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.


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


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.