Modeling of supersolidus liquid phase sintering: I. Capillary force
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I.
INTRODUCTION
THE capillary force plays an important role during liquid phase sintering (LPS). Many models have been proposed for the calculation of capillary force.V-6] Based on a model geometry with two identical spherical particles connected by a liquid bridge, Kingerytq derived the first quantitative expression for the driving force arising from the curvature of the liquid meniscus. Heady and Cahn[2] showed that in addition to the meniscus curvature, the liquid-vapor surface tension also contributed to the capillary force. Gessinger et al.t4] pointed out that these force calculations are only applicable to the initial stage of sintering, i.e., with no coalescence. They calculated the capillary force by considering particle coalescence; however, the geometrical change of the particles as coalescence occurs is neglected in their model. In classic LPS, the amount of liquid phase is predetermined by the lever rule. In supersolidus liquid phase sintering (SLPS), the volume fraction of liquid increases with temperature above the solidus. Therefore, capillary force analysis in SLPS should take into account the simultaneous occurrence of particle coalescence and liquid volume increase, tn In this respect, there is a fundamental difference between the capillary force models in classic LPS as compared to SLPS. The initial stage of SLPS involves the nucleation of the liquid phase along the grain boundaries, in the interparticle neck region, and within the grains depending on the starting powder microstructure. A typical microstructure prior to grain fragmentation is shown in Figure l(a), which is a micrograph of a nickel-base alloy quenched from 1130 ~ Adjacent to the neck region, where particle coalescence has occurred, the spherical geometry of the particles is not retained. A more clear representation of particle distortion is shown in Figure l(b), which is a three dimensional (3-D) micrograph of bronze particles sintered above the solidus YIXIONG LIU, Research Associate, RAJIV TANDON, Research Assistant, and RANDALL M. GERMAN, Brush Chair Professor in Materials, are with the Engineering Science and Mechanics Department, The Pennsylvania State University, University Park, PA, 16802-6809. Manuscript submitted October 10, 1994. METALLURGICALAND MATERIALS TRANSACTIONS A
(a)
(b) Fig. 1--Scanning electron micrographs of (a) nickel-base alloy quenched from 1130 ~ showing the nucleation of the liquid phase along the grain boundaries, and (b) bronze powder sintered at 860 ~ for 15 min showing particle distortion.
temperature for 15 minutes. Hence, models must account for the particle shape change as shrinkage occurs. Viscous flow modeling of the sintering process has shown that a VOLUME 26A, SEPTEMBER 1995--2415
sition. Mathematically, an oval function y2(x) with an adjustable major axis satisfies the two boundary conditions. By solving Eq. [1] (Appendix A), the major axis a can be expressed by the following relation: (R -
a
Fig. 2 - - A schematic drawing of the two-particle model geometry showing the parameters use
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