A stereological model of the degree
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I.
INTRODUCTION
THE degree of contact between grain boundaries and particles or pores is an important parameter in modeling second-phase inhibited grain growth. In the original Zener model, m this contact was implicitly assumed to be random. Aigeltinger and Exner I21 later developed a stereological method for quantifying this degree of contact, defined as the ratio of the length of triple line of intersection between grain boundaries and particle or pore surfaces to the length of intersection expected for random contact. For sintered copper, they found twice the amount of contact expected from random intersection throughout all stages of sintering. Patterson and coworkers 13,41 altered Aigeltinger and Exner's equation to remove a bias and again found the degree of contact to be constant throughout all stages of sintering, although having slightly lower values and ranging from 1.7 to 5.7 for different materials including copper, tungsten, UO2, and A1203 .13'41 It is rather surprising and not intuitively apparent why the degree of contact should remain constant throughout the large changes in pore morphology and volume fraction in the different stages of sintering. The objective of this study has been to develop a simple geometric model of grain and pore structure to explore via numerical analysis to understand the experimentally observed constancy of grain boundary-pore contact. The following model, suggested by Handwerker, tSl is based upon a tetrakaidecahedral grain structure with cylindrical grain edge pores to model intermediate stage sintering and with spherical pores on grain comers, edges, and surfaces to represent the microstructure during final stage sintering. Variation in pore radius, volume fraction, shape, location, and grain size via numerical simulation enables monitoring the effects of these variables Y. LIU, Graduate Student, and B.R. PATTERSON, Professor, are with the Materials Science and Engineering Department, The University of Alabama at Birmingham, Birmingham, AL 35294. Manuscript submitted July 9, 1992. METALLURGICAL TRANSACTIONS A
on the degree of contact, R, for intermediate and final stage sintering and the transition between the two stages. Tests of the model employing experimentally observed frequencies of pore locations and other microstructural parameters show constant degrees of contact throughout intermediate and final stage sintering. Analysis of the model explains the required morphological changes for the degree of contact to remain constant. It should be emphasized that the model cannot be expected to yield precise values of the various stereological parameters calculated, due to simplifying assumptions of geometric grain and pore shapes. Rather, the assumed geometric structure is used as a means for determining the microstructural changes required during sintering for the degree of grain boundary-pore contact to remain constant, as observed experimentally. In this vein, the particular geometric grain or pore shape used in the model, e.g., tetrakaidecahedra or cylinders, is ar
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