Mean field behavior

A more rigorous approach consists of considering that electron hopping between fixed redox sites is fundamentally a percolation problem, each redox center being able to undergo a bounded diffusion motion.16 If these are fast enough, a mean-field behavior is reached in which (4.24) applies replacing d2 by d2 + 3 Ad2, where Adr is the mean displacement of a redox molecule out of its equilibrium position. 

There are other scenarios for an apparent mean-field criticality [15, 17]. The most likely one is crossover from asymptotic Ising behavior to mean-field behavior far from the critical point, where the critical fluctuations must vanish. For the vicinity of the critical point, Wegner [43] worked out an expansion for nonasymptotic corrections to scaling of the general form... 

Figure 3. Effective exponent / eff of Pitzer s system n-hexyl-triethylammonium n-hexyl-triethylborate + diphenylether. Curves a and b are derived from Singh and Pitzer s data presuming asymptotic mean-field behavior and asymptotic Ising behavior [35], respectively. Curve c is derived from the data of Wiegand et al. [96].

Perhaps a more decisive discrimination between Ising and mean-field behavior could be provided by the investigation of weak anomalies [6] as predicted for the specific heat. Such weak anomalies are absent in the mean-field case (cf. Table I). Except for the diameter anomalies already mentioned, no thermodynamic investigations of weak anomalies were reported so far. However, dynamical properties such as the shear viscosity and electrical conductance may show weak anomalies as well. 

Another claim for an apparent mean-field behavior of ionic fluids came from measurements of heat capacities. The weak Ising-like divergences of the heat capacities Cv of the pure solvent and CPtx of mixtures should vanish in the mean-field case (cf. Table I). The divergence of Cv is firmly established for pure water. Accurate experiments for aqueous solutions of NaCl... 

The observation of crossover has later been substantiated by several other studies. In particular, Jacob et al. [165] performed light scattering measurements on the system 3-MP + water + NaBr. The data indicate comparatively sharp crossover in the range 10-4 salt concentration. It is intriguing to characterize this crossover by a suitably defined crossover temperature Tx, defined here by the point of inflection in the T-dependence of the effective exponent yeff. Figure 8 shows fx as a function of the amount of added NaBr. Eventually, plain mean-field behavior is obtained in a solution containing about 16.8 mass% NaBr. 

While the early work on molten NH4CI gave only some qualitative hints that the effective critical behavior of ionic fluids may be different from that of nonionic fluids, the possibility of apparent mean-field behavior has been substantiated in precise studies of two- and multicomponent ionic fluids. Crossover to mean-field criticality far away from Tc seems now well-established for several systems. Examples are liquid-liquid demixings in binary systems such as Bu4NPic + alcohols and Na + NH3, liquid-liquid demixings in ternary systems of the type salt + water + organic solvent, and liquid-vapor transitions in aqueous solutions of NaCl. On the other hand, Pitzer s conjecture that the asymptotic behavior itself might be mean-field-like has not been confirmed. 

We recall that comparatively sharp and even nonmonotonous crossover from Ising to mean-field behavior has been deduced from experiments for a diversity of ionic systems. We note that this unusually sharp crossover is a striking feature of some other complex systems as well we quote, for example, solutions of polymers in low-molecular-weight solvents [307], polymer blends [308-311], and microemulsion systems [312], Apart from the fact that application of the Ginzburg criterion to ionic fluids yields no particularly... 

Extension of the classical Landau-Ginzburg expansion to incorporate nonclassical critical fluctuations and to yield detailed crossover functions were first presented by Nicoll and coworkers [313, 314] and later extended by Chen et al. [315, 316]. These extensions match Ginzburg theory to RG theory, and thus interpolate between the lower-order terms of the Wegner expansion at T -C Afa and mean-field behavior at f Nci-... 

For more complex fluids, one expects < 0. Then, mean-field behavior can result from two different processes. First, the long-range nature of the intermolecular forces may cause u to be small, while A is not small. Second, may be large or even diverging. Then, A and Nqi will be small, while u is not necessarily small. This case is expected to give a sharp or even nonmonotonous crossover, because a second length scale is present. 

From a global assessment of these results, it seems inescapable to conclude that mean-field behavior does not remain valid asymptotically close to the critical point. Rather, ionic systems seem to show Ising-to-mean-field crossover. Such a crossover has been a recurring result observed near liquid-liquid consolute points in Coulombic electrolyte solutions, in ternary aqueous electrolyte solutions containing an organic cosolvent, and in binary aqueous solutions of NaCl near the liquid-vapor critical line. 

The diffusion coefficient D is plotted in Fig. 3 as a function of the reduced temperature e. The upper x-axis shows the correlation length = o -0 63 with = 1.5nm. The short downward arrow marks the approximate locus of the transition from the asymptotic critical to mean field behavior at Nl/ 2RS [4], Below this value, at smaller values of e and larger correlation lengths, the data are compatible with the asymptotic scaling law of (9). For large values of the slope continuously increases due to the transition to the mean field exponent and the growing influence of thermal activation [81]. 

Figure 35 shows the transition temperature T and the domain size r (= 2n/q ) as a function of Ts for different N. Note that both T and r are averaged over the ensemble of generated sequences ( 106). If Ts is too high, the sequences tend to become random. In contrast, when Ts is too low, the evolutionary algorithm leads to the trivial diblock sequence. In this case, one observes a typical mean-field behavior T ex N and r oc N1/2 (for long... 

One may however look at the data in Table 1 from a different point of view (see also ref 22). It is seen that a transition from Ising to mean-field behavior is observed, if the dielectric constant decreases, the transition regime occuring somewhere near D = 4.5. Could the dielectric constant be the major parameter controlling criticality In the latter case specific interactions play an indirect role They shift and thus enabling phase separations to occur... 

Crossover. Generally, crossover from an Ising-like asymptotic behavior to mean-field behavior further away from the critical point [86, 87] may be expected. Such a behavior is also expected for nonionic fluids, but occurs so far away, that conditions close to mean-field behavior are never reached. Reports about crossover [88] and the finding of mean field criticality [14—16] suggest that in ionic systems the temperature distance of the crossover regime from the... 

Thus Ginzburg analysis just described does not provide an explanation for the observed crossover and apparent mean field behavior in ionic systems. At present, the search for a solution to this problem is focused on the possibility of a tricritical point [96]. Crossover may be controlled by the approach toward a real or virtual tricritical point which in d = 3 is mean-field-like [4, 5]. 

The picture presented above is not complete as it neglects non-mean field behavior of polymer blends in the temperature range close to Tc [149]. The Ising model predicts phase diagrams of thin films, which are more depressed and more flattened than those yielded by mean field approach (as marked in Fig. 31d). Both effects were shown by Monte Carlo simulations performed by Rouault et al. [150]. In principle, critical regions of phase diagrams cannot be described merely by a cross-over from a three- to two-dimensional (for very thin films) situation. In addition, a cross-over from mean field to Ising behavior should also be considered [6,150]. 

The interesting aspects of the physical properties of systems with valence instabilities from the point of view of critical phenomena have also been discussed (44). A mean-field behavior has been found to hold for the systems CeTh and Snij Gd S. Further investigations along these lines will improve our understanding of the role of electronic and elastic effects in alloys (43). 

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