Impingement factor in the case of phase transformations governed by spatially correlated nucleation

The Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory fails to treat nonrandom nucleation and overgrowth processes. However, the very tractability of their solution in describing experimental data caused researchers to slightly modify KJMA’s differential equation so as to extend the applicability of the model. In doing this a phenomenological parameter is introduced, named as the impingement factor. Here we analyze, in depth, the limits within which the phenomenological approach is suitable for sidestepping the difficulty of nonrandom nucleation. In particular we tackled two cases: instantaneous cluster growth where cluster overgrowth prevails and the constant nucleation rate of spatially correlated nuclei according to the hard-core model. In the last part of the paper we show that Avrami’s general set theory is equivalent to the statistical mechanics of rigid disks. This permits a deeper appreciation of the KJMA theory and the nonrandom nonsimultaneous kinetics.

Figure 1

Schematic representation of allowed nucleation centers. Dots a and b are both possible nucleation centers. In the figure two configurations are shown for cluster radii $R_1$ and $R_2$. For $R=R_2$ the pre-existing nucleation center b might become a phantom nucleus, while a has a chance to start growing. Because of the correlation $\left(R=R_{\text{hc}}\right)$, no nuclei can lay within the correlation zone; i.e., a nucleation event at c is not allowed.

Figure 2

(Color online) Kinetics of random sequential adsorption of rigid disks in the coverage domain. The adsorption rates $dN/dt$, as obtained by numerical simulation, are shown as full squares. The behavior of kinetics equation (20), in terms of the impingement factor, is reported as a solid line. The short dashed line is the approximate solution computed by means of the $f$-series expansion, Eq. (6), up to the second-order term. The diamond symbols (in red) show the kinetics computed by using the correlation functions, Eq. (7), up to the second-order term in the argument of the exponential. In this expansion the lowest-order term of the two-dot correlation function has been retained. The approximate kinetics obtained by decoupling the integrals on the pair-correlation function is also displayed as a long dashed line (blue).

Figure 3

(Color online) Fraction of transformed surface in the case of correlated nucleation (solid lines) as a function of the dimensionless time $\tau$ and for several values of the $\eta$ parameter. The nucleation and growth rates are constants. Each of the three bunches of curves displays the KJMA solution [Eq. (23); black dashed-dotted lines), phenomenological equation (3) (blue solid lines), and the exact kinetics [Eq. (22); red dashed lines]. From right to left, the parameter values of each bunch are: (a) $\eta =1$, $\gamma =0.25$; (b) $\eta =5$, $\gamma =0.2$; and (c) $\eta =15$, $\gamma =0.2$.

Figure 4

(Color online) Modeling the phase transition in the case of nonrandom nucleation by employing the Poissonian extended surface in the KJMA expression. The nucleation and growth rates are constants. The exact solutions are displayed as solid lines (blue); the dashed lines are the kinetics computed in the framework of the Poisson process, i.e., KJMA approach. From right to left, the parameter values of each bunch are: (a) $\eta =1$, (b) $\eta =5$, and (c) $\eta =15$.

Figure 5

The Avrami exponent $n$ for the kinetics in Fig. 4 is displayed. In panel (A) $n$ is plotted as a function of $\tau$ and it is a “universal” curve. In panel (B) it is plotted as a function of surface coverage for (a) $\eta =1$, (b) $\eta =5$, (c) $\eta =15$, and (d) $\eta =100$. In the limit $\eta \to \infty$, $n=3$.

Figure 6

(Color online) (a) Schematic representation of the configurations of a pair of overlapping disks of radii $R_1$ and $R_2$ as a function of relative distance $r$. The values defining the integration domains of the $I_i$ integrals are also shown. For $r<R_1-R_2$, the smaller nucleus is a phantom. (b) Behavior of the ${\eta }_{1,2}\left(r\right)$, ${\eta }_{2,1}\left(r\right)$, and ${\phi }_{1,2}\left(r\right)$ functions which enter the $I_i$ integrals. The minimum of the ${\eta }_{1,2}\left(r\right)$ function is marked. The integration paths, after the order of integration has been inverted, are displayed through arrows. The roots of the equations $\xi ={\eta }_{1,2}\left(r\right)$, $\xi ={\eta }_{2,1}\left(r\right)$, and $\xi ={\phi }_{1,2}\left(r\right)$, required to invert the order of integration, are also shown. Specifically, they are two for ${\eta }_{1,2}$ and one for both ${\eta }_{2,1}$ and ${\phi }_{1,2}$.