Yasaman Madraki

Sarah Hormozi

Guillaume Ovarlez

Élisabeth Guazzelli

Olivier Pouliquen

Enhancing Shear Thickening

Keywords: numbers: 47, 57, Gc,83, 80, Hj

A cornstarch suspension is the quintessential particulate system that exhibits shear thickening. By adding large non-Brownian spheres to a cornstarch suspension, we show that shear thickening can be significantly enhanced. More precisely, the shear thickening transition is found to be increasingly shifted to lower critical shear rates. This influence of the large particles on the discontinuous shear thickening transition is shown to be more dramatic than that on the viscosity or the yield stress of the suspension.

I. INTRODUCTION

A shear thickening fluid is one in which the viscosity increases with the rate of shear [see e.g. [START_REF] Barnes | Shear-Thickening (Dilatancy) in Suspensions of Nonaggregating Solid Particles Dispersed in Newtonian Liquids[END_REF][START_REF] Mewis | Colloidal Suspension Rheology[END_REF][START_REF] Brown | Shear thickening in concentrated suspensions: phenomenology, mechanisms and relations to jamming[END_REF]. A classical example is a suspension of cornstarch in water which is often used in demonstrations to exhibit the counterintuitive behavior of shear thickening materials. It indeed behaves as a normal liquid if stirred slowly whereas it acts as a solid when agitated or struck forcefully. A person may even walk on a large pool of cornstarch without sinking provided the walking steps are quick and strong enough to cause the shear thickening phenomenon. Cornstarch suspensions exhibit both continuous (CST) and discontinuous shear thickening (DST). At low cornstarch concentration φ cs , rheological measurements show a smooth and continuous increase in viscosity with increasing shear rate. As φ cs becomes larger, the increase becomes steeper and eventually leads, at a critical shear rate γ0 c , to an order-of-magnitude discontinuous jump in viscosity or even directly to jamming. DST is observed in the dense suspension regime for φ cs 0.36.

Despite a sustained research attention since the mid-twentieth century, the origin of shear thickening is still not deciphered and remains a matter of active debate [see e.g. [START_REF] Hoffman | Discontinuous and dilatant viscosity behavior in concentrated suspensions. II. Theory and experimental tests[END_REF][START_REF] Fall | Shear Thickening of Cornstarch Suspensions as a Reentrant Jamming Transition[END_REF][START_REF] Wagner | Shear thickening in colloidal dispersions[END_REF][START_REF] Fall | Shear Thickening and Migration in Granular Suspensions[END_REF][START_REF] Brown | The role of dilation and confining stresses in shear thickening of dense suspensions[END_REF]. A new promising idea points to a transition from a frictionless to a frictional state of the suspension [START_REF] Fernandez | Microscopic Mechanism for Shear Thickening of Non-Brownian Suspensions[END_REF][START_REF] Seto | Discontinuous Shear Thickening of Frictional Hard-Sphere Suspensions[END_REF][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 non-Brownian suspensions[END_REF]. In this scenario, the shear thickening behavior stems from the existence of short-distance repulsive forces between particles. At low shear rate, the repulsion prevents frictional contacts between particles and the suspension behaves as a suspension of frictionless particles. When the shear rate is increased, the stresses on the particles increases and may overcome the repulsive forces. The friction is mobilized at contact and a transition to a frictional rheology then takes place. Within this picture, the discontinuous shear thickening transition occurs when the shear stress imposed to the suspension reaches a critical stress which depends upon the repulsive forces and the particle size. This new theory of friction-induced shear thickening is supported by recent experimental measurements [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] Hermes | Unsteady flow and particle migration in dense, non-Brownian suspensions[END_REF] which validate model and simulations. Even with these new advances in the understanding of shear thickening, there is still a compelling need to control more precisely the DST transition as this is an important challenge for industries that handle such fluids that can become solid-like, in particular in applications with dampening and shock-absorption such as found in armor composite material [START_REF] Lee | The ballistic impact characteristics of Kevlar woven fabrics impregnated with a colloidal shear thickening fluid[END_REF] and for curved-surface shear-thickening polishing [START_REF] Li | Shear-thickening polishing method[END_REF]. This paper presents a novel method to enhance shear thickening by adding large non-Brownian spheres to the cornstarch suspension.

More generally, adding particles to a fluid has been shown to enhance its shear viscosity. This is known for a Newtonian fluid since the seminal work of Einstein demonstrating that the viscosity of the mixture is increased above that of the suspending fluid [START_REF] Einstein | Über die von der molekularkinetischen Theorie der WŁrme geforderte Bewegung von in ruhenden Flssigkeiten suspendierten Teilchen[END_REF] and this gradually more with increasing particle concentration [START_REF] Stickel | Fluid mechanics and rheology of dense suspensions[END_REF]. As the jamming transition is approached, steric hindrance becomes dominant. Particle fluctuating motions then become more intense and collective, leading to a divergence of the viscosity [START_REF] Lerner | A unified framework for non-Brownian suspension flows and soft amorphous solids[END_REF][START_REF] Andreotti | Shear Flow of Non-Brownian Suspensions Close to Jamming[END_REF]. The enhancement effect is also observed when adding non-Brownian particles to a non-Newtonian matrix. In the case of a viscoplastic fluid, adding spherical particles induces an increasingly enhancement of the viscosity (or more precisely the consistency) and the yield stress [START_REF] Chateau | Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids[END_REF][START_REF] Dagois-Bohy | Rheology of dense suspensions of noncolloidal spheres in yield-stress fluids[END_REF][START_REF] Ovarlez | Flows of suspensions of particles in yield stress fluids[END_REF]. The intensification of these rheological quantities has been also seen for concentrated colloidal dispersions exhibiting a weak shear-thinning followed by continuous shear thickening [START_REF] Cwalina | Rheology of non-Brownian particles suspended in concentrated colloidal dispersions at low particle Reynolds number[END_REF], as well as for shear-thinning polymer solutions and shearthickening cornstarch suspension (but only investigated in the regime of CST) [START_REF] Liard | Scaling laws for the flow of generalized Newtonian suspensions[END_REF]. Theoretically, the rheological enhancement caused by adding large particles can be addressed by homogenization approaches [see e.g. 22]. The addition of large particles to a fluid increases locally the shear rates in the fluid, an effect sometimes refers to as a lever effect and described by introducing a lever function relating the magnitude of the local shear rate to the macroscopic shear rate imposed to the whole suspension mixture. The objective of the present work is to study how adding large spherical particles to a dense cornstarch suspension leads to a progressive shift of the DST transition to lower critical shear rates and to investigate how this effect is related to the rheological enhancement also observed.

II. MATERIALS AND METHODS

A. Particles and fluid

The cornstarch (73% Amylopectin and 27% Amylose from Sigma Aldrich, USA) used in the experiments consisted of irregularly-shaped particles ranging from 5 -20 µm with an average diameter d cs ≈ 13 µm and a density 1.68 g•cm -3 , see figure 1(b). The suspending fluid, a 54 wt% solution of cesium chloride (Cabot high purity grade from Sigma Aldrich, USA) in distilled water, was chosen to match the density of these particles. The cornstarch suspensions were prepared by carefully mixing the cornstarch particles with the suspending fluid at different volume fractions φ cs ranging from 0.10 to 0.44. Large non-Brownian particles with volume fraction φ p ranging from 0.05 to 0.35 were then added to this cornstarch suspension. We used φ cs = V cs /V t (1φ p ) and φ p = V p /V t with V cs , V p and V t the volumes of cornstarch, of large particles, and of the whole suspension, respectively. These definitions for the concentrations differed from those usually adopted for bidisperse suspensions. The large particles were Polymethyl methacrylate (PMMA) spheres (Cospheric, USA) of diameter d s ≈ 106 -125 µm, see figure 1(b). These PMMA particles were silver coated to ensure a density of 1.34 g•cm -3 which was closest as possible to that of the cornstarch suspension in order to avoid significant sedimentation effect. These large particles were insensitive to Brownian motion and colloidal interactions. Additional experiments were performed using large particles of different sizes (of diameters ranging from d p ≈ 45 -355 µm) but for non-coated PMMA particles (density 1.20 g•cm -3 ).

B. Rheological methods

The rheological measurements were conducted with a DHR-3 rotational rheometer (TA Instruments) using a 40 mm (diameter) serrated parallel plate geometry with 90 degree V-shaped grooves of height 0.5 mm and width 1 mm. This geometry eliminated wall slip effect, see figure 1(a). In a typical experiment, the desired amount of material was sandwiched between the two crosshatched plates and the rheological device delivered the torque C and the rotation rate Ω yielding to the determination of the maximum shear stress τ = 2C/πR 3 and the maximum shear rate γ = ΩR/h, and consequently to the effective viscosity η = τ/ γ.

