Effects of Domain Overlap

At finite concentrations, the frequency dependence of G and G — (aris can be examined directly without scaling the coordinates as in preceding figures. However, measurements at different temperatures may be combined by reduction to a standard temperature To G and G — coris are multiplied by Toco/Tc and u is multiplied by a shift factor aj which is given by rjo Vs)ocT/ t]o — Vs)coTq. Here r]o is the steady-flow (vanishing shear rate) viscosity and the subscript 0 otherwise refers to the reference temperature. This is the method of reduced variables which will be discussed fully in Chapter 11 with explanation of its rationale, and affords an extension of the effective frequency range. 

Examples of experimental data at finite concentration are shown in Fig. 9-21 for three polystyrene solutions with sharp molecular weight distribution. In 1, the curves for a sample with M = 267,000 and c = 0.0286 g/cm conforms rather closely in shape to the Zimm theory (c/. Fig. 9-8-II). In II, the same polymer at c = 0.124 conforms more nearly to the Rouse theory cf. Fig. 9-8-1). In III, a sample of higher molecular weight (1,700,000) but the same concentration as I also conforms rather closely to the Rouse theory. 

The results illustrate principles which have been observed repeatedly. At low finite concentrations, the form of the frequency dependence approaches the Zimm theory, especially in a 0-solvent. At somewhat higher concentrations, the shape of the frequency dependence changes progressively toward the Rouse theory. The higher the molecular weight, and the better the solvent, the lower the concentration at which the change in behavior is accomplished. 

A different index of the change from Zimmlike to Rouselike behavior with increasing concentration can be obtained from the limiting behavior at low frequencies. This does not involve differences in frequency dependence, since G and G — uvs are proportional to and co respectively in this region for all theories instead, it depends on the magnitudes of the respective proportionality constants, which are Ao and vo Vs- In connection with Table 9-II, it was noted that the parameter 52/5f varies from 0.2 to 0.4 for the change from free draining to dominant hydrodynamic interaction. For finite concentrations this parameter can be identified with 

De Gennes and Adam and Delsanti have discussed the properties of solutions in the concentration range of coil overlap in terms of a critical concentration c which is defined as M/Aoi where s is the radius of gyration and A o is Avogadro s number. This is similar in magnitude to the concentration defined by c[r)] = 3. Above the c, the solution is termed semidilute. 

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