Critical demixing

If we look at a binary fluid mixture instead of a liquid-vapor equilibrium, the variable p is the concentration or mixing ratio instead of the density. In both cases, for temperatures T below the critical or demixing critical temperature T, two bulk phases with p = Pi and p = P2 >pi can coexist with each other, and this is the situation we assume now. 

We consider now a polymer dissolved in a poor solvent, for which demixing may occur. The general features of the demixing critical point were discussed in Chapter IV. Here our aim is to analyze die cooperative diffusion Dcoop- We follow the classical description of Kawasaki and Ferrell for simple binary mixtures, incorporating the (few) special features required for polymer systems. 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

Groves, J. T., Boxer, S. G. and McConnell, H. M. (1998) Electric field-induced critical demixing in lipid bilayer membranes. Proc. Natl. Acad. Sci. USA, 95, 935-938. 

As can be seen from Fig. 6, liquid-liquid demixing clearly precedes crystallization in case Cl. Moreover, crystallization in this case occurs at a higher temperature than in cases C2 and C3. Apparently, the crystallization takes place in the dense disordered phase (which has a higher melting temperature than the more dilute solution Fig. 5). In case C2, the crystallization temperature is close to the expected critical point of liquid-liquid demixing, but higher than in case C3. This suggests that even pre-critical density fluctuations enhance the rate of crystal nucleation. 

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)...

Using the standard procedures to calculate the critical point for liquid-liquid demixing, Eq. (1.114) yields the critical point to be... 

It seems to be a kind of transition somewhat akin to such phenomena as phase inversion observed in oil/water/ surfactant systems(24), and critical demixing (i5j. 26). 

The phase-behaviour of such a system thus may change completely with variations in pressure, temperature, molar mass, and chemical composition of the polymer(s). Thus, polymer systems may show upper critical, lower critical, or hourglass-type demixing behaviour. 

Taking into account changes in concentration of four layers, Williams and Mason (50) have shown that if a > 0, enrichment in the component with the lower heat of sublimation is enhanced compared to that found for ideal solutions. Near the critical temperature of demixing, the dependence of surface concentration on bulk would be highly reminiscent of this dependence for temperatures lower than the critical temperature. 

Macrophase separation after microphase separation has been observed in an AB block copolymer/homopolymer C blend (Hashimoto et al 1995). Blends of a PS-PB starblock copolymer (75wt% PS) and PVME homopolymer were prepared by solvent casting. Binary blends of PS and PVME exhibit a lower critical solution temperature (LCST), i.e. they demix at high temperatures. The initial structure of a 50% mixture of a PS-PB diblock and PVME shown in Fig. 6.20(a) consists of worm-like micelles. Heating led to macrophase separation as evident... 

Apart from liquid-liquid transitions, liquid-vapor transitions in aqueous electrolyte solutions have played a crucial role in debates on ionic criticality [142-144], The liquid-vapor transition is usually associated with a mechanical instability with diverging density fluctuations, while liquid-liquid transitions are associated with a material instability with diverging concentration fluctuations. This requires, however, that both regimes are well-separated. Their interference can lead to complex phase behavior with continuous transitions from liquid-liquid demixing to liquid-gas condensation [9, 145, 146]. It is then not trivial to define the order parameter [147-149]. 

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. 

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