Phase behavior defined

A general prerequisite for the existence of a stable interface between two phases is that the free energy of formation of the interface be positive were it negative or zero, fluctuations would lead to complete dispersion of one phase in another. As implied, thermodynamics constitutes an important discipline within the general subject. It is one in which surface area joins the usual extensive quantities of mass and volume and in which surface tension and surface composition join the usual intensive quantities of pressure, temperature, and bulk composition. The thermodynamic functions of free energy, enthalpy and entropy can be defined for an interface as well as for a bulk portion of matter. Chapters II and ni are based on a rich history of thermodynamic studies of the liquid interface. The phase behavior of liquid films enters in Chapter IV, and the electrical potential and charge are added as thermodynamic variables in Chapter V. 

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

In a somewhat wider sense, one can define amphiphiles as molecules in which chemically very different units are linked together. For example, the structures formed by A B block copolymers in demixed A and/or B homopolymer melts and their phase behavior are very similar to those of classical amphiphiles in water and/or oil [13,14]. Copolymers are used not only to disperse immiscible homopolymer phases in one another, but also to create new, mesoscopically structured materials with unusual and interesting properties [15]. 

Here the functions g(0) and /(0) are defined in a suitable way to produce the desired phase behavior (see Chapter 14). The amphiphile concentration does not appear expHcitly in this model, but it influences the form of g(0)— in particular, its sign. Other models work with two order parameters, one for the difference between oil and water density and one for the amphiphile density. In addition, a vector order-parameter field sometimes accounts for the orientional degrees of freedom of the amphiphiles [1]. 

Of course, LC is not often carried out with neat mobile-phase fluids. As we blend solvents we must pay attention to the phase behavior of the mixtures we produce. This adds complexity to the picture, but the same basic concepts still hold we need to define the region in the phase diagram where we have continuous behavior and only one fluid state. For a two-component mixture, the complete phase diagram requires three dimensions, as shown in Figure 7.2. This figure represents a Type I mixture, meaning the two components are miscible as liquids. There are numerous other mixture types (21), many with miscibility gaps between the components, but for our purposes the Type I mixture is Sufficient. 

The synthesis and phase behavior of the model polydiethylsiloxane networks have also been studied. The networks were made by hydrosilylation of well-defined vinyl and allyl telechelic siloxanes obtained by kinetically controlled polymerization of cyclic trisiloxane.314 The effects of molecular weight between the cross-linkings on segment orientation in polydiethylsiloxane elastomers were studied.315... 

M. W. De Jager, G. S. Gooris, M. Ponec, and J. A. Bouwstra. Lipid mixtures prepared with well-defined synthetic ceramides closely mimic the unique stratum corneum lipid phase behavior. J. Lipid Res. 46 2649-2656 (2005). 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

LIQUID-PHASE BEHAVIOR. The liquid phase contains dissolved substances and contacts the solid phase. For our purposes, the liquid phase is used synonymously with aqueous phase , and all processes discussed in this section take place in aqueous solutions. The dissolved monomers of the solid phase are formed in equilibrium with their uncomplexed components. Such components may be uncomplexed ions (which are charged atoms or molecules) free in solution or ionic complexes in equilibrium with dissociated ions. Concentrations of the uncomplexed ions, therefore, depend upon the concentrations of all chemical substances competing for binding interactions with them. Each complex-ation reaction is defined by either a solution equilibrium constant ... 

This method requires knowledge of the characteristics values for the oil and alcohol effects, which is not always the case, in particular if some natural ill-defined product like a petroleum refinery cut is used. Alternatively, it might be impossible to attain three-phase behavior in the feasible experimental range, for instance the salinity that satisfies Eq. 4 might be too high to be attainable in practice. In such a case, another variable should be changed to keep the optimum value of the scan in the feasible range, for instance the introduction of another alcohol, which would alter the value of/(A). However, this tends to introduce inaccuracies. 

Until now we have discussed only insoluble monolayers. Although their behavior is complex, they have the conceptual simplicity of being localized in the interface. It has been noted, however, that even in the case of insoluble monolayers, the substrate should not be overlooked. The importance of the adjoining bulk phases is thrust into even more prominent view when soluble monolayers are discussed. In this case the adsorbed material has appreciable solubility in one or both of the bulk phases that define the interface. 

Therefore, polyrotaxanes can be simply defined as polymeric materials containing rotaxane units. They are different from conventional linear homopolymers because they always consist of two components, a cyclic species mechanically attached to a linear species. They also differ from polymer blends as the individual species are interlocked together and from block copolymers since the two components are noncovalendy connected. Thus new phase behavior, mechanical properties, molecular shapes and sizes, and different solution properties are expected for polyrotaxanes. Their ultimate properties depend on the chemical compositions of the two components, their interaction and compatibility. This review is designed to summarize the syntheses of these novel polymers and their properties. 

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]. 

The heavy line in Fig. 6(a)-(c) is the dilution line constraint m = mf or Pj = LnPo- As discussed already, we are interested mainly in systems whose mean composition lies on this line. [With x as an extra variable, this line becomes a plane (p1 — LNp0, %) in the space (pj, p0, x)-] Not all of the phase behavior shown in Fig. 6 is then accessible. The extremities of phase separation on the pj = LnP0 line are points where phase separation just starts to occur, and the locus of these points in the (pj = Zwp0, l) plane defines the... 

Figure 6. Conventional two-component phase behavior in poly disperse Flory-Huggins theory, shown in the (p, p0) plane for three values of % As in Fig. 5, the parent has Ln = 100 and Ly/ = 150 (hence a = 2). Along the y-axis, we plot L//p0 rather than p0 so that the dilution line p = LNp0, shown as the thick solid line in (a-c), is simply along the diagonal. With x considered as an additional variable, the dilution line constraint defines a plane (p, = L/vPq, x)- The last plot, (d), shows the cut by this plane through the phase behavior in (a-c) the solid line is the cloud point curve, and the dashed line is the spinodal stability condition.

Co is called the spontaneous curvature. The spontaneous curvature is a more general parameter than the surfactant parameter Ns, defined by Eq. (12.4). It makes it easier to discuss the phase behavior of microemulsions because we get away from the simple geometric picture. 

The quality of separation depends on several factors, among them the partition coefficients of biomolecules between the phases, phase behavior, temperature, and the type and concentration of both salt and polymer. The partition coefficient of species X between the top and bottom phases is defined by Eq. (8.67). 

Proteins are both colloids and polymers. Therefore, attempts have been made to understand the phenomenon of protein aggregation with the help of models from the polymer and colloid fields such as DLVO theory, describing the stability of colloidal particles, or phase behavior and attraction-repulsion models from polymers (De Young, 1993). For faster progress, more phase diagrams for equilibrium protein precipitation, in both the crystalline and the non-crystalline state, as well as more data on observations of defined protein oligomers or polymers, are required. 

Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. 

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