Mean field approximation expressions

Using Eqs. (75) [or Eq. (76)] and (74), we can easily obtain the adsorption isotherm assumed for the adsorption energy distribution. In the framework of the mean field approximation expressions for any thermodynamic quantity (e.g. internal energy, heat capacity) can be readily derived [234]. Adsorption on randomly heterogeneous surfaces has been studied in terms of the above-described approach. It has been demonstrated that this mean-field-type theory was valid only at very high temperatures. Below the critical two-dimensional temperature, the predictions of theory seriously underestimate the heterogeneity effects on phase transitions in adsorbed monolayers [12,234],... 

CO Stripping Chronoamperometiy Before discussing experimental results, let us examine what the LH mechanism predicts for the chronoamperometric response of an experiment where we start at a potential at which the CO adlayer is stable and we step to a final potential E where the CO adlayer will be oxidized. We will also assume that the so-called mean field approximation applies, i.e., CO and OH are well mixed on the surface and the reaction rate can be expressed in terms of their average coverages dco and qh- The differential equation for the rate of change of dco with time is... 

The average probability, /, that a lattice site is occupied by a segment of one of the (i -1) preceding molecules at random is given by the number of vacant sites 1-fi =(N -ri)/N. Using this mean field approximation is replaced by /, , even though the former is somewhat smaller than the latter. The expression for v +1 then becomes... 

The general expression for the BWG (mean field) approximation then gives... 

On another hand, differentiation of equality (15) associated with expression (16) and the mean field approximation (ni nq) enables to... 

The Mean-Field Approximation. The rate of a reaction when there are lateral interactions does not only depend on the reactants and temperature, but also on the occupation of the sites surrounding the sites where the reactants are found. As a consequence exact reactions rate equations contain probabilities of the occupation of clusters with many sites. We have already seen this for CO desorption in eqn. (6). To use this equation we have to express the 5-site probability on the right-hand-side in terms of 1-site probabilities i.e., the coverages). The simplest way to do this is to approximate a multisite probability as a product of 1-site probabilities. This is called a mean-field approximation. For the 5-site probability in eqn. (6) this would mean... 

Analytical expressions for the probabilities may be obtained from the maximum-entropy principle, but it may be necessary to make additional assumptions. For example, eqn. (25) is not sufficient to get the mean-field approximation. This can be seen as follows. Instead of eqn. (26) we get... 

This perturbative expression for the attractive force shift is derived from a van der Waals mean field approximation (23). Although the predictions of this model have been found to agree with numerous high pressure vibrational frequency shift measurements (23,25,28), a non-linear attractive force model has recently been suggested to be appropriate for some systems (26,27). 

A four-state Ising-Potts model is applied to the [Mn(taa)] system to elucidate thermodynamic relations [21]. An [Mn(taa)] molecule is assumed to take four different microscopic states state 0 is the LS state and states 1-3 are the HS states with the elongation axis parallel to x, y, and z, respectively. Interactions are assumed only between nearest neighbor molecules. Under a mean-field approximation [22], the internal energy of the [Mn(taa)] system is expressed in terms of populations Pi (i = 0,1,2,3) of four microscopic states,... 

We use now the results of the foregoing section to discuss the electronic transport properties of our model in some limiting cases for which analytic expressions can be derived. We will discuss the mean-field approximation and the weak-coupling regime in the electron-bath interaction as well as to elaborate on the strong-coupling limit. Furthermore, the cases of ohmic (s = 1) and superohmic (s = 3) spectral densities are treated. 

Finally we can write the expression of the interaction flux using a mean-field approximation for the conductance and using the value of the average individual spike reception rate o(t ) previously given, the complete expression of the imposed flux is ... 

Two infinite-size plates are immersed in a semidilute solution of polyelectrolyte in a good solvent which also contains the small ions of a salt. One of the plates is located at x=0 and the other one at x=D. The system is considered to be in contact with a large reservoir, which contains a polyelectrolyte/salt solution. In addition to the electrostatic interactions, the segments of the polymer have a van der Waals interaction —UkT with the plates. In the mean field approach, the intra- and interchain interactions together with the electric field induced by the surface charges of the plates and polyelectrolyte molecules are expressed as an external potential. Within the mean-field approximation, the free energy of the system with respect to that in the reservoir can be expressed as the sum of three contributions, the polymer contribution Fpol, the salt ions contribution Fion and the electrostatic field contribution Fels,... 

In the next section we first introduce the main ingredients of a FT by mimicking as far as possible what is done in QFT. To be illustrative, in Sec. 3 we show how it is possible to derive the virial expression for the pressure in a FT. In Sec. 4 we show that we have at our disposal some relations between field-field correlation functions that are obtained from symmetry arguments or from the fact that fields are dummy variables. In Sec. 5 on two examples, we derive some results starting from Dyson-like equations. Finally, in Sec. 6, leaving the purely microscopic level we introduce a model showing the existence of a demixion transition at a metal-solution interface using a mean field approximation. Brief conclusions are presented in Sec. 7. 

The form of the operators of the pseudospin is given in expressions (66)— (68). The pseudospin operators obey rules (69), which represent the switching relations in this situation studied. The total Hamiltonian for the dipoles in the domain in the mean field approximation is retained in the form... 

The lateral interactions between chemisorbed particles or between these particles and activated complexes should be taken into consideration describing the chemisorption kinetics on the solid surface. This interaction is determined by a distance between neighbouring active sites (r). The account for the lateral interaction is possible on a single crystal surface in quasi- chemical or mean-field approximations [30,32]. Let us write the common expression for chemisorption kinetics on the homogeneneous surface with the account for lateral interactions between chemisorbed particles ... 

The activity coefficients are affected from differences in size, the short-range lateral interactions among the adsorbed particles and in the case of adsorbed ions from the repulsive Coulombic interactions among these ions. Analytical expressions for the activity coefficients may be obtained either from statistical mechanical models or experiment. Thus, an approximate expression, arising from monolayer models under mean field approximation, is the following [12,13,21] ... 

It is found that the scaling exponent a presents universality properties, in the sense that its behavior is identical for any value of A , for the different topographies considered, for different thermodynamical quantities (i.e., adsorption isotherm and differential heat of adsorption) and for different reference curves, even a theoretical one expressed, for example, through a mean field approximation for the bp topography like ... 

As we are again interested in determining the phase behavior of the binary mixture in confinement and near solid interfaces, we are essentially confronted with the same problem already discussed in Section 4.5, namely finding minima of the grand potential for a given set of thermodynamic (T, /x) and model parameters [see Eqs. (4.125)]. To obtain expressions for u> that are tractable, at least numerically, we resort again to a mean-field approximation. That... 

Obviously the enthalpy expression x a b in Eq. (3) neglects any correlation effects in the occupancy of lattice sites the probability that on neighboring lattice sites A-B-pairs occur is simply taken as the product < >A< >B of the respective volume fractions. This is a special case of a mean-field approximation (MFA), which is known to yield a critical behavior described by the Landau theory of phase transitions [100], which differs from the correct critical behavior expected [73,74] in the universal regime close to the critical point Xcrm 4>cri in Fig. 2. We shall discuss these various types of critical behavior in Sect. 2.2. 

By virtue of their simple structure, some properties of continuum models can be solved analytically in a mean field approximation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is investigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-paiameter model are described in the review by Gompper and Schick [76]. A very interesting feature of these models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a functional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

An equation for a is obtained by eliminating Ap between equations 1 and 2b. In mean-field approximation the following expression for a cluster that consists of a single, isolated, long-chain macromolecule in swelling equilibrium was derived by Flory (4) ... 

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