Application of percolation theory in predicting shape distortion during liquid-phase sintering
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I. INTRODUCTION
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. Recently, German[2] introduced percolation concepts to describe liquid-phase sintering processes and shape distortion. It is postulated that if the solid phase is connected macroscopically, then shape retention is possible, otherwise the component distorts. Liquid formation during sintering usually enhances densification. Traditional liquid-phase sintering begins by mixing two or more powders of differing compositions.[3] On heating, the powder with the lower melting point melts or reacts to form a liquid between the particles that engulfs the morerefractory phase. If the particle size is small, then the capillary forces from the wetting liquid enhance densification.[4] Microgravity liquid-phase sintering provided a means to examine sintering distortion and microstructural changes under the condition of minimized gravity. At low solid contents, the compacts distort and spheroidize, since the surface tension exceeds all other forces in microgravity liquidphase sintering.[5] In ground-based sintering, gravity provides a progressive stress gradient on the powder compact that induces grain contact, settling, and anisotropic deformation (evident as shape distortion).[6,7,8] An “elephant-foot” shape is frequently observed for distorted W-Ni-Fe powder compacts. In addition, the solid and the liquid seldom have the same density. Hence, gravity induces microstructural separation gradients that result in nonuniform microstructures and properties in the sintered product.[7] The final microstructure consists of gradients which reflect the large density difference between the solid and liquid phases.[9] Gravity produces a systematic JIANXIN LIU, Researcher, Mechanical Properties, ANISH UPADHYAYA, Researcher, Materials Development, and RANDALL M. GERMAN, Brush Chair Professor in Materials, are with The Pennsylvania State University, University Park, PA 16802-6809. Manuscript submitted October 14, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
change in solid content, contiguity, connectivity, and grain size over relatively small distances.[10–13] Shape distortion will be associated with microstructural parameter changes in this percolation mode. The dominant feature is the connection or bonding between neighboring grains. A sufficient level of bonding is necessary to form a rigid skeleton and inhibit shape distortion during liquidphase sintering. This study addresses a critical condition via calculations based on percolation theory, to judge if a rigid skeleton is formed to resist shape distortion. The model is used to predict shape distortion under both microgravity and ground-based liquid-phase sintering conditions. II. ELEMENTS OF PERCOLATION THEORY The foundation of percolation theory is well explained in various references.[14–20] F
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