Fabien Gerbal
email: fabien.gerbal@upmc.fr

Yuan Wang

-Olivier Sandre

Fabien Montel

-Jean-Claude Bacri

Extensive characterization of magnetic microrods observed by optical microscopy

Keywords: characterization | micro, nanorod | micro, nanowire | susceptibility | rigidity | stiffness | superparamagnetic

for non-interacting and uniformly magnetized nanoparticles such as used here, the magnetization (and therefore the susceptibility) is simply proportional to the volume of magnetic matter.

ABSTRACT

The usage of micro or nanorods is steadily increasing in various applications from fundamental research to industry. Therefore their geometrical, mechanical and eventually magnetic properties need to be well determined. Here, using an optical microscope equipped with magnetic tweezers, we report an experimental procedure to obtain all these information on a single magnetic rod. In particular, we measured the magnetic susceptibility χ by analyzing the deformation of a rod submitted to a uniform magnetic field. To do so, we refined a theoretical model which takes into account the variation of χ with the internal field. We prove experimentally that this model yields consistent measures, at any value of field strength and incidence angle. From the combination of the different measurements, we also deduced the number of iron oxide nanoparticles which are embedded within the polymer matrix of the superparamagnetic rods under study.

A very broad research activity concerns micro and nanoscale magnetic rods which show fundamental interest in many applications such as surface coating reagents, actuators, bio-medical probes and so on [START_REF] Devan | One-dimensional metal-oxide nanostructures: Recent developments in synthesis, characterization, and applications[END_REF]. Depending on their size, various strategies have been developed to synthesize them: near-micron radius or larger wires are made for instance from rapid quenching [START_REF] Zhukova | Thin magnetically soft wires for magnetic microsensors[END_REF], whereas the probably most common strategy for preparing rods around tens of nanometers diameter consists in filling the pores of an etched template by electrochemical deposition of one or several metals [START_REF] Fert | Magnetic nanowires[END_REF].

Other groups realized nanorods by metallizing bacteria [START_REF] Liang | Magnetic and mechanical properties of micro/nano particles prepared by metallizing rod-shaped bacteria[END_REF], by chaining particles together [START_REF] Singh | Synthesis of flexible magnetic nanowires of permanently linked core-shell magnetic beads tethered to a glass surface patterned by microcontact printing[END_REF], or by embodying nanoparticles into an elastomer gel such as PDMS through lithographic technics [START_REF] Tavacoli | The fabrication and directed self-assembly of micron-sized superparamagnetic nonspherical particles[END_REF].

Many applications for which these rods are designed require a well-characterized calibration of their properties, often at the level of the individual object rather than for a broad distribution.

We report the measurements of their length L, radius r, flexural rigidity C, and magnetic susceptibility χ performed on single individual rods by using a conventional optical microscope equipped with magnetic tweezers (ESI †, note 1 shows the complete scheme of all measurements). Whereas the measurement presented in the paper could apply to many types of magnetic rods, we studied L ∼10-80 micron-long superparamagnetic rods with r ∼200-400 nm made of iron oxide nanoparticles (diameter∼13 nm) embedded into a polymer matrix [START_REF] Colin | Rotational microrheology of Maxwell fluids using micron-sized wires[END_REF][START_REF] Fresnais | Electrostatic co-assembly of iron oxide nanoparticles and polymers: Towards the generation of highly persistent superparamagnetic nanorods[END_REF][START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF][START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF][START_REF] Yan | Magnetic nanowires generated via the waterborne desalting transition pathway[END_REF]. We initially assessed each measurement independently: the super-resolutive optical model used to derive r was compared to Atomic Force Microscopy (AFM) scans performed on the same rods; the measured value of C was derived from the analysis of the thermal fluctuations [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF]; χ was obtained from the measurement of the tip deflection of a magnetically-bent rod. To do so, we used a theoretical model (which takes into account the variation of χ(H) = M (H) H with the field strength H [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF]) to describe the rod magnetoelastic deformation. We present an improved version of this model which applies well to rods of relatively low magnetic susceptibility such as the ones studied here. By comparison of the resulting χ value with the susceptibility χ v of the particles which compose the rods (measured independently by vibrating sample magnetometry (VSM)), we deduced the volume fraction φ of the magnetic nanoparticles. We also report experiments which yield constant values for φ, independently of the intensity and incidence angle of the field, thereby demonstrating the robustness of the model. Finally, the full analysis performed on 10 different rods showed consistent and relatively low dispersion of the values found for φ ∼ 10% -30% considering the broad disparity of χ ∼ 2.0 -7.4 (at µ 0 H =4 mT) and

C ∼ 10 -18 -10 -20 Jm.
The γ-Fe 2 O 3 (maghemite) iron oxide nano-particles which compose the rods were synthesized by the aqueous coprecipitation route also called the Massart procedure, followed by a size sorting procedure [START_REF] Bacri | Magnetic colloidal properties of ionic ferrofluids[END_REF]. Their nanometric size yields their superparamagnetic properties (each particle has a single Weiss magnetization domain) and their size disparity accounts for the divergence between the Langevin law and the magnetization curve of the particle ferrofluid suspension [START_REF] Elmore | The Magnetization of Ferromagnetic Colloids[END_REF]. This result was confirmed by our own magnetization curves (ESI †, Fig. S2) of a ferrofluid suspension of negatively charged (citrated) nanoparticles (volume fraction φ v = 3.9%) obtained by VSM using a home-made apparatus, from which was deduced the log-normal distribution of the particle diameters (10-16 nm range, mean=13 nm) [START_REF] Sandre | Assembly of microscopic highly magnetic droplets: Magnetic alignment versus viscous drag[END_REF]. Following an already published protocol [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF], the rods were then prepared by dialysis of a solution of the nanoparticles mixed with positively charged polymers (poly(diallyldimethylammonium chloride)) while being exposed to a ∼ 250 mT magnetic field. We used a microscopic setup equipped with magnetic tweezers designed to induce a uniform magnetic field on the sample (Fig. 1). We carefully calibrated the field, and checked that the influence of the gradient was negligible [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF]. To prepare the samples, we flowed a 100x diluted rod solution into an observation chamber, so that some of the rods were found in a cantilevered configuration, i.e. with a small part of the rod stuck on an inner coverslip while the longest part remained far from any solid surface (Fig. 1B). After performing the measurements, we checked that the clamped boundary condition was fulfilled by controlling that the anchorage resisted to a large deformation induced by a magnet brought at vicinity of the sample. The rods were observed with a DMIRB inverted Leica microscope equipped with an apo plan 100x NA 1.3 objective and a Photometrics fx-Coolsnap cam-era (pixel size (4.65 µm) 2 ) either by bright field or by reflection microscopy (Fig. 1C). Home made softwares were written to automatically pilot the magnetic tweezers and the camera, track noisy motion of the sample and digitize the rod shape with a resolution of 2-20 nm [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF].

Before performing the complete set of measurements on individual rods, we checked the validity of each of them. To characterize the rod geometry, the measure of L ∼ 30-80 µm was straight forward on the images taken with the microscope, but the measure of r ∼ 300 nm was more challenging because of the optical diffraction limit. We thus developed a theoretical model to compute the rod reflection intensity profile:

I r (x) = I 0 + I 1 r -r exp -(s-x-x 0 )) 2 2σxy 2 2r 1 -( s r ) 2 ds
, where I 0 is the background intensity, I 1 an effective intensity, σ xy is a function of the wavelength and the objective numerical aperture, and x the abscissa along the rod cross section (ESI †, note 1). The value of r was then obtained from the fit of this function to the experimental intensity measured at each rod cross-sections (Fig. 1C). We checked our model by comparing its result to the value of r measured by AFM on the same sample rods (ESI †, note 1 and Fig. S3). We found that the uncertainty of this optical measurement was of ∼30 nm, which is also the typical standard deviation of r along the rod.

