Phase diagrams, solid-fluid equilibrium

Let us assume some small number n of lipid molecules can form a relatively stable solid phase cluster when the temperature and composition of the lipid mixture is such that, according to the phase diagram, solid phase can exist in equilibrium with the fluid phase. (For example, we later assume that n 10.) Let us further assume that (1) the temperature and composition of the lipid mixture is such that X is small, X 1, and (2) all the solid phase present is in the form of clusters of n molecules each. If the clusters are randomly distributed in the plane of the membrane, then each cluster will be surrounded by a number of fluid molecules of the order of magnitude of N n/X. The area occupied by the surrounding fluid phase molecules is then NA0 where, A0 60A2. Let us now calculate lower limit on X, Xmin, such that each molecule in... 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

Fig. 2.37. Phase diagram for Ca0-Na20 Si02-(Al203)-H20 system in equilibrium with quartz at 400°C and 400 bars. Plagioclase solid solution can be represented by the albite and anorthite fields, whereas epidote is represented by clinozoisite. Note that the clinozoisite field is adjacent to the anorthite field, suggesting that fluids with high Ca/(H+) might equilibrate with excess anorthite by replacing it with epidote. The location of the albite-anorthite-epidote equilibrium point is a function of epidote and plagioclase composition and depends on the model used for calculation of the thermodynamic properties of aqueous cations (Berndt et al., 1989).

Figure 3.10. Phase diagrams of attractive monodisperse dispersions. Uc is the contact pair potential and (j) is the particle volume fraction. For udk T = 0, the only accessible one-phase transition is the hard sphere transition. If Uc/hgT 0, two distinct scenarios are possible according to the value of the ratio (range of the pair potential over particle radius). For < 0.3 (a), only fluid-solid equilibrium is predicted. For % > 0.3 (b), in addition to fluid-solid equilibrium, a fluid-fluid (liquid-gas) coexistence is predicted with a critical point (C) and a triple point (T).

Fig. 2. Phase diagram describing lateral phase separations in the plane of bilayer membranes for binary mixtures of dielaidoylphosphatidylcholine (DEPC) and dipalmitoyl-phosphatidylcholine (DPPC). The two-phase region (F+S) represents an equilibrium between a homogeneous fluid solution F (La phase) and a solid solution phase S presumably having monoclinic symmetry (P(J. phase) in multilayers. This phase diagram is discussed in Refs. 19, 18, 4. The phase diagram was derived from studies of spin-label binding to the membranes.

Fig. 6. Schematic model of two-phase system, with solid domain S in equilibrium with fluid domain F. The relative proportions, and compositions of F and S are determined from, for example, the phase diagram in Fig. 2. The 13C nuclear resonance spectra of lipids in the solid phase S are broader than the nuclear resonance spectra of lipids in the fluid phase F the spectra of lipids in the fluid phase F can be broadened if they diffuse to, and stick to, the solid phase. For a detailed theoretical calculation of line widths expected for this geometry, see Ref. 4. The bilayer length is L, and the domain boundary is b - b. [Reprinted with permission from P. Brulet and H. M. McConnell, J. Am. Chem. Soc., 98, 1314 (1977). Copyright by American Chemical Society.]...

Figure 14.10 The five types of (fluid + fluid) phase diagrams according to the Scott and van Konynenburg classification. The circles represent the critical points of pure components, while the triangles represent an upper critical solution temperature (u) or a lower critical solution temperature (1). The solid lines represent the (vapor + liquid) equilibrium lines for the pure substances. The dashed lines represent different types of critical loci. (l) [Ar + CH4], (2) [C02 + N20], (3) [C3H8 + H2S],...

Solubilities of meso-tetraphenylporphyrin (normal melting temperature 444°C) in pentane and in toluene have been measured at elevated temperatures and pressures. Three-phase, solid-liquid-gas equilibrium temperatures and pressures were also measured for these two binary mixtures at conditions near the critical point of the supercritical-fluid solvent. The solubility of the porphyrin in supercritical toluene is three orders of magnitude greater than that in supercritical pentane or in conventional liquid solvents at ambient temperatures and pressures. An analysis of the phase diagram for toluene-porphyrin mixtures shows that supercritical toluene is the preferred solvent for this porphyrin because (1) high solubilities are obtained at moderate pressures, and (2) the porphyrin can be easily recovered from solution by small reductions in pressure. 

This is designated as Li = V + L2. An L point arises when a low-density liquid (Li) and a high-density liquid (L2) become critically identical in the presence of a vapor phase. An L point is designated as Li = L2 -f V. Tricritical points, where three phases in equilibrium are also critically identical, are designated as Li = L2 = V. Such critical points, while present in phase diagrams and phase projections, are rarely observed in practice. At low temperatures, solid phases such as asphaltenes and wax can, and frequently do, coexist with the fluid phases noted here and are discussed in later sections. 

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