Adsorption model mobile-localized

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

In contrast to localized adsorption, mobile adsorption models assume that molecules can diffuse freely on the surface. One of the most popular equations used to describe mobile adsorption is that proposed by Hill and de Boer [105] as an analogue of the FG isotherm. This equation can be obtained by combining the two-dimensional form of van der Waals equation with the Gibbs adsorption isotherm. Note that the pre-exponential factors for localized and mobile adsorption are different. In the case of localized adsorption, the pre-exponential factor Kq takes into account the vibrations of adsorbing molecules in X, y and z direction, whereas the factor for the mobile adsorption contains only the partition functions for vibration in the z-direction and the transnational partition function describing mobility of adsorbing molecules in the (x,y)-plane. 

A considerable element of the model is the assumption connected with the possibility of the kinetic motion of adsorbed molecules. When the motion of molecules in the z direction is restricted but molecules are able to move freely in the (x,y) plane, the process is classified as mobile adsorption. However, if the lateral translation is also hindered, the process is classified as localized adsorption. The motion of admolecules is controlled by the energetic topography of the surface, molecular interactions, and thermal energies. The adsorbed molecule is considered as localized on a surface when it is held at the bottom of a potential well with a depth that is much greater than its thermal energy. Except for extreme cases, adsorption is neither frilly localized nor frilly mobile and can be termed partially mobile [8]. Because temperature strongly affects the behavior of the system, adsorption may be localized at low temperatures and become mobile at high temperatures. 

Experimental results have elearly shown that during the formation of the monolayer, a change from nonlocalized to locahzed adsorption occurs. Several theoretical studies have been made of the so-called partially mobile or partially localized adsorption models [11,60,161,164,228,230]. These theories must explain the phase transitions in the adsorbed monolayers and may also be useful in describing surface transport phenomena [11]. 

Hill ) idealized lateral transitions by considering the surface as "homogeneous" but with a sinusoidally varying potential energy. At low temperature, the adsorbed molecules can only vibrate in the minima, but with increasing temperature the fraction that can pass the maxima increases. In this model there is a gradual transition between localized and mobile adsorption over the entire (homogeneous) surface. 

Let us. before giving illustrations, discuss some aspects of the statistical foundations. Model assumptions regarding the mode of adsorption (patchwise or random mobile or localized mono- or multilayer with or without lateral interaction ) are reflected in the natures of the local partition functions and in the way they combine to Q[N,N, T). Again, no general solution can be given models of different degrees of sophistication can be developed. 

The matter discussed in sec. 2.3 concerned the phenomenology of adsorption from solution. To make further progress, model assumptions have to be made to arrive at isotherm equations for the individual components. These assumptions are similar to those for gas adsorption secs. 1.4-1.7) and Include issues such as is the adsorption mono- or multlmolecular. localized or mobile is the surface homogeneous or heterogeneous, porous or non-porous is the adsorbate ideal or non-ideal and is the molecular cross-section constant over the entire composition range In addition to all of this the solution can be ideal or nonideal, the molecules may be monomers or oligomers and their interactions simple (as in liquid krypton) or strongly associative (as in water). 

It may be good to note here that various molecular cross-sections have now been considered. In the treatment of adsorption on solid surfaces was introduced. Interpreting this area in terms of lattice models is not a property of the adsorptive molecule but of the adsorbent. It is possible to imagine a situation where greatly exceeds the real molecular cross-section. On the other hand, for mobile monolayers on homogeneous surfaces is the real molecular cross-section or, for that matter, it is the excluded area per molecule. To avoid an undue abundance of symbols we have used the same symbol for both situations, for instance in table 3.3 in sec. 3.4e. It is to be expected that a and a, obtained by compression of monolayers, are more similar to the a s for adsorbed mobile monolayers on homogeneous substrates than to those for localized monolayers. 

The periodic adsorption potential of surfaces and the potential barriers between adjacent sites lower than the desorption energy must result such that the state of adsorbates is intermediate between the ideal mobile and localized models. The lateral migration across the surface must make a positive contribution to the entropy of the adsorbate. 

As it can be seen from Fig. 5.22, when moving from the localized adsorption towards the mobile model, we can expect smooth decrease in the entropy of desorption. The entropy of the adsorbate which experiences lateral diffusion was discussed, in particular, by Patrikiejew, et al. [95]. They approached the problem by assuming that a fraction of the molecules is in completely mobile state, while the others are completely localized. Then they suggested that the canonical partition function (9ml... 

