Beyond the constraints underlying Kolmogorov-Johnson-Mehl-Avrami theory related to the growth laws

The theory of Kolmogorov-Johnson-Mehl-Avrami for phase transition kinetics is subjected to severe limitations concerning the functional form of the growth law. This paper is devoted to sidestepping this drawback through the use of the correlation function approach. Moreover, we put forward an easy-to-handle formula, written in terms of the experimentally accessible actual extended volume fraction, which is found to match several types of growths. Computer simulations have been performed for corroborating the theoretical approach.

Figure 1

Pictorial view of a random ensemble of actual (dots 1–3) and phantom (dots 4 and 5) nuclei. The definitions of both phantom-included ($\widehat{X_e}$) and actual ($X_e$) extended fractions also are reported. The (extended) volume of the $i$th nucleus is denoted as $|X_i|$, and the transformed fraction is denoted as $X$. The symbol $\left|A\right|$ denotes the cardinality of the set $A$.

Figure 2

The integration domains and the correlation circles are depicted for the integrals over the $f_2$ function. The cases of KJMA-compliant and KJMA-non-compliant growths are reported in panels (a) and (b), respectively. In the drawing, $R_{1,2}\equiv R(t_1,t_2),R_i\equiv R(t,t_i)(i=1,2)$, and $t_2<t_1$. In the case of KJMA-compliant growths, (a) $R_1+R_{1,2}\le R_2$, and the hard-core circle is within the integration domain $R_2$. In the case of KJMA-non-compliant growths, (b) $R_1+R_{1,2}>R_2$, and the hard-core disk overcomes the integration domain of the older nucleus.

Figure 3

Integration domains and correlation hard disks for the integral over the $f_3$ function. In the drawing, $R_{i,j}\equiv R(t_i,t_j)$ and $R_i\equiv R(t,t_i)$. In the case of KJMA-compliant growth, all the circles are entirely within the $R(t,t_3)$ circle.

Figure 4

Computer simulations of phase transformations ruled by KJMA-non-compliant growths. The surface fraction $X$ is shown as a function of the extended fraction $X_e$ for the power law ($R\approx t^n$) where $n=1/2$, 1/3, and 1/4 in (a)–(c), respectively. The kinetics with (w) and without (wo) the inclusion of phantoms are displayed.

Figure 5

Contribution of phantom overgrowth to the kinetics reported in Fig. 4 [displayed is the difference between curves (w) and (wo)]. The area beneath the curves, normalized to the area of the kinetics of Fig. 4, are 0.053, 0.038, and 0.02 for $n=1/4,n=1/3$, and $n=1/2$, respectively.

Figure 6

(a) Sketch of the overgrowth process in the space domain. $r_p$ denotes the location of the phantom, which starts growing at time $\overline{t}$ when the size of the actual nucleus is $R\left(\overline{t}\right)$. The phantom overtakes the actual nucleus at time $t_o$ when the size of the actual nucleus is $R\left(t_o\right)$. (b) Graphical solution of Eq. (22) for the determination of $\eta =\frac{t_o}{\overline{t}}>1$.

Figure 7

Behavior of the fitting parameters $a$ and $b$ of Eq. (21) as a function of growth exponent $n$ for KJMA-non-compliant growths.

Figure 8

Kinetics of the actual surface fraction as a function of the actual extended surface fraction for several values of $n$. In the graph, the kinetics for $n=1/2$ is compared with the KJMA-compliant growth with $n=1,n=2$, and $n=3/2$ (from the top, respectively). In the inset, the kinetics for $n=1$ and $n=1/2$ are displayed together with the truncated KJMA series expansions, Eqs. (8) and (18), respectively.

Figure 9

Kinetics of the actual surface fraction as a function of the actual extended surface for the nonconstant actual nucleation rate (left scale). The best fit of the function, Eq. (21), to the $X\left(X_e\right)$ kinetics has been shown as a dashed line. The correlation coefficient of the fit is, in fact, 1 for the parameters $a=1.014$ and $b=0.0382$. The actual nucleation rate as a function of $X_e$ also is reported on the right scale in $nuclei\times {10}^6/site$ units.