Intrinsic Viscosity and Overlap Concentration

The overlap concentration can be estimated as the concentration at which the number of coils per unit volume, v, times the volume pervaded by a single coil, R, is roughly unity, where Rg = R )q /Vb is the radius of gyration of the polymer coil. Now v = cNa/M, where c is the mass per unit volume of polymer in solution and Na is Avogadro s number. Therefore, in a theta solvent (a solvent at a temperature where polymer excluded-volume interactions are negligible see Section 2.3.1.2), the overlap concentration, c, is given by 

Even below c, the viscoelastic properties of a solution vary somewhat from those of a dilute solution, due to occasional coil overlaps (Ferry 1980). To obtain true dilute-solution properties from measurements made at a series of low, but finite, concentrations, one can subtract from those measurements any solvent contribution and then extrapolate the results to zero concentration. The intrinsic steady-state shear viscosity, for example, is one such extrapolated quantity it is defined as 

The intrinsic viscosity can be related to the overlap concentration, c, by assuming that each coil in the dilute solution contributes to the zero-shear viscosity as would a hard sphere of radius equal to the radius of gyration of the coil. This rough approximation is reasonable as a scaling law because of the effects of hydrodynamic interactions which suppress the flow of the solvent through the coil, as we shall see in Section 3.6.1.2. The Einstein formula for the contribution of suspended spheres to the viscosity is 

Now c is defined as the concentration at which the spheres begin to overlap. We expect this to occur roughly when 4 0.40. Thus from Eq. (3-7), we obtain 

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Derive th following general relation between intrinsic viscosity and overlap concentration ... 

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