Anaïs Gauthier

Guillaume Ovarlez

Annie Colin

Shear thickening in presence of adhesive contact forces: the singularity of cornstarch

Keywords: shear thickening, adhesion, cornstarch, non-Brownian suspensions, normal stress measurement, phase separation. I

Hypothesis: A number of dense particle suspensions experience a dramatic increase in viscosity with the shear stress, up to a solid-like response. This shear-thickening process is understood as a transition under flow of the nature of the contacts-from lubricated to frictional -between initially repellent particles. Most systems are now assumed to fit in with this scenario, which is questionable. Experiment: Using an in-house pressure sensor array, we provide a spatio-temporal map of the normal stresses in the flows of two shear-thickening fluids: a stabilised calcium carbonate suspension, known to fit in with the standard scenario, and a cornstarch suspension, which spectacular thickening behavior remains poorly understood. Findings: We evidence in cornstarch a unique, stable heterogeneous structure, which moves in the velocity direction and does not appear in calcium carbonate. Its nature changes from a stress wave to a rolling solid jammed aggregate at high solid fraction and small gap width. The modeling of these heterogenities points to an adhesive force between cornstarch particles at high stress, also evidenced in microscopic measurements. Cornstarch being also attractive at low stress, it stands out of the classical shear-thickening frame, and might be part of a larger family of adhesive and attractive shear-thickening fluids.

INTRODUCTION

Many industrial processes, such as waste water treatment, oil drilling or mixing pastes require the manipulation of liquids containing a high solid fraction in particles [START_REF] Blanco | Conching chocolate is a prototypical transition from frictionally jammed solid to flowable suspension with maximal solid content[END_REF][START_REF] Feys | Fresh self compacting concrete, a shear thickening material[END_REF]. However, the flows of dense suspensions are particularly complex. One of their most striking features is shear thickening, where the viscosity can increase by more than one order of magnitude with the shear stress [START_REF] Barnes | Shear-thickening ("dilatancy") in suspensions of nonaggregating solid particles dispersed in newtonian liquids[END_REF][START_REF] Brown | Shear thickening in concentrated suspensions: phenomenology, mechanisms and relations to jamming[END_REF][START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF][START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Guy | Towards a unified description of the rheology of hard-particle suspensions[END_REF][START_REF] Wagner | Shear thickening in colloidal dispersions[END_REF][START_REF] Brown | Generality of shear thickening in dense suspensions[END_REF]. For the densest suspensions, the material undergoes a liquid-solid transition: it is sometimes detrimental, causing pipe clogging or the breaking of blades in mixing systems [START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF]. But it can also be conveniently used to engineer smart fluids [START_REF] Gürgen | Shear thickening fluids in protective applications: A review[END_REF][START_REF] Ness | Shaken and stirred: Random organization reduces viscosity and dissipation in granular suspensions[END_REF][START_REF] Niu | Tunable solidification of cornstarch under impact: How to make someone walking on cornstarch sink[END_REF][START_REF] Zarei | Application of shear thickening fluids in material development[END_REF], shock absorber for batteries [START_REF] Liu | A shear thickening fluid based impact resistant electrolyte for safe li-ion batteries[END_REF], damping devices [START_REF] Zhang | The rheology of shear thickening fluid (stf) and the dynamic performance of an stf-filled damper[END_REF] or body armors [START_REF] Mahfuz | Enhanced stab resistance of armor composites with functionalized silica nanoparticles[END_REF][START_REF] Majumdar | Development of soft composite materials with improved impact resistance using kevlar fabric and nano-silica based shear thickening fluid[END_REF][START_REF] Xu | Stabbing resistance of body armour panels impregnated with shear thickening fluid[END_REF][START_REF] De Goede | Highvelocity impact of solid objects on non-newtonian fluids[END_REF].

In this context, and in particular to propose new formulations [START_REF] Gürgen | Shear thickening fluids in protective applications: A review[END_REF][START_REF] Jiang | Study of the particles' structure dependent rheological behavior for polymer nanospheres based shear thickening fluid[END_REF][START_REF] Hsu | Exploring the roles of roughness, friction and adhesion in discontinuous shear thickening by means of thermo-responsive particles[END_REF], a good understanding of the microscopic mechanisms at the origin of shear thickening is essential. In this field, the work of Wyart and Cates [START_REF] Wyart | Discontinuous shear thickening without inertia in dense non-brownian suspensions[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in nonbrownian suspensions[END_REF][START_REF] Lin | Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions[END_REF][START_REF] Royer | Rheological signature of frictional interactions in shear thickening suspensions[END_REF][START_REF] Clavaud | Revealing the frictional transition in shear-thickening suspensions[END_REF] is central. Shear thickening is explained by the presence of a short-ranged repulsive force between the particles. When the normal force exceeds the repulsive force [START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF], the particles switch from lubricated to frictional contacts, which leads to an increase of the viscosity.

In the Wyart and Cates model, the suspension has a purely viscous behavior, with a viscosity η depending on the ratio ϕ/ϕ j only (with ϕ the particle volume fraction and ϕ j the jamming volume fraction) and diverging * anais.gauthier@univ-rennes.fr, tel: +33223237300 at ϕ/ϕ j = 1. For the frictionless suspension, ϕ j is the random close packing ϕ rcp ; for the frictional suspension, ϕ j = ϕ m < ϕ rcp . In the model, ϕ j depends on the fraction of particles in frictional contact; ϕ j decreases from ϕ rcp to ϕ m when the applied stress increases. With these ingredients, a critical solid volume fraction ϕ dst < ϕ m emerges and three regimes are found. For ϕ < ϕ dst , the system shows continuous shear thickening (CST), characterized by a smooth transition from a low viscosity plateau (frictionless state with ϕ j = ϕ rcp ) to a high viscosity plateau (frictional state with ϕ j = ϕ m ) when increasing the shear stress σ. For ϕ dst < ϕ < ϕ m the flow curve is sigmoidal, which points to hysteretic discontinuous shear thickening (DST) between the low viscosity state and the high viscosity state. For ϕ m < ϕ < ϕ rcp the material flows at low shear stress only and is jammed at high shear stress, a behavior coined shear jamming (SJ), as no flow is possible without dilatancy for the frictional material. Above ϕ rcp , the material cannot flow without dilatancy. This framework has been successful in capturing qualitatively the main features of shear thickening [START_REF] Morris | Lubricated-to-frictional shear thickening scenario in dense suspensions[END_REF] for a wide range of suspensions, which now makes this description the standard explanation for shear thickening. Most systems are now assumed to fit in with this scenario, and focus is now made on the details of the transition.

