Concentration blob

Equation (22) has been confirmed by a variety of techniques including neutron scattering, dynamic light scattering, and osmotic pressured measurements [23]. As concentration increases the concentration blob decreases in size until the Kuhn length is reached and the coil displays concentrated or melt Gaussian structure. The coil accommodates concentrations between the overlap and concentrated through adjustment of the concentration blob size. 

For the tensile blob, thermal blob, and concentration blob we find that the coil accommodates external stress (thermal, concentration, or force) through a scaling transition that leads to two regimes of chain scaling. This directly impacts the free energy of the chain, the mechanical response, and the coil size. 

We should immediately note that the picture described here in the context. of scaling or renormalization theory leads to the important, concept of concentration blobs. (See Sect. 9.1.)... 

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... 

The concentration blob model of scaling theory has been introduced in [DCF+75, Far7G], where it has been shown to qualitatively explain a lot of experiments, A comprehensive review is given in [dG79]. 

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. 

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... 

When polymer chains overlap, they take conformations that are different from those of isolated individual chains because monomers interact in a different way. Figure 2.22 schematically shows the conformation of a polymer chain in a concentration well above the overlap concentration. It has a structure like a pearl-necklace a train of blobs (called a concentration blob) made up of groups of monomers connected in sequence. The size of each concentration blob is called the correlation length f. The monomers in the blob are directly in contact with the solvent so that they swell by the excluded-volume effect, in the same way as an isolated random coil does. 

Let be the number of monomers in a concentration blob. Since the number density inside the blob must be equal to the concentration (p/a of the solution, we find... 

The chain section of length f is unperturbed by interactions with other chains, that is, obeys SAW statistics and comprises g (if/fl) monomer units. The section of the chain of length g constitutes a concentration blob. 

The radial decay in polymer density corresponds to a radial decrease in local stretching of its arms, dr/dn -/[r c(r)]". At the same time, the local stretching of the branches controls the elastic tension t and, thereby, the size of the elastic blob, feiastic-feBT/tSfl(adn/dr) V /( l Within the blob approach, feiastic(r)-f(r). Hence, the radial increase in the size of the concentration blob i r) ensures also the decrease in local tension in the arms of the star, tlkeT-f lr. 

The main line of criticism for the inverse Daoud and Cotton model concerns the fact that the concentric blob structure does not represent the most stable state of the concave brush. On the other hand, the assumption that all chain ends are positioned at the end of the layer in the Sevick model is obviously not accurate and tends to become completely unjustified as we move closer to the confined brush regime. 

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