Before the experiments were carried out, different tests were performed. A first important test was to check that the rheological measurements were independent of the size of the gap between the plates. This gap dependency was found to be very sensitive to the roughness of the plates as well as to whether the surplus of material around the plates was removed or not. We found that using crosshatched plates and trimming carefully the surplus of material around the plates produced gap-independent measurements in agreement with the experiments of [START_REF] Fall | Shear Thickening of Cornstarch Suspensions as a Reentrant Jamming Transition[END_REF], as exhibited in figure 1(c) for a cornstarch suspension at φ cs = 0.40 (see the next paragraph for the description of the rheological curve). A gap size of h = 1.5 mm was then chosen for most of the experiments.

The second test concerned the reproducibility of the rheological properties of the suspensions. We obtained identical rheological measurements within an accuracy of 15% using different batches of suspensions prepared in the same way or the same batch for different tests but the same day. Last, we checked the time dependency of the experiments by varying the time of the shear rate ramp. No significant differences were observed for a total time of the experiments ranging between 60 and 360 s. This test was also performed when large particles were added to the cornstarch suspensions.

A total time of 180 s was chosen for all the experiments for which the influence of sedimentation was found to be inconsequential.

III. EXPERIMENTAL RESULTS

A. Rheological observations

The rheological results for pure cornstarch suspensions are presented in figure 1(c) for φ cs = 0.40 and in figure 2(a) and 2(b) for 0.33 φ cs 0.44. At low γ, the rheology presents a plateau in shear stress, which can be interpreted as a yield stress, τ y , as evidenced in the inset of figure 1(c).

In terms of viscosity, the response is shear thinning as the viscosity η is seen to decrease with increasing γ until it reaches a minimum plateau η p , see figure 2 of large particles, see figure 3(a), the effective viscosity of the mixture η is overall increased, see figure 3(b), as seen previously [START_REF] Cwalina | Rheology of non-Brownian particles suspended in concentrated colloidal dispersions at low particle Reynolds number[END_REF][START_REF] Liard | Scaling laws for the flow of generalized Newtonian suspensions[END_REF]. More strikingly, the DST transition is moved to lower shear rate with increasing particle volume fraction φ p , see figure 3(b). This behavior is systematically observed in the range 0.36 φ cs 0.41. The critical shear rates γ p c which characterize the onset of DST for the mixture of large particles and cornstarch suspensions at different φ p are plotted versus φ cs in figure 3(c). This graph clearly evidences the shift of the DST transition to lower shear rate with increasing φ p with an upper bound curve given by that for the pure cornstarch suspension (φ p = 0 in red) taken from figure 2(c), i.e. the curve γ0 c (φ cs ). In the following, we first provide a systematic investigation of the shift in critical shear rate and then discuss the effect of adding particles on the critical shear stress.

B. The shift in the critical shear rate

We examine first the critical shear rates γ p c for the mixture of silver-coated PMMA spheres (of diameter d s ≈ 106 -125 µm) and a cornstarch suspension at different φ cs and φ p . Normalizing the critical shear rate γ p c for the mixture of large particles and cornstarch suspensions by the critical shear γ0 c for the pure cornstarch suspension leads to a good collapse of the data onto a single master curve as evidenced in figure 4(a) where γ p c / γ0 p is plotted against φ p . The dynamics of the dense suspension mixture thus mainly depends on the large particle volume fraction φ p . To complete this rheological analysis and study the role of the aspect ratio between the large spheres and the cornstarch particles, experiments were also performed using large particles of different sizes.

It was however not possible to obtain coated PMMA particles of different sizes. Therefore, we used non-coated PMMA particles (of diameters ranging from ≈ 49 to 327 µm) with the drawback of having a larger mismatch in density. In figure 4(b), the critical shear rates γ p c for the mixture normalized by the critical shear γ0 c for the pure cornstarch suspension are plotted as a function of the volume fraction for four different sizes, showing a clear influence of the size d p of the large particles. The shift to lower shear rate when adding particles is more pronounced when the size of the large particles is decreased, i.e. when the ratio of the large particles to the cornstarch particles is diminished. This influence of the particle size ratio may be related to loosening effects in suspensions consisting of polydisperse particles, i.e. the change in maximum packing volume fraction (φ m ) of large particles (PMMA or silver-coated PMMA) due to presence of small particles (cornstarch grains) [START_REF] Chateau | Series in Civil and Structural Engineering[END_REF]. If we naively consider that two large particles are always separated by a cornstarch grain even in the closest packing, the new maximum packing volume fraction becomes

φ m = φ m d 3 p /(d p + d cs ) 3 .
When the data of figure 4(b) are replotted by rescaling the volume fraction φ p using φ m , the curves obtained for different particle sizes collapse onto a master curve as evidenced in figure 4(c). This collapse seems to substantiate the idea that the dynamics of the dense suspension mixture is mainly controlled by steric constraints.

C. Influence on the critical shear stress

Whereas adding large particles to a cornstarch suspension dramatically shift the critical shear rate to lower values, the effect is of lesser importance on the critical shear stress, as evidenced in figure 3(a). The DST is observed to occur at a critical stress ≈ 20 Pa. This critical value is in agreement with the experiments of [START_REF] Fall | Shear Thickening of Cornstarch Suspensions as a Reentrant Jamming Transition[END_REF] for pure cornstarch suspensions similar to those used in the present study. Note that a lower value of ≈ 5 Pa is found by [START_REF] Hermes | Unsteady flow and particle migration in dense, non-Brownian suspensions[END_REF] using similar Sigma Aldrich cornstarch but suspended in a (not so closely density matched) fluid obtained by adding glycerol into the water phase. We systematically measured the critical stress at which the DST transition occurs for different cornstarch concentrations φ cs , different large particle concentration φ p , and different particle size ratio. The resulting data are recapitulated in figure 5. The critical stress τ p c normalized by the critical value for pure cornstarch τ 0 c is plotted for all data as a function of the rescaled volume fraction φ p /φ m . The experimental data are seen to collapse onto a single curve.

The critical stress is approximately constant and equal to the critical stress for pure cornstarch at low volume fractions but begins to increase when φ p /φ m 0.5. This observation can be rationalized by the following understanding of the rheology of non colloidal rigid suspensions. At low volume fraction, while it is enhanced by the particles, the bulk stress of a suspension remains purely of hydrodynamic nature and is entirely carried by the suspending-fluid phase. The average stress experienced by the interstitial cornstarch is thus equal to the suspension stress. Assuming that the DST transition occurs at a constant stress in the cornstarch then means that DST occurs at a constant stress for the whole suspension, at least at low particle volume fraction φ p . However, at larger volume fraction, contact can occur between the large particles, and the bulk stress is now carried in part by the large-particle phase and not solely by the suspending-fluid phase. This may explain the increase of the critical stress for DST at larger volume fraction, as in this regime the stress experienced by the cornstarch suspension is now smaller than the total stress applied to the mixture.

The observation of a quasi constant critical shear stress provides a simple interpretation for the lowering of the critical shear rate for the DST transition. Adding large particles enhances the viscosity, implying that the critical shear rate decreases in order to keep the critical stress constant.

However, as discussed in the following section, the interpretation of the DST in terms of a constant

IV. INTERPRETATION IN TERMS OF A LOCAL SHEAR RATE

To analyze and interpret the rheology of suspensions in non-Newtonian fluids, a useful concept is that of local shear rate γlocal . The interstitial fluid between the particles experienced a very fluctuating velocity field, and the typical magnitude of the shear rate, called the local shear rate γlocal in the following, is larger than the macroscopic shear rate γ applied to the suspension because of the indeformability of the large particles. This amplification can be described by introducing a lever function F (φ p ) relating the local to the macroscopic shear rate such as γlocal = γF (φ p ) with the underlying assumption that F only depends on φ p [START_REF] Lerner | A unified framework for non-Brownian suspension flows and soft amorphous solids[END_REF][START_REF] Chateau | Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids[END_REF][START_REF] Dagois-Bohy | Rheology of dense suspensions of noncolloidal spheres in yield-stress fluids[END_REF].