Dense suspensions of cornstarch particles are the most emblematic shear-thickening systems, used both for scientific outreach [START_REF] Brown | Shear thickening in concentrated suspensions: phenomenology, mechanisms and relations to jamming[END_REF] and in experiments designed to study the shear-thickening process [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF][START_REF] Niu | Tunable solidification of cornstarch under impact: How to make someone walking on cornstarch sink[END_REF][START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Boyer | Drop impact of shear thickening liquids[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Texier | Surface-wave instability without inertia in shearthickening suspensions[END_REF]. Most cornstarch suspensions show shear jamming [START_REF] Fall | Macroscopic discontinuous shear thickening versus local shear jamming in cornstarch[END_REF][START_REF] Han | Stress controlled rheology of dense suspensions using transient flows[END_REF], with an apparent viscosity η increasing by more than one order of magnitude at a critical shear rate. The rheological behavior of cornstarch has been fitted to the Wyart and Cates model in multiple occasions [START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Texier | Surface-wave instability without inertia in shearthickening suspensions[END_REF][START_REF] Han | Shear fronts in shear-thickening suspensions[END_REF][START_REF] O'neill | Liquid migration in shear thickening suspensions flowing through constrictions[END_REF]. A qualitative agreement was found close to the shearthickening transition. The low shear-rate region, however, shows a yield stress [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF], which is not accounted for by the standard framework. Other comparisons with the Wyart-Cates theory have been performed with the help of phase diagrams in the (ϕ, σ) parameter plane [START_REF] Fall | Macroscopic discontinuous shear thickening versus local shear jamming in cornstarch[END_REF][START_REF] Han | Stress controlled rheology of dense suspensions using transient flows[END_REF], with a relative success. In cornstarch, the flow at high stress has been the object of a particular scrutiny in the past years. Unstable flows with large fluctuations are observed, with spatial and temporal heterogeneities of the velocity concentration and stress field in the gap of rheology cells [START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF]. To this day, the exact nature and stability of these objects is still the object of research.

We propose here to revisit the question of the nature of the flow of cornstarch and of its origin. We focus in particular on the origin of the fluctuations. To do so, we have developed a new experiment that allows us to probe the normal stress -a component of the flow that is rarely measured locally -at the millimeter scale. We vary systematically the fraction of particles and gap size, which helps us understand the nature of the structure that is formed during the shear-thickening transition. We compare the local flow of cornstarch to that of another suspension (calcium carbonate), which microscopic and macroscopic properties follows the Wyart and Cates model. The comparison with this model fluid gives a novel insight on the shear-thickening process of cornstarch, which is shown to differ fundamentally from the often admitted frictionless to frictional mechanism.

II. MATERIALS AND METHODS

In this work, we compare the behavior of two fluids. The first one is the well-known dispersion of cornstarch particles (Merk). Here, and following Refs [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF], the particles are placed in a density-matched liquid made of water and salt (Cesium Chloride, 55% in weight). Cornstarch particles are polydisperse, with a mean diameter of 15 ± 7 μm (see Supplementary Figure 1a). They are porous, so that a volume fraction cannot be determined without additional assumptions [START_REF] Han | Measuring the porosity and compressibility of liquid-suspended porous particles using ultrasound[END_REF]. Following Refs. [START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF], we characterize the suspensions using the weight percentage in cornstarch ϕ w = mp mp+ms (with m p the mass of particles and m s the mass of solvent) instead of a volume fraction. In our experiments, ϕ w is varied between 34% and 42%. The second fluid is a calcium carbonate suspension (Eskal 500, KSL Staubtechnik GmbH) in a water-glycerol mixture. This system shares similarities with cornstarch: it is a polydisperse industrial powder, with particles having a rhomboidal-like shape and a mean size of 4 μm (see picture in Supplementary Figure 1b). To our knowledge, stabilized calcium carbonate suspensions are one of the rare non-Brownian suspensions where two viscosity plateaus (at low and high stress) are actually observed [START_REF] Richards | Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction[END_REF], together with silica suspensions [START_REF] Royer | Rheological signature of frictional interactions in shear thickening suspensions[END_REF][START_REF] Etcheverry | Capillarystress controlled rheometer reveals the dual rheology of shear-thickening suspensions[END_REF]. In particular, the flow curves can be quantitatively described by the Wyart and Cates model, over a wide range of shear rates and volume fraction [START_REF] Richards | Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction[END_REF]. Here, following Richards et al. [START_REF] Richards | Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction[END_REF], the particles are dispersed in a 50-50% water-glycerol mixture containing 0.05 w/w% of polyacrylic acid. The poly-acrilic acid (PAA) removes the adhesive contacts between the grains: in presence of PAA, they interact as purely frictional hard particles with a short range repulsive force [START_REF] Richards | Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction[END_REF], similarly to model systems. Contrary to cornstarch, calcium carbonate particles are not porous: we thus characterize the suspensions using the classical volume fraction ϕ, defined as the volume of the solid particles divided by the total volume of the particles and solvent. It is varied between 35% and 52%.

We characterize the flow of the suspensions by combining classical rheology measurements with a mapping of the normal stresses at the millimeter scale, using an in-house sensor array. As shown in Figure 1a, the sensor array is placed on the bottom plate of a conventional rheometer (DHR-2, TA instruments) in a parallel-plate geometry with a gap height h varied between 0.5 mm and 3.0 mm. It consists of 25 regularly spaced piezocapacitive sensors, built together in a 5 × 5 array of total width 4 cm. The details of the sensor array are shown in Figure 1a. It is a sandwich of three main layers: i) an electrode array (the bottom part), which sets the position and the size of the measurement points, ii) a 25 μm thick Mylar grid which separates the top and the bottom parts and iii) a soft piezo-capacitive layer (made from a solid polymeric foam developed previously in our lab [START_REF] Pruvost | Polymeric foams for flexible and highly sensitive low-pressure capacitive sensors[END_REF]), which is covered with a soft silver electrode. When a pressure is applied on the foam, it partially collapses and bends onto the grid, which impacts its local capacitance. By measuring the capacitance between the top and the bottom electrodes, we can thus map the vertical component of the stress in the fluid, with a spatial resolution of 4.5 × 4.5 mm 2 (the size of the electrodes), a precision of 2 Pa and a temporal resolution of typically 50 Hz [START_REF] Gauthier | A new pressure sensor array for normal stress measurement in complex fluids[END_REF].

During an experiment, the pressure of the nine central sensors (in red in Figure 1b), which are fully covered by the geometry, is recorded. In a typical experiment, the shear stress σ is imposed: it is increased by constant steps of duration 50 to 250 s, between 0.05 and 1000 Pa. The rheometer records the shear rate γ and the average vertical stress P m through the built-in force sensor. Simultaneously, the sensor array on the bottom plate records the local normal stress P as a function of time t. All these data are aggregated to obtain both a global and a local view of the shear-thickening process.

III. RESULTS AND DISCUSSION

When discussing the flow curves of cornstarch, we use the term discontinuous shear-thickening to refer to situations where the flow curve is vertical or sigmoidal. We call continuous shear thickening the situation where the viscosity increases smoothly with the shear rate.

A. Local flow in cornstarch suspensions

The flow curve of a cornstarch suspension with weight fraction ϕ w = 40 % is presented in Figure 2a. Three different regions are distinguished: a shear-thinning region at low shear rates γ < 3 s -1 , a continuous shearthickening region for 5 s -1 < γ < 40 s -1 and a discontinuous shear-thickening region for γ ≃ 40 s -1 . At low shear rates, the strong shear thinning is consistent with a small yield stress of order 0.1 Pa, which matches previous measurements [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Fall | Shear thickening of cornstarch suspensions[END_REF]. As shown in Figure 2b, the mean pressure P m measured by the force sensor of the rheometer is slightly negative and decreasing with γ in the first two regions of the flow curve. It switches to a positive value, and increases strongly with the shear stress in the discontinuous shear-thickening regime. This is characteristic of all the cornstarch suspensions that we have considered here, with with 35% < ϕ w < 42 % (Supplementary Figure 2a). At these weight fractions, fluctuations of the macroscopic normal stress also appear in the discontinuous shear thickening region (see Supplementary Figure 2b andc) and their amplitude increases with the applied stress.

