Concentration dependence of the interaction parameter

Illustrative values of the interaction parameter, measured as a function of the volume fraction of polymer for several nonaqueous systems, are shown in Fig. 3.6. These results clearly indicate that the interaction parameter can be a strong function of the volume fraction of polymer. Usually Xi increases with the volume fraction of polymer (e.g. for polyisobutylene in benzene) but this behaviour is by no means universal (e.g. Xi for polystyrene in toluene decreases with increasing polymer concentration). 

Orofino and Flory (1957) have suggested that the concentration dependence of the interaction parameter can be expressed by the perturbation expansion 

The problem is to determine the values of the higher order interaction parameters Xi, Z3. etc. Some guide to these values can be obtained in the following manner. 

It has been shown (Evans and Napper, 1977) that the interaction between polymer chains, which display a concentration dependent interaction parameter, vanishes when x, = /( +1) for all positive integral values of i. This relationship sug ts that near to the 0-point Xt=h Z2 = l/3=(2/3)xi and X3 = U4=(i)xi- In these circumstances, we mi t guess that the simplest volume fraction dependence might be 

These relationships provide convenient mathematical expressions for the concentration dependence of the interaction parameter that can be used to incorporate this dependence into the theory for steric stabilization. 

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The first report on the gel-gel transition was presented in September 1967 at the 1st Prague Microsymposium on Marcomolecules [3]. A paper was submitted to the Journal of Polymer Science and was published in 1968 [4], One of the referees wrote that it was questionable whether a paper should be published on a phenomenon which could hardly be observed experimentally and recommended a reduction of the manuscript to about 50%. To meet, at least partly, his wishes, we reduced the manuscript to about 70% by removing all speculations about the possible concentration dependences of the interaction parameter. 

For binary data at fixed temperature and pressure, there are two independent measurements, mutual solubility data alone, all of the parameters in the Koningsveld-Kleintjens expression [Equation (2F-6)] for the concentration dependency of the interaction parameter, g12. 

A prime objective of early studies of polymer stationary phases was to assess the reliability of GC-detived activity coefficients and interaction parameters. Conventional static methods of measurement are difficult and time-consuming, and data on relatively few systems are available for the purpose of comparison. The facility with which such data can be obtained by gas duromatc r hy diould remedy this situation. In order to achieve a meaningful comparison, results from static methods must be extrapolated to infinite dilution, owing to the concentration dependence of the interaction parameter. 

The experimental work reported up to this point has dealt exclusively with solutes at infinite dilution in the polymer stationary phase. Recent theoretical advances (S8, 70) have made it possible to extend the ran of gas chromatc t hy to flnite concentration of the solute in the stationary phase. In the caM of polymers arch studies should provide a direct insight into the concentration dependence of the interaction parameter and allow for comparison with static results without recourse to extrapolations to infinite dilution. 

The concentration dependence of the interaction parameter adds a restriction on plotting the left-hand side of Eq 3.40 versus (j) to obtain a single value for 2 [Kwei and Frisch, 1978 Plans et al, 1984 Walsh et al, 1985]. 

Fig. 3.6. Experimental plots of the concentration dependence of the interaction parameter for (1) poly(isobutylene) in benzene, (2) poly(dimethylsiloxane) in benzene, (3) poly(isobutylene) in cylcohexane, and (4) polystyrene in toluene. 

Fig. 3.10. Comparison of the predictions of the equation-of-state theory (full line) with the results ( ) of experiment for the concentration dependence of the interaction parameter of poly(isobutylene) in benzene at 2S °C (after Eichinger and Flory, 1968).

Fig. 3.12. The experimental concentration dependence of the interaction parameter for aqueous solutions of (1) poly(vinyl pyrrolidone), PVP and (2) poly(vinyI methoxyacetal), PVMA.

The disagreement between the values 3.12 and 2.27 points to the fact that Equation 45 is incorrect owing to the neglect of the concentration dependence of the interaction parameter. The chief result of Vrij (1974) is in the fact that near the spinodal maximum, macromolecules have the statistical coil conformation with sizes slightly different from those in the 0 state at c —> 0 (at Iceist, for the polystyrene-t-cyclohexane system). Generally, however, this may not be the case (Vrij and van den Esker, 1972). 

The concentration dependence of the interaction parameter x> obtained from experimental data according to Equation 92, matches that of the (linear PDMS)- -benzonc system (sec subsection 3.6.2, Figure 3.79). 

I he CPCs of the type shown in Figure 3.102 have been observed experimentally on polymer mixtures with a narrow MWD (see Figure 3.95). Thus, multiple equilibrium and the existence of multiple critical points are caused by either a substantial MWD asymmetry or a strong concentration dependence of the interaction parameter g, or both. 

A three-phase equilibrium is observed in the system P-I-LMWL, if the polymer has a very asymmetric MWD, or is represented by two polymer homologues with molec ular weight,s differing more than ten-fold, or there is a square (at least) concentration dependence of the interaction parameter. 

The concentration dependence of the interaction parameter was derived by Koningsveld, based on the concept of the Huggins theory, in the following form ... 

One potential problem with this technique is that x i known to be a function of concentration and the polymer Hildebrand parameter is determined at infinite dilution of solvent. For a number of binary systems, the change in x with solvent weight fraction is the largest as w 0. The concentration dependence of the interaction parameter can be modeled using methods given in the chapter on interaction parameters in this Handbook. 

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