The present study of the shift of the critical shear rate at which DST occurs provides a direct estimation of the lever function. The DST should be obtained when γlocal reaches the critical values for pure cornstarch γ0 c and therefore when the macroscopic shear rate γ reaches γ0 c /F (φ p ).

The lever function estimated from DST is then simply given by the ratio F DST = γ0 c / γ p c . These data are plotted in figure 6 as a function of the normalized volume fraction of large particle φ p /φ m (solid lines and symbols). The good collapse of all data for different large particles and different cornstarch concentrations seems to substantiate the idea that the dynamics of the dense suspension mixture is mainly controlled by steric constraints and thus only depends on the large particle volume fraction φ p .

In addition, the present rheological measurements provide two others ways to estimate the lever function from the shift in rheological properties prior DST, in particular the shift in plateau viscosity η p (φ p ) and the shift in yield stress τ y (φ p ). The first estimate is derived from an energetic argument stipulating that the dissipation in the suspension is equal to the dissipation in the suspending fluid (which means that contact between the large particles is supposed to be negligible), i.e. η p (φ p ) γ2 = (1φ p )η p (0) γ2 local . This lead to the estimate of the lever function based on the viscosity:

F η p (φ p ) = η p (φ p )/[η p (0)(1 -φ p )].
The second estimate is obtained again by writing the dissipation at low shear rate when the stress is given by the yield stress: τ y (φ p ) γ = (1φ p )τ y (0) γlocal leading to the lever function based on the yield stress: F τ y (φ p ) = τ y (φ p )/τ y (0)(1φ p ). We have computed these two estimates of the lever function from the rheological curves, η p (φ p ) being inferred as the minimum plateau viscosity and τ y (φ p ) as the shear stress at γ = 0.04 s -1 . These two alternative estimates of F (φ p ) are also plotted in figure 6 (dashed and dotted lines). They are both in very good agreement and again confirm the sole dependance on the large particle volume fraction φ p .

While these three evaluations of F (φ p ) present an increase with increasing large particle volume fraction φ p , they do not match completely. Both estimates of the lever function from viscosity shift (dashed lines) and yield stress shift (dotted lines), F η p (φ p ) and F τ y (φ p ) respectively, are similar whereas the predictions from the DST shift (solid lines and symbols), F DST (φ p ), are systematically larger. The influence of the large particles on the DST is clearly more dramatic than that on the viscosity or the yield stress. This discrepancy shows that a simple mean field argument based on a single scalar estimate of the local shear rate is not sufficient to capture the observed rheology and that the DST may be controlled by the extreme values of the local shear rate distribution. It is indeed sufficient to have a percolating jamming network within the interstitial fluid to block macroscopically the mixture.

V. CONCLUSION

In conclusion, we have shown that adding large non-Brownian particles to a cornstarch suspension moves the DST transition to lower critical shear rates providing a method for controlling shear thickening properties by simply varying the concentration of the large particles. The effective plateau viscosity of the mixture prior DST is observed to increase as seen previously [START_REF] Cwalina | Rheology of non-Brownian particles suspended in concentrated colloidal dispersions at low particle Reynolds number[END_REF][START_REF] Liard | Scaling laws for the flow of generalized Newtonian suspensions[END_REF].

The critical stress for the DST transition is much less affected as it is seen to be approximately constant and equal to that for pure cornstarch in a relative wide range of concentrations of large particles before presenting some increase for larger concentrations.

Interpreting our results in terms of a local shear rate, i.e. stipulating that the presence of large particles enhances the shear rate in the suspending fluid, exhibits a difference between the influence of the large particles on the DST and on the bulk rheology. The local shear rate estimated from the shift in the DST transition is larger than the local shear rate estimated from the shift in viscosity. A simple interpretation in terms of a single estimate of the local shear is thus not appropriate, which may suggest that the DST transition mobilizes the extreme values of the local shear rate distribution while the plateau viscosity (or more generally the bulk rheology prior DST) involves the averaged value. These observations raise questions that require theoretical studies going beyond mean field approaches in order to tackle thoroughly these nonlinear rheological systems.

2 FIG. 1 :

 21 FIG. 1: (a) Sketch of the plate-plate rheometer exhibiting the manufactured grooves. (b) Micrograph of PMMA and cornstarch particles. (c) Viscosity η (Pa • s) versus shear rate γ (s -1 ) for different gap sizes h with a cornstarch suspension at φ cs = 0.40. The inset shows the corresponding shear stress τ (Pa) versus shear rate γ (s -1 ) for the same gap sizes.

6 φFIG. 2 : 4 FIG. 3 :FIG. 4 :

 62434 FIG. 2: (a) Shear stress τ (Pa) and (b) viscosity η (Pa • s) versus shear rate γ (s -1 ) for pure cornstarch suspensions at different φ cs . (c) DST critical shear rate γ0 c (s -1 ) versus φ cs . The inset shows the same plot in semi-log scale.

dFIG. 5 :

 5 FIG. 5: Critical stress τ p c (φ p ) at the DST transition normalized by the averaged critical stress for pure cornstarch ( τ 0 c = 20.8 Pa) as a function of the rescaled volume fraction φ p /φ p for data corresponding to various φ cs and d p .

FIG. 6 :

 6 FIG. 6: Estimate of the lever function F (φ p ) based on the DST shift F DST (φ p ) = γ0 c / γ p c (solid lines and symbols), the viscosity shift F η p (φ p ) = η p (φ p )/[η p (0)(1φ p )] (dashed lines), and yield stress shift F τ y (φ p ) = τ y (φ p )/τ y (0)(1φ p ) (dotted lines) versus φ p versus φ p /φ m .

ACKNOWLEDGMENTS This work is undertaken under the auspices of the ANR project 'Dense Particulate Systems' (ANR-13-IS09-0005-01), the 'Laboratoire d'Excellence Mécanique et Complexité' (ANR-11-LABX-0092), and the 'Initiative d'Excellence' A * MIDEX (ANR-11-IDEX-0001-02) as well as the NSF Grant No. CBET-1554044-CAREER. We thank Dr. David F. J. Tees (Department of Physics and Astronomy, Ohio University) for his assistance on the suspensions micrography.


Franco Tapia

Saif Shaikh

Jason E Butler

Olivier Pouliquen

Élisabeth Guazzelli

Rheology of concentrated suspensions of non-colloidal rigid fibres

Pressure and volume-imposed rheology is used to study suspensions of non-colloidal, rigid fibres in the concentrated regime for aspect ratios ranging from 3 to 15. The suspensions exhibit yield-stresses. Subtracting these apparent yield-stresses reveals a viscous scaling for both the shear and normal stresses. The variation in aspect ratio does not affect the friction coefficient (ratio of shear and normal stresses), but increasing the aspect ratio lowers the maximum volume fraction at which the suspension flows. Constitutive laws are proposed for the viscosities and the friction coefficient close to the jamming transition.

Introduction

The rheological properties of viscous Newtonian fluids containing rigid fibres remains relatively unexplored as compared to suspensions of spherical particles, and a consensus on even the qualitative description of the rheology is still lacking for concentrations beyond the dilute limit. As one example, the steady values of the shear stresses should, for suspensions of fibres that are large relative to colloidal scales and free of external body forces, follow a Newtonian law [START_REF] Dinh | A rheological equation of state for semiconcentrated fiber suspensions[END_REF]. However, many experimental studies find yield stresses and a nonlinear scaling of the shear stresses with the rate of shear, where these non-Newtonian effects become more prominent with increasing concentration [START_REF] Ganani | Suspensions of rodlike particles: Literature review and data correlations[END_REF][START_REF] Powell | Experimental analysis of the porosity of randomly packed rigid fibers[END_REF]. Different explanations have been proposed to explain the departure from a Newtonian response. This includes arguments that the fibres were not rigid under the imposed conditions [START_REF] Powell | Experimental analysis of the porosity of randomly packed rigid fibers[END_REF][START_REF] Sepehr | Rheological properties of short fiber model suspensions[END_REF] or that the fibres are not force-free. An example of the latter is the assertion that adhesive forces [START_REF] Mongruel | Shear viscosity of suspensions of aligned non-Brownian fibres[END_REF][START_REF] Chaouche | Rheology of non-Brownian rigid fiber suspensions with adhesive contacts[END_REF]Bounoua et al. 2016b) can exist between the fibres, even though their size is large compared to typical colloidal scales.

