Critical unmixing

The critical unmixing condition is obtained from equations 3.201 and 3.202 by substituting for the left-side terms the algebraic form of the excess function appropriate to the case under consideration. For a simple mixture, we have, for instance (cf eq. 3.144),... 

From Eqs. (24)-(27) we have seen that the standard mean-field power laws describing the singularities near the critical unmixing point can hold only for f = (jf//crit - 1) 1. Of course, when / -> xait at fixed N, we do expect that mean-field theory breaks down due to the neglect of thermal fluctuations, and in reality a crossover to the critical behavior described by the Ising model universality class [34, 35, 36] sets in. Thus, very close to the critical point, we expect the following critical exponents [35, 36]. 

In the next section, we briefly describe some results obtained [82, 85] for lattice models of critical unmixing of polymers, which have some relevance for the theories summarized in the previous sections. We do not give a more detailed account of the simulation techniques applied in those papers, however, because they are well documented in the literature [70, 71, 72, 73, 77, 78, 81, 82, 85, 91]. The extensions of these techniques needed to cope with the complications due to competing order parameters (as they occur in polymer+solvent systems when both liquid-vapor unmixing and fluid-fluid unmixing is possible) are deferred to Sect. 5. 

The binary polymer blend exliibits a second-order unmixing transition. Close to the critical temperature the... 

It has been proposed to define a reduced temperature Tr for a solution of a single electrolyte as the ratio of kgT to the work required to separate a contact +- ion pair, and the reduced density pr as the fraction of the space occupied by the ions. (M+ ) The principal feature on the Tr,pr corresponding states diagram is a coexistence curve for two phases, with an upper critical point as for the liquid-vapor equilibrium of a simple fluid, but with a markedly lower reduced temperature at the critical point than for a simple fluid (with the corresponding definition of the reduced temperature, i.e. the ratio of kjjT to the work required to separate a van der Waals pair.) In the case of a plasma, an ionic fluid without a solvent, the coexistence curve is for the liquid-vapor equilibrium, while for solutions it corresponds to two solution phases of different concentrations in equilibrium. Some non-aqueous solutions are known which do unmix to form two liquid phases of slightly different concentrations. While no examples in aqueous solution are known, the corresponding... 

Figure 3,10 Solvus and spinodal decomposition fields in regular (B) and subregular (D) mixtures. Gibbs free energy of mixing curves are plotted at various T conditions in upper part of figure (A and C, respectively). The critical temperature of unmixing (or consolute temperature ) is the highest T at which unmixing takes place and, in a regular mixture (B), is reached at the point of symmetry.

One of the Interesting features of these binary solutions, and of many microemulsions, is their tendency to unmix at higher temperature. For example triethylamine-water mixtures unmix into nearly pure triethylamine and nearly pure water at 18.5 C similarly 2-butoxyethanol has a lower critical solution temperature at 49 C. 

On the left-hand side of Fig. 4 we have the normal phase diagram of the binary BE-H2O system. The addition of DEC shifts the two phase equilibria to higher BE concentrations and to lower temperatures. If the addition of a third component is continued beyond the cloud point, eventually three distinct phases appear. Unfortunately the cloud point technique gives us the initial concentration or temperature where unmixing begins but is not suitable to distinguish between the coexistence of two phases and three phases. Also the three phase region depends quite critically on temperature. 

Blend solutions. Solutions of blends comprising immiscible polymers Pj and P2 in a nonselective solvent have miscibility gaps as shown schematically in Fig. 14. When the polymer concentration increases by solvent evaporation the polymer coils start to interpenetrate above a certain concentration. As a consequence, interactions between the polymers become operative and phase separation must start above a critical polymer concentration p. The composition of the new phases will be situated on the branches of the coexistence curve. Finally, the unmixing process is arrested owing to enhanced viscosity. This simple scheme reveals the factors directing morphology evolution in blend solutions ... 

For the polymer blend with symmetric walls, approaching the bulk critical temperature from the disordered region there occurs a completely smooth and gradual formation of surface enrichment layers on both walls (Fig. lc), without any phase transition. The unmixing phase transition involves formation of con-... 

Fig. 55. Schematic phase diagram of a binary mixture with an unmixing transition in the bulk (miscibility gap from composition 4 (7 ) to cJoixOT) ending in a critical point Tc, . ,) and a first-order wetting transition at Tv at the surface of the mixture and a wall. For T > rw, a (thick) layer of concentration with the other branch of the coexistence curve, cSx(T ) is adsorbed at the wall. The prevvetting line ending in a surface critical paint Tcs is also shown. After Cahn (1977).

This can be explained as follows. It is known that PP and EPDM are immiscible materials and they exhibit a lower critical temperature (LCST) phase diagram (19). During mixing, especially at high shear rates, the LCST curve elevates with temperature and shear-induced mixing takes place. Thus, in the process of dynamic vulcanization, PP and EPDM can be considered as miscible materials under high shear rates. As a result, after the cross-linking reaction, the unmixed EPDM component forms the dispersed domain, while the matrix consists of mixed PP (dominant) and EPDM (minor) components connected by chemical cross-links (20). 

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