Interparticle Friction Leads to Nonmonotonic Flow Curves and Hysteresis in Viscous Suspensions

Hysteresis is a major feature of the solid-liquid transition in granular materials. This property, by allowing metastable states, can potentially yield catastrophic phenomena such as earthquakes or aerial landslides. The origin of hysteresis in granular flows is still debated. However, most mechanisms put forward so far rely on the presence of inertia at the particle level. In this paper, we study the avalanche dynamics of non-Brownian suspensions in slowly rotating drums and reveal large hysteresis of the avalanche angle even in the absence of inertia. By using microsilica particles whose interparticle friction coefficient can be turned off, we show that microscopic friction, conversely to inertia, is key to triggering hysteresis in granular suspensions. To understand this link between friction and hysteresis, we use the rotating drum as a rheometer to extract the suspension rheology close to the flow onset for both frictional and frictionless suspensions. This analysis shows that the flow rule for frictionless particles is monotonous and follows a power law of exponent $\alpha =0.37\pm 0.05$, in close agreement with the previous theoretical prediction, $\alpha =0.35$. By contrast, the flow rule for frictional particles suggests a velocity-weakening behavior, thereby explaining the flow instability and the emergence of hysteresis. These findings show that hysteresis can also occur in particulate media without inertia, questioning the intimate nature of this phenomenon. By highlighting the role of microscopic friction, our results may be of interest in the geophysical context to understand the failure mechanism at the origin of undersea landslides.

Popular Summary

A layer of grains starts flowing at an angle larger than that at which it stops flowing. This hysteresis of the flow onset is a major feature of granular materials such as sand and rocks, with implications in catastrophic phenomena such as earthquakes or landslides. However, despite years of investigations on the physics of granular flows, the origin of hysteresis remains mysterious. Here, we experimentally show that hysteresis arises from friction between particles.

So far, researchers have mainly investigated hysteresis in granular materials that are above ground. The general belief was that inertial effects, such as shocks between grains or self-induced acoustic noise, were at the origin of this phenomenon. To address this hypothesis, we investigate the flow onset of submarine granular avalanches in which inertial effects are completely negligible. Remarkably, we find that large hysteresis of the avalanche angle can still be observed under such conditions, ruling out the role of inertia.

To elucidate the origin of hysteresis, we turn to a specific suspension of silica microparticles, which allows us to tune the interparticle friction coefficient. While we observe large hysteresis when the grains are frictional, we find that hysteresis completely disappears when there is no friction between particles.

Our results show that hysteresis in particulate media stems from microscopic friction, while inertia is not required. These findings provide new clues to understand the mechanisms at the origin of hysteresis in granular media. They also open the door to describing, in a unified framework, both above-ground and submarine landslides.

Figure 1

Evidence of hysteresis in overdamped granular suspensions: (a) Sketch of the experimental setup. (b) Picture of the large frictional glass beads ($d=490\pm 70\mu \mathrm{m}$, ${\rho }_p=2500\mathrm{kg}·{\mathrm{m}}^{-3}$). (c) Angle of avalanche $\theta$ versus time for large glass beads immersed in pure water ($\mathrm{St}=4$) or in a mixture of water and Ucon oil ($\mathrm{St}=6\times {10}^{-2}$); the rotation rate of the drum is $\omega =5\times {10}^{-3}°·{\mathrm{s}}^{-1}$. (d) Difference between two images taken just prior to and after an avalanche event.

Figure 2

Interparticle friction triggers hysteretic avalanches: (a) Picture of the silica particles ($d=23.46\pm 1.06\mu \mathrm{m}$, ${\rho }_p=1850\mathrm{kg}·{\mathrm{m}}^{-3}$, $\mathrm{St}=2\times {10}^{-2}$). (b) Schematic depicting how interparticle friction is tuned by varying the solvent ionic concentration. (c) Time-averaged avalanche angle $\overline{\theta }$ versus ionic concentration [NaCl]. (d) Avalanche angle $\theta$ versus time for frictionless (black) and frictional (green) particles. (e) Mean hysteresis amplitude $\overline{\mathrm{\Delta }\theta }$ (averaged over 15 avalanches) versus salt concentrations. Square markers correspond to experiments performed with a water/glycerol mixture ($\mathrm{St}=2\times {10}^{-3}$). All measurements were performed for $\omega ={10}^{-3}°·{\mathrm{s}}^{-1}$ except for the water/glycerol mixture where $\omega ={10}^{-4}°·{\mathrm{s}}^{-1}$.

Figure 3

Frictional and frictionless rheologies extracted from steady avalanches [(a)–(c)] and transient relaxations [(d)–(f)]. (a) Sketch defining the parameters used to extract the rheology for steady avalanches. (b) Avalanche angle $\theta$ versus time for frictional and frictionless silica particles ($[\mathrm{NaCl}]={10}^{-1}$ and ${10}^{-6}\mathrm{mol}·{\mathrm{L}}^{-1}$, respectively) measured by successively decreasing the rotation rate $\omega$ of the drum. (c) Corresponding suspension effective friction coefficient $\mu$ versus viscous number $J$; the color code corresponds to the rotation rate of the drum. (d) Sketch defining the parameters used to extract the rheology from transient relaxations. (e) Relaxation of the angle of avalanche versus time for frictional and frictionless silica particles ($[\mathrm{NaCl}]={10}^{-1}$ and ${10}^{-6}\mathrm{mol}·{\mathrm{L}}^{-1}$, respectively). The preparation rotation rates are $\omega =5\times {10}^{-2}°·{\mathrm{s}}^{-1}$ and $\omega ={10}^{-3}°·{\mathrm{s}}^{-1}$, respectively; the drum is stopped ($\omega =0$) at $t=0$. (f) Corresponding $\mu$ versus $J$. The large markers correspond to the steady-state preparation, while the small markers are obtained from the relaxation where $J=\eta |\dot{\theta }|R^2/(h^{2}P)$ is computed from $\dot{\theta }$ measured in panel (e). The data were smoothed using time windows of variable widths, and $\dot{\theta }(t)$ was computed using the finite-difference method. The same color code as in panel (e) is used.

