Semi-dilute solutions screening effects

A comparison with Burchard s first cumulant calculations shows qualitative agreement, in particular with respect to the position of the minimum. Quantitatively, however, important differences are obvious. Both the sharpness as well as the amplitude of the phenomenon are underestimated. These deviations may originate from an overestimation of the hydrodynamic interaction between segments. Since a star of high f internally compromises a semi-dilute solution, the back-flow field of solvent molecules will be partly screened [40,117]. Thus, the effects of hydrodynamic interaction, which in general eases the renormalization effects owing to S(Q) [152], are expected to be weaker than assumed in the cumulant calculations and thus the minimum should be more pronounced than calculated. Furthermore, since for Gaussian chains the relaxation rate decreases... 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

The same criterion does not apply in the semi-dilute region because, then, screening effects must be taken into account. Let us thus consider a semi-dilute solution of Brownian chains. The distance 0 between Brownian chains, the so-called overlap Brownian length , is given by (13.2.4) and, for d - 3,... 

Thus, the transition between poor and good solvent in a semi-dilute solution corresponds to a concentration C = C at which the screening effect begins to... 

In a semi-dilute solution, the interactions between polymers are screened but, if in a semi-dilute solution of monodisperse polymers, a very long polymer is introduced, the screening for this polymer cannot be complete. Indeed, for a scale of the same order as the size of the very long polymer, the semi-dilute solution may look practically homogeneous, but as was noticed by Flory51 and de Gennes,52 the very long chain is subjected to an excluded volume interaction our aim here is to calculate this effective interaction. 

Of course, in a semi-dilute solution, because of screening effects, the influence of the interactions on the swelling diminishes. In other words, in the vicinity of the tricritical point, a chain is nearly Brownian with reduced logarithmic corrections. In first approximation, we can deduce an expression of the swelling from (14.6.54) by using for h the value given by (14.6.69). In this way, we obtain... 

In entangled solutions screening becomes effective at a characteristic distance, called the screening-length , and denoted s- Figure 3.7 provides an experimental example of the evidence. We see the scattering intensity measured for a semi-dilute solution of polystyrene in CS2, in a plot of versus 5/3 interested in the single chain structure factor in the semi-dilute... 

Another important implication of the screening of the hydrodynamic interactions is that the effective friction of the monomers on the solvent is concentration dependent in a semi-dilute solution. The finite concentration of the surrounding monomers renormalizes the friction constant... 

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. 

Fig. 64. Single-chain behavior in semi-dilute PDMS/d-chlorbenzene solutions. Line-shape parameter (3 as a function of Q at the concentration c = 0.18 and c = 0.45, indicating the occurance of two crossover effects, as predicted by the concept of incompletely screened hydrodynamic interactions. (----), (---) asymptotic Zimm and Rouse behavior, respectively. (Reprinted with per-...

This point of view was too narrow. In fact, until 1970, experimentalists felt no necessity to study the structure function of polymer solutions and the screening effects which it reveals. As a consequence, the physicists have been deprived of the facts that could have induced them to conceive and recognize the existence of the semi-dilute state. On the other hand, a strong interest was shown in osmotic pressure, but always in the framework of the simple-tree approximation. As a consequence, great pains were taken to let the osmotic pressure in the semi-dilute range appear to be proportional to the square of the concentration. The dangers of such a preconceived idea are now well-known. 

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