Chain Model and Naive Two Parameter Theory

The two parameter hypothesis assumes that physical properties of dilute polymer solutions depend only on the two parameters 

Even for d 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

It turns out that this shift of f.ip(n)/n has a good physical reason. It identifies the chemical potential per segment of an infinitely long chain and thus is essential for reaching the critical limit Rc — oo. 

We indeed encountered additive renormalization before. In our calculation of f.ip(n) (Sect. 5.4.3) we had to subtract the single chain part from the one loop correction, before we could replace the sum over discrete segments by the Debye function, which is calculated for a continuous chain. (See Eq. (3.22).) In the result (5.6 ) this introduced the term -n/ c (f) to shift the chemical potential. All the 

In two panmieter theory we ignore the microscopic length Lscale i. It is thus intimately related to the continuous chain limit 0, where the discrete sequence of segments is replaced by a continuous space curve. On the level of the configurational probability JF exp [—17] this leads to the Edwards 

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Continuous Chain Model and Naive Two Parameter Theory... 

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