Long-range critical fluctuations

As a result of the long-range critical fluctuations, the local density will actually be a function of the position r. In the classical theory of fluctuations such a spatial dependence is accounted for by the presence of a gradient term (Vp) in the local Helmholtz-energy density which has an expansion of the form [11]... 

Aqueous solutions of many nonionic amphiphiles at low concentration become cloudy (phase separation) upon heating at a well-defined temperature that depends on the surfactant concentration. In the temperature-concentration plane, the cloud point curve is a lower consolution curve above which the solution separates into two isotropic micellar solutions of different concentrations. The coexistence curve exhibits a minimum at a critical temperature T and a critical concentration C,. The value of Tc depends on the hydrophilic-lypophilic balance of the surfactant. A crucial point, however, is that near a cloud point transition, the properties of micellar solutions are similar to those of binary liquid mixtures in the vicinity of a critical consolution point, which are mainly governed by long-range concentration fluctuations [61]. 

The most provocative prediction of the PRISM-MSA theory was a non-classical relation between the critical temperature for pha% separation, and degree of polymerization, N, of the form T,. oc This unexpected result corresponds to a massive stabilization of the mixed phase (via a long range concentration fluctuation process) relative to Flory-Huggins mean field theory which predicts T oc N. Moreover, in the high temperature limit, XiNc/Xo oc thereby implying that the Flory-Huggins form is not re-... 

At the time of writing, the only evidence for critical fluctuations near the consolute point known to us comes from the work of Damay (1973). The thermopower of Na-NH3 plotted against T at the critical concentration is shown in Fig. 10.21. We conjecture that this behaviour is due to long-range fluctuations between two metallic concentrations, and that near the critical point, where the fluctuations are wide enough to allow the use of classical percolation theory, the... 

It should be emphasized that the comparatively large change obtained in more recent work is mainly caused by the application of finite-size scaling. Under these circumstances, one certainly needs to reconsider how far the results of analytical theories, which are basically mean-field theories, should be compared with data that encompass long-range fluctuations. For the van der Waals fluid the mean-field and Ising critical temperatures differ markedly [249]. In fact, an overestimate of Tc is expected for theories that neglect nonclassical critical fluctuations. Because of the asymmetry of the coexistence curve this overestimate may be correlated with a substantial underestimate of the critical density. 

Second, predictions of p are substantially improved when account is made for ion pairs. The increase of the critical density is easily understood A certain free-ion density is needed for driving criticality. If pairs are formed, this free-ion density can only be achieved at a higher overall ion density. Nevertheless, all theories yield too low values if assessed by the more recent MC data. As mentioned, one reason for low critical densities may result from comparison with MC data that encompass long-range fluctuations. It will, however, be shown in the subsequent section that all available analytical theories seem to overestimate the degree of dissociation. Such an overestimate almost invariably leads to an underestimate of the critical density. 

At first, one would tend to reconsider conventional crossover due to mean-field criticality associated with long-range interactions in terms of the refined theories. Conventional crossover conforms to the first case mentioned—that is, small u with the correlation length of the critical fluctuations to be larger than 0. However, in the latter case one expects smooth crossover with slowly and monotonously varying critical exponents, as observed in nonionic fluids. Thus, the sharp and nonmonotonous behavior cannot be reconciled with one length scale only. 

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