Binary coexistence curve

Figure 1, Liquid-liquid equilibria in a polymer/solvent solution ethylene-polyethylene binary coexistence curve (constant T and molecular weight)...

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

B. Ternary Phase Diagram from Binary Data 1. Coexistence Curve... 

In our discussion of the coexistence curve, we tacitly assumed that at least one liquid phase can exist for any composition. However, since component 2 is a supercritical gas, the physically allowable liquid compositions are limited by the solubility of the gas in the binary solvent mixture. As the pressure rises, the solubility of the gas increases, thereby enlarging the composition range which permits the existence of a liquid phase. 

Since industrial separation processes operate in the Li L2 region, it is important to determine how the Margules parameters affect the shape of the coexistence curve and the slope of the tie lines. For any liquid-liquid region to exist, at least one of the binary Margules constants must be greater than 2RT(on y positive values are considered here) this is a consequence of the... 

Typically, the liquidus lines of a binary system curve down and intersect with the solidus line at the eutectic point, where a liquid coexists with the solid phases of both components. In this sense, the mixture of two solvents should have an expanded liquid range with a lower melting temperature than that of either solvent individually. As Figure 4 shows, the most popular solvent combination used for lithium ion technology, LiPFe/EC/DMC, has liquidus lines below the mp of either EC or DMC, and the eutectic point lies at —7.6 °C with molar fractions of - 0.30 EC and "-"0.70 DMC. This composition corresponds to volume fractions of 0.24 EC and 0.76 DMC or weight fractions of 0.28 EC and 0.71 DMC. Due to the high mp of both EC (36 X) and DMC (4.6 X), this low-temperature limit is rather high and needs improvement if applications in cold environments are to be considered. 

Liquids. The translational absorption profiles of a 2% solution of neon in liquid argon have been measured at various temperatures along the coexistence curve of the gas and liquid phases [107]. Figure 3.8 shows the symmetrized spectral function at four densities. At the lowest density (479 amagat for T = 145 K curve at top) the profile looks much like the binary spectral function seen in Fig. 3.2, especially the nearexponential wing for frequencies v > 25 cm-1. With increasing density the intercollisional dip develops at low frequencies, much like the dips seen at much lower densities in Fig. 3.5 - only much broader. 

Table 3.3. Spectral moments of the neon-argon liquid mixture along the coexistence curve measurement [107] compared with binary values calculated from first principles. (Calculated ternary moments are given in Table 3.2 above.)...

Figure 6 Coexistence curves of binary Ising lattice (Yan et al., 2004).

Figure 9 Coexistence curve of binary polymer solutions with different chain lengths. From bottom to top the chain lengths are 1, 2, 4, 8,16, 32, 64,100, 200, and 600, respectively.

Figure 10 Coexistence curve of binary polymer solutions with chain length r2 = 18 and 60. Open squares MC data solid line this work dash line Flory-Huggins theory dotted line Freed theory.

Binary-Liquid Option. As an alternative to this study of critical behavior in a pure fluid, one can use quite a similar technique to investigate the coexistence curve and critical point in a binary-liquid mixture. Many mixtures of organic liquids (call them A and B) exhibit an upper critical point, which is also called a consolute point. In this case, the system exists as a homogeneous one-phase solution for all compositions if Tis greater than... 

Coexistence of binary systems. Coexisting phases are characterized by different figures of the order parameter M. In pure fluids, one identifies M with the density difference of the coexisting phases. In solutions, M is related to some concentration variable, where theory now advocates the number density or the closely related volume fraction [101]. At a quantitative level, these divergences are described by crossover theory [86,87] or by asymptotic scaling laws and corrections to scaling, which are expressed in the form of a so-called Wegner series [104], The two branches of the coexistence curve are described by... 

Wiegand S., Briggs M.E., Levelt Sengers, J.M.H., Kleemeier, M., and Schroer, W. Turbidity, light scattering, and coexistence curve data for the ionic binary mixture triethyl n-hexyl ammonium triethyl n-hexyl borate in diphenyl ether. 

The binodal for binary mixtures coincides with the coexistence curve, since for a given temperature (or A%) with overall composition in the two-phase region, the two compositions that coexist at equilibrium can be read off the binodal. Any overall composition at temperature T within the miscibility gap defined by the binodal has its minimum free energy in a... 

Note that the spinodal and binodal for any binary mixture meet at the critical point (Fig. 4.8). For interaction parameters x below the critical one (for xhomogeneous mixture is stable at any composition 0 < < 1 For higher values of the interaction parameter (for x > Xc) there is a miscibility gap between the two branches of the binodal in Fig. 4.8. For any composition in a miscibility gap, the equilibrium state corresponds to two phases with compositions

coexistence curve at the same value of x-... 

For binary mixtures, the binodal line is also the coexistence curve, defined by the common tangent line to the composition dependence of the free energy of mixing curve, and gives the equilibrium compositions of the two phases obtained when the overall composition is inside the miscibility gap. The spinodal curve, determined by the inflection points of the composition dependence of the free energy of mixing curve, separates unstable and metastable regions within the miscibility gap. 

Fig. 1. Temperature T vs volume fraction phase diagram of a binary polymer blend. Solid line denotes the coexistence curve (binodal) while the me dashed line marks the spinodal line. Binodal connects with spinodal at the critical point (( )c> Tc)...

It is of interest to trace the temperature dependence of these two types of observables. The locus of (jq and as a function of temperature yields the coexistence curve in the composition-temperature plane for the studied binary blend. A related data set on the interfacial width is plotted as a function of temperature. 

Correlate, using a single parameter per binary system, binary polymer-solvent VLE and 1.1 -E (UCST). Of particular interest is the fact that the vdW 13oS predicts a much flatter coexistence curve than the FH and other related models, in agreement with experimental evidence. 

Fig. 40. Schematic description of unstable thermodynamic fluctuations in the two-phase regime of a binary mixture AB at a concentration cb (a) in the unstable regime inside the two branches tp of the spinodal curve and (b) in the metastable regime between the spinodal curve tp and the coexistence curve The local concentration c(r) at a point r = (x. y, z.) in space is schematically plotted against the spatial coordinate x at some time after the quench. In case (a), the concentration variation at three distinct times t, ti, u is indicated. In case (b) a critical droplet is indicated, of diameter 2R , the width of the interfacial regions being the correlation length Note that the concentration profile of the droplet reaches the other branch ini, of the coexistence curve in the droplet center only for weak supersaturations of the mixture, where cb -

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).

Some binary systems show a minimum at a lower critical-solution temperature a few systems show closed-loop two-phase regions with a maximum and a minimum.) As the temperature is increased at any composition other than the critical composition x = x, the compositions of the two coexisting phases adjust themselves to keep the total mole fraction unchanged until the coexistence curve is reached, above which only one phase... 

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