Quasi-binary solution

Fig. 21. Average molecular weight MJ, dependence of T 0 for the quasi-ternary system containing two xan-than samples with (N, Nj) = (0.54,4.5) in 0.1mol/l NaCl [144], Solid curves represent the Mv dependence of T 0 for quasi-binary solutions of xanthan at 0.1 mol/1 NaCl...

This rather simple scheme of solution behavior is complicated by various interesting aspects. It is not at all clear that the number of dissociated or non-condensed counterions remain constant with added salt. Likewise, at high salt concentrations, the polyelectrolyte solution may no longer represent a quasi binary solution but rather a ternary mixture with the added salt representing a third component with distinctly different solvation properties as compared to water. This scenario is supported by the fact that for some salts the ordinary salting out is reversed if the inert monovalent salt concentration is further increased. In other words the chains become redissolved at an even higher salt concentration [21, 22]. Consequently, this redissolution is denoted as salting in . 

Graessley s theory [9] predicts that eq 2.28 holds for tracer diffusion coefficients from experiments with quasi-binary solutions if the term l/N is replaced by N/P, ... 

Thirdly, the solution chosen by Wheeler et al, containing chemically dissimilar polymers, was thermodynamically more complex than quasi-binary solutions as investigated by Kim et al. and Nemoto et al. In the former, thermodynamic interactions with the solvent should be different for the tracer and matrix polymers, while, in the latter, there is no such difference. 

We consider a quasi-binary solution of q + 1 components. When q = 1, the system will be referred to as binary. We designate the solvent as component 0 and q monodisperse homologous homopolymers as components 1, 2,..., q in the order of increasing relative chain length. Here, the relative chain length Pi of component i is defined by... 

We denote AG per unit volume of solution by AG. Thermodynamics tells us that AG of a quasi-binary solution is a function of T, p, and 4>i. For a given polymer species it also may depend on the distribution of relative chain length in the polymer mij re. Thus, when the pressure effect is not considered, the basic variables for AG of quasi-binaiy solutions are T, , and /(P) as the basic variables. For a paucidisperse polymer, q is finite and /(P) is represented by q delta functions with different strength, while for a truly poly-disperse polymer, q is infinitely large and /(P) becomes a continuous function of P. Thus, AG for quasi-binary solutions of a truly polydisperse polymer is a functional with respect to /(P), and requites sophisticated mathematics for its treatment. 

The derivation of the phase relationships for quasi-binary solutions is given in textbooks of polymer thermodynamics [1,2]. Here, the results necessary for the subsequent discussion are summarized. 

Now that the P dependence of g for binary solutions has become apparent, it is clear that this factor for quasi-binary solutions has to be treated as a function of the chain length distribution /(P). Historically, the /(P) dependence of g is not a recent finding but has become gradually recognized through analyses of observed phase relationships. In what follows, we outline some typical contributions which played a role in this process. 

The top of a cloud-point curve is called the threshold point. The critical point on the T — plane for a binary solution is situated at the threshold point, but this is no longer the case for quasi-binary solutions. The critical point on the T — < > plane for a quasi-binary solution should appear somewhere on the... 

These features of the critical point exemplify a great variety of phase equilibria of quasi-binary solutions, which are associated with the multiplicity of polymer components and the dependence of g on T, , and /(P). Among many others, Sole [8] made a most detailed theoretical study on the effects arising from the many-component nature of quasi-binary solutions. Since he assumed for simplicity that g depends only on temperature, his findings are of limited use for quantitative purposes. A good summary of Sole s theory can be seen in Kurata s monograph [2]. 

The work of Shultz and Flory showed clearly that g cannot be a function of T only, and it was natural that attempts toward a more accurate description of phase equilibria in quasi-binary solutions began by choosing (j> as a second variable of this factor. Koningsveld and Staverman [6] showed for the first time that if g depends on both T and , the spi nodal and critical state for quasibinary solutions are given by eq 1.20 and 1.22. These general equations make the following important predictions. 

The findings of Scholte and Derham et al. revealed that g for quasi-binary solutions ought to depend not only on T and (j>, but on at least one more variable. It is not evident whether, in the early 1970s, one had a clear recognition that this third variable should be the chain length distribution f P) or, to be equivalent, that g should depend on the concentrations of the individual polymer components ((/> ,, [Pg.307]

In the late 1970s when they looked over the existing literature, Hashizume et al. noticed that even for the simplest of quasi-binary solutions, i.e., ternary solutions containing two homologous polymers in a single solvent, systematic phase equilibrium data were virtually lacking. Since such data seemed essential for testing any proposed expression for g for quasi-binary solutions, they [22] undertook extensive measurements of cloud-point curves, binodals, and critical points on mixtures of two narrow-distribution polystyrenes f4 and flO dissolved... 

Empirical Form of the Apparent Second Virial coefRcient For a quasi-binary solution eq 2.16 should read... 

The phase relations for quasi-binary solutions outlined in Section 1 are general and exact under the basic assumptions made. However, the computational work with them becomes exponentially difficult as the number of components increases. In fact, it is virtually impossible to solve the phase equilibrium equations for solutions of actual synthetic polymers, which contain an almost infinite number of components. We thus need a novel approach to analyze phase equilibrium data on such systems. The discipline called continuous thermodynamics has emerged to meet this requirement. It deals with mixtures of molecules whose physical properties such boiling point, molecular weight, and so forth vary continuously, and is the correct method for treating solutions of a truly polydisperse polymer (see Section 1.1 of this chapter for its definition). 

Both pi and are functions of T and p that thermodynamics cannot evaluate. With eq 9A-5 and 9A-6 introduced into eq 1.11 we obtain the phase equilibrium equations for polymers 1 and 2. Since p, and pi°° drop out of both sides, these equations contain only F as an unspecified function. It can be shown that the same holds for quasi-binary solutions containing more than two polymer components. In conclusion, the phase equilibrium behavior of any quasi-binaty solution is completely determined if F for the chemical potential of the solvent component is specified as a function of T, 4>, and /(P). 

One can see the very complicated behavior of quasi-binary solutions where the phase boundary is given by a cloud-point curve and where an infinite number of coexistence curves exists (one pair for each starting concentration, i.e., each cloud-point). The cloud-point is a point in the T-W2- or the P-W2-diagram where a homogeneous solution of concentration Wq2 begins to demix (where the firsf droplet of the second phase occurs, T(2) in Figure 4.4.19). If W02 is smaller than the critical concentration, the cloud-point belongs to the sol-phase, otherwise to the gel-phase. 

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