Molecular Weight Dependence of Viscosity of Polymers Melts

Abstract

A new approach to problems of the so-called inter-chain relaxation is presented and the dependence of viscosity of polymer Melts on molecular weight, M, is discussed. In the measurement of steady viscosity, the time scale of observation is generally long. Hence the intra-chain motion with shorter relaxation times will relax almost completely and reach a state of dynamic equilibrium, while the inter-chain relaxation mechanism of longer time scale may still be effective. In the present theorry, each chain molecule is expressed only by the position of its center of mass and its velocity. The intermolecular force between the two molecules is assumed to be the sum of two kinds of force: the one is a force derived from potential like that in simple liquids, and the other is a frictional force due to the entanglements between the two molecules. On the basis of the transport theory of liquids by Born and Green, the general expression of viscosity is derived from the fundamental equation for the velocity distribution function of a molecule. The expression is factorized into some factors such as mutual potential, friction constant.
The M dependence of each factor is roughly estimated under the following assumptions; (i) each chain molecule is composed of N freely orienting segments, (ii) the spacial distribution of segments of a molecule is Gaussian about its center of mass since an applied velocity gradient is very small, (iii) the mutual potential energy between the two segments is approximated by a three dimensional square well potential, (iv) the effective friction constant between the two molecules is proportional to the co-volume occupied in common by them, and (v) the equilibrium distribution function for the two molecules is approximated in a simple form.
In conclusion we obtain the viscosity η as follows:
η=AN+BN³,
where A and B are constants independent of N, and it is expected that A>>B. This theoretical prediction agrees semi-quantitatively with the experiment.