 9th ed. (American Society for Metals, Metals Park, OH, 1985) p. 123.Google Scholar
- 34.J. C. RUSS, “The Image Processing Handbook,” 2nd ed. (CRC Press, Boca Raton, 1995).Google Scholar