Stefano Villa
email: stefano.villa@ds.mpg.de

Domenico Larobina

Antonio Stocco

Christophe Blanc

Massimiliano M Villone

Gaetano D'avino

Maurizio Nobil
email: maurizio.nobili@umontpellier.fr

Dynamics of Ellipsoidal Particles in the Vicinity of an Air-Water Interface

We present the tracking of ellipsoidal micrometric particles close to an air-water interface performed with Dual Wave Reflection Interference Microscopy. Particle's position and angles with respect to the interface are simultaneously measured as a function of time. From the measured Mean Square Displacement five particle mobilities (3 translational and 2 rotational) and two translationalrotational cross-correlations are extracted and compared with our simulations, obtained imposing either slip and no-slip boundary conditions. The comparison reveals an agreement with no-slip boundary conditions prediction for orthogonal translation and out-of-plane rotation, and with slip ones for parallel translations and in-plane rotation. We rationalize these evidences in the framework of surface incompressibility at the interface.

Introduction

The dynamics of spherical particles dispersed in a Newtonian fluid near an interface has been widely studied in the past both experimentally and theoretically. Although for a fluid-fluid interface the particle dynamics is expected to be governed by slip boundary conditions (BC) [1][2][3] , experiments often deviates from expectations. Indeed, while all experimental results relative to the movement parallel to the interface approaches predictions for full slip BC [4][5][6] , for orthogonal dynamics the measured mobilities range from slip to no-slip BC [6][7][8] . Deviation from slip boundary condition has been recently rationalized considering the presence of small concentration of interfacial active agents determining surface incompressibility 8 6 . In this perspective, our group has shown that surface incompressibility at air-water interface peculiarly affects Brownian dynamics of spherical particles, with movement normal to the interface approaching no-slip BC predictions, while the one parallel being in agreement with full-slip BC 6 .

Differently from spherical particles, the dynamics of colloids with complex morphology is still almost unexplored [9][10][11][12][13] . Ear-lier studies have mostly focused on ellipsoidal particles 14 , as this morphology is the simplest after the spherical one. Because of anisotropy, particle hydrodynamic interactions with the confining interface result in a complex tensorial mobility, which demands the knowledge not only of the position but also of the orientation and the aspect ratio of the ellipsoid. Studies of ellipsoids in the presence of a fluid-fluid interface concerned mainly the absorption 15,16 and the 2D confined dynamics at the interface 17 . Very little is known on the dynamics of an ellipsoidal particle near an air-liquid interface. This lack of knowledge contrasts with the essential role played by the mobility of micrometric non-spherical entities close to fluid interfaces. Many systems of practical interest, like the motion of bacteria in biofilm formation 18,19 , the draining dynamics in flotation processes 20 , and the fabrication of Pickering emulsions 21 , demand indeed the knowledge of the near interface particle mobilities.

In this paper, we study the dynamic behavior of micrometric ellipsoids near an air-water interface. Using Dual Wave Reflection Interference Microscopy (DW-RIM) 22 , we are able to measure the dynamics of five (3 translational and 2 rotational) degrees of freedom of the particle, along with the particle aspect ratio and its distance from the interface. In particular, we here address how the surface incompressibility affects the ellipsoid rotational degrees of freedom and the translational-rotational cross correlations. The experimental results are compared with the outcomes of our numerical simulations.

Sample preparation and experimental set-up

Prolate ellipsoids are produced at the desired aspect ratio (A), i.e. the ratio between the major and the minor axes, in the range 5 to 12 by stretching with a custom developed stretching device sulfate latex beads of diameter 2a = 8.7 ± 0.9 µm (MolecularProbes TM ) embedded in a Poly(vinyl alcohol) matrix and heated at 100 • C 23 . For each particle, we measure the time evolution of the three translation degrees of freedom (x, y, and z) together with the zenithal angle (θ ), which the ellipsoid's major axis forms with the air-water interface plane xy, and the azimuthal angle (φ ) of rotation of the major axis projection in the xy plane (see Fig. 1).

Samples are prepared dispersing ellipsoids in a cell made of a hollow glass cylinder enclosed on the top by a microscope slide and exposed to air on the bottom. Water pinning at the lower cylinder edges prevents water falling due to gravity. Ellipsoids number concentration is chosen low enough to allow neglecting inter-particles hydrodynamic couplings (mutual distance at the interface vicinity larger than 100 µm). Sample cell is then located on the sample stage of the DW-RIM set-up which, in brief, consists of a custom inverted microscope working in reflection and mounting a 50X long working distance objective * . The set-up is designed to simultaneously illuminate with two low coherence monochromatic light sources † and to collect the interference signal from the reflections at the particle-water and at the air-water interfaces with an RGB camera. The choice of an RGB camera is motivated by the exigence of simultaneously recording in different channels the light incoming from the two light sources. From the interference pattern information on the motion orthogonal to the interface is accessed, while the superposition of the two wavelengths allows to unequivocally determine the particle-interface distance 6,22 . All precautions are taken to reduce evaporation, vibrations and particles drifts 24 . Because of gravity, ellipsoids sediment towards the air-water interface until they reach an equilibrium gap distance at few hundreds of nanometers where gravity is balanced by DLVO repulsion 22 . At equilibrium, ellipsoids major axis is on average parallel to the interface, but because of thermal agitation θ fluctuates in a range of few degrees (less than 5 • ) around zero. After the equilibrium position is reached, data are acquired for few minutes at 30 fps in order to recover trajectories of 5000-10000 frames length.

