Mean-field description

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

Let us begin with the one-mode electron-transfer system. Model IVa, which still exhibits relatively simple oscillatory population dynamics [205]. SimUar to what is found in Fig. 5 for the mean-field description, the SH results shown in Fig. 13 are seen to qualitatively reproduce both diabatic and adiabatic populations, at least for short times. A closer inspection shows that the SH results underestimate the back transfer of the adiabatic population at t 50 and 80 fs. This is because the back reaction would require energetically forbidden electronic transitions which are not possible in the SH algorithm. Figure 13 also shows the SH results for the electronic coherence which are found to... 

The evolution of emulsions through coalescence can be characterized by a kinetic parameter, >, describing the number of coalescence events per unit time and per unit surface area of the drops. Following the mean field description of Arrhenius,... 

The regime governed by coalescence was examined in more detail. The process of film rupture is initiated by the spontaneous formation of a small hole. The nucleation frequency. A, of a hole that reaches a critical size, above which it becomes unstable and grows, determines the lifetime of the films with respect to coalescence. A mean field description [19] predicts that A varies with temperature T according to an Arrhenius law ... 

A model for the changes in reactivity for a reaction on a catalytic surface in the presence of adsorbed inactive atoms. The model is based on a mean field description of the formation of partly disordered structures for the adsorbed atoms. 

In spite of simple theoretical formalism (for example, mean-field descriptions of certain aspects) structural aspects of the systems are still explicitly taken into account. This leads to the results which are in a good agreement with computer simulations. But the stochastic model avoids the main difficulty of computer simulations the tremendous amount of computer time which is needed for obtaining good statistics for the reliable results. Therefore more complex systems can be studied in detail which may eventually lead to better understanding of real systems. In the theory discussed below we deal with a disordered surface. This additional complication will be handled in terms of the stochastic approach. This is also a very important case in catalytic reactions. 

In our discussion of the transition-state theory with static solvent effects, it was noticed that it is a mean field description where the effects of dynamical fluctuations in the solvent molecule positions and velocities were excluded. 

Also the second class of methods include very different approaches however, in all of them we can individuate a common aspect, namely the use of a mean-field description for the part of the system encircling the subsystem of real interest. In the application of this class of methods to the study of liquid solutions, the most important mean-field approach is represented by continuum models. In such models, the solute is assumed to be inside a cavity of proper shape and dimension within an infinite continuum dielectric mimicking the solvent. 

In the mean field description, the potential 0(x) is assumed to obey the PB equation... 

Equation 6.36 for the adiabatic potential is exact within the framework of the mean field description. However, the structure of the electric part P1 is too complex to disclose its analytic properties. Here we examine the adiabatic potential numerically following the Carlson theory of elliptic integrals [15-21], To proceed with numerical computation, it is necessary to enter a set of parameters designed to describe an experimental situation. It will not surprise the reader who has made it this far that we use values of the chemically fixed parameters specified by the n-butylammonium vermiculite gels [22], namely m+ = 74 mp and m = 36 mp. The average density n0 of the small ions is given by... 

When the solvency is better the quadratic term in [5.2.15] may be dominant. The transition occurs when the solvent quality is marginally good this regime may therefore be denoted as the marginal (m) regime (see fig. 5.3). In very good solvents (i.e., better than marginal) the mean-field description breaks down, as... 

In the concentrated and marginal regimes of fig 5,3 a mean-field description, which neglects any spatial fluctuations, is appropriate. In these regimes, the solution is homogeneous and there is no chain-length dependence. Neither does the persistence p of the chains play a role since the Floiy-Huggins expressions do not contain the chain flexibility. This is so because the flexibility is assumed to be the same in the solution and in the reference state, so that p cancels in the entropy difference between the two states. 

Dickman [60], who was the first to study the mean-field description of the ZGB model, used a different approach in order to circumvent the problem of AB pairs. He split up the adsorption reactions in the ZGB model, thereby differentiating between adsorption adjacent to different surface species. For example, when an A molecule adsorbs next to a Bads> immediate reaction will occur. Thus, the reaction becomes A(g) -I- - - Bads —> AB(g) - - 2. This leads in the MF approximation to differential equations with fourth-order terms for the surface coverages of A and B. The infinite rate constant kr is absent from these equations. 

We have also cast the DMC model in a set of ordinary differential equations, thus translating it to a mean-field approach with the site-approximation. Only the kinetic oscillations can be modeled in this way. To model the spatio-temporal pattern formations, diffusion terms would have to be added to the mean-field description, in order to account for the spatial dependence of the reactant concentrations. 

For situations of overlapping chains, where lateral fluctuations in the segment concentration become rather small, mean-field descriptions become appropriate. The most successful of this type of theoiy is the lattice model of Scheutjens and Fleer (SF-theoiy). In chapter II.5 some aspects of this model were discussed. This theory predicts how the adsorbed amount and the concentration profile 0(z) depend on the interaction parameters and x and on the chain length N. From the statistical-thermodynamic treatment the Helmholtz energy and, hence, the surface pressure ti can also be obtcdned. When n is expressed as a function of the profile 0(z), the result may be written as ... 

As a particular case we can consider a mean field description, where the diffusion and convection terms disappear. That is equivalent to assuming that changes in the size of the particles are only caused hy coalescence and aggregation processes that also implies that ds/dt=0. The J term is obtained by summing the interactions between particles over all possible sizes. The resulting equation is the generalized Smoluchowski discrete equation [12], which describes the time evolution of the number of clusters of size s, N at time t... 

This energy equation is a mean-field description of all binary regular mixtures regular solutions, polymer solutions, and polymer blends. 

The PT model is frequently used as a minimalistic approximation for more complex models. For instance, it is the mean-field version of the Frenkel Kontorova (FK) model as stressed by D. S. Fisher [29,83] in the context of the motion of charge-density waves. The (mean-field) description of driven, coupled Josephson junctions is also mathematically equivalent to the PT model. This equivalence has been exploited by Baumberger and Carol for a model that, however, was termed the lumped junction model [84] and that attempts to... 

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