Calculation of the product phase grain edge length and quadruple points per unit volume during solid state transformatio
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Vvex
= 0.155
(Sv~,)2 Exp(-Vv~,) dVv~
"0
and,
71" ~Vve-t av = 9-6 J0
(Sv,x)3 Exp(-Vv~) dVv,x
where Vvex is the extended volume fraction of the product phase, and Svex is the total extended product phase-matrix interfacial area per unit volume. It is assumed that the spatial distribution of the product phase nuclei is random. The analysis is applicable to any arbitrary time dependent nucleation rate, and any arbitrary time and/or size dependent growth rate, provided that the product phase particles have an equiaxed shape in the extended structure. The analysis is applicable to isothermal transformations as well as nonisothermal and continuous cooling transformations. It is shown that L v~a~ and Qv are basically determined by the path of microstructural evolution described by the variation of product phase-matrix interfacial area per unit volume with the volume fraction of the product phase. The ASTM grain size number of the transformed microstructure and average grain shape can be calculated from these results.
I.
grain boundary
INTRODUCTION
S O L I D state transformations occurring by a nucleation and growth process involve a sequence of microstructural changes. Transformation begins with the nucleation of particles of the product phase (say fl-phase) in the matrix of the parent phase (say a-phase). The size distribution of fl-phase particles changes with time due to the growth of existing particles and further nucleation. The increase in the volume fraction of product phase leads to impingement, as discrete particles come into mutual contact. The pairwise impingement of fl-phase particles leads to the formation of tiff grain boundaries or product phase grain boundaries (see Figure 1). As the transformation proceeds, the extent of impingement increases, and this leads to formation of flflfl grain edges or product phase grain edges (see Figure 2). The total length of these grain edges and their number increase with time. Eventually the grain edges are also impinged upon by other particles, leading to formation of flflflfl quadruple points or "product phase quadruple points." A product phase quadruple point is a junction point of four product phase grain edges and six product phase grain boundaries. Complete transformation of a-phase to fl-phase in solid state gives rise to a microstructure consisting of A . M . GOKHALE, on leave from Indian Institute of Technology, Kanpur, India, is Visiting Associate Professor, School of Materials Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245. Manuscript submitted February 22, 1988. METALLURGICAL TRANSACTIONS A
,l'y~.,, .,
'.'.'.'"2 L
.:..,.;j ,"
\
"
/
triple line Fig. 1 --Pairwise impingement of two fl-particles.
polyhedral grains of fl-phase, tiff grain boundaries, flflfl grain edges, and flflflfl quadruple points. Thus, the calculation of total product phase grain boundary area, grain edge length, and the number of quadruple points per unit volume is of interest. The procedure for calculation of the total VOLUME 20A, MARCH 1989--349
IB
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