## Tuesday, December 13, 2011

### A Derivation of the Cox-Merz Rule

For reasons I've never understood, the most common search term that consistently, year-in, year-out brings people to this blog is "Cox-Merz rule" or some variation thereof. So I'm going to add fuel to the fire with some new research results about this important rheological relationship.

The Cox-Merz rule is a empirical observation that h($\dot \gamma$) = h*(w) when $\dot \gamma$= w.

The importance of this is that it allows a tremendous amount of lab work to have significance in non-lab (production) settings. In the lab, viscosity is normally measured by imposing sinusoidal stresses or strains on a sample and observing the mechanical response. Sharp observers will question how a sinusoidal deformation can be useful in a production setting which has unidirectional flow (or something close to it). The Cox-Merz rule is able to jump that gap - as long as the magnitude of the oscillatory frequency matches the magnitude of the steady-state shear rate, then the viscosities will be the same. This was first observed empirically and holds for many many materials, thus greatly simplifying the equipment and time spent in the lab.

But once one learns of the rule, the questions then arise as to when it won't work. Again, empirical observations have shown to be cautious with "networky" type materials such as filled polymers or systems with complex hydrogen bonding for example.

David Mead has a fine magnum opus published in Rheologica Acta (free access until the end of the year!) in which he is able to derive the Cox-Merz rule for polydisperse materials. This has been previous done for the idealistic case of monodisperse polymers, so this is a real advancement. The kicker however, is the last line in the article (actually, the last line in the Appendix):
"The Cox-Merz rule is effectively a 'suspicious' coincidence' based on identical dimensionless frequency transitions and correct asymptotic scaling behavior rather than anything predicated on fundamental polymer physics."
All this work and our basics thoughts aren't changed.

Mead, D. (2011). Analytic derivation of the Cox–Merz rule using the MLD “toy” model for polydisperse linear polymers Rheologica Acta, 50 (9-10), 837-866 DOI: 10.1007/s00397-011-0550-5