The mechanical characterization of the rod was previously fully described [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF] and consisted of acquiring ∼900 images of the rod to analyze its thermal fluctuations, from which were deduced the rod persistence length Lp, its bending modulus C = L p /k B T (k B T is the thermal energy) and its Young modulus Y = C 4πr 4 . The magnetic susceptibility χ of the rod material (defined as χ = M H , where M is the magnetization and H the field inside the material) was deduced from magnetoelastic experiments in which we analyzed the deflection of the rod deformed by a uniform field. During such magnetic deformation, the magnetic torque per unit volume Γ m = || M ∧ B 0 || ( B 0 is the uniform induction field) acting on the rod is counter-balanced by the elastic restoring torque C R , R being the local radius of curvature. As shown in ESI †, note 2, Γ m = ∆χ sin(2α) πr 2 B 2 0 2µ 0 , where µ 0 is the vacuum permeability, α the angle of B 0 with respect to the rod main axis and ∆χ a function of χ and of the local components of the field H in the rod. Thus, the measurement of the rod deflection δ (Fig. 2A) may yield either one of the parameter C or χ when the other one is known. However, a difficulty arises from the fact that χ depends on H. This can be directly observed on the magnetization curve of the nanoparticles before they were embedded in the rods (ESI † Fig. S2) which shows deviation from linearity even at very low field (χ v varies by more than a factor of 2 between 2 and 10 mT). Consistently, previously detailed [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF] viscous drag versus magnetic torque (VD/MT) experiments (in which the magnetically driven rotational kinetics of free rods in solution is analyzed), also showed that the rod material susceptibility χ depends on H and is proportional to χ v , the constant ratio χ/χ v being simply the ratio of the magnetic nanoparticle volume fraction within the rod and in the initial ferrofluid φ/φ v . * Thus, to analyze the magnetoelastic experiments, we modified a theoretical model [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF] to study the magnetoelastic deformation of low susceptibility rods. Although this model is only detailed here (ESI †, note 2), we showed that it matched accurately the shape of the deformed rods [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF]. This model takes into account the dependence of χ with H, a variability which is often neglected [START_REF] Chevry | Sub-piconewton force detection using micron-size wire deflections[END_REF][START_REF] Fresnais | Electrostatic co-assembly of iron oxide nanoparticles and polymers: Towards the generation of highly persistent superparamagnetic nanorods[END_REF][START_REF] Frka-Petesic | Dynamics of paramagnetic nanostructured rods under rotating field[END_REF][START_REF] Yan | Magnetic nanowires generated via the waterborne desalting transition pathway[END_REF].

Here, we also report experiments aimed at controlling that this model yields a constant measure of φ independently of the field incidence and strength (Fig. 2). Fig. 2B shows experiments on a 33.1 µm long rod oriented at various incidence angles θ 0 from 0 to 60 • in a constant applied field. The deflection δ(θ 0 ) (Fig. 2A) varies accordingly to our theoretical model: when θ 0 varies, so does the internal field H (ESI †, Fig S4). and we found that the variation of the measured χ(H) matches the data χ v (H). Consequently the comparison yielded a fairly constant value for φ (within ±2%).

This experiment and the theoretical curves shown on Fig. 2A also suggested that the best values at which these deflection measurements should be performed were θ 0 ∼ 30 -50 • : at these angles, the deflection δ reaches a maximum plateau when B 0 is below the magnetoelastic buckling field [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF]. Next, at a constant incidence θ 0 = 35 ± 0.1 • , we studied the effect of the field intensity on the rod deflection (Fig. 2C). We monitored 130 deformations of a 46.4 µm long rod submitted to an incrementally increased intensity of the external induction field from 0 to 7 mT. The fields was successively turned on during 500 ms (a sufficient time to reach the equilibrium configuration) and off for the same amount of time to monitor the rod returning to its undeformed state. Excepted the thermal fluctuations, the deformations were reproducible. They did not increase incrementally, an effect which could have occurred in the case of remanent magnetization, thus confirming the superparamagnetism of the rod. During this experiment, the rod deflected by a maximum of 1.6 µm (corresponding to a deviation of 2 • ), and the induced longitudinal field µ 0 H 0 ranged from 0 to 4 mT. The analysis of the deflection according to the theoretical model showed that χ(H 0 ) varied by more than a factor of two, but remained proportional to χ v (H 0 ). The values found for χ below 2 mT (for which no VSM data are available) also appeared consistent with the expected shape of the magnetic susceptibility curve. The deduced measure of φ was found to vary by less than 10% and yielded a mean value of < φ >= 15% ± 0.8. † In contrast, if χ is assumed constant, φ varies by a factor of two when deduced from these measurements.

We then proceeded to the full characterization of ten individual rods. All came from the same synthesis batch. Once L, r and C were determined as described above, the measurement of χ consisted in submitting the rod to ∼ 1 s long pulses of magnetic field applied at a 35 • incidence.

The field intensity was varied from ∼2 to ∼8 mT to check the constancy of φ for a given rod. At each field intensity, the measurements were repeated 5 times to averaged out the errors mostly due to the thermal fluctuations (∼ 1/10 of the magnetic deflections) (ESI †, Fig. S5A and Movie 1). The deflections always remained small (δ ∼ 1 µm, θ L 1 • ), which also ensured that the system was probed in the linear elastic regime: for larger deformations, we observed a plastic behavior (ESI †, Fig. S5b). To monitor small deformations, we took advantage of our super-resolutive image analysis algorithm [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF] to measure the rod centerline from the microscopy images with a precision of a few tens of nanometers (Movie 1).

Table 1 shows the results for 10 rods selected over nearly 100 studied samples. Reasons for rejections were insufficiently straight or badly anchored rods, or uncontrolled vibrations of the sample (noise) which prevented the detection of the thermal fluctuations. The selected rods were homogeneous in length because we chose the longest to allow a fine detection of the thermal fluctuations.

This selection might also explain the small dispersion in thickness as r varies only from 0.3 to 0.4 µm. By contrast the flexural rigidity was extremely variable, ranging from 21.5 to 536 ×10 -21 Jm, corresponding to a persistence length from 5 to 130 m and a Young modulus of 2 to 60 MPa. The measured susceptibility (for B 0 =4 mT) was found to vary by a factor of 3 (<χ>=3.8±1.7) as the volume fraction (<φ>=14.7±6.7) and the mean particle number per rod cross-section: 144±59. ‡

In these measurements, the final precision of only 25-50% on χ and φ mostly arises from the uncertainty on C, due to the difficulty to detect the small fluctuations of the rods (∼ 10 -100 nm [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF]). However, this precision was sufficient to display the large disparity of the rod characteristics, unreported by the previous study on the stiffness of such rods [START_REF] Chevry | Sub-piconewton force detection using micron-size wire deflections[END_REF] which contained a single measure.

This surprisingly high dispersion is corroborated by several observations: (i) several previous papers concerning this type of rods also reported a large distribution of their properties such as their lognormal distribution of lengths [START_REF] Frka-Petesic | Dynamics of paramagnetic nanostructured rods under rotating field[END_REF] or as their susceptibility [START_REF] Fresnais | Electrostatic co-assembly of iron oxide nanoparticles and polymers: Towards the generation of highly persistent superparamagnetic nanorods[END_REF] which was plotted on a logarithmic † The standard deviation of 0.8% only accounts for the variability of the measures and not for the uncertainty of the other previously independently determined parameters (see Table 1).

‡ The uncertainty for these numbers is the standard deviation of the measures on the 10 rods.

scale; (ii) the VD/MT experiments performed on ten other rods gave a similar range of values for χ = 4.6±1.2 (for B 0 =4 mT), which consistently yielded a similar result for < φ >= 17.8 ± 4.5% [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF];

(iii) the alternative analysis of the magnetoelastic experiments (in which χ is set from the VD/MT experiments and C is looked for) confirmed the values of C determined by the thermal fluctuation method [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF], and in particular (iv), the fact that despite their large variation and the complete independence of their measurements, the combination of values found for C and χ yielded consistent results for φ = 14.7 ± 6.7% which falls into a narrow acceptable range of values: from electron micrographs [START_REF] Yan | Magnetic nanowires generated via the waterborne desalting transition pathway[END_REF], it has to be above 10% but also below 50%, the fraction of randomly close-packed spheres. By comparison, a small angle neutron scattering study on isotropic aggregates made of different kind of polymers and smaller nanoparticles yielded a mean value for φ = 38% [START_REF] Fresnais | Universal scattering behavior of coassembled nanoparticle-polymer clusters[END_REF]. In the present study, the large dispersion of values highlights the advantage of fully characterizing each rod individually: if we had only known a mean value for C from a distribution of rods, the magnetoelastic experiments would have yielded results for χ ranging from 0.6 to 13 and inconsistent values of φ from

to 60%!

This study shows that the number of nanoparticles per rod cross-section is relatively small (Fig. 3), thus indicating that rods with a higher concentration of particles could be synthesized to obtain a greater magnetic susceptibility. This information on the microscopic structure also suggests that the rod stiffness is mostly dictated by the polymer matrix. A putative explanation of the rod stiffness variability is that fluctuations in numbers of particles from one rod to another yield even larger fluctuations of the concentration in polymer chains c, which adsorb onto the particles. This must induce variable elastic properties according to the scaling law of a polymer mesh Y ∝ c 2.2 [START_REF] Mackintosh | Elasticity of Semiflexible Biopolymer Networks[END_REF]. For future uses, a sorting or a more selective method to produce the rods more homogeneously needs to be developed. Meanwhile, individual rod characterization as described in the paper may be required for many applications. We also believe that insights could be gained on numerous other types of magnetic microrods from the techniques we have introduced here. 1) randomly distributed in a cross-section of a rod of radius r=300 nm. Measured parameters of 10 microrods from the same batch. The standard deviation for L indicates the uncertainty of the measurement (estimated as a pixel size at each rod end). For r, the value is the mean±SD of the variation of the measure along the rod (the precision of a single measure being 30 nm). For C, χ and φ the value is provided with the experimental uncertainty of the measurements. From these values, we estimate the total number of particles in each rod and per cross-section. The last two lines are the mean and SD of each parameter for the 10 analyzed rods. field is increased, Red: field is decreased (refer to the left-axis for the scale). Blue data: magnetic susceptibility χ v = M H deduced from the magnetization values (scale on the right axis). The full cycle, shown in the grey inset, confirms the known surperparamagnetism of such a suspension [START_REF] Elmore | The Magnetization of Ferromagnetic Colloids[END_REF].