Figure 4

Reduced macroscopic friction coefficient $\mathrm{\Delta }\mu =\mu -{\mu }_0$ versus viscous number $J$ for frictional and frictionless particles immersed in solutions of ionic concentration ($[\mathrm{NaCl}]={10}^{-1}$ and ${10}^{-6}\mathrm{mol}·{\mathrm{L}}^{-1}$, respectively). Large markers were obtained from the steady-state measurements presented in Figs. 3. Small markers correspond to transient relaxations measurements from different initial steady states as described in Figs. 3. Solid lines are fits (in grey) by $\mathrm{\Delta }\mu =(J/J_0{)}^{\alpha }$, with $\alpha =0.37\pm 0.05$ and $J_0\approx 2.27\times {10}^{-2}$, for frictionless particles and (in green) by $\mathrm{\Delta }\mu =[(J-J_c)/J_0{]}^{\beta }$, with $\beta =0.7\pm 0.3$, $J_0\approx 8.69\times {10}^{-3}$ and $J_c\approx {10}^{-6}$, for frictional particles.

Figure 5

Average amplitude of hysteresis $\overline{\mathrm{\Delta }\theta }$ versus Stokes number St. We show the comparison between data obtained by Courrech du Pont et al. (23) (in black) and in the present study. Red star markers correspond to the large glass beads in water and in the mixture of Ucon-oil and water [Fig. 1]. Red circle (resp. square) markers correspond to the silica particles in different salt concentration in water (resp. water/glycerol) [Fig. 2]. For the present study, hysteresis amplitudes were measured at a rotation rate of $\omega ={10}^{-3}°·{\mathrm{s}}^{-1}$, except for the silica particles in the water/glycerol mixture where $\omega ={10}^{-4}°·{\mathrm{s}}^{-1}$ and for the large glass beads where $\omega =5\times {10}^{-3}°·{\mathrm{s}}^{-1}$.

Figure 6

(a) Flowing thickness $h/d$ of the avalanche in the steady states measured at the front wall of the drum using particle image velocimetry, as a function of the reduced avalanche angle $\mathrm{\Delta }\tan \theta$ for frictionless (grey circles, $[\mathrm{NaCl}]={10}^{-6}\mathrm{mol}·{\mathrm{L}}^{-1}$) and frictional (green triangles, $[\mathrm{NaCl}]={10}^{-1}\mathrm{mol}·{\mathrm{L}}^{-1}$) particles (silica spheres of diameter $d=23.46\pm 1.06\mu \mathrm{m}$ and density ${\rho }_p=1850\mathrm{kg}·{\mathrm{m}}^{-3}$). Dashed lines correspond to average values $h/d=25$ (frictional grains) and $h/d=50$ (frictionless grains). Note that the flowing depth for frictionless particles is always much smaller than the critical depth ($h_c/d=400$) for which the weight of the grains overcomes the interparticle repulsive force so that contacts become frictional [10]. (b) Rheology $\mu (J)$ extracted from the stationary regimes using the averaged value of $h/d$ or the measured values shown in Fig. 3 for computing the viscous number $J=\eta \omega R^2/(h^{2}P)$ [grey symbols are frictionless grains and green symbols are frictional grains; we use the same experimental conditions as in Fig. 3].

Figure 7

Values of $\tan \theta$ when the flow stops for frictionless (top, $[\mathrm{NaCl}]={10}^{-6}\mathrm{mol}·{\mathrm{L}}^{-1}$) and frictional (bottom, $[\mathrm{NaCl}]={10}^{-1}\mathrm{mol}·{\mathrm{L}}^{-1}$) particles (silica spheres of diameter $d=23.46\pm 1.06\mu \mathrm{m}$ and density ${\rho }_p=1850\mathrm{kg}·{\mathrm{m}}^{-3}$). The first column is obtained by fitting the data in steady rotations using a power law with a threshold. The other columns correspond to the angles measured at the end of the relaxations for different initial steady rotation rates.

Figure 8

Prediction of the avalanche model in the rotating drum. (a) Rheologies for the frictionless [grey curve, Eq. (d3)] and frictional [green curve, Eq. (d4)] suspensions as deduced from the experiments. The symbols correspond to the steady states for the constant rotation rates given in panel (b). (b) Time evolution of the avalanche angle for decreasing steps of rotation rates. (c) Relaxation of the avalanche angle when the drum stops ($\omega =0$) at $t=0$ after a continuous rotation. (d) Reduced macroscopic friction coefficient $\mathrm{\Delta }\mu =\mu -{\mu }_0$ versus viscous number $J$ showing that, for a given $\mathrm{\Delta }\mu$, the viscous number for the frictional suspension is higher than for the frictionless suspension, thereby explaining the faster relaxations observed for the frictional case.