Improved analytical model for isochronal transformation kinetics

Abstract

Analytical model for isochronal phase transformation kinetics attracts much attention for its advantages and importance. However, the simple but exact analytical formula of the isochronal transformation is unavailable because of the so-called temperature integral, and the asymptotic expansions have to be adopted to obtain approximate results. Here a generally used asymptotic expansion was proved divergent, and a reasonable approximation was proposed to obtain a more precise description as compared numerically to the previous one. Based on the proposed approximation, an analytical model for isochronal transformation kinetics was developed, which was proved more effective than the previous analytical model when the transformation occurs in a narrow temperature range and exhibited an identical form to the previous model when in a wide temperature interval.

Appendix A: Deducing the expression for the isochronal transformation kinetics model

The extend volume Eq. 12 can be calculated upon Eq. 22:

$$ \begin{aligned} & V^{\rm e}\left[{T(t)}\right] = \frac{VgN_0 \upsilon _0 ^{d/m}}{\Upphi ^{(d/m){+1}}}\, \hbox{exp}\, \left({-\frac{(d/m)Q_{\rm G}} {RT(t)}} \right)\left({\frac{RT(t)^{2}}{Q_{\rm G}}} \right)^{d/m}\cdot \int\limits_{T_0 }^{T(t)}{\, \hbox{exp}\, \left({-\frac{Q_{\rm N}}{RT(t)}} \right)\left\{ {1-\, \hbox{exp}\, \left[{\frac{Q_{\rm G} [T(\tau) -T(t)]}{RT(t)^{2}}}\right]} \right\}^{d/m}{d}T(\tau )} \\ & = \frac{VgN_0 \upsilon _0 ^{d/m}}{\Upphi ^{(d/m){+1}}}\, \hbox{exp}\, \left({-\frac{Q_{\rm N} +(d/m)Q_{\rm G}}{RT(t)}} \right)\left({\frac{RT(t)^{2}}{Q_{\rm G} }} \right)^{d/m}\cdot \int\limits_{T_0 }^{T(t)} {\, \hbox{exp}\, \left[{\frac{Q_{\rm N} [T(\tau )-T(t)]}{RT(t)^{2}}}\right]\left\{ {1-\, \hbox{exp}\, \left[{\frac{Q_{\rm G} [T(\tau )-T(t)]}{RT(t)^{2}}}\right]} \right\}^{d/m}{d}T(\tau )}\\ \end{aligned} $$

(A1)

The power term in (A1) can be expanded as binomial series [14, 31]:

$$ \left\{ {1-\hbox{exp}\, \left[{\frac{Q_{\rm G} [T(\tau) -T(t)]}{RT(t)^{2}}}\right]} \right\}^{d/m} =\sum_{i=0}^\infty {\left({{\begin{array}{l} d/m \\ i \\ \end{array} }} \right)(-1)^{i}\, \hbox{exp}\, \left[{\frac{iQ_{\rm G} [T(\tau )-T(t)]}{RT(t)^{2}}}\right]} $$

(A2)

where $\left(\begin{array}{l} d/m\\ i\\ \end{array}\right)=\frac{\prod\limits_{j=0}^{i-1}\left((d/m)-j\right)} {i!}, \left(\begin{array}{l} d/m\\ 0\\ \end{array}\right)=1,$ is the binomial coefficients. Then the integral (A1) can be calculated as:

