Abdoulaye Fall

François Bertrand

Guillaume Ovarlez

Daniel Bonn

Shear Thickening of Cornstarch Suspensions

We study the rheology of cornstarch suspensions, a non -Brownian particle system that exhibits discontinuous shear thickening. Using Magnetic Resonance Imaging (MRI), the local properties of the flow are obtained by the determination of local velocity profiles and concentrations in a Couette cell. For low rotational rates, we observe shear localization characteristic of yield stress fluids. When the overall shear rate is increased, the width of the sheared region increases. The discontinuous shear thickening is found to set in at the end of this shear localization regime when all of the fluid is sheared: the existence of a non-flowing region thus seems to prevent or delay shear thickening. Macroscopic observations using different measurement geometries show that the smaller the gap of the shear cell, the lower the shear rate at which shear thickening sets in. We thus propose that the discontinuous shear thickening of cornstarch suspensions is a consequence of dilatancy: the system under flow attempts to dilate but instead undergoes a jamming transition because it is confined. This proposition is confirmed by an independent measurement of the dilation of the suspension as a function of the shear rate. It is also explains the MRI observations: when flow is localized, the non-flowing region plays the role of a "dilatancy reservoir" which allows the material to be sheared without jamming.

I. Introduction:

Shear thickening has been observed for a wide variety of suspensions [START_REF] Barnes | Shear-Thickening ('Dilatancy') in Suspensions of Non aggregating Solid Particles Dispersed in Newtonian Liquids[END_REF]]. The phenomenon is frequently encountered during the processing of concentrated dispersions in various industries where it has a strong impact on energy consumption. Shear thickening can be defined as an increase in the steady-state shear viscosity η of a fluid with the shear rate γ when the latter exceeds some critical value c γ . The detailed mechanism of shearthickening is still under debate [see e.g. [START_REF] Boersma | Shear thickening (dilatancy) in concentrated dispersions[END_REF], [START_REF] Franks | Effect of interparticle forces on shear thickening of oxide suspensions[END_REF], [START_REF] Hoffman | Discontinuous and Dilatant Viscosity Behavior in Concentrated Suspensions. I. Observation of a Flow Instability[END_REF], [START_REF] Macias | On the shear thickening flow of dilute CTAT worm-like micellar solutions[END_REF], [START_REF] Laun | [END_REF], [START_REF] Hoffman | Discontinuous and dilatant viscosity behavior in concentrated suspensions. II. Theory and experimental tests[END_REF], [START_REF] Foss | Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation[END_REF], [START_REF] Ackerson | Shear induced order and shear processing of model hard sphere suspensions[END_REF] and [START_REF] Chen | Rheological consequences of microstructural transitions in colloidal crystals[END_REF]. The majority of investigations of shear thickening were conducted on colloidal suspensions [START_REF] Boersma | Shear thickening (dilatancy) in concentrated dispersions[END_REF], [START_REF] Franks | Effect of interparticle forces on shear thickening of oxide suspensions[END_REF], [START_REF] Hoffman | Discontinuous and Dilatant Viscosity Behavior in Concentrated Suspensions. I. Observation of a Flow Instability[END_REF]]. [START_REF] Bender | Reversible shear thickening in monodisperse and bidisperse colloidal dispersion[END_REF] and [START_REF] Maranzano | Flow-small angle neutron scattering measurements of colloidal dispersion microstructure evolution through the shear thickening[END_REF] attribute the phenomenon to the shear-induced formation of hydrodynamic clusters -transient concentration fluctuations that are driven and sustained by the applied shear field. The viscosity rise is continuous at low volume fractions, but can also be discontinuous at higher ones [START_REF] Macias | On the shear thickening flow of dilute CTAT worm-like micellar solutions[END_REF], [START_REF] Laun | [END_REF], [START_REF] O'brien | Stress components and shear thickening of concentrated hard sphere suspensions[END_REF], [START_REF] Bertrand | From shear thickening to shear-induced jamming[END_REF]], probably because of aggregation of hydroclusters creating a jammed network [START_REF] Hoffman | Discontinuous and dilatant viscosity behavior in concentrated suspensions. II. Theory and experimental tests[END_REF], [START_REF] Foss | Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation[END_REF]]. In the latter case, the clustered shear thickened state may be a long-lived metastable state characterized by a large yield stress, as shown by [START_REF] Cates | Dilatancy, jamming, and the physics of granulation[END_REF]. In the picture of [START_REF] Bender | Reversible shear thickening in monodisperse and bidisperse colloidal dispersion[END_REF], the formation of flow-induced hydroclusters results in an increased dissipation of energy and, consequently, the viscosity increases. The commonly accepted picture for shear thickening in Brownian suspensions is now the formation of shear-induced hydroclusters [see e.g. the recent summary by [START_REF] Wagner | Shear thickening in colloidal dispersions[END_REF]]. This results in large hydrodynamic stresses in rapid flowing suspensions, and there is simulation and other evidence that these are the dominant stresses in the shear-thickened regime [START_REF] Bender | Reversible shear thickening in monodisperse and bidisperse colloidal dispersion[END_REF], [START_REF] Phung | Stokesian dynamics simulation of brownian suspensions[END_REF]]. However, some of these simulations are limited to relatively modest volume fractions ( 49 . 0 ≤ ϕ ); experiments at much higher concentrations suggest that instead a thermodynamic mechanism may take place [START_REF] O'brien | Stress components and shear thickening of concentrated hard sphere suspensions[END_REF]]. In any case it seems highly unlikely that the 'static jamming' observed by [START_REF] Bertrand | From shear thickening to shear-induced jamming[END_REF] is due to hydrodynamic interactions alone, since in the persisting solid phase there is no macroscopic flow to provide those interactions. Also, as far as we know, hydrodynamic models for shear thickening offer no immediate explanation of nonmonotonic regions of the flow curve [START_REF] Holmes | Glass transitions and shear thickening suspension rheology[END_REF]]. Thus it seems clear that mechanisms other than pure hydrodynamics are at work in sufficiently dense shear-thickening suspensions [START_REF] O'brien | Stress components and shear thickening of concentrated hard sphere suspensions[END_REF], [START_REF] Bertrand | From shear thickening to shear-induced jamming[END_REF]]. Indeed, it has already been emphasised [Ball and Melrose (1995), Melrose and Ball (1995;2004a,b)] that deviations from pure lubrication forces can dominate the physics of any hydrodynamically clustered state [START_REF] Holmes | Glass transitions and shear thickening suspension rheology[END_REF]].