Data acquisition and analysis

Dedicated algorithms have been designed specifically for the tracking of the DW-RIM interference pattern of ellipsoidal particles (Fig. 2). The distance z and the zenithal angle θ are collected analyzing, for each frame, the interference (see Fig. 2). Simultaneously, the movement in the plane parallel to the airwater interface along the axes parallel (x) and orthogonal (y) to the ellipsoid main axis and the azimuthal angle φ are accessed by tracking the interference pattern center of mass and its orienta-tion (see Supplementary Information).

With this procedure the obtained resolutions are 100 nm for the ellipsoid x and y positions, few nanometers for z, 0.3 • for φ and about 10% for θ . A more detailed description of the optical system and of experimental technique can be found in Ref. 22 .

The data reported in the present work refer to 11 different ellipsoids with average aspect ratio 8.4 ± 1.9 and mean distance from the air-water interface in the range 787 ± 103 nm. In figure 3, typical time evolution of the measured positions (Fig. 3a) and orientations (Fig. 3b) is shown for one ellipsoid.

From the tracking the Mean Square Displacement (MSD) (Fig. 3c) and the Mean Square Angular Displacement (MSAD) (Fig. 3d) are evaluated:

MSD k (τ) = [k (t + τ) -k (t)] 2 t ,
where k is x, y, z, θ and φ (for notation simplicity in equations we name MSD k both the MSD and the MSAD). From the MSD and MSAD the diffusion coefficient D k along the freely diffusive degrees of freedom x, y and φ are obtained with a linear fit. As it can be seen in Figs. 3a and3b, the motion along z and the rotation along θ are not free but confined in the DLVO potential due to the colloid and the air-water interface interactions 22 . As a consequence, MSD z and MSD θ are both characterize by sub-diffusive behavior with a plateau for large lag times. They are therefore fitted with the analytic solution of the Langevin equation assuming an harmonic potential 25,26 , whose free fitting parameters are the diffusion coefficient and the potential stiffness. Translational and rotational mobilities M k are obtained as M k = D k /k B T , where k B is the Boltzmann constant and T the temperature.

Simulations

Because of the lack of exact models for ellipsoids at close distance to the interface, we compare experimental results with numerical simulations. The latter are designed considering a free-buoyant Fig. 1 Geometry of the system where measured and simulated parameters are reported:reference systems solidal to the laboratory (x ′ , y ′ , z ′ ) and to the ellipsoid (x, y, z), ellipsoid's center of mass coordinates (x 0 , y 0 , z 0 ), zenithal angle θ between ellipsoid major axis and x axis, azimuthal angle φ in the xy plane. The simulation is performed within a cube of lateral size L and with boundary surfaces Γ p (particle surface), Γ w (all the external boundaries of the computational domain except the air-water interface) and Γ s (the air-water interface). BC have been chosen based on previous results on the Brownian diffusion of spherical particles 6 where the surface incompressibility condition results in a no-slip BC for particle diffusion along the z-axis and quasi-slip BC for the parallel one. The same no-slip BC is expected to hold for an ellipsoid diffusing perpendicularly to an incompressible interface (see Supplementary Informations). On the ellipsoid surface, no-slip boundary conditions are imposed. A schematic representation of the investigated system is shown in Fig. 1. A Cartesian reference frame with the origin at the airwater interface and with the z-axis orthogonal to the interface is considered. The particle is positioned with its center of mass on the z-axis. The system of partial differential equations along with the BC are reported in the Supplementary Informations.

We solve the governing equations by applying a single non-zero component of the combined force/torque vector (F, T ) and compute the six components of the translational and rotational velocity vector (u, ω). By varying the imposed non-zero component of the force/torque vector, in six iterations we can compute the full mobility matrix M 27 :

u ω = M • F T (1)
that is symmetric and positive definite and can be partitioned as:

M = M a M b M T b M c (2)
with both the submatrices M a and M c being symmetric. The submatrices M a and M c account for the translational and rotational coupling, whereas M b accounts for the translation-rotation coupling.

Due to the symmetry, for any particle-interface distance z, all the configurations obtained for different azimuthal angles φ (rotations around the z-axis) are equivalent. Consequently, the particle-interface hydrodynamic interactions and, in turn, the mobility matrix are functions of z and θ only, namely, M = M(z, θ ). Because of the invariance for rotations around z-axis, it is sufficient to compute the mobility matrix only for configurations with the major axis of the spheroid lying in the xz-plane; the mobility or friction matrices for orientations outside the xz-plane are simply obtained by rotations around z-axis.

Results and Discussions

We compare the measured mobilities M meas k with simulations obtained using slip and no-slip BC M sim k . To this aim, for each experiment, we run a simulation having as input the measured values of a, b and average z for the particle at hand, with θ kept fixed to zero § . For the five mobilities we define the mobility relative difference as: with their error bars represent the averages and standard deviations on the 11 measured particles. As expected the experimental translational mobilities parallel to the interface (Figs. 4a,4b) are close to the slip BC, while the translational mobility orthogonal to the interface (Fig. 4c) approach the no-slip BC. In Fig. 4d-e, we also report the in-plane and out-of-plane rotational mobility relative difference, again for slip (full circles) and no-slip (open circles) BC. Consistently with the translations, the average value of the in-plane rotational mobility (Fig. 4d) is close to slip BC, while the one of the out-of-plane rotational mobility is near to no-slip BC (Fig. 4e). Actually, in the case of the out-of-plane rotational mobility the two boundary conditions give comparable results within the experimental error. This is not surprising because, for small θ , the rotation is equivalent to the superposition of two orthogonal motions in opposite directions localized on the ellipsoid tips. In such an idealized picture, the interfacial velocity field has the symmetry of a dipolar field having both radial and non-radial components. Since slip BC applies for the non-radial component and no-slip BC for the radial one, we expect similar differences in both cases. Please note that for both BC the measured significant deviations from the simulated values can result from the large experimental noise on θ ¶ .

