Screening of Excluded Volume Forces

In the previous chapter, we considered the structures of single chains in the dilute regime. Now we may inquire how these become altered in semidilute solutions. Discussions can be based on the pair distribution function of the individual chains, thereby focussing on the structure of single chains in states where chains overlap and interpenetrate. We choose for this intramolecular pair correlation function a symbol with a hat, g(r), to distinguish it from the general pair distribution function g(r), which includes monomers from all chains. 

For g(r), we can assess the behavior for both limits, dilute solutions and the melt. As explained earlier in Sect. 2.3.2, we find for isolated expanded chains g oc for distances in the range r Rp. On the other 

S q) can be measured if the dissolved polystyrene includes a small fraction of deuterated molecules. Due to the large difference in the scattering length of protons and deuterium, the deuterated chains dominate the scattering pattern, which then indeed may be described as 

For a dilute solution, one observes the scattering function of expanded chains, I oc which corresponds to the straight line shown previously in Fig. 2.18. 

Now we notice a change at low scattering vectors q, indicative of a cross-over from the scattering behavior of an expanded chain to that of an ideal one, with oc = (o / ) / . The cross-over occurs around a certain q, related to 4s by 7 4  

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The swollen fractal dimension is due to solvent around the big cluster.The solvent may consist of unreacted molecules and of smaller clusters, so that the excluded volume forces are screened on small scales but not on scales involving larger parts of the infinite cluster than a typical solvent cluster. Screening of excluded volume forces on the scale of distances between two crosslinks is due to the presence of unreacted chains. Thus one can assume that the chains between crosslinks are still Gaussian and = Df -h 2 is still valid. 

Envisage now the situation given in the melt. In contrast to an isolated polymer molecule, the monomer concentration here is constant. For the excluded volume force onto a monomer, it is irrelevant whether the contacting other monomers are parts of the same chain or of other chains. The determining quantity is the total concentration and the latter does not vary. Hence, no forces arise and the polymer chain does not expand. In the literature one often finds a particular formulation for addressing this effect. As the concentration gradient given for an isolated chain is compensated for by the presence of monomers from the other chains, one says that the latter ones screen the... 

A major difference of branched molecules from chain molecules is that more units are bound together and compressed into a very narrow space around the center of gravity. Hence, an immediate supposition is that in order for the monomer-monomer interaction to balance with the monomer-solvent interaction and the entropy force, and for the excluded volume effects to vanish, more attractive force between monomers are needed than is the case of chain molecules. Now we will focus our attention on concentrated systems such as non-solvent systems. An interesting idea is the influence of solvenf size on the osmotic pressure (screening effect) [19]. 

This produces a net force for all non-imiform density distributions so that for the bell-shaped distribution in Figure 1.3 there will be a net force of expansion of the chain. When the melt is considered, every chain is surrounded by a chain of the same type, so the concentration is constant in all directions (the dotted line in Figure 1.3). No distinction is drawn between repeat units on the same or different chains. (As noted above, there will be interpenetration of chains in all but dilute solutions.) The result is that there is no gradient in potential and there are no forces of expansion. In effect, the polymer chain in the melt behaves as if the forces of expansion due to excluded volume were screened from each chain and the dimensions are those for the unperturbed chain. 

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