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

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Original entry: Sujit S. Datta, APPHY 225, Fall 2009. $Insert formula here$

## 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 $\gamma_{0}$ and frequency $\omega$, 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 $G*(\omega)$. As with any response function, $G*$ has a real in-phase component and an imaginary out-of-phase component: $G*(\omega)=G'(\omega)+iG(\omega)$, where the elastic modulus $G'$ characterizes the energy stored in the material while the viscous modulus $G$ characterizes energy dissipation.

Somewhat surprisingly, a wide variety of soft materials consisting of densely packed, disordered, "soft" components seem to exhibit qualitatively similar $G*(\omega)$. In particularly, $G*$ tends to be fairly insensitive to frequency. As a function of strain, on the other hand, $G'$ and $G$ 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, $G(\gamma)$ 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 $\dot{\gamma_{0}}=\gamma_{0}\omega$. The time scale of this relaxation is phenomenologically given by $1/\tau(\dot{\gamma_{0}})=1/\tau_{0}+K\dot{\gamma_{0}}^{\nu}\sim K\dot{\gamma_{0}}^{\nu}$ when the intrinsic relaxation time of the material $\tau_{0}$ 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 $\dot{\gamma_{0}}$ (instead of constant $\omega$ or constant $\gamma_{0}$, as is typically done), one can test whether or not the timescale of the peak in $G$ 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 $\nu \sim 0.9-1$, in agreement with theoretical predictions.