∆M k /M sim k = (M meas k -M sim k )/M sim k . In
We also evaluated the cross correlations between different degrees of freedom, comparing their trend with simulated cross mobilities. To this aim we calculate the rotational-translational crosscorrelations C v z ,ω and C v x ,ω (at time delay 0) between the angular velocity of the out-of-plane rotations ω and two components of ¶ As error on θ propagates from the error on b (see Supplementary Information) an improved characterization of the ellipsoids semi-axes should be required for improving the measurement of θ . the particle velocity: v z , along the interface normal, and v x , along the ellipsoid main axis.

C v i ,ω (θ ) = ⟨v i (θ ,t) • ω (θ ,t + τ)⟩ t v 2 i (θ ,t) t • ω 2 (θ ,t) t τ=0 (3)
In Eq. ( 3), v i is either v z or v x and ⟨...⟩ t indicates the time average on t. Angular dependency is obtained splitting the trajectories in 5 different subsets depending on the instantaneous value of θ and evaluating Eq. (3) for each interval of θ . In Fig. 5a are reported the cross correlations averaged over all the measured ellipsoids, where the abscissa reports the average angle θ in each subset of data.

In the case of C v z ,ω for negative θ we observe that v z and ω are anti-correlated, while for positive θ they are correlated. To rationalize the observed cross-correlation, note that as soon as θ deviates from zero one extremum of the ellipsoid is farther from the interface than the other. In this condition, if the particle approaches the interface (negative v z ), the ellipsoid extremum closer to the interface will experience a greater drag, independently from the assumed BC. Consequently the ellipsoid is subjected to a torque restoring the condition θ = 0. Therefore, if θ > 0 a v z < 0 induces an ω < 0, while v z > 0 induces an ω > 0, and thus in both cases a positive cross correlation holds. Viceversa, if θ < 0, a v z < 0 induces an ω > 0, while a v z > 0 induces an ω < 0, and thus in both cases a negative cross correlation. It is worth noticing that the observed cross correlation is not a consequence of the ellipsoids-interface potential well, which also tries to restore the θ = 0 condition. Indeed, in the limit of a harmonic potential, a rotation of the ellipsoid along θ would generate only a restoring torque, with no force in the direction of the z axis.

For C(v x , ω), the negative sign observed indicates that the behaviour is anti-correlated for both θ > 0 and θ < 0, as in systems where parallel drag decreases approaching the boundary [START_REF] Lee | [END_REF] .

In order to verify whether the cross-mobilities have the same trend as the cross-correlations, we simulated and reported in Fig. 5b the cross mobilities M b,zy (blue curves) and M b,xy (red curves) versus θ : where M b,zy corresponds to C(v z , ω), while M b,xy cor-responds to C(v x , ω). For each mobility both no-slip (dashed line) and slip (continuous line) BC are considered. The crossmobilities are simulated for an ellipsoid of aspect ratio A = 8.6 at a particle-interface distance z c /2b = 1.167, corresponding to the average aspect ratio and distance of the experiments. Shadowed regions in Fig. 5b account for the inter-sample aspect ratio variance ∆A = ±0.6. || The trend of the simulated mobilities is the same of the experimental cross correlations, for both full-slip and no slip BC.

Conclusions

We report the first systematic study of mobilities of a micrometric ellipsoidal particle in the close vicinity of an air-water interface. Five different ellipsoidal mobilities (3 translational and 2 rotational) are measured and compared with dedicated simulations for slip and no-slip boundary conditions. Moreover, the outof-plane and the in-plane roto-translations cross-correlations are calculated and compared with simulated cross-mobilites. Analogously to the spherical particles 6 , surface incompressibility rules the translational mobilities: parallel ones approach predictions for slip BC, while the orthogonal one is close to no-slip BC. The rotational degrees of freedom reinforces the importance of surface incompressibility on the mobility of the ellipsoid: in fact, the in-plane rotation mobility is governed by the slip BC, while the one of out-of-plane rotation is controlled by the non-slip BC. The roto-traslations cross-correlations are found in a qualitatively agreement with the simulated cross-mobilites.

We believe that our findings will be relevant to predict the motion of micro and nano entities close to liquid interfaces in a variety of fields ranging from biophysics to material processing. In particular, the relevance of the present results can be perceived considering the strong geometrical analogy of the present system with the one of a living bacteria moving close to an air water interface. Indeed, recent papers report how anomalies on the motion of Escherichia coli 12 and Pseudomonas aeruginosa 29 results from surface incompressibility, thus making the classical theoretical assumption of free air-water interface far from reality in such systems. Pointing out the additional effect of surface incompressibility on the rotational dynamics, we hope to help systematize the study of surface incompressibility effects on the dynamics of micrometric organisms in water and stimulating further investigations in the field.

Notes and references

Fig. 2

 2 Fig. 2 Typical interference pattern of an ellipsoidal particle of aspect ratio 6 as it is recorded from the camera (a) and after binarization (b). Data relative to the red cannel. Please note that the high intensity lateral wings are not part of the interference pattern but consequences of multiple reflection and refraction inside the ellipsoid.

Fig. 3

 3 Fig. 3 Data relative to the trajectory of one of the tracked ellipsoids after sedimented to the air-water interface. a. Translational measured degrees of freedom x (red line), y (orange line) and z (blue line). b. Rotational measured degrees of freedom θ (light-blue line) and φ (magenta line). c. Mean Square Displacement along x, y and z (same colors of a). Dashed lines are linear guidelines representing the expected behaviour for a free Brownian diffusion. d. Mean Square Angular Displacement along φ and θ (same colors of b). Dashed lines are linear guidelines representing the expected behaviour for a free Brownian diffusion.

