Abstract
The process by which sheared suspensions go through a dramatic change in viscosity is known as discontinuous shear thickening. Although wellcharacterized on the macroscale, the microscopic mechanisms at play in this transition are still poorly understood. Here, by developing new experimental procedures based on quartztuning fork atomic force microscopy, we measure the pairwise frictional profile between approaching pairs of polyvinyl chloride and cornstarch particles in solvent. We report a clear transition from a lowfriction regime, where pairs of particles support a finite normal load, while interacting purely hydrodynamically, to a highfriction regime characterized by hard repulsive contact between the particles and sliding friction. Critically, we show that the normal stress needed to enter the frictional regime at nanoscale matches the critical stress at which shear thickening occurs for macroscopic suspensions. Our experiments bridge nano and macroscales and provide long needed demonstration of the role of frictional forces in discontinuous shear thickening.
Introduction
Dense suspensions of solid particles immersed in a Newtonian solvent display complex flow properties. Shear thickening, where the resistance to flow increases when the dispersion is stirred is one of the most intriguing behaviour^{1,2}. In the extreme situation of discontinuous shear thickening, shear viscosity increases by orders of magnitude at a given shear rate. Under impact, the formation of a dynamic jamming front makes the fluid so resistant that a person can run on it^{3,4,5}. This phenomenon, well documented^{6,7} yet not really understood, is essential from the perspective of applications and materials. Flows of concentrated dispersions are ubiquitous in nature and industry: water or oil saturated sediments, muds, crystalbearing magma^{8}, concrete, silica suspensions, cornflour mixtures, latex suspensions and clays are examples of shear thickening systems. Industrially, these systems can have disastrous effects by damaging mixer blades or clogging pipes. Discontinuous shear thickening may also be harnessed and desirable when engineering composite materials, for shockabsorbing materials or soft body armour^{9}.
Despite an extensive characterization of discontinuous shearthickening transitions at the macroscale, there is still no clear understanding of the microscopic mechanisms at play in this transition, principally owing to the challenges associated with quantitative frictional measurements at the nanoscale^{10}, especially for pairs of particle^{11}.
A longstanding view is that thickening is driven by hydrodynamic and Brownian forces^{12}. At large shear rate, these forces create highly dissipative transient clusters of particles, due to the singular lubrication flows between the particles. When normal elastohydrodynamic contact forces are taken into account (without solid friction), these simulations capture continuous shear thickening and large increase in suspension viscosity^{13}. However, this model predicts shear rateindependent rheology for nonBrownian systems and broader transition that observed experimentally.
To get around this issue, recent works^{14,15} propose a new picture that neglects thermal fluctuations and put forward the role of (nanoscale) repulsive and frictional forces. At low pressure, neighbouring particles are separated by a gap filled with solvent and interact via hydrodynamic forces. At high pressures, repulsive forces are overcome, leading to frictional contacts and shear thickening.
At this stage, this picture and the role played by frictional forces have been validated through numerical simulations but only indirectly through experiments at the level of the suspension^{2,6,16}.
In the present work, we bridge this gap between nano and macroscales. Taking advantage of quartztuning fork based atomic force microscope with subnanonewton resolution, we build upon state of the art procedures^{17,18} to measure the pairwise force profile and the frictional interactions between pairs of particles and correlate these measurements to the macroscale rheology of suspensions. We study two wellknown shearthickening systems: polyvinyl chloride (PVC) suspended in various solvents^{19,20} and cornstarch particles suspended in water^{21,22}. Our measurements provide a clear view of discontinuous shearthickening transition as the breakdown of lubricated contact between particles, at a critical normal force.
Results
Experimental setup
We present in Fig. 1a the schematic of the experimental setup. Briefly, we glue an electrochemically etched tungsten tip of ∼50 nm end radius to a millimetric quartz tuning fork, which serves as our force sensor. Using an inhousebuilt nanomanipulator in a scanning electron microscope (SEM), we then glue one individual particle to the end of the tungsten tip (Fig. 1b).
During a typical experiment, the attached particle is immersed in solvent and brought into contact to another bead fixed on the substrate, while monitoring the force profile (Fig. 1a). Further characterization of substrate roughness and topography are shown in Supplementary Figs 1 and 2.
