Simulation of percolation structure of grain bonding in liquid-phase sintering by three-dimensional grain structure reco
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I. BACKGROUND
MODELS that predict the properties of disordered media have emerged from percolation concepts first introduced by Broadbent and Hammersley.[1] Percolation theory predicts when a system is macroscopically connected. This macroscopic connectivity is important to many phenomena. Classical percolation theory centers around two problems.[2,3] In the bond percolation problem, the bonds of the network are either occupied, randomly and independently of each other with probability p, or are vacant with probability (1 2 p). For a large network, this assignment is equivalent to removing a fraction (1 2 p) of all bonds at random. Two sites are called connected if there is at least one path between them consisting solely of occupied bonds. A set of connected sites bound by vacant bonds is termed a cluster. If the network is of very large extent and if p is sufficiently small, the size of any connected cluster is small. But if p is close to 1, the network should be entirely connected, apart from occasional small holes. At some well-defined value of p, there is a transition in the random network structure from a macroscopically disconnected structure to a connected one; this value is called the bond percolation threshold, pcb. It is the smallest fraction of randomly occupied bonds below which there is no infinite cluster of occupied bonds. Similarly, in the site percolation problem, sites of the network are occupied with probability p or vacant with probability (1 2 p). Two nearest-neighbor sites are called connected if they are both occupied, and connected clusters on the network are again defined in the obvious way. There is a site percolation threshold pcs above which an infinite cluster JIANXIN LIU, Senior Materials Engineer, is with the Prometal Division, Extrude Hone Corporation, Irwin, PA 15642. YANG LIU, Director of Computer Simulation, and RANDALL M. GERMAN, Brush Chair Professor in Materials and Director, are with the Center for Innovative Sintered Products, The Pennsylvania State University, University Park, PA 16802-6809. Manuscript submitted August 18, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS A
of occupied sites spans the network. Percolation problems generally depend on site and bond properties simultaneously. Computer simulations of the percolation structure often employ ordered lattice networks. In site percolation, all lattice sites are initially empty, and then filled randomly to a desired probability. Nearest neighbors are considered perfectly connected or bonded. The derivation of the exact values of the bond percolation threshold, pcb, and site percolation threshold, pcs, has been possible for regular lattices.[3,4] For some three-dimensional (3-D) networks, percolation thresholds have been calculated by Monte Carlo simulations and other techniques.[3,4] Percolation in a topologically random network during liquid-phase sintering, in which the grain size and position are randomly distributed and the coordination number varies from site to site, is of great interest. In a liquid-phase sint
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