Revealing the frictional transition in shear-thickening suspensions

Steady Avalanches.

A simple way to probe the frictional behavior of a suspension is to measure its quasi-static avalanche angle in a rotating drum using nonbuoyant particles (20, 21). For the sake of clarity, we compared the avalanche angle of a standard Newtonian suspension made of large frictional glass beads of diameter d= 500 μm (Fig. 1A, Left), to that of a typical shear-thickening suspension made of potato starch particles (d= 25 μm) immersed in water (Fig. 1A, Right). In both cases, the particle density ρp is larger than that of the suspending fluid ρf and the fluid viscosity ηf is chosen so that the Stokes number St is low and inertial effects can be neglected (St=ρpΔρgd3/18ηf∼ 10−2 where Δρ=ρp−ρf and g stands for gravity) (21). Both suspensions are placed within a rotating drum as shown in the experimental setup illustrated in Fig. 1B, Center. By imposing a slow and constant rotation speed ω, the nonbuoyant grains at the surface of the pile flow under their own weight, forming a steady avalanche of angle θ on top of a region experiencing a rigid rotation with the drum (20). In such a configuration, the confining pressure acting on the flowing layer of grains is P=ϕΔρghcos⁡θ, where h is the height of the flowing layer and ϕ its volume fraction; the corresponding tangential stress is τ=ϕΔρghsin⁡θ. In the steady state, the macroscopic friction coefficient of the suspension μ is directly given by the avalanche angle because by definition μ=τ/P=tan⁡θ. The avalanche angle θ thus provides access to the macroscopic friction coefficient of the suspension μ, which itself depends monotonically on the particle friction coefficient μp (22). For frictional grains, the macroscopic friction coefficient μ≃ 0.4 has only a weak dependence on μp and yields a typical avalanche angle θ≃ 25∘ (21). However, when the interparticle friction μp becomes very small (below 0.1), the macroscopic friction μ sharply drops. However, because of steric constraints, μ remains finite as μp→ 0. For frictionless spherical particles (μp= 0), discrete simulations predict a quasi-static macroscopic friction μ= 0.105, corresponding to a nonzero avalanche angle θ= 5.76∘ (23). Therefore, measuring the pile angle of steady avalanches constitutes a simple, yet decisive way to probe the interparticle friction coefficient in suspensions. Moreover, in this rotating drum configuration, the slope of the avalanche is set by the flowing layer of grains that is located near the free surface of the pile. For the low rotation speeds investigated here, the thickness of the layer h is of the order of a few particle diameters (h∼ 10d) (24). This means that the measure of the avalanche angle gives access to the frictional state of the grains under very low confining pressure. Typically, for the potato starch particles, the confining pressure within the flowing layer is P∼ 10ϕΔρgd≃ 1 Pa.

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Fig. 1.

Steady avalanches in (Left) Newtonian and (Right) shear thickening suspensions. (A) Picture and rheograms (viscosity η vs. shear-rate γ˙) of (Left) a Newtonian suspension of large glass beads (ϕ= 50%) and (Right) a shear-thickening suspension of potato starch particles (ϕ= 40%). Rheograms were obtained in the configuration shown, using density-matched suspensions. (B, Center) Sketch of the rotating drum. Shown are pictures of a typical steady avalanche for (B, Left) the glass beads immersed in a mixture of Ucon oil and water and (B, Right) the potato starch immersed in water. (C) Angle of avalanche θ vs. time for (Left) the glass beads and (Right) the potato starch. (C, Insets) Steady-state avalanche angle θs vs. drum rotation speed ω (see Materials and Methods for the detailed description of particles, fluids, and experimental protocol).

For the Newtonian suspension of large glass beads, the avalanche angle θ shows classical hysteretic fluctuations (20) around a time-averaged angle θs≃ 25∘ (picture in Fig. 1B, Left and data in Fig. 1C, Left). This angle corresponds to a suspension friction coefficient μ≃ 0.47, which is a usual value for frictional particles. The striking result is that, under the same flowing conditions, the shear-thickening suspension of potato starch particles yields a much lower avalanche angle (picture in Fig. 1B, Right and data in Fig. 1C, Right). The suspension can flow steadily with an angle of avalanche as small as θs≃ 8.5∘. This angle corresponds to a suspension friction coefficient μ≃ 0.15, showing that the friction coefficient between particles is nearly vanishing. The suspension friction coefficient (μ≃ 0.15) here is slightly larger than the expected value for frictionless spheres (μ= 0.105) because potato starch particles are prolate, which geometrically increases the macroscopic friction coefficient (25, 26). Importantly, in the range of drum rotation speeds ω investigated, the avalanche angles reported here do not depend on ω (Fig. 1C, Left and Right Insets). They thus characterize the frictional properties of the suspension in its quasi-static regime, i.e., when the suspension reaches its critical jamming state (9).