The classical theory of the Gibbs adsorption isotherm is based on the use of an equation of state for the adsorbed phase hence it assumes that this adsorbed phase is a mobile fluid layer covering the adsorbent surface. By contrast, in the statistical thermod)mamic theory of adsorption, developed mainly by Hill [15] and by Fowler and Guggenheim [12], the adsorbed molecules are supposed to be localized and are represented in terms of simplified physical models for which the appropriate partition function may be derived. The classical thermodynamic fimctions are then derived from these partition fimctions, using the usual relationships of statistical thermodynamics. 

The conditions of validity of this isotherm model are the same as those of the competitive Langmuir isotherm, ideal behavior of the mobile phase and the adsorbed layer, localized adsorption, and equal column saturation capacities of both t3q>es of sites for the two components. The excellent results obtained with a simple isotherm model in the case of enantiomers can be explained by the conjunction of several favorable circumstances [26]. The interaction energy between two enantiomeric molecules in solutions is probably very close to the interaction energy between two R or two S molecules and their interactions with achiral solvents are... 

In addition, the combination of these EBL-fabricated Pd model catalysts and UHV molecular beam system has allowed the first experimental confirmation that surface diffusion over the whole particle must be taken into account to describe the global kinetics, and thus support the theoretically predicted communication effects between different facets on a nanoparticle [4, 146, 148, 149]. These experiments also allowed measurement of the surface mobility of oxygen under reaction conditions [83,150]. On a given nanoparticle, regions with locally different adsorption or reaction properties (such as different crystallographic orientations, i.e., facets) can communicate. 

Up to this point the essential equations have been presented. Now, it is possible to analyze the work carried out in connection with the classical thermodynamic approach. The first systematic study of a thermodynamic adsorption quantity was perhaps the work done by de Boer and coworkers [10] on the determination, interpretation and significance of the enthalpy and entropy of adsorption. Their papers analyzed almost all aspects of the experimental determination of the entropy and how to interpret the values obtained in terms of two extreme models, i.e., those of mobile and locahzed adsorption, which today have lost much of their usefulness. To catalog the behavior of the adsorbed film as localized or mobile is a very simplistic solution and it has been demonstrated [9] that in most cases the adsorbed film is neither completely localized nor completely mobile. This approach also is somehow outdated because numerical simulations provide a better microscopic interpretation of the system s behavior. 

Bhatia [39] studied the transport of adsorbates in microporous random networks in the presence of an arbitrary nonlinear local isotherm. The transport model was developed by means of a correlated random walk theory, assuming pore mouth equilibrium at an intersection in the network and a local chemical potential gradient driving force. The author tested this model with experimental data of CO2 adsorption on Carbolac measured by Carman and Raal [40]. He concluded that the experimental data are best predicted when adsorbate mobility, based on the chemical potential gradient, is taken to have an activation energy equal to the isosteric heat of adsorption at low coverage, obtained from the Henry s law region. He also concluded that the choice of the local isotherm... 

The most complete mathematical model of a nonuniform adsorbed layer is the distributed model, which takes into account interactions of adsorbed species, their mobility, and a possibility of phase transitions under the action of adsorbed species. The layer of adsorbed species corresponds to the two-dimensional model of the lattice gas, which is a characteristic model of statistical mechanics. Currently, it is widely used in the modeling of elementary processes on the catalyst surface. The energies of the lateral interaction between species localized in different lattice cells are the main parameters of the model. In the case of the chemisorption of simple species, each species occupies one unit cell. The catalytic process consists of a set of elementary steps of adsorption, desorption, and diffusion and an elementary act of reaction, which occurs on some set of cells (nodes) of the lattice. 

Several other theoretical models [47-49] have attempted to give a more realistic description than the Langmuir and BET models of the gas-surface interactions that lead to physical adsorption. The variable parameters in these models are the interaction potential, the structure of the adsorbed layer (mobile or localized monolayer of multilayer), and the structure of the surface (homogeneous or heterogeneous, number of nearest neighbors). 

The first such solutions were carried out by Ross and Olivier [1, p. 129 6,7]. Using Gaussian distributions of adsorptive potential of varying width, they computed tables of model isotherms using kernel functions based on the Hill-de Boer equation for a mobile, nonideal two-dimensional gas and on the Fowler-Guggenheim equation [Eq. (14)] for localized adsorption with lateral interaction. The fact that these functions are implicit for quantity adsorbed was no longer a problem since they could be solved iteratively in the numerical integration. 

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