Concentration, overlap

Let us first find the overlap concentration for various temperature regions. In the high-temperature region where Flory s law R = Rp = ax l n l holds, we find from (2.163) that 

The overlap is indicated by the symbol. The superscript indicates that the property is in the high-temperature region. For high-molecular weight polymers, the number n of the repeat units is so large that the overlap concentration is small. For example, it is approximately 4 = 0.1% for n = 10 .  

Change in the structure of a polymer solution with varied concentration. 

Alternatively, the overlap concentration can be defined as the concentration at which the number of chains contained in the volume occupied by one random coil chain is just one. If R is considered to be the hydrodynamic radius / h, this number is 

Similarly, in the theta region where R = Rg= arO-/ holds, we find 

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In HOPC, a concentrated solution of polymer is injected. The concentration needs to be sufficiently higher than the overlap concentration c at which congestion of polymer chains occurs. The c is approximately equal to the reciprocal of the intrinsic viscosity of the polymer. In terms of mass concentration, c is quite low. For monodisperse polystyrene, c is given as (4)... 

Overlap concentration in benzene calculated from the external diameter in benzene. 

Let us remark that relation (6) is given for polymer concentration c lower than the critical overlapping concentration c above which higher terms in c must be considered. In fact, the concentration practically used ( around 10 3 g/cm3) corresponds to the semi-dilute regim for which the behavior is not well known in the case of polyelectrolytes. We have however kept relation (6) by introducing for K a mean apparent value determined from our experiments ( K - 1 )... 

The relevant part of the phase diagram (x > 0) is shown in Fig. 38. The c-x-plane is divided into four areas. The dilute regime I and I are separated from the semi-dilute regimes III and II, where the different polymer coils interpenetrate each other, by the so-called overlap concentration... 

Efforts at synthesis and studies of temperature-dependent solution behaviour of these chemically hydrophobized polyacrylamides are now in progress. However, it is reasonable to point out that in this case, contrary to the hydrophilization of the hydrophobic precursor, the problems associated with additional swelling of the globular core (as the modification proceeds) are absent however, the problem of the choice of working concentration for the precursor is still present since above the coil overlapping concentration the intermolecular aggregation processes at elevated temperatures can compete with the intramolecular formation of core-shell structures. 

A rough measure for the overlap concentration 4>ov is that volume fraction of polymer at which close-packed spheres with radius rg just touch. Then 4> =0.74 rl /(4Trr /3). Taking r — A((J> - 0) ... 

This means that the expansion factor depends only upon the end-to-end length R2(Ms) of chains of molecular weight Ms relative to the chain in 6 conditions. We can derive an effective concentration of the chains since this is related to the overlap concentration of the polymer of molecular weight Ms. ... 

Non-dilute solutions also allow for theoretical descriptions based on scaling theory [16, 21]. When the number of polymer chains in the solution is high enough, the different chains overlap. At the overlapping concentration c , the long-scale density of polymer beads becomes uniform over the solution. Consequently c can be evaluated as... 

Tbe discussion of the semi-chlute properties remains confined mainly to the osmotic modulus which in good solvents describes the repulsive interaction among the macromolecules as a function of concentration. After scaling the concentration by the overlap concentration c = A2M.Yf) and normalizing the osmotic modulus by the molar mass, universal masteS" curves are obtained. These master curves differ characteristically for the various macromolecular architectures. The branched materials form curves which lie, as expected, in the range between hard spheres and flexible linear chains. 

Branched polymers can also be dissolved at fairly high concentrations. Because of the higher segment density in the isolated macromolecules the overlap concentration will also be increased. For this reason the semi-dilute regime of branched polymers may in some cases be larger than for linear chains, say about 20% or more. Clearly, however, a full interpenetration, as was assumed for flex-... 

Most important, however, was the discovery by Simha et al. [152, 153], de Gennes [4] and des Cloizeaux [154] that the overlap concentration is a suitable parameter for the formulation of universal laws by which semi-dilute solutions can be described. Semi-dilute solutions have already many similarities to polymers in the melt. Their understanding has to be considered as the first essential step for an interpretation of materials properties in terms of molecular parameters. Here now the necessity of the dilute solution properties becomes evident. These molecular solution parameters are not universal, but they allow a definition of the overlap concentration, and with this a universal picture of behavior can be designed. This approach was very successful in the field of linear macromolecules. The following outline will demonstrate the utility of this approach also for branched polymers in the semi-dilute regime. 

So far, the effects of semi-dilute solutions are qualitatively clear. Ambiguity comes in, however, when the overlap concentration has to be defined quantitatively. This ambiguity arises from the fact that the volume of a macromolecule cannot be uniquely defined. Because of the segment mobility the shape of a macromolecule varies in time such that only a statistical description can be made. As... 

The differences in the overlap concentrations are instructively demonstrated by Fig. 31 for amylopectin fractions [144], which are representatives for hyper-branched structures. 

Fig. 31. Molar mass dependencies of four differently defined overlap concentrations c (ml/g) for amylopectin fractions in O.Smol/lNaOH [144]. Reprinted with permission from [144]. Copyright [1996] American Society...

In order to resolve these challenges, it is essential to account for chain connectivity, hydrodynamic interactions, electrostatic interactions, and distribution of counterions and their dynamics. It is possible to identify three distinct scenarios (a) polyelectrolyte solutions with high concentrations of added salt, (b) dilute polyelectrolyte solutions without added salt, and (c) polyelectrolyte solutions above overlap concentration and without added salt. If the salt concentration is high and if there is no macrophase separation, the polyelectrolyte solution behaves as a solution of neutral polymers in a good solvent, due to the screening of electrostatic interaction. Therefore for scenario... 

Thus in salt-free semidilute solutions, the fast diffusion coefficient is expected to be independent of both N and c, although the polyelectrolyte concentration is higher than the overlap concentration. This remarkable result is in agreement with experimental data [31, 33, 34] discussed in the Introduction. Upon addition of salt, Df decreases from this value as given by the above formulas. 

The crossover between the Kirkwood-Riseman-Zimm behavior and the Rouse behavior requires a better understanding, in terms of the contributing factors for the occurrence of a maximum in the plot of reduced viscosity against polyelectrolyte concentration at low salt concentrations. A firm understanding of the structure factor of polyelectrolyte solutions at concentrations comparable to the overlap concentration is necessary. 

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