  Fig. 4a-c, we report ∆M k /M sim k for the translational motion, in the case of slip (full circles) and no-slip (open circles) BC. The reported values with respect to the position of the interface. § We checked that the experimental fluctuations in z around its mean value and of θ around 0 • only slightly affect the simulated mobilities.

Fig. 4

 4 Fig. 4 Relative difference of mobilities between data and simulations for full slip (filled circles) and no-slip (empty circles) boundary conditions. The reported data refer to: (a) M a,xx , (b) M a,yy , (c) M a,zz , (d) M c,yy , (e) M c,zz . In the corresponding insets, sketches depict the motion direction or the rotation with respect to the interface (laying on the xy plane).

Fig. 5 a

 5 Fig. 5 a. Cross-correlations vs. theta: blue circles correspond to C vz,ω (θ ), red squares corresponds to C vx,ω (θ ). b. Cross mobilities M b,zy (blue) and M b,xy (red) vs. theta: full line represent simulation with slip BC, while dashed line the simulation with no-slip BC.

| 1-5

| 1-5

H. Brenner, Chem. Eng. Sci., 1961, 16,

2 G. Perkins and R. Jones, Physica A: Statistical Mechanics and its Applications, 1991, 171, 575. 3 A. V. Nguyen and G. M. Evans, Journal of colloid and interface science, 2004, 273, 262-270. 4 G. M. Wang, R. Prabhakar and E. M. Sevick, Phys. Rev. Lett., 2009, 103, 248303.|| In the experimental ranges, simulations are more sensitive to the aspect ratio than to particle-interface distance.


Lucas Prévost

Anke Lindner

Olivia Du Roure

Pitch Controls the Flexibility of Helical Ribbons

Keywords: Helices, spring, microfabrication, elastic strips, force-extension measurements

Helical objects are often implemented in electronic or mechanical micro-systems, requiring a precise understanding of their mechanical properties. While helices formed by cylindrical filaments have been intensely investigated, little is known about the role of the cross-section of the filament at the basis of the helical shape. We study experimentally the force-extension response of micro-helices fabricated from ultra-thin PMMA ribbons. Leveraging newly achieved control on the helix geometry, the influence of the helical pitch is quantified and a significant stiffening of the helical ribbons with increasing pitch is highlighted. Two phenomena are identified: a mechanical transition from a regime dominated by twisting of the ribbon at small pitch to a bending-dominated regime at high pitch and a purely geometrical effect, specific to helical ribbons. Excellent agreement is found with a previously established analytical model of inextensible elastic strips.

Introduction

Helices display very interesting mechanical properties. They are typically formed by slender filaments rolled into a helical shape, with the radius of the helix being much larger compared to the filament thickness. Thanks to their tightly packed geometry, they store a high amount of elastic energy in a small volume. Due to the scale separation between the helix radius and the filament thickness, high stretch ratios of the overall geometry can be achieved with comparatively very little material strain. Helices are common in nature, spanning several orders of magnitude in length, including double-stranded DNA, α-helix in proteins, bacterial flagella [START_REF] Lauga | The hydrodynamics of swimming microorganisms[END_REF], cholesteric crystals [START_REF] Zastavker | Self-assembly of helical ribbons[END_REF], or plant tendrils [START_REF] Gerbode | How the cucumber tendril coils and overwinds[END_REF]. Synthetic helices are widespread at the macroscale: helical springs can absorb shocks, measure forces, or serve as actuators. In recent years, considerable efforts have been dedicated to implement such structures at sub-millimetric scales and integrate them within electronic or mechanical micro-systems [START_REF] Huang | Helices in micro-world: Materials, properties, and applications[END_REF][START_REF] Ren | A review of helical nanostructures: growth theories, synthesis strategies and properties[END_REF][START_REF] Liu | Helical metallic micro-and nanostructures: fabrication and application[END_REF][START_REF] Wan | Helical structures mimicking chiral seedpod opening and tendril coiling[END_REF]. In regards to these applications, it is critical to accurately characterize the mechanical properties of helices.

Many studies have investigated theoretically, numerically, or experimentally the mechanics of helices and in particular the force-extension behavior [START_REF] Love | A treatise on the mathematical theory of elasticity[END_REF][START_REF] Smith | Tension-induced straightening transition of self-assembled helical ribbons[END_REF][START_REF] Pham | Stretching of assembled nanoparticle helical springs[END_REF][START_REF] Bell | Fabrication and characterization of threedimensional ingaas/gaas nanosprings[END_REF][START_REF] Chen | Mechanics of a carbon nanocoil[END_REF][START_REF] Gao | Superelasticity and nanofracture mechanics of zno nanohelices[END_REF][START_REF] Khaykovich | Thicknessradius relationship and spring constants of cholesterol helical ribbons[END_REF][START_REF] Wada | Stretching helical nano-springs at finite temperature[END_REF]. But, despite some natural and many synthetic helices being constituted of ribbon-like filaments [START_REF] Zastavker | Self-assembly of helical ribbons[END_REF][START_REF] Bell | Fabrication and characterization of threedimensional ingaas/gaas nanosprings[END_REF][START_REF] Pham | Highly stretchable nanoparticle helices through geometric asymmetry and surface forces[END_REF][START_REF] Grützmacher | Ultra flexible sige/si/cr nanosprings[END_REF][START_REF] Zhang | Artificial bacterial flagella: Fabrication and magnetic control[END_REF][START_REF] Li | Superelastic metal microsprings as fluidic sensors and actuators[END_REF][START_REF] Zhang | Dynamic switching of helical microgel ribbons[END_REF], studies rarely investigated the specific case of helical ribbons and generally referred to cylindrical filaments, using classical rod models. Recent works have highlighted that the mechanical properties of a ribbon, i.e. a filament of highly anistropic cross-section, are qualitatively and quantitatively different to that of a rod, i.e. a filament of istropic or near-isotropic cross-section (e.g. circle, square), and thus requires specific investigation [START_REF] Audoly | Buckling of naturally curved elastic strips: the ribbon model makes a difference[END_REF][START_REF] Kumar | An investigation of models for elastic ribbons: Simulations & experiments[END_REF]. Pham et al. [START_REF] Pham | Stretching of assembled nanoparticle helical springs[END_REF][START_REF] Pham | Deformation and shape of flexible, microscale helices in viscous flow[END_REF] probed experimentally the mechanical response of tightly coiled micron-sized helical ribbons of various materials and the work of Smith et. al [START_REF] Smith | Tension-induced straightening transition of self-assembled helical ribbons[END_REF] has measured experimentally the spring constant of cholesteric helical ribbons. However, only the analytical and numerical work of Starostin et al. [START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF] has specifically investigated the mechanical properties of helical ribbons and has systematically addressed the influence of the pitch on their mechanical response but experimental validation is still lacking.