Shear thickening is also observed in non-Brownian suspensions with larger particle sizes [START_REF] Williamson | Some properties of dispersions of quicksand type[END_REF]; [START_REF] Fall | Shear thickening in cornstarch suspension is a re-entrant jamming transition[END_REF]2010); van der Werff and de [START_REF] Van Der Werff | Hard-sphere colloidal dispersion: The scaling of rheological properties with particle size, volume fraction, and shear rate[END_REF]; [START_REF] Sellitto | Shear-Thickening and Entropy-Driven Reentrance[END_REF] Berthier, et al (2000); [START_REF] Brown | Dynamic jamming point for shear thickening suspensions[END_REF]2010)]; here the mechanisms at play are less clear. Recently, [START_REF] Brown | Dynamic jamming point for shear thickening suspensions[END_REF] ). However [START_REF] Fall | Shear thickening and migration in granular suspension[END_REF], in a non-Brownian particle suspensions similar that those of Brown and Jaeger show that the intrinsic behavior (from local MRI measurements) shows only viscous (n = 1) or granular scaling (with n = 0.5) and that shear thickening simply corresponds to the transition between the two regimes. Note that, we call this regime granular; however it is important to realize that the scaling law with n = 0.5 can correspond to any inertial flow, including that of Newtonian fluids, and is not necessarily a granular scaling. The MRI data showed that in steady-state such systems are heterogeneous due to particle migration, and that consequently the macroscopic stress-strain rate relationship cannot be directly related to the local constitutive behavior and thus in particular to the shear thickening.

Here, we compare local and global measurements for what is perhaps the best -known example of a shear thickening suspension: cornstarch particles suspended in water. We show that the shear thickening can in fact be viewed as a re -entrant solid transition in this system: (i) at rest the material is solid because it has a (small) yield stress; (ii) for low shear rates, shear banding (localization) occurs, and the flowing shear band grows with increasing shear rate, the shear thus liquefies the material; (iii) shear thickening happens at the end of the localization regime, where all of the material flows, subsequently it suddenly becomes ''solid'' again. In addition, (iv) we find a pronounced dependence of the critical shear rate for the onset of shear thickening on the gap of the measurement geometry, which can be explained by the tendency of the sheared system to dilate. This is confirmed by an independent measurement of the dilation of the suspension as a function of the shear rate. It also explains the MRI observations: when flow is localized, the non-flowing region plays the role of a "dilatancy reservoir" which allows the material to be sheared without jamming.

This paper follows up on our earlier work on shear thickening of cornstarch [START_REF] Fall | Shear thickening in cornstarch suspension is a re-entrant jamming transition[END_REF]], but is much more detailed in that here we present also the MRI measurements of the concentration, more detailed measurements of the velocity profiles, plate-plate measurements, oscillation measurements and more detailed measurements of the variation of the gap of the plate-plate cell under an imposed normal stress. In order for these new data to be comprehensible we do have to repeat some of the earlier data and discussion. In this way we obtain a more complete picture of the shear thickening behavior.

II. Materials and methods

The cornstarch particles (from Sigma Aldrich) are relatively monodisperse particles with, however, irregular shapes [Figure 1]. Suspensions are prepared by mixing the cornstarch with a 55 wt% solution of CsCl in demineralized water. The CsCl allows one to perfectly match the solvent and particle densities [START_REF] Merkt | Persistent holes in a fluid[END_REF]]. We study suspensions of volume fraction ranging between 38% and 46%, and focus here mainly on the behavior of a 44% cornstarch suspension that is representative of the rest. The effect of changing the volume fraction will be discussed in detail in Sec.III.6. Experiments are carried out with a vane-in-cup (inner cylinder radius Ri = 12.5 mm, outer cylinder radius Re = 18.5 mm, height H = 45 mm) or parallel plate geometry on a commercial rheometer (Bohlin C-VOR 200) that imposes either the torque or the rotational velocity (with a torque feedback). The vane geometry is equivalent to a cylinder with a rough lateral surface which reduces wall slip [START_REF] Larson | The structure and rheology of complex fluids[END_REF]]. The inside of the cup is also covered with the granular particles using double sided adhesive tape. For the parallel plate geometry, the upper plate is of 40 mm diameter; both plates are roughened. Velocity profiles in the flowing sample were obtained with a velocity controlled magnetic resonance imaging (MRI) rheometer from which we directly get the local velocity distribution in a Couette geometry with a gap of 1.85 cm. We investigated the stationary flows for inner cylinder rotational velocity Ω ranging between 0.2 and 10 rpm, corresponding to overall shear rates between 0.04 and 2.35 s -1 [START_REF] Raynaud | Direct determination by NMR of the thixotropic and yielding behavior of suspensions[END_REF], [START_REF] Rodts | From "discrete" to "continuum" flow in foams[END_REF]), Bonn et al.(2008)]. All the measurements were done in the same laboratory with a controlled humidity of 40%; cornstarch is likely to take up some water from the atmosphere.

III. Experimental results

Typical macroscopic behavior

Let us first present the typical behavior observed when shearing a cornstarch suspension (Figure 2 S hear rate (s -1 )

Figure 2: Apparent viscosity vs. shear rate when performing a step stress test in a vane in cup geometry.

The viscous properties at higher stresses were measured with step stress tests. The basic trends in ( ) γ η are similar for all measured volume fractions: at low stresses, η decreases with increasing applied stress, reflecting shear- thinning behavior. As the stress increases further, η increases, reflecting shear-thickening behavior, and then reaches a plateau. This abrupt increase in viscosity observed is characteristic of "discontinuous" shear thickening. To better understand this behavior, we performed MRI measurements in a Couette geometry.

Local rheology: velocity and concentration profile measurements

In Figure 3, we plot the dimensionless velocity profiles for the steady flows of a cornstarch suspension, for various rotational velocities ranging from 0.2 to 9 rpm. The MRI measurements of the velocity profiles first, show that there is no slip to within the experimental uncertainty. The data shown is normalized on the speed of the rotating inner cylinder, and extrapolates to unity, showing that slip is negligible. Second, the flowing part of the sample occupies only a small fraction of the gap at low rotation velocities: we observe shear localization. The velocity profiles are composed of two regions: the part close to the inner cylinder is moving, and the rest is not. For the lower rotation speeds, since the part of the material that does not move is subjected to a stress, this implies that the suspension has a yield stress. The yield stress can be determined from the critical radius c r at which the flow stops: the shear stress at a given radius r as a function of the applied torque T and fluid height H follows from momentum balance, and thus the yield stress at c r follows immediately as . The yield stress turns out to be on the order of 0.3 Pa. Although it seems obvious that concentrated suspensions that show shear thickening also have a yield stress, we have not found literature comparing the prethickening flow behavior to a Bingham model as we do here, with the exception of the recent work of Brown and Jaeger [START_REF] Brown | On the generality of shear thickening in suspensions[END_REF] where the pre-thickening behavior was compared to a Herschel-Bulkley model. This is probably due to the fact that the yield stress is low. We can detect it relatively easily here because we use the MRI data.