The slight shift between the increasing and decreasing curves ascribes to a known thermal drift of the Hall probe gaussmeter used to measure the magnetic induction.

ESI, note 1: Optical and Atomic Force Microscopy measures of the rod radius Much attention is necessary for the precise measurement of the radius r, on which depends the further determination of other parameters such as the Young modulus Y (Y = 4 Π r -4 C for cylindrical rods [START_REF] Landau | Theory of Elasticity[END_REF]), the magnetic susceptibility χ ∝ C.r -2 (ESI †, note 2), and hence the particle volume fraction φ ∝ χ. We therefore searched for a precise direct determination of the radius of individual rods. Unfortunately, the typical diameter size (200-800 nm) was in the order of the Abbe's resolution limit ρ = 0.61λ N A ∼250 nm (N A is the numerical aperture of the objective and λ the wave length). We thus developed a method to measure the rod radius inspired by the measurement of the length of sub-micron large bacteria by fluorescent microscopy [START_REF] Zhang | Automated super-resolution detection of fluorescent rods in 2D. Biomedical Imaging: Nano to Macro[END_REF] and driven by the assumption that in reflection microscopy, individual nanoparticles of the rod behave as individual point sources, like fluorescent particles in biological samples. We neglected the possible interference between them as ascribed from weak coherence of the light source. We also adopted the simple viewpoint where each diffuser reemits light with the same intensity. With the further hypothesis that at a given crosssection the nanoparticle concentration is homogeneous, the back scattered light intensity is simply dictated by the geometry and should be proportional to 2r 1 1 -( x r ) 2 , where x denotes for the abscissa along the orthogonal axis to the rod, r the radius in the transverse direction of the optical axis and r 1 the radius along the optical axis (r 1 = r for an elliptic cross-section). In this view, the reflected light is therefore the point spread function of the microscope convoluted by the emitted light and should be:

I r (x) = I 0 (r) r -r J 1 [a(s -x) 2 ] 2r 1 1 -( s r ) 2
ds where J 1 is the Bessel function of the first kind and a = 2πN A λ . A common simplification consists in replacing J 1 by a Gaussian law with standard deviation σ xy = .21 λ N A [START_REF] Thomann | Automatic fluorescent tag detection in 3D with super-resolution: Application to the analysis of chromosome movement[END_REF]. To test this model, we thus compared the expected intensity

I r (x) = I 0 + I 1 r -r exp -(s-x-x 0 )) 2 2σxy 2 2r 1 -( s r ) 2
ds with many measures of grey intensity profiles from reflection images of the various rods (Fig. 1C andESI †, S2). I 0 , I 1 , x 0 and r where adjustable parameters. In all cases, the theoretical curve fitted well the measured curves. We could thus automatize the procedure to probe the rod along its entire length (every two pixels=91 nm, ESI †, Fig. S3).

To check the accuracy of these optical measures, we compared them with measures obtained by an AFM scan performed on the same optically analyzed rod. The AFM used was a MFP-3D-BIO from Azylum Research, mounted on an inverted Olympus optical microscope. The AFM cantilevers were OMCL AC160TS R3 probes from Olympus with nominal stiffness of 26 N/m, nominal tip height of 14 µm, and nominal tip radius of 7 nm. The resonance frequency was 271 kHz and the scan rate was 1 Hz. The imaging was performed in the soft tapping regime. The images were acquired with lateral sizes ranging from 5 to 20 µm. To measure the radius by analysis of the optical images, we used the same equipment as for the rest of the experiments (camera, objective and beam splitter).

To avoid the difficulties of AFM on water immersed samples, we partially allowed the sample to dry before the scan. It is clear that the procedure somewhat damaged the rods (as shown by images taken before, during and after rehydratation) but we found that a greater variability of the rod cross-section was actually more suitable than homogeneous rods to compare both radius measurement methods.

After AFM analysis, the samples were gently rehydrated with a pipette, without moving the sample from the microscope stage. Images taken before and after the procedure showed that no observable modification occurred during the procedure. The AFM scans were done in tapping mode in the soft tapping regime from which we derived the height and radius of the rod profiles from the AFM scans (ESI †, Fig. S3) taking into account the geometry of the tips. The height was found to be smaller than the width diameter, indicating that their lower surface may have flattened on the coverslip while drying. In all cases, we found that the variation of the height and the width were strongly correlated. The model for the optical analysis also holds for elliptic cross-section (or half-cut elliptic section) through the adjustment of the fit variable I 1 .

We performed this analysis respectively on 15 rods made from the 13 nm (and also 8 nm) average diameter nanoparticle fractions. The graphs of ESI †, Fig. S3 show the strong similitude between the profiles from the two methods for three rods. However, we find that rather than being stochastic, the sign of the difference between the two curves is persistent over micron-long distances. Our interpretation is that, given the low number of particles (see Table 1 and Fig. 3), the AFM tip scans the polymer hairy shell whereas the optical method accounts for the presence of the inorganic nanoparticles. The AFM precision could thus be affected by the presence of sticky blobs of polymer or nanoparticles onto the tip, or the presence of a heterogeneous water layer around the rod. Despite these differences, we found that the mean difference (averaged over the rod length) between AFM and optical measures were respectively 1.5, 1.9 and 12 nm and that the standard deviation was ranging from 20 to 30 nm. This latter value of 30 nm also appeared to be the typical standard deviation of the optical radius measured along a water-immersed rod and was retained to be the uncertainty of our optical measurement. Curvilinear abscissa (microns) height or radius (nm)

A 1 A 2 A 3 B 2 B 3 B 1
Fig. S3 Comparison of rod radius measurement by AFM and optical reflection image analysis for three different rods. A 1 -A 3 grey intensity map of the height measured by AFM. For each panel, the inset shows the reflection optical image of the same rod after the sample has been rehydrated (bars=2 µm). B 1 -B 3 each panel corresponds to the A images and shows the AFM-measured height (black line), the AFM-measured radius deduced from the width (blue line), the radius derived from the optical analysis (red line) and the absolute difference between the two blue and red measures (green line). For each curve the sampling was every 2 camera pixels=93 nm.

volume Γ m = M ∧ B 0 . The contributions from the and ⊥ yield its algebraic amplitude:

Γ m = Γ m + Γ m⊥ = ∆χ sin(2α) πr 2 B 2 0 2µ 0 (1)
where

∆χ = χ(H ) - χ(H ⊥ ) 1 + χ(H ⊥ )/2 (2)
depends on both components of the internal magnetic field.

Secondly, we now assume that the cylinder is a Hookean deformable solid of Young modulus Y and bending modulus C = π 4 r 4 Y . According to classical elasticity equations [START_REF] Landau | Theory of Elasticity[END_REF], the restoring torque per unit length of a bent cylinder is

Γ b = C d 2 θ
dl 2 . The local torque balance equation (which may be obtained through minimization of the energy functional even when χ depends on H [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF]) is therefore:

C d 2 θ(l) dl 2 + πr 2 ∆χB 2 0 2µ 0 sin [2θ(l)] = 0 (3)
where l designates the curvilinear abscissa along the rod and θ the angle with respect to the orthogonal direction to the field (see Fig. 2A). However, this equation relies on the supporting hypothesis of equation ( 1) which holds for a single straight rod, not for a deformed cylinder that changes direction (i.e. bends) relatively to the external field. It remains approximatively valid if each section of the rod magnetically responds to the external field independently of what happens in the rest of the rod. We thus designate it as the "independent model". We recently introduced an alternative so-called "axial model" [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF], in which we hypothesized that for large χ the magnetization is mostly longitudinal and also approximatively constant along the rod main axis. In the same paper, using the dipolar approximation, we discussed of the validity of each model and showed that the "independent model" is actually true when χ 2 because with this low susceptibility, the main contribution to the internal field H in any direction is the external field rather than the magnetic self-induction of the rod. Thanks to the demagnetizing field, this is also true in the transverse direction whatever the value of χ > 0. But in the axial direction, quite the contrary happens when χ 2: each cross-section of the rod is influenced by both the close and distant magnetized parts of the rod. As a conclusion, our experiments demonstrated that the "axial model" describes more accurately the shape of nickel wires with χ > 100 and also of microrods described here, although χ ∼ 2 [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF]. It also better predicted the

