Renormalized excluded-volume

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

We must note also a second important restriction of the continuous chain model. As we will see. by construction it deals with infinitely long chains n — oo. infinitesimally close to the -point , 5C — 0. Thus naive two parameter theory is valid only very close to the -temperature. In later chapters we will see how further renormalization leads to a theory of excluded volume effects valid for all /%, > 0. 

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

In Chap. 6 we learned that in the excluded volume limit ftc > 0,n —> oo, the cluster expansion breaks down, simply because it orders according to powers of z = j3enef2 —> oo. To proceed, we need a new idea, going beyond perturbation theory. The new concept is known as the Renormalization Group (RG), which postulates, proves, and exploits the fascinating scale invariance property of the theory. 

We can now state clearly what in the context of the renormalization group we mean by the excluded volume limit... 

By definition the excluded volume limit is reached if under renormalization the coupling constant approaches the fixed point so closely that we can replace it by 8. ... 

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

To corroborate these qualitative considerations in Sect, 8/2 we use our perturbative results for df to construct a simple realization of the renormalization group, first concentrating on the excluded volume limit. In Sect. 8.3 wc show how the crossover among 0-point and excluded volume limit can be treated in principle. 

Fig. 9.3. The scaling function R (N,Cp)/R (A, Cj, — 0) (excluded volume limit, monodisperse) hh function of s = CpRg N,Cp = 0). The full line is the renormalization group result. (See Chap. 18.) The broken lijies give approximations as motivated bv the blob model...

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