Interactions and the Average Molecular Extension

The perturbation of the configuration of the polymer chain caused by its internal interactions may also be considered from the somewhat different viewpoint set forth qualitatively in Chapter X, Section 3. There it was indicated that, because of the obvious requirement that two segments shall not occupy the same space, the chain will extend over a larger volume than would be calculated on the basis of elementary random flight statistics. As a matter of fact, the overwhelming majority of the statistical configurations calculated without regard for this requirement are found to be unacceptable, on this account, to 

Proceeding in a manner paralleling the derivation of the excluded volume for a pair of molecules, we consider the polymer molecule to consist of a swarm of segments distributed on the average about the molecular center of gravity in accordance with the Gaussian formula (see Eq. XII-51). This spatial distribution in the unperturbed molecule, as it would exist on the average in the total absence of inter- 

Introducing Eq. (XII-43 ) (which merely stipulates the inevitable proportionality between this chemical potential and the square of the concentration in dilute solutions) 

From Eq. (3), since v j is small compared with unity 

The evaluation of the elastic free energy AFei rests on the assumption that the root-mean-square distance between the ends of the chain is distorted by the same factor a representing the linear expansion of the spatial distribution. As in the treatment of the swelling of network 

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