An Analytical Approach to Model Heterogonous Recrystallization Kinetics Taking into Account the Natural Spatial Inhomoge
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UCTION
THE classical model
for recrystallization kinetics is based on the concepts originally formulated by Johnson and Mehl,[1] Avrami[2] and Kolmogorov[3] in the 1930s, and is now often called the JMAK model, in which a random distribution of nucleation sites is assumed, and then the fraction recrystallized XV can be derived as XV ¼ 1 expðktn Þ:
½1
This simple equation just includes two parameters: the rate parameter k and the Avrami exponent n. This exponent is usually determined by plotting the equation in a double logarithmic form and taking n to be the slope of a log(ln(1/(1 XV)) vs log(t) plot, the latter also called JMAK plot. In practice, Eq. [1] is widely used as an empirical equation to analyze the measured kinetics of recrystallization and other reactions involving both nucleation and growth in steels and other alloys.[4–9] In the classical JMAK theory, Eq. [1] is actually derived from the concept of extended volume, Xvex, i.e., the fraction of material that would have recrystallized if the phantom nuclei were real.[4] In the case of nucleation and growth kinetics in three dimensions are both time-dependent, the fraction recrystallized is:
HAIWEN LUO, Professor, is with the State Key Laboratory of Advanced Metallurgy, and also with the School of Metallurgical and Ecological Engineering, University of Science and Technology Beijing, 3 Xue Yuan Lu 30, Beijing 100083, P.R. China. Contact e-mail: [email protected] SYBRAND VAN DER ZWAAG, Professor, is with the Faculty of Aerospace Engineering, Delft University of Technology, 2629 HS Delft, The Netherlands. Manuscript submitted December 6, 2014. Article published online November 5, 2015 METALLURGICAL AND MATERIALS TRANSACTIONS A
XV ¼ 1 expðXVex Þ ( Z 3 Z t Gðt0 Þdt0 ¼ 1 exp fs s
)
s 0
Nðs Þds
0
;
½2
0
where G(t) and N(t) are growth and nucleation rates, respectively; fs is the geometrical factor, for an example, it is 4p/3 for the spherical nuclei. By the comparison of Eqs. [1] and [2], it is known that the rate parameter k could be influenced by many factors, such as nucleation kinetics, growth kinetics, the geometrical shape of nuclei and dimensions of growth. In contrast, the Avrami exponent n only depends on the time dependence of nucleation and growth rate, rather than the rate itself; Therefore, n is a stable inherent physical parameter related to the underlying mechanisms of nucleation and growth. For the three-dimensional growth, Eq. [2] indicates that n is equal to 3 in the case of site saturation and a constant growth rate, 4 when both the rates of nucleation and growth are constant. However, such high values for n have only been measured for the recrystallization of the lightly deformed fine-grained metals; for example, Gordon[10] studied the recrystallization of copper and obtained an Avrami exponent of approximately 4, identical to the value predicted by the JMAK model. Surprisingly, the majority of experimental studies on the recrystallization kinetics of steels and alloys yielded Avrami exponents lower than 2, witho
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