Viscosity of low-molecular-weight chain liquids

Once the chains are long enough to reach the asymptotic characteristic ratio discussed in Chapter 2, it is found that the measured shear viscosity increases linearly with molecular weight for many polymer liquids. This phenomenon can be explained in terms of the dynamics of individual polymer chains as long as the system is below the entanglement limit. The basic theory is due to Rouse and is described in detail by Ferry.i  

The chain is modeled as a system of beads and springs undergoing Brownian motion in a viscous medium. The other polymer chains provide the viscous medium for any individual chain. The inherent dynamics can be represented in terms of N relaxation modes, where N is the number of statistical subunits in the chain. The shear relaxation modulus G(f) is given by  

There is a certain circularity in this relationship, since the shear viscosity is defined as the integral of the shear relaxation modulus. A useful relationship for the shear viscosity is in terms of the relaxation modulus and the average shear relaxation time  

This means that each molecule contributes N terms of to the relaxahon modulus. Each relaxation mode depends on the overall molecular weight of the chain to the first power, explicitly, and on the viscosity itself, which depends on the first power of the molecular weight. 

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