Compaction and Dilatancy.

Another robust way to probe the frictional behavior of a suspension is to investigate compaction and dilatancy effects (21). The protocol is the following: Particles are first suspended entirely within the drum before being allowed to sediment (Fig. 2A). The sediment is then compacted by gently hitting the drum with a rubber-head hammer Ntaps times. The volume fraction ϕ of the sediment (the ratio of the particle volume to the total volume of the sediment) is measured throughout this compaction process. Finally, the drum is quickly rotated by a fixed angle θs+ 10∘ above the steady-state avalanche angle, to generate a transient avalanche whose angle θt is measured vs. time.

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Fig. 2.

Compaction and transient avalanches in (Left) Newtonian and (Right) shear-thickening suspensions. (A) Sketch of the experimental protocol to study compaction and dilatancy effects. (B) Volume fraction of the sediment ϕ vs. number of taps Ntaps (Left) for the large glass beads and (Right) for the potato starch particles. (C) Angle of the transient avalanche θt vs. time obtained for different initial compactions of the sediment (Left) for the large glass beads and (Right) for the potato starch particles.

A conspicuous feature of frictional systems, such as a pile of large glass beads, is that it compacts under vibrations. As shown in Fig. 2B, Left, the packing fraction of the glass beads sediment, which right after sedimentation starts from a loose state (ϕ= 0.56<ϕc), progressively increases with the number of taps to eventually reach a dense state (ϕ= 0.61>ϕc). Remarkably, within this frictional system, very different transient avalanches are observed, depending on the initial preparation of the sediment (Fig. 2C, Left). For an initial loose packing (0 tap), the avalanche rapidly flows until its angle relaxes to an angle much lower than θs (the steady-state avalanche angle measured earlier). Conversely, for an initially compact sediment (120 taps), one observes a long delay during which the avalanche stays still. It then slowly flows, relaxing to θs. Such a drastic change in the avalanche dynamics with the packing of the initial sediment relies on a well-known pore pressure feedback mechanism (27⇓–29), which applies only for dilatant (i.e., frictional) systems (23). As first described by Reynolds (30), deformation of a dense granular packing of frictional grains requires its dilation. Thus, as the drum is tilted for the avalanche to flow, the initially compact sediment must dilate. However, dilation in the presence of an incompressible interstitial fluid induces a fluid flow: Some fluid is sucked within the granular network. This fluid flow presses the grains against each other, thereby enhancing friction. As a result, for a compact pile of frictional grains, the avalanche is strongly delayed and then flows slowly, relaxing to θs (28, 31). Conversely, a loosely packed granular bed tends to compact when it deforms. This time, the interstitial fluid is expelled from the granular packing, thereby resulting in a fluidization of the grains: The avalanche flows rapidly and relaxes to an angle smaller than θs.

The compaction and dilatancy effects observed for the large glass beads are the phenomenological signature of frictional grains. For frictionless particles, the situation is markedly different because there is only one possible state of compaction (ϕc=ϕrcp) and no dilatancy effects are thus expected under shear (23). This is precisely what we observe with the shear-thickening suspension of potato starch particles: Tapping the settled bed of potato starch particles does not modify its packing fraction (Fig. 2B, Right) and hardly changes the dynamic of the transient avalanches that all relax to θs (Fig. 2C, Right). These experiments again show that, under low confining pressure, potato starch particles behave as if they were frictionless. Note that the maximum packing fraction of the potato starch particles, ϕ= 49%, may seem small. However, this low value can be explained by the anisotropic shape of the particles and also their tendency to swell when immersed in water (32).

Tuning Microscopic Friction Using a Model Suspension.

