Competing Timescales Lead to Oscillations in Shear-Thickening Suspensions

Competing timescales generate novelty. Here, we show that a coupling between the timescales imposed by instrument inertia and the formation of interparticle frictional contacts in shear-thickening suspensions leads to highly asymmetric shear-rate oscillations. Experiments tuning the presence of oscillations by varying the two timescales support our model. The observed oscillations give access to a shear-jamming portion of the flow curve that is forbidden in conventional rheometry. Moreover, the oscillation frequency allows us to quantify an intrinsic relaxation time for particle contacts. The coupling of fast contact network dynamics to a slower system variable should be generic to many other areas of dense suspension flow, with instrument inertia providing a paradigmatic example.

Figure 1

Imposed-stress rheology. (a) Flow curves: dimensionless stress vs dimensionless average shear rate, ${\mathrm{\Sigma }}_E(\dot{\mathrm{\Gamma }})$, and absolute, ${\sigma }_E(\dot{\gamma })$, at given weight fractions, $\phi$. Lines: WC model for ${\phi }_m=0.457$, ${\phi }_{\mathrm{rcp}}=0.546$, $\beta =0.94$ ($\Rightarrow {\phi }_{\mathrm{DST}}=0.445$), and ${\sigma }^{*}=5.1\mathrm{Pa}$ at given $\phi$ (dashed) and $\phi =0.45$ (dotted). Blue, $\phi <{\phi }_{\mathrm{DST}}$; grey, ${\phi }_{\mathrm{DST}}<\phi <{\phi }_m$; red, $\phi >{\phi }_m$. Symbols: ${\mathrm{\Sigma }}_E$ vs time-averaged $\dot{\mathrm{\Gamma }}$ for cornstarch suspensions in a 50 wt.% glycerol-water mixture. Error bars denote standard deviation from three up-sweeps. (b)–(d) Time-dependent experimental shear rate, $\dot{\mathrm{\Gamma }}(t)$, for $\phi =0.48$, showing respectively: (b) steady flow below the onset of shear thickening, (c) periodic shear-rate oscillations, and (d) aperiodic flow at high stress. (e) Rheometric geometry: infinite plates, separation $h$, velocity $u$ ($\dot{\gamma }=u/h$), areal density ${\rho }_A$, and applied stress ${\sigma }_E$. (f) Experimental geometry: rotating plates, radius $R$, gap height $h$, relative angular velocity $\mathrm{\Omega }$ ($\dot{\gamma }=\mathrm{\Omega }R/h$), rotational inertia $I$ and applied torque ${\mathcal{T}}_E={\sigma }_E\pi R^3/2$. Equivalent ${\rho }_A=2I/\pi R^4$, using Eq. (5).

Figure 2

Limit-cycle behavior for Eqs. (6) and (7). (a) Phase-plane schematic for ${\phi }_{\mathrm{DST}}<\phi <{\phi }_m$. Red line, $\dot{\mathrm{\Gamma }}$ nullcline; red hatched shading, $g_1>0$; blue line, $f$ nullcline; blue shading, $g_2>0$. Fixed point, FP. On the black rectangle, trajectories point inwards, indicating the existence of a limit cycle if FP is unstable. (b) Critical stability criterion value, ${\epsilon }_c$, from Eq. (9). Solid black lines ${\epsilon }_c=0$; grey shading, shear jammed. (c) Limit cycle for $\mathsf{S}$-shaped flow curve, $\phi =0.455$ and ${\mathrm{\Sigma }}_E=3.0$, WC model parameters from Fig. 1. Black line, numerical solution for $\epsilon ={10}^{-9}$; shading as in (a). (d) Limit cycle for SJ flow curve, $\phi =0.475$ and ${\mathrm{\Sigma }}_E=3.0$, grey shading, shear jammed; other parameters and shading as in (c).

Figure 3

Tuning shear-rate oscillations for $\phi \gtrsim {\phi }_{\mathrm{DST}}$. (a)–(b) Low viscosity: cornstarch in a 50 wt.% glycerol-water mixture, $\phi =0.47$; relaxation oscillations. Red, experimental data; black, model: $\epsilon =2.7\times {10}^{-6}$, flow curve parameters from Fig. 2. Traces aligned by eye. (c)–(e) Medium viscosity: cornstarch in a 67 wt.% glycerol-water mixture, ${\eta }_s=15\mathrm{mPa}\mathrm{s}$ at $\phi =0.45\gtrsim {\phi }_{\mathrm{DST}}\approx 0.44$, damped oscillations in a narrow range of stress. (f) High viscosity: cornstarch in an 85 wt.% glycerol-water mixture, $\phi =0.44\gtrsim {\phi }_{\mathrm{DST}}\approx 0.44$, DST with no relaxation oscillations. (g) $4\mu \mathrm{m}$ silica spheres in an 87 wt.% glycerol-water mixture (${\eta }_s=151\mathrm{mPa}\mathrm{s}$ and $\epsilon =1\times {10}^{-4}$) at $\varphi =0.574\gtrsim {\varphi }_m=0.57$, DST with no large shear-rate oscillations. (h)–(i) Silica in dimethyl sulfoxide-water mixture (${\eta }_s=3.4\mathrm{mPa}\mathrm{s}$ and $\epsilon =2\times {10}^{-7}$) at $\varphi =0.58$, shear-rate oscillations.

Figure 4

Oscillation frequency, $\nu$, vs applied stress ${\sigma }_E$. Points: $\nu$ for cornstarch in a 50 wt.% glycerol-water mixture, from Fourier transform of 30 s upwards stress sweep (excluding first 2 s) in steps of 0.796 Pa: $\phi =0.47$ (square), $\phi =0.48$ (circle). Lines are model predictions for $\epsilon =2.7\times {10}^{-6}$ (see legend). Inset: effect of additional time-dependent relaxation on oscillation shape.