An analysis for estimating the probability of particle coalescence in liquid phase sintered systems
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C O M P L E T E wetting of the solid by the liquid is believed necessary for full densification during liquid phase sintering of materials containing a high volume fraction of solid phase. A particularly striking example of such wetting is afforded in the studies of Huppmann and Riegger I on the initial stages of sintering in Ni-W alloys in which the original polycrystalline tungsten powder dissolved into single crystals due to penetration of liquid nickel along the solid grain boundaries. However, after longer sintering periods, ~,3microstructural studies clearly indicate that some of the tungsten single crystals move about within the liquid and subsequently contact and coalesce into a single particle. The coalescence phenomena are somewhat complicated in the Ni-W system in that some of the coalescing grains have a thin film of liquid between them while others apparently do not. However, similar observation on the dissolution and subsequent coalescence phenomena have been made in the Cu-Fe system4 and in this case no film of liquid is observed between the two grains.4,5 Moreover, recent work on the Fe-Cu system5, has shown that the frequency of such coalescence is sufficiently great so that the coalescence phenomena may well account for a significant fraction of the microstructural coarsening that occurs in high Fe content alloys sintered for long times. Complete wetting implies that 2"r ~ ~'sswhere "/sLis the solid-liquid surface energy and ~ss the "average" solid-solid grain boundary energy. That some particles coalesce implies the existence of certain solid-solid orientations for which the wetting relationship does not hold. These orientations are most likely simple tilt, T. H. COl IRTNI~Y and ]. K. I.F.E are Prnfe~nr and A~i~tant Professor, respectively, Department of Metallurgical Engineering, Michigan Technological University, Houghton, MI 49931. Manuscript submitted August 27, 1979.
twist, or tilt plus twist boundaries of several degrees misorientation. Such low angle boundaries have a deep cusp in the boundary energy-misorientation relationship in that the boundary energy goes to zero at zero misorientation. Other boundaries (special boundaries) of greater misorientation also have energy cusps but the boundary energy at the cusp positions is not zero. To obtain a complete solution for the probability of two solid particles coming together in such a manner as to satisfy the condition that Yss ~ 2ysL where 7ss is the particular boundary energy at the contacting point (or area) of the particles is a formidable problem. Such an analysis would not only consider the probabilistic aspects of particle collisions at certain orientations but would also include detailed considerations of boundary energy anisotropy and the form of the boundary energy-misorientation relationship for all possible misorientations, In this paper we shall instead attempt to make a reasonable estimate of this coalescence probability utilizing geometrical probabilities and a physically plausible model of the energy-misorientation relationsh
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