Examination of an analytical phase-transformation model
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comparison between results of a recently published quasi-exact solution of the temperature integral used for the Avrami model of isochronal phase transformations and an analytical phase-transformation model in relation to exact solutions from numerical integration has been performed. The results for the transformed fraction from the quasiexact solution are more precise than the corresponding results of the analytical model, whereas the results for the transformation rate from both models are sufficiently precise for the nucleation mode of site saturation or continuous nucleation. It has been further shown that an analytical solution of the transformation rate cannot be obtained using a quasi-exact solution of the temperature integral in case of mixed nucleation, and that the results of the corresponding solution with the analytical model substantially alleviate the influence of the approximated temperature integral. By this method, an analytical approach of modeling, which can substantially alleviate the deviation (of model prediction) arising from approximations to the temperature integral, has been developed. The proposed approach has been successfully applied to experimental data of the crystallization of bulk amorphous Pd-Ni-P-Cu alloys.
I. INTRODUCTION
Solid-state phase transformations are important means for the adjustment of the microstructure and thus are the tuning of the properties of materials. To exploit this tool to its full extent, much effort is spent on the modeling of phase transformations.1–20 More recently, an analytical phase-transformation model21–23 was proposed that incorporates a choice of nucleation and growth mechanisms, as well as impingement modes, and has been successfully applied to experimental results.24,25 The model leads to equations for the degree of transformation that have the structure of the Johnson-Mehl-Avrami (JMA) equation but with time-dependent kinetic parameters n(t), Q(t), K0(t) (isothermal transformation) or temperature-dependent kinetic parameters n(T), Q(T), K0(T) (isochronal transformation), i.e., for a mixture of site saturation and continuous nucleation (or mixed nucleation). Given an impingement mode due to random nuclei dispersion, a relation between the transformed fraction, f, and the extended transformed fraction, xe, is obtained,1–5 f ¼ 1 expðxe Þ :
ð1Þ
See details in Sec. II. A.
a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2009.0194 J. Mater. Res., Vol. 24, No. 5, May 2009
To derive an analytical model, the term below, called the temperature integral, cannot be solved analytically and has to be approximated.22,26–31 Z1 0
df ¼ Fð f Þ
ZT
Q dT K0 exp RT
;
ð2Þ
T0
where F(f) is a kinetic function that depends on the reaction mechanism, K0 is the pre-exponential factor for rate constant, R is the gas constant, and T0 (t = 0, the starting temperature for annealing) and T(t)= T0 + Ft with F (=dT/dt) are the heating rates. The fitting of the analytical model to the experimental data [f or df/dT as fu
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