Semi-dilute solutions concentration dependence

Nevertheless, the three-body osmotic parameter h is not really a constant for a semi-dilute solution, it depends on the area S and on the concentration C... 

Taking into account the relevance of the range of semi-dilute solutions (in which intermolecular interactions and entanglements are of increasing importance) for industrial applications, a more detailed picture of the interrelationships between the solution structure and the rheological properties of these solutions was needed. The nature of entanglements at concentrations above the critical value c leads to the viscoelastic properties observable in shear flow experiments. The viscous part of the flow behaviour of a polymer in solution is usually represented by the zero-shear viscosity, rj0, which depends on the con-... 

For semi-dilute solutions, two regimes with different slopes are similarly obtained the powers of M, however, can be lower than 1.0 and 3.4. Furthermore, the transition region from the lower to the higher slope is broadened. The critical molar mass, Mc, for polymer solutions is found to be dependent on concentration (decreasing as c increases), although in some cases the variation appears to be very small [60,63]. 

The second position assumes that in semi-diluted solutions the polymeric chains are as much strong intertwined that the all thermodynamic values, in particular the osmotic pressure, achieve the limit (at N —>oc) depending only on the concentration of monomeric links, but not on the chain length. 

Accordingly to (19) the osmotic compressibility dlt / dc into diluted solutions does not depend on the concentration of macromolecules (dft / dc = RT) on the contrary, in semi-diluted solutions it becomes (as it follows from (25)) as linear function of relative concentration ... 

In that way, the thermodynamic approach with the use of conformational term of chemical potential of macromolecules permitted to obtain the expressions for osmotic pressure of semi-diluted and concentrated solutions in more general form than proposed ones in the methods of self-consistent field and scaling. It was shown, that only the osmotic pressure of semi-diluted solutions does not depend on free energy of the macromolecules conformation whereas the contribution of the last one into the osmotic pressure of semi-diluted and concentrated solutions is prelevant. 

Fig. 8.5 A sketch of the dependence R(c), where R is the size of the coil and c is concentration. Notice that concentration on the abscissa must be in the Log scale, otherwise the region of dilute solutions, c < c c A/-Y5 would look so narrow as not to be visible at all. The interval between c and c corresponds to semi-dilute solution.

This relationship can help us to learn about the viscosity of a polymer solution. In particular, we can use it to figure out how the viscosity depends on the number of monomer units At in a chain (in the limit N 1). Presumably, we need to know first how E and t depend on N. So we should venture a little investigation. Let s consider E and r separately, and concentrate on the case of a pol3Tner melt, to make it easier. In principle, the same sort of logic should apply to concentrated and semi-dilute solutions. 

At describing the viscosity properties of diluted solution one usually proceeds from the linear dependence of an increment in viscosity on the polymer solution concentration. However, in the case of polar polymers to which CHT belongs there is a possibility of the occurrence of reversible agglomeration process which can take place not only in the area of semi-diluted solutions but even in the area of diluted ones. In this case the contribution to viscosity is made not by separate particles with V volume but by their aggregates whose volume V(n) depends not only on the number of particles constituting it, but also on their density characterized by fraction dimensions D [3] ... 

The size of the blob ranges from the size of monomers to the whole chain, depending upon polymer concentrations. Therefore, the dynamic scaling law for the single short chain in the semi-dilute solutions is to insert between To and Xr. In other words, the 2/3 scaling segment is inserted before the 1/2 segment, as illustrated in Fig. 5.4. 

The aim of the theories developed for polymer solutions has been to explain experimental observations such as deviations from Raoult s law and the molecular weight dependence of solubility at any given temperature. Indeed, it is necessary to account for the phase diagram for polymer solutions in general. The more recent theories deal with semi-dilute and concentrated solutions. No single theory currently explains ail the experimental observations for polymer solutions. Some of the more significant are [8-16] ... 

The zero shear viscosity scales with Nf" to contrast Af dependence for isotropic polymers [20] So far, we have examined the dynamics of rod-Uke macromolecules in isotropic semi-dilute solution. For anisotropic LCP solutions in which the rods are oriented in a certain direction, the diffusion constant increases, and the viscosity decreases, but their scaling behavior with the molecular weight is expected to be unchanged [2,17], Little experimental work has been reported on this subject. The dynamics of thermotropic liquid crystalline polymer melts may be considered as a special case of the concentrated solution with no solvent. Many experimental results [16-18] showed the strong molecular weight dependence of the melt viscosity as predicted by the Doi-Edwards theory. However, the complex rheological behaviors of TLCPs have not been well theorized. 

In a semi-dilute solution, each macromolecule can be described by a series of unperturbed sections, separated by regions tangled with other chains. This has been schematised in Fig. 3.10, in which the chains mark out a 3-dimensional lattice the nodes of the lattice represent points where the chains tangle. The lattice fluctuates in space and time and the mean mesh size clearly depends on the polymer concentration Cp. In fact. 

Eq. (VII. 11) is the basis of the photon beat studies at long wavelengths, summarized in Section III.2.2. Our aim in this section is to generalize eq. (VII. 11) to semi-dilute solutions. We show that there still exists a coefficient Dcoop (< )> (dependent on concentration) which describes the relaxation of concentration fluctuations. The reason for the subscript "coop" (cooperative) is explained below. 

Figure 3.9 presents experimental results obtained from small angle X-ray scattering ( SAXS -) experiments on semi-dilute solutions of polystyrene in toluene, choosing three different concentrations. They agree with Eq. (3.62) and enable a determination of the concentration dependence of to be made. 

Thus, interpretation of the experimental evidence on adsorption from mixture is complicated by the thermodynamic quality of the solvent, and for either pol3rmer, it differs from that for the other pol3rmer and varies with the concentration of the latter. If the problem of adsorption of mixtures is approached on the basis of the concept of a dilute and semi-dilute solutions, then it is obvious that conditions of aggregation and adsorption should also depend on the critical concentration of overlapping of coils, i.e., on the concentration at which coils of the component macromolecules start touching and overlapping one another. It is clear that the values C for individual components in the solvent cannot be used here, since in the solution of a mixture, the thermodynamic quality of the mixed solvent differs from that of the pure solvent and depends on the concen-... 

The semi-dilute regime constitutes a sizeable concentration range vl V2 1, where u is the volume fraction corresponding to the concentration c. The semi-dilute solution case is in fact the simplest to describe. Molar mass effects are highly suppressed but still the solution is diluted. The solution activity and the osmotic pressure are independent of molar mass but dependent on the distance between the entanglement points in the loose network. The latter is controlled simply by the volume fraction of polymer V2), and the osmotic pressure is given by des Cloiseaux law ... 

Early MC-simulations studied successfully the cross-over regime from dilute to semi-dilute solutions of charged rigid rod-like polyions [80, 81]. In highly diluted solution (Cp < c ) the rod-like particles resemble pointlike objects and the solution properties are similar to those of charged spheres. With increasing concentration the distribution function g(r, u, U2) increasin y depends on the... 

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