Shadow curves

Figure 5.1-2. Phase behaviour of ethylene LDPE mixtures [10]. Thick line, cloud-point curve dotted line, shadow curve thin lines, co-existence curves total polymer concentration a, 6.1 wt.% b, 11.4 wt.% c, 18.6 wt.% d, 28.0 wt.% e, 36.5 wt.%.

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. 

We now consider the properties of the moment free energy with the (chain number) density p0 retained as an extra moment. This provides additional geometrical insight into the properties of polydisperse chains (while for the numerical determination of the CPC and shadow curve, the above one-moment free energy is preferable). To construct the two-moment free energy, we proceed as before. The family (39) is now... 

Figure 8. Phase behavior in (p, = curve and (b) p0-projected shadow curve.

As a result of polydispersity effects, the composition of the incipient 13-phase segregated at the cloud point is located on a shadow curve, outside the cloud-point curve (point (3 in Fig. 8.4). The effects of polydispersity on phase diagrams and phase compositions may be found in specialized reviews (Tompa, 1956 Kamide, 1990 Williams et al., 1997). Because < )Mo < ( M,crit(xcp), the incipient (3-phase, which is richer in the modifier, will be dispersed in the a-phase, which is richer in the growing thermosetting polymer. The opposite occurs when < )M0 > M,crit(xcp)- It has been shown both theoretically (Riccardi et al., 1994 and 1996 Williams et al., 1997), and experimentally (Bonnet et al., 1999) that... 

FIGURE 16.9 The cloud-point and shadow curves for poly(ethylene-octene)-hexane at450 K from experiment (points) and the SAFT equation of state (curves). The dotted curve shows the composition of the incipient phase at the cloud point. Results for monodisperse and polydisperse polymer are included. (From Jog, P.K. et al., Ind. Eng. Chem. Res., 41(5), 887, 2002. With permission.)... 

Figure 15.5 Effect of molecular weight dispersity ( )) (formerly known as polydispersity) using schematic Gibbs triangle diagrams for polymer-solvent system, generation of cloud point curve and shadow curve in temperature-composition diagram.

Figure 1. Demixing diagrams for PS in 0-solvents and poor solvents (schematic). The variable X might be pressure, Mw D/H ratio in solvent or solute, etc. See text for a further discussion, (a, top left) PS in a 0-solvent (monodisperse approximation). For X=Mw - the X=0 intercepts of the upper and lower heavy lines drawn through the minima or maxima in the demixing curves define 0Land0u, respectively, (b, top right) PS in a poor solvent (monodisperse approximation). The heavy dot at thecenterlocates the hypercritical (homogeneous double critical) point. (c, bottom right) The effect of polydispersity. BIN=binoda] curve, CP=cloud point curve, SP=spinodal, SHDW=shadow curve. See text Modified from ref. 6 and used with permission.

Even if the pseudo-component approach does not fit the whole CPC it is possible to use this model to adjust the branch poor in CO, i.e. that of practical interest, leading to a dispersed phase rich in CO and generating a shadow curve (dotted line in Fig. 14). The interaction parameter resulting from the fitting was [78]... 

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. 9-20. Calculated cloud-point curves (solid lines), shadow curves (dashed lines), and loci of critical points (dot-dashed lines) for PS f4 + PS flO (upper panel) and PS f4 + f40 (lower panel) in CH [29]. 

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. 

Calculation of Cloud-Point Curve, Shadow Curve... 

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]. 

As discussed above, further treatment depends on how G depends on WafrB, Yb). If this dependence is provided by functionals only, the problems of the unknown distribution functions and the unknown scalars can be separated as discussed above for the problem of cloud-point and shadow curves. 

The application of the Stockmayer distribution function is especially advantageous in the framework of continuous thermodynamics. When cloud-point and shadow curves are calculated, the double integrals in Eqs. (40)-(42) can be evaluated analytically (in contrast the double integrals in Eqs. (50)-(52) can be computed... 

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