Modulus of Elasticity and the Intermediate Length

Initially, the elasticity of concentrated polymer systems was ascribed to the existence of a network in the system formed by long macromolecules with junction sites (Ferry 1980). The sites were assumed to exist for an appreciable time, so that, for observable times which are less than the lifetime of the site, the entangled system appears to be elastic. Equation (1.44) was used to estimate the number density of sites in the system. The number of entanglements for a single macromolecule Z = M/Me can be calculated according to the modified formula 

The length of a macromolecule between adjacent entanglements Me is used as an individual characteristic of a polymer system. Table 1 contains values of Me for certain polymer systems. The more complete list of estimates of the quantity Me can be found in work by Aharoni (1983, 1986). One can compare expressions (6.52) and (6.54) for the value of the modulus on the plateau to see that the length of a macromolecule between adjacent entanglements Me is closely connected with one of the parameters of the theory 

We should note, recalling the interpretation of x as the ratio of the doubled intermediate length to the size of the coil discussed previously (formula (5.8)), that the length Me, determined in the usual way, is actually related to the intermediate length . Expression (6.52) can be rewritten in a form which is identical to the relation by Doi and Edwards (1986) 

Note that the squared diameter of the Doi-Edwards tube relates to our intermediate length as follows 

The intermediate length (tube diameter) 2 can be estimated from the modulus with the aid of the above equations. Comparison of values of the intermediate length found from dynamic modulus and from neutron-scattering experiments was presented by Ewen and Richter (1995). They found the values to be close to each other, though there is a difference in the temperature dependence of the values of intermediate length found by different methods. 

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