Solid phase crystallization under continuous heating: kinetic and microstructure scaling laws

The kinetics and microstructure of solid-phase crystallization under continuous heating conditions and random distribution of nuclei are analyzed. An Arrhenius temperature dependence is assumed for both nucleation and growth rates. Under these circumstances, the system has a scaling law such that the behavior of the scaled system is independent of the heating rate. Hence, the kinetics and microstructure obtained at different heating rates differ only in time and length scaling factors.Concerning the kinetics, it is shown that the extended volume evolves with time according to alpha_ex=[exp(kappa Ct)]^m+1, where t' is the dimensionless time. This scaled solution not only represents a significant simplification of the system description, it also provides new tools for its analysis. For instance, it has been possible to find an analytical dependence of the final average grain size on kinetic parameters. Concerning the microstructure, the existence of a length scaling factor has allowed the grain-size distribution to be numerically calculated as a function of the kinetic parameters.


Introduction
Crystallization of amorphous materials and other solid state transformations usually involve random nucleation and growth. Under this assumption, the phase transformation is described by the Kolmogorov-Johnson-Mehl-Avrami theory (KJMA) [1][2][3][4][5][6]. The transformed fraction, α , is related with the extended transformed fraction, ex α , through the so-called KJMA relation: (1) ex α would be the transformed fraction if grains grew through each other and overlapped without mutual interference, i.e.: where I is the nucleation rate per unit volume and v ex (u,t) is the extended volume transformed at time t by a single nucleus created at time u In Eq. (3), 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] (e.g., m=3 for three dimensional, 3D, growth).
Unfortunately, owing to the dependence of G and I on temperature, general exact solutions do not exist for non-isothermal conditions. Accordingly, a number of published works have developed different theoretical and numerical approaches to analyze non-isothermal phase transformations within the framework of KJMA theory . Recently, a quasi-exact solution of the KJMA theory was obtained under continuous heating conditions [31].
A useful approach to investigate the kinetics and grain morphology consists of finding a scaling law such that the system behavior is universal. This method has been successfully used for the isothermal case [32]. In this case the time, τ , and length, λ , scaling factors are [33]: When time is scaled in Eq. (4), one gets a universal solution (independent of I and G): where, and τ is the dimensionless time.
In this paper we will show that a similar scaling law applies for transformations at a constant heating rate (Sec. 2). For a given ratio between the activation energies of I and G, there exists an approximate scaled solution independent of the heating rate.
Accordingly, the kinetics and microstructure for any heating rate can be obtained from this scaled solution simply by multiplying the dimensionless time and length values by the corresponding scaling factors. In Sec. 3 we obtain the scaled solution for the transformation kinetics, ) ' (t α , which represents a significant simplification when compared to the quasi-exact solution recently published [31].
Apart from the transformation kinetics it would be very useful to know the resulting material's microstructure because many of the material's physical properties are microstructure-dependent. Surprisingly, work related to the microstructure obtained under continuous heating conditions is very scarce. As far as we know, only Crespo et al. [34] have addressed this problem for a particular case. In Section 4, and thanks to the simplicity of the scaled solution, an analytical expression is obtained for the average grain size. Additionally, we numerically analyze the dependence of the grain size distribution on the ratio between the nucleation and growth activation energies. Finally, in Section 5 the limits of thermally activated nucleation are analyzed. It will be shown that when the activation energies of nucleation and growth are significantly different, the model of pre-existing nuclei is more adequate. A scaled exact solution for preexisting nuclei is also included.

