Adrien Bouty

Laurent Petitjean

Julien Chatard

Rachid Matmour

Christophe Degrandcourt

Ralf Schweins

Florian Meneau

Paweł Kwasńiewski

François Boué

Marc Couty

Jacques Jestin
email: jacques.jestin@cea.fr

Interplay between Polymer Chain Conformation and Nanoparticles Assembly in Model Industrial Silica/Rubber Nanocomposites

The question of the influence of nanoparticles (NP) on chain dimensions in polymer nanocomposites (PNC) has been treated mainly through the fundamental way using theoretical or simulation tools and experiments on well-defined model PNC. Here we present the first experimental study about the influence of NP on the polymer chain conformation for PNC designed to be as close as possible to industrial systems employed in tire industry. PNC are silica nanoparticles dispersed into a Styrene-Butadiene-Rubber (SBR) matrix whose NP dispersion can be managed by NP loading with interfacial coating or coupling additives usually employed in the manufacturing mixing process. We associated specific chain (d) labeling, and the so-called Zero Average Contrast (ZAC) method, with SANS, in-situ SANS and SAXS/TEM experiments to extract the polymer chain scattering signal at rest for non-cross linked and under stretching for cross-linked PNCs. NP loading, individual clusters or connected network, as well as the influence of the type, the quantity of interfacial agent and the influence of the elongation rate have been evaluated on the chain conformation and on its related deformation. We clearly distinguish the situations where the silica is perfectly matched from the unperfected matching by direct comparison of SANS and SAXS structure factor. Whatever the silica matching situation, the additive type and quantity and the filler content, there is no thus significant change in the polymer dimension for NP loading up to 15% v/v within a range of 5%. One can see an extra scattering contribution at low Q, as often encountered, enhanced for non-perfect silica matching but also visible for perfect filler matching. This contribution can be qualitatively attributed to specific h or d chains adsorption onto the NP surface inside the NP cluster that modifying the average scattering neutron contrast of the silica cluster. Under elongation, NP act as additional cross-linking junction preventing chain relaxation giving a deformation of the chain with NP closer to theoretical phantom network prediction than for pure matrix.

I. Introduction

Progress in the understanding of the mechanisms that govern reinforcement of polymer matrices by NP is of a strategic interest to improve the design and the industrial processes of conception of new materials, specifically in tires industries. One of the critical breakthroughs is to be able to understand and describe the local structure of the filler (individual dispersion, connected network, aggregates or agglomerates…) as well as the one of the polymer chain (unmodified, stretched or compressed…) and to correlate them with the macroscopic mechanical behaviors of the materials like reinforcement, flow behavior or non-linear deformation [1,2]. Despite of the lack of a general model to describe the filler contribution (NP or assembly of NP), many progresses have been performed recently especially to dissociate the filler-filler interactions from the filler-matrix ones in reinforcement on both well-defined model [3][4][5][6][7][8][9][10][11] and simplified industrial systems [12][13][14][15][16][17][18]. On the contrary, the question of the matrix chain conformation and its modification due to the presence of NP has been, up to now, only addressed from a fundamental point of view. Theories have been developed using mainly simulations [19][20][21][22][23][24][25] considering the polymer radius of gyration R g and the NP radius R p ratio (R g /R p ). They lead to contradictory conclusions, either swelling or decreasing of R g as a function of the NP volume fraction, mostly when the polymer radius of gyration R g is equal or superior to the NP radius R p . For example, Termonia et al. and Sung et al. [19,20] reported a significant swelling of the chains for R g /R p > 1 while Vacatello et al. [21,22] reported a contraction of the chains. Sharaf et al. [23] have reported a swelling of the chain but for R g /R p < 1. The situation has been also probed from an experimental view on welldefined model systems [26][27][28][29][30][31][32][33][34] using SANS and conventional polymers for which the labeling (d) forms can be conventionally accessible. These studies also yield controveral behaviors whose origins can be discussed according to different issues. A first possible origin is the difficulty to extract the experimental signal of the chain without ambiguous matching or undefined extra scattering. Nakatani et al. [26], Mackay et al. [27] (for the C60/PS system) and also Botti et al. [28] extracted the conformation of the polymer chain from a complex scattering signal in which the interactions between labeled (d) and

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View Article Online DOI: 10.1039/C5FD00130G non-labeled (h) chains have to be taken into account. Sen et al. [29], Nusser et al. [30] and Genix et al. [31] (on latex/silica systems) used ZAC method, which cancels inter and intra-chains correlations, under unperfected matching conditions [29,30] or in the presence of extra scattering signal coming from chain diffusion [31]. Jouault et al. [32] have demonstrated with a random (h/d) matrix who's matched the silica NP the unambiguous validity of the ZAC method while also observing an unresolved extra scattering contribution. An unambiguous contrast matched method used is also the one of Tuteja et al. [33] with cross-linked PS particles as fillers but concerns a very specific case of PNCs filled with soft fillers. Another contribution comes from the role played by the NP dispersion into the polymer matrix.

