Large amplitude oscillatory shear (LAOS) in model colloidal suspensions and glasses: frequency dependence

Abstract

We investigate the effect of frequency on the non-linear large amplitude oscillatory shear (LAOS) response of concentrated colloidal suspensions of model soft and hard spheres at various concentrations, both below and above the glass transition. We show that the anharmonic response in the stress increases with frequency for liquid-like samples but decreases with frequency for solid-like samples. We argue that for samples below the glass transition, higher frequencies involving higher maximum shear rates promote shear thinning and increase anharmonicity. On the other hand, solid-like samples deform plastically at low frequencies as they are subjected to low shear rates within the period. Higher frequencies (higher average shear rates) lead to viscous flow over a larger fraction of the period thereby decreasing anharmonic behavior. We also demonstrate that LAOS experiments in strain-controlled rheometry at moderately high frequencies (ω > 5 rad/s) have to be very carefully interpreted, due to the superharmonic instrumental resonance effects.

Appendix

Oscillatory rheometry at high frequencies

We have measured the LAOS response of all samples at a relatively limited range of frequencies (0.1–5 rad/s). Measuring at higher frequencies is essential to enter the domain of non-linear viscoelasticity and gain a more complete understanding of the processes involved. For these samples, the stress-controlled Anton-Paar MCR 501 rheometer does not produce perfect sinusoidal large-amplitude deformations through the direct strain control (DSO) feedback loop, introducing additional artificial higher harmonics in the stress signal. Hence, in order to perform high-frequency (ω > 5 rad/s) LAOS measurements, a strain-controlled ARES rheometer with a 100FRTN1 force rebalance transducer was also used. However, as we will show below, the interpretation of non-linear data was not possible due to the transducer resonance effect that introduced higher harmonics even at moderate frequencies, above about 10 rad/s.

Dynamic time sweeps were performed at different amplitudes and frequencies on the solid-like star-like micelle samples. The normalized third and fifth harmonics, I₃/I₁ and I₅/I₁, are plotted in a Pipkin-type diagram (I_n/I₁ vs amplitude γ ₀ and frequency ω). The resulting contour plots are shown in Fig. 7. Both I₃/I₁ and I₅/I₁ show a well-defined peak in the Pipkin space. The third harmonic peak is at γ ₀ = 100 % and ω = 65 rad/s and the fifth harmonic peak at the same amplitude γ ₀ = 100 %, but lower frequency ω = 40 rad/s. Furthermore, at the peak, the non-linearity is extremely large; it reaches values of 48 % for the third harmonic and 24 % for the fifth which are quite unusual. In comparison, the non-linearity in the low-frequency regime saturates at I ₃/I ₁ = 28 % (Fig. 2c). The Lissajous figures for selected data points can be seen in Fig. 8 in a Pipkin representation. It is clear that the shape drastically changes at around ω = 60 rad/s. The maximum stress that follows strain reversal is dominating the response, and it is followed by a stress minimum close to maximum shear rate.

Fig. 7

Normalized amplitude of the third and fifth Fourier harmonics of the stress response of star-like micelles at ϕ _eff = 3.1. A peak in I_n/I₁ can be observed at an angular frequency of 60 rad/s in the case of the third harmonic and at 40 rad/s in the case of the fifth harmonic

Fig. 8

Pipkin plot showing Lissajous figures (stress vs strain) for selected data. The star-like micelle sample is at ϕ _eff = 3.1. Note the unconventional shapes at ω = 60 rad/s

In Fig. 9, we show the variation of the third, fifth, and seventh harmonic with frequency at an amplitude γ ₀ = 100 %. All three normalized harmonics can be fitted with a damped, driven harmonic oscillator equation:

$$ \frac{I_n}{I_1}=\frac{\alpha }{\sqrt{{\left({\omega}^2-{\omega}_0^2\right)}^2+{\beta}^2{\omega}^2}} $$

where ω ₀ is the resonance frequency, β is the damping parameter, and α is the normalized amplitude. The lines in Fig. 9 indicate least-squares fitting of the data with the resonance equation, which gives excellent agreement and a resonance frequency of 66, 40, and 28 rad/s for the third, fifth, and seventh harmonics, respectively. In the inset of Fig. 9, we show that the resonance frequencies are proportional to 1/n (where n is the number of the harmonic). This is typical of non-linear or superharmonic resonance effect which corresponds to resonance of the higher harmonics at integer fractions of the natural (or fundamental) frequency of a driven non-linear oscillator. Extrapolation of the line (inset, Fig. 9) to where 1/n is equal to one gives a fundamental resonance frequency of 200 rad/s.

Fig. 9

Fit of the normalized amplitude of higher harmonics. Lines represent fit using the damped, driven harmonic oscillator equation. Inset: resonance frequency ω ₀ vs 1/n, where n is the harmonic number

In order to verify this resonance effect, we also used PMMA hard spheres at ϕ = 0.6. A similar resonance peak is visible at a lower amplitude of 20 % but crucially at the same frequency of 65 rad/s. The fact that resonance appears at the same frequencies for samples as different as star-like micelles and PMMA hard spheres, as well as the unnaturally high values of non-linearity, points out to an instrumental origin of the non-linear resonance effect.

It is important to realize that the transducer of the rheometer has a feedback loop that keeps the measurement geometry in place, and the torque measured is just proportional to the reaction torque that drives the tool back to the original position. This feedback loop has a characteristic time constant in the tens of milliseconds range (Dullaert and Mewis 2005) and can thus be driven to resonance at frequencies around hundreds of rad/s. Thus, a fundamental frequency of 200 rad/s is consistent with the view that the superharmonic resonance effect is caused by the transducer and not by the sample.