To measure simultaneously normal and tangential force profiles between the two approaching particles, we simultaneously excite the tuning fork via the piezodither at two distinct resonance frequencies f_{N}≈31 kHz and f_{T}≈17 kHz, corresponding to the excitation of both normal (blue arrows) and shear modes of the tuning fork (red arrows), as shown in Fig. 1c. Both modes correspond to symmetric excitation of the prongs, leading to negligible motion of the centre of mass and highquality factor of the oscillator^{23}. Monitoring changes in the resonance of each modes (Fig. 1d) allows us to measure respectively the normal and tangential force profile between the two objects.
We now detail how to obtain force profile for each normal (N) and tangential (T) modes (Fig. 1d). When excited close to a resonance frequency, the system can be modelled as a massspring resonator^{24}, whose resonance profile will be modified by the interacting forces. First, position dependent forces F(x, z) will lead to shifts in the resonance frequency of the oscillator (comparing black and red resonance curves in Fig. 1d). Hence, the spatial force gradient ∇F (N m^{−1}) is proportional to the shift in the resonance frequency δf^{i} (Hz) according to^{24}:
where (Hz) and (N m^{−1}) are the spring constants, respectively, associated with normal (N) and tangential (T) oscillation modes, e_{i} corresponds to the unit vector along direction i.
We can then express the frictional forces applied on the bead as the sum of viscouslike friction force (proportional to speed) and solid friction force (independent of speed and corresponding to sliding friction for tangential motion, i=T):
where v ( m s^{−1}) is the oscillating speed, γ (kg s^{−1}) is a viscouslike friction coefficient and the second term F_{s} [N] characterizes the solid friction force, only dependent of the sliding direction v_{i}/v_{i}. The friction forces F_{s} are assumed to be velocity independent as experimentally confirmed a posteriori. Over one oscillation period, all friction forces in equation (2) have zero mean, only leading to changes in the height and broadening of the resonance peak (comparing black and red resonance in Fig. 1d). It is noteworthy that we can not measure static friction forces with our setup. However, we expect static friction to appear concurrently to sliding dynamic friction.
Keeping a constant oscillation amplitude a_{i} at resonance requires the external excitation force F_{ext} (due to the piezo dither) to be equal to the to the sum of all dissipative forces F_{D}. As the external force F_{ext} is directly proportional to the excitation voltage E_{ext}, the sum of all dissipative forces F_{D} is directly measured by tracking the excitation voltage necessary to keep a constant oscillation amplitude (see Methods for further details, Supplementary Figs 4–6 and Supplementary Note 1).
Nanoscale force profile
We show in Fig. 2 the typical force profile measured between two approaching PVC beads in good solvent. Monitoring changes in the resonance (Fig. 1d) for the two oscillating modes (Fig. 1c) allows us to characterize pairwise interparticles interactions through the normal dissipation (Fig. 2b), the projection of the normal force gradient along the normal direction—∂F_{N}/∂z (Fig. 2c) and the tangential friction force (Fig. 2d).
We first show in Fig. 2b the inverse of the normal dissipative force 1/. The red line is a linear fit of the inverse dissipation, showing that normal dissipation is characterized by hydrodynamic drainage and Stokes law as the beads are separated from each other:
where η≈40 mPa s^{−1} is solvent viscosity (see Supplementary Section S2.3), v=aω (m s^{−1}) is the typical speed of the oscillating particle, z (m) is the distance between the two noshear planes and R (m) is an equivalent bead radius. The intersection of the red line with the horizontal axis defines the hydrodynamic zero (z=0), which defines the absolute position of the noshear planes between the two objects (vertical dotted lines Fig. 2a). This zero defines two domains, corresponding to (i) z>0, hydrodynamic lubrication (light blue) and (ii) z<0 hard contact (light red). It is noteworthy that for confinement below ≈ 3 nm, we observe a deviation from Stokes hydrodynamics with a regularization of the hydrodynamic divergence, possibly stemming from elastohydrodynamic interactions^{13,25,26}. The dissipative normal forces measured for z<0 may be due to viscoelasticity of the PVC particles.
