Adsorption of binary mixtures on two-dimensional triangular lattices

Highlights

• Interacting binary mixtures adsorbed on triangular lattices were studied.

• A lattice-gas model was developed.

• The calculations were performed by Monte Carlo (MC), quasi-chemical (QCA) and cluster-exact approximation (CA).

• The presence of ordered structures in the adsorbed layer was observed.

• An excellent agreement was found between MC and CA.

• QCA fails to reproduce the low temperature ordered structures.

Abstract

The adsorption of interacting binary mixtures on triangular lattices is studied by combining theory and Monte Carlo (MC) simulations. Two theoretical approximations are used in the present work: (i) the cluster approximation (CA), based on exact counting of adsorption states on small lattices; and (ii) an extension of the standard quasi-chemical approximation (QCA) that includes two adsorbed species (a and b). In the case of CA, an own algorithm is developed to obtain the configurational grand partition function for small cells. Repulsive lateral couplings between adsorbate-adsorbate species are incorporated in the lattice-gas framework. Theoretical (CA and QCA) total and partial adsorption isotherms are compared with MC simulations. Quantitative and qualitative differences are shown and discussed, being CA the more accurate approach in all cases.

Introduction

The study of adsorbed films on regular substrates in the presence of adsorbate-adsorbate interactions is of great interest in surface science [1], [2], [3] due, particularly, to the rich variety of ordered structures that appear on the surface. These ordered phases have applications on catalyst surfaces, microelectronics fabrication, surfaces undergoing corrosion, chemical sensors and electrodes [4], [5], etc.

Based on the lattice-gas model, several analytical methods have been proposed to investigate monolayer adsorption [1], [2], [6], [7], [8], [9], [10], [11], [12]. For non-interacting ideal adsorbates, simple solutions can be obtained [1]. Excluded volume interactions (or rigid-sphere interactions) can also be analytically treated. However, for the simple case of pairwise adsorbate-adsorbate interactions, the statistical problem becomes difficult, and a very few cases can be analytically solved [13], [14], [15].

The interacting particles problem is even more complex when the adsorbate is a multicomponent phase and, consequently, most of the studies on adsorption of interacting adsorbates are devoted to single gases. The theoretical difficulties in the treatment of multicomponent statistics are mainly associated with the large number of parameters involved: inter- and intra- species couplings, molar fractions, chemical potentials, etc. [16]. All these factors have made difficult to obtain analytical expressions for the adsorption thermodynamic functions of multicomponent adsorbates.

The mixed-gas adsorption problem is also a demanding challenge from the experimental perspective [17], [18], [19]. In the case of single components, the adsorbed amount can be precisely measured from the variation in the weight of the adsorbent sample. The same does not happen for multicomponent gases, where additional measurements must be carried out to estimate the composition of the sample.

One way to overcome the theoretical and experimental complications is to employ Monte Carlo (MC) simulation technique [19], [20], [21], [22], [23], [24], [25], [26], [27], complemented by classical approaches such as Bragg-Williams (BWA) and quasi-chemical (QCA) approximations [1], and exact calculations in finite systems [or cluster approximation (CA)] [28], [29], [30], [31]. Each one showing its advantages/disadvantages. While BWA has demonstrated to be the simplest way to include pairwise lateral interactions in lattice-gas models, QCA has always provided better results (in comparison with MC data) than those derived from BWA [15], [32], [33]. In the case of CA, this approach is a straightforward calculation of a statistical sum, with the restrictions imposed by the small size of the studied system (for example, it is well-known that no phase transition develops in a finite system). However, and even though MC method is much superior, recent papers have proven the applicability of CA as a simple programing tool to treat mixture adsorption systems with very low time-consuming computation [34], [35], [36], [37], [38].