A striking observation is that the shear rate at the interface between sheared and unsheared regions is different from zero, in contrast with what is observed in simple yield stress fluids [START_REF] Bonn | Yield stress fluids slowly yield to analysis[END_REF], Moller et al. (2009), [START_REF] Ovarlez | Wide gap Couette flows of dense emulsions: Local concentration measurements, and comparison between macroscopic and local constitutive law measurements through magnetic resonance imaging[END_REF][START_REF] Ovarlez | Investigation of shear banding in three-dimensional foams[END_REF]. We indeed observe that the slope of the velocity profile at that point [Figure 3, Insert] is equal to 0.2s -1 . This implies that below 1 2 . 0 - ≈ s yield γ there is no stable flow. The existence of a critical shear rate for yield stress materials has been discussed previously in detail, and requires that the system be (slightly) thixotropic. The critical shear rate associated with the yielding of thixotropic materials has been discussed in detail elsewhere [START_REF] Moller | Yield stress and thixotropy: on the difficulty of measuring yield stresses in practice[END_REF][Moller et al. ( ), (2008[Moller et al. ( ), (2009))]. In our case, this is likely to be due to competition between slight sedimentation or creaming, i.e., a density matching that is not perfect and shear-induced resuspension [Coussot et al., (2002a, b); [START_REF] Fall | Yield stress and shear banding in granular materials[END_REF]]. Upon increasing the rotation rate, a larger part of the fluid is sheared, and for the highest rotation speeds the sheared region occupies the entire gap. We are unable to go to higher rotation rates in the MRI since the shear thickening sets in immediately when the shear band occupies the entire gap of the Couette cell, and when it does the motor of the rheometer is no longer sufficiently strong to rotate the inner cylinder: shear thickening is observed as an abrupt increase of the measured torque on the rotation axis. We therefore see the onset of shear thickening at the first shear rate for which all of the material is sheared; we will show below that this is likely to happen because the presence of a non-flowing region delays and attenuates the shear thickening. From the velocity profiles ) (r v , we can determine the constitutive behavior. In the flowing part, the local shear rate within the Couette gap can be given as ( ) The MRI also allows us to measure the particle concentration; to within the experimental uncertainty of ±0.2% in volume fraction the particle concentration is homogeneous throughout the gap. Thus, during the flow and at the onset of shear thickening, the suspension remains homogeneous within the gap. As a result, no migration of particles is observed to the limit of the transition to the shear thickening regime. This is perhaps surprising in the light of recent measurements of shear thickening in suspensions of spherical particles, where shear thickening is always accompanied by particle migration [START_REF] Fall | Shear thickening and migration in granular suspension[END_REF]. In Couette flows, the consequence of migration is an excess of particles near the outer cylinder [Leighton and Acrivos (1987a, b), [START_REF] Ovarlez | Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging[END_REF], [START_REF] Huang | Viscosity of a dense suspension in Couette flow[END_REF]]. However our measurements do not completely rule out particle migration; we estimate, from the MRI data that the maximum gradient in particle concentration, if any, is around 0.1%. This is indeed very small when compared to migration for suspensions of spherical particles, where gradients of several percents are observed in the same MRI Couette cell.

r v r v r - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = γ . By combining ( )
The critical shear rate is due to the existence of a yield stress and is in principle decoupled from the onset of shear thickening. However, in the Couette geometry used for the MRI experiments, the existence of a yield stress makes that part of the material flows, and another part does not because the stress it is subjected to is smaller than the yield stress. As shown by the plate-plate presented below, the existence of a non-flowing region influences the onset of shear thickening, and so the critical shear rate is indirectly coupled to the shear thickening phenomenon. To the contrary, for the critical stress both the dilation and the measurements as a function of volume fraction show that the onset of shear thickening is directly dependent on the stress.

The emergence of shear thickening at the end of the localization regime then suggests that the non-flowing part plays an important role in the shear thickening. To assess what this role is, classical rheology measurements were conducted.

Role of a dead zone

In order to investigate in detail the influence of the non-flowing region in the Couette cell on the observed shear thickening, parallel plate geometry is used. This geometry has the additional advantage that there is no reservoir of particles present, as is the case of the Couette. If we need to, a 'non-flowing' region can be created in this geometry by simply leaving a few milliliters of paste around the gap and in contact with the sample between the plates [Figure 5]. In addition this geometry allows us to measure the normal stresses. Figure 6 shows the evolution of viscosity and normal stress with the applied shear rate. For low shear rates, a typical behavior γ σ / 1 ∝ of a shear thinning fluid is observed. At a certain shear rate, a very abrupt increase in viscosity is observed. However, this abrupt increase in viscosity is only observed when the surplus of paste around the plates is carefully removed. If a few milliliters of suspension are left on the bottom plate in contact with the paste between the two plates, the shear thickening is much attenuated: there is no abrupt increase in viscosity. Defining the critical shear rate as the first shear rate for which the apparent viscosity goes up, the surplus of paste increases the critical shear rate very significantly.

We also observe that the shear thickening is accompanied by the emergence of large normal stresses. For low shear rates, the normal stresses are very small. However, from the critical shear rate on, grows and becomes very important. The critical shear rate for which normal stresses appear is very similar to that for which the viscosity increases. We note also that if a surplus of paste is left around, the shear thickening is accompanied by much lower normal stresses.

These results provide a possible explanation for the MRI experiments in which shear thickening only happens when all of the material flows in the Couette geometry. When only part of the material flows, the dead zone plays a role analogous to the surplus in the parallel plate experiments, which delays and weakens the shear thickening, as is observed here in the parallel plate geometry. At the end of the shear localization regime, the Couette system is suddenly analogous to the parallel plate geometry without surplus; the critical shear rate is thus suddenly lowered and the material jams. In conclusion, it seems evident that, indeed, the presence of a non-flowing region delays and attenuates the shear thickening. We can note that these findings are consistent with a picture that hydroclusters formation leads to discontinuous shear thickening and jamming when the hydroclusters percolate the structure [START_REF] Wagner | Shear thickening in colloidal dispersions[END_REF]; [START_REF] Cates | Dilatancy, jamming, and the physics of granulation[END_REF]].

Dilation effect

These data suggest that, in both measurement geometries, the dead zone plays the role of a "reservoir" of dilation which helps the material to flow without jamming. Indeed, the principal information obtained from the normal stress measurement is their on-off behavior, which is quantitatively linked with the onset shear rate of shear thickening, as was verified by studying different volume fractions [START_REF] Fall | Shear thickening in cornstarch suspension is a re-entrant jamming transition[END_REF]]. The normal stresses are reminiscent of the shear induced dilatancy of dry granular matter: when sheared, it will dilate in the normal direction of the velocity gradient. Dilatancy is a direct consequence of collisions between the grains: to accommodate the flow, the grains have to roll over each other in the gradient direction, and hence the material will tend to dilate in this direction. However, in our system without a surplus, the grains are confined, both between the plates and in the solvent. The latter provides a confining pressure that is mainly due to the surface tension of the solvent, making it impossible to remove grains from the suspension. As suggested by [START_REF] Cates | Dilatancy, jamming, and the physics of granulation[END_REF] and [START_REF] Fall | Shear thickening in cornstarch suspension is a re-entrant jamming transition[END_REF], the confinement pressure associated with this should be on the order of the surface tension over the grain size,

Pa R P c 7000 / ≈ = γ
. This confinement pressure is in the same order of magnitude as the typical normal stresses measured near the onset of shear thickening [Figure 6]. In addition, this gives a maximal dilation that is on the order of 1 particle diameter ( m μ 20 ≈

); compared to the radius of the parallel plate cell this gives a maximum dilation of about 0.1%, too small to be detected by our MRI density measurements.

