The role of rigidity constraints in the rheology of dilute polymer solutions

This paper investigates a class of bead–rod and bead–spring models which have been proposed to describe the dynamics of an isolated macromolecule in a flowing solution. Hassager (1974 a ) has pointed out a surprising result in regard to these models: the statistical conformation of a molecule (and hence its influence on the flow) apparently depends upon whether a very stiff-springed model structure or a rigid one is used. This paradox is examined and resolved. It is shown that a unique answer is obtained by regarding the system as the classical limit of a quantum-mechanical one. The extent of the quantum influence can be characterized by a dimensionless group Q . For a ‘hot’ or ‘large’ system (for which Q → 0) the classical (stiff spring) results are recovered. The effects of the parameter Q on the size of the molecules and the rheology of the solution are calculated in detail for a simple model, and the gross features are identified for a more realistic Rouse chain model, each in both weak and strong flows. A final section considers weak, rapidly varying flows. It is shown that, within the context of classical (non-quantum) physics, for sufficiently rapid changes any model structure will tend to move with the applied flow, and therefore exert no stresses on the fluid. This explains the theoretical observation of Fixman & Evans (1976) that, in regard to the particle stress, the limits of rigidity and infinitely high frequency do not commute.