We have recently developed a fabrication method for highly flexible micron-sized helical ribbons with full control of the geometrical parameters (helical radius, total length and pitch) [START_REF] Prévost | Shaping nanoscale ribbons into microhelices of controllable radius and pitch[END_REF]. In this work, we leverage this unprecedented control to study experimentally the influence of the pitch on the force-extension behavior of helical ribbons. To do so we clamp the helices between a microcapillary and a cantilever. With the help of a micromanipulator the capillary is moved successively, imposing a well controlled extension of the micro-helix, whereas the deformation of the cantilever is used to detect the applied force with a nano-Newton resolution. We report a significant stiffening of helical ribbons as the pitch is increased (going from closed-coiled to open-coiled). Our experimental results are compared to the analytical model of Starostin et al. [START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF], developed specifically for helical ribbons, finding excellent agreement.

Experimental Methods

General Principles

The helical ribbons are produced using a two-step fabrication method, described in detail in a previous publication [START_REF] Prévost | Shaping nanoscale ribbons into microhelices of controllable radius and pitch[END_REF]. The first step relies on the spontaneous formation of highly flexible helical ribbons driven by surface tension [START_REF] Pham | Highly stretchable nanoparticle helices through geometric asymmetry and surface forces[END_REF]. Nanometer-thick ribbons are prepared on a flat sacrificial layer through an evaporative assembly method [START_REF] Kim | Nanoparticle stripes, grids, and ribbons produced by flow coating[END_REF]. As a result of the fabrication process, the ribbons display a near-triangular cross-section of width w (typically 0.5 -5 µm) and thickness t (typically 5 -50 nm), such that the aspect ratio of the cross-section verifies t/w 1. Upon release of the ribbons into a liquid, they quickly form a tightly coiled helical geometry. The helical ribbon geometry is shown in fig. 1.a, along with the relevant geometrical parameters. The pitch angle is defined such that tan α = p/2πR, and is found to be close to zero after the initial helix formation [START_REF] Pham | Highly stretchable nanoparticle helices through geometric asymmetry and surface forces[END_REF]. The total filament length L is directly controlled during the fabrication process and can be further tuned by cutting the ribbons prior to release in liquid. The helical radius R is determined by a balance between surface tension and elasticity and scales as R ∼ Et 2 /γ, with E being the Young's modulus and γ the surface tension [START_REF] Pham | Highly stretchable nanoparticle helices through geometric asymmetry and surface forces[END_REF]. This affords control over the radius through modification of the ribbon thickness, either during ribbon fabrication or by their subsequent etching [START_REF] Choudhary | Controlled processing of polymer nanoribbons: Enabling microhelix transformations[END_REF].

Control of the pitch angle α is achieved in a second step by leveraging the creep properties of the material to shape the helical ribbon into the desired geometry. Sizable creep effects are not expected in bulk for the materials we use but are enabled by the nanoscale confinement in the ribbon [START_REF] Paeng | Molecular motion in free-standing thin films of poly (methyl methacrylate), poly (4-tert-butylstyrene), poly (α-methylstyrene), and poly (2-vinylpyridine)[END_REF][START_REF] Bansal | Quantitative equivalence between polymer nanocomposites and thin polymer films[END_REF]. A stress is applied to the material by extending the helix for a long period of time, typically several minutes, after which the helix is allowed to relax. During this application of uniform stress, achieved by endloading the helix, an irreversible uniform increase in the pitch angle is observed. This process, executed in-situ, is termed the 'stretching treatment' [START_REF] Prévost | Shaping nanoscale ribbons into microhelices of controllable radius and pitch[END_REF].

We take advantage of this in-situ procedure to investigate the influence of the pitch angle on the force-extension response, keeping all other parameters fixed. In the work presented here, we alternate, for a given helical ribbon, a mechanical characterization and a stretching treatment, thus obtaining the evolution of the helix mechanical properties as the pitch angle is increased. By working with a single ribbon throughout an experiment, the ribbon thickness t and width w, which are difficult to accurately measure or control due to the small size, will remain unchanged, as well as the total filament length L. The radius R may vary slightly as a by-product of stretching but as the influence of the radius on the pulling force is well established (F ∝ 1/R 2 ) [START_REF] Smith | Tension-induced straightening transition of self-assembled helical ribbons[END_REF][START_REF] Pham | Highly stretchable nanoparticle helices through geometric asymmetry and surface forces[END_REF][START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF], these variations are easily corrected.