Using the sensor array, the flow is probed at the millimeter scale. For clarity, we present in Figure 2c the signal of two sensors (in red and blue) placed symmetrically under the geometry (see the schematic in the inset of Fig. 2b) as a function of time. The signal measured by all nine sensors in the continuous shear thickening region (Supplementary Movie 1) and in the discontinuous shear thickening region (Supplementary Movie 2 and 3) is available in the Supplementary Materials. In the shearthinning and in the continuous shear-thickening regions of the flow curve (σ = 15 Pa, top panel), the local normal stress matches the mean normal stress: the pressure measured by each of the sensors is constant and slightly negative. In the discontinuous shear thickening region (σ = 80 Pa and σ = 130 Pa), however, the two sensors present pressure peaks of short duration in phase opposition. As evidenced in Supplementary Movie 2, it corresponds to a unique pressure wave that rotates within the geometry in the velocity direction. The peak pressure is typically 4 to 5 times higher than the mean normal stress measured by the rheometer, and, between two events, the pressure is close to zero and slightly negative. When the shear stress is increased, for example from σ = 80 Pa (middle panel) to σ = 130 Pa (bottom panel), the amplitude of the stress signal increases, but we still observe a unique wave. This regular local normal stress wave is not evidenced by the force sensor of the rheometer, as shown in Supplementary Figure 2c andd. The mean normal stress only presents some noise around a constant value, significantly smaller than the local peak pressure. When increasing the applied stress, both the mean normal stress P m and the amplitude of the noise increase, while, at the small scale, the amplitude of the normal stress peak increases.

This single, stable and regular stress wave is observed in all cornstarch suspensions with a weight fraction ϕ w > 35% and for gap widths varied between 0.5 mm and 2 mm. The regularity of the signal is lost in the situations of large gap width (h ≥ 2 mm) or small weight fraction (35% ≤ ϕ w < 37%), as shown in Supplementary Figure 3. Finally, the signal completely disappears for ϕ w ≤ 34%, when the shear thickening is continuous (Supplementary Figure 4).

The pressure wave being very regular, we now focus on its velocity. In Figure 3a we plot the angular velocity of the signal Ω a (compared with the angular velocity Ω of the geometry) as a function of the applied shear stress σ. Interestingly, Ω a /Ω is independent of σ: it keeps a constant value Ω a /Ω = 1.6 ± 0.1 for h = 1.0 mm (in red) and Ω a /Ω = 0.4 ± 0.1 for h = 0.5 mm (in blue). What is remarkable here is that the two signals obtained for h = 1.0 mm and h = 0.5 mm, that could seem similar from a distance, surely correspond to very different physical situations. Indeed, the normal stress wave is faster than the geometry for h = 1.0 mm (Ω a > Ω): it is thus very unlikely that it originates from a solid object. However, for h = 0.5 mm, Ω a ≃ 0.5 Ω: the angular velocity of the signal is close to the mean velocity of the fluid within the geometry, which could indicate the presence of a rolling solid aggregate. The shape of the signal is also different in both situations: when Ω a /Ω > 1, the signal is peaked as in Figure 3b. For Ω a /Ω < 1 (Figure 3c, h = 0.5 mm), the signal is more noisy and the width of the peak is larger.

To get a better picture of this phenomenon, we var- ied systematically the gap thickness h (0.5 mm < h < 2.0 mm, Figure 3d) and the weight fraction in particles ϕ w of the suspensions (37% ≤ ϕ w ≤ 42%, Figure 3e). In both cases, we evidence an abrupt transition between a slow pressure wave (with Ω a /Ω constant and close to 1/2 of the velocity of the geometry) and a fast wave (with a signal faster than the geometry: Ω a /Ω > 1). The slow signal is measured when the suspension is confined (h < 1 mm ) and dense (ϕ w ≥ 41%), while the fast one is seen at lower weight fraction and large gap. The transition between the two regimes is sharp: for ϕ w = 40%, the velocity of the signal is multiplied by 3 when the gap width increases from 0.95 mm to 1.0 mm (Figure 3d); moreover, this velocity decreases by a factor ≃ 3 when switching from ϕ w = 40% to ϕ w = 41% for h = 1 mm. It is also associated with a visible change in the shape of the signal recorded by the sensors, as evidenced in Supplementary Figure 5: similarly to what is shown in Figure 3c, the "slow" pressure signal is more noisy and of longer peak extent than the fast one.

B. Comparison with calcium carbonate suspensions

In order to know whether the presence of a stable normal stress wave is specific to cornstarch or not, we have considered another shear-thickening fluid: a calcium carbonate suspension in a water-glycerol mixture, stabilized by a very small quantity of poly-acrylic acid. As shown by Richards et al. [START_REF] Richards | Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction[END_REF] and Figure 4a, these suspensions behave as in the theory and the simulations of an ideal hard sphere system [START_REF] Wyart | Discontinuous shear thickening without inertia in dense non-brownian suspensions[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in nonbrownian suspensions[END_REF]. By contrast with cornstarch, calcium carbonate suspensions are not shear thinning: only two regions are distinguished, and the shear thickening happens between a low viscosity plateau and a highviscosity plateau.

Here, and contrary to cornstarch, we can clearly define the random close packing volume fraction ϕ rcp and the jamming volume fraction ϕ m from the flow curve. This is done in Figure 4b, where we report the relative viscosity η r = η/η 0 in the low (in blue) and high (in red) viscosity plateaus, with η 0 = 23 mPa s the viscosity of the suspending fluid. The value of η r in the low-viscosity region is obtained by averaging all values of η before shear thickening (typically for γ < 0.2 s -1 ), and on the last 4 points after shear thickening for the high viscosity region.

In Figure 4b, the viscosity η r (ϕ) is fitted to a Krieger-Dougherty model [START_REF] Krieger | A mechanism for nonnewtonian flow in suspensions of rigid spheres[END_REF] η r = (1 -ϕ/ϕ j )

-β , with β = 2.5. From this, we can extract the low-shear jamming volume fraction (ϕ j = ϕ rcp = 0.62) and the high-shear jamming volume fraction (ϕ j = ϕ m = 0.48), which are respectively materialized in Figure 4b by a vertical blue and pink line, and correspond to the divergence of the viscosity in the frictionless and frictional regions of the flow curve. From this, and similarly to Richards et al. [START_REF] Richards | Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction[END_REF], we expect calcium carbonate to be in the continuous and discontinuous shear-thickening regime for ϕ < ϕ m (filled circles in Figure 4a andc) and in the shear jamming regime at higher volume fractions (open circles). This is further confirmed in Figure 4c, when plotting the mean normal stress. For ϕ < 47%, P m is positive and increases with γ. For ϕ > 47%, P m is first positive and suddenly decreases (in the region of the apparent high viscosity plateau). Following [START_REF] Richards | Turning a yield-stress calcite suspension into a shear-thickening one by tuning inter-particle friction[END_REF] and [START_REF] Dhar | Signature of jamming under steady shear in dense particulate suspensions[END_REF], we interpret the drop away from the model in the high viscosity plateau in Figure 4b along with the sudden decrease of the normal stress as a fracturing of the jammed suspension for ϕ ≥ 47% at high shear rate.

