Modification of the Kolmogorov-Johnson-Mehl-Avrami rate equation for non-isothermal experiments and its analytical solution

Avrami's model describes the kinetics of phase transformation under the assumption of spatially random nucleation. In this paper we provide a quasi-exact analytical solution of Avrami's model when the transformation takes place under continuous heating. This solution has been obtained with different activation energies for both nucleation and growth rates. The relation obtained is also a solution of the so-called Kolmogorov-Johnson-Mehl-Avrami transformation rate equation. The corresponding non-isothermal Kolmogorov-Johnson-Mehl-Avrami transformation rate equation only differs from the one obtained under isothermal conditions by a constant parameter, which only depends on the ratio between nucleation and growth rate activation energies. Consequently, a minor correction allows us to extend the Kolmogorov-Johnson-Mehl-Avrami transformation rate equation to continuous heating conditions.


Introduction
Phase transitions are among the most important topics in materials science.
Crystallization of amorphous materials and other solid state transformations usually involves nucleation and growth. These transformations are generally described by Kolmogorov-Johnson-Mehl-Avrami [1][2][3][4][5] Although this equation is obtained from the isothermal solution (eqn. (1)), it constitutes the basis for analyzing non-isothermal experiments [6][7][8]. This is because the transformation rate "seems" to depend only on temperature (through k) and on the transformed fraction. However, depending on the thermal history (e.g. the heating rate), a given value of α will correspond to a different state and consequently it will evolve at a different rate. Indeed, the KJMA rate equation is valid for non-isothermal transformations only when very particular conditions are met (see Section 3.a). Despite these severe limitations, non-isothermal experiments are commonly interpreted within the KJMA rate equation. As pointed out by several authors [7][8][9][10], analytical methods based on the KJMA rate equation have been developed regardless of its validity. In particular, the widespread Kissinger method [11] can be applied to any transformation described by the KJMA rate equation [6]. Even though one would expect erroneous conclusions from this incorrect use of the KJMA rate equation, the fact is that good agreement with other exact methods is often obtained. This is a strong indication that many properties of the exact solution are shared by the KJMA rate equation.
In this work a "quasi-exact" solution of Avrami's model for the continuous heating case is obtained by imposing only an Arrhenian temperature dependence for both nucleation and growth rate. Our solution proves to be the exact solution of a KJMA rate equation in which the kinetic constant, k, is slightly modified with respect to the isothermal case.
So the validity of the KJMA rate equation is extended beyond the severe limitations of the isothermal KJMA rate equation.

The isothermal KJMA rate equation
For the transformations involving nucleation and growth and assuming that the nuclei of the new phase are randomly distributed, Avrami [2,3] obtained the following relation where α ex is the extended transformed fraction, i.e. the resulting transformed fraction if nuclei grow through each other and overlap without mutual interference N is the nucleation rate and v(τ,t) is the volume transformed at time t by a single nucleus σ is a shape factor (e.g. σ =4π/3 for spherical grains), G is the growth rate and m depends on the growth mechanism [7,9,12] (e.g. m=3 for three dimensional growth).
Eqns. (3)(4)(5) show the kinetics of the transformation under very general assumptions about the rate constants (any time or temperature dependence) and for any thermal history. The KJMA relation (eqn. (1)) is the particular solution for isothermal conditions provided that both G and N do not depend on time. The overall rate constant is given by: In most practical situations it is possible to assume an Arrhenian temperature dependence for both N and G [12,13] ) where E N and E G are the activation energies for nucleation and growth respectively, and K B is the Boltzmann constant. Substitution of eqn. (7) into eqn. (6) gives 4 where we have defined the overall activation energy as: As mentioned in the introduction, differentiation of eqn. (1) leads to eqn. (2) which will be referred to as the isothermal KJMA rate equation for the rest of the paper.

The non-isothermal case
Although non-isothermal experiments can use any arbitrary thermal history, the most usual experiments performed in thermal analysis involve heating at a constant rate, β=dT/dt. Therefore, and for the rest of the paper, we will deal with this particular nonisothermal condition.

3.a The isokinetic case (E N =E G ).
The KJMA rate equation can be applied to the non-isothermal case when the transformation rate depends exclusively on temperature and on the degree of transformation [6,8,9] and not on the thermal history. This condition is fulfilled in particular cases such as "site saturation", where nucleation is completed prior to crystal growth [9,14], or the singular "isokinetic" situation where N and G have the same activation energy [2].
For a constant heating rate, introducing eqn. (7) into eqn. (5) gives the volume transformed by a single nucleus: where T=T 0 +βt, T'=T 0 +βτ, (T 0 is the initial temperature) and the function p(x) is defined as (see Appendix A): Accordingly, the extended transformation fraction, α ex , can be deduced after substituting eqn. (10) into eqn. (4), and assuming that the transformation rate is negligible at T 0 : and, substituting α ex into eqn. (3) gives, finally, the transformed fraction: Derivation of eqn. (14) with respect to time shows that it is an exact solution of the isothermal KJMA rate equation. Consequently, this equation is valid for non-isothermal conditions provided that E N =E G (isokinetic case). The literature shows [10,15] that the solution, α(t), for the "site saturation" case also obeys the isothermal KJMA rate equation (eqn. (2)).