Previous rheological studies have focused on suspensions at relatively small volume fractions. Identifying measurements of rheology for volume fractions, φ, above 0.1 is difficult for fibres of large aspect ratios, A = L/d, where L and d are the fibre length and diameter, respectively. The lack of data is attributable, at least in part, to the difficulty of preparing and measuring the rheology of suspensions at high concentrations for large aspect ratios. Even for aspect ratios as high as 17 or 18, measurements are available for volume fractions of only up to φ = 0.15 or 0.17 (Bounoua et al. 2016a;[START_REF] Bibbó | Rheology of semiconcentrated fiber suspensions[END_REF]; measurements as high as φ = 0.23 were made by [START_REF] Bibbó | Rheology of semiconcentrated fiber suspensions[END_REF] for smaller aspect ratios of A = 9. As a result, the rheological properties of suspensions of rigid fibres remains to be characterised in the limit of large concentrations where mechanical contacts are expected to matter [START_REF] Sundararajakumar | Structure and properties of sheared fiber suspensions with mechanical contacts[END_REF][START_REF] Petrich | Interactions between contacting fibers[END_REF][START_REF] Snook | Normal stress differences in suspensions of rigid fibres[END_REF]. Likewise, the volume fraction at which the shear stresses diverge, and the flow of the suspension ceases (i.e. becomes jammed), has not been determined previously for non-colloidal fibres, though such measurements have been made for shear-thickening suspensions of colloidal fibres [START_REF] Egres | The rheology and microstructure of acicular precipitated calcium carbonate colloidal suspensions through the shear thickening transition[END_REF][START_REF] Brown | Shear thickening and jamming in densely packed suspensions of different particle shapes[END_REF].

Here, a custom-built rheometer has been used to explore the shear stresses and normal forces in suspensions of non-colloidal, rigid fibres for concentrations exceeding φ = 0.23. The rheometer [START_REF] Boyer | Unifying suspension and granular rheology[END_REF][START_REF] Dagois-Bohy | Rheology of dense suspensions of non-colloidal spheres in yield-stress fluids[END_REF] measures the stresses in both a pressure and volume-imposed configuration. The measurements indicate the presence of yield stresses in the tested suspensions, but also a viscous scaling wherein the stress grows linearly with the rate of shear. The unique rheometer design facilitates the study of these highly concentrated suspensions, and the volume fractions at which the stresses diverge are measured. The scaling of the stresses near this jamming transition are found to differ substantially from that of a suspension of spheres. These measurements are reported in § 3, after presenting the experimental materials and techniques in § 2; conclusions are drawn in § 4.

Experiments

Fibres and fluids

Four batches of rod-like particles were used in the experiments. They were obtained by using a specially-designed device to cut long cylindrical filaments of plastic (PLASTINYL 6.6) that were supplied by PLASTICFIBRE S.P.A. (http://www.plasticfibre.com). Images of typical fibres from each batch are shown in figure 1 (b). The length and diameter of over 100 fibres were measured with a digital imaging system. The distributions of lengths and diameters were found to be approximately Gaussian for all aspect ratios. The mean value and standard deviation of the fibre aspect ratio A = L/d, length L, and diameter d are shown in table 1. Note that batches (II) and (III) have very different lengths and diameters, but roughly the same aspect ratio of A ≈ 6 -7.

The rigid fibres were suspended in a Newtonian fluid that had a matching density of ρ f = 1056 kg/m 3 . The suspending fluid was a mixture of water (10.72 wt%), Triton X-100 (75.78 wt%), and Zinc Chloride (13.50 wt%). The fluid viscosity of η f = 3 Pa•s and the density were measured at the same temperature (25 • C) at which the experiments were performed. The suspensions were prepared by adding the fibres to the fluid, where both quantities were weighed, and gently stirring. Little to no settling or creaming was observed.

The rheological measurements were performed at a maximum shear rate of γ ≈ 3 s -1 , ensuring that a maximum Reynolds number (ρ f γL 2 /µ f ) of 0.04 was achieved. The fibres can be considered non-colloidal, owing to their large size, and rigid under the conditions of the experiment. Regarding the latter, the buckling criterion has been characterised by

Experimental techniques

The experiments were conducted using a custom rheometer that was originally constructed by [START_REF] Boyer | Unifying suspension and granular rheology[END_REF] and then modified by [START_REF] Dagois-Bohy | Rheology of dense suspensions of non-colloidal spheres in yield-stress fluids[END_REF]. This rheometer, sketched in figure 1 (a), provides measurements of both shear and normal stresses. The shearing cell consists of (i) an annular cylinder (of radii R 1 = 43.95 mm and R 2 = 90.28 mm) which is attached to a bottom plate that can be rotated and (ii) a top cover plate that can be moved vertically. This top plate is porous, enabling fluid to flow through it, but not particles. The plate was manufactured with holes of sizes 2 -5 mm and then was covered by a 0.2 mm nylon mesh (see figure 1 (c)). The parallel bottom and top plates have also been roughened by positioning regularly-spaced strips of height and width 0.5 mm onto their surfaces. A transparent solvent trap covers the cell, hindering evaporation of the suspending fluid.

In a typical experiment, the annular cell was filled with suspension and the porous plate was lowered into the fluid to a position h. This height, measured independently by a position sensor , ranges between 10.8 to 18 mm, corresponding to 13 to 25 fibre diameters depending on the fibre batch. The height measurement enables calculation of the fibre volume fraction, φ. The bottom annulus was rotated at a rate Ω by an asynchronous motor (Parvalux SD18) regulated by a frequency controller (OMRON MX2 0.4 kW), while the torque exerted on the top plate was measured by a torque transducer (TEI -CFF401). The shear stress τ was deduced from these torque measurements after calibration with a pure fluid to subtract undesired contributions resulting from the friction at the central axis and the shear in the thin gap between the top plate and the cell walls; the calibration method is decribed by [START_REF] Dagois-Bohy | Rheology of dense suspensions of non-colloidal spheres in yield-stress fluids[END_REF]. A precision scale (Mettler-Toledo XS6002S) was placed on a vertical translation stage driven by a LabVIEW code in order to measure the apparent weight of the top plate. This measurements, after correcting for buoyancy, provided the determination of the normal force that the particles exert on the porous plate in the gradient direction. Dividing by the area of the plate gives the gradient component of the normal stress, which is referred to simply as the particle pressure, P . A normal viscosity in the gradient direction can be defined as P/η f γ, as was done by [START_REF] Morris | Curvilinear flows of noncolloidal suspensions: The role of normal stresses[END_REF].

The rheometer can be run in a pressure-imposed mode or in a volume-imposed mode, and measurements were recorded as a function of the mean shear rate, γ = Ω(R 2 + R 1 )/2h, once steady state was achieved. In pressure-imposed rheometry, the particle pressure P is maintained at a set value that is measured by the precision scale; the volume fraction φ and the shear stress τ are measured as a function of the shear rate γ and pressure P . In volume-imposed rheometry, the height h, and consequently the volume fraction φ, are maintained at a fixed value, while the shear stress, τ , and particle pressure, P , are measured as a function of the shear rate, γ. Errors in the measurements of τ , P , and φ for the suspensions depend upon the calibration experiments, the preparation of the suspension samples, and the precision of the height, torque, and scale measurements. Estimates, based upon tests with independently created samples of suspension, suggest errors of ±6 Pa, ±5 Pa, and ±0.005 for τ , P , and φ, respectively.

Rheological measurements

Rheological observations

Typical rheological data for the apparent relative shear and normal viscosities, τ /η f γ and P/η f γ, are plotted against volume fraction, φ, in figures 2 (a) and (b). The data was collected for fibres of batch (II) using pressure-imposed and volume-imposed measurements. As expected, both quantities increase with increasing φ. However, multiple values of the apparent viscosities are measured for any given φ. Plotting the shear stress, τ , and the particle pressure, P , against the shear rate for different values of φ demonstrates that τ and P are linear in γ, but have a non-zero value at γ = 0, see figures 3 (a) and (b). This seems to suggest that a yield-stress exists for both the shear stress and the particle pressure, τ 0 and P 0 , respectively. Their values can be determined using a linear fit of the stress and pressure data as a function of γ, as indicated by the lines in figures 3 (a) and (b). Both yield-stresses, τ 0 and P 0 , increase with increasing φ as shown in figures 3 (c) and (d) for all four batches of fibres. The growth in τ 0 and P 0 with respect to φ is more pronounced for larger aspect ratios A.