In dilatancy [START_REF] Reynolds | On the dilatancy of media composed of rigid particles in contact, with experimental illustrations[END_REF]], the volume of a collection of particles must increase upon shearing to enable flow. This has been suggested as a possible mechanism for jamming in concentrated colloidal suspensions [START_REF] Cates | Dilatancy, jamming, and the physics of granulation[END_REF]]. Dilation within a fixed volume of suspending liquid involves the formation of force transmitting 'hydroclusters', whose growth eventually causes particles to encounter the air-liquid interface. This generates large capillary forces at the free surface, which can then balance the normal inter-particle forces and resist further motion. The particles can thus form spanning hydroclusters in close contact, jamming the sample. This may then fracture into millimetre-scale 'granules' [START_REF] Cates | Dilatancy, jamming, and the physics of granulation[END_REF]]. Jamming of colloids is also seen in pipe and channel flows [START_REF] Isa | Velocity oscillations in microfluidic flows of concentrated colloidal suspensions[END_REF]; [START_REF] Haw | Two-Fluid Behaviour, and 'Self-filtration' in Concentrated Particulate Suspensions[END_REF]]; here free surfaces are not present. As the upper plate is retracted, a filament forms which narrows and eventually breaks. An elongational flow, in contrast to more conventional shear and pipe geometries, therefore implies an increase of the interfacial area during flow. Although purely elongational flow can be achieved by exponential plate separation, a constant separation speed is closer to fiber-spinning and other industrial processes. In these, a purely tensile loading evokes a mixed flow combining elongation and shear. Recent studies [START_REF] Bischoff White | Extensional rheology of a shear-thickening cornstarch and water suspension[END_REF]] have demonstrated that extensional rheometry can successfully be performed on colloidal suspensions. Bischoff White et al measured the tensile stresses of a (φ ~ 0.355) cornstarch solution. They observed a flowable filament at low extension rates but beyond this found a transition to brittle fracture. The interactions in this system are poorly characterized but clearly attractive (see Figure 1 of [START_REF] Bischoff White | Extensional rheology of a shear-thickening cornstarch and water suspension[END_REF]]), presumably due to strong van der Waals forces. Such interactions could strongly influence the flow behavior as they do in strongly aggregating colloids at lower densities [START_REF] Bischoff White | Extensional rheology of a shear-thickening cornstarch and water suspension[END_REF]].

We further investigate the role of dilation by doing oscillatory rheology (Figure 7). We find that, the material shows a linear elastic behavior G G ′ ′ ≈ ′ 10 at low shear stresses, i.e. it behaves as a "solid" material. Increasing the stress, a "solid-liquid" transition is found characterized by a yield stress at which " ' G G ≈ ; then the suspension begins to flow and " ' G G < . These observations are consistent with the behavior observed in steady shear (see above).

Increasing the applied stress even more, the 'liquid' regime quickly ends by the shear thickening of the system (characterized by an abrupt increase of G' and G''), for a critical shear strain It is clear from Figure 7 that the nonlinearity between stress amplitude and strain amplitude becomes more and more significant as the stress amplitude increases. Moreover, the appearance of nonlinear behavior is accompanied by non-sinusoidal responses in the oscillatory shear experiments. As the higher harmonic signals emerge, the material functions such as G' and G'' lose their original physical meaning. Considerable efforts [Yu et al. 2009;Cho et al. 2005;Ewoldt et al. 2008;Klein et al. 2007] have been made towards obtaining useful and desirable material information from such LAOS (Large Amplitude Oscillatory) experiments. For our purposes, the nonlinear oscillatory experiments are useful for characterizing the onset of shear thickening as well for determining the time scales required to generate the shear thickening response. Indeed, in the oscillatory experiments, the first natural interpretation of the observation of shear thickening would be that the shear rate f πγ γ 2 = applied during the oscillations is equal to the critical shear rate c γ observed in the continuous shear experiments when c γ γ ≈ . With a frequency f =1Hz, the critical oscillatory shear rate at the onset of thickening is ≈ 6.5 s -1 , a value 3 times higher than the critical shear rate observed during continuous shear. To understand this difference, we have performed the same experiments for several frequencies. We observe, in Figure 7(b), that the critical shear strain 1 ≈ c γ at the onset of thickening is constant; it is also independent of the gap sized, in contrast with c γ (see below). The conclusion is that 1 ≈ c γ is indeed the relevant physical quantity in the oscillatory experiments. This observation means that, even for high shear rates, thickening cannot take place if a two neighboring grains did not experience a relative motion of order of particle diameter; this is consistent with the major role of dilation evidenced above.

Confinement effect

It is therefore tempting to see whether the shear thickening phenomenon itself can be attributed entirely to the confinement: if the cornstarch is confined in such a way that the grains cannot roll over each other, this could in principle lead to an abrupt jamming of the system.

Experimentally, instead of setting the gap size in the rheometer for a given experiment, we can impose the normal stress and vary the gap size in order to reach a target value of the normal stress. If this is done for different shear rates, and the target value for the normal stress is taken to be zero, we can obtain the dependence of the gap variation on shear rate. A typical measurement is shown in Figure 8 (a), where we impose a constant shear rate and measure the gap and viscosity as a function of time. This shear rate and initial gap combination are beyond the shearthickening transition in shear rate, and thus the viscosity (shear stress) starts to strongly increase, as do the normal stresses. The latter then leads, through a feedback loop, to an increase in the gap, allowing the system to dilate, until the shear thickening disappears altogether: the shear stress is back to low values (less than 10 Pa). This unambiguously demonstrates that the shear thickening is a dilation effect, and that taking away a confining factor makes the thickening disappear altogether. This can be compared to the parallel-plate experiments in which the gap was systematically varied. Figure 10(a) shows the measured apparent viscosity as a function of shear rate for different gaps. For low shear rate, again a shear thinning behavior is observed. At the critical shear rate, a very abrupt increase in viscosity is observed; the main point here is that this critical shear rate increases with increasing gap. Comparison between parallel plate, cone and plate, and Couette cells showed identical critical shear rates to within the experimental uncertainty showing that the shear rate gradient present in our parallel plate geometry does not strongly affect our results. We then define the critical shear rate as the shear rate for which both the apparent viscosity increases and for which non-zero normal stresses are first observed: both values are very similar and, more importantly, increase linearly with the gap [Figure 10]. Again, if a few milliliters of suspension are left on the bottom plate in contact with the paste between the two plates, the critical shear rate strongly increases and becomes roughly independent of the gap size [Figure 10].

The critical shear rate with a surplus of paste present is, in addition, the same as that found in the large gap Couette cell, in which there is also a reservoir of particles present in the non-flowing region. The flow curve of Figure 3 shows that in the MRI experiments the critical shear rate is ~ 4 s -1 ; as soon as this shear rate is exceeded, the system shear thickens. We therefore conclude that the critical shear rate for thickening obeys Moreover, this linear evolution of a critical shear rate with the gap size presents a striking similarity with the dilation results shown in Figure 9. Indeed, we can observe that the value of = α 4.95 is roughly consistent with the value found in Figure 9 ( 66 .