The bent rod can be approximated by several contiguous ellipsoids.

threshold of the field intensity at which the rod buckles. The constant axial part of the magnetization is determined by the part of the rod which is the most influenced by the external field,i.e. the section most aligned with the external field. In the cantilevered configuration studied here, this is thus the tip of the rod. Using H = H 0 (the demagnetizing factor is assumed to be null along the rod main axis as in the infinite cylinder), this writes M = χ(H )H = χ(H ) sin [θ(L)] H 0 . We also neglected Γ m⊥ in front of Γ m because mathematically ∀χ, χ ⊥ < 2 when n ⊥ = 1/2. With these conditions the "independent model" described by equation ( 3) is replaced by:

C d 2 θ(l) dl 2 + πr 2 M B 0 cos [θ(l)] = 0 (4)
This "axial model" also presents a very convenient advantage: with the constancy of M = M , follows the constancy of H and χ (see below). But for an accurate measure of χ we improve for this paper the "axial model" to take into account Γ ⊥ which may account for∼ 40% of Γ m when χ is low. If neglected, the experiments described in the main text done for variable field strength and directions (Fig. 2) does not yield constant φ. Since H ⊥ always follows the "independence rule", the expression of Γ m⊥ in equation ( 1) is valid and the torque balance of the complemented axial model becomes:

C d 2 θ(l) dl 2 + χ(H ) sin(θ L ) - χ(H ⊥ ) 1 + χ(H ⊥ )/2 sin [θ(l)] cos [θ(l)] πr 2 B 2 0 µ 0 = 0 (5)
Without the assumption that χ(H) is constant, neither equation (3), nor (5) may be integrated.

To circumvent these difficulties we performed very small deformation measurements (deflection δ ∼ L/50 ∼ 1 µm) in order to keep almost constant the orientation of the rod with respect to the field.

In this condition, we approximate in equation ( 5): sin [θ(l)] sin [θ(L)] * * . This also allowed to consider H ⊥ as constant in the rod, and thus also χ(H ⊥ ) (Fig. S4). With these approximations, equation ( 5) may be integrated and yields the shape of the rod:

x(l) = 2λ 1 - sin(θ 0 ) sin(θ L ) ) -1 - sin [θ(l)] sin(θ L ) (6) * * sin [θ(l)] sin [θ(L)] + cos(θ L )(θ(l) -θ L ). With θ L -θ 0 3 • θ 0 = 55 • (see further
), the approximation is better than 4%. Alternatively, replacing sin [θ(L)] by sin [θ(l)] yields the equation of the independent model (Eq. ( 3)). Indeed, with a small deformation, the infinite straight cylinder model holds to compute the internal magnetization. Thus, analytical integration of equation ( 3) is feasible and yields somewhat different expressions which are: x(l) = κ arcsin( sin θ(l) sin θ L ) -arcsin( sin θ0 sin θ L ) , y(l) = κ arccosh( cos θ0 cos θ L ) -arccosh( cos θ(l) cos θ L ) and L = κ . Although this model does not match the rod shape as well as the "axial model" [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF], they are very similar for small deflections. In this case, both models yield similar values for the volume fraction φ. . From equations ( 6) and [START_REF] Fresnais | Electrostatic co-assembly of iron oxide nanoparticles and polymers: Towards the generation of highly persistent superparamagnetic nanorods[END_REF], we deduce the relation between L and θ L :

L = λ θ L θ 0 dθ 1 -sin(θ)
sin(θ L ) [START_REF] Fresnais | Universal scattering behavior of coassembled nanoparticle-polymer clusters[END_REF] as well as the deflection of the tip: δ = -sin(θ 0 )x(L) + cos(θ 0 )y(L)

From the experimental measurements of L and δ, we numerically solved equations ( 8) and ( 9) to find θ L and λ from which follows ∆χ. Equation ( 8) always admits a solution for θ L θ 0 > 0. But in the case θ 0 = 0, a solution exists only if L 2λ. This is the mathematical translation of the magnetoelastic buckling instability which occurs at the critical field B c = 1 rL 2µ 0 C π∆χ

1 rL 2µ 0 C πχ(H 0 L )
in the axial model [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF].

From ∆χ and the geometrical parameters, one can compute χ(H ) and χ(H ⊥ ), each of them being proportional to the magnetic susceptibility of the nanoparticles which compose the rods, i.e. χ(H (⊥) ) = φ φ v χ v (H (⊥) ). Using the expression of ∆χ and the data χ v (H), φ could thus be obtained by solving numerically the set of equations: [START_REF] Gerbal | Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods[END_REF] based on these latter equations, Fig. S4 shows the variation of the magnetic susceptibilities in a infinitely-long cylinder as a function of the field components, which depends themselves on the induction field angle with respect to the cylinder.

H = H 0 H ⊥ = H 0⊥ 1 + φ φ v χ v (H ⊥ )/2 (10)
φ = φ v ∆χ χ v (H 0 ) -χ v (H ⊥ ) H ⊥ H 0⊥ (

Fig. 1 Fig. 2

 12 FIGURES

Fig. 3

 3 Fig.3Simulation of 144 nanoparticles (ø=13 nm) (mean value found for the 10 analyzed rods detailed Table1) randomly distributed in a cross-section of a rod of radius r=300 nm.

Fig.

  Fig. S2 Magnetization curves of the nanoparticles used to prepare the rods. The magnetization curves M (H) were obtained by vibrating sample magnetometry on a 3.9% volume fraction ferrofluid suspension of non-aggregated citrated particles. At this low concentration, the magnetic interaction between the particles is negligible and thus, no demagnetizing field takes place. Data have been extrapolated as if the solution volume fraction were 100%. The main figure is a plot of the magnetization cycle for the low values of the field at which the flexural experiments were performed. Green:

√

  sin 2 θ L -sin 2 θ with κ = µ0C πr 2 ∆χB 2 0

  Fig. S5. Lower left bar=10 µm. The direction of the field is indicated by the upper right arrow (θ 0 = 35 • ) and its intensity written on the movie. The dark line drawn on the middle of the rod image is its centerline automatically digitized by our software.

Table 1 -

 1 Results for 10 rods

	N.	L (µm)	r (µm)	C (x10 -21 Jm)	Y(MPa)	χ@4mT (x1)	φ (%)	Part./rod (x1000) Part./Cr.-Sec.(x1)
	1	34.36±0.05 0.316±0.027	21.5±4	2.7±1.5	2.4±0.9	9.2±3.4	108±59	82±45
	2	26.83±0.05 0.302±0.001	38.1±16	5.8±3.1	4.1±1.7	15.8±6.6	132±66	128±64
	3	44.92±0.05 0.366±0.024	398±74	28.3±12.6	3.0±1.0	11.4±4.0	234±115	135±67
	4	35.94±0.05 0.372±0.027	76.7±14.3	5.1±2.4	2.00±0.7	7.7±2.7	130±67	94±49
	5	54.17±0.05 0.410±0.080	42.8±8.0	1.9±1.9	2.5±1.4	9.75±5.2	303±281	145±135
	6	42.41±0.05 0.320±0.013	448±84	54±19	7.4±2.4	28.5±9.1	422±174	259±107
	7	55.70±0.05 0.330±0.040	536±100	58±39	5.3±2.4	20.3±9.4	420±299	196±140
	8	25.52±0.05 0.314±0.024	154.0±29	20±10	3.4±1.3	13.4±5.0	115±62	117±64
	9	42.31±0.05 0.301±0.014	30.3±5.7	4.7±1.8	2.6±0.9	10.0±3.6	131±60	81±37
	10	36.41±0.05 0.331±0.037	158±30	16.8±11	5.5±2.3	21.0±8.7	285±185	204±132
	mean	39.86	0.336	190.4	19.8	3.8	14.7	228	144
	SD	10.15	0.035	196	21	1.7	6.7	124	59

ACKNOWLEDGMENTS

The authors thank M. Devaud for fruitful discussions and his patient proof-reading of the manuscript.

ESI, note 2: Theory of the magnetoelastic experiments

In this section, we detail the theoretical model used to deduce the magnetic susceptibly χ from the magnetoelastic bending of a cantilevered rod submitted to a uniform external field B 0 = µ 0 H 0 (µ 0 is the vacuum permeability). As described in the main text, we assume that the rod geometrical parameters -the radius r, the length L, and the deflection δ of the rod tip -as well as its bending modulus C are known from independent measurements. As shown on (Fig. 2A), we note H the field inside the rod and M its magnetization. The symbols and ⊥ applied to any vector ( M , B 0 , H...) denote respectively their longitudinal and orthogonal projections on the rod. In absence of remanent field (the rods are superparamagnetic), the bulk rod material is characterized by a magnetic susceptibility which is defined as χ(H) = M H . We assume the material to be homogeneous and isotropic so that this relationship is the same in any direction and we may write: M = χ(H) H. § .

