Random sequential adsorption on Euclidean, fractal, and random lattices

Irreversible adsorption of objects of different shapes and sizes on Euclidean, fractal, and random lattices is studied. The adsorption process is modeled by using random sequential adsorption algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension $d$ between 1 and 2, and on Erdős-Rényi random graphs. The number of sites is $M=L^d$ for Euclidean and fractal lattices, where $L$ is a characteristic length of the system. In the case of random graphs, such a characteristic length does not exist, and the substrate can be characterized by a fixed set of $M$ vertices (sites) and an average connectivity (or degree) $g$. This paper concentrates on measuring (i) the probability $W_{L\left(M\right)}\left(\theta \right)$ that a lattice composed of $L^d\left(M\right)$ elements reaches a coverage $\theta$ and (ii) the exponent ${\nu }_j$ characterizing the so-called jamming transition. The results obtained for Euclidean, fractal, and random lattices indicate that the quantities derived from the jamming probability $W_{L\left(M\right)}\left(\theta \right)$, such as ${(d{W}_L/d\theta )}_{\text{max}}$ and the inverse of the standard deviation ${\mathrm{\Delta }}_L$, behave asymptotically as $M^{1/2}$. In the case of Euclidean and fractal lattices, where $L$ and $d$ can be defined, the asymptotic behavior can be written as $M^{1/2}=L^{d/2}=L^{1/{\nu }_j}$, with ${\nu }_j=2/d$.

Figure 1

The log-log plots of ${(d{W}_L^{\prime }/d\theta )}_{\max }$ and ${\mathrm{\Delta }}_L$ as a function of $L$ for the following systems: (a) linear $k$-mers ($k=2$) on 1D lattices; (b) linear $k$-mers ($k=2$) on 2D square lattices, linear $k$-mers ($k=2$) on 2D square lattices in the presence of impurities ($\rho =0.15$), linear $k$-mers ($k=10$) on 2D triangular lattices, and $k^2$-mers ($k=4$) on 2D square lattices; and (c) linear $k$-mers ($k=2$) on 3D simple cubic lattices, $k^2$-mers ($k=24$) on 3D simple cubic lattices, and $k^3$-mers ($k=4$) on 3D simple cubic lattices. According to Eq. (4), the slope of each line corresponds to $1/{\nu }_j$ [or to $-1/{\nu }_j$ in the case of Eq. (5)].

Figure 2

(a) Basic units of the Sierpinski carpets studied in the present paper. (b) Sierpinski carpet with $l=3$ and $n=4$. Accordingly, $L=l^n=81$ and $d_f=ln(4l-4)/ln\left(l\right)=1.89279$. (c) Snapshot corresponding to a typical configuration of dimers (solid circles joined by lines) on a Sierpinski carpet with $l=3$ and $n=4$. Gray squares represent accessible sites in all the figures.

Figure 3

(a) Curves of the jamming probability $W_L^{\prime }$ as a function of the fraction of occupied sites $\theta$ for a typical case: $3\times 3$ pattern and different values of $n$ between 2 and 5. Accordingly, the lattice sizes are $L=9,27,81,243$. (b) A log-log plot of ${(d{W}_L^{\prime }/d\theta )}_{\max }$ as a function of $L$. (c) A log-log plot of ${\mathrm{\Delta }}_L$ as a function of $L$ for the data in (a).

Figure 4

Values of $2/{\nu }_j$ as a function of the lattice dimension $d$ for Euclidean 1D, 2D, and 3D lattices (open circles) and Sierpinski carpets with $l$ ranging between 3 and 8 (closed squares).

Figure 5

(a) Curves of the jamming probability $W_M^{\prime }$ as a function of the fraction of occupied sites $\theta$ for a typical Erdős-Rényi case with $M=10000,25000,50000,100000$. A log-log plot of (b) ${(d{W}_M^{\prime }/d\theta )}_{\max }$ and (c) ${\mathrm{\Delta }}_M$ as a function of $M$ for different cases, as indicated.

Figure 6

(a) A log-log plot of ${(d{W}^{\prime }/d\theta )}_{\max }$ as a function of $M$ for different RSA processes on Euclidean, fractal, and random lattices, as indicated. (b) Same as (a) but for ${\mathrm{\Delta }}_M$. The symbol references are the same for both panels.