Solid-phase crystallization under continuous heating: Kinetic and microstructure scaling laws
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The kinetics and microstructure of solid-phase crystallization under continuous heating conditions and random distribution of nuclei are analyzed. An Arrhenius temperature dependence is assumed for both nucleation and growth rates. Under these circumstances, the system has a scaling law such that the behavior of the scaled system is independent of the heating rate. Hence, the kinetics and microstructure obtained at different heating rates differ only in time and length scaling factors. Concerning the kinetics, it is shown that the extended volume evolves with time according to ␣ex ⳱ [exp(Ct⬘)]m+1, where t⬘ is the dimensionless time. This scaled solution not only represents a significant simplification of the system description, it also provides new tools for its analysis. For instance, it has been possible to find an analytical dependence of the final average grain size on kinetic parameters. Concerning the microstructure, the existence of a length scaling factor has allowed the grain-size distribution to be numerically calculated as a function of the kinetic parameters.
I. INTRODUCTION
Crystallization of amorphous materials and other solid-state transformations usually involve random nucleation and growth. Under this assumption, the phase transformation is described by the Kolmogorov– Johnson–Mehl–Avrami theory (KJMA).1–6 The transformed fraction, ␣, is related with the extended transformed fraction, ␣ex, through the so-called KJMA relation: ␣共t兲 = 1 − exp关−␣ex共t兲兴 .
(1)
␣ex would be the transformed fraction if grains grew through each other and overlapped without mutual interference, i.e., ␣ex共t兲 =
兰 I共u兲v t
ex共u,t 兲du
0
,
(2)
where I is the nucleation rate per unit volume and vex(u,t) is the extended volume transformed at time t by a single nucleus created at time u: vex共u,t兲 =
冉兰
t
G共z兲dz
u
冊
m
.
(3)
In Eq. (3), is a shape factor (e.g., ⳱ 4/3 for spheri-
cal grains), G is the growth rate, and m depends on the growth mechanism7 [e.g., m ⳱ 3 for three-dimensional (3D) growth]. For the particular case of isothermal transformations, where growth and nucleation rates are constant in time, Eqs. (2) and (3) have an analytical solution: ␣ex共t兲 =
冉 冊
IGm m+1 t . m+1
(4)
Unfortunately, because of the dependence of G and I on temperature, general exact solutions do not exist for nonisothermal conditions. Accordingly, a number of published works have developed different theoretical and numerical approaches to analyze non-isothermal phase transformations within the framework of KJMA theory.8–30 Recently, a quasi-exact solution of the KJMA theory was obtained under continuous heating conditions.31 A useful approach to investigate the kinetics and grain morphology consists of finding a scaling law such that the system behavior is universal. This method has been successfully used for the isothermal case.32 In this case the time, , and length, , scaling factors are33 = 共IGm兲−1 Ⲑ 共m+1兲
and =
冉冊 G I
I Ⲑ 共m+1兲
.
(5)
a)
Address all correspondence to this author. e-mail: jordi.farjas@udg
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