Using the normal stress sensors, we now focus on the variations of the local normal stress P with time t along the flow curves of calcium carbonate, measured for two sensors placed symmetrically below the geometry (see the insert of Fig. 4c). The top panel of Figure 4d shows a typical signal in the continuous shear-thickening regime (ϕ = 45%, pink square) and the middle panel shows it in the discontinuous shear-thickening regime (ϕ = 50%, light blue). In both cases, the pressure P does not vary with time, a striking difference with what is observed in cornstarch suspensions. Here, P is constant, positive and of the order of 10 Pa, in good agreement with the mean normal stress P m (figure 4c). The bottom panel of Figure 4d shows the signal in the fracturing -shear-jammedregime (dark blue square). In this region, the two sensors measure a different signal, which varies with a timescale of ∼ 10 s. This is a bit larger than the period of rotation of the geometry, and could indicate slow rearrangements of the fracture within the jammed suspension. Here also, the signal is fundamentally different from anything that we have detected in cornstarch.

C. Discussion

The results obtained with the calcium carbonate suspension show that flow instabilities and fluctuations are not present in all shear-thickening fluids, despite being often reported [START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF][START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF][START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF][START_REF] Rathee | Localized stress fluctuations drive shear thickening in dense suspensions[END_REF]. A stable flow in the shear-thickening regime, as seen for calcium carbonate is fully in line with theory and simulations, which predict that both homogeneous or inhomogeneous flows are possible [START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Olmsted | Perspectives on shear banding in complex fluids[END_REF][START_REF] Chacko | Dynamic vorticity banding in discontinuously shear thickening suspensions[END_REF]. In the case of inhomogeneous flows, what seems expected from the Wyart and Cates model is unstable bands moving in the vorticity direction [START_REF] Chacko | Dynamic vorticity banding in discontinuously shear thickening suspensions[END_REF].

In the following discussion, we will focus on the local flow and the shear-thickening process in cornstarch. We will first consider the nature and origin of the stress wave. We will then see how this new result fits in with the current knowledge on cornstarch, by carrying out a critical analysis of the bibliography. We will finally see that cornstarch seems to belong to another family of shear-thickening fluids, which calls for an additional shear-thickening mechanism.

On the nature of the stress wave in cornstarch

In previous experiments, large temporal fluctuations of the shear rate or the viscosity have been reported [START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF][START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF][START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF][START_REF] Boersma | Timedependent behavior and wall slip in concentrated shear thickening dispersions[END_REF][START_REF] Pan | S-shaped flow curves of shear thickening suspensions: Direct observation of frictional rheology[END_REF][START_REF] Bossis | Discontinuous shear thickening in the presence of polymers adsorbed on the surface of calcium carbonate particles[END_REF], associated with an heterogeneous flow [START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF][START_REF] Saint-Michel | Uncovering instabilities in the spatiotemporal dynamics of a shear-thickening cornstarch suspension[END_REF][START_REF] Rathee | Localized stress fluctuations drive shear thickening in dense suspensions[END_REF][START_REF] Rathee | Localized transient jamming in discontinuous shear thickening[END_REF]. Here, we evidence in cornstarch something of the same line: with the help of the pressure sensors, we clearly detect in the parallel-plane geometry a unique wave, moving in the velocity direction, which presence generates macroscopic fluctuations of the shear rate and normal stress. An original aspect of our work is to reveal that there are two categories of pressure signals, depending on the confinement and on the fraction of particles in the suspension. Interestingly, the existence of two different local flows reconciles two previous experiments performed separately on cornstarch, by Ovarlez et al [START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF] and Rathee et al. [START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF].

Indeed, using X-ray radiography in a 41% cornstarch suspension, Ovarlez et al. see a single and regular wave moving at a constant speed Ω a < Ω in the velocity direction, associated with an object denser than the rest of the fluid. This signal is fully consistent with our experimental observations of the "slow" stress wave. Using the same suspension and the same gap width as in Ref.

[10] (h = 0.5 mm; in a parallel-plate geometry instead of a Couette cell) we measure a normal stress signal with a velocity Ω a = 0.40 Ω, which is close to the value reported in Ref. [START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF], with Ω a = 0.30 Ω. By combining our results, we deduce that the signal that we detect at large weight fractions and small gap width is most likely associated with the motion of a high solid fraction object, associated with a local jamming of the suspension. This rolling floc would naturally move at a velocity Ω a = 0.5 Ω or more slowly if it partially slides: this is exactly what we observe in our experiments (Figure 3) and what is shown in Ref. [START_REF] Ovarlez | Density waves in shearthickening suspensions[END_REF].

By contrast, at low weight fraction ϕ w ≤ 40% and large gap height (h > 1 mm), the signal that we detect is faster than the geometry, which is not consistent with the motion of an assembly of particles. We infer that, in this region, the signal is rather the signature of a density wave. This observation is compatible with the recent results of Rathee et al. [START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF]: by combining boundary stress microscopy and particle tracking, they evidenced (at a smaller scale than our experiment) the presence of stress waves associated with propagating fronts. Similarly to our experiment, the stress waves are moving in the velocity direction at a velocity Ω a ≃ Ω or slightly higher. However, a quantitative comparison is not possible: in Ref [START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF], the authors add glycerol in the solution, which has been shown to modify the interparticle forces [START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF].

How can we now explain the change in the nature of the signal with the experimental conditions? Log-rolling flocs, which could correspond to what we see at high fraction of particles and small gap, are often seen in dilute suspensions of weakly attractive or adhesive particles sheared between parallel planes. The conditions of formation and of stability of these objects has been studied recently by Varga et al. [START_REF] Varga | Hydrodynamics control shear-induced pattern formation in attractive suspensions[END_REF]. They have shown that log-rolling flocs can only appear when the viscous drag force acting on them (which tends to break them) is smaller than the total interaction force between the particles within the floc. Thus, a stable phase separation in a system of attractive or adhesive particles is only possible at low enough shear rate and gap height [START_REF] Varga | Hydrodynamics control shear-induced pattern formation in attractive suspensions[END_REF]. This is a feature that we also see in cornstarch, as evidenced in Figure 5. In this phase diagram, we show the region in which the possible aggregate is seen (in red) as a function of the shear rate and dimensionless gap height h/a, with a the mean particle size. The green dots show the cases where a stress wave is detected. There is a clear transition between the two regimes, the aggregate being only present at low enough γ and h/a. These results suggest that cornstarch particles are attractive and/or adhesive at high stress. A way to verify this hypothesis, and to estimate the typical interaction force between the particles is to determine the stability region of the flocs, in our experiment. To do so, we follow the approach of Varga et al. [START_REF] Varga | Hydrodynamics control shear-induced pattern formation in attractive suspensions[END_REF]. At the boundary between the stress waves and the aggregates, we expect the objects to have the same size as the gap, and roll at half the speed of the top plate. In a shear flow, an aggregate of diameter h is subjected of a torque M = 6πη γh 3 that destabilizes it. It is stable only if the yield torque Γ associated with its internal structure is capable of resisting it. Γ is directly proportional to the microscopic force F associated with the attractive or adhesive bonds between the particles. It writes Γ = n l F a, with n l ∼ (h/a)

d f -1
the number of links in a plane of the fractal structure and a the size of a particle. Thus, the condition of stability of the particles M < Γ writes 6πη γh 3 < (h/a) d f -1 aF , which can be expressed as a function of the Mason number: Mn < 2.5 h a d f -4 , with Mn = 6πη γa 2 F . d f is taken to be 2.6 in the Varga et al. [START_REF] Varga | Hydrodynamics control shear-induced pattern formation in attractive suspensions[END_REF] experiments, so the transition criterion is Mn < 2.5 h a -1.4 .

