Jamming and percolation properties of random sequential adsorption with relaxation

The random sequential adsorption (RSA) model is a classical model in statistical physics for adsorption on two-dimensional surfaces. Objects are deposited sequentially at random and adsorb irreversibly on the landing site, provided that they do not overlap any previously adsorbed object. The kinetics of adsorption ceases when no more objects can be adsorbed (jamming state). Here, we investigate the role of post-relaxation on the jamming state and percolation properties of RSA of dimers on a two-dimensional lattice. We consider that, if the deposited dimer partially overlaps with a previously adsorbed one, a sequence of dimer displacements may occur to accommodate the new dimer. The introduction of this simple relaxation dynamics leads to a more dense jamming state than the one obtained with RSA without relaxation. We also consider the anisotropic case, where one dimer orientation is favored over the other, finding a non-monotonic dependence of the jamming coverage on the strength of anisotropy. We find that the density of adsorbed dimers at which percolation occurs is reduced with relaxation, but the value depends on the strength of anisotropy.

Figure 1

Typical jamming state configuration of the dimers on a $64\times 64$ square lattice for the RSA model (a) with and (b) without relaxation. The dimers oriented in the horizontal and vertical directions have been painted in red and blue colors, respectively. The single vacant sites are represented by white color.

Figure 2

Plot of the binned data of the relaxation time distribution $D\left(T\right)$ for the entire process of adsorption on a log-lin scale using the lattice sizes $L=256$ (black), 512 (red), 1024 (green), and 2048 (blue). The data points are averages over samples ranging from $2\times {10}^6$ for $L=256$ to 12000 for $L=2048$.

Figure 3

Plot of the percolation threshold $p_c\left(L\right)$ against $L^{-1/\nu }$ with $1/\nu =0.756\left(6\right)$ for the lattice sizes $L=512,1024,2048,$ and 4096. The asymptotic value of the percolation threshold in the limit $L\to \infty$ has been estimated to be 0.5140(1). The data points are averages over samples ranging from $9\times {10}^5$ for $L=512$ to 8900 for $L=4096$.

Figure 4

For $L=1024$, the jamming state coverage $p_j\left(p_v\right)$ has been plotted against the selection probability $p_v$ of the vertically oriented dimers. The data points are averages over (at least) ${10}^5$ samples.

Figure 5

Plot of the deviation of the percolation threshold $p_c\left(p_v\right)-p_c(1/2)$ against $p_v-1/2$, $p_v$ being the selection probability of the vertical dimers, on a log-log scale with $p_c(1/2)=0.5140\left(1\right)$ and 0.5619(1) for the RSA with (open circles) and without (filled circles) relaxation, respectively. For each value of $p_v$, the $p_c\left(p_v\right)$ in the limit of $L\to \infty$ has been obtained using the values of $p_c(p_v,L)$ for $L=256,512,1024,2048,$ and 4096 and an extrapolation given by Eq. (1). The slopes of fitted straight lines have been measured as 2.05(6) and 2.07(7), respectively. The data points are averages over samples ranging from (at least) $1.6\times {10}^6$ for $L=256$ to 5000 for $L=4096$.

Figure 6

Right at the jamming state for the anisotropy parameter $p_v=1/2$, (a) the binned data for cluster size distribution $D\left(s\right)$ of the vertically oriented dimers has been exhibited on a semilog scale for $L=256$ (black), 512 (red), 1024 (green), 2048 (blue), and 4096 (magenta); (b) the average size of the largest cluster $\langle s_{max}(p_v,L)\rangle$ for the same values of $L$ has been plotted against $L$ on a lin-log scale. The data points fit considerably well with a straight line indicating the logarithmic growth of the largest cluster. The results are averages over samples ranging from $6.1\times {10}^6$ for $L=256$ to 6300 for $L=4096$.

Figure 7

(a) For $L=256$ (black), 512 (red), and 1024 (blue) the percolation probability $\mathrm{\Pi }(p_v,L)$ has been plotted with the probability of selection $p_v$ of the vertically oriented dimers. (b) Scaling plot of the same data as in (a). A plot of $\mathrm{\Pi }(p_v,L)$ against $(p_v-p_{vc})L^{1/\nu }$ using $p_{vc}=0.5577\left(5\right)$ and $1/\nu =0.754\left(5\right)$ exhibits a nice data collapse. The data points are averages over samples ranging from (at least) $2.4\times {10}^6$ for $L=256$ to ${10}^5$ for $L=1024$.

Figure 8

(a) For $L=256$ (black), 512 (red), and 1024 (blue) the scaled second moment $M_2^{\prime }(p_v,L)/L^2$ has been plotted against the selection probability $p_v$ of the vertically oriented dimers. (b) Finite-size scaling of the same data as in (a). Plot of the rescaled second moment $M_2^{\prime }(p_v,L)L^{-\gamma /\nu }$ with the scaling variable $(p_v-p_{vc})L^{1/\nu }$ using $p_{vc}=0.5577\left(5\right)$, $1/\nu =0.754\left(5\right)$, and $\gamma /\nu =1.795\left(5\right)$ exhibits a nice data collapse. The data points are averages over samples ranging from (at least) $2.4\times {10}^6$ for $L=256$ to ${10}^5$ for $L=1024$.