Shadow curves phase coexistence

In fact, the condition just described holds whenever all but one of a set of coexisting phases are of infinitesimal volume compared to the majority phase. This is because the density distribution, p (cr), of the majority phase is negligibly perturbed, whereas that in each minority phase differs from this by a Gibbs-Boltzmann factor, of exactly the form required for (10) we show this formally in Section III. Accordingly, our projection method yields exact cloud point and shadow curves. By the same argument, critical points (which in fact lie at the intersection of these two curves) are exactly determined the same is true for tricritical and higher-order critical points. Finally, spino-dals are also found exactly. We defer explicit proofs of these statements to Section III. 

Given f P), these q I equations can, in principle, be solved for Tp, ", and f Py as functions of . The resulting relation between Tp and gives the cloud-point curve, while that between Tp and " gives a line called the shadow curve. The latter cannot be determined experimentally, since the second phase is too small in volume to be analyzed for the composition. It can be shown that the cloud-point and shadow curves coincide with the conjugate coexistence curves when and only when the solution is strictly binary. This fact is important, because some authors make no distinction between cloud-point curve and coexistence curve in describing phase equilibria of polydisperse solutions. 

Fig. 9-22. Calculated and observed phase diagrams for PS f4 + PS fl28 (the weight fraction of the latter is 0.05) + CH on the T — ( > plane. Thick solid line, calculated cloud-point curve. Dot-dashed line, calculated shadow curve. Dashed lines, calculated two-phase conjugate coexistence curves for the indicated polymer volume fractions. Thin solid line, three-phase coexistence curve. Unfilled circle, calculated critical point. Filled circles, measured cloud points. Filled triangles, measured polymer volume fractions in three separated phases. 

For solutions of polydisperse polymers, such a procedure cannot be used because the critical concentration must be known in advance to measure its corresponding coexistence curve. Additionally, the critical point is not the maximum in this case but a point at the right-hand side shoulder of the cloud-point curve. T wo different methods were developed to solve this problem, the phase-volume-ratio method, e.g., Koningsveld, where one uses the fact that this ratio is exactly equal to one only at the critical point, and the coexistence concentration plot, e.g. Wolf, where an isoplethal diagram of values of tp 2 and tp 2 vs. (p 2 gives the critical point as the intersection point of cloud-point and shadow curves. 

Under the conditions of T, P, and ipB where phase separation occurs in the miscibility gap, i.e., below (UCST behavior) or above (LCST behavior) doud-point conditions, two phases may be observed. They are characterized by distribution functions, WB(rB, Yb) and W (rB, Yb), which differ from them in the initial homogeneous phase. This effect leads to T(ipb), T(ipS) or P(tpb)> P( Pb) curves, which are different from the cloud-point curve and the shadow curve. There is an infinite number of such differing sets of coexisting curves, because they depend on the value of ips in the initial homogeneous phase. Thus, each different ips leads to a different coexistence curve, and the cloud-point curve and the shadow curve are the envelopes of these coexistence curves [16, 29]. 

The coexistence curves are usually no closed curves but rather divided into two branches beginning at corresponding points of the cloud-point curve and of the shadow curve. Only if the composition of the initial homogeneous phase equals... 

Polymers in coexisting phases show different molar-mass distributions which are also different from that of the initial homogeneous system (Figure 10.15). This effect is called fractionation effect and can be used for the production of tailor-made polymers [2, 39, 40], The phase with a lower polymer concentration contains the major part of the polymers with a lower molar mass. The cloud-point curve always corresponds to the molar-mass distribution of the initial polymer, but the first droplets of the formed coexisting new phase never do so (with the exception of the critical solution point) and, hence, they are not located on the cloud-point curve but on the shadow curve. 

Figure 3.12. Schematic of the quasibinary section of a ternary (multicomponent) system, Tp, is tile temperature of the precipitation threshold CPC is the cloud-point curve (tlw phase boundary). / stands for the phase coexistence curves at the total polymer concentration V cj and 4. SL is the shadow line. The concentrations of the coexisting...

The spinodal curve and the critical points (including multiple critical points) only depend on few moments of the molar-mass distribution of the polydisperse system while the cloud-point curve the shadow curve and the coexistence curves are strongly influenced by the whole curvature of the distribution function. The methods used that include the real molar-mass distribution or an assumed analytical distribution replaced by several hundred discrete components have been reviewed by Kamide. In the 1980s continuous thermodynamics was applied, for example, by Ratzsch and Kehlen to calculate the phase equilibrium of a solution of polyethene in supercritical ethene as a function of pressures at T= 403.15 K. The Flory s model was used with an equation of state to describe the poly-dispersity of polyethene with a a Wesslau distribution. Ratzsch and Wohlfarth applied continuous thermodynamics to the high-pressure phase equilibrium of ethene [ethylene]-I-poly(but-3-enoic acid ethene) [poly(ethylene-co-vinylace-tate)] and to the corresponding quasiternary system including ethenyl ethanoate [vinylacetate]. In addition to Flory s equation of state Ratzsch and Wohlfarth also tested the Schotte model as a suitable means to describe the phase equilibrium neglecting the polydispersity with respect to chemical composition of the... 

BOR Borchard, W., Frahn, S., and Fischer, V., Determination of cloud-point, shadow and other coexistence curves in multicomponent systems from measurements of phase volume ratios, Macromol. Chem. Phys., 195, 3311, 1994. 

Fig. 1 Schematic liquid-liquid phase diagram for a polydisperse polymer in a solvait Solid line cloud-point curve, broken line shadow curve, dotted line spinodal curve, star critical point, thick lines coexisting curves, solid line with squares tie isie,Xp iy r segment fraction of polymer...

Fig. 9. Phase diagrams of quasi-ternary systems containing two different molecular weight samples a PHIC-toluene with (Ni, N2) = (4.46,0.38) [73] b schizophyllan-water system with (Nt, N2) = (0.930, 0.0765) [75,76]. (O, A) experimental coexisting isotropic phase ( , , ) experimental coexisting anisotropic phase dashed segments, experimental tie lines the shadowed triangular region, the IAA triphasic region thick full curves, theoretical binodals thin full segments, theoretical tie lines...

The distribution function of the shadow phase, phase II, is also of the Stockmayer type (Eq. (84)) having the same parameters kB and Eb as in the initial homogeneous phase I. Evidently, this does not hold true for the coexistence curves, compare Eqs. (50)-(52). 

When ip < Pc the left-hand branch of the phase cright-hand branch goes out of the CPC and finishes at the temperature of the left-hand branch break and at the polymer concentration in the second phase when the amount of this phase is infinitesimal. A set of endpoints of this right-hand branch of the pha.se coexistence curve (PCC) forms a shadow line (SL). 

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