Analysis of early-stage sintering with simultaneous surface and volume diffusions
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INTRODUCTION
E A R L Y - s t a g e sintering can take place by several mechanisms (Figure 1). II-61 Sintering by simultaneous mechanisms in the early stage has been noted in the literature, u,6-1~ and it has been studied using computer simulation tH] or by applying numerical techniques to differential equations, u~ However, analytical solutions are very much desired. Although analytical analyses have been used in certain simultaneous sintering cases, I91 satisfactory analytical solutions have yet to be developed. Works by German and Munir [~~ and Rockland I9] are worth mentioning here, partially because they represent the better-known works on simultaneous sintering and partially because both this work and their works use the same general approach of assuming that the neck growth rates are directly additive. German and Munir ~l~ investigated two cases using a semianalytical technique (numerical solutions to differential equations). The first case was a combination of nondensifying volume diffusion (path 4 in Figure 1) and surface diffusion. The second case was a combination of nondensifying volume diffusion and grain-boundary diffusion. (There are two types of volume diffusion: densifying and nondensifying, depicted by paths 5 and 4 in Figure 1, respectively. Although German and Munir t1~ did not state which volume diffusion they were analyzing, a careful examination of the references cited by them indicates that the nondensifying type was analyzed.) In each case, the numerical solutions for neck growth and the kinetic exponent were given graphically. Herring's exponent [~2] and densification were not treated. Rockland t9] investigated two cases analytically. The first case was a combination of surface diffusion and nondensifying volume diffusion. The second case was a combination of grain-boundary diffusion and densifyingtype volume diffusion. In the first case, neck growth was treated, but no formal treatment was given to the second case. The ratio of the instantaneous neck growth rates between the simultaneous mechanisms was very briefly touched upon. Also touched upon were the relative contributions to neck size by the individual mechanisms which, if not incorrect, were severely oversimplified. No treatment was given to the kinetic exponent, Herring's exponent, or densification. JAMES C. W A N G , formerly with Kemet Electronics Corporation, Greenville, SC, is Research Staff Member with Lanxide Corporation, Newark, DE 19714-6077. Manuscript submitted December 16, 1988. METALLURGICAL TRANSACTIONS A
Since the mathematical approach to simultaneous sintering is still very much in its infancy, before developing a comprehensive model which includes all the possibly simultaneous mechanisms, a simple model seems appropriate. This author chooses the simple case which involves only two simultaneous mechanisms: surface diffusion (path 3 in Figure 1) and volume diffusion from the grain boundary (i.e., the densifying type, path 5 in Figure 1). To the author's knowledge, no one single analytical model capable of provid
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