The data of figures 3 (a) and (b) demonstrate that the stresses scale linearly with the rate of shear, as expected. Furthermore, the slopes of τ and P with γ increase with φ, which is evidence of the increase of the shear and normal viscosities with φ. These shear and normal viscosities can be collapsed into a single function of φ by removing the yield stresses. Figures 2 (c) and (d) show the results of (τ -τ 0 )/η f γ and (P -P 0 )/η f γ as a function of φ. In all of the following analysis, the yield stresses are subtracted systematically from the raw data.

Constitutive laws

Figures 4 (a) and (b) show η s = (τ -τ 0 )/η f γ and η n = (P -P 0 )/η f γ, the relative shear viscosity and relative normal viscosity, for all of the fibre batches. Both quantities increase with φ and seem to diverge at a maximum volume fraction that depends on the aspect ratio A. The influence of the aspect ratio is also seen on the rheological functions as η s (φ) and η n (φ) shift toward lower values of φ with increasing A. An interesting observation is that the data for batches (II) and (III), corresponding to similar values of A but different sizes, collapse onto the same curve. This indicates that finite size effects are not significant. Also, the decrease of η n is much stronger than that of η s for φ 0.35.

An alternative representation of the rheological data plots the friction coefficient µ = η s /η n and the volume fraction φ as a function of the dimensionless shear rate, J = η f γ/(P -P 0 ) [START_REF] Boyer | Unifying suspension and granular rheology[END_REF]; note that J = 1/η n and is a function of φ as shown in figure 4b. The rheology is then described by the two functions µ(J) and φ(J) as shown in figures 4 (c) and (d) for the same data as in figures 4 (a) and (b). A striking result is that a complete collapse of all the data is observed for µ(J), indicating that the friction coefficient is independent of the aspect ratio A. The volume fraction φ is a decreasing function of the dimensionless number J. There is a clear shift of φ(J) toward the lower values of φ when A is increased. The data for batches (II) and (III), having similar aspect ratios, again collapse onto the same curve.

This frictional approach is particularly well suited to study the jamming transition, as it circumvents the divergence of the viscosities. From the semi-logarithmic plot of φ(J), shown in the inset of figure 4 (d), the critical (or maximum flowable) volume fraction φ m can be determined from the limiting value of φ as J goes to zero. Similarly, the semilogarithmic plot of µ(J) in the inset of figure 4 (c) shows that the friction coefficient tends to a finite value µ s at the jamming point.

The critical values φ m and µ s are plotted against the fibre aspect ratio A in figures 5 (a) and (b), respectively. Again, the similar results for batches (II) and (III) indicate that confinement is not influencing the measurements, and the values obtained by [START_REF] Boyer | Unifying suspension and granular rheology[END_REF] for suspensions of poly(methyl methacrylate) spheres are also plotted on these graphs (for A = 1, although strictly speaking a sphere is not a cylinder of aspect ratio one). Clearly, φ m decreases with increasing A. This follows the general trends of a decrease in volume fraction with the aspect ratio for processes such as dry packing, as shown in figure 5 (a). A comparison is also made in figure 5 (a) between the values of φ m and estimates from simulations [START_REF] Williams | Random packings of spheres and spherocylinders simulated by mechanical contraction[END_REF] of the maximum concentration at which the orientation distribution remains random. The critical friction µ s does not vary significantly with A in the explored range and its value (≈ 0.47) is larger than that obtained for spheres (≈ 0.32) by [START_REF] Boyer | Unifying suspension and granular rheology[END_REF].

Figure 6 displays the same data as figure 4, but with φ scaled by φ m . This simple rescaling leads to a good collapse of the data for all of the fibre batches, indicating that the aspect ratio principally impacts the maximum volume fraction, φ m . Another remarkable result is that the relative shear and normal viscosities, η s and η n , diverge near the jamming transition with a scaling close to (φ m -φ) -1 , as clearly evidenced by the insets of figures 6 (a) and (b). This starkly contrasts with the divergence of (φ m -φ) -2 observed for suspensions of spheres [START_REF] Boyer | Unifying suspension and granular rheology[END_REF]. A constitutive law for µ can be generated by fitting the data to a linear combination of powers of (φ m -φ)/φ,

µ(φ) = µ s + α φ m -φ φ + β φ m -φ φ 2 , (3.1)
as was done by [START_REF] Dagois-Bohy | Rheology of dense suspensions of non-colloidal spheres in yield-stress fluids[END_REF]. The red curve in figure 6 (c) shows the result, with µ s = 0.47, α = 2.44, and β = 10.20. As noted previously, the value for µ s is larger than that obtained for suspensions of spheres (µ s = 0.3). The values for α and β also differ from those obtained for suspensions of spheres (α = 4.6 and β = 6). The best fit for η s was found to be

η s (φ) = 14.51 φ m -φ φ m -0.90 , (3.2)
as seen in figure 6 (a). Note that the best-fit exponent is -0.9, rather than -1. The rheological law for η n is then just given by

η n (φ) = η s (φ)/µ(φ), (3.3)
which is represented by the red curve in figure 6 (b). The variation of φ with J can be deduced from this last law since J = 1/η n (φ); this result is shown in figure 6 (d).

Discussion and Conclusions

Using a custom rheometer [START_REF] Boyer | Unifying suspension and granular rheology[END_REF][START_REF] Dagois-Bohy | Rheology of dense suspensions of non-colloidal spheres in yield-stress fluids[END_REF], we have performed pressure and volume-imposed measurements of the rheology of non-colloidal rigid fibres suspended in a Newtonian fluid. Measurements for the shear stress and particle pressure have been obtained in the dense regime and for aspect ratios between 3 and 15, and the volume fractions at which the rheology diverges has been characterised as a function of the aspect ratio.

The suspensions exhibit yield-stresses which increase with increasing volume fraction, φ, and are more pronounced for larger aspect ratios. Yield-stresses have been reported previously for rigid fibres suspended in Newtonian fluids, and the yield stresses have been attributed to adhesive contacts (see e.g. [START_REF] Mongruel | Shear viscosity of suspensions of aligned non-Brownian fibres[END_REF][START_REF] Chaouche | Rheology of non-Brownian rigid fiber suspensions with adhesive contacts[END_REF] despite the relatively large size of the fibres. A recent model (Bounoua et al. 2016b), which considered attractive interactions between fibres in the dilute regime, predicted simple Bingham laws for both the shear stress and the first normal stress difference, with the apparent shear and normal yield stresses proportional to φ 2 and φ 3 , respectively. The present data also follows Bingham laws, but the yield stress, τ 0 , and pressure, P 0 , increase with higher power laws in φ than predicted. This can be seen in the insets of figures 3 (c) and (d), where it is also demonstrated that the data for all aspect ratios collapses onto single curves by rescaling φ by φ m .

It is unclear whether, for the large fibres used here, colloidal forces are responsible for the yield-stresses. Finite-size effects close to the jamming point can also be advocated, particularly since lubrication forces are inefficient at preventing mechanical contacts between elongated particles [START_REF] Sundararajakumar | Structure and properties of sheared fiber suspensions with mechanical contacts[END_REF]. Close to jamming, since the system has a finite size, percolating jamming network of particles can exist. While it is transient phenomenon, it may impact the averaged rheological measurements which consequently may exhibit apparent yield stresses. Clearly, more work is necessary to elucidate the origin of the yield stresses.

Subtracting the apparent yield-stresses reveals a viscous scaling for both the shear stresses and particle pressures, wherein both grow linearly with the rate of shear. The aspect ratio of the fibres does not affect the friction coefficient, µ, but does impact the maximum flowable volume fraction, φ m . Rescaling the volume fraction, φ, by this maximum volume fraction, φ m , leads to an excellent collapse of all the data on master curves for the shear and normal viscosities. Hence, we argue that the aspect ratio principally affects the maximum volume fraction at which the suspensions can be sheared. Similar collapse of the rheological data across multiple aspect ratios has been observed previously for shear-thickening suspensions of colloidal fibres [START_REF] Brown | Shear thickening and jamming in densely packed suspensions of different particle shapes[END_REF].

Using the data presented here, constitutive laws in the form of expansions in (φ m -φ) have been generated for the rheology of dense suspensions of rigid fibres. An important product of the present study is the examination of the rheology close to the jamming transition. At jamming the friction coefficient is found to be constant and to be larger than that found for suspensions of spheres. Both shear and normal viscosities present a similar algebraic divergence in ≈ (φ m -φ) -1 in stark contrast to that in (φ m -φ) -2 observed for suspensions of spheres near the jamming point. The maximum volume fraction φ m is seen to decrease with increasing aspect ratio, similar to the dry packing of rigid fibres found in experiments (Rahli et al. 1999), see figure 5 (a). However, no inferences about the general structure of the suspension at jamming is possible for A < 15, as comparisons with estimates of maximum random packing [START_REF] Williams | Random packings of spheres and spherocylinders simulated by mechanical contraction[END_REF] do not clearly indicate that the orientation distribution has organised. The comparsion does indicate that the structure is organised for A = 15, though direct observations, or simulations, of the structures need to be developed in future work to conclusively resolve this question. The experimental data are available as supplementary material for future comparison.