We have shown that potato starch particles immersed in water produce a shear-thickening rheology and are frictionless under low confining pressure. These results are consistent with the frictional transition scenario for shear thickening presented in the Introduction (7, 8). They suggest the existence of a short-range repulsion force or a microscopic pressure-dependent friction between the starch particles. However, the origin of this force and more generally the surface physico-chemistry of starch remain unclear (6, 33, 34). To vary the interaction force between particles and investigate the frictional transition within a well-controlled system, we turn to a model suspension composed of silica beads immersed in water (Fig. 3A, Top). We use non-Brownian silica particles of diameter d= 24μm comparable in size to the starch particles. When immersed in water, such silica particles spontaneously develop negative surface charges, which generates an electrostatic repulsive force between the grains (35⇓–37). Moreover, this surface charge can easily be screened by dissolving electrolytes (salt) within the solvent. Increasing the salt concentration decreases the range of the repulsive force, i.e., the Debye length λD (Fig. 3A, Bottom), except at very high molarities not considered here (38). This model system is thus particularly appealing to test the recent frictional transition scenario and clarify the link between microscopic interactions between particles, friction, and the suspension macroscopic rheology.

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Fig. 3.

Steady avalanches, compaction, and dilatancy effects in suspensions of silica beads in (black) pure water or (green) ionic solution. (A) Picture of the silica particles and sketch of a silica bead immersed in an ionic solution. Silica particles spontaneously carry negative charges on their surface when immersed in water. The range of the resulting repulsive force, i.e., the Debye length λD, can be tuned by changing the ionic concentration of the solvent. (B–D) Pile slope avalanche angle θ vs. time (B), relative packing fraction of the sediment Δϕ=ϕ(Ntaps)−ϕ(Ntaps= 0) vs. number of taps (C), and transient avalanche angle θt vs. time (D) for the silica beads in (green) a sodium chloride solution with [NaCl] = 0.1 mol⋅L⁻¹ and in (black) pure water.

We first use the rotating drum to systematically investigate steady avalanches, compaction, and dilatancy effects on two suspensions of silica beads: one with silica beads immersed in pure water and the second one with the beads immersed in a solution of water and NaCl with a large concentration of salt ([NaCl] = 0.1 mol⋅L⁻¹) to fully screen the Debye layer. Here again inertia is negligible (St∼ 5× 10−3) and the flowing regime is quasi-static (ω→ 0). As illustrated in Fig. 3 B–D (green data), in the presence of a large concentration of salt that screens the repulsive force, the suspension behaves as frictional. The steady-state avalanche angle is large, θs≃ 27.5∘; the packing fraction of the sediment evolves from a loose packing right after sedimentation to a dense packing after 60 taps; and the dynamics of the transient avalanche strongly depend on the initial packing, showing features related to the dilatancy effects discussed earlier. [The absolute volume fraction is not reported in Fig. 3 because here it depends on the system size. This size effect can be explained by the large value of the Debye length in pure water (λD≃ 1 μm), which is not negligible relative to the silica particle diameter (39). Here, particles may be thought of as a hard core of diameter d surrounded by a soft crust of thickness equal to the Debye length. The maximum packing of such a system, when the confining pressure P→ 0, is ϕmax≈ϕrcp(1− 3λD/d)≈ 0.48. However, due to the hydrostatic pressure within the granular bed that depends on the system size, grains deep in the bed must be closer to each other than those at the top. The absolute volume fraction may thus lie anywhere between the latter value and 0.64 (random close packing), depending on the system size.] Conversely, the silica beads in pure water behave as frictionless particles (Fig. 3 B–D, black data). The steady-state avalanche angle is θs≃ 6∘. This value is remarkably close to the quasi-static macroscopic friction angle obtained numerically for ideal frictionless spheres θs= 5.76∘± 0.22 (23). Additionally, no compaction of the granular bed and no discernable effect of the initial packing on the transient avalanches are observed. These results clearly demonstrate that the presence of a short-range repulsive force can lead under low stress to a frictionless behavior of the particles.

