StrainRate Frequency Superposition (SRFS)
A new approach to oscillatory rheology for soft materials
Hans Wyss, Kunimasa Miyazaki, Johan Mattsson, Zibing Hu, David Reichman, David Weitz
Soft materials such as suspensions, emulsions, or foams are of tremendous importance in a variety of industrial applications. While many of these applications rely on accurate control of the complex rheology of these systems, the physical mechanisms that determine this rheological behavior are still not fully understood.
Surprisingly, the rheological behavior of almost all of these soft materials are very similar, despite the fact that the colloidal building blocks that make up these materials have very different mechanical properties. These similarities exist in both the linear and the nonlinear viscoelastic response: Rheological hallmarks of soft materialsLinear viscoelastic behavior
Typical linear viscoelastic response as a function of frequency. Both the storage and the loss modulus depend only weakly on frequency. Within the accessible frequency range, the storage modulus is larger than the loss modulus. Often, a minimum in the loss modulus is observed. Nonlinear viscoelastic behavior
Typical nonlinear viscoelastic response as measured in a "strain sweep" at fixed oscillation frequency as a function of strain amplitude. The soft materials we are studying show a distinct yielding behavior, where with increasing strain amplitude the loss modulus rises towards a pronounced peak before decreasing again at the largest strains. The storage modulus, however, decreases monotonically with increasing strain amplitude.
Origin of the peak in the loss modulusWe show that the linear and the nonlinear rheological response of soft materials can be described in a single physical picture. The viscoelastic response of these systems is dominated by slow structural relaxation processes that mostly occur at time scales to low to be accessed in conventional oscillatory measurements. However, it is known that the time scale of such a relaxation process will depend on an applied strain rate. When the system is subjected to shear, it becomes easier for the system to relax and the relaxation time becomes faster. We could expect for instance the following behavior for the strain rate dependent time scale $\tau(\dot{\gamma})$:$$\frac{1}{\tau(\dot{\gamma})}=\frac{1}{\tau_0} + K \dot{\gamma}^\nu$$ , where $\tau_0$ is the natural relaxation time at rest, $K$ is a prefactor, and the exponent $\nu$ is close to unity. At low strain rates the relaxation time remains at its natural time scale, while at very large strain rates the relevant time scale is given by the applied strain rate. 
Colloidal glass (Mason 1995)
Oilinwater emulsion
Aqueous foam (Gillette Foamy) 
The following movie schematically illustrates the influence of strain rate on the viscoelastic response and the reason why a peak in the loss modulus is observed in a strain sweep at constant frequency (shown on the right hand side of the movie): 
"Constant rate frequency sweep" Expected viscoelastic response at increasing levels of shear rate (corresponding to the strain rate applied in the strain sweep on the right). 
"Strain sweep" Strain sweep performed at a fixed frequency of 1 rad/s (marked as a red line in the frequency sweep on the left). As the applied strain amplitude is increased, the amplitude of strain rate increases accordingly, causing a shift in the structural relaxation time. The observed peak in the loss modulus is a direct consequence of the structural relaxation process and its dependence on strain rate. 
StrainRate Frequency Superposition (SRFS)We take advantage of the strain rate dependence of the structural relaxation time and use it to study and understand the linear and nonlinear viscoelastic behavior of a wide range of soft materials.
(a) 'Constantrate frequency sweeps'The natural scale of strain rate in oscillatory measurements is the strain rate amplitude, the product of the strain amplitude and the oscillation frequency. We study the influence of strain rate on the viscoelastic response experimentally by performing frequency sweeps at constant strain rate; as the frequency is increased, we decrease the strain amplitude in order to maintain a constant amplitude of the strain rate. By performing a series of these constant rate frequency sweeps at different strain rates, we can study the dependence of the slow structural relaxation process on the strain rate amplitude. 
from low frequency and high amplitude to high frequency and low amplitude.. 
(b) Measured viscoelastic response as a function of strain rate amplitudeWhile the shape of the viscoelastic response remains surprisingly similar, the features of the response are shifted towards higher and higher frequencies, as the applied strain rate amplitude is increased. The inset shows the applied strain amplitude as a function of frequency. The strain rate increases from the grey to the green, red, and blue curves. 
Series of constant rate sweeps at different strain rates: The system used here is a suspension of hydrogel particles which consist of an interpenetrated network of PolyNIPAM and PolyAcrylicAcid. 
(c) Master curve of all constant rate measurementsTo show the similarities in the shape of the viscoelastic response, we superimpose all curves measured at different strain rate amplitudes onto a single master curve. The curves are shifted in both amplitude and frequency. The inset shows the amplitude shiftfactor and the frequency shiftfactor as a function of strain rate. While the amplitude shiftfactor a changes only by a factor of two over 4 order of magnitude in strain rate, the frequency shiftfactor b shows the behavior expected for the strain rate dependence of the structural relaxation time: At low strain rates we observe a plateau, where the structural relaxation time is essentially independent of strain rate. At high strain rates the shiftfactor b increases as a power law with an exponent close to unity. This indicates that here the time scale is entirely determined by the applied shear, scaling as the inverse of the strain rate amplitude. 
Master curve of constant rate measurements: The good overlap shows that the shape of all curves taken at different strain rates remains essentially unchanged. The inset shows the amplitude and frequency shiftfactors a and b as a function of strain rate amplitude. The frequency shiftfactor b is a direct measure of the rate dependence of the structural relaxation time. 
(d) Contributions to the viscoelastic response that are independent of strain rateThe good overlap in the above master curve shows that the shape of the viscoelastic response does not depend strongly on the strain rate amplitude.Nevertheless, deviations from the master curve are observed at the higher end of the frequency spectrum. It turns out that these deviations are in fact very systematic  the curves initially measured at the highest strain rates show the strongest deviations. The points that deviate were thus initially (before shifting) measured at the largest oscillation frequencies. This suggests that there could be a contribution to the viscoelastic response at high frequencies that does not depend on the applied strain rate amplitude. In the case of the studied system of hydrogel particles, this contribution scales as the square root of the oscillation frequency. 

(e) We can isolate the response due to structural relaxationTo isolate the response that is due to structural relaxation, we subtract the contributions that are independent of strain rate (the highfrequency linear response) from the constant rate measurements, as shown on the left side of the figure figure at the right.The shape of all curves is now remarkably similar, as shown by the right side of the figure, where all curves are superimposed on a master curve which now reflects the shape of the viscoelastic response that is due to structural relaxation. 

Our results suggest that we can access the shape of the structural relaxation of soft materials from nonlinear viscoelastic measurements, even if this relaxation is too slow to be accessed by conventional oscillatory measurements.
Moreover, the observed scaling suggests that the linear and the nonlinear viscoelastic response of soft materials can be described within a single physical picture. Our approach thus allows us to:
We are currently studying a variety of systems, including foams, emulsions, and hard sphere suspensions.
