Lattice, gases energy

As an example of a multilayer system we reproduce, in Fig. 3, experimental TPD spectra of Cs/Ru(0001) [34,35] and theoretical spectra [36] calculated from Eq. (4) with 6, T) calculated by the transfer matrix method with M = 6 on a hexagonal lattice. In the lattice gas Hamiltonian we have short-ranged repulsions in the first layer to reproduce the (V X a/3) and p 2 x 2) structures in addition to a long-ranged mean field repulsion. Second and third layers have attractive interactions to account for condensation in layer-by-layer growth. The calculations not only successfully account for the gross features of the TPD spectra but also explain a subtle feature of delayed desorption between third and second layers. As well, the lattice gas parameters obtained by this fit reproduce the bulk sublimation energy of cesium in the third layer. 

Note that wq cannot be fixed by detailed balance, the reason being that it contains the information about the energy exchange with the solid which is not contained in the static lattice gas Hamiltonian. However, by comparison with the phenomenological rate equation (1) we can identify it as... 

The fact that detailed balance provides only half the number of constraints to fix the unknown coefficients in the transition probabilities is not really surprising considering that, if it would fix them all, then the static (lattice gas) Hamiltonian would dictate the kind of kinetics possible in the system. Again, this cannot be so because this Hamiltonian does not include the energy exchange dynamics between adsorbate and substrate. As a result, any functional relation between the A and D coefficients in (44) must be postulated ad hoc (or calculated from a microscopic Hamiltonian that accounts for couphng of the adsorbate to the lattice or electronic degrees of freedom of the substrate). Several scenarios have been discussed in the literature [57]. 

The simplest model is the lattice-gas or Ising model. The whole space is divided into a lattice of N sites, and on each site two different states are possible a crystalline state denoted by the variable 5, = 1 and a gaseous state by Sj = -. The variable s denotes the degree of crystalline order. The cohesion of nearest-neighboring solid atoms leads to the following interaction energy... 

Determination of Lattice Breakup Energies from Experimental Data The process of lattice breakup can be split into individual steps for which the energies can be measured. Thus, breaking up the NaCl lattice to form free ions in the gas phase can be described (with a Born-Haber cycle) as... 

From this equation and Eqs. (2) and (4) the energy of the system can be obtained. The entropy is more difficult to derive, and we refer to the literature [4,6]. Generally, the quasichemical gives better results than the mean-field approximation, since it allows for local order. We note that for the three-dimensional lattice gas no exact analytical solution exists. 

As noted above, the capacity of liquid-liquid interfaces depends on the nature of the ions dissolved in the two adjoining phases. In some cases the capacity is related o the free energy of transfer of the ions involved, but in other cases quite strong dependencies are observed which can be explained by a tendency to form ion pairs at the interface. Evidence for this effect, which was first discussed by Hajkova et al. [14], was obtained in a paper by Cheng et al. [15], who observed marked changes in the capacity when they varied the composition of the aqueous phase. Further examples were provided by Pereira et al. [16], who also performed explicit calculations for ion pairing based on the lattice-gas model. 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

What is next Several examples were given of modem experimental electrochemical techniques used to characterize electrode-electrolyte interactions. However, we did not mention theoretical methods used for the same purpose. Computer simulations of the dynamic processes occurring in the double layer are found abundantly in the literature of electrochemistry. Examples of topics explored in this area are investigation of lateral adsorbate-adsorbate interactions by the formulation of lattice-gas models and their solution by analytical and numerical techniques (Monte Carlo simulations) [Fig. 6.107(a)] determination of potential-energy curves for metal-ion and lateral-lateral interaction by quantum-chemical studies [Fig. 6.107(b)] and calculation of the electrostatic field and potential drop across an electric double layer by molecular dynamic simulations [Fig. 6.107(c)]. 

The thermodynamic functions of fc-mers adsorbed in a simple model of quasi-one-dimensional nanotubes s adsorption potential are exactly evaluated. The adsorption sites are assumed to lie in a regular one-dimensional space, and calculations are carried out in the lattice-gas approximation. The coverage and temperature dependance of the free energy, chemical potential and entropy are given. The collective relaxation of density fluctuations is addressed the dependence of chemical diffusion coefficient on coverage and adsorbate size is calculated rigorously and related to features of the configurational entropy. 

For a lattice gas, the 0 and 1 stand for an empty and filled site respectively. Consequently, u, n is the attractive potential energy between two gas molecules when they occupy adjacent sites on the lattice, and WOO = 1013 = o. 

The lattice-gas model allows to use it for studying the effect of the lateral interactions between the adspecies on the surface process rate or, in other words, to consider the non-ideality of the reaction system in the surface process kinetics. In the lattice-gas model the interaction of adspecies / and j in sites / and g at the distance r is set by the energy parameter sjg(r). In the homogeneous lattice systems such distances can be conveniently determined with the use of the numbers of the (c.s.) where site g is located relative to site /. In this case in the parameter y(r) the index r runs a discrete series of values from 1 to R, where R is the interaction radius 1 1) = 0, one deals with the nearest-neighbors... 

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