Branch counting probability approach to random sequential adsorption

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Abstract

In a previous Letter we derived analytic expressions to describe the adsorption kinetics of trimers on two-dimensional arrays. As a prerequisite to find the average saturation coverage at the jamming limit (〈θJL〉) we had to solve first the kinetic rate equation, which is a lengthy and troublesome task. This motivated us to derive a faster and easier procedure to determine 〈θJL〉. The method allows the exact determination of 〈θJL〉 when a small two-dimensional lattice is bombarded by objects that occupy β  2 contiguous lattice sites, and is applied on a 3 × 5 lattice ceaselessly bombarded by linear trimers.

Introduction

The solution of problems related to the random filling of space with sets of geometrical objects that occupy β (β  2) contiguous lattice sites is of considerable interest. However, the exact solution of such problems in more than one dimension presents serious difficulties and most of the results available are for one-dimensional problems. It is well known that the methods of statistical mechanics are inappropriate when we come to consider the kinetic aspects of the adsorption of molecules that can occupy more than one site in a lattice space. In such systems there is configurational correlation in the sense that if a compartment is occupied, then at least one of its neighbors is also occupied. Thus, there is no random distribution of occupied compartments. The neglect of such configurational correlation has led to the failure of the Bethe approximation and similar statistical-mechanical techniques to predict nonunity values for the saturation coverage [1], [2], [3], [4], [5], [6], [7], [8].

In a previous Letter we derived analytic expressions to describe the adsorption kinetics of trimers on rectangular two-dimensional arrays [9]. Results are relevant to understand experimental information. The thermal conversion of acetylene to vynilidene on Rh{1 1 1} has been studied by King and co-workers [10] by laser-induced thermal desorption. The authors conclude that a one-step mechanism with a single activated complex involving the surface-assisted migration of one H atom across the C–C bond can explain the experimental data. Their experimental results were in full agreement with earlier conclusions of Silvestre and Hoffman [11], and Kang and Anderson [12], who suggested that vynilidene sits preferentially on a μ/π site, with one carbon atom strongly bonded on the twofold bridge site and a π bond to the third metal atom.

More recent experiments of H2 adsorption on Pd(1 1 1) have questioned the classical Langmuir picture of second-order adsorption kinetics at high surface coverage requiring pairs of empty sites for dissociative chemisorption [13], [14]. Experiments show that at least three empty sites are needed. Through density functional theory, it has been found that H2 dissociation is favored on ensembles of sites that involve a Pd atom with no direct interaction with adsorbed hydrogen. Such active sites are formed by aggregation of at least three H-free sites revealing the complex structure of the ‘active sites’.

As a prerequisite to find the average saturation coverage at the jamming limit 〈θJL we had to solve first the kinetic rate equation, which is a lengthy and troublesome task [9]. This has motivated us to derive a faster and easier procedure to determine the average coverage at the jamming limit. Therefore, in the present Letter, we derive an alternative route to find the average saturation coverage at the jamming limit without the complication of solving first the kinetic rate equation.

The theoretical model developed in Section 2 can be applied to determine either the average coverage or the average number of attempts until the jamming state 〈θJL and 〈m〉, respectively, of a random sequential adsorption of molecules that require more than one neighboring site in order to be adsorbed. The method can be applied to any lattice size or particle length in an easy and straightforward procedure that allows investigating its value for a considerably large lattice size. Once the branching structure formed by the interconnected microstates is determined the method allows an easy evaluation of the probability of observing the different jammed microstates. The simplification becomes especially important for bending molecules or large lattices as the number of microstates strongly increases. In the present Letter, without loss of generality, the particular case β = 3 is considered in detail.

Exact solutions are available in a few cases, typically for dimers in one dimension, whereas only numerical or approximate results exist for more complex objects, and fractal or high dimensional media [15], [16], [17], [18].

In Section 3, for comparison reasons, Monte Carlo simulations are presented on lattices of different size and our conclusions are summarized in Section 4.

Section snippets

Average coverage at the jamming limit

By following a procedure analogous to the one outlined in [9], we first look for all the possible arrangements when trimers are placed on a 3 × 5 array, see Fig. 1. Periodic boundary conditions are introduced in the model.

Fig. 1 shows the 18 different microstates (A, B, C,  O, P*, Q, R*) that can be obtained on a 3 × 5 array when the lattice is randomly bombarded by trimers until the jamming state. The asterisk indicates a jammed microstate. That figure also shows the evolution of every microstate

Simulations

We have simulated the random sequential filling of a two-dimensional lattice space with periodic boundary conditions by trimers. Table 1 shows the average coverage at the jammed state 〈θJL, its standard deviation σ and the average number of Monte Carlo steps until the jamming state is attained, 〈MCβM×N, obtained from 103 independent simulations for different matrix sizes.

The simulation results for different lattice sizes are shown, and the theoretical expression for a 3 × 5 lattice, derived

Discussion and conclusions

We have derived a method to find the average saturation coverage at the jamming limit without the complication of solving first the kinetic rate equation, which could be useful to understand recent experiments of conversion of acetylene to vynilidene on Rh{1 0 0} [10], [11], [12] or H2 adsorption on Pd(1 1 1) [13], [14].

In the first example acetylene and vynilidene require two and three surface sites, respectively, to be adsorbed. No investigations have been performed to determine whether product

Acknowledgements

This work was financially supported by the Consejo Nacional de Investigaciones Cientı́ficas y Técnicas, CONICET, Universidad Nacional de La Plata, UNLP, Fundación Antorchas, and Agencia Nacional de Promoción Cientı́fica y Tecnológica, ANPCyT.

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