Figure 1 :

 1 Figure 1: (a) Sketch of the experimental apparatus. (b) Images of the plastic fibres. (c) Image of the top plate (the inset is a blowup of the image showing the nylon mesh).

Figure 2 :

 2 Figure 2: Apparent relative (a) shear (τ /η f γ) and (b) normal (P/η f γ) viscosities as well as relative (c) shear [(τ -τ 0 )/η f γ] and (d) normal [(P -P 0 )/η f γ] viscosities (after subtraction of the yield-stresses) versus volume fraction, φ, for the fibres of batch (II) in pressure-imposed ( ) and volume-imposed ( ) configurations.

Figure 3 :

 3 Figure 3: (a) Shear stress (τ ) and (b) particle pressure (P ) versus shear rate, γ, for the fibre suspension of batch (II) at different φ values of 0.26 (lightest grey shade), 0.30, 0.35, 0.38, and 0.41 (black). The lines represent the linear fit for each different φ value. Yield-stress (c) for the shear stress (τ 0 ) and (d) particle pressure (P 0 ) versus φ for fibres of batch (I), (II), (III), and (IV) shown using the symbols , , ♦, and , respectively (see table 1). The insets of graphs (c) and (d) are log-log plots versus φ/φ m where φ m is the maximum flowable volume fraction given in figure 5 (a).

Figure 4 :Figure 5 :

 45 Figure 4: Rheological data: (a) η s = (τ -τ 0 )/η f γ and (b) η n = (P -P 0 )/η f γ versus φ as well as (c) µ = η s /η n and (d) φ versus J = η f γ/(P -P 0 ), for fibre batches (I), (II), (III), and (IV) as represented by the symbols , , ♦, and , respectively (see table 1). The insets of graphs (c) and (d) are log-log and semi-logarithmic plots.

Figure 6 :

 6 Figure 6: Rescaled rheological data: (a) η s = (τ -τ 0 )/η f γ, (b) η n = (P -P 0 )/η f γ and (c) µ = η s /η n versus φ/φ m as well as (d) φ/φ m versus J = η f γ/(P -P 0 ), for all the data of the different batches (I), (II), (III), and (IV) shown using the symbols , , ♦, and , respectively (see table 1). The insets of graphs (a), (b), and (d) are log-log plots. The red solid curves correspond to the rheological laws given by equations (3.1), (3.2), and (3.3).

Table 1 :

 1 Properties of each batch of fibres. Data shown includes the mean value and standard deviation of the aspect ratio A, fibre length L, and fibre diameter d. Values of the dimensionless number S p , characterising the relative strengths of the viscous and elastic forces, are also reported.

	Fibre label Symbol	A	L (mm)	d (mm)	Sp
	(I)		14.5 ± 0.8 5.8 ± 0.1 0.40 ± 0.01 5 • 10 -3
	(II)		6.3 ± 0.4 2.5 ± 0.1 0.40 ± 0.01 2.4 • 10 -4
	(III)	♦	7.2 ± 0.4 5.8 ± 0.2 0.81 ± 0.02 3.9 • 10 -4
	(IV)		3.4 ± 0.3 2.8 ± 0.1 0.81 ± 0.03 2.7 • 10 -5

Acknowledgments

We thank Simone Dagois-Bohy for his generous assistance with the experiments. This work is undertaken under the auspices of the ANR project 'Dense Particulate Systems' (ANR-13-IS09-0005-01), the 'Laboratoire d'Excellence Mécanique et Complexité' (ANR-11-LABX-0092), and the Excellence Initiative of Aix-Marseille University -A * MIDEX, a French "Investissements d'Avenir programme", and COST Action MP1305 'Flowing Matter'. FT benefited from a fellowship of CONICYT and SS from a fellowship of the A * MIDEX Excellence Academy -PhD Collegium. The plastic filaments were donated by PLASTICFIBRE S.P.A. This work was also supported by the National Science Foundation (grants #1511787 and #1362060).


D Chevallier

M Albert

P Devillard

Probing Majorana and Andreev Bound States with Waiting Times

Keywords: 02, 50, Ey -Stochastic processes 72, 70, +m -Noise processes and phenomena 74, 50, +r -Superconductivity-Tunneling phenomena

We consider a biased Normal-Superconducting junction with various types of superconductivity. Depending on the class of superconductivity, a Majorana bound state may appear at the interface. We show that this has important consequences on the statistical distribution of time delays between detection of consecutive electrons flowing out of such an interface, namely the waiting time distribution. Therefore, this quantity is shown to be a clear fingerprint of Majorana bound state physics and may be considered as an experimental signature of its presence.

Introduction. -During the last two decades, Majorana fermionic states in condensed matter physics, have received a lot of interest because of their exotic properties such as non-Abelian statistics, that open the perspective of using them for quantum computation. These exotic states have been studied extensively in various systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] with among them, a conceptually simple one made with a semiconducting nanowire of InAs or InSb, with strong spin-orbit coupling, subjected to an external Zeeman field and in the proximity of an s-wave superconductor (SC) [3,4,19,20]. In this situation, a Majorana Bound State (MBS) may appear at the interface of a normal/superconducting junction, under proper conditions, and strongly affects the electronic conduction properties (see Fig. 1). Several experiments have reported the observation of a zero-bias conductance peak in such physical setups, which are in good qualitative agreement with all theoretical predictions based on Majorana physics so far but still not fully consistent with the predicted conductance and magnetic field value needed for the existence of a MBS [21][22][23][24]. Therefore, several works have been conducted in order to understand these inconsistencies based on alternative interpretations by including other physical processes [25][26][27][28][29]. However, a clear consensus is still lacking mostly because of the absence of an experimental smoking gun for Majoranas. Generally these MBS appear in hybrid junction by tuning one of the parameter of the system (i.e. phase difference or Zeeman field) in a topological phase. Along this transition, these states mutate from Andreev Bound State (ABS) in the non-topological phase to MBS in the topological one and understanding their differences is thus of fundamental importance in order to distinguish them. So far, many efforts have focused on the relation and the evolution of ABS onto MBS by tuning the system parameters [8,30,31] but less on their own properties [20,[32][33][34] and the consequences on physical observables which is the purpose of this contribution.

Recently, an intriguing feature due to MBS was identified in Ref. [35] and named selective equal-spin Andreev reflection (SESAR). The presence of a MBS drastically modifies Andreev reflection and leads to a spin polarization of the current as well as to interesting correlations between different spin components which are visible in the zero frequency noise [36] for instance. However, such fingerprints are based on the possibility to observe fine quantitative differences between spin resolved current-current cross correlations which seems to be complicated experimentally in the present situation. In this letter, we show that a very clear qualitative difference is visible in the Waiting Time Distribution (WTD) of electrons flowing out of the interface making it an interesting and alternative signature of MBS. The WTD is the statistical distribution of time delay between the detection of two consecutive electrons and has been shown to be a very informative and powerful quantity for understanding correlations in mesoscopic quantum conductors [37][38][39][40][41][42][43][44][45][46][47][48][49][50].

