Kinetic description for solid-state transformation using an approach of summation/product transition

Abstract

In a recent work by the present authors, an analytical kinetic model has been derived for solid-state phase transformations on the basis of summation/product transition. In the present paper, this approach is extended to models that describe transformations involving various time synchronous and time asynchronous sub-processes. According to the newly proposed models, several interesting kinetic phenomena are revealed: negative sub-processes are responsible for the decreasing tendency of Avrami exponent; the evolution of overall kinetic parameters during time synchronous transformation reflects the relative contributions from different sub-processes; the mechanism of transformation subject to Avrami nucleation can be regarded as a combination of various time synchronous sub-processes; a process due to continuous nucleation is equal to the total effect of infinitely many sub-processes due to site saturation starting at different times; and abnormal results of Avrami plot in time asynchronous transformation are due to a sudden change in transformation mechanisms. Finally, the time asynchronous model is applied successfully to describe the crystallizations of Mg–Cu–Y and Zr–Cu–Al amorphous alloys, measured by differential scanning calorimetry.

Appendices

Appendix 1: Iso-kinetic relationship

Upon recent study [25], applying the SPT for transformation assuming mixed nucleation of site saturation and continuous nucleation, the overall mechanism is consistent with iso-kinetic relationship, and additivity rule can be used to extract isochronal data from isothermal experiments, and vice versa [25]. For the time synchronous model, it should be noted that constant kinetic parameters n _z , Q _z , and K _0z hold for each sub-process (see “General model for time synchronous transformation” section), but lead to versatile variation subjected to the SPT. Whether the overall mechanism is still consistent with the iso-kinetics should be examined.

Following a general definition of iso-kinetic reaction by Christian [1], a transformation process is called iso-kinetic, if the instantaneous reaction rate may be written in the form of a separable differential equation:

$$ \frac{{{\text{d}}f}}{{{\text{d}}t}}{ = }h\left( T \right)g\left( f \right), $$

(50)

where h(T) and g(f) are functions only of T and f, respectively. On this basis, it will be herein proved that for a transformation involving i sub-processes, the overall mechanism would be still consistent with iso-kinetic relationship, using mathematical induction.

For isothermal transformation involving two sub-processes, substitution of Eq. (15) with i = 2 into Eq. (4), leads to

$$ \frac{{{\text{d}}f}}{{{\text{d}}t}} = n_{1}^{ * } K_{01}^{*} \exp \left( { - {{Q_{1}^{*} } \mathord{\left/ {\vphantom {{Q_{1}^{*} } {RT}}} \right. \kern-0pt} {RT}}} \right)\left( {1 - f} \right)\left( { - \ln \left( {1 - f} \right)} \right)^{{1 - \frac{1}{{n_{1}^{ *} }}}}. $$

(51)

This case has been proved to be consistent with iso-kinetic relationship [25].

When i = k (≥2), the transformed fraction can be expressed as

$$ f{ = }1 - \exp \left( { - \left[ {K_{{0\left( {k - 1} \right)}}^{*} \exp \left( { - \frac{{Q_{k - 1}^{ *} }}{RT}} \right)t} \right]^{{n_{k - 1}^{ *} }} } \right) $$

(52)

with the kinetic parameters as described by Eq. (23). Supposing for this case ${{{\text{d}}f} \mathord{\left/ {\vphantom {{{\text{d}}f} {{\text{d}}t}}} \right. \kern-0pt} {{\text{d}}t}} = n_{k - 1}^{ * } K_{{0\left( {k - 1} \right)}}^{ *} \exp \left( { - {{Q_{k - 1}^{*} } \mathord{\left/ {\vphantom {{Q_{k - 1}^{*} } {RT}}} \right. \kern-0pt} {RT}}} \right)\left( {1 - f} \right)\left( { - \ln \left( {1 - f} \right)} \right)^{{1 - \frac{1}{{n_{k - 1}^{ *} }}}}$ does hold. This subsequently leads to

$$ \frac{{d{ \ln }\left( { - \ln \left( {1 - f} \right)} \right)}}{{d{ \ln }t}}{ = }n_{k - 1}^{ *} $$