In the absence of magnetic dipolar interaction between the nanoparticles [START_REF] Gazeau | Anisotropy of the structure factor of magnetic fluids under a field probed by small-angle neutron scattering[END_REF], we also have χ(H) = φ φ v χ v (H) where φ and φ v are respectively the volume fractions of the particles in the rod and in the ferrofluid of the same nanoparticles on which the measurement of χ v was performed by vibrating sample magnetometry (VSM).

We first consider a non-deformable paramagnetic rod of size L and radius r, submitted to B 0 oriented at an angle α = π 2 -θ with respect to the rod main axis (Fig. 2A). When L r, the cylindrical rod may be approximated by a degenerate ellipsoid for which the Maxwell equations have an analytical solution ¶ . In our case where the material is uniform and isotropic, the computation yields the demagnetizing fields and the magnetization in both directions:

where

is the effective susceptibility for each canonic directions and n (⊥) the respective demagnetization factors. For L/r ∼ 100, as in the experiments, the infinite cylinder is an excellent approximation which gives n = 0 (there is no demagnetizing field in the axial direction and thus H = H 0 ) and n ⊥ = 1/2 [START_REF] Gerbal | A refined theory of magnetoelastic buckling matches experiments with ferromagnetic and superparamagnetic rods[END_REF], with a relative error of ∼ 7 × 10 -3 .

In these conditions, each section of the cylinder is submitted to a magnetic torque per unit § see Landau and Lifchitz, Electrodynamics of continuous media, (Second edition, Pergamon press, Oxford 1984), chapter IV p106 Eq. (29.10) and (29.11)

¶ see Demagnetizing factors of the General ellipsoid, (1945) Osborn, Phys. Rev. 11 and 12, p351. The calculation holds when the ellipsoid is homogeneous, isotropic (our case) and magnetically linear. Our weaker condition of a non-constant χ(H) is nevertheless sufficient: in the ellipsoid, a solution exists for any susceptibility χ in which H and M are constant throughout the ellipsoid. The solution verifies M = χ H everywhere. Thus, there is a valid solution for our specific χ(H) = χ = M/H. By unicity of the solution of the Maxwell equations, it is the only one.


Eric Lajeunesse

Olivier Devauchelle

F Lachaussée

P Claudin

Bedload Transport in Laboratory Rivers: the Erosion-Deposition Model

Keywords: erosion and deposition in bedload transport, laboratory experiments, alluvial morphology, tracer particles

This chapter is a review of the erosion-deposition model of bedload transport, first proposed by Charru et al. (2004). This model, established on the basis of laboratory experiments, proposes a simplified theoretical framework describing the exchange of particles between the sediment bed and the layer of entrained grains. The latter is treated as a uniform reservoir of independent particles moving at velocity V . The exchange of particles between the sediment bed and the bedload layer sets the surface concentration of moving particles, n, and determines the sediment transport rate, q s = nV .

For a steady flow over a wavy topography, the sediment flux adjusts to a change of shear-stress over a characteristic deposition length. This relaxation plays a crucial role in selecting the wavelength of bedforms such as ripples, rhomboid patterns and bars (Andreotti et al., 2012; Charru, 2006; Charru et al., 2013; Charru and Hinch, 2006; Devauchelle et al., 2010b).

The equations describing bedload transport and the development of bedforms are similar in laminar and turbulent flows. This analogy explains why most alluvial morphologies formed in nature also form at a smaller scale when a laminar flow interacts with a sediment bed (Paola et al., 2009).

A preliminary investigation indicates that the erosion-deposition model also accounts for the propagation of a plume of tracers entrained by bedload transport. It suggests that, after a short transient, the plume reaches asymptotically an advection-diffusion regime, in which it advances with a constant velocity while its variance increases linearly with time.

The results discussed in this chapter are based on idealized laboratory experiments (uniform grain size, constant flow and sediment discharges). Rivers, and in particular gravel-bed rivers, often exhibit high grain size heterogeneities and fluctuating flow discharges. Recent progress in extending the erosion-deposition model to the transport of mixed grain sizes should allow us to investigate more realistic configurations (Houssais and Lajeunesse, 2012; Houssais et al., submitted).

I. INTRODUCTION

When the shear stress exerted by a river on its bed exceeds a threshold, superficial sediments are entrained by the flow. For moderate values of the shear stress, they move by rolling, sliding and bouncing while gravity maintains them close to the bed surface (Bagnold, 1973). This so-called "bedload transport" deforms the bed, and the resulting interaction between flow and sediment transport generates a beautiful variety of river shapes and coastal morphologies (Church, 2006;Gomez, 1991;Seminara, 2010). Understanding this process requires a sound theory of bedload transport.

Since the pioneering work of Shields (1936), a lot of field, experimental and theoretical work has been devoted to bedload (e.g. Ancey (2010); Ancey et al. (2008); Ancey and Heyman (2014); Ashida and Michiue (1973); Bagnold (1956); Bridge and Dominic (1984); Charru et al. (2004); Durán et al. (2012); Durán et al. (2014); Einstein (1950); Engelund and Fredsoe (1976); Fernandez-Luque and Van Beek (1976); Furbish et al. (2012a,b,c); Gomez (1991); Gomez and Church (1989); Lajeunesse et al. (2010a); Meyer-Peter and Müller (1948); Nino and Garcia (1994); Roseberry et al. (2012)). Most studies focus on the prediction of the sediment transport rate at a specific time and location, q s (x, t). The net exchange of sediment between the bed and the moving layer is then deduced from mass conservation, sometimes referred to as the "Exner equation" in the context of fluvial geomorphology (Seminara, 2010),

∂h ∂t + δv ∂q s ∂x = 0 (1)
a) lajeunes@ipgp.fr FIG. 1. A plume of dyed sediments in an experimental channel (1 m-long and 3 cm-wide). Left : snapshots of the sediment bed; Right : intensity profiles (dots) averaged along the channel width (levels of grey) used as a proxy for the average tracer concentration. The bed of sediments (plastic grains of size 720 ± 120µm and density 1520 ± 50kg.m -3 ) is sheared by a mixture of water and glycerol (viscosity 10 -2 Pa s and density 1155 kg m -3 ). The flow and sediment discharges, respectively 0.82 l min -1 and 0.14 g min -1 , are constant during the experiment. The ratio of the concentration of moving particles to the concentration of static particles is α ∼ 8 10 -2 . The plume of tracers (bed sediments dyed in black) is deposited in the channel once the sediment bed has reached steady-state. Solid line: numerical solution of equations ( 23) and ( 24) for α = 0.08. The parameters V = 0.046m s -1 and ts = 0.081s are adjusted to the experimental data.

where we introduce the streamwise coordinate x, the time t, the volume of a sediment particle, δv, and the bed elevation h(x, t). The sediment flux, q s (x, t), is expressed in number of particles per unit river width and time [L -1 T -1 ].

To investigate bedload transport, we can use a laboratory channel filled with a bed of plastic sediments (Figure 1). A pump located at the upstream end of the channel sets a steady, uniform and laminar flow entraining the bed sediments. A sediment feeder imposes a constant sediment flux at the channel inlet. After a few hours, the sediment bed reaches a steady-state characterized by a constant slope. We then deposit a plume of dyed sediments at the bed surface and monitor its evolution by means of a camera placed above the channel. We observe that the plume spreads as it is transported downstream (Figure 1). This indicates that the dyed particles move with varied velocities. In particular, some tracer particles initially in motion settle on the bed where they remain trapped. These trapped particles are replaced by sediments freshly entrained from the bed to keep the sediment discharge constant.

These observations suggest a continuous exchange of particles between the sediment bed and the layer of moving grains, referred to as the "bedload layer". Equation (1) accounts for the evolution of the bed topography. However, it does not explicitly describe the exchange of particles between the bed and the bedload layer which occurs even when the sediment flux is uniform. Yet, this exchange is at the heart of several processes such as the emplacement of alluvial deposits, the determination of the residence time of sediments in a drainage basin and the propagation of sediment tracers in a river bed (Carretier et al., 2007;Gayer et al., 2008;Willenbring and von Blanckenburg, 2010). Sayre and Hubbell (1965) were the first to experiment sediment tracing in a natural stream. In November 1960, they monitored the propagation of about 18 kg of radioactive sand deposited on the bed of a sandy river in Nebraska (USA). Their data exhibit the same qualitative behaviour as our laboratory experiment : the plume of radioactive sand spreads as it is entrained downstream by the flow (Figure 2).

