An examination of the interparticle contact area during sintering of W-0.3 Wt Pct Co
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I.
INTRODUCTION
A prediction of the densification rate in a compact is the ultimate outcome of a sintering model and depends on the sintering stress, which is the sintering force divided by the interparticle contact area. For this reason, being able to correctly predict the interparticle contact area is essential for developing accurate sintering equations. A two-sphere geometry has often been used to model densification occurring through various sintering mechanisms.[1–11] Figure 1(a) shows a two-sphere geometric model, where both powder particles are perfect spheres and are equal in size. The linear shrinkage DL/L0 is related to the center to center approach h and to the grain size G. DL/L0 5 2h/G
[1]
Several researchers[2–5] have approximated the relationship between the center to center approach, the neck size, and the grain size, by applying the Pythagorean triangle equation to the two-sphere geometry. Once only the largest term is considered, the following relation is obtained: h 5 x2/G
[2]
where x is the interparticle neck size. Combining Eqs. [1] and [2] results in an expression for the neck area between two contacting particles as a function of shrinkage and grain size:
D. MITLIN, formerly Graduate Student with the Department of Engineering Science and Mechanics, P/M Lab, The Pennsylvania State University, is Graduate Student with the Center for Advanced Materials, Lawrence Berkeley National Laboratory, and Department of Materials Science and Engineering, University of California, Berkeley, CA 94720. R.M. GERMAN, Brush Chair Professor in Materials, is with the Department of Engineering Science and Mechanics, P/M Lab, The Pennsylvania State University, University Park, PA 16802-6809. Manuscript submitted July 28, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
p x2 5
pG 2 (DL/L0) 2
[3]
where the linear shrinkage is related to the relative density r and the relative green density r0. DL/L0 5 1 2 (r0 /r)1/3
[4]
The reported coordination number of particles in a green compact is either 6[12] or 7.[13] Choosing 6 as the coordination number results in an expression for the interparticle contact area per particle, a. a 5 6p x2 5 3p G 2 (DL/L0)
[5]
Equation [5] is the contact area per particle assuming the lowest possible coordination number. The maximum possible coordination number of a powder particle in a compact is 14.[14] This coordination number is present once the individual particles in a compact assume the tetrakaidecahedron geometry. The corresponding upper limit to the interparticle contact area per particle is then defined by Eq. [6]: a 5 14p x2 5 7p G 2 (DL/L0)
[6]
Equations [5] and [6] will be referred to as the twosphere models, with a coordination number of 6 and 14, respectively. In real sintering situations, the particle coordination number changes with increasing compact density.[15–18]. Arzt[15] developed an analytic expression for the coordination number of a particle as a function of density, by assuming a random dense packing with a simple radial distribution function. Acco
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