Generalized additivity rule for the Kolmogorov–Johnson–Mehl–Avrami kinetics

Abstract

The validity of the classical additivity rule for the non-isothermal KJMA kinetics is studied. In the case of linear temperature variation, it is demonstrated that, with an exception for the isokinetic range, the kinetics is not additive. Yet, a generalized form of the additivity rule holds true with a weighting function, W, which depends upon actual temperature, heating rate, and time. The exact expression of the weighting function is computed, in closed form, for integer and half-integer Avrami exponents. The behavior of W strongly depends on the ratio between activation energies for nucleation and growth. Its effect on the rate of the transformation, as well as on transformed volume fraction, is investigated in connection with the classical additivity rule. The present results are also discussed in relation to approximate computations previously proposed in the literature.

Appendices

Appendix 1

By differentiating $\eta = \frac{{S_{m - 1} }}{{S_{m}^{m/m + 1} }}$, one gets

$$ \frac{{{\text{d}}\eta }}{{{\text{d}}\tau }} = \frac{1}{{S_{m}^{m/m + 1} }}\frac{{{\text{d}}S_{m - 1} }}{{{\text{d}}\tau }} - \frac{m}{m + 1}\frac{{S_{m - 1} }}{{S_{m}^{(2m + 1)/(m + 1)} }}\frac{{{\text{d}}S_{m} }}{{{\text{d}}\tau }}. $$

(18)

Use of Eq. 9a in Eq.18 gives ($m \ne 1$),

$$ \frac{{{\text{d}}\eta }}{{{\text{d}}\tau }} = \frac{m}{\tau }e^{a\tau /(1 + \tau )} \left[ {\frac{{S_{m - 2} }}{{S_{m}^{m/(m + 1)} }} - \frac{{S_{m - 1}^{2} }}{{S_{m}^{(2m + 1)/(m + 1)} }}} \right] $$

or

$$ \frac{{{\text{d}}\eta }}{{{\text{d}}\tau }} = \eta \frac{m}{\tau }e^{a\tau /(1 + \tau )} \left[ {\frac{{S_{m - 2} }}{{S_{m - 1}^{{}} }} - \frac{{S_{m - 1}^{{}} }}{{S_{m}^{{}} }}} \right]. $$

(19)

Since $W = \eta /\eta_{i} = \eta e^{{\frac{{(a - b)}{\tau }}{{(m + 1)}{(1 + \tau) }}}}$, with $\eta_{i} (T_{0} ,\tau ) = e^{{ - \frac{{(a - b)}{\tau }}{{(m + 1)}{(1 + \tau )}}}}$, the integration of Eq. 19 leads to

$$ W(T_{0} ,\tau ) = e^{{\frac{(a - b)\tau }{(m + 1)(1 + \tau )}}}\times \exp \left[ {m\int\limits_{0}^{\tau } {{\text{d}}\tau '\frac{1}{\tau '}} e^{a\tau '/(1 + \tau ')} \left( {\frac{{S_{m - 2} (\tau ')}}{{S_{m - 1}^{{}} (\tau ')}} - \frac{{S_{m - 1}^{{}} (\tau ')}}{{S_{m}^{{}} (\tau ')}}} \right)} \right]. $$

Appendix 2

At temperature T, the isothermal kinetics is

$$ X(t) = 1 - \exp \left[ { - \left( {A^{{}} e^{{ - {\left(\frac{ma + b}{m + 1}\right)}\frac{{T_{0} }}{T}}} } \right)^{m + 1} \frac{{t^{m + 1} }}{m + 1}} \right]. $$

(20)

The time $t_{F} (T)$ needed to complete about 95 % of the transformation at T (i.e., $X(t_{F} ) = 1 - e^{ - 3}$) is equal to $t_{F} (T) = \frac{{[3(1 + m)]^{1/(m + 1)} }}{{Ae^{{ - {\left(\frac{ma + b}{m + 1}\right)}\frac{{T_{0} }}{T}}} }}$, from which

$$ A = \frac{1}{{t_{F} (T)}}[3(1 + m)]^{{\frac{1}{m + 1}}} e^{{{\left(\frac{ma + b}{m + 1}\right)}\frac{{T_{0} }}{T}}} . $$

(21)

In order to estimate A, in Eq. 21 we set $T = T_{0}$ and $t_{F} \approx 6000{\text{ s}}$.