Weibull function to describe the cumulative size distribution of clumps formed by two-dimensional grains randomly arranged on a plane

Many manifestations of natural processes give rise to interesting morphologies; it is all too easy to cite the corrugation of the Earth's surface or of planets in general. However, limiting ourselves to 2D cases, the morphology to which crystal growth gives rise is also intriguing. In particular, it is interesting to study some characteristics of the cluster projection in 2D, namely the study of the shapes of the speckles (fractal dimension of their rims) or the distribution of their areas. Recently, for instance, it has been shown that the size cumulative distribution function (cdf) of “voids” in a corrole film on Au(111) is well described by the well known Weibull distribution. The present article focuses on the cdf of cluster areas generated by numerical simulations: the clumps (clusters) are generated by overlapping grains (disks) whose germs (disk centers) are chosen randomly in a $2000\times 2000$ square lattice. The obtained cdf of their areas is excellently fitted to the Weibull function in a given range of surface coverage. The same type of analysis is also performed for a fixed-time clump distribution in the case of Kolmogorov-Johnson-Mehl-Avrami (KJMA) kinetics. Again, a very good agreement with the Weibull function is obtained.

Figure 1

An example of morphology and cumulative counts (full symbols) of clump areas (grain area units) for grains generated at random for $\mathrm{\Theta }=0.45$, $r=40;$ and (a) $\infty$ overlaps, (b) one overlap, and (c) two overlaps. Continuous line represents the best fit of simulation results to the Weibull function (see Table 1). First part of the curve is dashed since the theoretical point (0,0) is not included in the fitting process, (see the text for details).

Figure 2

KJMA kinetics morphology for (a) instantaneous (Dirac $\delta$) nucleation at time t = 28 and $\mathrm{\Theta }=0.46$; (b) linear nucleation at time t = 36 and $\mathrm{\Theta }=0.45$. Full symbols are the corresponding cumulative counts (arbitrary units) of clump areas fitted tothe Weibull function (continuous line), (see Table 3 and Table 4). First part of the curve is dashed since the theoretical point (0,0) is not included in the fitting process, (see the text for details).

Figure 3

Panels (a)–(c) show a graphical representation of the functions $f^{\left(G\right)}(x,y)$, $f^{\left(ss\right)}(x,y),$ and $f^{\left(s^2\right)}(x,y)$. Panels (d)–(f) report an example of morphology and the cumulative counts (arbitrary units) of clump areas when respective functions are introduced (see text for details); $\mathrm{\Theta }=0.45$, $r=40$.

Figure 4

Schematic graphical representation to show how the periodic boundary conditions are taken into account in the simulations (see text for details).