Mean-field solutions temperatures

Figure A2.5.27. The effective coexistence curve exponent P jj = d In v/d In i for a simple mixture N= 1) as a fimction of the temperature parameter i = t / (1 - t) calculated from crossover theory and compared with the corresponding curve from mean-field theory (i.e. from figure A2.5.15). Reproduced from [30], Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 358, figure 3, by pennission of Elsevier Science.

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

To answer the above question new results have been obtained by the study of very fast protolytic reactions in aqueous solution. These were carried out during the last few years by means of relaxation methods (sound absorption, dispersion of the dissociation field effect, temperature jump method) (for a survey cf. [3]). The neutralization reaction HgO+ -j- OH- - (Ha0)8 is the most characteristic example. It was possible to determine the rate constant of this reaction by measuring the time dependence of the dissociation field effect of very pure water of specific conductivity of 6 7 10-8 (at 25°C). 

The above reasoning can easily be generalized to the case of a phase transition to a spatially modulated ground state, characterized by non-zero magnetization Mq. The corresponding mean-field value of the ordering temperature Tc(q) is given by the solution of the equation (Diet et al. 1999)... 

The appearance of the scattering maximum in SANS was observed by several workers particularly for polyelectrolyte solutions [83]. The mean field theories used to predict such scattering curves were presented by Borne et al. [84] and Joanny et al. [85]. Recently, such a peak was also observed in weakly ionized polymer gels as well by Schosseler et al. [78, 79]. Let us discuss the origin of the peak at high temperatures. 

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. 

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