In Figure 5, we compare our experimental results to the model (shown with a dotted gray line) with a = 15 μm. In the flow curves of cornstarch, the shear-thickening regime (where the pressure waves are present) appears at a critical shear rate γ, which value decreases sharply with the weight fraction ϕ w (Supplementary Figure 2a). For this reason, the transition between a stress wave and an aggregate when increasing the weight fraction at a fixed gap height h (Figure 3e) directly translates as a γ transition. On figure 5, a transition at high shear rates correspond to low weight fractions, while a transition at low shear rates is associated with a high fraction of particles. In this ( γ, h/a) phase diagram, the model fits nicely our experimental measurements, by taking an interparticle force at contact F = 20 nN, a value consistent with an earlier measurement of the force between two particles using atomic force microscopy [START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF]. This result has two important consequences: first, it shows that the results of Varga et al. [START_REF] Varga | Hydrodynamics control shear-induced pattern formation in attractive suspensions[END_REF] can apply to dense suspensions, where the aggregates cannot be visually observed, but can be detected by other means. In addition, and more importantly, it shows the existence of an attractive or an adhesive force between the cornstarch particles at high shear. As discussed later, other papers lift the uncertainty on the nature of the force, and clearly point towards an adhesive force (present after contact between the particles) instead of an attractive force that could be measured before contact. To our knowledge, this is the first time that an interparticle force can be indirectly measured in the flow. Indeed, such forces are usually measured using atomic force microscopy (AFM) [START_REF] Hsu | Exploring the roles of roughness, friction and adhesion in discontinuous shear thickening by means of thermo-responsive particles[END_REF][START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF][START_REF] James | Interparticle hydrogen bonding can elicit shear jamming in dense suspensions[END_REF], by approaching a particle glued to the tip of the AFM to another particle fixed to a substrate. This method is very accurate but it is limited: it cannot fully mimic what happens under shear, since the dynamics of the approach and retraction between two particles in the flow are often too fast to be reproduced with this tool.

Microscopic and macroscopic behavior of cornstarch

The presence of adhesion at high stress that we evidence here is particularly meaningful when it is combined to what is already known about cornstarch.

Cornstarch particles are made of two biopolymers (amylose and amylopectin) that are partially soluble in water [START_REF] Buleon | Starch granules: structure and biosynthesis[END_REF][START_REF] Green | Which starch fraction is water-soluble, amylose or amylopectin?[END_REF]. When dispersed in water, the granules are thus surrounded by a soft polysaccharide layer, which is evidenced when probing the microscopic interaction between particles. This has been done with quartz-tuning fork atomic force microscopy [START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF] and classical atomic force microscopy (AFM) [START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF]. Before contact, the interaction force between the starch grains is very small [START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF][START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF]. The force profiles do not evidence any attraction, and repulsive forces, if any, are smaller than 2 nN. In the Wyart and Cates model [START_REF] Wyart | Discontinuous shear thickening without inertia in dense non-brownian suspensions[END_REF], a 2 nN repulsion force would imply a critical stress for the shear-thickening of order 1 Pa: this is more than one order of magnitude lower than the 20 Pa stress measured experimentally. This points to a first discrepancy with the Wyart and Cates scenario: if cornstarch was following the classical model, the expected repulsive force would be clearly observable, as e.g. for PVC particles [START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF]. Furthermore, while there is no measurable attractive force (that would be visible when approaching the particles), a strong adhesive force between the particles is measured. When the particles are put in contact with an indentation force of the order of 100 nN, an hysteresis is observed at the retraction, which indicates the presence of an adhesive force of the order of 10 -20 nN (in water) [START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF]. This force extends over typically 100 nm after contact. When separating the particles, a series of "pulling events" have been evidenced, corresponding to small and sudden jumps in the force profile [START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF]. Such events are seen when high density dangling polymers (in a poor solvent) disentangle, and corresponds to the breaking of bonds during the retraction [START_REF] Yu | Cosolvency-induced switching of the adhesion between poly (methyl methacrylate) brushes[END_REF]. These results indicate that cornstarch particles interact through adhesive forces at high stress. They are most likely slightly swollen, [START_REF] Belitz | Food chemistry[END_REF] and surrounded by a collapsed polymer brush, a microscopic configuration that is known to generate adhesion [START_REF] Hsu | Exploring the roles of roughness, friction and adhesion in discontinuous shear thickening by means of thermo-responsive particles[END_REF].

The absence of a significant repulsive force implies that cornstarch particles are already in solid contact at low stress, in opposition to what is expected from the Wyart and Cates model. This is supported by three other observations. First, cornstarch suspension is a yield stress fluid: it has been evidenced in particular by Fall et al. [START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF], which measured the velocity profiles of a cornstarch suspension in a Couette Cell. From the shear-localization of the flow, they measured a yield stress τ y of the order of 0.1 Pa. This points to a small attractive force between the particles at low shear, of the order of τ y π(a/2) 2 ≃ 0.03 nN, far below the resolution on the force measurements of [START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF][START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF]. Second, the avalanche angle measurements performed by Clavaud et al. [START_REF] Clavaud | Revealing the frictional transition in shear-thickening suspensions[END_REF], which have been successful in showing that potato starch and silica particles are in a frictionless state at low shear stress, failed to show the same feature for cornstarch. The results are not shown in Ref. [START_REF] Clavaud | Revealing the frictional transition in shear-thickening suspensions[END_REF] but can be found in the PhD dissertation [START_REF] Clavaud | Rhéoépaississement des suspensions denses: mise en évidence de la transition frictionnelle[END_REF], where aggregate formation and high value of the avalanche angle are reported (see e.g. Fig. 2.10 in [START_REF] Clavaud | Rhéoépaississement des suspensions denses: mise en évidence de la transition frictionnelle[END_REF]), suggesting that the low shear stress state is not frictionless. Third, it should be mentioned that pressure-imposed rheology, which has been successful in demonstrating the frictionless/frictional transition in potato starch and silica beads when increasing the particle pressure [START_REF] Etcheverry | Capillarystress controlled rheometer reveals the dual rheology of shear-thickening suspensions[END_REF], failed to demonstrate the same behavior for cornstarch. The results are not shown in Ref. [START_REF] Etcheverry | Capillarystress controlled rheometer reveals the dual rheology of shear-thickening suspensions[END_REF] but can be found in the PhD dissertation [START_REF] Etcheverry | Le capillarytron: un nouveau rhéomètre pour sonder la rhéologie frictionnelle des suspensions colloïdales[END_REF] where high values of the macroscopic friction coefficient are reported for dense slow flows at low particle pressure (see eg. Fig. 4.3 in [64]), suggesting again that the low shear stress state is not frictionless.