Since this pioneering work, sediment tracing has become a popular tool to investigate the properties of bedload transport in natural streams (Bradley et al., 2010;Ferguson and Wathen, 1998;Haschenburger and Church, 1998;Reid et al., 1985;Wilcock, 1997). The recent development of radio-frequency identification (RFID) allows the insertion of passive integrated transponders in pebbles (Lamarre et al., 2005;Nichols, 2004). These small electronic tags provide the means to track individual pebble (Figure 3). RFID-tracking is used to study bedload threshold at a fixed location, particle vertical mixing rates, sediment travel lengths, bed and bedforms mobility and particle storage in the sediment bed (Ferguson et al., 2002;Habersack, 2001;Haschenburger and Wilcock, 2003;Hassan et al., 2013;Hodge et al., 2011;Nathan Bradley and Tucker, 2012;Nikora et al., 2002;Phillips and Jerolmack, 2014;Phillips et al., 2013). The interpretation of these observations requires that we understand the exchange of particles between the bed and the bedload layer.

The erosion-deposition model of bedload transport, first proposed by Charru et al. (2004), is based on the description of the exchange of particles between the bed and the bedload layer. This heuristic model, inspired by laboratory experiments, describes how this exchange sets the number of moving particles and determines the sediment transport rate (Charru et al., 2004;Lajeunesse et al., 2010a;Seizilles et al., 2014). It is versatile enough to describe the formation of various bedforms, the transport of mixed grain sizes and the propagation of tracers (Charru, 2006;Charru and Hinch, 2006;Devauchelle et al., 2010a,b;Houssais and Lajeunesse, 2012;Lajeunesse et al., 2013). This chapter is a review of the erosion-deposition model. We begin with a derivation of the model equations based on recent laboratory experiments. We discuss the role of turbulence and the predictions of the model in terms of bedforms development. The chapter ends with a preliminary investigation of the propagation of a plume of tracer sediments.

II. THE EROSION-DEPOSITION MODEL A. Laboratory observations

Particles entrained by the flow move by rolling, sliding and bouncing until they settle back on the bed. In this paper, we restrict ourselves to the case where the shear stress is close to the entrainment threshold. Image analysis shows that, in this regime, the bedload layer is approximately one grain-diameter thick (Figure 4). This confinement implies that bedload transport is controlled by the characteristics of the flow near the sediment bed. In particular, experiments show that bedload transport is primarily controlled by the ratio of the flow-induced shear stress to the weight of a particle, usually referred to as the Shields parameter (Meyer-Peter and Müller, 1948;Shields, 1936):

θ = τ (ρ s -ρ)gd s (2)
where we introduce the shear stress exerted on the bed τ , the grain size d s , the sediment density ρ s , the fluid density ρ and the gravity g. When the Shields parameter exceeds a threshold value, θ t , the shear stress is strong enough to overcome the weight of a particle and to entrain it in the bedload layer. As the Shields stress increases, bedload transport intensifies (Meyer-Peter and Müller, 1948). The ensemble-averaged sediment transport rate, q s [L -1 T -1 ], can be expressed in terms of the particles velocity distribution,

q s = n ∞ 0 F (v) v dv, (3)
where F (v) is the probability distribution function of the particle velocities and n 1 is the surface concentration of the bedload layer, i.e. the number of moving particles per unit bed area (Ballio et al., 2014;Furbish et al., 2012b).

Using a camera placed a few centimetres above our experimental channel, we track the grain trajectories and compute their streamwise velocities (Seizilles et al., 2014). The resulting velocity distribution exhibits a peak centred around v = 0 with an asymmetric tail on its right side (Figure 5a). Since the camera is placed above the channel, the free surface waves cause a spurious motion which affects even resting particles. We calibrate this noise by plotting the distribution of apparent velocities for static particles (red curve on figure 5a). The result suggests that we may distinguish two populations of particles: (i) particles at rest on the bed corresponding to the central peak on Figure 5a; (ii) particles belonging to the bedload layer corresponding to the distribution tail. Discarding the central peak, an exponential function approximates reasonably the velocity distribution in the bedload layer (figure 5a), in accordance with previous experiments involving either laminar or turbulent flows (figure 5b) (Charru et al., 2004;Furbish et al., 2012b;Lajeunesse et al., 2010a;Seizilles et al., 2014). This observation suggests that the streamwise velocity distribution of particles sheared by a flow is the sum of two contributions (Figure 5a),

F (v) = 1 V e -v/V if v > 0 δ(v) otherwise (4)
where δ(v) is the Dirac function. Equation ( 4) distinguishes between two reservoirs : (i) the particles at rest represented by a Dirac distribution and (ii) the bedload layer represented by an exponential distribution.

A complete description of sediment transport requires a theoretical formulation of the velocity distribution of particles F (v) and its dependency on the flow conditions (Furbish et al., 2012b). The erosion-deposition model adopts a simplified approach by treating the bedload layer as a uniform reservoir of independent particles moving at velocity V (Charru et al., 2004). Accordingly, the bedload flux equation (3) becomes

q s = nV, (5)
To calculate the bedload transport rate, we need to relate the average particle velocity, V , and the surface concentration, n, to the flow conditions.

B. Average particle velocity

The expression of the average particle velocity depends on the flow regime. In a viscous flow, we expect the particles to move at about the velocity of the fluid that entrains them. The average particle velocity should therefore scale with the fluid velocity in the bedload layer, i.e. about one grain diameter above the bed surface,

V ∝ d s ∂u ∂z (6)
where we introduce the flow velocity u and the vertical coordinate z. In a viscous flow, the shear rate ∂u/∂z is proportional to the shear stress, τ , so that

V ∝ d s η τ ∝ V s θ (7)
where η is the fluid viscosity and V s is the settling velocity of the particle (Charru et al., 2004;Seizilles et al., 2014). Equation ( 7) accords with experiments (Charru et al., 2004;Lajeunesse et al., 2010a). In a turbulent flow, one expects the particle velocity to depend on the shear velocity, u = (τ /ρ) 1/2 = V s θ 1/2 . Laboratory experiments and numerical simulations show that where a and b are dimensionless coefficients (Figure 6a) (Durán et al., 2012;Lajeunesse et al., 2010a).

V = V s a(θ 1/2 -θ 1/2 t ) + b . (8
In both laminar and turbulent flows, the average particle velocity, V , scales like the particle settling velocity, V s , which depends on the flow regime. In a viscous flow, it is the Stoke velocity V s = (ρ s -ρ)gd 2 s / 18η, whereas it reads V s = ((ρ s -ρ)gd s /ρ)

1/2 in a turbulent one. In a laminar flow, the particle velocity tends towards a finite value near threshold (Figure 6b) (Seizilles et al., 2014). This question is still a matter of debate for turbulent flows (Durán et al., 2014;Lajeunesse et al., 2010b).

C. Surface concentration of moving particles

The bedload layer results from a constant exchange of particles with the bed. The associated balance reads (figure 7): where we define the erosion rate, ṅe , as the number of bed particles set in motion per unit time and surface. Similarly, we introduce the deposition rate ṅd defined as the number of bedload particles settling on the bed per unit time and surface. The mass balance needs to be complemented with expressions for the erosion and deposition rates.

∂n ∂t + V ∂n ∂x = ṅe -ṅd (9)
Experimental observations suggest that a particle entrained by the flow bounces on the bed at a rate proportional to d s /V s (Charru et al., 2007;Lajeunesse et al., 2010a;Seizilles et al., 2014). Each time the particle hits the bed, it can be trapped with a finite probability. Assuming a uniform probability, the deposition rate scales like the concentration of moving particles times the bouncing frequency:

ṅd ∝ n V s d s . ( 10
)
Similarly, laboratory experiments suggest that the erosion rate increases with the excess Shields stress, θ-θ t (Charru et al., 2004;Lajeunesse et al., 2010a;Seizilles et al., 2014). At first order, we also expect it to be proportional to the concentration of particles at the bed surface, σ ≈ 1/d 2 s :

ṅe ∝ f (θ -θ t ) d 2 s 1 t s ( 11
)
where t s = d s /V s and f is a growing function. The form of f can be deduced from steady-state experiments. In this configuration, the mass balance (9) reduces to f (θ -θ t ) ∝ n d 2 s . Measurements of the concentration of bedload particles (see Figure 8) show that f is about linear in both laminar and turbulent flows (Charru et al., 2004;Lajeunesse et al., 2010a;Seizilles et al., 2014). The erosion rate is therefore:

ṅe ∝ 1 d 2 s V s d s (θ -θ t ). (12)
D. Bedload transport rate Equations ( 5), ( 6), ( 9), ( 10) and ( 12) describe bedload transport according to the erosion-deposition model. The resolution of these equations for given flow conditions provides the erosion, deposition and bedload transport rates.