$$ \begin{aligned} & V^{\rm e}\left[{T(t)}\right]= \frac{VgN_0 \upsilon _0 ^{d/m}}{\Upphi ^{(d/m){+1}}}\, \hbox{exp}\, \left({-\frac{Q_{\rm N} +(d/m)Q_{\rm G}} {RT(t)}} \right)\left({\frac{RT(t)^{2}}{Q_{\rm G} }} \right)^{d/m} \\ &\times \int\limits_{T_0 }^{T(t)} {\, \hbox{exp}\, \left[{\frac{Q_{\rm N} [T(\tau )-T(t)]}{RT(t)^{2}}}\right]\sum_{i=0}^\infty {\left({{\begin{array}{l} d/m \\ i \\ \end{array} }} \right)} (-1)^{i}\, \hbox{exp}\, \left[{\frac{iQ_{\rm G} [T(\tau )-T(t)]}{RT(t)^{2}}}\right]} {d}T(\tau ) \\ & = \frac{VgN_0 \upsilon _0 ^{d/m}}{\Upphi ^{(d/m){+1}}}\, \hbox{exp}\, \left({-\frac{Q_{\rm N} +(d/m)Q_{\rm G}}{RT(t)}} \right)\left({\frac{RT(t)^{2}}{Q_{\rm G}}} \right)^{d/m}\sum_{i=0}^\infty {\left({{\begin{array}{l} d/m \\ i \\ \end{array} }} \right)} (-1)^{i}\int\limits_{T_0 }^{T(t)} {\, \hbox{exp}\, \left[{\frac{\left({Q_{\rm N} +iQ_{\rm G}} \right)[T(\tau) -T(t)]}{RT(t)^{2}}}\right]{d}T(\tau )} \\ & = \frac{VgN_0 Q_{\rm G}\upsilon_0^{d/m}}{\Upphi ^{(d/m){+1}}}\, \hbox{exp}\, \left({-\frac{Q_{\rm N} +(d/m)Q_{\rm G} }{RT(t)}} \right)\left({\frac{RT(t)^{2}}{Q_{\rm G}}} \right)^{(d/m){+1}}\sum_{i=0}^\infty {\left({{\begin{array}{l} d/m \\ i \\ \end{array} }} \right)} \frac{(-1)^{i}}{Q_{\rm N} +iQ_{\rm G}}\left\{ {1-\, \hbox{exp}\, \left[{\frac{\left({Q_{\rm N} +iQ_{\rm G} } \right)[T_0 -T(t)]}{RT(t)^{2}}}\right]} \right\} \\ & = \frac{VgN_0 \upsilon _0 ^{d/m}}{Q_{\rm G}{}^{d/m}}\, \hbox{exp}\, \left({-\frac{Q_{\rm N} +(d/m)Q_{\rm G}}{RT(t)}} \right)\left({\frac{RT(t)^{2}}{\Upphi }} \right)^{(d/m){+1}}\psi \left({T_0 , T(t), d/m} \right) \\ \end{aligned} $$

(A3)

where

$$ \psi \left({T_0 , T(t), d/m} \right)=\sum_{i=0}^\infty {\left({{\begin{array}{l} d/m \\ i \\ \end{array} }} \right)} \frac{(-1)^{i}}{Q_{\rm N} +iQ_{\rm G}} \left\{ {1-\hbox{exp}\, \left[{\frac{\left({Q_{\rm N} +iQ_{\rm G}} \right) \left[{T_0 -T(t)}\right]}{RT(t)^{2}}}\right]} \right\} $$

(A4)

For the interface-controlled growth, d/m is integer, the binomial coefficients would be truncated at i = d/m, and then (A2) becomes:

$$ \psi \left({T_0 , T(t), d/m} \right)=\sum_{i=0}^{d/m} {\left({{\begin{array}{l} d/m \\ i \\ \end{array} }} \right)} \frac{(-1)^{i}}{Q_{\rm N} +iQ_{\rm G} } \left\{ {1-\, \hbox{exp}\, \left[\frac{\left({Q_{\rm N} +iQ_{\rm G} } \right) \left[{T_0 -T(t)}\right]}{RT(t)^{2}}\right]} \right\} $$

(A5)

For the diffusion-controlled growth, d/m is semi-integer except d/m = 2/2, and then ψ(T₀, T(t), d/m) has infinite terms. However, ψ(T₀, T(t), d/m) converges quickly [14]. Therefore, a reasonable approximation can be available by truncating at the primary terms. The expressions for ψ(T₀, T(t), d/m) are given in Table 1.