To further inquire about the microscopic origin of this frictionless behavior under low stress, we measured the steady avalanche angle θs as a function of the salt concentration [NaCl]. Fig. 4A shows that the suspension transits progressively from frictionless with θs≃ 6∘ in pure water to frictional with θs≃ 30∘ when the salt concentration [NaCl] is increased. In addition to this macroscopic measurement, we also characterized the particle roughness with an atomic force microscopy (AFM) and found that the peak roughness of the silica particles is ℓr= 3.73±0.80 nm. Because the Debye length is entirely set by the salt concentration, λD= 0.304/[NaCl] nm (at T= 25 °C) (36), we can plot the steady avalanche angle θ_s as a function of ℓr/λD (Fig. 4A, Top Inset). Interestingly, the transition from frictionless to frictional occurs for ℓr/λD∼ 1. This result strongly supports the idea that the frictionless state arises from the interparticle repulsive force caused by the electrostatic double layer. When its range is smaller than the particle roughness, this force becomes ineffective to prevent the grains from touching and the system is frictional. According to the scenario described in the Introduction (7, 8), the frictional transition should occur when the confining pressure P exerted by the weight of the granular layer equals the critical pressure Pc that the short-range repulsive forces can sustain. Assuming that the repulsive force follows an exponential decay as Frep(z)=F0exp(−z/λD) (36), where z is the distance between the particles surfaces, the critical pressure at contact z=2ℓr writes Pc∼Frep(2ℓr)/(πd2/4)∼(4F0/πd2)exp(−2ℓr/λD). Matching it to the confining pressure P∼ 10ϕΔρgd predicts a transition when ℓr/λD∼−(1/2)log(10πϕΔρgd3/4F0). Using F0/d∼ 1 mN/m as reported for silica surfaces in NaCl electrolytes (35) yields ℓr/λD≃ 1.9, in fair agreement with the transition observed in Fig. 4A.

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Fig. 4.

Frictional transition in silica suspensions when varying the range of the repulsive force and link with macroscopic rheology. (A) Steady avalanche angle θs vs. salt concentration [NaCl]. (A, Top Inset) Steady avalanche angle θs vs. particle roughness normalized by the Debye length ℓr/λD. (A, Bottom Inset) AFM scan of a silica particle surface. The peak roughness is ℓr= 3.73±0.80 nm. (B) Rheograms of silica bead suspensions obtained for different volume fractions in solutions of water and NaCl with salt concentration (Top) [NaCl] = 10⁻⁴ mol⋅L⁻¹ and (Bottom) [NaCl] = 0.1 mol⋅L⁻¹. The effective viscosity is ηeff=αΓ/(2πΩL3), where Γ is the torque, Ω is the revolution speed, and α≃ 0.483 is a calibration constant. (B, Inset) Sketch of the experimental configuration used to obtain the rheograms.

Finally, we investigate whether for this model suspension, the existence of a frictionless state under low confining pressure leads to a shear-thickening rheology and whether the elimination of this state (by screening repulsive forces) restores a Newtonian behavior. Rheograms of the silica suspensions were obtained using the configuration sketched in Fig. 4B, Inset. Because silica particles are denser than the aqueous suspending fluid, we used a double helix with tilted blades shearing the entire sample to avoid sedimentation and maintain the homogeneity of the suspension during the measurement. We then define an effective viscosity ηeff=αΓ/(2πΩL3) given by the ratio of the effective stress Γ/L3, where Γ is the torque and L the helix diameter, to the effective shear rate 2πΩ given by the revolution speed Ω of the helix. The constant α≃0.483 is set to ensure that the effective viscosity matches the actual viscosity for a Newtonian fluid. We performed rheological measurements on two suspensions of silica particles immersed in water with a salt concentration [NaCl] = 10⁻⁴mol⋅L⁻¹ and [NaCl] = 10⁻¹mol⋅L⁻¹, respectively, i.e., just before and after the frictional transition observed in Fig. 4A. First, we find that the suspension that has a frictionless state under low confining pressure displays all of the features of a shear-thickening suspension (Fig. 4B, Top): Continuous shear thickening is observed at moderate volume fractions, whereas larger volume fractions lead to a dramatic increase of its effective viscosity (by about four orders of magnitude). Second, the striking result is that when the repulsive force is screened, so that no frictionless state exists under low confining pressure, the same suspension no longer shear thickens (Fig. 4B, Bottom). These measurements corroborate previous rheological characterizations using Brownian silica microspheres (17, 40, 41) and confirm the link between the existence of a repulsive force, friction, and shear-thickening rheology.