The scaling law
In most practical situations where continuous nucleation takes place, it is possible to assume an Arrhenius temperature dependence for both I and G [9][10][11][12]21,25,35]: where E N and E G are the respective activation energies for nucleation and growth, k B is the Boltzmann constant and T is the temperature. Under this assumption, Eqs. (1)- (3) have a quasi-exact solution [31]: where ( ) where The time, P τ , and length, P λ , scaling factors we propose here are inspired by the isothermal case, Eq. (5). Since I and G depend on time through temperature for constant heating, we define the scaling factors using the values of I and G for a particular temperature. A logical choice is the well-defined peak temperature, T P , i.e., the temperature at which the transformation rate is maximum: where T P , is given by Eq. (A2) (see Appendix A).
Under the approximation that the crystallization takes place in a relatively narrow temperature range, the dimensionless growth and nucleation rates become (see Appendix A): Therefore kinetics and microstructure can be obtained from the scaled system simply by multiplying the dimensionless time and length by P τ and P λ , respectively.
Equation (9) is obtained under the assumptions that the critical nuclei size, the transformation rate at the initial temperature, T 0 , and the incubation time for nucleation are negligible. The first assumption relies on the fact that the average grain size is usually much larger than the critical nuclei size. Thus, this approximation only affects the very early stages of crystallization. Concerning the second assumption, it is based on the fact that, in well designed experiments, T 0 is low enough to ensure that the experimental results do not depend on T 0 . Finally, the existence of a finite incubation time would modify Eq. (9). However, as the incubation time is linked to the crystallization kinetics, in many cases an approximate relation equivalent to (A2) is expected and the scaling law is still valid. For instance, we have verified the validity of the scaling law for the case of crystallization of a-Si where the activation energy of the incubation time is similar to that of crystallization [36].

Scaled approximate solution for the transformation kinetics
In this section, we will find a scaled expression for ) (t α (i.e., independent of β ) which virtually coincides with the quasi-exact solution. Let us rewrite the non- and show that, under the approximation of Eq. (12), it is scalable with time. With Eq.
Once k(T) is substituted in Eq. (14), a scaled equation results: after imposing that 1 = ex α at the peak temperature [31] (i.e., at t'=0). Finally the scaled solution for the transformed fraction is obtained after combining Eqs. (1) and (17): Alternatively, Eq. (17) can be obtained after integration of Eqs.  Table I).
The coincidence for both heating rates and the scaled solution is excellent. The In Fig. 2 we see that this value departs only slightly from the exact one when E N is very different from E G , the discrepancy being higher for Let us highlight the formal simplicity of the scaled solution [Eq. (18)] when compared with the quasi-exact solution [Eq. (9)]. This simplicity has been reached without any significant loss of accuracy in the range of transformed fractions of practical interest (say, In contrast with the isothermal case [Eq. (6)] we see that the scaled solution for continuous heating is not universal (independent of G and I) but depends on the particular value of the ratio between the activation energies E G and E N (through the parameter C). This dependence has important consequences for the microstructure development, which will be analyzed in the next section.

Grain size morphology
In this section we will verify the length scaling law proposed in Sec. 2 [Eq. (11)] and analyze the dependence of the final microstructure on the kinetic parameters.
To characterize the final microstructure we have calculated the grain size distribution, where the grain size of an individual grain, i, is defined as: and i v is the actual grain volume. The numerical algorithm used for the calculation of the grain size distribution is described in [32]. To characterize the grain size distributions from the numerical simulations, we have calculated the average grain size, > < r , the mean grain radius, r and its standard deviation, r σ , defined as: where N is the final number of grains. The average grain size after 3D crystallization as a function of

Analytical solution for the average grain size
Before looking at the grain size distributions in detail, let us take advantage of the scaled solution for α [Eq. (18)] which allows us to find an analytical expression for the average grain size. According to Ref. [32], > < r can also be calculated from the number of grains formed after complete crystallization: where N can be obtained from In Appendix B, the scaled solution has been substituted in Eq. (23) and an analytical expression for > < r has been obtained: where Γ is the gamma function [37]. It has been plotted in Fig. 2 , the distribution coincides with the isothermal distribution obtained in ref.
[32]. This is as expected because G and I have the same temperature dependence and their ratio is constant. Consequently, the temperature has an effect on the rate at which the transformation proceeds but not on the microstructure. In this particular case, the grain size distribution is independent of the thermal history. For , the grainsize distribution departs progressively from the isothermal one, and when the ratio G N E E / is far from unity, the distributions have characteristic shapes which can be readily understood.
, during the first stages of the transformation, nucleation dominates over growth. Consequently, the nuclei density is higher when compared to the isothermal case. Thus, when G N E E / diminishes, the average grain is reduced (Fig.   3). Concerning the bell-shaped grain-size distribution for Fig. 4(a)], it can be explained by the fact that most nuclei are formed at a temperature range where they are not allowed to grow significantly. This means that they grow together at higher temperatures leading to a narrow distribution of grain sizes. In Fig. 5 we see that, indeed, the standard deviation diminishes drastically for In contrast, when 1 / > G N E E during the first stages of crystallization, growth dominates and the nucleation rate increases progressively as crystallization proceeds.
Since the time left for growing is lower for the nuclei that appear later, the density of small grains will be higher than for larger grains [as shown in Fig. 4(b)]. In fact, from Fig. 4(b), one can infer that the slow initial nucleation results in the formation of a small quantity of large grains. Moreover, this initial low nucleation rate results in a reduction of the transformation rate which is manifested in Fig. 2

