Determination of the 0-point

Determination of the theta-point It was shown above that at the 0-point the second virial coefficient vanishes. This allows the 0-point to be located by light scattering or osmotic pressure measurements. It can also be found by the rather time-consuming determination of the critical solution temperature extrapolated to infinite molecular weight. 

Quicker, but less soundly based, methods have been developed by Elias (1959 1961), Comet and van Ballegooijen (1966) and Talamini and Vidotto (1967). These allow the rapid determination of either the 0-temperature or 0-composition. In these methods, the temperature of phase separation, for example, over a range of polymer concentrations (10 V2 10 ) is measured for a given polymer sample. The polymer may even be polydisperse in molecular weight. A plot of the reciprocal temperature vs the logarithm of the vffiume fraction of polymer is usually linear (see Fig. 3.4). Extrapolation to log V2 =0 yields the value of 1/0. This method is, of course, inapplicable to polymers that crystallize on phase separation. Note that Elias (1977) has reviewed in an extremely comprehensive manner the rapid methods of determining the 0-point. 

If a polymer solution undergoes phase separation into two liquid phases then the solution cannot be too far removed from the 0-point. It follows that the Flory-Huggins theory, which was derived for moderately concentrated solutions, can be applied to both of the polymer solution phases, one of which is dilute and the other concentrated. The chemical potentials of both the polymer and the solvent must be equal in the two co-existing phases  

Since vf 1, we may expand the logarithm as a power series so that 

On substituting the Flory-Huggins expressions for chemical potential into equation (3.38), we find that 

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