In the case of a steady and spatially uniform flow above a flat topography, the erosion and deposition rates balance each other. In this equilibrium regime, the sediment flux reads

q s,eq ∝ V d s (θ -θ t ) ( 13
)
where the index "eq" denotes equilibrium. The average particle velocity V is given by either ( 7) or ( 8) depending on the nature of the flow. Near the threshold of entrainment, the particle velocity scales as the settling velocity so that the sediment transport rate becomes a linear function of the excess Shields stress,

q s ∝ V s d s (θ -θ t ), ( 14
)
in agreement with measurements (Figure 9). The erosion-deposition model proves reasonable over a broad range of shear stress (Ancey, 2010;Charru et al., 2004;Lajeunesse et al., 2010b). At high shear stress, however, the bedload layer becomes vertically stratified (Durán et al., 2012;Recking et al., 2009). Equation ( 5) then breaks down, and must be replaced with more detailed diphasic models (e.g. Chiodi et al. (2014); Revil-Baudard and Chauchat (2013)).

III. DEPOSITION LENGTH AND BEDFORMS

Rivers exhibit a beautiful variety of bedforms such as ripples, dunes and bars (Charru et al., 2013;Seminara, 2010). These alluvial morphologies result from the feedback between the flow and the bed geometry : (i) the shape of the sediment bed sets the fluid velocity and, therefore, the sediment transport rate; (ii) sediment transport modifies in its turn the shape of the bed.

The mathematical description of this feedback requires an expression for the sediment flux above a wavy bed. When the bed evolves slowly enough, we may consider that the flow is stationary (quasistatic approximation). The shear stress then depends on the bed's shape only, and so does the sediment flux. Assuming that particles have negligible inertia, their velocity V adapts instantaneously to the shear stress. Equations ( 5), ( 6), ( 9), ( 10) and ( 12) thus reduce to

d ∂q s ∂x = q s,eq -q s , (15)
where q s,eq is the equilibrium sediment flux predicted from equation ( 13) and d is the average length travelled by a particle before it settles back on the bed ( d ∼ V t s ∼ V d s /V s ). According to equation ( 15), the sediment flux adjusts to the local shear-stress over the typical distance d . This non-locality of sediment discharge is consistent with other models proposed in the literature (Bell and Sutherland, 1983;Phillips andSutherland, 1990, 1989). In most of these studies however, non-locality is introduced heuristically. In the erosion-deposition model, it results spontaneously from the exchange of particles between the bed and the bedload layer: entrainment and deposition depend on local parameters through equations ( 10) and ( 12), and the sediment flux adapts with a delay described by the relaxation equation ( 15). The length d is analogous to the saturation length introduced in the context of aeolian sand transport where it results from grain inertia and scales like d s ρ s /ρ, regardless of the wind velocity (Andreotti et al., 2002(Andreotti et al., , 2010;;Charru et al., 2013;Claudin and Andreotti, 2006;Hersen, 2005;Narteau et al., 2009;Pähtz et al., 2013;Sauermann et al., 2001).

Bedload transport generates instabilities: a small perturbation on a flat bed can cause the sediment flux to further increase its size. Charru (2006) successfully describes the early stages of ripple growth in laboratory experiments with a linear stability analysis. The relaxation mechanism embedded in equation ( 15) introduces a cutoff length which stabilizes the short wavelengths (Fourriere et al., 2010;Kouakou and Lagrée, 2005;Lagrée, 2003;Lagrée et al., 2009). As a result, the size of the most unstable ripples scales with the deposition length d , in agreement with experiments (figure 10 a). Similarly, the deposition length seems to control the size of the rhomboid pattern and of alternate bars (figure 10b) (Andreotti et al., 2012;Devauchelle et al., 2010a,b).

Laminar free-surface flows are certainly distinct from turbulent ones in many regards. For instance, the shape of the vertical velocity profile differ between the two, and so does the friction law. However, these quantitative differences do not affect the structure of the associated shallow-water equations (Devauchelle et al., 2007). Similarly, the nature of the flow changes the velocity of bedload particles (equations ( 7) and ( 8)), but does not qualitatively alter the mechanism by which the bed exchanges particles with the bedload layer. This analogy explains why many alluvial structures found in nature also form in small laboratory experiments (Coleman and Eling, 2000;Devauchelle et al., 2010a;Guerit et al., 2014;Lajeunesse et al., 2010c;Malverti et al., 2008;Metivier and Meunier, 2003;Paola et al., 2001Paola et al., , 2009;;Seizilles et al., 2013).

IV. SPREADING OF A PLUME OF TRACERS

Even in steady state, the bedload layer constantly exchanges particles with the static bed. To illustrate this mechanism, we now consider a plume of tracer particles entrained by bedload transport. We separate the particles into two groups with the same physical properties, hereafter referred to as "marked" and "unmarked". For the sake of simplicity, we furthermore assume that sediment transport is uniform and steady. In this configuration, equations ( 5), ( 6), ( 9), ( 10) and ( 12) predict that the Shields stress (and consequently the mean particle velocity), the erosion and deposition rates and the surface concentration of moving particles are constant. Equation ( 13) gives the resulting transport rate.

Let us define φ as the proportion of marked grains in the moving layer. Similarly, ψ is the proportion of marked grains on the bed surface. The concentration of marked grains, defined as the ratio of the number of marked particles Coleman and Melville (1996). Solid line : prediction of the erosion-deposition model (Charru, 2006). (b) Opening angle α of a rhomboid pattern formed by a viscous flow shearing a bed of glass beads. Dots: comparison between a linear stability analysis and observations. Inset: example of rhomboid pattern. After Devauchelle et al. (2010b).

to the surface concentration of particles, is

c = nφ + σψ n + σ = α α + 1 φ + 1 α + 1 ψ (16)
where we introduce α = n/σ ∼ nd 2 s , the ratio of the surface concentration of moving particles, n, to the concentration of static particles, σ (this ratio is smaller than one). The total amount of tracers,

M = ∞ -∞ c dx, (17)
is conserved. The mass balance for marked grains in the moving layer reads

∂ ∂t (φ n) + ∂ ∂x (φ n V ) = ṅe ψ -ṅd φ. (18)
In steady-state, erosion compensates deposition:

ṅe = ṅd = n t s , (19)
where t s is the characteristic settling time (t s ∼ d s /V s , see equation 10). Combining equations ( 18) and ( 19), we find the evolution equation for the proportion of marked grains in the moving layer:

∂φ ∂t + V ∂φ ∂x = 1 t s (ψ -φ). (20)
We further assume that the flow can only entrain grains from the bed surface. In other words, the active layer is one-grain-diameter thick. This hypothesis is consistent with a small departure from threshold. Balancing the fluxes of marked grains, we find

σ ∂ψ ∂t = ṅd φ -ṅe ψ (21)
which, when combined with (19), leads to the evolution equation for the proportion of marked grains on the bed surface:

∂ψ ∂t = - α t s (ψ -φ) . (22)
FIG.

11. Numerical computation of the concentration of tracer particles as a function of the dimensionless distance X, for α = 0.01. The initial length of the plume is L = 10. The concentration is normalized so that the total amount of tracers is 1.

Complemented with initial and boundary conditions, equations ( 20) and ( 22) describe the evolution of the plume of marked particles.

In dimensionless form, equations ( 20) and ( 22) read

∂φ ∂ t + ∂φ ∂ x = (ψ -φ) (23) ∂ψ ∂ t = -α (ψ -φ). ( 24
)
where t = t/t s and x = x/(V t s ) are dimensionless variables. A single parameter controls this system of equations: the ratio of surface densities α, which characterizes the average distance between grains in the bedload layer. Since the erosion-deposition model assumes that the behavior of an individual particle is independent from other particles, we can only expect it to be valid when moving particles are sufficiently far away from each other, that is when α is small or, equivalently, when the Shields parameter is near threshold. We did not measure the ratio of surface densities for the experiment of figure 1. However, its value can be estimated from the sediment flux and the particle settling velocity : α ∼ 8.10 -2 . We solved equations ( 23) and ( 24) FIG. 12. Mean position < X > (solid line, left axis) and variance σ 2 X (dashed line, right axis) of a plume of tracers as a function of time for the numerical run of Figure 11. After a short transient, the average position and the variance of the plume increase linearly with time.

for α = 8.10 -2 with finite volumes (Lajeunesse et al., 2013). The agreement with experimental data suggests that the erosion-deposition model reasonably represents the plume dynamics.

We now turn our attention to the behaviour of the sediment plume at long times. To do so, we solve numerically equations ( 23) and ( 24), starting from an arbitrary configuration, and let it evolve for a long period (Figure 11). As expected, the plume spreads as it is transported downstream by the flow. Initially, its shape is skewed towards the direction of propagation. This skewness later disappears, and the plume resembles a normal distribution at long times.