(53)

and then

$$ \frac{{{\text{dln}}\left( {K_{{0\left( {k - 1} \right)}}^{*} } \right)^{{n_{k - 1}^{*} }} }}{{{\text{dln}}t}} - \frac{1}{RT}\frac{{{\text{d}}\left( {n_{k - 1}^{ *} Q_{k - 1}^{ *} } \right)}}{{{\text{dln}}t}}{\text{ + ln}}t\frac{{{\text{d}}n_{k - 1}^{ *} }}{{{\text{dln}}t}}{ = }0. $$

(54)

Please note that Eq. (54) is an essential condition for a successful application of mathematical induction.

When i = k+1, substitution of Eq. (22) into Eq. (4) leads to

$$ \begin{gathered} \ln \left( { - \ln \left( {1 - f} \right)} \right) = \ln \left[ {\left( {\left( {K_{{0\left( {k - 1} \right)}}^{*} } \right)^{{n_{k - 1}^{*} }} \left( {1 \pm {{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}} \right)} \right)^{{\frac{1}{{1 \pm {{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}}}}} \left( {K_{{0\left( {k + 1} \right)}}^{{n_{k + 1} }} \left( {\left( {{{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}} \right)^{ - 1} \pm 1} \right)} \right)^{{\frac{1}{{1 \pm \left( {{{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}} \right)^{ - 1} }}}} } \right] \\ - \frac{{\frac{{n_{k - 1}^{*} Q_{k - 1}^{*} }}{{1 \pm {{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}}} + \frac{{n_{k + 1} Q_{k + 1} }}{{1 \pm \left( {{{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}} \right)^{ - 1} }}}}{RT} + \left[ {\frac{{n_{k - 1}^{*} }}{{1 \pm \left( {{{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}} \right)}} + \frac{{n_{k + 1} }}{{1 \pm \left( {{{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}} \right)^{ - 1} }}} \right]\ln t \hfill \\ \end{gathered}. $$

(55)

Taking the derivation of Eq. (55) with respective to ln t strictly and simplifying, it thus leads to

$$ \frac{{{\text{dln}}\left( { - \ln \left( {1 - f} \right)} \right)}}{{{\text{dln}}t}} = n_{k}^{ *} { + }\frac{1}{{1 \pm {{r_{2k} } \mathord{\left/ {\vphantom {{r_{2k} } {r_{2k - 1} }}} \right. \kern-0pt} {r_{2k - 1} }}}}\left[ {\frac{{{\text{dln}}\left( {K_{{0\left( {k - 1} \right)}}^{*} } \right)^{n*} }}{{{\text{dln}}t}} - \frac{1}{RT}\frac{{{\text{d}}\left( {n_{k - 1}^{ *} Q_{k - 1}^{ *} } \right)}}{{{\text{dln}}t}}{ + }\ln t\frac{{{\text{d}}n_{k - 1}^{ *} }}{{{\text{dln}}t}}} \right] $$

(56)

which, in combination with Eq. (54), can be given further as

$$ \frac{{{\text{dln}}\left( { - \ln \left( {1 - f} \right)} \right)}}{{{\text{dln}}t}} = n_{k}^{ *} $$

(57)

and thus

$$ \frac{{{\text{d}}f}}{{{\text{d}}t}} = n_{k}^{ * } K_{0k}^{*} \exp \left( { - {{Q_{k}^{ *} } \mathord{\left/ {\vphantom {{Q_{k}^{ *} } {RT}}} \right. \kern-0pt} {RT}}} \right)\left( {1 - f} \right)\left( { - \ln \left( {1 - f} \right)} \right)^{{1{ - }\frac{1}{{n_{k}^{ *} }}}} = h\left( T \right)g\left( f \right). $$

(58)

Until now, it can be concluded that for a transformation involving i different sub-processes, the resulted mechanism is still consistent with iso-kinetic relationship. Using mathematical induction, analogous treatment which is also based on Eq. (51) can be performed for non-isothermal transformations. It should be noted that for transformations due to a single sub-process, Eq. (51) holds exactly for the isothermal case, while it is proved to be an excellent approximation for the non-isothermal case at a constant heating rate [57].