Experimental Set-up

The ribbons are prepared on a water-soluble sacrificial layer, following the method established by Lee et al. [START_REF] Lee | Macroscopic nanoparticle ribbons and fabrics[END_REF], then released into a pool of water (see fig. prepared on a sacrificial layer of poly(acrylic acid) (PAA, M w = 1800). An open glass capillary tube is connected to a syringe pump and fixed to a micromanipulator to allow capture and release of the helices by pumping or expelling liquid. A carbon fiber cantilever is similarly fixed to a second micromanipulator to afford force measurements. The experimental set-up is mounted on an inverted optical microscope connected to a numerical camera.

Prior to experiments, the bending modulus of the carbon fiber cantilever is calibrated at B cant = (1.55 ± 0.02) × 10 -11 Pa • m 4 . When the cantilever is submitted to a force F perpendicular to its direction, the tip deflection is given by δ = F L 3 cant /3B cant . With typical cantilever length L cant ∼ 1 cm and an optical resolution of a few microns, sub-nanonewton forces can be measured.

Experimental Workflow

The experimental workflow is illustrated in fig. 1.c. After release and coiling of the helices, a helix is selected and one of its ends is caught and clamped by the open glass capillary. The cantilever is approached and contact is made between the cantilever tip and the other end of the helix. Non-specific contact forces afford strong adhesion between the cantilever and the helix. Two steps are then repeated until the end of the experiment: first measuring the force-extension response at a given geometry and then increasing the pitch angle using the stretching treatment. Both steps are further detailed in the following.

To measure the force-extension response, a series of increasing extension steps is imposed to the helix by successively displacing the capillary. The capillary is quickly displaced from one position to another (typically 6 to 8 positions, with typical displacement ∼ 10 -20 µm ) and is held still for a few seconds at each position (typically 5 seconds). At each position and for the few seconds the capillary is held still, the geometry of the deformed helix and the cantilever tip position are recorded. From the experimental images and knowing the reference position of the cantilever tip (i.e. when no force is applied), the cantilever tip displacement δ and the helix axial length H are extracted.

We show in fig. 1.d a typical measurement for δ and H obtained when measuring a force-extension response. As expected, as the axial length is further increased, the cantilever tip is further deflected: the force applied to stretch the helix is recorded by the cantilever. We observe that noise can be significant, which probably originates from ambient flow in the fluid pool combined with the very long length of the cantilever fiber, thus creating fluctuations in the cantilever tip position. As the helix is held between the cantilever tip and the capillary, these fluctuations also impact the measurement of the helix axial length. To mitigate this noise, the cantilever tip deflection δ and the helix axial length H are averaged over the 5 seconds recordings. The tension force F is then computed from the cantilever length and bending modulus:

F = 3δB cant /L 3 cant .
Before and after each force-extension measurement, the helix is released from the open glass capillary and its resting axial length is measured. Due to the creep properties of the material (leveraged to control the pitch angle) an increase in the resting axial length is observed. We mitigate these effects by keeping the experiment duration as short as possible: typically ∼ 30 -40 s for a given forceextension curve (the typical timescale of creep deformations is several minutes). As a result, the difference between the two measurements of the resting axial length is always below ∼ 2 % of the total length L, which corresponds to a variation in pitch angle of less than 1°, which we neglect. The helix reference axial length H 0 , taken as the average over the two measurements, is thus considered constant for a given force-extension curve. The reference pitch angle is simply obtained from the geometrical relation H 0 /L = sin α 0 .

We then apply the stretching treatment to increase the reference pitch angle α 0 . The helix is clamped at its two ends (one by the cantilever and one by the capillary), an axial extension (typically ∼ 50 -100 µm) is imposed for several minutes by displacing the capillary, and the helix is finally let to relax. This treatment results in an irreversible increase in the reference pitch angle. We show in fig. 1.e the typical evolution of the reference pitch angle over a full experiment.

A new force-extension curve is then measured for this new reference geometry and so on. At the end of the experiment, the helix is completely stretched to measure the total filament length L. This experimental workflow yields a series of force-extension curves, each corresponding to an increasing reference pitch angle α 0 .

Experimental Results

Force-Extension Curves

We show in fig. 2.a the measured force F , computed from the cantilever tip deflection δ, as a function of the rescaled axial length H/L, for a given helix as the helix geometry is varied. Each curve on the plot corresponds to a different reference pitch angle α 0 . The corresponding reference axial length H 0 for each geometry is represented at the bottom of the plot.

We observe that the helical ribbon stiffens as the reference pitch angle is increased. Indeed, for the light-colored curves, which correspond to higher reference pitch angles, a comparatively higher force is necessary to extend the helix than for the dark-colored ones. The same observation is confirmed for other helical ribbons, as shown in the SI. To explain this stiffening, the relation between the helix geometry and the force-extension response must be understood.

As shown in fig. 2.a, the imposed extension can be significant, the axial length reaching up to ∼ 95 % of the total length L i.e. almost to full extension. In this high stretch regime, close to full helix extension (corresponding to H/L = 1), the force-extension relationship is not linear. Due to the specific geometrical properties of the helical object, such significant global deformation is obtained while remaining in the material elastic regime [START_REF] Prévost | Shaping nanoscale ribbons into microhelices of controllable radius and pitch[END_REF]. The nonlinearity of the force-extension relationship thus does not originate from the material itself but from the helical geometry. The analytical work of Starostin et al. [START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF] has examined in the general case the elastic deformation of a helical ribbon submitted to an axial end-loading. Under the assumption that the deformed helix remains a uniform helix (uniform radius and angle), they obtained the following expression for the force-extension relationship

