Theories of Entanglement Friction

The combined viscosity and diffusion measurements on concentrated systems by Bueche et al. (33), were described in Section 2. The results suggest that viscosity behavior above Mc can be explained simply in terms of an enhanced Stokes friction for the molecules. Bueche (7, 224) suggested that since polymer molecules in concentrated systems are looped through the coils of neighboring molecules, relative motion must be resisted by the need for these loops to slip around one another. Each molecule is assumed to participate in an average of E such temporary couples, creating (for E 1) a temporary network 

The induced motions in the surroundings are estimated by examining the effects of a systematic velocity v imposed on one of the molecules, called here the central molecule. The couples of the central molecule lead to JV, other molecules which move in response to the motion of the central molecule. Likewise, the Ni first order partners are coupled to N2 second order partners, and so on through a hierarchy of orders order / contains Nj molecules, coupled for the first time to the central molecule through a sequence of / couples. The induced velocity of each order is assumed to be a constant fraction s of the velocity of the preceding order. The formal expression for the force resisting the motion of the central molecule becomes therefore  

Redundancy (multiple coupling between molecules or between members of different orders, coupling between molecules of the same order, etc.) reduces Nj below its maximum possible value, ( — If -1. Bueche asserts that the number of effective /-order couples is given by  

The product v is the total number of coupling units per unit volume in the system and Qj(r) is the concentration of potential /-order coupling sites at a distance r from the central molecule  

Bueche s result (for E l, and with some harmless approximations to simplify integration) is  

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