We now turn to the normal force gradient −∂F_{N}/∂z (N m^{−1}) shown in Fig. 2c. For the two approaching particles, we observe an increasing repulsive normal force gradient (−∂F_{N}/∂z>0) before contact between the two particles (z>0, blue zone). These repulsive forces vary steadily and smoothly with distance, whereas normal dissipation is dominated by hydrodynamics during the approach (Fig. 2b). We thus interpret these repulsive forces as a signature of the entropic repulsion between polymer brushes forming at the surface of the PVC beads, due to the effect of the plasticizing solvent^{19,27} (Fig. 2a). We can characterize the steepness of this repulsive profile right before contact by an exponentiallike law F≈exp(−z/λ) with λ≈4 nm (Fig. 2c, red dotted line)^{28}. Upon contact, the steepness of the repulsive profile increases slightly.
We show in Fig. 2d the tangential dissipative friction force (tangential mode T, Fig. 2c). Before contact (blue zone (i)), tangential forces are below 1 nN, consistently smaller that the normal hydrodynamic dissipative forces which are of the order of 5–10 nN. Upon contact (red zone, (ii)), we observe a clear increase of frictional forces. We note that, depending on the respective surface states of the beads, contact can also occur before the hydrodynamic zero (for z>0), due to the presence of asperities on one of the bead surface (see Supplementary Fig. 5). Finally, we note that the fact that we recover solvent viscosity η in the dissipative normal force and that there is low tangential lubrication forces before contact are a signature of the absence of brush interpenetration in the probed experimental conditions^{29}.
Critical load friction
We now turn in Fig. 3 to the nature of the frictional profile, as uncovered by the two regimes shown in Fig. 2. We plot in Fig. 3a the typical form of the tangential dissipative force as a function of the normal load F_{N}, obtained by integrating the normal force gradient^{30}. In the first regime of hydrodynamic lubrication (blue zone, (i)), tangential frictional forces are small and arise purely from hydrodynamic interactions, whereas a normal load F_{N} can be sustained due to entropic repulsion of the brushes. This situation results in a friction coefficient as low as μ≈0.02, as observed in previous friction studies on polymer brushes in SFA^{29,31,32,33,34}.
On a critical normal load corresponding to the force necessary to completely compress the polymer layers and reach hard contact, the system switches to a second state characterized by a sharp increase in friction (Fig. 2d, red zone (ii)). This second regime is well characterized by Amontons–Coulomb laws, with a proportionality between tangential frictional forces and normal load: , where is the tangential viscous dissipation right before contact. Moreover, as shown in Fig. 3b, the friction coefficient between two beads is independent of the sliding speed for tangential speeds above 200 μm s^{−1}, a clear characteristic of solidlike friction (relative speed is changed here through the oscillation amplitude a_{0}). The independence of sliding friction with speed validates a posteriori our choice for the form of equation (2).
Finally, we show in Fig. 3c the distribution of friction coefficient obtained over 30 different pairs of beads. As characterized in Fig. 3b, the friction coefficient is a welldefined property of each particle interactions (Fig. 3b) but also depends on the local physicochemical, geometrical, mechanical and roughness surface state of the two sliding beads. We find a mean interparticle friction coefficient μ=0.45±0.2, in very good agreement with the macroscopic friction coefficient of PVC on PVC^{35}.
Discussion
We now come back to the macroscale behavior of nonBrownian suspensions.
As shown in Fig. 4a,b, we measured using standard rheometry the flow curves (that is, the relation between the applied shear stress σ and the measured shear rate ) for various solutions of PVC particles in mixtures of Dinch and mineral oil^{19,20}, and cornflour particles in water^{21,22} at various solid fractions (see Supplementary Fig. 3 for details). All these suspensions exhibit a discontinuous shear thickening transition above a critical shear stress σ_{C} (defined as the critical shear stress above which the viscosity starts to increase as a function of the shear rate^{2}). As found previously, σ_{C} (contrary to the corresponding shear rate ) does not depend upon the solid fraction of particles φ^{2} for high solid fractions (Fig. 4b).
Following recent models, σ_{C} corresponds to the shear stress required to obtain a particule normal stress high enough to overcome the repulsive forces and to transit from lubricated to frictional contacts^{15,36}. If the shearthickened state is characterized by frictional interactions between the particles, there should be a correlation between the macroscale critical shear stress σ_{C} and the critical load uncovered in Fig. 3. To verify this correlation, we measured the microscopic critical load for each of the systems of Fig. 4a (typical distribution of normal critical force is shown in Supplementary Fig. 7).