In Refs. [34], [35], the adsorption of $a-b$ interacting mixtures on homogeneous substrates was investigated by theoretical modeling and MC simulations. Adsorbate-adsorbate interactions were included through pairwise additive lateral interactions. Later, the study was generalized to non-additive couplings [38]. For this purpose, it was supposed that the interaction linking a given adsorbed particle with any of its nearest-neighbor sites strongly depends on the occupancy state in the first coordination sphere of such an adparticle. The scheme introduced in Refs. [34], [35] was also extended to include the effects of non-homogeneous surfaces. In this line, a heterogeneous cell composed by m sites with n different adsorption energies was used to obtain the grand partition function of the system [36], [37]. Non-interacting and [36]interacting [37] adsorbates were considered. In both cases, CA predictions were compared with experimental data and results obtained from the classical Ideal Adsorbed Solution Theory (IAST) [39].

In a pioneer work [40], Votyakov and Tovbin studied the adsorption of interacting mixed species on homogeneous and heterogeneous surfaces. The authors performed a complete analysis of the ground-state diagrams (temperature $T=0$) for heterogeneous regular and patchwise surfaces, showing that on the regular surfaces, phase transitions occur on a whole surface, while on the patchwise ones each homogeneous patch works independently with its own contribution to the summary diagram of the ground states. Later, by using MC simulations and transfer matrix technique, Fefelov et al. have deeply studied the adsorption of $a-b$ binary mixtures on square surfaces [41], [42], [43]. The authors carried out an exhaustive analysis, exploring the phase behavior of the system for a wide range of lateral interactions between nearest adsorbed molecules. The study was complemented with transfer matrix calculations. By this method, the ground state of the model was investigated, showing that the phase behavior of the model is maintained at finite temperatures.

The analysis developed in Refs. [34], [35], [36], [37], [38], [40], [41], [42], [43] was restricted to the case of square substrates. In contrast to the problem of non-interacting particles, the arrangement degeneracy of interacting particles strongly depends on the structure/geometry of the lattice. Consequently, it is of value and of interest to investigate how a specific surface geometry affects the thermodynamic functions corresponding to adsorbed binary mixtures. Of special importance is the case of lattices with triangular geometry, since the (111) facet is representative of many metallic-oxide surfaces and is also the most frequently exposed surface in supported metallic nanoparticles [44].

In this context, the main idea of this article is (i) to extend previous scheme (MC, QCA and CA) [34], [35], [36], [37], [38], [41], [42], [43] to triangular lattices and (ii) to study how the lattice structure affects the adsorption properties of interacting mixtures. For this purpose, (1) an extension of the standard QCA is applied to describe a two-component mixture adsorbed on a triangular substrate; and (2) a triangular (rhombus-shaped) cluster composed by $m=l\times l$ sites is used to obtain the grand canonical partition function of the system. In both cases, partial and total adsorption isotherms are calculated for different values of the adsorbate-substrate and adsorbate-adsorbate adsorption energies. We focus on the case of repulsive lateral interactions, where a wide diversity of structural orderings is observed in the adsorbed layer. The validity of the theoretical solutions is tested by comparison with simulation data.

The present study is a natural extension of our previous paper in Ref. [45], where a system of multicomponent gases adsorbed on triangular lattices was studied only by using MC simulations. Here, we will introduce new developments based on (i) exact calculation of adsorption states on small lattices, and (ii) the configuration-counting procedure of the QCA. On the basis of these simple theoretical models, we hope to contribute to a better understanding of a system, whose experimental realization is rather difficult.

As a background to our work, the problem of interacting particles (one species) adsorbed on triangular lattices has been extensively studied [7], [8], [9], [10]. Using QCA, Campbell and Schick [7] investigated the case of repulsive nearest-neighbor and attractive second nearest-neighbor interactions. In Refs. [8], [9], the same system was studied by means of MC simulations. Notorious discrepancies between theoretical [7] and simulation [8], [9] data were reported. Later, Votyakov and Tovbin [10] showed that these differences were just because the lateral interaction parameters in MC and QCA methods differed for compared phase diagrams. With a correct inclusion of the lateral interaction values, an excellent agreement between MC and QCA results was obtained in Ref. [10].

The organization of the paper is as follows. In Section 2, the model is given along with the basic definitions. Theoretical approximations (QCA and CA) and MC simulation methodology are described in Sections 3 and 4, respectively. In Section 5, the results are presented and discussed. Finally, the conclusions are drawn in Section 6.