Application of microstructure modelling to the kinetics of recrystallization
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the product phase in the microstructure, and let V v , S v , and M v be the corresponding extended measures. Wl~en the nucleation sites are randomly distributed in space, the relationships between V v , S v, and M v and their extended values are given by~ d V v = (1 or
V v = l - exp ( - Vv.,)
[2]
S v = (l -
[3]
Vv)Sv.
and 7 M v = (1 -
Vv)
All models for phase transformations occurring by nucleation and growth must include a consideration of impingement unless the volume fraction of the particulate phase is small throughout the evolution process. Otherwise, the values of the global properties derived from any model are overestimates of the measured properties. The overestimates result because in a modelled structure, the particles ignore each other's presence, and grow right through one another. The resulting structure, in which the particles of the growing phase freely overlap, is defined as the extended structure. The geometrical properties of such a structure are called extended properties. Let V V, S v , and M V be the volume fraction, surface area per unit volume, and integral mean curvature of A. M. GOKHALE is Lecturer, Indian Institute of Technology, Kanpur, India. C. V. ISWARAN is Graduate Student, and R. T. DEHOFF is Professor, Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611. Manuscript submitted October 15, 1979.
q72S2v,, ]
Mv."
~
[4]
The following phenomenological equation has been suggested for a nonrandom impingement.8 d V V = (1 -
IMPINGEMENT
[1]
Vv)dVv.
[5]
Vv)idVv~ x
The exponent i depends on the deviation of the spatial distribution of nuclei from randomness; i is greater than one if the nucleation sites are clustered and it is less than one if they are ordered. Starting from Eq. [5] i t can be shown that 9 S v = (1 -
[6]
Vv)iSv..
If the local velocity of a moving interface is v, then the "surface-area-averaged interface velocity" vs is given by 1
Vs = f f v d S f fdS
1 dVv - Sv dt
[71
Similarly for the extended structure, (Vs)ex = f f vd Sex _
f f dSex
1
d Vv,
Svo
at
[81
(Vs)ex can be calculated from the nucleation and growth models (for example, see Ref. 3), and vs can be
ISSN 0360-2133/80/0811-1377500.75/0 METALLURGICAL TRANSACTIONS A 9 1980 AMERICAN SOCIETY FOR METALS AND THE METALLURGICAL SOCIETY OF AIME
VOLUME 11A, AUGUST 1980--1377
calculated from the experimental measurements of V v and S v . Equations [1], [3], [7], and [8] give the following result for random impingement. vs = (Vs)ex
[9]
Equations [5], [6], [7], and [8] give the same result for nonrandom impingement. It can be shown that 9 as long
as the impingement probability depends only on the volume fraction and the spatial distribution of nuclei, vs is equal to (Vs))ex. RECRYSTALLIZATION Recrystallization is a particularly interesting process from the point of view of quantitative metallography because the path of microstructural change covers the entire range of volume fraction from zero to one. The recrystallized regions, although not a second pha
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