Nucleation and growth in one dimension. I. The generalized Kolmogorov-Johnson-Mehl-Avrami model

Motivated by a recent application of the Kolmogorov-Johnson-Mehl-Avrami (KJMA) model to the study of DNA replication, we consider the one-dimensional (1D) version of this model. We generalize previous work to the case where the nucleation rate is an arbitrary function $I\left(t\right)$ and obtain analytical results for the time-dependent distributions of various quantities (such as the island distribution). We also present improved computer simulation algorithms to study the 1D KJMA model. The analytical results and simulations are in excellent agreement.

Figure 1

Definitions. In the KJMA model, a hole is the liquid domain between the growing solid domains (island). The island-to-island is defined as the distance between the centers of two adjacent islands.

Figure 2

Kolmogorov’s method. (a) Spacetime diagram. In the small square box, the probability of nucleation is $I_0\Delta x\Delta t$, where $I_0$ is the nucleation rate. In order for the point $X$ to remain uncovered by islands, there should be no nucleation in the shaded triangle in spacetime. (b) Kinetic curve for constant nucleation rate $I_0$: $f\left(t\right)=1-\exp (-I_{0}v{t}^2)$.

Figure 3

Spacetime diagram. The hole-size distribution ${\rho }_h(x,t)$ is proportional to the probability $p_0(x,t)$ for no nucleation event occurs in the shaded parallelogram $ABCD$ (see text).

Figure 4

Constraint plane $S:(i_1+i_2)∕2+h=x$.

Figure 5

Comparison of simulation times for the three algorithms discussed in the text [$I\left(t\right)={10}^{-5}t$ and $v=0.5$]. Circles are used for the lattice-model algorithm, squares for the double-list algorithm, and triangles for the phantom-nuclei algorithm. For each system size, the number of Monte Carlo realizations ranges from 5 to 20, and the lines connect the average simulation times. The double-list algorithm is two to three orders of magnitude faster than the lattice algorithm, while the phantom-nuclei algorithm ranges from three to five orders of magnitude faster, depending on the number of time points at which one records data. The solid triangles show the fastest case, with only one time point requested, while the open triangles show the slowest case, where data are recorded at each intermediate time step.

Figure 6

Schematic description of the double-list algorithm. (a) Basic setup for lists $\left\{i\right\}$ and $\left\{h^{\prime }\right\}$. Note that $\left\{h^{\prime }\right\}$ records cumulative lengths. (b) Nucleation. (c) Coalescence due to growth.

Figure 7

Schematic description of the phantom-nuclei algorithm. The figure shows the distribution of potential initiation sites in the spacetime plane. The open circles denote sites that do initiate, while the “phantom” solid circles, lying in the “shadow” of the open circles, do not initiate.

Figure 8

(Color online). Theory and simulation results for $I\left(t\right)={10}^{-5}t$ and $v=0.5$. Size distributions are calculated at these time points: $t=50$, 75, and 100. (a) Hole-size distribution ${\rho }_h(x,t)$. (b) Island distribution ${\rho }_i(x,t)$. The inset plots $f\left(t\right)$ vs $t$, with the dot at $t=75$ $(f=0.5)$. (c) Island-to-island distribution ${\rho }_{i2i}(x,t)$. The analytical curves have been obtained by Eq. (20). There is a crossover of the decay constant slightly after $t=75$ $(f=0.5)$ (see text). The inset shows ${\rho }_h(x,t)$ and ${\rho }_{i2i}(x,t)$ for $t=50$. All figures including insets have the same vertical range of ${10}^{-7}–{10}^{-3}$ (log scale).

Figure 9

(Color online). Decay constants $\left(t^{-1}\right)$ of ${\rho }_h(x,t)$, ${\rho }_i(x,t)$, and ${\rho }_{i2i}(x,t)$ for $I\left(t\right)={10}^{-5}t$ and $v=0.5$. The symbols are simulations, and the solid lines are theory.