For the Fraenkel dumbbell , where L is the length of the rigid dumbbell when . For the solution of Fraenkel dumbbells, the thermal conductivity may become arbitrarily large, when the spring is "tightened up." It is found [5f] that the major contribution to the thermal conductivity in dilute solutions is that of . We can make a similar comparison for the zero-shear-rate rheological properties:
Hooke:Fraenkel: (3.7; 3.8)
Hooke:Fraenkel: (3.9; 3.10)
Inasmuch as the product is independent of H, it is apparent that "tightening up" the springs in the Fraenkel model will have no effect on the rheological properties of the solution.
The thermal conductivity of the dilute solution of Rouse chains has also been worked out (not a simple problem) and one finally gets for a chain of N beads in a solvent at rest:
The analogous problem for a chain with Fraenkel springs has not been worked out.
The energy equation can be written in terms of temperature, and this is a simple exercise for Newtonian fluids (BSL, p. 337). For a solution of Rouse chains,