Appendix 2: Non-isothermal time asynchronous transformation

Corresponding to Eq. (27), the evolution of sub-process extended transformed fraction (with kinetic parameters as n _z , Q _z , and K _0z ), x _ez , with temperature during a non-isothermal transformation can also be given in a piecewise function

$$ x_{{{\text{e}}z}} = \left[ {\int\limits_{{T_{z}^{*} }}^{T} {K_{0z} \exp \left( { - \frac{{Q_{z} }}{RT\left( t \right)}} \right)} \frac{{{\text{d}}T\left( t \right)}}{\varPhi }} \right]^{{n_{z} }} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{rect}}\left( {\frac{{T - {{\left( {T_{z}^{*} + T^{\prime}_{z} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{z}^{*} + T^{\prime}_{z} } \right)} 2}} \right. \kern-0pt} 2}}}{{T^{\prime}_{z} - T_{z}^{*} }}} \right) + C_{z} H\left( {T - T^{\prime}_{z} } \right) $$

(59)

with $C_{z} = \left[ {\int\limits_{{T_{z}^{*} }}^{{T^{\prime}}} {K_{0z} \exp \left( { - \frac{{Q_{z} }}{RT\left( t \right)}} \right)\frac{dT\left( t \right)}{\varPhi }} } \right]^{{n_{z} }}$, rect(± 1/2) = 0 and H(0) = 1. In this equation, it encounters the so-called “temperature integral,” which cannot be solved analytically and has to be approximated, i.e., for $T_{z}^{ * }$ < T < T _z ′ (rect = 1 and H = 0), Eq. (59) can be rewritten as

$$ \begin{gathered} x_{\text{ez}} = \left[ {\int_{{T_{z}^{*} }}^{T} {K_{0z} \exp \left( { - \frac{{Q_{z} }}{{RT^{\prime}}}} \right)\frac{{dT^{\prime}}}{\varPhi }} } \right]^{{n_{z} }} \hfill \\ = \left[ {\int_{0}^{T} {K_{0z} \exp \left( { - \frac{{Q_{z} }}{{RT^{\prime}}}} \right)\frac{{dT^{\prime}}}{\varPhi }} - \int_{0}^{{T_{z}^{*} }} {K_{0z} \exp \left( { - \frac{{Q_{z} }}{{RT^{\prime}}}} \right)\frac{{dT^{\prime}}}{\varPhi }} } \right]^{{n_{z} }} \hfill \\ \approx \left[ {\frac{{K_{0z} }}{{Q_{z} }}\exp \left( { - \frac{{Q_{z} }}{RT}} \right)\frac{{RT^{2} }}{\varPhi } - \frac{{K_{0z} }}{{Q_{z} }}\exp \left( { - \frac{{Q_{z} }}{{RT_{z}^{*} }}} \right)\frac{{R\left( {T_{z}^{*} } \right)^{2} }}{\varPhi }} \right]^{{n_{z} }}. \hfill \\ \end{gathered} $$

(60)

If $T_{z}^{*} \ll T$ holds, the second term in the square bracket is so small that can be neglected, compared with the first term [7]. But in present treatment, such assumption ceases to be valid. Therefore, assuming that the two terms in the square bracket represent two reverse parts, application of the SPT indicated in “Extended method” section to Eq. (60) leads to

$$ x_{{{\text{e}}z}} = \left[ {K^{\prime}_{0z} \exp \left( { - \frac{{Q_{z} }}{RT}} \right)\frac{{RT^{2} }}{\varPhi }} \right]^{{n^{\prime}_{z} }} $$

