Mean-field-like criticality

In the literature there have been repeated reports on an apparent mean-field-like critical behavior of such ternary systems. To our knowledge, this has first been noted by Bulavin and Oleinikova in work performed in the former Soviet Union [162], which only more recently became accessible to a greater community [163], The authors measured and analyzed refractive index data along a near-critical isotherm of the system 3-methylpyridine (3-MP) + water -I- NaCl. The shape of the refractive index isotherm is determined by the exponent <5. Bulavin and Oleinikova found the mean-field value <5 = 3 (cf. Table I). Viscosity data for the same system indicate an Ising-like exponent, but a shrinking of the asymptotic range by added NaCl [164],... 

We turn now to theories of ionic criticality that encompass nonclassical phenomena. Mean-field-like criticality of ionic fluids was debated in 1972 [30] and according to a remark by Friedman in this discussion [69], this subject seems to have attracted attention in 1963. Arguments in favor of a mean-field criticality of ionic systems, at least in part, seem to go back to the work of Kac et al. [288], who showed in 1962 that in D = 1 classical van der Waals behavior is obtained for a potential of the form ionic fluids with attractive and repulsive Coulombic interactions have little in common with the simple Kac fluid. 

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. 

One option to explain classical critical behavior and unusual crossover, is the existence of a tricritical point, which in d 3 is mean-field like [4, 5], In ternary systems, tricritical behavior is generally obtained, if three phases have a common critical point. 

Note that all these formulas also contain the result for the limiting case of short chains dynamics described by the Rouse model [139,140] if we formally put Ne N in these equations. As will be discussed later (Sect. 2.5), there occurs a crossover in the static critical behavior from mean-field-like behavior where ocR e-1/2 with e= 1 — x/X rit> Scon(0)ccN e to the nonclassical critical behavior with Ising model [73, 74] critical exponents cce-v, S, ii(0) oceT, vw0.63, 1.24. This crossover occurs, as predicted by the Ginzburg... 

These data are replotted in a different form in Figure 12, on the assumption that the order parameter (the coexistence density gap) for the LJ system should behave in an Ising-like manner. This is reflected in the nearly straight-line behavior of much of the data very close to the critical points the data deviate from linearity, becoming mean-field-like because of the limitation on fluctuations in a finite system. The precision of the results puts us in position to study this finite-size crossover and also other nonuniversal properties of the critical behavior of fluid phase transitions. 

Here we have very briefly introduced the (lattice) SAW model of linear polymers, their configurational statistics and the (lattice) percolation model of disodered media. Approximate mean field-like and scaling arguments have been forwarded to indicate that the SAW critical behaviour on disordered lattices, percolating lattice in particular, could be significantly different from those of SAWs on pure lattices. More careful analysis, as we will see in the following chapters, show even more subtle effects of disoder on the polymer conformation statistics. Also, as we will see, such effects are not necessarily confined only to the cases of extreme disorder like percolating fractals. 

Experiments and modem physics [3] has shown that the way KT and other thermodynamic properties diverge when approaching the critical point is described in a fundamentally wrong way by all classical, analytical equations of state like the cubic equations of state and is path dependent. The reason for this is that these equations of state are based on mean field theory,... 

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... 

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... 

A quantitative comparison between the mean field prediction and the Monte Carlo results is presented in Fig. 15. The main panel plots the inverse scattering intensity vs. xN. At small incompatibility, the simulation data are compatible with a linear prediction (cf. (48)). From the slope, it is possible to estimate the relation between the Flory-Huggins parameter, x, and the depth of the square well potential, e, in the simulations of the bond fluctuation model. As one approaches the critical point of the mixture, deviations between the predictions of the mean field theory and the simulations become apparent the theory cannot capture the strong universal (3D Ising-like) composition fluctuations at the critical point [64,79,80] and it underestimates the incompatibility necessary to bring about phase separation. If we fitted the behavior of composition fluctuations at criticality to the mean field prediction, we would obtain a quite different estimate for the Flory-Huggins parameter. 

Howrever, the reader should note that the latter feature is not correct with respect to corresponding experimental observations w here the critical density is usually shifted to higher values and the coexistence curve of the confined fluid turns out to be narrower with respect to its bulk counterpart [31]. This reflects the fact that, with regard to mean densities, gas- and liquid-like confined phases aic more alike than in the bulk. The absence of a shift in critical density in the theoretical curves is caused by the fact that within the context, of the current perturbational approach the density dependence of the free energy remains the same in both confined and bulk fluids [see, for example, Eq. (4.26)], which shows that confinement effects are solely restricted to the density-independent van der Waals parameter ap(0- How cvor, on the positive side, we are now- equipped with equations of state for both the confined fluid [sckj Eq. (4.28)] and its bulk counterpart [see Eq. (4.29)]. Together these equations of state enable us to revisit the Thommes Findenegg experiment at mean-field level. 

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