Kuhnian limit

However, in the Kuhnian limit, the renormalized partition function 2W(ES,--., CS) depends only on the size X of the isolated polymer. Thus... 

Formula (12.2.27) describes the behavious of flfi in the whole concentration domain which extends from dilute to semi-dilute solutions, but, in this domain, it is not universal, and therefore it gives only an approximate image of reality. On the contrary, when z -+ oo and g- g (large swelling but X is kept fixed), the Kuhnian limit is reached... 

The semi-dilute regime corresponds to the domain CXd 1. In this domain Ilfi is expected to depend only on the monomer concentration. As was pointed out in Section 2.1, the number of monomers of a chain is proportional to X11", and consequently, in this Kuhnian limit, the monomer concentration must be defined as the Kuhnian concentration. 

In order to determine the behaviour of the solution in the Kuhnian limit, we must re-express 17ft and Cin terms of renormalized quantities. Consequently, we write... 

One-loop approximation of the osmotic pressure in the Kuhnian limit... 

We showed that, in the Kuhnian limit, the osmotic pressure n given by J(f) can be expressed in the form of a scaling law, provided / and M[f) are properly renormalized (see Section 2.5.2). 

Then, we observe that CXd remains finite and retains a meaning in the Kuhnian limit, whereas, on the contrary, bCS2 becomes infinite and loses any meaning in this limit. Accordingly, f and J,(f) renormalize in a manner which is highly non-trivial consequently, we must abandon this approach contenting ourselves with the summation of fewer diagrams ... 

Our definition of the structure function H q) being quite simple (see 13.2.105), for a good solvent (Kuhnian limit), we can directly write the following scaling law... 

It is always useful to check for the number of independent parameters. For the Brownian chain there is one parameter only, namely Rq. For the expanded chain in general, we find two parameters, Rp and but in the Kuhnian limit 0 we return again to the simple one-parameter case. 

On the other hand, one can carry out the passage to the Kuhnian limit z oo as realized in good solvents. Theory shows that there is a well-defined limiting value, h z — oo), and a corresponding limiting function, which now depends on X only... 

The value of the expansion coefBcient in Eq. (3.22) in the Kuhnian limit, h(z —> oo), can be calculated using renormalization group methods, with the result... 

Equation (3.30) formulates an interesting result. It reveals that the increase in the osmotic pressure over the ideal behavior, as described in lowest order by the second term on the right-hand side, may be understood as being caused by repulsive hard core interactions between the polymer chains which occupy volumes in the order of h z)R. To see it, just compare Eq. (3.30) with Eq. (2.73) valid for a van der Waal s gas. For a gas with hardcore interactions only, i.e. a = 0, the second virial coefficient equals the excluded volume per molecule 6/Nl. Therefore, we may attribute the same meaning to the equivalent coefficient in Eq. (3.30). Our result thus indicates that polymer chains in solution behave like hard spheres, with the radius of the sphere depending on Rf, and additionally on z, i.e. on the solvent quality. For good solvents, in the Kuhnian limit h z —> 00) = 0.353, the radius is similar to iJp-... 

For good solvents as represented by the Kuhnian limit z oo we obtain... 

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