Coarsening of SiO 2 particles in copper and MnS inclusions in steel
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I.
INTRODUCTION
RELATIONSHIPS to describe the kinetics of growth and dissolution or coarsening of second phase particles have been derived by many workers. Their contributions can be classified by the following groups: 1. dissolution or growth of a single particle,~-4 2. dissolution or growth of particles with a large population assuming no interaction among the diffusion fields surrounding the particles, 5-8 3. dissolution or growth of particles with a large population considering one dimensional diffusion with a linear solute distribution in the matrix between nearby particles, 9'1~and 4. the growth of particles within a group of particles considering each particle as a point potential in a diffusion field which satisfies the Laplace equation. 1~'12 In the third group, previous workers have derived the following equation to describe coarsening:
-~3
-
-~3
=
8 OCeqorV2 9 RgT
t
[1]
where R and R0 are the average radii of particles at time t and zero time, respectively; D is diffusivity; Ceq is the solubility in the matrix; tr is the particle-matrix interfacial energy; Vm is the molar volume; Rg is the gas constant, and T is absolute temperature. Equation [1] is called the Liftshiftz-Slyozov-Wagner equation (i.e., the L.S.W. equaT. FUJII is Senior Researcher, Kawasaki Steel Corporation, Research Laboratories, l, Kawasaki-cho, Chiba 260, Japan; D. R. POIRIER is Associate Professor, Department of Metallurgical Engineering, University of Arizona, Tucson, AZ 85721; and M. C. FLEMINGS is Toyota Professor of Materials Processing and Head, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted January 25, 1982. METALLURGICAL TRANSACTIONS A
tion) and has been applied to the coarsening of precipitates in some binary metallic systems ~3'14 where the kinetics of coarsening are limited by diffusion through the matrix. Calculations of particle sizes using the L. S. W. equation have also been compared to measurements of the coarsening of silica in copper. 15 Included among the second group is an expression given by Greenwood9 who considered a system of N particles, in which the growth of a particle of radius R is given by N
4zrpR ~
=
D C e q - - . 2 .
RgT ,=, "x ,
-
[2]
where p is the density, Ri is the radius of the i-th particle, and (A/x)i is the area-to-length ratio of the effective diffusion path between the particles with radii R and Ri. The basic concept behind Eq. [2] is that diffusion is one dimensional and at steady-state between any two particles, with potentials given by the Thomson-Freundlich equation according to their respective radii. Weins and Cahn u'j2 pointed out a number of difficulties with Eq. [2]. The most serious drawback is that onedimension does not, in fact, describe diffusion between particles in three-dimensional space, and so the effective area-to-length ratio cannot be evaluated. They give an alternative approach in which diffusion among the particles is taken to be diffusion among point potentials distribute
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