On the rule of additivity in phase transformation kinetics
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I.
INTRODUCTION
THE most commonly used tool for tracking volume fraction kinetics of diffusive transformations and recrystallization phenomena is the Johnson-Mehl-Avrami-Kolmorgorov (JMAK) equation:[1,2,3] F(t) 5 1 2 e2k t n
[1]
where k and n are empirical parameters and F is the product volume fraction which varies with time t. This equation was derived from idealized micromechanical considerations under isothermal conditions. As is clear from the C-shaped curves of time-temperature-transformation (TTT) diagrams, the value for k varies with hold temperature since n is assumed to be a constant. This model has also served as a basis for considering phase transformations that occur under nonisothermal conditions through an appeal to the rule of additivity—first presented by Avrami in a landmark contribution to computational metallurgy.[4] The valid application of this rule was subsequently extended by Cahn[5] to rate relations of the form z F 5 ˆj (u )hˆ (F)
[2]
where the dot indicates a time derivative and u is the temperature. Such reactions are called general isokinetic. Cahn’s result allows a wide range of nonisothermal reactions to be accurately predicted from isothermal kinetics using the rule of additivity. For instance, the JMAK equation with n constant can be recast as a general isokinetic relation. Although Cahn’s argument is valid with respect to the rule of additivity and such general isokinetic reactions, he further stated that the rule holds for an even broader class of kinetic relations. Specifically, Cahn said that the rule of additivity can be applied to any kinetic equation of the form z F 5 g (F, u )
[3]
However, it will be shown that not all such processes, here referred to as rate independent, obey the rule of additivity. Our motivation for considering the range of validity of the rule of additivity stems from a recently advanced approach to modeling phase transformation kinetics using a principle of microbalance,[6,7] which is fundamentally different than the many generalizations of JMAK theory.[8–16] In the microbalance model, based on the foundational work
MARK LUSK, Assistant Professor, and HERNG-JENG JOU, Postdoctoral Research Fellow, are with the Division of Engineering, Colorado School of Mines, Golden, CO 80401. Manuscript submitted July 8, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
of Fried and Gurtin,[17] a balance postulate leads naturally to a differential kinetic relation. For a particular class of constitutive assumptions, that relation can be written in the form of Eq. [3]. We advocate solving such kinetic equations numerically in full coupling with the heat equation. However, in comparing the solutions obtained in especially simple cases with those obtained using the rule of additivity, we became aware of the discrepancy brought forth in this article. The rule of additivity can be mathematically derived in just a few lines. Consider a thermally induced phase transformation with a single product phase whose volume fraction is denoted by F. Let F be governed by the r
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