Surface atoms statistics

Van Hardeveld R, Hartog E. 1969. Statistics of surface atoms and surface sites on metal crystals. Surf Sci 15 189-230. 

Figure 23 Cancellation of lateral forces between two surfaces. The atoms in the top layer, represented by circles, experience forces that are dependent on the position of the atom with respect to the periodic substrate. The arrows on the atoms indicate the magnitude and direction of these forces. For contacts lacking commensurability that are contain a sufficiently large number of surface atoms or irregularities, these forces will cancel in a statistical sense.

Quite different site densities are obtained if these assumptions are changed. Perez et al.13 have calculated the surface site statistics using a computer model which can simulate incomplete layers by removing atoms from complete shells. The atoms removed are those which have the smallest number of first and second nearest neighbours. Many more types of site are considered in the models used by Perez et al. However, one of the most interesting results of their calculations is to demonstrate that for all sites, apart from B2 sites, there are very pronounced oscillations in number as the particle size is increased. Figure 2 shows the variation in the number of B2, B3, and B4 sites, and Figure 3 shows the ratio of B3/B4 sites as a function of particle size. Any reaction which is controlled by this ratio will show activity maxima for particle diameters of 0.8 and 2.0 nm. On the other hand B2 and B2 sites are the ones most likely to catalyse structure insensitive reactions. 

Recently, Do and co-workers [6-10] have proposed a very simple method but it does reveal the mechanistic pictures of what are occurring in pores of different size. The process of adsorption in miy pore is viewed as follows. Molecules in pore are constantly in motion but statistically there is a spatial distribution of these molecules due to the interactive forces between them and the surface atoms. We treat this spatial distribution as a step function uniformly high density near the surface and uniformly low density in the inner core of the pore. Due to the long range interaction of the surface, the pressure of the fluid in the inner core is not the same as that in the bulk phase. Assuming a Boltzmann distribution, the pressure of the inner core is related to the bulk fluid as... 

The more complex selective 1,3-butadiene (BD) hydrogenation was also examined [56, 57]. Butadiene hydrogenation produces 1-butene, tranx-2-butene, cti-2-butene, and n-butane, with 1-butene as the desired product. Pd-Al Oj model catalysts with mean particle diameters of 2-8 nm were applied to examine size effects. The abihty to accurately determine the relative abundance of specific surface sites (such as terrace, edge, interface atoms, etc. cf. surface site statistics in Table 1, 2 of [51]) is a tremendous advantage of model catalysts. Knowledge of the exact number and type of available surface sites allows the calculation of more accurate turnover frequencies. 

The current burst model is potentially powerful in providing explanations for many mechanistic and morphological aspects involved in the formation of PS. However, as recognized by Foil et al. themselves, it would be extremely difficult for such a unified model to be expressed in mathematical form because it has to include all of the conditional parameters and account for all of the observed phenomena. Fundamentally, all electrochemical behavior is in nature the statistical averages of the numerous stochastic events at a microscopic scale and could in theory be described by the oscillation of the reactions on some microscopic reaction units which are temporally and spatially distributed. Ideally, a single surface atom would be the smallest dimension of such a unit and the integration of the contribution of all of the atoms in time and space would then determine a specific phenomenon. In reality, it is not possible because one does not know with any certainty the reactivity functions of each individual atoms. The difficulty for the current burst model would be the establishment of the reactivity functions of the individual reaction units. Also, some of the assumptions used in this model are questionable. For example, there is no physical and chemical foundation for the assumption that the oxide covering the reaction unit is... 

Considerable efforts have been made over the years to discover if specific activity (and, where relevant, selectivity) are dependent on the size of the metal crystallites in a supported catalyst, which might be anticipated for several reasons. With use of regular shapes, such as the cubo-octahedron, as crystallite models the fractions of surface atoms with various co-ordination numbers can be calculated as a function of crystaUite size and show marked variation in the size range below The statistics of surface atoms and... 

Adsorption requires the O2 molecule to find a pair of adsorption sites and desorption requires two adsorbed G atoms to be adjacent. Hill [15] and Kisliuk [18,19] discuss lattice statistics and the probability of find-Ing pairs of sites in two-dimensional arrays presented by the regular arrangement of surface atoms illustrated in Figure 5.16. Boudart and pjega-Mariadassou [3],and Hayward andTrapnell [13], describe how the probability of finding pairs of sites is used to develop rate expressions on surfaces. When a bimolecular surface reaction occurs, such as dissociative adsorption, associative desorption, or a bimolecular surface reaction, the rate in the forward direction depends on the probability of finding pairs of reaction centers. This probability, in turn, depends... 

R. van Hardeveld and F. Hartog. The Statistics of Surface Atoms and Surface Sites on Metal Crystals. Surf. Sci. 15 189 (1969). 

Surface diffusion has so far been discussed in terms of a single surface atom. However, on a real surface many atoms diffuse simultaneously and in most diffusion experiments the measured diffusion distance after a given diffusion time is an average of the diffusion lengths of a large, statistical number of surface atoms. A statistical thermodynamic treatment in terms of macroscopic parameters leads to the... 

It has often been observed that catalytic samples having the same qualitative composition (for instance, platinum on alumina) but differing in preparation mode, show pronounced dissimilarities in catalytic behavior. Different methods of preparation will yield catalysts differing in crystallite size, crystallite size distribution and shape of the metal crystals. Since 1969, an important work on the statistics of surface atoms and surface sites on metal crystals was undertaken by van Hardeveld and Hartog [14]. In the mean time, a considerable amount of work started to determine the specific rates, rates per unit surface area of metal, for various reactions and in parallel, improvement of chemisorption techniques to determine the surface area of a metal component was achieved. 

Catalytic Alkane Reforming on Nanometallic Particles j 537 Table 17.3 Statistics of surface atoms and surface sites in an incomplete fee cubooctahedron. 

E is the adsorption energy, y is a constant Qg and Qp are the adsorption heat on flat surface and in pores, respectively z = p/po i n is the number (noninteger) of statistical mono-layers of adsorbate molecules and its maximal value for a given is equal to (Rp - (Tgl2)ltn, is the colhsion diameter equal to the average size of nitrogen and surface atoms and Rg is the gas constant. Desorption data were utilized to compute flie / pj distributions with Equations (38.2)-(38.4) and the modified regularization procedure [27]... 

Similar arguments may be used to show that pairs of Co atoms (with their axis in the film plane) in excess of a random distribution can be formed during bias sputtering. With this technique an adatom bonded to a similar surface atom is expected to have a higher resputtering probability than an adatom bonded to a dissimilar atom, since Co-Co and Gd-Gd bonds are energetically less favoured than Gd-Co bonds. Consequently the statistical probability of Gd-Co pairs oriented perpendicular to the film plane will be higher than that of Gd-Co pairs oriented parallel to the film plane, while the opposite holds for Co-Co pairs. 

Figure 6-1. Surface statistical consequences of the subdividing of a cube with 16 atoms on a side. N = total atoms n = surface atoms. The structural, electronic, and chemical consequences of the presence of a high proportion of surface and near-surface atoms predominate in the more highly dispersed material.

The difference in surface energies of the metal components that form the alloy help to determine the level of enrichment with the low surface energy component at coordinatively unsaturated surface atom sites. The enrichment depends exponentially on the degree of coordinative unsaturation of the surface atoms. For the statistical mechanical ideal solution model of an alloy, one can derive the surface composition, Xg, to be... 

The statistics of surface atoms and surface sites on metal crystals. 

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