Cox–Merz rules from general rigid bead-rod theory

The value of this work is in its macromolecular explanations of both Cox–Merz rules, thus of when to expect them to work. For polymeric liquids and their solutions, the measured values of the steady shear viscosity and the magnitude of the complex viscosity often equate, within experimental error, when compared at common shear rate (in units of $t − 1$⁠) and angular frequency (in units of $rad t − 1$⁠). Called the first Cox–Merz rule, this remarkable empiricism, with one exception, has defied most macromolecular explanations. This one exception is the suspension of multi-bead rods and its special case of rigid dumbbells. The second Cox–Merz rule equates approximately the slope of the first derivative of steady shear viscosity with respect to shear rate with the real part of the complex viscosity when compared at common shear rate (in units of $t − 1$⁠) and angular frequency (in units of $rad t − 1$⁠). In this paper, we explain both Cox–Merz rules for all axisymmetric macromolecules, be they prolate or oblate, of almost any lopsidedness. Furthermore, through the lens of general rigid bead-rod theory, we define under what conditions these rules do not apply. Specifically, the first Cox–Merz rule fails when the macromolecules are too oblate.