Gap-size distribution functions of a random sequential adsorption model of segments on a line

We performed extensive simulations accompanied by a detailed study of a two-segment size random sequential model on the line. We followed the kinetics towards the jamming state, but we paid particular attention to the characterization of the jamming state structure. In particular, we studied the effect of the size ratio on the mean-gap size, the gap-size dispersion, gap-size skewness, and gap-size kurtosis at the jamming state. We also analyzed the above quantities for the four possible segment-to-segment gap types. We ranged the values of the size ratio from one to twenty. In the limit of a size ratio of one, one recovers the classical car-parking problem. We observed that at low size ratios the jamming state is constituted by short streaks of small and large segments, while at high values of the size ratio the jamming state structure is formed by long streaks of small segments separated by a single large segment. This view of the jamming state structure as a function of the size ratio is supported by the various measured quantities. The present work can help provide insight, for example, on how to minimize the interparticle distance or minimize fluctuations around the mean particle-to-particle distance.

Figure 1

Coverage dependence on the size ratio for equal depositing fluxes of each segment size. (a) Plot of the coverage up to 100 time units for various values of the size ratio, namely, 1, 1.1, 2, 4, and 10. (b) Plot showing the jamming coverage dependence on the size ratio.

Figure 2

In both plots, one has the $AA$, $AB$, and $BB$ gap types represented by circles, squares, and triangles, respectively. (a) Fraction of available empty space, at the jamming state. (b) Normalized population of gaps at the jamming state. In the insets the size ratio varies between $1$ and 20. Please, refer to the text for further details.

Figure 3

Gap-size distribution functions at the jamming state: (a) For a size ratio of one, the solid curve represents $P_0\left(x\right)$, while the remaining $AA$, $AB$, and $BB$ gap types are identical, and therefore fall onto a single curve. (b) For a size ratio of 2.0, one observes the splitting of the various gap types. Please, refer to the text for further details.

Figure 4

Plots involving cumulants up to the fourth order of the gap-size distribution functions for each gap type as a function of the size ratio. We use the same legend as in Fig. 2 (a) Average distance between pairs of segments. (b) Dispersion of the distance between pairs of segments. (c) Skewness. (d) Kurtosis.