Journal of the Ceramic Society of Japan 103 [2] (1995) Paper Sintering and Grain Growth Rates of Two Spheres with Different Radii Hidehiko TAN

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Journal of the Ceramic Society of Japan 103 [2] 138-143 (1995) Paper Sintering and Grain Growth Rates of Two Spheres with Different Radii Hidehiko TANAKA National Institute for Research in Inorganic

1 Journal of the Ceramic Society of Japan 103 [2] (1995) Paper Sintering and Grain Growth Rates of Two Spheres with Different Radii Hidehiko TANAKA National Institute for Research in Inorganic Materials, 1-1, Namihi, Tsuhuba-shi, Ibaraki 305 [Received July 18, 1994; Accepted November 22, 1994] Rate equations for sintering and grain growth have been proposed on the two-sphere model that simulates shrinkage processes during heating of a ceramic pow der compact. In the model it was assumed that total ex cess energy associated with surfaces and grain bounda ry drives mass transport in sintering and grain growth. The rate equations formulated here show that sintering and grain growth are activated by the same kinds of driving force and processes. They show quan titatively that the sintering rate increases with a decrease in grain boundary energy and with a decrease in the grain size difference. It is found also that the grain growth of a larger grain is much slower than a smaller grain and that the growth rate of a smaller grain is accelerated remarkably when its grain size becomes smaller compared with a lager grain. Key-words: Sintering, Grain growth, Rate equation, Free energy theory, Two-sphere model Fig. 1, Geometry of two spherical grains for the sintering model. It is assumed that the sintering proceeds by expanding of grain boundary and grain radius with constant volume of each grain. As shown by the thick solid lines, the two grains contact each other with the grain boundary. When sintering proceeds, the boundary area expands, but the volumes of the two grains keep constant, which is shown by the hairlines.

2 @ Journal of the Ceramic Society of Japan 103 [ 2 ]

3 Fig. 2. Geometry of two spherical grains for the grain growth model. The grain growth is defined as the growth of large grain and shrinkage of small grain with constant grain boundary area and constant total volume. As shown by the thick solid lines, the two grains contact each other with the grain boundary. When grain growth proceeds, the large grain grows and the small grain shrinks, but the boundary area remains constant, which is shown by the hairlines.

4 @ Journal of the Ceramic Society of Japan 103 [ 2 ] Fig. 3. Total excess free energy function of the Eq. (6), ƒµ, for the sintering. The ƒµis plotted against X1 under the conditions of 2 ƒ =0.5 and R0= in (a) and (b), and R0=0.5 and 2ƒ =0-1 in (c). X1 is a rela tive distance between the center of sphere and the grain boundary, ƒô1/ƒá1. 2ƒ and R0 are defined as a surface to grain boundary energy ra tio, 2ƒÃgb/ƒÃs, and a radius ratio of two grains, ƒá02/ƒá01, respectively (refer to Figs. 1 and 2). Fig. 4. The function, f(x1), in the Eq. (10) for the grain growth. The f (X1) is plotted against X1 under the conditions of 2 ƒ =0.5 and R0= in (a) and (b), and R0=0.5 and 2ƒ =0-1 in (c). Fig. 5. Sintering rate, d(ƒ L/L0)/dt, as a function of a and R0. The sintering rate is plotted against X1 under the conditions of 2 ƒ =0.5 and R0= in (a) and (b), and R0=0.5 and 2ƒ =0-1 in (c). Fig. 6. Grain growth rates, dr1/dt for large grain and dr2/dt for small grain, as a function of a and R0. The grain growth rates are plotted against X1 under the conditions of 2ƒ =0.5 and R0= in (a) and (b), and R0=0.5 and 2ƒ =0-1 in (c). The dr1/dt and the dr2/dt are referred as Ri and R2 in legends of the figures. The dr1/dt is very much smaller than the dr2/dt and it is shown to be nearly 0 in the figures.

5 Fig. 7. Dependence of (a) sintering rate and (b) grain growth rate on grain size ratio, R0. The two rates are plotted against R0 with different X1 under the condition of 2ƒ =0.5.

6 @ Journal of the Ceramic Society of Japan 103 [2] ) Y. Inomata, Proc. First Int. Symp. on Ceramic Components for Engine, Ed. by S. Somiya, E. Kanai and K. Ando, KTK Scientific Publ., Tokyo (1984) pp ) W. D. Kingery, H. K. Bowen and D. R. Uhlmann, "Introduc tion to Ceramics, Second Edition", John Wiley and Sons, 4) 5) M. F. Ashby, Acta Metall., 22, (1974). F. B. Swinkels and M. F. Ashby, Acta Metall., 29, (1981). 6) R. L. Coble, J. Am. Ceram. Soc., 56, (1973). 8) I. M. Stephenson and J. White, Trans. J. Br. Ceram. Soc., 66, (1967). 9) W. J. Huppmann and H. Riegger, Acta Metall., 23, (1975). 10) H. F. Fischmeister, E. Arzt and L. R. Olsson, Powder Metall., 4, (1978). 12) M. Blanc and A. Mocellin, "Sintering Processes, Materials Science Research", Vol. 13, Ed. by G. C. Kuczynski, Ple num Press, New York (1980) pp ) R. M. German, "Liquid Phase Sintering", Plenum Press, New York (1985) pp Fig. A1. Relation between X1 and X2. Two parameters are defined as a relative distance between sphere's center and grain boundary, ƒô1/ƒá1 and ƒô2/ƒá2, respectively.

Studies of Foot Form for Footwear Design (Part 9) : Characteristics of the Foot Form of Young and Elder Women Based on their Sizes of Ball Joint Girth

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