Calculation of the product phase grain boundary area during solid state transformations
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_2fvvo Av = 3 .Io
Sve~ " exp(-Vv~) ' dVv~
where Vv.~ is the extended volume fraction of the product phase, and Sv,~ is the total extended product pKase-matrix interfacial area per unit volume. If the growth rate depends on particle size or particle size and time, then,
3fvv. Av = 5 3o
Sv, x " exp(-Vv~) 9 dVvex
The results are applicable to any arbitrary functional form of nucleation rate. The result for size dependent growth rate is approximate; however, the error involved in this approximation is less than - 1 0 pct. The analysis demonstrates that A v, and also the grain size of transformed structure, are basically determined by the path of microstructural evolution described by the variation of product phase-matrix interracial area per unit volume with the product phase volume fraction, and do not explicitly depend on any other variables. The analysis is also applicable to nonisothermal and continuous cooling transformations.
I.
INTRODUCTION
A large number of solid state transformations occur by nucleation and growth process. Nuclei of the product phase form and grow, producing a population of discrete convex particles in the matrix of the parent phase. The size distribution of these particles evolves with time due to growth of the existing particles and further nucleation. As the transformation proceeds, the impingement eventually occurs; i.e., the discrete product phase particles come into mutual contact. The impingement leads to the formation of new microstructural features. The pairwise impingement of the product phase particles leads to the formation of the product phase grain boundary surfaces. The number of such grain boundaries is equal to the number of interparticle contacts. The number and total surface area of the product phase grain boundaries increase with time. The complete transformation of the parent phase to the product phase yields the microstructure consisting of the polyhedral grains of the product phase ,;eparated by the grain boundaries. Part of this sequence of microstructural changes may be expected to occur during any solid state transformation. Thus, the theoretical calculation of the total product phase grain boundary area per unit volume is of interest. Meijring ttl has calculated the product phase grain boundary area per unit volume at the end of complete transformation of one phase to another in the solid state, under the following assumptions: A.M. GOKHALE, on leave from the Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur-208016, India, is Visiting Associate Professor, School of Materials Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245. Manuscript submitted October 5, 1987. METALLURGICALTRANSACTIONS A
(1) Random spatial distribution of nuclei. (2) Spherical particle shape. (3) Constant growth rate of the product phase particles. (4) Site saturation or constant nucleation rate. However, in general, the nucleation rate can vary with time, and the growth rate can be a function of particle size and time. It is the purpose of
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