Random sequential adsorption of self-avoiding chains on two-dimensional lattices

Random sequential adsorption of extended objects deposited on two-dimensional regular lattices is studied. The depositing objects are chains formed by occupying adsorption sites on the substrate through a self-avoiding walk of $k$ lattice steps; these objects are also called “tortuous $k$-mers.” We study how the jamming coverage, ${\theta }_{j,k}$, depends on $k$ for lattices with different connectivity (honeycomb, square, and triangular). The dependence can be fitted by the function ${\theta }_{j,k}={\theta }_{j,k\to \infty }+B/k+C/k^2$, where $B$ and $C$ are found to be shared parameters by the three lattices and ${\theta }_{j,k\to \infty }$ ($>0$) is the jamming coverage for infinitely long $k$-mers for each of them. The jamming coverage is found to have a growing behavior with the connectivity of the lattice. In addition, ${\theta }_{j,k}$ is found to be higher for tortuous $k$-mers than for the previously reported for linear $k$-mers in each lattice. The results were obtained by means of numerical simulation through an efficient algorithm whose characteristics are discussed in detail. The computational method introduced here also allows us to investigate the full-time kinetics of the surface coverage ${\theta }_k\left(t\right)$ $[{\theta }_{j,k}\equiv {\theta }_k(t\to \infty )]$. Along this line, different time regimes are identified and characterized.

Figure 1

(a) Available configurations for tetramers ($k=4$) adsorbed on honeycomb lattices. (1) and (2) correspond to $U$-shaped and step-shaped, respectively. Black spheres, joined by thick lines, represent adsorbed $k$-mer units. Open circles correspond to empty sites. (b) The same as part (a) for tetramers ($k=4$) adsorbed on square lattices. $\left(1\right),\left(2\right),\left(3\right),\left(4\right)$ correspond to linear, step-shaped, $U$-shaped, and $L$-shaped tetramers, respectively. (c) The same as (a) for trimers ($k=3$) adsorbed on triangular lattices. $\left(1\right),\left(2\right),\left(3\right)$ correspond to linear, ${120}^{\circ }$, and ${60}^{\circ }$ trimers, respectively.

Figure 2

Example of systematic generation of SAW configurations with $k=4$ in the square lattice. The initial site (head) is showed in yellow, and the tail is represented by gray (from light to dark).

Figure 3

(a) Surface coverage $\theta$ as a function of time $t$ (solid black circles) for one sample corresponding to SAW $k$-mers with $k=13$ deposited on a square lattice with $L=1300$. The results were obtained by following the acceptance-rejection deposition scheme presented in Sec. 2 (stage 1) with $t_1\approx 74$ MCSs. The value of the surface coverage after stage 1 $[{\theta }_1=\theta (t=t_1)]$ is indicated. (b) Surface coverage $\theta$ (left axis, solid red circles) and remaining $k$-uples in list $N_R$ (right axis, open blue circles) as a function of the number of steps for the same case in (a). The data were obtained by following the rejection-free deposition scheme presented in Sec. 2 (stage 2). The value of the jamming coverage (${\theta }_j$) is indicated. (c) Surface coverage $\theta$ as a function of time $t$ ($\theta$ axis in linear scale, $t$ axis in log scale) for the same sample shown in (a) after the complete deposition process. The red (blue) line represents results obtained in stage 1 (2).

Figure 4

Jamming coverage ${\theta }_{j,k}$ vs the lattice size $L/k$ for SAW $k$-mers adsorbed on a square lattice and $k=5$. The error bars are included.

Figure 5

Jamming coverage ${\theta }_{j,k}$ as a function of $k$ for tortuous $k$-mers on honeycomb (hexagons), square (squares), and triangular (triangles) lattices. Symbols represent simulation results, and dashed line corresponds to the fitting function.

Figure 6

Surface coverage as a function of time for tortuous $k$-mers on square lattices. The curves have been obtained for different values of $k$ as indicated.

Figure 7

Semilogarithmic plot for the difference between jamming coverage and coverage at time $t$ for square lattices and different chain lengths as indicated. For a better view of all the curves on the same plot, we plotted again the normalized time $t/\tau$. The dashed line is the pure exponential decay $exp(-t)$ for $k=1$. The value of $\tau$ was selected following the theoretical prediction in Eq. (3). Solid lines correspond to our results, and square symbols correspond to previous results in Ref. [45].

Figure 8

Jamming coverage ${\theta }_{j,k}$ as function of $k$ for linear $k$-mers on honeycomb (hexagons), square (squares), and triangular (triangles) lattices with $k$ between 2 and 20. Solid symbols correspond to results obtained in this work, and open symbols to data previously reported in the literature as indicated.