It should finally be noted that most papers that report an agreement between the rheological behavior of cornstarch and the Wyart and Cates model only fit the flow curves in a very limited range of shear rates, close to the shear thickening. The low shear-rate region is often disregarded, as well as the high shear rate limit. Both regions actually show a shear stress plateau as reported by Fall et al. [START_REF] Fall | Macroscopic discontinuous shear thickening versus local shear jamming in cornstarch[END_REF], who show the flow curves for a wide range of shear rates and volume fractions.

From all these previous results combined, we point out the presence of attractive forces at low stress in cornstarch and the absence of a frictionless/frictional transition with increasing stress. Our experimental results add another important information to this: cornstarch particles are also in contact at high stress. In this region of the flow curve, the adhesive force is of the order of 20 nN, which is 1000 times higher than the attractive force at the origin of the yield stress.

Cornstarch thus appears as a singular shear-thickening fluid, in which particles are in contact all along the flow curve, due to attractive or adhesive forces. For this reason, it cannot fit in with the Wyart-Cates framework, which assumes repulsive short-range forces between the particles to explain shear thickening. Macroscopically, the flow curves also differ from the model. Indeed, Most papers that report an agreement between the rheological behavior of cornstarch and the Wyart and Cates model only fit the flow curves in a very limited range of shear rates, close to the shear thickening. The low shear-rate region is often disregarded, as well as the high shear rate limit. Both regions actually show a shear stress plateau as reported by Fall et al. [START_REF] Fall | Macroscopic discontinuous shear thickening versus local shear jamming in cornstarch[END_REF], who show the flow curves for a wide range of shear rates and volume fractions.

Shear thickening in presence of adhesive forces

From the previous discussion, we see that another interpretation has to be found to explain the flow curve of cornstarch. We suggest here that cornstarch particles are very weakly attractive in suspensions in CsCl brine (to account for the yield stress), and strongly adhesive once the normal stresses pushes them into contact. More precisely, the initial contact between particles induces a small yield stress. When subjected to stress, the fragile bonds break and the suspension turns to a liquid, exhibiting a strongly shear thinning behavior. Beyond a critical stress, the particle pressure pushes the particles in contact and involves an important indentation. The polysaccharide chains surrounding the starch granules become entangled. An adhesive force coming from hydrogen bonds is set up. In this regime, there is a competition between the time for the polysaccharide to disentangle and the contact time (∝ 1/ γ) between the particles. When the first is larger than the latter, the system thickens.

IV. CONCLUSION

To better understand the mechanism of shear thickening of cornstarch solutions, we study the structure of the suspensions under flow, with a focus on the high shear region. Using in-house sensors, we probe at the millimeterscale, in a new way, the normal stresses that develop in two shear-thickening fluids: calcium carbonate and cornstarch. We show that the model suspensions of calcium carbonate particles (which are known to interact repulsively at low stress) remains homogeneous at all shear stresses, with a macroscopic flow curve that matches the Wyart and Cates model. In cornstarch, however, the sensors evidence a regular normal stress signal in the shear thickening regime. The nature of this signal changes with the constraining conditions: at low weight fractions ϕ w ≤ 40% and large gap widths (h ≥ 1 mm) the signal is consistent with a stress wave moving 1 to 1.5 times faster than the geometry. At high weight fractions ϕ w ≥ 41% and small gap (h < 1 mm), the signal arises from the passage of a solid aggregate rolling between the plates. This transition is explained by taking into account the presence of adhesive forces between the particles at high stress. The presence these forces, combined with the existence of a yield stress (showing that the particles are also in contact at low stress) clearly indicates that cornstarch particles are in contact all along the flow curve.

To the best of our knowledge, this is the first time that the behavior of cornstarch suspensions at high shear stress is quantitatively described as a function of the gap height, shear rate and fraction of particles in the parameter plane (h/a, γ). By mapping the boundary of the phase separation region as a function of the gap width and shear rate, we are able to use the presence of the solid aggregate as an original way to estimate the force of adhesion between the particles at high shear stress. The adhesion force measured at the transition is in perfect agreement with AFM measurements [START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF].

Our work evidences the need for a new mechanism in the current research on shear thickening. Indeed, a known effect of attractive forces is to reduce or hide the shear thickening [START_REF] Brown | Generality of shear thickening in dense suspensions[END_REF][START_REF] Pednekar | Simulation of shear thickening in attractive colloidal suspensions[END_REF]. To our knowledge, a shearthickening transition in presence of adhesive forces is not mentioned in the abundant literature on non-Brownian suspensions [START_REF] Brown | Shear thickening in concentrated suspensions: phenomenology, mechanisms and relations to jamming[END_REF][START_REF] Fall | Shear thickening of cornstarch suspensions as a reentrant jamming transition[END_REF][START_REF] Guy | Towards a unified description of the rheology of hard-particle suspensions[END_REF][START_REF] Niu | Tunable solidification of cornstarch under impact: How to make someone walking on cornstarch sink[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in nonbrownian suspensions[END_REF][START_REF] Lin | Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions[END_REF][START_REF] Royer | Rheological signature of frictional interactions in shear thickening suspensions[END_REF][START_REF] Clavaud | Revealing the frictional transition in shear-thickening suspensions[END_REF][START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF][START_REF] Morris | Lubricated-to-frictional shear thickening scenario in dense suspensions[END_REF][START_REF] Hermes | Unsteady flow and particle migration in dense, non-brownian suspensions[END_REF][START_REF] Rathee | Structure of propagating high-stress fronts in a shear-thickening suspension[END_REF] or hydrocolloids [START_REF] Wagner | Shear thickening in colloidal dispersions[END_REF][START_REF] Jamali | Microstructure and rheology of soft to rigid shear-thickening colloidal suspensions[END_REF][START_REF] Maranzano | Flow-small angle neutron scattering measurements of colloidal dispersion microstructure evolution through the shear thickening transition[END_REF]. Singh et al. [START_REF] Singh | Shear thickening and jamming of dense suspensions: the "roll" of friction[END_REF] show that cohesive suspensions can present a shear thickening flow curve, but the simulations also include a repulsive force that we do not see in cornstarch. Indeed, the current understanding of the mechanism is that a repulsive force must somehow ex-ist and be overcome during the flow to lead to an increase of the viscosity [START_REF] Wyart | Discontinuous shear thickening without inertia in dense non-brownian suspensions[END_REF][START_REF] Mari | Shear thickening, frictionless and frictional rheologies in nonbrownian suspensions[END_REF][START_REF] Morris | Lubricated-to-frictional shear thickening scenario in dense suspensions[END_REF]. While this vision works for a number of shear-thickening fluids [START_REF] Guy | Towards a unified description of the rheology of hard-particle suspensions[END_REF][START_REF] Lin | Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions[END_REF][START_REF] Royer | Rheological signature of frictional interactions in shear thickening suspensions[END_REF][START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF][START_REF] Richards | Competing timescales lead to oscillations in shear-thickening suspensions[END_REF][START_REF] Singh | Shear thickening and jamming of dense suspensions: the "roll" of friction[END_REF], we show here that it does not explain all occurrences of shear thickening. Strikingly, fluids such as the most famous one, namely cornstarch, in which particles are always in contact, can also exhibit typical shear-thickening flow curves.