(61)

with

$$ n^{\prime}_{z} = \frac{{n_{z} }}{{1 - r^{\prime}_{z} }}. $$

(62a)

$$ K^{\prime}_{0z} = \left( {\frac{{K_{0z} }}{{Q_{z} }}} \right)^{{1 - r^{\prime}_{z} }} \left( {1 - r^{\prime}_{z} } \right)\; \times \;\left[ {\exp \left( { - \frac{{Q_{z} }}{{RT_{z}^{*} }}} \right)\frac{{R\left( {T_{z}^{*} } \right)^{2} }}{\varPhi }\left( {\left( {r^{\prime}_{z} } \right)^{ - 1} - 1} \right)} \right]^{{\left( { - r^{\prime}_{z} } \right)}}. $$

(62b)

$$ r^{\prime}_{z} = {{\left[ {\exp \left( { - \frac{{Q_{z} }}{{RT_{z}^{*} }}} \right)\left( {T_{z}^{*} } \right)^{2} } \right]} \mathord{\left/ {\vphantom {{\left[ {\exp \left( { - \frac{{Q_{z} }}{{RT_{z}^{*} }}} \right)\left( {T_{z}^{*} } \right)^{2} } \right]} {\left[ {\exp \left( { - \frac{{Q_{z} }}{RT}} \right)T^{2} } \right]}}} \right. \kern-0pt} {\left[ {\exp \left( { - \frac{{Q_{z} }}{RT}} \right)T^{2} } \right]}}. $$

(62c)

Note that due to the effect of $T_{z}^{ * }$, the modified sub-process kinetic parameters are marked with the symbol “′”.

Corresponding to Eq. (28), for a common temperature range between T _l and T _l+1, in which the transformation mechanism remains unchanged, the total x _e can be expressed as

$$ \begin{gathered} x_{\text{e}} = \sum\limits_{z = 1}^{i} { \pm \left[ {K^{\prime}_{0z} \exp \left( { - \frac{{Q_{z} }}{RT}} \right)\frac{{RT^{2} }}{\varPhi }} \right]^{{n^{\prime}_{z} }} {\kern 1pt} {\kern 1pt} {\text{rect}}\left( {\frac{{T - {{\left( {T_{z}^{*} + T^{\prime}_{z} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{z}^{*} + T^{\prime}_{z} } \right)} 2}} \right. \kern-0pt} 2}}}{{T^{\prime}_{z} - T_{z}^{*} }}} \right) + C_{z} H\left( {T - T^{\prime}_{z} } \right)} \hfill \\ + \sum\limits_{k = i + 1}^{i + j} { \pm \left[ {K^{\prime}_{0k} \exp \left( { - \frac{{Q_{k} }}{RT}} \right)\frac{{RT^{2} }}{\varPhi }} \right]^{{n^{\prime}_{k} }} {\kern 1pt} {\kern 1pt} \times {\text{rect}}\left( {\frac{{T - {{\left( {T_{k}^{*} + T^{\prime}_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{k}^{*} + T^{\prime}_{k} } \right)} 2}} \right. \kern-0pt} 2}}}{{T^{\prime}_{k} - T_{k}^{*} }}} \right) + C_{k} H\left( {T - T^{\prime}_{k} } \right)} \hfill \\ = \sum\limits_{z = 1}^{i} { \pm C_{z} } + \sum\limits_{k = i + 1}^{i + j} { \pm \left[ {K^{\prime}_{0k} \exp \left( { - \frac{{Q_{k} }}{RT}} \right)\frac{{RT^{2} }}{\varPhi }} \right]^{{n^{\prime}_{k} }} } \hfill \\ \end{gathered}, $$

(63)

where the two terms represent the contributions from i ended sub-process and from j active sub-processes, respectively.

