The semidilute limit

Let us now consider the semidilute limit. The power law (1.7) derives from the assumption that JJ for a strongly overlapping solution depends on api n only via the segment concentration c = cpn. We thus write... 

Similar problems are abundant as soon as we leave the region of small momenta and isolated chains. As a final example we consider the semidilute limit. Using the unrenormalized loop expansion in Sect, 5.4.3 we have calculated the first order correction to fip(n). We found a correction of order where c is the segment concentration. The form of this term is due to screening and has nothing to do with the critical behavior treated by renormalization and -expansion. It thus should not be expanded in powers of e. We can trace it back to the occurrence of the size of the concentration blobs as an additional length scale. 

In Fig. 13.3 we also plotted lines / = 0.9,/ = 0.1, which bound the excluded volume or the 0-region, respectively. In the dilute limit w —> 1 these lines reduce to z = const. In the semidilute limit they behave as z s, corresponding to... 

In the semidilute limit renormalized perturbation theory verifies the law (cf. Eq. (9.14))... 

If compared to Eq. (9 17) the second equation shows that N/Nr is to be identified with the number of segments n per concentration blob, whereas Nr — Nfn is the number of blobs per chain. The first equation shows that we find a smooth crossover from the dilute limit w — 1 to the semidilute limit w — 0. In the latter limit Eqs. (14 13), (14.14) yield the expected power law... 

In the semidilute limit Eqs. (13.30)-(13.32) show that / is a function of c and T, only. Since the same holds for CpSz cpN2 we find the expected... 

In the dilute limit of course equals the radius of gyration. We thus consider the semidilute limit, where is of the order of the screening length. (As will be discussed in Chap. 19, it is not strictly identical to the screening length.) Equation (14.35) reduces to... 

Similar problems are abundant as soon as we leave the region of small momenta and isolated chains. As a final example we consider the semidilute limit. Using the unrenormalized loop expansion in Sect, 5.4.3 we have calculated the first order correction to We found a correction of order... 

The interpretation of Eq. (135) is simply that the number of unfavorable contacts vanishes in the semidilute limit, the chains becoming more and more compatible because locally they do not see each other although they are still strongly interpenetrating. 

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