F = C R 2 cos α 0 cos α + C B sin α 0 sin α cos 2 α + C B sin 2 α 2 sin(α -α 0 ) cos α (1)
with α = arcsin H/L pitch angle of the deformed helix (supposedly uniform along the filament length), which thus describes the deformation of the helix. Because of the multiplicative term 1/ cos α, the force diverges as the helix is fully extended (corresponding to α → 90°), which describes the helix finite extensibility. For a flat triangular cross-section, we have B = Ewt 3 /36 and C = µwt 3 /12 [START_REF] Gloumakoff | Torsion of Bars With Isosceles Triangular and Diamond Sections[END_REF] with µ shear modulus and hence C/B = 3/2(1 + ν) i.e. independent of the ribbon thickness and width. The Poisson's ratio ν for bulk PMMA is usually estimated within the range ν = 0.35 -0.4 [START_REF] Gilmour | Elastic moduli of glassy polymers at low strains[END_REF]. We consider that the Poisson's ratio is not affected by the strong material confinement due to the ribbon vanishing thickness and we take ν = 0.375.

In theory, the twisting modulus C could be measured prior to experiments by measuring the ribbon thickness and width. But in practice, accurate measurement of the ribbon nanoscale thickness prior to experiments is very difficult. Furthermore, determining the material shear modulus µ would require further investigation as the bulk value is most likely not relevant: nanoscale thicknesses have been shown to influence the mechanical properties of materials [START_REF] Stafford | Elastic moduli of ultrathin amorphous polymer films[END_REF][START_REF] Torres | Elastic modulus of amorphous polymer thin films: relationship to the glass transition temperature[END_REF][START_REF] Bay | Mechanical properties of ultrathin polymer nanocomposites[END_REF]. We thus use C as a fitting parameter.

We show in fig. 2.b the force-extension curves (same data as in fig. 2.a) fitted using eq. ( 1). The successive curves are fitted independently, each fitting yielding an estimate of the twisting modulus C. The curves are shifted by an arbitrary amount to better distinguish between the different curves. For all reference pitch angles, the proposed expression accurately fits the experimental data and describes the non-linearity of the force-extension curves. The experimental protocol is repeated for three other helical ribbons (4 in total): we present in the SI the fitted force-extension curves for the other helical ribbons. Excellent agreement with the theoretical prediction is obtained for all four, using in all cases C as the single fitting parameter. Our experimental results thus provide strong validation for the analytical relationship obtained by Starostin et al. [START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF]. To further verify the consistency of these results, we examine the obtained values of C.

Measurements of the Twisting Modulus

At each reference pitch angle α 0 , the fitting yields one estimate of the ribbon twisting modulus C. Naturally, this parameter is expected to remain constant for a given helix throughout the experiment. We show in fig. 3.a the successive measurements for C as the reference pitch angle α 0 is increased, for the four different helical ribbons. In all four cases and within the experimental errors, no change is observed for the twisting modulus. Overall, the reference pitch angle spans a wide range ∼ 10 -70°. These results further validate the model proposed by Starostin et al. [START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF]: the stiffening observed as the reference pitch angle increases is accurately captured by the proposed expression.

The work of Pham et al. [START_REF] Pham | Deformation and shape of flexible, microscale helices in viscous flow[END_REF] previously studied the mechanical response of PMMA helical ribbons, with the same experimental system. The twisting modulus C was measured for several PMMA helical ribbons of various radii R. These results were obtained for helices in their initial configuration after coiling i.e. with vanishing reference pitch angles α 0 . At small reference angle (α 0 1) and within the small deformation regime (α -α 0 1), the force-extension relationship eq. ( 1) simplifies to F = (C/R 2 ) * (∆H/L). Following the common

Discussion

In order to explain the stiffening of the helical ribbons, it is beneficial to write the linear limit of the forceextension relationship i.e. when α -α 0 1. Within this limit, the expression reads

F = 1 R 2 1 1 C cos 2 α 0 + 1 B sin 2 α 0 cos 2 α 0 ∆H L (2)
In the case of filaments with isotropic or near-isotropic cross-sections i.e. rods, the force-extension relationship has been found to be [8]

F iso = 1 R 2 1 1 C cos 2 α 0 + 1 B sin 2 α 0 ∆H L (3)
The only difference between the ribbon case and the rod case is thus the multiplicative term 1/ cos 2 α 0 . The common term 1/ 1 C cos 2 α 0 + 1 B sin 2 α 0 denotes a mechanical transition from a regime dominated by twisting of the ribbon (C being the relevant modulus) at small α 0 (i.e. closed-loop helices) to a bending-dominated regime (B being the relevant modulus) at high α 0 (i.e. open-loop helices). In the case of a circular cross-section, we have C/B = 1/(1 + ν) and hence C/B ∼ 0.7 for most materials (since usually ν ∼ 0.3 -0.5 ): this term drives a weak stiffening with increasing α 0 . In our experimental conditions we have C/B = 3/2(1 + ν) ∼ 1.1: this term alone would thus drive a very weak softening as α 0 increases. Therefore, the significant stiffening observed for helical ribbons stems from the second term 1/ cos 2 α 0 , which dominates the other and is not found in the isotropic case. We conclude that this strong stiffening behavior is specific to helical ribbons. Figure 4 3) (i.e. without the 1/ cos 2 α 0 term) for a flat triangular filament (C/B ∼ 1.1) to illustrate the weak softening induced by the twisting-to-bending mechanical transition in our experimental conditions. We call this fictive case the 'flat triangular rod'.

