Cell-size distribution and scaling in a one-dimensional Kolmogorov-Johnson-Mehl-Avrami lattice model with continuous nucleation

The Kolmogorov-Johnson-Mehl-Avrami (KJMA) growth model is considered on a one-dimensional (1D) lattice. Cells can grow with constant speed and continuously nucleate on the empty sites. We offer an alternative mean-field-like approach for describing theoretically the dynamics and derive an analytical cell-size distribution function. Our method reproduces the same scaling laws as the KJMA theory and has the advantage that it leads to a simple closed form for the cell-size distribution function. It is shown that a Weibull distribution is appropriate for describing the final cell-size distribution. The results are discussed in comparison with Monte Carlo simulation data.

Figure 1

Sketch of the 1D KJMA lattice model dynamics.

Figure 2

Fraction of activated sites (transformed phase), $w=W\left(t\right)/L$, as a function of time. Continuous curves are computer simulation results obtained for lattices with $L={10}^8$ sites and for various nucleation probabilities $p$, averaged on $Q=100$ realizations. Blue squares are the prediction of the KJMA theory, while the red dashed line is the result given by the MF theory.

Figure 3

Time needed for saturation, $t_{\mathrm{sat}}$, as a function of the nucleation probability, $p$. Computer simulation results (circles) are obtained on lattice sizes $L$ and averaged on $Q$ realizations as indicated in the legend. Black lines represent the prediction of the KJMA theory given by Eq. (14). Red lines indicate our MF prediction given in Eq. (35).

Figure 4

Saturation time, $t_{\mathrm{sat}}$, as a function of the system size. Simulation results (symbols) are for different $p$ values indicated in the legend. The data is averaged on many configurations $Q$ which is changing with the system size $L$ (the same $Q$ values are used as in case of Fig. 3). The dashed line indicates the upper bond given by Eq. (35).

Figure 5

Mean cell size as a function of the $p$ nucleation probability. Computer simulations were realized on lattices with different sizes $L$ and averaged on $Q$ realizations as it is indicated in the legend. The continuous black line represents the prediction of the KJMA theory [Eq. (13)] and the red dashed line is our MF prediction given in Eq. (30).

Figure 6

Final cell-size distributions, $\rho \left(x\right)$, obtained for different $p$ nucleation probabilities (as indicated in the legend) and $L={10}^8$ system size. The continuous curve shows a Weibull fit with $\alpha =1.78$ for the collapsed data. Simulation results are generated from $Q=100$ different runs, which result in $N_{\mathrm{cells}}$ individual cells (indicated in the legend).

Figure 7

Scaling properties related to the final cumulative cell-size distribution. The linear trend on the log-log plot for $-ln[1-\mathrm{\Omega }(x\left)\right]$ suggests that the Weibull distribution function with $\alpha =1.78$ describes well the computer simulation data. Symbols indicate computer simulation results for different nucleation probabilities, as specified in the legend. The continuous red line is a power law with exponent 1.78. Simulation results were obtained on lattices with $L={10}^8$ sites and calculated from $Q=100$ different runs which result in $N_{\mathrm{cells}}$ individual cells (indicated in the legend).

Figure 8

Results in comparison with the ones numerically computed and plotted by Axe and Yamada [18]. Black dots are our simulation results for the $\rho \left(x\right)$ density function, and the circles are the corresponding numerical results of Axe and Yamada. Filled black squares are MC simulation results for the cumulative cell-size distribution function, $\mathrm{\Omega }\left(x\right)$, and the empty squares are the results calculated form the work of Axe and Yamada. Filled triangles are the $R\left(x\right)$ values computed for our simulation results from Eq. (54) and the empty triangles are the $R\left(x\right)$ values calculated from the numerical results of Axe and Yamada.