3.b The general case (E N ≠ E G ).
When E N ≠ E G , the integral of eqn. (12) has no analytical solution. For most experiments E/K B T>>1, thus p(x) is usually approximated by its first term in a series of 1/x [7,10,16- with this first order approximation, a number of authors [18][19][20][21] obtained an identical solution of eqn. (12).
We will follow a different approach to solve the integral of eqn. (12). The fact that the arguments of the exponential functions and p(x) are different makes it impossible to solve it analytically. We overcome this problem by replacing these arguments by a common averaged argument (see Appendix B). With this approximation, eqn. (12) can be solved and the corresponding transformed fraction is (see Appendix C): where C is a constant that depends on m, E N and E G : It is worth noting that our approximate solution coincides with the exact isokinetic solution (eqn. (14)), except for the constant C. As a consequence, the approximate solution for the general non-isothermal case is also a solution of the isothermal KJMA rate equation with the overall rate constant k multiplied by the constant C: In the rest of the paper, we will refer to eqn. (18) as the non-isothermal KJMA rate equation. As expected when E N =E G , C reduces to unity and our solution (eqn. (16)) coincides with the exact solution for this particular limit (eqn. (14)).

3.c Accuracy of the non-isothermal KJMA rate equation
We will analyze the accuracy of our solution by comparing it to the exact solution that results from the numerical integration of Avrami's model (eqn. (3)(4)(5)). We will also show that it is much more accurate than: (a) the solution of the isothermal KJMA rate In Fig. 1, Δα/α has been plotted versus E N /E G for different "normalized" temperatures Fig. 1 it can be concluded that in any condition our solution is the most accurate one. For α=0.5, Fig. 1 tells us that the relative error of our solution is lower than 0.07 for E/K B T greater than 20. The absolute error ∆α is thus lower than 0.035. In most practical situations E/K B T is greater than 30, thus ∆α would be even lower (<0.01).
We conclude that with the accuracy of experimental data, our solution can barely be distinguished from the exact solution. It is worth mentioning that Fig. 1  In the limit where E G >> E N , nucleation takes place before growth [13] and the "site saturation" approximation is the most appropriate description. Conversely, when E N >> E G either heterogeneous nucleation dominates and the "site saturation" approximation is also the appropriate description, or crystallization is driven by epitaxial growth.
Another way to test the accuracy is by plotting the crystallization rate as a function of temperature. This has been done for the particular G and N values of amorphous silicon crystallization [22]. The result has been plotted in Fig. 2  (16)) and equating it to zero: by substituting p(x) by its first order approximation in the last term, one gets: which tells us that at the peak temperature, α ex =1, (compare eqn. (23) with eqn. (16)) and consequently: The previous result does not depend on any parameter and it was proposed by Henderson [6] as a test for the applicability of the KJMA model. Moreover, ln .
Deviations of α P from 0.632 are negligible in real situations (E/K B T P > 20) [8]. The prediction of the peak temperature for the non-isothermal KJMA rate equation is plotted in Fig. 3 and compared with the exact solution. For a wide range of heating rates (1<β<100 K/min), discrepancies vary from 0.11 to 0.14 K. Consequently, within experimental accuracy, eqn. (18) can be considered exact.
The Coats-Matusita method [23,24] can be worked out from eqn. (16) by taking twice the logarithm: By substituting p(x) by its first order approximation, we obtain It can be easily verified numerically that the plot of ( )  Most of the abovementioned isoconversional methods rely on replacing p(x) at a given stage by the first term of its series (eqn. (15)). However, more accurate isoconversional methods do not use any mathematical approximation [26] or are based on a more precise approximation [8,17,27,28]. Nonetheless, since our solution is an exact solution of the KJMA rate equation, their applicability is automatically extended to the general non-isothermal case.
Ozawa's method [29] is a widely used exact method, and can be inferred by taking twice the logarithm of eqn. (16) [ ]

Conclusion
We  Table I summarizes the main results obtained in this paper.

Appendix A: the function p(x)
The function is related to the exponential integral E 2 (x) [30] according to: and, from the asymptotic expansion of E 2 (x) [30], p(x) can be developed as bearing in mind relation A.3, the latter integral can be solved and is the right hand side term of B.1.

Avrami's model
If we integrate by parts, eqn. (12) becomes: Where a ≡ E N /E G . After a second integration by parts, α ex is expressed as:  If a is replaced by E N /E G the later expression can be rewritten as:

Appendix D. Approximate solution of Vázquez et al.
The solution of Vázquez et al. [18]: can be deduced from our solution (eqn. (16)) simply by substituting p(x) by its first order approximation: and rewriting C as (see Appendix E): One can easily come to the same conclusion for the solutions obtained by other authors [19][20][21]. The identity is then established if we can prove that: Expansion of the right hand side term of (D.1) results in ) ( Thus relation D.2 and consequently the identity are proved.

Appendix F. Kissinger plot.
Consider a rate equation with the general form: where g is an arbitrary function. For a constant heating rate experiment ( ) the peak temperature T P is determined by the condition: which leads to the relationship: where g'(α P ) is the first derivative of g with respect to α evaluated at the maximum of