We hope that this new insight will help to get a more complete picture of the shear-thickening process. This also brings to light a central issue: if cornstarch is far from being a canonical system, it should be used with care especially when its macroscopic flow is compared to the standard models and to simulations. In this line of work, our results also open new roads in suspensions formulation [START_REF] Gürgen | Shear thickening fluids in protective applications: A review[END_REF][START_REF] Zarei | Application of shear thickening fluids in material development[END_REF], as they highlight the possibility of engineering a new kind of adhesive shear thickening suspensions, which viscosity can be controlled through surface chemistry [START_REF] Hsu | Exploring the roles of roughness, friction and adhesion in discontinuous shear thickening by means of thermo-responsive particles[END_REF]. To finish with, our work highlights the considerable importance of the microscopic interac-tion between the particles. In the past years, new measurements methods and original processes have been developed to control and measure the inter-particle force [START_REF] Hsu | Exploring the roles of roughness, friction and adhesion in discontinuous shear thickening by means of thermo-responsive particles[END_REF][START_REF] Comtet | Pairwise frictional profile between particles determines discontinuous shear thickening transition in non-colloidal suspensions[END_REF][START_REF] Gálvez | Dramatic effect of fluid chemistry on cornstarch suspensions: Linking particle interactions to macroscopic rheology[END_REF][START_REF] James | Interparticle hydrogen bonding can elicit shear jamming in dense suspensions[END_REF]. In the future, a better knowledge of both the microscopic forces and the dynamics between particles seem necessary to understand the details of the flow at all scales.

FIG. 1 .

 1 FIG. 1. a. Normal stress measurement setup. A piezocapacitive pressure sensor array is placed at the bottom of a parallel-plate rheometer. The diameter of the geometry is 4 cm and the gap thickness is typically 1 mm. The sensor is a sandwich of many layers (see legend). Its core is a soft house-made dielectric foam, which capacitance (measured in 25 points) depends on the local pressure. b. Top view of the experiment, where each square represent a different pressure sensor. The blue circle shows the geometry. Here, the pressure is measured in the nine positions shown in red.

FIG. 3 .

 3 FIG. 2. a. Flow curve (viscosity η as a function of the shear rate γ) of a cornstarch suspension with weight fraction ϕw = 40%. b. Mean vertical pressure Pm as measured by the force sensor of the rheometer. c. Local normal stress as a function of time t, for increasing shear stresses σ. The inset of b. shows the position of the two measurement points in the sensor array: the red and blue sensors are placed symmetrically with respect to the center of the geometry. The position of the measurements in the flow curves are indicated with colored triangles.

FIG. 4 .

 4 FIG. 4. a. Flow curve of calcium carbonate suspensions, for volume fractions ϕ varied between 35% and 52%. The full circles correspond to the continuous shear-thickening regime, and the empty circles to the discontinuous shear-thickening regime. b. Relative viscosity ηr = η/η0 in the low (in blue) or high (in red) plateau (with η0 = 23 mPa s the viscosity of the suspending fluid). The blue and the red curves are fits of Krieger-Dougherty form [46] ηr = (1 -ϕ/ϕj) -β with β = 2.5. ϕj = ϕrcp = 0.62 for the low viscosity plateau and ϕj = ϕm = 0.48 for the high viscosity plateau. The vertical pink line materializes the divergence of the viscosity of the high viscosity branch at ϕ = ϕm; the vertical blue line shows the divergence of the viscosity of the low viscosity branch at ϕ = ϕrcp. c. Mean vertical stress Pm, the color code is the same as in a. d. Stress signal measured by the sensor array, of 2 sensors placed symmetrically below the geometry. The position of the measurements in the flow curves are indicated with the colored squares.

FIG. 5 .

 5 FIG.5. Phase diagram showing the two different normal stress signals (in red, the signal associated with a possible aggregate and in green, with a normal stress wave) as a function of the shear rate γ and the gap width h (compared with the particle size a = 15 μm. The dotted line corresponds to the prediction of Ref.[START_REF] Varga | Hydrodynamics control shear-induced pattern formation in attractive suspensions[END_REF] on the stability of log-rolling flocs, without any adjustable parameter.

Scientific Reports 9, 1250 (2019).

ACKNOWLEDGMENTS

We thank Vikram Rathee, Jeffrey Urbach, Daniel Blair and Joia Miller for fruitful discussions, and Mickaël Pruvost for his help in the fabrication of the sensors and the acquisition system.

Credit authorship contribution statement

Anaïs Gauthier: Investigation, Formal analysis, Writing -original draft, review & editing. Guillaume Ovarlez: Conceptualization, Writing -review & editing. Annie Colin: Conceptualization, Writing -review & editing, Resources, Project administration


Anaïs Gauthier

Focus on Fluids 1 FoF Header and Title image Slippery bounces

Keywords: Capillary flows, drops, drag reduction

The complex hierarchical texture covering the lotus leaf is at the source of two of its extraordinary properties. While its water-repellent properties are now emblematic, the lotus is much less known for its extreme slipperiness. And for good reason: until the recent work of Martouzet et al., the effect of slippage on drop impact dynamics had never been demonstrated. This remarkable study unveils a complex interplay between wetting and friction, with counter-intuitive consequences. Hierarchical structures, which minimize the contact between the substrate and the droplets, are less efficient at repelling viscous liquids than simpler systems, because of the slip! A clever and original approach, based on a scaling analysis of the spreading time, is used to disentangle the different physical phenomena at stake during drop impact. This is an important step towards a better understanding of the crucial problem of drop impact dynamics on both wetting and non-wetting substrates.

Introduction

By observing nature on a misty morning, one will notice that a number of plants are naturally engineered to repel water. Tiny droplets appear as shiny marbles which glide at the faintest quivering of the leaves. This remarkable feature is derived from the presence of a micrometer or nanometer-scaled hydrophobic roughness on the surface [START_REF] Neinhuis | Characterization and distribution of water-repellent, selfcleaning plant surfaces[END_REF]. In presence of water, large pockets of air remain entrapped between the textures: the rough material thus behaves as if it had a hybrid surface made of both solid and gas. Air is thus at the heart of two highly unusual attributes of the so-called superhydrophobic materials: extreme water-repellency and large-scale slipperiness. While water-repellency is captured by the contact angle of droplets, typically higher than 160 • (figure 1a), the amplitude of the slippage is measured through the slip length b -the extrapolated distance where the liquid tangential velocity vanishes (figure 1b). This length b, which is typically molecular on hydrophilic materials and of the order of 10 nm on hydrophobic surfaces, is multiplied by a factor 1000 on superhydrophobic substrates. The two distinct properties of these materials attracted the attention of different communities, drawn either by the unusual dynamics of drops deposited on the surfaces [START_REF] Blossey | Self-cleaning surfacesvirtual realities[END_REF] or by the drag-reduction potential of the immersed solids [START_REF] Rothstein | Slip on superhydrophobic surfaces[END_REF]. While a nano-structure alone is enough to promote excellent water repellency, most plants and insects possess a hierarchical structure. The celebrated lotus leaf, for example, is covered with 20 µm tall papillae, themselves covered with waxy nanostructures at the scale of 100 nm. On the one hand, the presence of a large texture can be seen as detrimental for the lotus leaf: it makes it sensitive to condensation [START_REF] Cheng | Is the lotus leaf superhydrophobic?[END_REF] and to droplet intrusion at impact [START_REF] Deng | Nonwetting of impinging droplets on textured surfaces[END_REF]). On the other hand, it does make the surface extremely slippery: tall pillars, separated by a relatively large distance generate giant slip lengths, of the order of 10 µm [START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF]). One might then wonder: what role does the slip play when the leaf is lashed by rain? This original question is the starting point of the work of Martouzet et al., whose approach also sheds new light into a much wider and debated issue on the role of friction on the impact dynamics of droplets on both wetting and non-wetting substrates.