3 1 = - β
), providing a quantitative check that indeed the dilatancy is responsible for the shear thickening.

Shear thickening as a Viscous/Granular transition

Another parameter that has a large effect on the critical shear rate is the volume fraction. At low volume fractions: 4 . 0 < ϕ , shear thickening is either less dramatic or absent [START_REF] Van Der Werff | Hard-sphere colloidal dispersion: The scaling of rheological properties with particle size, volume fraction, and shear rate[END_REF]]. In Figure 11 [START_REF] Fall | Shear thickening and migration in granular suspension[END_REF]]. In this picture, for low stresses, the viscosity of the interstitial fluid lubricates the contacts and one recovers a viscous behavior; this agrees with the observed Bingham behavior provided the stress on the system is much larger than the yield stress so that the latter can be neglected. However for higher stresses, a Bagnold-type dry granular rheology would be expected. The hallmark of this behavior is that the stress in the Bagnold regime, i.e., beyond shear thickening scales inertially [START_REF] Fall | Shear thickening and migration in granular suspension[END_REF], [START_REF] Lemaître | What do dry granular flows tell us about dense non-Brownian suspension rheology?[END_REF], [START_REF] Mills | Apparent viscosity and particle pressure of a concentrated suspension of a non-Brownian hard spheres near the jamming transition[END_REF]], implying 2 γ σ ∝ . However, for the cornstarch, this cannot be verified directly because the system beyond shear thickening has a very high viscosity making measurement over a large range of shear rates impossible. In addition, for the highest shear rates one observes instabilities in both the parallel plate cell and the Couette cell that are probably due to the normal stresses, and that make that the fluid is expelled from the gap.

However remaining around the onset shear rate, we can verify that in the second, shear thickening regime, the system behaves similarly to a dry granular material, which is exactly what is at the basis of the lubricated to inertial transition. If the system is a frictional one (as a dry granular material should be), due to the steric interaction between the particles, the local shear stress σ induces a local normal stress Figure 12 shows the evolution of shear stress as a function a normal stress for different gap sizes of parallel plate geometry. We found that the normal stresses change linearly with the shear stresses. This linearity can be used to define the macroscopic friction coefficient of the suspension; we find a value 0.62±0.01: this value is similar to that found in dry granular materials [START_REF] Da Cruz | Rheophysics of dense granular flows: Discrete simulation of plane shear flows[END_REF]] and other shear-thickening fluids [START_REF] Lootens | Giant stress fluctuations at the jamming transition[END_REF][START_REF] Lootens | Dilatant flow of concentrated suspensions of rough particles[END_REF]; [START_REF] Brown | The role of dilation and confining stresses in shear thickening of dense suspensions[END_REF]]. This suggests that in the shear thickening regime, the interstitial fluid plays no more role as lubrication forces become negligible: contacts behave like "dry" contacts. In this case, shear thickening can therefore again be considered as a transition from lubricated regime to a frictional regime under flow. The shear thickening transition is then due to direct contacts between particles induced by the shear.

Shear thickening as a re-entrant jamming transition

In our situation, the yield stress is smaller than the critical stress and consequently the sample yields and flows before thickening. If the critical stress is the smallest one, the thickening behavior may even disappear altogether [START_REF] Gopalakrishnan | Effect of attractions on shear thickening in dense suspensions[END_REF], [START_REF] Brown | Dynamic jamming point for shear thickening suspensions[END_REF]]. Our data show that for the cornstarch suspensions there are two critical stresses for which the apparent viscosity becomes infinite: first, upon approaching the yield stress from above, the viscosity diverges, in agreement with the MRI observations that the flow behavior is close to that of a Bingham fluid. Second, at the critical stress for thickening, an almost discontinuous jump of the viscosity is observed. These observations suggest a "solid-liquid-solid" transition. This is in addition qualitatively the same picture as that obtained from by the oscillatory shear experiments (Figure 7). Figure 13: Solid-liquid-solid transition; apparent viscosity vs. stress when performing a step stress test. This is a typical example of discontinuous shear thickening, in which the region with positive slope of ) (γ σ occurs in a stress range that is nearly independent of packing fraction as mentioned above in Figure 11. This slope increases with packing fraction, approaching σ η ∝ corresponding to a discontinuous stress/shear-rate relation. These results are similar to theory that suggests that shear thickening is due to a re-entrant jamming transition [Sellito and Kurchan (2005)]. It has been suggested for glassy systems that applying a shear is equivalent to increasing the effective temperature with which the system attempts to overcome energy barriers [Berthier et al. (2000)]. If now a system has a re-entrant ''solid'' transition as a function of temperature, the ''solid'' phase may also be induced by the shear, leading to shear thickening [Sellito and Kurchan (2005); [START_REF] Fall | Shear thickening in cornstarch suspension is a re-entrant jamming transition[END_REF]], as is also observed here (Figure 13). Indeed, the sample is "solid" in the sense that G'>>G'' (Figure 7) and the torque becomes too large for the MRI rheometer to turn. In the convential rheology, there are instabilities due to the large elasticity that expel the starch from the gap of the measurement geometry, making determination of an apparent viscosity difficult.

IV. Conclusion

We have studied the flow behavior of dense suspensions of non-colloidal particles, cornstarch particles in water, by coupling local velocity and concentration measurements through MRI techniques, and macroscopic rheometric experiments in Couette and parallel plate geometries. The MRI data reveals that the flow exhibits shear-banding at low rotation velocities of the inner cylinder. The MRI also showed that the material is homogeneous and no migration is observed, contrary to what happens for non-Brownian suspensions of spherical particles [START_REF] Fall | Shear thickening and migration in granular suspension[END_REF]].

Classical rheology then shows that the critical shear rate for the onset of shear thickening depends on the confinement: it evolves linearly with the gap size. This linear evolution is a direct consequence of the dilatancy observed in this suspension under shear. Indeed, from dilation measurements, we have shown that the application of a shear rate higher than the critical shear rate for thickening directly leads to a dilation of the suspension. Conversely, taking away the confinement attenuates the shear thickening behavior, or even makes it disappear altogether. For the MRI observations this implies that when flow is localized, the nonflowing region plays the role of a "dilatancy reservoir" which allows the material to be sheared without undergoing a jamming transition.

This convincingly shows that the dilation is at the origin of the discontinuous thickening. In addition, in oscillatory experiments we found a critical shear strain for shear thickening that appears independent of the frequency. The value of the critical shear strain ( 1 ≈ c γ ) also supports the idea that discontinuous shear thickening is a direct consequence of shear induced dilatancy that is hindered by the confinement.

