Edwards excluded volume parameters

The bij are proportional the usual Edwards excluded volume parameters vij (notice however that bij is not homogeneous to a volume). 

For most polymer-solvent combinations i > 0 because of a net segment-segment attraction. This implies that the energy effect opposes the entroplcally driven dissolution of the polymer in the solvent. Such solutions are still thermodynamically stable unless x becomes too high see sec. 5.2e. If > 0 the excluded volume is smaller than the real volume (= ] of a segment. FloryH, Edwards ), De Gennes ) and others have shown that the excluded volume may be written as v . where the dimensionless excluded volume parameter v is defined as... 

Fig. 10.2. The segment density distribution function for two flat plates stabilized by tails for an excluded volume parameter (u=5) according to Dolan and Edwards (1975). The dashed line is for zero excluded volume ( =0).

Predictions of the Dolan and Edwards theory. The dependence predicted by Dolan and Edwards for the free energy per tail as a function of the distance between the two parallel flat plates to which they are attached is shown in Fig. 11.6. The curves span three different values of the excluded volume parameter u, which is given in classical Flory-Huggins terms by... 

Fig. 11.6. Comparison of the predictions ( ) of the theory of Dolan and Edwards (1975) with those of Hesselink, Vrij and Overbeek (1971) (full line) for the distance dependence of the increase in free energy per tail for three different values of the excluded volume parameter u (after Hesselink, 1977).

As mentioned above, at the theta temperature, because of the compensation between attractive and repulsive parts of the potential, the random walk model gives an adequate description of a chain in three-dimensional space [1-6]. Actually, there are still logarithmic corrections, but they may be neglected. In two dimensions, a chain at theta temperature is still not equivalent to a random walk [18]. In what follows, we will be concerned with solutions in a good solvent It was realized by Edwards [10] that the exact shape of the potential is not important and that it could be described by a parameter w(T), where T is the temperatiue, called the excluded volume parameter, defined as... 

The natural approach initiated by the classic woikers (Kuhn, Hetmans, Flory, etc.) and formalized later by Edwards —Is based on the idea of a self-consistent field. We describe it for a typical case where I) all monomers are chemically identical, and 2) the interactions are repulsive and local (no long range forces). We write the interaction between monomers (i) and O ) in the form vT8(xy), where v is the excluded volume parameter defined in eq. (III. 10). This form is adequate for uncharged molecules in semi-dilute (or dilute) solutions with good solvents. 

A two-parameter model [15] predicts that a is a imiversal function of an excluded volume parameter, z, which is a dimensionless excluded volume, and the theory of Edwards and Singh [16] leads to the same conclusion. Their self-consistent model is based on the assumption of imiform expansion, i.e., that the expansion of the chain can be represented by an increase in the effective bond length. This affects the size of the molecule and thus the value of a. Edwards and Singh developed the following expKdt relationship between a and z for large N ... 

---