We report in Fig. 4b the macroscale shear stress σ_{C}, versus the nanoscale critical force for each discontinuous shearthickening systems. For both PVC and cornstarch systems, we see a clear correlation between those two nanometric and macroscopic quantities, with the critical stress at the macroscale varying proportionally to the critical force needed to enter into frictional contact at the nanoscale:
We find here a proportionnality coefficient β_{PVC}≈0.006 and β_{Cornstarch}≈0.02, in relatively good agreement with predictions from simulations performed on smooth particles (β_{simu}≈0.05 for a friction coefficient μ=1 (ref. 36)). Those coefficients characterize stress transmission from the suspension to the particles level and depend on the microscopic friction coefficient^{36} for both static and sliding friction, as well as particle shape and roughness. It is noteworthy that in the simulations, the values of the static and dynamic friction are assumed to be the same, which may not be the case in our situation. Macroscopic roughness may also block the particles and affect the value of β found by numerical simulations.
Let us underline that does not depend upon the relative tangential velocities between particles in the range of experimental data. Moreover, both and τ_{C} are measured for approximatively the same range of relative velocities between the two particles. In the macroscopic experiments, the relative velocity between two particles can be approximated as at the onset of the shear thickening transition and varies between 4 and 200 μm s^{−1}. In the AFM experiments, the normal root mean square (RMS) speed is ∼30–200 μm s^{−1} and the tangential RMS speed 150–800 μm s^{−1}.
Our unique experimental setup allowed for the first time to characterize frictional interactions between pairs of particles from discontinuous shearthickening suspensions. We explain unambiguously discontinuous shearthickening transition as stemming from the breakdown of lubrication between particles and the onset of hard frictional contacts. Even though normal elastohydrodynamic forces appear for very small separation distances (Fig. 2b, z<2 nm), they do not contribute to the observed shearthickening transition. Indeed, we find that the critical shear stress σ_{C} at the shearthickening transition varies as a function the contact forces (Fig. 4c). This is opposite to predictions of numerical simulations proposed by Jamali et al.^{13}, for which the shearthickening transition is found to be independent of particle elastic modulus if stemming from elastohydrodynamic forces.
Our measurements pave the road for direct simulation of the rheological properties of dispersions. In particular, the distribution of friction coefficients, contact forces and size distribution of the dispersion should be taken into account in future numerical simulations to capture at which solid fraction the dispersion evolves to a continuous or discontinuous shear thickening transition.
Going further, our measurements provide for the first time a firm microscopic description for the rheological behaviour of suspensions of particles. Similar measurements could be extended to the regimes of high shear and high load, and might explain the shearthinning regimes often measured after the shearthickening transition. Finally, rheological behaviors as diverse as shear thinning^{37}, selffiltration^{38} or particle migration could be disentangled and better understood through similar approaches.
Methods
PVC suspensions, plasticizer and mineral oil
The mean particle radius of PVC particles, defined as is 1 μm. The size distribution is lognormal and the s.d. estimated using the volume distribution is 45%. We measured a RMS roughness of 2.2 nm for PVC particles.
As a plasticizer for PVC particles, we use 1,2cyclohexane dicarboxylic acid diisononyl ester (Dinch) supplied by BASF. This organic liquid Dinch slightly disolves the outer part of the particles and creates a polymer brush around them. This brush enables stabilization of the suspension due to steric repulsion^{19}. At high temperature (T≥100 °C), Dinch can dissolve the PVC particles^{39}. At room temperature, this process is much slower and takes >1 year.
Degree of plasticization can be changed by using a mix of mineral oil and plasticizer^{20}, because mineral oil has a low affinity with PVC. Mineral oil viscosity standards were provided by Paragon Scientific Ltd. Same viscosity as Dinch (41.1 mPa s^{−1} at 25 °C) was achieved by mixing two different viscosity standards (55.7 and 29.0 mPa s^{−1}). Resulting viscosity was checked in a shear rate range from 1 to 100 s^{−1} for different temperatures. Same viscosity at 25 °C and same temperature dependence of viscosity (range 20–25 °C) were found between plasticizer and obtained mineral oil.
Concerning PVC, we report results (both AFM and rheological experiments) for three different plasticizing liquids: (1) 100 vol.% plasticizer, (2) 90 vol.% plasticizer+10 vol.% mineral oil, and (3) 67 vol.% plasticizer and 33 vol.% mineral oil.