Overview

Martouzet et al. focus on one of the most emblematic property of non-wetting surfaces: their ability to repel impacting drops (figure 1c). An important parameter here is the time the droplets spend in contact with the substrate, which determines the amount of mechanical and thermal exchanges with the solid. In practical problems, such as the freezing of airplane wings, it is crucial to make the contact time as short as possible.

Recent studies have demonstrated that the addition of well-chosen textures dramatically decrease the recoiling time, leading to a contact time reduction by a factor 2 or more [START_REF] Bird | Reducing the contact time of a bouncing drop[END_REF][START_REF] Gauthier | Water impacting on superhydrophobic macrotextures[END_REF][START_REF] Liu | Pancake bouncing on superhydrophobic surfaces[END_REF]. It is much harder, however, to reduce the duration of the spreading. Numerous experiments and simulations show that the first stage of impact is only very hardly influenced by the surface wetting properties, or by the impact velocity of the droplet. Despite its huge interest, the physical parameters that limit the spreading are still discussed. More importantly, conflicting theoretical models [START_REF] Clanet | Maximal deformation of an impacting drop[END_REF][START_REF] Roisman | Inertia dominated drop collisions. ii. an analytical solution of the navierstokes equations for a spreading viscous film[END_REF][START_REF] Eggers | Drop dynamics after impact on a solid wall: theory and simulations[END_REF] show reasonably good agreement with both simulations and experiments (see [START_REF] Josserand | Drop impact on a solid surface[END_REF] for a review). In their recent paper, Martouzet et al. tackle experimentally and theoretically this complex problem. Through a clever choice of surfaces, they clearly evidence the role of a lubricating air layer on the impact. They present their results in a global phase diagram in terms of Reynolds (Re) and Weber (We) numbers (which compare inertial forces respectively to viscous and surface tension forces), an original approach which reconciles the theoretical models to the spreading dynamics.

To disentangle the effects of wetting and friction, Martouzet et al. compare drop impact dynamics on two surfaces with identical non-wetting properties but with very different textures. Electron microscope images of the substrates are presented in Figure 2: the first surface (figure 2a) has a hierarchical structure similar to the lotus leaf made of tall pillars covered with nanometer-sized hairs. The large texture confers to the surface an extreme slipperiness, with a slip length b = 35 µm. The second surface (figure 2b) is covered with a unique texture at a smaller scale, associated with a moderate slip (b = 8 µm). An originality of the work of Martouzet et al. is to focus on the spreading time τ , which is essential in impact dynamics. Their experiment, presented in Figure 2 gives a surprisingly counter-intuitive result: the spreading lasts longer on the hierarchical surface than on the nanotextured one. The surface which is covered by the larger, thicker air pockets is thus less fast at repelling drops! The difference, of the order of 20% at high Reynolds numbers Re, is the first clear evidence of the influence of the friction on impact dynamics. The experiment therefore validates the idea that spreading is here stopped by the growth of a viscous boundary layer at the solid/liquid interface [START_REF] Roisman | Inertia dominated drop collisions. ii. an analytical solution of the navierstokes equations for a spreading viscous film[END_REF][START_REF] Eggers | Drop dynamics after impact on a solid wall: theory and simulations[END_REF][START_REF] Lastakowski | Bridging local to global dynamics of drop impact onto solid substrates[END_REF]. The spreading time τ on a slippery surface asymptotically increases with b as τ = τ 0 (1 + α), with α ∼ 2b 5R Re 1/5 (with R the drop size and τ 0 the spreading time in absence of slip). Martouzet et al. interestigly consider these results in light of the much more general question of drop impact dynamics on wetting and non-wetting substrates. The authors propose a clever scaling analysis based on the comparison of characteristic timescales associated either with inertia (τ i = R/U with U the drop impact velocity), capillarity (τ c = R U W e 1/2 ) and viscous friction (τ v = R U Re 1/5 ). They evidence three main regimes in a Re -We phase diagram, corresponding to different impact dynamics. In each regime, the spreading is governed either by the growth of a boundary layer (in the "viscous" region), by the rim dynamics (in the "capillary" region) or by the drop deceleration (in

Figure 1 .

 1 Figure 1. Some properties of superhydrophobic surfaces. a. Non-wetting: droplets deposited on the surface keep the spherical shape their have in air. b. Large-scale slipperiness: surfaces covered with water are extremely slippery, with a slip length b of ∼ 10 µm (defined as the distance below the top of the pillars for which the velocity profile of a flowing liquid vanishes) 1000 times higher than on hydrophobic materials. c. Drop bouncing on a non-wetting surface.

Figure 2

 2 Figure 2. a. SEM image of the slippery hierarchical surface, with b = 35 µm b. Image of the nanotextured surface, with b = 8 µm. c. Effect of the slipperiness on the non-dimensional spreading time τ /τ0 on two superhydrophobic surfaces, with different slip lengths: in blue, b = 35 µm and in red b = 8 µm. τ0 is the spreading time on a smooth hydrophilic silicone substrate. From Martouzet et al. (2020)

the "inertial" region). As demonstrated by the authors, the effect of the slip is thus only visible for viscous impacts, a region defined by spreading times τ = τ v < τ c , i.e. for Re < We 5/2 . This global approach disentangles the existing models of drop spreading and provides a general framework in the analysis of drop impact dynamics. More generally, it demonstrates that the regime of impact should be considered with care, as it might dramatically alter the physics of the spreading.

Future

The work of Martouzet et al. shows that the subtle interplay between wetting and slip generates unexpected effects, such as an effective reduction of water-repellency by hierarchical textures, which are the most lubricated superhydrophobic materials. Is it then all lost for the hierarchical lotus leaf? Not necessarily: typical raindrops impact at high Reynolds number, typically Re > 1000. The spreading dynamics of the drops is then much more likely governed by the rim formation than by friction. And indeed, the authors evidence an opposite effect in this region of the phase diagram: the spreading time on the hierarchical surface is reduced by 30% compared with a smooth surface. This interesting and surprising additional observation surely deserves a more in-depth analysis. On another note, this original approach at the interface between wetting and drag reduction opens many questions. One might wonder, for example: what is the impact of the slip on the receding and take-off of droplets? Would a larger slip length facilitate the evacuation of viscous rolling droplets, which are usually much harder to repel than water?