Size of a single chain

In dilute solutions of good solvent, the size of a polymer is much larger than that at the 0 condition due to the excluded volume interaction. How does the potymer size behave in the concentrated srdution To answer this question let us suppose that added to the concentrated polymer solution is a test polymer of the same chemical structure and the same molecular weight. If the conformation of the test polymer is R , the 

The first term is the energy of the host polymer, which is given in the previous section, the second and third terms are the energy of the test polymer, and the last term denotes the interaction between the test polymer and the host polymer. In terms of c, eqn (5.47) is written as 

The potential v(r) denotes an effective potential between the segments of the test polymer. The effective potential consists of a strong repulsive part (ud(r)) of very short interaction range, and a weak attractive part (-U exp(-r/ )/4 r V) of interaction range . In the length-scale larger than I, Uiese two parts cancel precisely indeed from eqn (5.53) it can be shown that 

Thus there is no excluded volume interaction among the segments whose mean separation is larger than. This effect is caUed the screening of the excluded volume interaction. 

As a consequence of the screening, the distribution of the conformation of the test polymer becomes Gaussian. Indeed given the effective exduded volume potential v(r), it is easy to calculate the mean size of the polymer by use of the method described in Section 2.5. For example, the straightforward perturbation calculation gives (see Appendix 5.m) 

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In order to be consistent with the semidilute concentration regime, the correlation length should be smaller than the size of a single chain, = at, where V 3/5 is the Flory exponent in good solvent conditions. This sets a lower bound on the polymer concentration in the bulk, c > c. ... 

To find the region of validity of the mean field theory we have to use the mean field exponents rather than the exact ones, so that we have to use the values in equation (16). The correlation length for vulcanization is certainly proportional to the typical size of a single chain of which the... 

In three dimensions p(R) becomes large (p - 1) only if JR N, but according to equation (30) this is much larger than the size of a single chain in the cluster. 

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