PVC substrate
To make the substrate, a given amount of PVC powder is introduced into a metallic mold laying on a glass slide. A countermould is used on top. The mould and countermould are then transferred into a hot press and compressed 5 min at 150 °C and 20 bars. Then, the sample is cooled down, resulting in a compact and transparent piece of PVC. Even if the pressing temperature is higher than the PVC glass transition temperature (T_{g}=80 °C), the original shape and surface topography of the particles is preserved (see Supplementary Fig. 1).
Cornstarch
Cornstarch was supplied by SigmaAldrich and used without further modification. It contains ∼73% amylopectin and 27% amylose with particle diameter around 14 μm (polydispersity 40% from static light scattering)^{22}. Owing to very low volume used and evaporationrelated problems, AFM measurements for cornstarch were done in pure water as a suspending liquid. Rheological measurements were also carried out in pure water for the cornstarch suspensions. Substrates were made by gluing cornstarch particles on a flat silicon substrate using cyanoacrylate glue (see Supplementary Fig. 2). We measured a RMS roughness of 14 nm for the cornstarch particles.
Particle gluing under SEM and particle size
PVC and cornstarch particles are glued to the etched tungsten tip of the tuning fork using SEMGLU from Kleindiek and a nanomanipulation station insitu SEM (FEI Nova NanoSEM 450) (see Fig. 1b). Measurements presented in the supplemental materials and main text were obtained for four different attached particles (radius of 0.6 and 0.5 μm for the two PVC particles, and 3.8 and 4 μm for the two cornstarch particles).
Measurement of frictional and dissipative forces
We consider below the equation of motion for the tuning fork close to its resonance frequency, in the presence of both viscous and sliding friction. We recall that the friction force can be expressed as the sum of a viscous like friction coefficient and a sliding friction force: . is the viscous force, with γ_{i} the viscous friction coefficient and the sliding friction force (equations (1) and (2)). In its complex form and for the tangential and the normal directions, the equation of motion becomes:
where ω is the excitation frequency, is the natural frequency of the oscillator, with m the equivalent mass and k_{i} the equivalent stiffness of the mode i∈{N, T}. is the complex oscillator amplitude, is the external piezoelectric excitation force and is the friction force. Upon approach, γ, and ∇F will vary with position. The factor 2/π stems from the first Fourier coefficient of the squarelike shape of the solid friction force. From equation (5), we obtain a fundamental relation between the amplitude at resonance a_{0}, the quality factor Q=mω_{0}/γ characterizing viscous dissipation and the solid friction force . Neglecting the force gradient compared to the very large stiffness of the tuning fork, we obtain:
The external excitation force necessary to keep a constant oscillation amplitude a_{0} can then write as the sum: . This external force is proportional to the piezoelectric excitation voltage =CV_{ext}, where C is a factor calibrated at the beginning of each experiment. When the two particles are distant, we get from a lorentzian fit of the resonance curve. Monitoring the excitation voltage necessary to keep a constant oscillation amplitude a_{0} gives us a direct measure of the sum of all forces acting on the tuning fork as ==CV_{ext} to obtain an amplitude a_{0} typically equal to 1–50 nm. Both shear and normal modes of the tuning fork can be excited at the same time by summing the two excitation voltage at each frequencies (Fig. 1c). Owing to the very large differences in resonance frequency, the two modes are uncoupled. This uncoupling can further be verified by checking that changes in the excitation voltage for one mode do not affect the resonance curve of the second mode. We take k_{N}=40 kN m^{−1} for the 32 kHz mode and k_{T}=12 kN m^{−1} for the 18 kHz mode. In practice, two phaselocked loops allow us to track the two centre resonance frequencies f_{N} and f_{T}. A proportionalintegralderivative controller (PID) keeps the oscillation amplitude a_{T} of the tangential mode constant, allowing a direct measure of the frictional forces by monitoring the amplitude of the excitation voltage E_{T}. A fixed amplitude of the excitation voltage E_{N} is applied to the normal mode and dissipation is measured by monitoring the oscillation amplitude a_{N}. The electronic lockin and phaselocked loops s are implemented using a Nanonis from (SPECS Zurich) and a HF2LI Lockin Amplifier (Zurich Instrument).