Limits of thermally activated nucleation
From a formal point of view, the analysis given in Sec. 2-4 for continuous nucleation can be applied for any arbitrary value of the ratio G N E E / . In the following, we will argue that, when this ratio is far from unity, the material will follow the kinetics of pre-existing nuclei (described in Appendix C), when nucleation is not thermally activated but a constant density of nuclei, n 0 , already exists before they grow. When G N E E << , nucleation takes place early and, eventually, its rate may vanish before the onset of particle growth (site saturated nucleation [21,38] [35,39,40], i.e., it is virtually impossible to prevent heterogeneous nucleation. In this case, again, one can also apply the model of preexisting nuclei provided that nuclei are randomly distributed [41,42]. In the case of heterogeneous nucleation, the latter condition can be jeopardized by a particular distribution of the external nucleation sites. However, in several practical situations and in the case of heterogeneous nucleation localized at the container walls it is possible to assume that nucleation sites are randomly distributed.
Thus, the problem can be solved by assuming an initial surface density of preexisting nuclei [43][44][45].
A universal scaled solution can also be obtained for the case of pre-existing nuclei (see Appendix C). Calculations, like those done for continuous nucleation in Sec.

Conclusions
In In addition to the crystallization kinetics, it has been shown that the grain size distribution can be scaled with a characteristic length. Again, for a given For the sake of completeness, the kinetics and grain size distribution have been calculated for the case of preexisting nuclei. It has been shown that it is also possible to find appropriate time and length scaling factors.
Finally, our analysis relies on the fact that the transformation is thermally activated and, consequently, that it takes place in a narrow temperature range. Indeed, many real transformations are thermally activated, thus we believe that our approach can by applied to a large number of transformations.

Appendix A. Dimensional scaling law for the case of continuous nucleation
The time, P τ , and length, P λ , scaling factors are defined in Eq. (11) where the peak temperature, T P , is given by: Substitution of Eqs. (9) and (10) in Eq. (A1) leads to the value of T P as the solution of an algebraic equation: The scaled system is universal (independent of β ) provided that the dimensionless growth and nucleation rates do not depend on β . Actually, the dimensionless growth and nucleation rates are: Unfortunately, the result does depend on β through the relationship between T and t. To suppress this dependence we will suppose that the temperature range where the crystallization takes place is relatively narrow:  T   T  T  T   T  T  T   T  T  T Furthermore, selecting a time scale requires selecting a scale factor as well as a time origin. This origin must correspond to an equivalent state for any dimensional system (any particular value of β ). Here again the natural choice is the time at which the transformation rate is maximum: Then, substitution of Eqs. (A5) and (A2) into Eq. (A4) gives: Thus, the dimensionless growth and nucleation rates become: Appendix B. Analytical calculation of > < r and HM t Δ for the scaled system The total number of grains N is given by Eq. (23). Combining Eqs. (23), (18) and (13), one gets: . Finally, > < r is obtained from substituting Eq. (B1) into Eq. (22): For the calculation of HM t Δ we first calculate the transformation rate from Eq.
and the transformation at the maximum is: consequently,

Appendix C. Universal scaled solution for the case of pre-existing nuclei
When nucleation is completed prior to crystal growth, the kinetics of the transformation is simpler because it is exclusively governed by the growth rate: and n 0 is the pre-existing nuclei density. Then, the corresponding peak temperature is given by: (C2) On the other hand, according to the scaling law for the isothermal case [32,46] Supposing again that the temperature range where the crystallization takes places is relatively narrow, one gets and a much simpler expression results for ex (C6) For pre-existing nuclei, the grain-size distribution f(r) coincides with the distribution obtained under isothermal conditions. In [32] it has been shown that, for 3D, it can be fitted to a Gaussian distribution (the square correlation coefficient is The transformation rate at the maximum is (t'=0):