Surface diffusion statistics

RuO2(110) exemplifies Langmuirian behaviour where the catalyst surface consists of equivalent sites statistically occupied by the reactants. This contrasts markedly with catalytic oxidation at metal surfaces, where oxygen transients, high surface mobility and island structures are dominant. The difference is in the main attributed to differences in surface diffusion barriers at metal and oxide surfaces. 

Kimmich and coworkers have studied the magnetic relaxation dispersion of liquids adsorbed on or contained in microporous inorganic materials such as glasses and packed silica (34-43) and analyze the relaxation dispersion data using Levy walk statistics for surface diffusion to build... 

By using statistical arguments and assuming contributions due to surface diffusion and momentum transfer to be additive to that of the gas phase diffusion. Fain and Roettger [1993] have derived the following expression for the flow rate of each gas, Fi ... 

The next section gives a brief overview of the main computational techniques currently applied to catalytic problems. These techniques include ab initio electronic structure calculations, (ab initio) molecular dynamics, and Monte Carlo methods. The next three sections are devoted to particular applications of these techniques to catalytic and electrocatalytic issues. We focus on the interaction of CO and hydrogen with metal and alloy surfaces, both from quantum-chemical and statistical-mechanical points of view, as these processes play an important role in fuel-cell catalysis. We also demonstrate the role of the solvent in electrocatalytic bondbreaking reactions, using molecular dynamics simulations as well as extensive electronic structure and ab initio molecular dynamics calculations. Monte Carlo simulations illustrate the importance of lateral interactions, mixing, and surface diffusion in obtaining a correct kinetic description of catalytic processes. Finally, we summarize the main conclusions and give an outlook of the role of computational chemistry in catalysis and electrocatalysis. 

Theoretical studies of the properties of the individual components of nanocat-alytic systems (including metal nanoclusters, finite or extended supporting substrates, and molecular reactants and products), and of their assemblies (that is, a metal cluster anchored to the surface of a solid support material with molecular reactants adsorbed on either the cluster, the support surface, or both), employ an arsenal of diverse theoretical methodologies and techniques for a recent perspective article about computations in materials science and condensed matter studies [254], These theoretical tools include quantum mechanical electronic structure calculations coupled with structural optimizations (that is, determination of equilibrium, ground state nuclear configurations), searches for reaction pathways and microscopic reaction mechanisms, ab initio investigations of the dynamics of adsorption and reactive processes, statistical mechanical techniques (quantum, semiclassical, and classical) for determination of reaction rates, and evaluation of probabilities for reactive encounters between adsorbed reactants using kinetic equation for multiparticle adsorption, surface diffusion, and collisions between mobile adsorbed species, as well as explorations of spatiotemporal distributions of reactants and products. 

For a brief introduction, we refer again to Fig. 25.5(a), which immediately reveals that it requires an activation energy of for an atom (or a molecule) to be transferred from a site A to another, geometrically identical site B on the same periodic surface. In the classical view, this two-dimensional diffusion process can be thought of as a sequence of individual and statistical hopping events of frequency v, each activated with an energy as pointed out, for example, by Roberts and McKee [35]. The inverse of this frequency then yields the residence time, t, of the particle in the respective site. For thermally equilibrated particles, the temperature dependence of the classical surface diffusion is described by the well-known Arrhenius relation... 

It has been shown that PEEM is a versatile instrument that can be used to determine the thermodynamic and kinetic properties of molecular systems. The preliminary results presented here are presently being analysed to yield statistically significant values for the molecular binding energy as well as for the molecular surface diffusion constant and other kinetic parameters of interest. 

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... 

To compute the entropy and heat capacity from statistical thermodynamics, one has to consider the type of adsorption, depending on how strongly the adsorbate binds to the surface. In general, adsorbates that bind weakly to surfaces, for example, closed-shell adsorbates such as H O and CO, have a low barrier for surface diffusion, which makes them highly mobile on the surface. On the other hand, strongly bound adsorbates have a high barrier for surface diffusion and are assumed to be immobile on the surface. As a result of differences in adsorption, the statistical... 

Statistical mechanical Monte Carlo as well as classical molecular dynamic methods can be used to simulate structure, sorption, and, in some cases, even diffusion in heterogeneous systems. Kinetic Monte Carlo simulation is characteristically different in that the simulations follow elementary kinetic surface processes which include adsorption, desorption, surface diffusion, and reactivity . The elementary rate constants for each of the elementary steps can be calculated from ab initio methods. Simulations then proceed event by event. The surface structure as well as the time are updated after each event. As such, the simulations map out the temporal changes in the atomic structure that occur over time or with respect to processing conditions. 

Diffusion of Carbon. When carbon atoms are deposited on the surface of the austenite, these atoms locate in the interstices between the iron atoms. As a result of natural vibrations the carbon atoms rapidly move from one site to another, statistically moving away from the surface. Carbon atoms continue to be deposited on the surface, so that a carbon gradient builds up, as shown schematically in Figure 5. When the carbon content of the surface attains the equihbrium value, this value is maintained at the surface if the kinetics of the gas reactions are sufficient to produce carbon atoms at least as fast as the atoms diffuse away from the surface into the interior of the sample. 

LEED patterns at 0 = 1 /4, but was identified at lower coverages in islands surrounded by mobile sulfur atoms at platinum, rhodium and rhenium surfaces. Sautet and co-workers42 have analysed the statistical correlations between the intensities of sulfur features in p(2 x 2) islands on rhenium surfaces and also of streaks in areas between islands, which they attribute to sulfur atoms diffusing under the tip (Figure 10.12). 

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