Similarly, if only one sub-process is prevailing, i.e. j = 1, using the same procedure in Eqs. (30)–(37) the rate equation can be given as

$$ \frac{df}{dT} = \left[ {\frac{{n^{\prime}_{i + 1} Q_{i + 1} }}{{1 \pm \left( {r_{2i + 1,2i + 2} } \right)^{ - 1} }}\frac{1}{{RT^{2} }} + \frac{{n^{\prime}_{i + 1} }}{{1 \pm \left( {r_{2i + 1,2i + 2} } \right)^{ - 1} }}\frac{2}{T}} \right]\left( {1 - f} \right)\left( { - \ln \left( {1 - f} \right)} \right) $$

(64)

with the ratio $r_{2i + 1,2i + 2} = {{\left[ {K_{0i + 1} \exp \left( { - {{Q_{i + 1} } \mathord{\left/ {\vphantom {{Q_{i + 1} } {RT}}} \right. \kern-0pt} {RT}}} \right)\left( {t - t_{i + 1}^{*} } \right)} \right]^{{n_{i + 1} }} } \mathord{\left/ {\vphantom {{\left[ {K_{0i + 1} \exp \left( { - {{Q_{i + 1} } \mathord{\left/ {\vphantom {{Q_{i + 1} } {RT}}} \right. \kern-0pt} {RT}}} \right)\left( {t - t_{i + 1}^{*} } \right)} \right]^{{n_{i + 1} }} } {\sum\nolimits_{z = 1}^{j} { \pm C_{z} } }}} \right. \kern-0pt} {\sum\nolimits_{z = 1}^{j} { \pm C_{z} } }}$. Equation (64) can be compared with the classical non-isothermal rate equation [26],

$$ \frac{{{\text{d}}f}}{{{\text{d}}T}} = \left[ {\frac{nQ}{{RT^{2} }} + \frac{2n}{T}} \right]\left( {1 - f} \right)\left( { - \ln \left( {1 - f} \right)} \right). $$

(65)

By comparing Eqs. (64) and (65), the expressions for kinetic parameters incorporating the effect of T ^* _z result,

$$ n_{i + 1}^{*} = \frac{{n^{\prime}_{i + 1} }}{{1 \pm \left( {r_{2i + 1,2i + 2} } \right)^{ - 1} }} $$

(66a)

$$ Q_{i + 1}^{*} = \frac{1}{{n_{i + 1}^{*} }}\left[ {\frac{{n^{\prime}_{i + 1} Q_{i + 1} }}{{1 \pm \left( {r_{2i + 1,2i + 2} } \right)^{ - 1} }}} \right]. $$

(66b)

Similarly, for the transformation with j active sub-processes in the range, it will give the general rate equation,

$$ \frac{{{\text{d}}f}}{{{\text{d}}T}} = \left[ {\frac{{n_{i + j}^{ * } Q_{i + j}^{ * } }}{{RT^{2} }} + \frac{{2n_{i + j}^{ * } }}{T}} \right]\left( {1 - f} \right)\left( { - \ln \left( {1 - f} \right)} \right) $$

(67)

with the recursive relation for the kinetic parameters,

$$ n_{i + j}^{*} = \frac{{n_{i + j - 1}^{*} }}{{1 \pm r_{2i + 2j - 1,2i + 2j} }} + \frac{{n^{\prime}_{i + j} }}{{1 \pm \left( {r_{2i + 2j - 1,2i + 2j} } \right)^{ - 1} }}. $$

(68a)

$$ Q_{i + j}^{*} = \frac{1}{{n_{i + k}^{*} }}\left[ {\frac{{n_{i + j - 1}^{*} Q_{i + j - 1}^{*} }}{{1 \pm r_{2i + 2j - 1,2i + 2j} }} + \frac{{n^{\prime}_{i + j} Q_{i + j} }}{{1 \pm \left( {r_{2i + 2j - 1,2i + 2j} } \right)^{ - 1} }}} \right]. $$

(68b)

$$ r_{2i + 2j - 1,2i + 2j} = \frac{{x_{{{\text{e}}\left( {i + j} \right)}} }}{{\sum\nolimits_{z = 1}^{i} { \pm C_{z} } + \sum\nolimits_{k = i + 1}^{i + j - 1} { \pm x_{{{\text{e}}k}} } }}. $$

(68c)

For K _0k ′ and r _k ′, see Eq. (62a–c).