Concentration blobs and screening

To give another example we consider the density autocorrelation function Ida (q) defined as Fourier transform of the segment density correlations within a chain (cf, Eq. (5.17)). According to Eq. (5.24) this function obeys the sum rule 

Other examples can be discussed in the same way. We note that all these results follow from simple dimensional analysis in terms of coil radius Rg, chain concentration cp, and (if present) momentum variables. The method is based on the assumptions that 

The latter assumption distinguishes scaling theory from straightforward dimensional analysis in the continous chain model. 

We further note that we necessarily have to take cp as measure of the concentration. The segment concentration c = cpn depends on our definition of the segment and thus is no uniquely defined observable. 

Scaling theory also derives such results in another, more intuitive way, based on some heuristic picture of the internal structure of the polymer solution. Consider some piece of length nB within a chain of length n. It is natural to assume that this piece forms a subcoil, a blob1, of typical extension R, which scales like the coil radius for a polymer molecule of segments Rb nB- Thus the local density of segments due to the blob is estimated as 

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For s 1. the Tit blob is smaller than the whole chain and the blob-concept starts to make sense. For large overlap in view of screening the number of concentration blobs per chain should not be important. Thus iJ should reduce to a function of the blob concentration only. In view of Eq. (9-11) we therefore expect U to become a function of c independent of n. With this assumption the scaling law (9.2) yields... 

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. 

Note that as concentration increases, the sizes of the blob and of the chain decrease. In the bulk, we recover Floiy s results the interaction completely screened, the size of the blob is the step length, and the chain is ideal. Thus the present model ensures a gradual cross-over from... 

Thus, the size of the electrostatic blob and the numba of monomas per blob inaease with increasing salt concentration (deaeasing the value of the Debye screening length). 

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

Most often, g increases with concentration which atte.sts to the contribution to the interaction parameter of not only the closest neighbours in the lattice (cf. Equations 3.1-28 and 1.3-4). In terms of screening length and blobs (see subsection 3.1.1), it is equivalent to a decrease in the blob sizes with increasing concentration. 

Some of the questions to be addressed in Section 1 are i) what are excluded volume and the 0 point ii) why are polymer coils expanded due to excluded volume, and what limits that swelling iii) what is meant by a semi-dilute solution iv) what is screening v) how do screening distance, radius of gyration, and osmotic pressure vary with concentration in the semi-dilute regime vi) what are scaling laws and how are they used and vii) what are blobs ... 

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