The critical shear rate associated with the yield stress is due to the existence of a yield stress and is in principle decoupled from the onset of shear thickening. However, in the Couette geometry used for the MRI experiments, the existence of a yield stress makes that part of the material flows, and another part does not because the stress it is subjected to is smaller than the yield stress. As also shown by the plate-plate measurements (Figure 6), the existence of a non-flowing region influences the onset of shear thickening, and so the critical shear rate is indirectly coupled to the shear thickening phenomenon. However both the dilation and the measurements as a function of volume fraction show that the critical stress for the onset of shear thickening is directly coupled to the onset behavior.

  have shown a transition between a shear thinning and a shear thickening regime where the shear thickening behavior is characterized by

Figure 1 :

 1 Figure 1: Micrograph of the cornstarch particles.

Figure 3 :

 3 Figure3: Dimensionless velocity profile in the gap obtained by MRI measurements. Insert: the shear rate at the interface between sheared and unsheared regions is given by the slope of the velocity profile at that point, taken of course in the moving part of the material.

Figure 4 :

 4 Figure 4: (a) Local shear stress as a function the local shear rate. The line is a fit to the Bingham model: γ + σ = σ k y

  each radial position r, for various rotational velocities Ω, one obtains the constitutive equation ( ) γ σ of the fluid. The locally measured constitutive equation of the cornstarch suspension is shown in Figure 4 (a). It is consistent with the observation that

Figure 5 :Figure 6 :

 56 Figure 5: example of a 'non-flowing' region or 'dead zone' in parallel plate geometry.

Figure 7 :

 7 Figure 7: (a) Elastic modulus (G') and loss modulus (G") in the 44% volume fraction suspension as a function of shear strain for an imposed shear stress (0.001 to 100 Pa) at 1 Hz in a vane geometry. (b) Critical strain vs. gap and frequency

Figure 9 :

 9 Figure 8: Time evolution for a

α

  Figure 10: (a) Apparent viscosity and normal stress as a function of shear rate for different gaps in mm. Measurements were made with a parallel plate rheometer (Bohlin C-VOR 200) with radius R =20 mm. (b) Evolution of the critical shear rate a function of the gap. The error bars correspond to the uncertainty (reproducibility) of the experiments.

  constant above; here h is the gap size, = α (4.95±0.44) (mm.s) -1 , M c γ the critical shear rate when the suspension is sufficiently confined and cI γ the intrinsic critical shear rate.

Figure 11 :

 11 Figure 11: Viscosity as a function a shear rate for different concentrations. (b) Critical shear stress of shear thickening vs. concentration in different measurement geometries. Inset: Critical shear rate of shear thickening vs. concentration.

Figure 12 :

 12 Figure 12: Proportionality between normal stress and shear stress in the shear thickening regime for the 44% volume fraction suspension


Hugo Jacquin

Frédéric Van Wijland

Brownian dynamics : from glassy to trivial

We endow a system of interacting particles with two distinct, local, Markovian and reversible microscopic dynamics. We find that while the first, standard one, leads to glassy behavior, the other one leads to a simple exponential relaxation towards equilibrium. This finding questions the intrinsic link that exists between the underlying, thermodynamical, energy landscape, and the dynamical rules with which this landscape is explored by the system.

The microscopic phenomena driving the dynamical arrest in supercooled liquids and in glasses are still controversial. One line of thought, which originates in the work of Adam and Gibbs, pictures a glass-forming liquid as a system whose energy landscape complexity accounts for the slowing down of its dynamics. The original Adam-Gibbs [1] theory relates viscosity -the momentum transport coefficient-to configurational entropy. The more recent Kirkpatrick, Thirumalai, and Wolynes [2] scenario is based on the study of the metastable configurations of the system and on the concept of nucleating entropic droplets. This is the Random-First-Order Theory (RFOT). The idea is that metastability arises from the many valleys of the energy landscape the system can be trapped in. RFOT gives a large set of quantitative predictions including non-mean field particle models [3]. These aspects of the physics of glasses were reviewed by Debenedetti and Stillinger [4] and more recently by Biroli and Bouchaud [START_REF] Biroli | The Random First-Order Transition Theory of Glasses : a critical assessment[END_REF].

In another line of thought no complex energy landscape needs to be invoked, and dynamically induced metastability alone is held responsible for the dynamics slowing down. This is a phenomenological approach in which at a coarse-grained scale local patches of activity are the only ingredients of the dynamics. This has led to the development of kineticallyconstrained models (see [START_REF] Ritort | [END_REF] for a review), which have the advantage of lending themselves with greater ease than molecular models to both numerical experiments and analytical treatment. The hallmark of the corresponding lattice models is the absence of any structure in the energy landscape (the equilibrium distribution is that of independent degrees of freedom). These somewhat simplistic descriptions are not directly built from the microscopics, and it is often argued that their glassylike properties, like the existence of dynamical heterogeneities, are almost tautological, but recent works [START_REF] Speck | Constrained dynamics of localized excitations causes a non-equilibrium phase transition in an atomistic model of glass formers[END_REF] have tried to bridge the gap from the microscopics to dynamic facilitation.

A central concept in both approaches is that of metastability. A metastable state can be viewed as an eigenstate of the evolution operator with a nonzero but small relaxation rate. The existence of such states can be induced by the dynamical evolution rules alone, as in KCM's, but physical intuition dictates that, in realistic systems, these originate from the deepest structures of the energy landscape. The method for catching and counting metastable structures from the statics alone was coined by Mézard and Parisi, thus providing the first first-principle calculation of some of the properties of glasses. Their idea is to help polarize two attractively coupled copies the system into one of these deep valleys and to investigate whether thermal fluctuations are enough to wash out any overlap between them. By contrast, catching dynamically induced metastable structures is done by examining the overlap of the system properties in the course of a long time interval. Implementing this program was done by exploiting Ruelle's thermodynamic formalism first on KCM and then in realistic systems [START_REF] Hedges | [END_REF].

Our purpose in this work is to show that a minute modification of the evolution rules of a glass-forming liquid can lead to dramatic differences in the dynamics. We will propose two possible dynamics for our system of interacting particles, which are both time-reversible and Markovian. However, one is known to induce dynamical arrest and glassy-like phenomenology, while we show that the other one is characterized by straight, one-step, exponential relaxation, thus bypassing any slowing down. We emphasize that time reversibility ensures that the equilibrium distribution is the same in both systems. They share the exact same energy landscape. In both cases, the complex energy landscape is sampled by means of local moves. A central difference with existing works is that ours focuses on interacting degrees of freedom, while KCM's, the system of oscillators studied earlier by Kawasaki and Kim [9], or the single oscillator of Whitelam and Garrahan [10], are noninteracting degrees of freedom.