Our investigation provides strong experimental validation to the model proposed by Starostin et al. [START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF]. This validation comes despite several differences between the assumptions of Starostin et al. and our experimental conditions. They assumed a force-controlled stretching while in our experiments the stretching is displacementcontrolled (the applied force being measured through the cantilever). This entails differences in the boundary conditions: in our conditions both helix ends are clamped and thus the applied force is not purely axial, a tangential component may also exist. This tangential component is not recorded, as it is applied parallel to the cantilever axis. Thus, the very good agreement between the predictions of Starostin et al. and our experimental results suggests that boundary conditions have only a weak impact on the helix global mechanical response. We strongly suspect that, similarly to what was found in the case of rods [START_REF] Kim | Deformation of a helical filament by flow and electric or magnetic fields[END_REF], this is due to the typical filament length being much larger than the typical helical radius. In the case of shorter helices, boundary conditions may have a more pronounced influence. Finally, Starostin et al. based their analytical model on the case of flat rectangular filaments and validated their model against numerical simulations of such helices, while in our experimental conditions the cross-section is neartriangular. Our results thus suggests validity of the model for ribbons of all cross-sectional shapes: flat triangle or flat ellipse for example.

Conclusion

In this work, we have characterized experimentally the mechanical properties of micron-sized helical ribbons by measuring the force-extension relationship. An axial extension was imposed, and the corresponding force was measured using a cantilever beam. Benefiting from newly achieved control on the pitch angle, the influence of this parameter was investigated experimentally in detail. At a given reference pitch angle, the force-extension response was found to be non-linear. As the reference pitch angle α 0 was increased, a strong stiffening of the helical ribbons was observed. We found that both the non-linearity and the stiffening originate from the helical geometry and are accurately described by the analytical model proposed by Starostin et al. [START_REF] Starostin | Tension-induced multistability in inextensible helical ribbons[END_REF]. Our results thus provide strong experimental validation for the obtained force-extension relationship, given by eq. ( 1). Comparison with a model previously established for filaments of isotropic cross-section [START_REF] Love | A treatise on the mathematical theory of elasticity[END_REF] highlighted an effect common to rods and filaments: a twisting-to-bending transition as the pitch increases. But, while this effect stiffens helices formed by rods, it tends to soften helical ribbons. Therefore, the observed strong stiffening effect is specific to helical ribbons. Modifying the filament cross-section can thus be used in the future as a design principle to fabricate helices with specific mechanical properties. We hope that these findings stimulate further studies of the intermediate case i.e. filaments whose cross-section is neither isotropic nor flat as to clarify the origin of this stiffening behavior and to explain the crossover between the two models proposed.

  Figure 1: a. Helical ribbon geometry and relevant geometrical parameters. Typical values range within R = 1 -100 µm, L = 100 -3000 µm, t = 5 -50 nm and w = 0.5 -5 µm b. Schematic of the experimental set-up. c. Experimental workflow for the mechanical characterization of helical ribbons. Samples are immersed in water: ribbons lift-off following the dissolution of the sacrificial layer and self-coil into a tight helical shape. A helix is selected and captured by the open glass capillary. Contact is made between the tip of the cantilever and the other helix end.The two next steps are repeated 10 to 15 times: firstly establishing the force-extension curve for a given helix geometry and secondly increasing the pitch angle using the stretching treatment method. The timeline illustrates the succession of these different steps. d. Typical raw measurements obtained when establishing a force-extension curve: the capillary is successively displaced (left plot), which extends the helix. The extension is recorded by measuring the axial length H (middle plot) and the force is obtained from the cantilever tip displacement δ (right plot). e. Typical evolution of the reference pitch angle α 0 of a given helical ribbon over a full experiment.

Figure 2 :

 2 Figure 2: Experimental results for one PMMA helical ribbon (with total length L = 480 µm and radius R = 5.0 µm) a. Pulling force F as a function of rescaled axial length H/L as the geometry is varied. Each curve is associated to a different reference axial length H 0 , shown at the bottom of the plot, and thus to a different reference pitch angle α 0 . The error bars are the standard deviations calculated over the 5 seconds recordings. b. Successive force-extension curves fitted with the analytical model proposed by Starostin et al. [24]. The curves are shifted by an arbitrary amount to better distinguish between the different curves.

Figure 3 :

 3 Figure 3: a Estimate of the twisting modulus C obtained from fitting the force-extension curves as a function of reference pitch angle for 4 different PMMA helical ribbons. The error bars represent the numerical uncertainties from the fitting. Data shown in fig. 2 correspond to the red squares. b. Twisting modulus C of each helical ribbon, averaged over all reference pitch angles, as a function of initial helical radius R. Colored dots show our experimental results while gray dots show the results of Pham et al. (also obtained for PMMA helical ribbons) [23].

  synthesizes these two different behaviors by representing the inverse of the normalized spring constant 1/k, with k defined such that F = k(α 0 ) C R 2 ∆H L , as a function of reference pitch angle α 0 for a flat triangular filament and for a circular filament. The dashed line represents eq. (

Figure 4 :

 4 Figure 4: Helix linear stiffness for a flat triangular filament and for a circular filament, represented by the inverse of the normalized spring constant 1/k as a function of reference pitch angle α 0 . The dashed line represents the fictive case of a flat triangular rod i.e. without the stiffening term. The Poisson's ratio was taken as ν = 0.4 (since for most materials ν ∼ 0.3 -0.5 ).

Acknowledgments

We thank Prof. A. J. Crosby for stimulating discussions and a critical reading of the manuscript. We thank Dr. D. M. Barber for help with the helix fabrication. AL and LP acknowledge funding by the European Research Council through a consolidator grant (ERC PaDyFlow 682367). This work received the support of Institut Pierre-Gilles de Gennes ( Équipement d'Excellence, "Investissements d'Avenir", Program ANR-10-EQPX-34).

Conflict of Interest

The authors declare no conflict of interest.