The average position of the plume, defined as

X = 1 M ∞ -∞ c x dx , (25)
increases with time and quickly reaches a constant velocity (figure 12). Similarly, the variance of the plume, defined as

σ 2 X = 1 M ∞ -∞ c(x -X ) 2 dx ( 26
)
increases linearly with time after a short transient, thus mimicking diffusion. We observe this "effective diffusion" behavior regardless of α. This suggests that, at long times, a plume of tracers advances at constant velocity and diffuses linearly, a behavior reminiscent of the linear advection-diffusion equation. This observation is consistent with the diffusive "intermediate range" regime introduced within the conceptual model of Nikora et al. (2002).

In the model presented here, the spread of the plume results only from the continuous exchange of particles between the bedload layer and the immobile bed. In reality, the distribution of particle velocities in bedload transport is continuous (Furbish et al., 2012a,c;Lajeunesse et al., 2010b). Therefore, moving grains in the bedload layer also disperse due to velocity fluctuations. This effect, in addition to the continuous exchange of particles between the bedload layer and the immobile bed, is likely to enhance the dispersion of the plume of tracers.

Although derived in the context of bedload transport, equations ( 23) and ( 24) apply to any steady uniform system in which some passive tracer is exchanged between two layers moving at different velocities. Among such systems are the Taylor dispersion, distillation and chromatography (James et al., 2000;Taylor, 1953).

V. CONCLUSION

Laboratory experiments reveal that, for moderate values of the shear stress, bedload transport involves a continuous exchange of particles between the sediment bed and a thin bedload layer of entrained grains (Charru et al., 2004;Fernandez-Luque and Van Beek, 1976;Lajeunesse et al., 2010b;Nino and Garcia, 1994;Van Rijn, 1984). Taking advantage of this observation and using experimental particle velocity distributions, the erosion-deposition model provides a simple description of bedload transport. The bedload layer is treated as a uniform reservoir of independent particles moving at velocity V (Charru et al., 2004). The exchange of particles between the sediment bed and the bedload layer sets the surface concentration of moving particles, n, and therefore the sediment transport rate, q s = nV .

For a steady flow over a varying topography, the sediment flux adjusts to a change of shear-stress over a characteristic deposition length. This relaxation plays a crucial role in selecting the wavelength of bedforms such as ripples, rhomboids and bars (Charru, 2006;Charru and Hinch, 2006;Devauchelle et al., 2010b). The equations describing bedload transport and the development of bedforms exhibit a strong mathematical analogy in laminar and turbulent flows, thus explaining the "unreasonable effectiveness of stratigraphic and geomorphic experiments" (Paola et al., 2009).

A preliminary investigation indicates that the erosion-deposition model also accounts for the propagation of a plume of tracer sediments entrained by bedload transport. It suggests that, after a short transient, the plume reaches asymptotically an advection-diffusion regime, in which it advances with a constant velocity while its variance increases linearly with time. A more quantitative assessment of the model through systematic comparison between theory and experiments is currently under progress.

The results discussed in this chapter are based on idealized laboratory experiments (uniform grain size, constant flow and sediment discharges). The discharge of most rivers, on the other hand, fluctuates significantly throughout the year, and the sediment they transport are often broadly distributed in size. This is specially true for gravel-bed rivers. Recent progress in extending the erosion-deposition model to the transport of mixed grain sizes should allow us to investigate more realistic configurations (Houssais and Lajeunesse, 2012;Houssais et al., submitted).

FIG. 2 .

 2 FIG.2. Evolution of the concentration of radioactive sand injected by Sayre and Hubbell in the North Loup River (Nebraska, USA) in November 1960. AfterSayre and Hubbell (1965) andBradley et al. (2010).

  FIG. 3. RFID-tagged pebbles in the Mameyes River (Luquillo Critical Zone Observatory, North East Puerto Rico). Distribution of tracers in (a) May 2010, (b) summer 2011, and (c) summer 2012 (Phillips and Jerolmack, 2014; Phillips et al., 2013). Courtesy of Colin Phillips and Douglas Jerolmack.

FIG. 4

 4 FIG. 4. (a) Side-view of bedload transport in the experimental flume of Figure 1. The sediment bed, made of plastic grains (size ds ≈ 720 ± 120µm, density ρs = 1520 ± 50kg.m -3 ) is sheared by a viscous flow of glycerol (viscosity η = 10 -2 Pa s, density ρ = 1155 kg m -3 , Re ≈ 50). (b) Average amplitude of the difference between two successive pictures. The bedload layer appears in white.

  FIG. 5. (a)Streamwise velocity distribution of plastic sediments (density ρs = 1520 ± 50kg.m -3 , size ds = 344±90µm) entrained by a laminar flow (Reynolds number ≈ 500). The velocity distribution (blue curve) is obtained by tracking dyed particles, including those resting on the bed. The spurious noise induced by free surface waves is characterized by the distribution of apparent velocities computed for static particles (red curve). The tail of the velocity distribution is reasonably well fitted by an exponential function (dashed line). AfterSeizilles et al. (2014). (b) Streamwise velocity distribution of sand particles (density ρs = 2500kg.m -3 , size ds = 500 ± 50µm) sheared by a turbulent flow (Reynolds number ≈ 30000). Particles are discarded when slower than a small cutoff velocity. The velocity distribution is fitted by an exponential function (dashed line). AfterRoseberry et al. (2012).

FIG. 6 .

 6 FIG. 6. (a) Average particle velocity, V /Vs, as a function of (θ 1/2 -θ 1/2 t ) for quartz grains in a turbulent flow. A fit of the data by equation (8) (dashed line) leads to a = 4.4 ± 0.2 and b = 0.11 ± 0.03. After Lajeunesse et al. (2010a). (b) Average particle velocity, V /Vs, as a function of the concentration of moving sediments, n d 2 s , measured near the threshold. Plastic grains (density ρs = 1520 ± 50kg.m -3 , size ds = 344 ± 90µm) are entrained by a laminar flow (Reynolds number ≈ 500). After Seizilles et al. (2014).

  FIG. 7. Granular bed sheared by a laminar flow. Moving grains settle on the bed, while static grains get entrained by the flow.

FIG. 8 .

 8 FIG. 8. Surface concentration of moving grains, n d 2 s , vs Shields number, θ. (a) Spherical acrylic beads (density ρs = 1180kg m -3 , size ds = 580 ± 100µm) entrained by a viscous flow of silicon oil (Reynolds number ≈ 40). After Charru et al. (2004). (b) Quartz grains (size ds = 2.2mm) entrained by a turbulent flow. After Lajeunesse et al. (2010a).

FIG. 9 .

 9 FIG. 9. Sediment transport rate vs Shields number close to the entrainment threshold. (a) plastic sediments (density ρs = 1520 ± 50kg.m -3 , size ds = 344 ± 90µm) entrained by a laminar flow (Re ≈ 500). After Seizilles et al. (2014). (b) Quartz grains (size ds = 2.2mm) entrained by a turbulent flow. After Lajeunesse et al. (2010a).

  FIG. 10. (a) Ripples wavelength as a function of Shields stress for sand grains in water (ds = 200µm). Dots : experimental measurements byColeman and Melville (1996). Solid line : prediction of the erosion-deposition model(Charru, 2006). (b) Opening angle α of a rhomboid pattern formed by a viscous flow shearing a bed of glass beads. Dots: comparison between a linear stability analysis and observations. Inset: example of rhomboid pattern. AfterDevauchelle et al. (2010b).

ACKNOWLEDGEMENT

  We are grateful to Colin Phillips and Douglas Jerolmack for the data of Figure2and to Francesco Ballio for his insightful review of the manuscript. We also thank F. Mtivier and F. James for fruitful discussions. Part of the experimental work was funded by the SYSTER program of the INSU CNRS.

  of bedload particle velocities d ∼ V ts ∼ V ds/Vs average flight length travelled by a particle before it settles back on the bed n surface concentration of the bedload layer in number of particles per unit bed area ṅe erosion rate in number of particles per unit time and surface ṅd deposition rate in number of particles per unit time and surface qs sediment flux in number of particles per unit river width and time qs,eq sediment flux in the case of steady and uniform transport t time ts ∼ ds/Vs characteristic settling time u shear velocity of the flow V average velocity of bedload particles Vs sediment settling velocity x downstream distance X average position of the plume of tracers α = n/σ ratio of the surface concentrations of the bedload layer to the bed surface η fluid viscosity φ proportion of tracers in the moving layer ψ proportion of tracers on the bed surface ρ fluid density ρs sediment density σ ∼ 1/d 2 s surface concentration of static particles at the bed surface σ 2 X variance of the plume of tracers τ shear stress exerted by the flow on the bed θ Shields number θt threshold Shields number

TABLE I .

 I List of symbols.