Our system is made of identical particles interacting via a two-body potential v(x-y) and we choose to describe its properties in terms of collective density modes ρ(x, t) with average density ρ 0 . In terms of these, the equilibrium distribution is the standard Gibbs weight P eq [ρ] ∝ e -βF [ρ] , where the density dependent effective free energy has the well-known expression

βF [ρ] = x ρ(x, t) ln ρ(x, t) ρ 0 -ρ(x, t) + 1 2 x,y ρ(x, t)βv(x -y)ρ(y, t), (1)
in which the first term in the rhs is of entropic origin, and the second one is the potential interaction energy. From a dynamical point of view, the particle density must locally be conserved, which constrains it to evolve according to a continuity equation

∂ t ρ + ∇ • j = 0 (2)
Our two choices of dynamics lie in the specific density dependence of the particle current j. We begin with a standard choice in which the particle current is given by

j L = -ρ∇ δF [ρ] δρ + 2T ρξ (3)
where ξ is a vector whose components are independent white noises with unit correlations

ξ α (x, t)ξ β (x ′ , t ′ ) = δ αβ δ (d) (x -x ′ )δ(t -t ′ )
, where d is the space dimension.

In an expanded form, the deterministic part of j L features a Fick diffusion term -T ∇ρ and a driving term ρF, where F(x, t) =y ∇v(xy)ρ(y, t) is the local fluctuating force field acting on particles located around x. We index the particle current with the letter L because this very expression of j L was shown by Dean [11] to exactly encode the dynamics of a fluid of interacting particles evolving through an overdamped Langevin equation. The parameter β = T -1 is the inverse temperature of the thermostat. Once the deterministic part of j L is set, time-reversibility (or detailed balance) forces the specific √ 2T ρ density dependence of the noise strength. Our second choice of dynamics for the collective modes is defined by another expression for the particle current, namely

j B = -ρ 0 ∇ δF [ρ] δρ + 2T ρ 0 ξ (4)
where the index B is reminiscent of model B dynamics of the Hohenberg and Halperin classification [12], as already defined for discussion purposes in [11]. Note that in (4) the coupling to the thermal bath is now independent of the density (noise is additive instead of being multiplicative as in ( 3)). Dynamics B can be obtained from dynamics L by thinking the fluctuations of the coupling to the thermostat have been turned off. We emphasize that the detailed balance property is satisfied also in this alternative dynamics, which ensures that in principle both j L and j B drive the system towards the same Gibbs equilibrium state P eq [ρ].

Our goal here is not to come up with a new approximation for the Langevin dynamics. A huge body of numerical and analytical literature is devoted to extracting signatures of glassiness from these dynamical rules [13]. We take them for granted. Instead, our purpose is to conduct an analytical study of our model B dynamics in which it will appear that no sign of dynamical arrest can be found. We search for an evolution equation for the density-density correlation function (otherwise called the intermediate scattering function). If a glassy behavior is to be found, then we must obtain a non vanishing long-time limit of this correlation function, indicating ergodicity breaking, or, if not, we should at least expect a multistep slow relaxation. As we shall now see, model B dynamics leads to a simple exponential relaxation for the densitydensity correlations G ρρ , as we conclude in (20). In order to demonstrate this property, we resort to a set of techniques coined by Kim and Kawasaki (KK) [14], originally designed to cope with Langevin dynamics, which we adapt to our model B evolution. The central idea of their work is a refined way to cast the dynamical rules in such a way that the detailed balance property is a manifest and linearly-realized symmetry of the evolution equation. While it was not clear whether and how time-reversibility and ergodicity breaking are related, KK showed that a certain type of (reversibility preserving) approximation on the dynamics, when applied to the two point density correlations, was fully equivalent to the well known Mode-Coupling approximation [15] (alternative methods were recently found [16]). Their work follows a number of earlier attempts [17] to deal with collective coordinates within a field-theoretic framework.

The field theory that describes the evolution of the density is written in terms of the fluctuations of the density field δρ around its mean value ρ 0 and a response field ρ :

Z = DρD ρ e -S[ρ, ρ] , (5)
S[ρ, ρ] = x,t i ρ∂ t δρ -i ρρ 0 ∆ δF [ρ] δρ -T ρ 0 (∇ρ) 2 (6)
In this theory, time-reversal (TR) is encoded in the following symmetry [START_REF] Janssen | Field-theoretic method applied to critical dynamics[END_REF] :

δρ(t) → δρ(-t) ρ(t) → -ρ(-t) + iβ δF [ρ] δρ(-t) (7)
We parallel the KK approach, and define θ = β δF δρ -K ⋆ δρ, with K ⋆ δρ(x, t) = δρ(x,t) ρ0 + y βv(xy)δρ(y, t) and θ is an auxiliary Lagrange multiplier field that enforces this expression for θ.

The introduction of the θ field was originally intended to deal with the non-linearity of the time-reversal symmetry in the Langevin dynamics described by the current given in (3). In our case, the time-reversal symmetry is a linearly realized symmetry, and use of an additional field is thus not necessary in that matter. However, introducing the θ field was shown by Kawasaki and Kim to be useful to properly take into account, non-perturbatively, the renormalized static properties of the system, and lead to simplifications that would be otherwise hard to detect in a standard Martin-Siggia-Rose field theory.

Grouping all four fields in a vector Φ = δρ, θ, ρ, θ , and introducing Ω(k) = ρ 0 T k 2 K(k), the action now reads :

S[Φ] = S g [Φ] + S ng [Φ], (8)
S g [Φ] = 1 2 k,ω Φ(-k, -ω)G -1 0 (k, ω)Φ(k, ω), (9)
S ng [Φ] = x,t iΦ 4 (x, t)f (Φ 1 (x, t)), (10)
where we have split the quadratic part S g from the higher order terms S ng , and f is a function of δρ given by f (x) = ln(1 + x/ρ 0 ) -x/ρ 0 . The bare propagator G -1 0 is given by :

G -1 0 (k, ω) =     0 0 -ω -iΩ(k) 0 0 0 -iρ 0 T k 2 -i ω -iΩ(k) -iρ 0 T k 2 2ρ 0 T k 2 0 0 -i 0 0     (11)
The TR invariance (7) now reads

       δρ(t) → δρ(-t) θ(t) → θ(-t) ρ(t) → -ρ(t) + iK ⋆ δρ(-t) + iθ(-t) θ(t) → θ(-t) -i∂ t ρ(-t) . ( 12
)
As was the case in KK, both S g and S ng are separately invariant under (12). We define the vertex functions Σ[G] as the functional inverse of the correlator matrix G = ΦΦ as :

G -1 = G -1 0 -Σ[G], (13)
We note in passing that the standard strategy to derive an approximation for G is based on performing an approximation on Σ[G] and inserting it back into (13), thus yielding a self-consistent equation for G alone. Here, however, no approximation is involved. TR symmetry, along with causality and time and space translational invariance, impose severe constraints on the form of the correlation functions of the theory, as well as on the vertex functions. The only difference between KK's field theory and ours lies in the non-Gaussian term S ng of the action in (10). Since our bare propagator and form of the time-reversal coincide with those in KK's work [14], it is easy to verify that all non-perturbative results that follow from the symmetries of the action coincide between the two theories. Differences will be found when determining the vertex functionals Σ[G].

