Analytical and numerical study of particles with binary adsorption

Electro-oxidation of ethanol represents a key process in fuel-cell technology. We introduce a generalization of the random sequential adsorption model to study the long time scale and large length scale properties of the electro-oxidation process. We provide an analytical solution for one dimension and Monte Carlo results in two dimensions. We characterize the coverage and percolation properties of the jammed state and unveil the influence of quenched impurities in the selectivity of oxidation products.

Figure 1

Schematic representation of the adsorption rules (1). The A species (red) can deposit either as dimers, on two empty sites, yielding B products (blue), or as monomers, on one empty site with at least one occupied neighbor, yielding C products (green). In ethanol oxidation, $A$ is the ethanol, $B$ stands for the two products (CH and CO), and C the acetyl.

Figure 2

Catalog of possible adsorption attempts of a dimer on a segment of empty sites with length $n=5$, which is part of a larger segment of length $\ell$ ($\ell ⩾n$). For adsorption events with both landing sites in segment $n$, a dimer is adsorbed (a). For adsorption events with a single landing site in the $n$ segment, either a dimer or a monomer is adsorbed, depending on the occupation state of the neighboring site (b–e).

Figure 3

Coverage $\theta \left(t\right)$ and rates of adsorption $s\left(t\right)$ as a function of time obtained through Monte Carlo simulations (symbols) and analytically (solid line) for equal rates of adsorption (a and d), preferential carbon adsorption (b and e), and different rates of adsorption ($k_d=0.5$ and $k_m=1$; c and f). Total coverage and rates of adsorption (open squares), dimers (filled squares), and monomers (open circles).

Figure 4

Coverage of dimers (open circles) and monomers (filled squares) for Monte Carlo simulations and obtained analytically (solid line), with the analytical solution (solid line), at the jamming limit as a function of the ratio between the rate of deposition of dimers and monomers $R$ for the one-dimensional case. Inset: Two-dimensional Monte Carlo results.

Figure 5

(a) Dependence on the ratio $R$ of the spanning probability $R_L$ for dimers (open) and monomers (filled). Square lattices have been considered with ${128}^2$ (squares), ${256}^2$ (circles), and ${512}^2$ (triangles) lattice sites, for the spanning probability $R_L$ and fraction of sites belonging to the largest cluster $P_{\infty }$ (inset). (b) Percolation threshold ($R_c$) as a function of the system sizes, for linear sizes of $L=\{32,64,128,256,512\}$, for dimers (filled circles) and monomers (filled squares).

Figure 6

(a) Largest cluster $s_{\max }$ and (inset) second moment of the cluster size distribution function $M_2$ at $R_c$ as a function of the system sizes, for linear sizes of $L=\{32,64,128,256,512\}$, for dimers (filled circles) and monomers (filled squares). Fractal dimension for monomers and dimers of $D_m=1.898\pm 0.008$ and $D_m=1.890\pm 0.009$, respectively. (b) Correlation function $g\left(r\right)$ for dimers (monomers in the inset) for a system of linear size $L=1024$ and averaged over 100 samples with a power-law exponent of $\eta =0.2101\pm 0.0002$ ($\eta =0.1361\pm 0.0002$).

Figure 7

Difference in the jamming coverage of monomers and dimers as a function of the impurity coverage, in two dimensions, for $R=0.1$ (open circles), $R=1$ (filled squares), and $R=10$ (filled circles).

Figure 8

Wrapping probability as a function of the ratio $R$, in two dimensions, for (a) monomers with impurity coverage, from right to left, of 0$\%$, 10$\%$, 20$\%$, 30$\%$ and, 40$\%$ and (b) dimers with impurity coverage, from left to right, of 0$\%$, 10$\%$, 20$\%$, and 30$\%$. Simulation of a system size of ${512}^2$. Inset: Fraction of sites belonging to the largest cluster $P_{\infty }$.