Characterization of PVC suspensions
We prepare our dispersions by weighting a given amount of PVC particles, a given amount of Dinch and a given amount of mineral oil. The solid fractions are then calculated knowing the density of PVC ρ_{PVC}=1.38 g cm^{−3}, the density of Dinch ρ_{Dinch}=0.95 g cm^{−3} and the density of mineral oil ρ_{oil}=0.84 g cm^{−3}. Suspensions are stirred 5 min at 1,000 r.p.m. using a Dispermat LC55 (VMA Getzmann), to ensure good dispersion state. Samples were freshly mixed for each experiment. This protocol was found to produce reproducible samples. The solid volume fraction of the suspension is defined as the volume of particles divided by the total volume: .
Rheology was measured using a stresscontrolled DHR3 rheometer (TA Instruments) equipped with a smooth cone (diameter D=40 mm, angle=2°, truncation gap=54 μm). A logarithmic stress sweep (50 s per point) between 1 and 1,000 Pa (10 points per decade) was performed to measure the flow curve. Flow curve obtained matches the viscosity found using ‘peak holds’ of shear rate (flow steady state is reached within 10 s). No wall slip was measured in these experiments. This was checked indirectly by measuring velocity profiles using ultrasounds in Couette cells.
Characterization of Cornstarch suspensions
Dispersions were also prepared by weighting a given amount of cornstarch and a given amount of water. The solid fractions are then calculated knowing the density of cornstarch ρ_{cornstarch}=1.63 g cm^{−3} and the density of water ρ_{water}=1.00 g cm^{−3}. The solid volume fraction of the suspension is also defined as the volume of particles divided by the total volume. Samples were freshly mixed for each experiment. Rheology was measured using a stress controlled DHR3 rheometer (TA Instruments) equipped with a hatched plate (diameter D=40 mm) similar to that in previous works^{22}. Contrary to PVC suspensions, both lower and upper plate are hatched to avoid wall slip. Flow curves were obtained with a logarithmic stress sweep from 0.1 to 100 Pa (10 points per decade). Each point was measured during 10 s, which was long enough to ensure equilibrium, while avoiding water evaporation and/or particles sedimentation. At high stresses (see Supplementary Fig. 3), both PVC and cornstarch suspensions exhibit discontinuous shear thickening where the gradient d(log η)/d(log σ) reaches 1 (vertical flow curve when plotting ). Within experimental uncertainty, shear thickening begins at a fixed onset stress, which depends only on the studied system and not on the volume fraction φ^{2} (see dotted lines on Supplementary Fig. 3).
For PVC particles in a suspending liquid made of 100 vol.% plasticizer, an onset stress of 75±5 Pa is found. For PVC particles in a suspending liquid made of 90 vol.% of plasticizer an onset of 38±5 Pa is measured and for PVC particles in a suspending liquid made of 67 vol.% of plasticizer an onset of 8±2 Pa is found. For cornstarch particles suspended in pure water, an onset stress of 3–4 Pa in found, in good agreement with previous works^{21,22}.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Comtet, J. et al. Pairwise frictional profile between particles determines discontinuous shear thickening transition in noncolloidal suspensions. Nat. Commun. 8, 15633 doi: 10.1038/ncomms15633 (2017).
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Acknowledgements
We thank Cyprien Poirier for fabrication of the PVC substrates and Mario S. Rodrigues for many fruitful discussions. J.C., A.N. and A.S. acknowledge funding from the European Union’s H2020 Framework Programme/ERC Starting Grant agreement number 637748—NanoSOFT. L.B. acknowledges support from the European Union’s FP7 Framework Programme/ERC Advanced Grant Micromegas and funding from a PSL chair of excellence. J.C., A.N., L.B. and A.S. acknowledge funding from ANR project BlueEnergy.
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J.C., G.C and A.C wrote the paper. J.C. performed the experiments and analyzed the data dealing with AFM. G.C performed the experiments and analyzed the data dealing with rheology. L.B., A.N. and A.S. discussed the experiments. A.C.,L.B,A.S. supervised the project. All authors discussed the results and commented on the manuscript.
Corresponding author
Correspondence to Annie Colin.
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The authors declare no competing financial interests.
Supplementary information
Supplementary Information
Supplementary Figures and Supplementary Note 1 (PDF 456 kb)
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