Writing down the inverse relation that exist between Σ and G, and combining them with the constraints arising from the time-reversal symmetry and causality, we can write exact nonperturbative evolution equations for G ρρ , G θρ and G θθ , the 11, 12 and 22 elements of the matrix G. Inspection of the equal-time singularities of the Schwinger-Dyson equations ( 13) also shows that :

Σθθ(k, 0) = 1 G ρρ (k, 0) -K(k), (14)
which give a renormalized value of the bare diffusion coefficient, in a way that is reminiscent of the renormalizationgroup approach to model B dynamics of magnetic systems [12] : the initial time value of G ρρ is by definition ρ 0 S(k), where S(k) is the static structure factor of the liquid, thus we have, using the expression of K given after [START_REF] Speck | Constrained dynamics of localized excitations causes a non-equilibrium phase transition in an atomistic model of glass formers[END_REF] :

Σθθ(k, 0) = 1 ρ 0 1 S(k) - 1 ρ 0 -βv(k) = -c(k) -βv(k), ( 15
)
where c is the full two-body direct correlation function of the liquid, related to S by

S(k) = (1 -ρ 0 c(k) -1 .
By Laplace transforming the four relevant members of the Schwinger-Dyson equations (13), one can deduce that the correlation functions G ρθ and G θθ are simply related to G ρρ :

G ρθ (k, τ ) = 1 G ρρ (k, 0) -K(k) G ρρ (k, τ ), (16)
G θθ (k, τ ) = 1 G ρρ (k, 0) -K(k) 2 G ρρ (k, τ ), (17)
Substitution of all these results in the Schwinger-Dyson equation (13) lead to a closed equation on the density-density correlation function :

∂ τ G ρρ (k, τ ) + T k 2 S(k) G ρρ (k, τ ) = (18)
- τ 0 dt Σ ρρ (k, τ -t) G ρρ (k, t) ρ 0 S(k) +Σ ρθ (k, τ -t)∂ t G ρρ (k, t)
This final non-perturbative result is identical to that obtained by Kawasaki and Kim, which therefore also applies to our modified dynamics. Starting from this closed equation, we now exploit the fact that our theory has only one non-quadratic term, that involves the field θ and any (greater or equal to two) number of δρ fields. Thus, when calculating perturbatively the functions Σ, we are bound to find, to all orders of perturbation :

Σ ρρ = Σ ρθ = 0 (19)
which shows that we have

∂ τ G ρρ (k, τ ) = - T k 2 S(k) G ρρ (k, τ ). (20)
The physical meaning of ( 20) is clear : relaxation towards equilibrium occurs in a single exponential step. This is the central analytical result of this letter. The relaxation rate T k 2

S(k)

is proportional to k 2 (due to the local conservation of particles) and only involves the additional knowledge of the static structure factor. Away from any thermodynamic phase transition, the structure factor displays no specific singularity and thus the relaxation rate has no reason to vanish.

In retrospect, the two dynamical rules considered so far are very similar to each-other. Only local moves are involved, they both preserve particle conservation and for both timereversibility is encoded in the dynamics. Closer inspection, however, shows potentially relevant differences. We denote by r[ρ] the interaction-dependent component of the rate at which the system leaves a given configuration of the density landscape (we leave out the component expressing independent local diffusive hops). For Langevin dynamics, this rate reads

r L [ρ] = β 2 4
x,y,z ∇v(xy)∇v(xz)ρ(x, t)ρ(y, t)ρ(z, t)

- β 2 x,y ρ(x, t)∇ 2 v(x -y)ρ(y, t) (21)
The average rate, as can be seen in the first term in the right hand side of ( 21), involves two and, more importantly threebody correlations. The knowledge of these correlations is necessary for the system to decide to which location of phase space it is going to evolve. Turning now to the model B dynamics, we find that

r B [ρ] = β 2 ρ 0 4
x,y,z ∇v(xy)ρ(y, t)∇v(xz)ρ(z, t) +βρ 0

x,y ∇ρ ρ (x, t)∇v(xy)ρ(y, t) (22) so that on average the system requires no more than two-point correlations to proceed with its evolution. The Langevin dynamics requires a more refined knowledge of the local organization than model B dynamics. We speculate that we may relate this mathematical observation to the physical picture of local caging. It is a well-known fact that the pair correlation function of a glass is no different from that of a liquid, but recent works [START_REF] Mayer | [END_REF]20] suggest that triplet correlations (and higher order correlations) may behave differently. Our result stands out as one in which the dynamics of a system of interacting particles can exactly be solved at the level of two-point functions. Yet is based on a coarse-grained formulation in which the collective density modes are treated as smoothly varying function of space. For softly repulsive potentials, the system will fall into a jammed state as the density is increased. Our approach is thus bound to reach its limitations at high densities. However, in the usual glassy regime where ergodicity is maintained, a standard two-step relaxation will never be observed with our choice of dynamics.

Furthermore, we have insisted on characterizing the dynamics via its density modes, but relating the dynamics of the modes to that of the particles is a non-trivial task. We can see the two dynamics, described by the currents j L and j B , as the coarse-grained continuum versions of two dynamics for interacting particles on a lattice. Denoting by i the lattice sites and n i the occupation number on this site, a particle on site i may hop on an adjacent site j with rate

n i e -β∆Ep/2
for Langevin dynamics n 0 ni nj +1 e -β∆Ep/2 for model B dynamics (23) where E p = 1 2 i =j n i v(i -j)n j and ∆E p is the potential energy variation due to the particle hopping. For Langevin dynamics, the reader is referred to [21] for a more complete discussion (with slightly different rates). Starting from these lattice dynamics, standard manipulations [22] allow one to write down a path integral representation for the two dynamics. In the coarse-grained limit, they coincide with the path integral representations of the stochastic equations Eq.( 3) and (4), obtained via standard procedures [23]. Going back and forth between particles and modes is simple in Langevin dynamics, but what would an effective particle system described by model B dynamics look like ? Would this effective particle dynamics still be local ? Answering that question would allow for the design of algorithms able to bypass metastable traps.

From a broader perpsective, our calculation points to the necessity of establishing criteria permitting to relate static energy-landscape-based considerations, to kinetic properties. A naive way to ask the question would be : What are the generic conditions on the effective diffusion constant coupling the energy gradient to the thermal bath that allow for a reliable correspondence between minima of the energy landscape, and metastable states ? This question echoes a recent work [24] in which it is shown that one could map a subclass of KCM models onto a subclass of spin-glasses, uncovering a non-trivial behavior of appropriately defined static quantities in these KCMs, which could mirror non-trivial dynamical transitions in the corresponding spin-glass systems. Once again, the traditional separation between static and dynamic framework does thus not seem to hold anymore, calling for a more refined treatment of the interplay between the two.

A quantitative characterization of the dynamic complexity, as expressed for example by the Lyapunov spectrum of our two dynamics could uncover the fundamental difference that exist between the two. More generally, what is the signature of an ergodicity breaking transition on the Lyapunov spectrum ? These are ongoing investigations.

We warmly acknowledge discussions with Kyozi Kawasaki, Bongsoo Kim, Vivien Lecomte, Kunimasa Miyazaki, Estelle Pitard, Peter Sollich, Grzegorz Szamel and Francesco Zamponi. H. J. PhD work is funded by a fondation CFM-JP Aguilar grant.