Model. -We consider two types of hybrid junctions as depicted on Fig. 1 is converted into an electron with opposite spin ↓ n (↑ n) and b) Normal-(topological)Superconducting junction: a hole with spin ↑ n is converted into an electron with same spin ↑ n and a hole with spin ↓ n is reflected as a hole also with same spin ↓ n. In the first case n denotes any possible direction whereas in the second case a Majorana bound state appears at the interface and sets a special spin direction n for scattering (see text). In both cases a bias voltage eV is imposed and brings the superconducting chemical potential µS above the Fermi energy EF of the normal metal with the restriction that eV ∆ the superconducting gap.

one is a normal metal(N)/s-wave superconductor(S) carrying an ABS at the interface and the second one is a N/topological superconductor(TS) where the topological junction is made of a Rashba nanowire in proximity with an s-wave superconductor and in presence of Zeeman field V z carrying a MBS at each boundary. However, we assume the nanowire to be long enough to decouple the two MBS. In both situations, the Fermi energy of the normal metal is E F , the superconducting gap is ∆ and the superconducting chemical potential µ S is biased in such a way that E F = µ S -eV and eV ∆ as shown on Fig. 1. As a consequence, a stream of non-interacting holes is approaching the interface from the Normal part where it is scattered as a coherent superposition of electron and holes. This incoming scattering state reads

|ψ in = k V k=0 c k,n c k,-n |0 , (1)
where c k,n is the destruction operator of electron with momentum k, energy E = hv F |k| and spin orientation n (or creation of holes with opposite properties), |0 stands for the Fermi sea filled with states of energies up to µ s and k V = eV /hv F with v F the Fermi velocity. So far, n denotes any unitary vector and not necessary ẑ or x for instance. Below we will connect it to the polarization axes of the MBS but up to now this is just a choice of basis. It is important to note that spin components ↑ z / ↓ z or more generally ↑ n / ↓ n are equally distributed namely the incoming quantum state is isotropic in spin space. The key difference between these two junctions is how Andreev reflection occurs. In the case of a NS junction, the usual Andreev reflection takes place, meaning that a hole with a given spin is reflected as an electron with an opposite spin leading to the presence of ABS in such a junction (See Fig. 1 a)). Replacing the s-wave superconductor by a topological one strongly changes scattering properties and especially the Andreev reflection. If the Zeeman field is strong enough to enter the topological phase (V z ≥ ∆), the p-wave pairing dominates and the Andreev reflection is spin selective [35] meaning that a hole with spin up is reflected as an electron with the same spin and a hole with spin down is normally reflected as a hole with spin down (See Fig. 1 b)). More precisely, the presence of a MBS in the latter case, leads to a spin scattering symmetry breaking. There is a special spin orientation n, called the Majorana's polarization, along which electrons or holes are totally Andreev reflected as a hole or electron respectively with spin conservation whereas particles with opposite spin are normally reflected. This is the essence of SESAR effect [35] that leads to spin polarized current in this kind of hybrid junction. However, this precise direction cannot be determined from first principles and, in general, incoming particles are not spin oriented along this direction which leads to formally more complicated scattering although everything can be understood by decomposing the state onto this spin basis.

In order to evaluate the WTD and use it as a tool to probe the scattering properties of ABS and MBS in hybrid junctions we now need the expressions of the different outgoing scattering states. In such a junction, the interface plays an important role on the transmission which gives a finite width to the states [51][52][53]. In Appendix A, we discuss this effect. However, for the sake of simplicity we focus on the zero temperature and perfect Andreev reflection limit and following Ref. [35,48] write down the different out-going quantum states.

Outgoing states for N/S junction. -In this case, the interface acts as a perfect Andreev mirror where all the holes with a given spin are Andreev reflected as electrons with opposite spin [48]

|ψ ABS out = k V k=0 c † -k,-n c † -k,n |0 . (2)
Again, this quantum state is isotropic in spin space and simply corresponds to a stream of one-dimensional free electrons with energies between µ s and µ s + eV and two possible spin states (two channels of free fermions).

Outgoing states for N/TS junction. -The presence of the MBS deeply affects the outgoing state. As mentioned before, the key point is that a hole with a spin n is totally reflected as an electron with the same spin whereas a spin -n hole is subjected to perfect specular reflection with the same spin as well. It is therefore simpler to write the outgoing state in the Majorana polarization frame which reads

|ψ MBS out = k V k=0 c † -k,n c -k,-n |0 , (3)
where it is now obvious that the outgoing stream of electrons is totally spin-polarized in the n direction. Therefore, if one were able to measure the electronic current in this spin direction, one would get the result of a perfect single quantum channel. On the contrary, if one measures it in a random direction, for instance ẑ, one would measure partition noise just because the spin operators Ŝz does not commute with Ŝn . In that sense, such an experiment is very similar to the so called Stern and Gerlach historical experiment as we will discuss later. Apart from this remark the many body state can be simply obtained from (3), by replacing the c † n with the proper linear combination of c † ↑ and c † ↓ of the right basis. This can be done using the scattering matrix obtained in [35] depending on the value of the Zeeman field V z . However, in the simple case where the Zeeman field is large compared to the superconducting gap, the Majorana is fully polarized in the z-direction which means that n and ẑ are the same. Another simple case is right at the topological transition when the Zeeman field is just above the gap where c

† -k,↑ n = 1 √ 2 (c † -k,↑ ẑ + c † -k,↓ ẑ ).
Finally, it is important to note that in the non-topological case, namely when V z ∆, the outgoing state behaves like in the swave junction (usual Andreev reflection on an ABS) [35] and along the transition, the scattering matrix is not continuous.

Waiting time distribution. -We now turn to the calculation of the WTD. To do so, we need to specify the detection process. A time-resolved single electron detector is placed far away from the interface and is assumed to be sensitive to electrons only with energy above the superconducting chemical potential µ S . In addition the detector can be spin selective or not. Following Ref. [46,48] the WTD is obtained from the Idle Time Probability (ITP), namely the probability of not detecting any electron during a time slot τ . The precise definition of it depends on the detector capabilities. Without spin filtering it reads

Π(τ ) = ψ out | : e -Q ↑,E>µs :: e -Q ↓,E>µs : |ψ out , (4)
where : • • • : stands for the normal ordering and

Q σ,E>µs = x0+v F τ x0 c † σ (x)c σ (x)Θ(E -µ S )
dx is nothing else than the probability of presence of a charge Q during a time slot τ with E > µ S and spin projection σ =↑ / ↓ along a given direction (eg x, ẑ...). In Appendix B, we discuss in more details the derivation of Q and Π depending on the applied filtering, energy or/and spin. In the case of spin filtered detection (for instance spin up with respect to a given direction), this quantity is

Π(τ ) = ψ out | : e -Q ↑,E>µs : |ψ out . (5)
In both cases, the WTD is obtained from the second derivative of the ITP with respect to τ , W(τ ) = τ d 2 Π(τ ) dτ 2 , where τ is the mean waiting time given by 1/ τ = dΠ dτ (τ = 0). Eq. ( 4) and ( 5) are evaluated numerically for both many-body scattering states (2) and (3) with the same method as Ref. [46,48]. However, before discussing our results it is useful to recall several established results on WTD in quantum coherent conductors. In Ref. [39,46], it was shown that for a single quantum channel with a voltage bias eV (spinless electrons), the scattering quantum state is a train of non interacting fermions whose WTD is approximately the Wigner Surmise

W W S (τ ) = 32 π 2 τ 2 τ 3 exp - 4 π τ τ 2 , (6)
with τ = h/eV is the average waiting time, which means that due to Pauli's exclusion principle, electrons are separated in time by τ on average. An important feature of this WTD is the fact that it vanishes for τ τ which is the hallmark of fermionic statistics. If now this stream of electron is partitioned by a scatterer with energy independent transmission coefficient T , this WTD is continuously modified until it reaches an exponential form T exp(-T τ /τ )/τ when T 1. This exponential shape is the signature of uncorrelated events since detected electrons are well separated in time and therefore uncorrelated. In this case, the mean waiting time is τ = τ /T and therefore the average current e/ τ = e 2 h V T in agreement with the so called Landauer's formula [54]. Finally, when spin 1/2 are considered there are two conducting channel at disposal and the WTD no longer vanishes for small waiting times. At perfect transmission, it is described by the generalized Wigner-Dyson statistics [46].

WTD without spin filtering. -We start with the simplest situation where the single electron detector is spin insensitive. In the absence of MBS, it was shown [48] that the situation reduces to a stream of one dimensional free electrons with two spin components. Indeed, at perfect Andreev reflection, all the incoming holes are converted into electrons (Andreev mirror) with spin flip and energy between µ S and µ S + eV . The WTD is therefore the one of two perfect and independent quantum channels and is described by the generalized Wigner-Dyson distribution [46] depicted on Fig. 2a. The average waiting time is τ = h/2eV or in other words the average current is 2e 2 h V . On the other hand, in the topological case, the SESAR effect selects only one spin species reducing the possibilities to a single perfect quantum channel (with spin orientation +n). As a consequence, the WTD boils down to the so called Wigner surmise [39] also depicted on Fig. 2a. The average waiting time is h/eV and the average current e 2 h V therefore twice smaller than for the topological case. This is in agreement with the common interpretation that a Majorana behaves as "half an electron" [55].

