# Strain-Rate Frequency Superposition: A Rheological Probe of Structural Relaxation in Soft Materials

Original entry: Sujit S. Datta, APPHY 225, Fall 2009.

## Reference

H. M. Wyss, K. Miyazaki, J. Mattson, Z. Hu, D. R. Reichman and D. A. Weitz, *Phys. Rev. Lett.* **98,** 238303 (2007).

## Keywords

Rheology, viscoelastic, glass

## Key Points

A common way of experimentally studying soft materials is using oscillatory rheology, in which the material being studied is subjected to a given oscillatory strain of peak amplitude <math>\gamma_{0}</math> and frequency <math>\omega</math>, and the resulting stress is measured, for example. For sufficiently small strain, the stress response is linear, with proportionality constant given by the complex shear modulus <math>G*(\omega)</math>. As with any response function, <math>G*</math> has a real in-phase component and an imaginary out-of-phase component: <math>G*(\omega)=G'(\omega)+iG*(\omega)</math>, where the elastic modulus <math>G'</math> characterizes the energy stored in the material while the viscous modulus <math>G*</math> characterizes energy dissipation.

Somewhat surprisingly, a wide variety of soft materials consisting of densely packed, disordered, "soft" components seem to exhibit qualitatively similar <math>G*(\omega)</math>. In particularly, <math>G*</math> tends to be fairly insensitive to frequency. As a function of strain, on the other hand, <math>G'</math> and <math>G*</math> exhibit some unusual, and seemingly universal features: for small strain, both are constant, while for very large strain, both exhibit characteristic power-law yielding. Just before this power-law decrease, <math>G*(\gamma)</math> shows a peak.

While a good deal of work has attempted to explain this ubiquitous dissipation peak in terms of various models, a phenomenological approach pioneered by Miyazaki and Reichman (2006) involves describing this as being characteristic of the glassy, disordered structure of the materials being studied. In particular, such a peak in dissipation can be thought of as representing a structural relaxation process driven by the shear rate <math>\dot{\gamma_{0}}=\gamma_{0}\omega</math>. The time scale of this relaxation is phenomenologically given by <math>1/\tau(\dot{\gamma_{0}})=1/\tau_{0}+K\dot{\gamma_{0}}^{\nu}\sim K\dot{\gamma_{0}}^{\nu}</math> when the intrinsic relaxation time of the material <math>\tau_{0}</math> is very large, as is characteristic of glasses.

The key achievement of the work by Wyss *et al.* is that it directly tests this prediction: if oscillatory rheological measurements are done with constant <math>\dot{\gamma_{0}}</math> (instead of constant <math>\omega</math> or constant <math>\gamma_{0}</math>, as is typically done), one can test whether or not the timescale of the peak in <math>G*</math> changes. Indeed, this is what Wyss *et al.* find for a range of soft glassy materials: a microgel paste, a densely-packed oil-in-water emulsion, an aqueous foam, and a densely-packed hard sphere colloidal suspension, with <math>\nu \sim 0.9-1</math>, in agreement with theoretical predictions.*