Modeling time dependence of the average interface migration rate in site-saturated recrystallization
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I.
INTRODUCTION
NEW strain-free grains will nucleate and grow at the expense of a cold-worked matrix when it is heated to a sufficiently high temperature. Such a thermally activated process is denominated primary recrystallization. A considerable amount of theoretical work has been devoted to modeling such a process. A detailed review of this subject has been published recently by Vandermeer.[1] One important issue[1,2] that arises from the analysis of recrystallization kinetics is that the average interface migration rate is often found to decrease with time.[2,3,4] The challenging aspect of this finding is that the same time dependence is observed over the entire temperature range of study. Vandermeer and Rath[2] found a time dependence of t20.38 and Speich and Fisher[4] of t21. Vandermeer and Rath[2] discuss two possibilities to explain this: concurrent recrystallization and recovery and nonuniform distribution of stored energy.[1,5–7] These two mechanisms offer physical reasons which qualitatively support the experimental fact that the interface migration rate decreases with time. However, neither mechanism is able at present to explain quantitatively the observed time dependence. In view of this, further development of phenomenological but quantitative models is justified. In this work, a modification of the classical model of Kolmogorov,[8] Avrami,[9] and Johnson and Mehl,[10] hereafter referred to as the KJMA model, is proposed to account for the time dependence of the average interface migration rate. The present analysis is restricted to site-saturated reactions. This is probably a good assumption for recrystallization.[1,2] II.
MODEL DEVELOPMENT
A. Derivation of the Fundamental Equation
XV 5 1 2 exp (2XVex)
[1]
Equation [1] relates the observed volume fraction transformed in ‘‘real’’ space, XV, to the extended volume fraction, XVex, obtained in ‘‘extended’’ space. This relationship allows one to model the kinetics in extended space, that is, ignoring impingement and the consequent change in shape of the growing grains. Equation [1] is a mathematically exact relationship for a spatially random distribution of nuclei.[8–12] The second part of the KJMA model would then be a model for the nucleation, shape, and growth of the extended grains. The most simple assumption about nucleation is that of site saturation. This means that at t 5 0, a fixed number of grains per unit volume (NV) start to grow, and no new grains appear after that. In the context of recrystallization, ‘‘nucleation’’ means only the beginning of growth and does not imply that the nucleation is due to fluctuations as in the classical nucleation theory. The most simple assumption about the shape of an extended grain is that its volume (Vex) is spherical, so that the extended volume fraction can be written as XVex 5 NV Vex 5 NV
~ ! 4p Rex3 3
[2]
It remains to define the growth rate of the extended grains and integrate it to obtain Rex and thus Vex so that the model is complete. It is widely accepted that the interface migration rate is
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