We therefore conclude that not only the WTD repro- duces well known differences about the average current but also exhibits a qualitative mismatch between the two situations. With MBS, the WTD is exactly zero at τ = 0 because of Pauli's exclusion principle whereas it is not in the ABS case since two channels are available [46]. However, this discrepancy must be visible in any statistical measure of the electronic current like noise, third cumulant and Full Counting Statistics (FCS) in the same way that it is between one and two channel standard mesoscopic conductors.

WTD with spin filtering. -We now turn to a richer situation where we assume the single electron detector to be spin sensitive along a direction d and therefore collects only electrons with spin projection ↑ d. The following discussion is basically equivalent to the interpretation of the famous Stern and Gerlach experiment. The key point is that quantum state ( 2) is spin isotropic whereas (3) is spin polarized in the n direction. In particular, all possible observables are totally independent of the detector spin orientation in the ABS case. This is in strong contrast with the MBS where, for instance, filtering spin along ±n leads obviously to orthogonal results. We illustrate this statement on the WTD but it is very important to note that it applies to any other observables such as the spin resolved average current [35] or noise [4] and FCS in general.

For ABS, the single particle detector, whatever its spin orientation d, filters one spin species, namely ↑ d. Since they are equally populated and independent, the outgoing state (2) reduces to a single perfect quantum channel and its WTD is therefore Wigner surmise (see Fig. 2b). This WTD is characterized by a single peak centered around τ = τ with broad fluctuations and exactly zero value at zero which is the hallmark of Pauli exclusion principle as already mentioned. When a MBS is present, the precise shape of the WTD crucially depends on the detector spin orientation. Although quite academical, we can start by setting it to n. In that case, the detector collects every electron coming from the interface and the WTD is also the one of a single quantum channel. In this situation both ABS and MBS yield the same spin resolved WTD. If we choose now d = -n the detector collects nothing and if d slightly deviates from -n only a few electrons are kept and the WTD is expected to be exponential with rate eV h P d where

P d is the overlap | ↑ d | ↑ n | 2 .
For arbitrary d, the detector will partition the single quantum channel according to spin. The situation is almost formally equivalent to the one of a spinless single quantum channel flowing across a Quantum Point Contact (QPC) with energy independent transmission probability [39]. Here this transmission probability will simply be given by the overlap P d between | ↑ d and | ↑ n . In the special case where d ⊥ n, the quantum state ( 3) is a balanced mixture of | ↑ d and | ↓ d and then will be filtered exactly like a single quantum channel across a QPC with transmission probability 1/2. The situation can be implemented experimentally either by tuning V z just above ∆ and setting d = ẑ or in the limit V z ∆ where n = ẑ and filtering spin along x or y. This is shown on Fig. 2b where we have evaluated Eq. 5 along ẑ right above the topological transition by brute force numerics (limited to a quite small number of basis state (thirteen here) which explains the small discrepancy) and compared it to the expected result with very good agreement.

At this point, it is important to give some explanations on the experimental feasibility. The substrate, namely the heterojunction, has already been fabricated during the quest for Majorana quasi-particle [21][22][23] and consists of a Rashba nanowire partially in contact with an s-wave superconductor and in presence of a Zeeman field. The crucial point is to detect reflected electrons one by one in the normal part. Although still quite challenging, single electron detection technology is progressing very fast and might become a routine in the near future as reviewed in [45,49]. Otherwise, partial information on waiting times can be extracted from the average current, shot noise or second order coherence function obtained from Hong-Ou-Mandel experiment [45,56,57].

Conclusion. -We have studied the consequences of the presence or not of a MBS at the interface between a normal and a superconducting conductor on the electronic WTD. When a single electron detector is placed far away from the interface and detects electrons above the superconducting chemical potential µ S without spin filtering we observe a clear qualitative distinction between the topological and non topological situations. In addition, we have shown that the non topological situation (ABS) is immune to spin filtering in sharp contrast with the topological one due to SESAR effect. This conclusion is valid for the WTD which makes it a clear fingerprint of MBS but is also true for other quantities like the average current or higher moments of the FCS which can be easier to measure in actual experiments. Extension of this work could be the study of the influence of Coulomb repulsion when the superconducting part is not grounded but floating or the poisoning by another Majorana [58,59] and temperature or disorder effects. * * *

We are grateful to G. Candela, G. Haack and J. Klinovaja for useful discussions and remarks. The research of D. C. was supported by the Swiss NSF and NCCR QSIT.

Appendix A: Finite width of the outgoing states.

-We study here the effect of a finite energy width of the Majorana bound state and show that it does not qualitatively change the waiting time distribution. Due to the finite hopping strength at the interface, the Majorana bound state located at zero energy has a finite width Γ. The main consequence of this is the energy dependence of the Andreev reflection coefficient at the interface which becomes [60]

R A (E) = Γ E + iΓ 2 . (7)
However, we can recover the same states (2) and (3) as in the main text when taking the limit eV Γ. Using this assumption, we can calculate the ITP and thus the WTD when the states have a finite width (see Fig. 3) using the energy dependent coefficients [46].

In fig. 3, we can see that the effect of the broadening of the states has no dramatic effects on the WTD. For this reason, we do not focus on this effect in the main text.

Appendix B: Derivation of the Idle Time Probability. -In this appendix we explain how to calculate the idle time probability, following the notations of Ref. [46]. It is important to note that the energy range of detected particles is assumed to be small enough that the dispersion relation of electrons or holes is linear E = hv F k. In that case, charge measurements over a time window ∆t are equivalent to charge measurements over a space window ∆x/v F thanks to Galilean invariance. The key point to calculate Q σ,E is to specify the detection procedure. This includes the possibility to only detect positive/negative energies, spin projection up/down with respect to a given quantization axis. In an actual experiment, the detection can be done by connecting the superconductor or the topological superconductor to two quantum dots instead of a normal metal. By doing so one can filter energy by applying an external gate on the two dots or select the spin by using interacting quantum dots with strong repulsion in order to get rid of the spin degeneracy [61,62] . Analytically these properties can be implemented easily in the definition of Q. This operator can be represented in the basis of the scattering states as

Q E>µs = t(k)t * (k ) e i(k-k )v F τ -1 i(k -k ) dk 2π dk 2π (8)
where the t(k) are the energy dependent transmission amplitudes of a scattering state and may be chosen in that case as t(k) = 1 if E(k) > µ s and t(k) = 0 if E(k) < µ s .

In order to compute the ITP, the transport window has to be discretized into N energy compartments of size eV /N with corresponding momentum intervals of size κ = eV N hv F with v F = hk F m is the Fermi velocity defined with m the electron mass. Using this discretization leads to the following matrix elements for Q E>µs in the large N limit

with m, n = 1, ...., N . From this definition of the ITP and when the average is taken over a Slater determinant of free fermions, the WTD can be cast as a determinant of the form [41,46] Π(τ ) = det(1 -Q τ ), (10) which can be evaluated with a computer. Then, it is straigthforward to extend this detection procedure to a spin selective one by setting spin dependent transmission coefficients.

Fig. 1 :

 1 Fig. 1: (color online) Schematic picture of the Andreev reflection processes in the two different junctions. a) Normal-(trivial)Superconducting junction: a hole with spin ↑ n (↓ n)is converted into an electron with opposite spin ↓ n (↑ n) and b) Normal-(topological)Superconducting junction: a hole with spin ↑ n is converted into an electron with same spin ↑ n and a hole with spin ↓ n is reflected as a hole also with same spin ↓ n. In the first case n denotes any possible direction whereas in the second case a Majorana bound state appears at the interface and sets a special spin direction n for scattering (see text). In both cases a bias voltage eV is imposed and brings the superconducting chemical potential µS above the Fermi energy EF of the normal metal with the restriction that eV ∆ the superconducting gap.

Fig. 2 :

 2 Fig. 2: (color online) WTDs versus τ /τ of electrons flowing out of the interface without spin filtering a) and with spin filtering orthogonal to the majorana polarization b). The solid gray line (resp. red line) corresponds to an NS junction without (resp. with a) MBS (see text) at perfect Andreev reflection. The dashed black line represents the WTD of a single channel normal conductor with transmission probability one (a)) and one-half (b)) for comparison [39].

Fig. 3 :

 3 Fig. 3: (color online) WTDs versus τ /τ of electrons flowing out of the interface of a N/S (gray line) and a N/TS (red line). The dashed and solid line correspond to two different width of the states as mentioned in the legend.

2 (

 2 [Q] m,n = κt * κm t κn π e -i 2 κ(n-m)v F τ sin (κn-κm)v F τ κnκm)