Despite of the general prediction proposed by Mackay et al. [27] that suggested that the particle dispersion is only governed by the R g /R p ratio , it has been also demonstrated that the particle dispersion strongly depend on the processing conditions [7,34]. Indeed, some of the previous studies suffer from a clear NP dispersion characterization that can render a confusing reading of the results if NPs are aggregated or form connected network. When reducing the discussion to the specific case for which the polymer radius of gyration is close to the NP radius, the situation is however relatively clear when most of the experiments have concluded that the chain conformation seems to be unaffected by the NPs [27,28,29,31,32,33]. Only two references [26,30] show some effects that arise only at high NP loading, up to 40%v/v. The absence of NP influence has been also confirmed recently by Crawford et al. [35] for PNCs with uniformly dispersed NPs. The previous examples examine the possible pre-strain of the chain induced by the NP. Another relevant contribution is to see whether the chain deformation is also affected by the NP under non-linear (large) deformations. This point has been addressed with simulations [36] or with indirect experiments like NMR [37]. A few experiments have probed the chain deformation scattering signal directly in presence of NP [38][39][40] on model PNC. Homogeneous overstrain has been established in triblock copolymer micelles of PI-PS-PI in cross-linked polyisoprene (PI) matrix where no overstrain effect have been measured in uncross-linked model silica/polystyrene PNC [40]. Beyond the fundamental interest of the previous studies mainly performed on model system, the question of the chain conformation inside a composite is also a main preoccupation of industrial actors.

Faraday Discussions Accepted Manuscript

In the current context of petroleum resources depletion, a new challenge of the tire industry is to develop new materials that can reduce the oil consumption well as the CO 2 emission while offering high mechanical performances (reinforcement, adhesion, tensile strength…). One possible way is to act on the NP, for example, by changing the classical carbon black filler by silica nanoparticles. Another possible way is to act on the rubber matrix by modifying the chain characteristics (molecular mass, polydispersity index, chemical functions…) to play with the polymer-NP interactions. Consequently, the question of the matrix chain conformation/deformation modification in PNC and its correlation with the mechanical performances of the material becomes a central question that we have addressed in the present paper.

We present here a complete study of model industrial silica/SBR systems using complementary SAXS/SANS/TEM experiments. We have previously demonstrated [18] that the silica NP dispersion can be controlled with coating (Octyltriethoxysilane -OCTEO) and coupling (Sulfurfunctional organosilane bis(triethoxysilyl)propyl tetrasulfide -Si69) additives into SBR for non-cross-linked PNCs similarly whatever the way of the sample preparation, solvent casting or internal mixing. Thanks to covalent bonds with the chains of the matrix, the coupling agent gives rise to smaller and denser aggregates than can form an interpenetrated percolating network when increasing the NP loading. The coating agent has a more repulsive action that reduce the network density. In other words, it is demonstrated that interfacial additives can tune the matrix chains confinement and appears thus to be a relevant way, as well as the NP loading, to evaluate the effect on the chain conformation. The synthesis of (d) labelled chain, restricted to low molecular mass, limits the capacity here to probe a broad range of NP dispersion leads to treat the case of NP organized as clusters. We varied the type and the quantity of interfacial additives for different NP loading from 1 to 15% vol. We applied the principle of ZAC by mixing labelled (d) and non-labelled (h) SBR chains to extract the single chain scattering signal and see whether or not the NP are potentially inducing some pre-straining effect on the chains. Clustering of NP is evaluated from SAXS/TEM analysis. To evaluate the effect of NP on the chain deformation at large non-linear elongation rate, we completed the study with in-situ SANS measurements on cross-linked silica NP/SBR PNCs for different elongation rates.

Faraday Discussions Accepted Manuscript

II. Material and methods

II.1. Synthesis of the (d) SBR chains

d8-Styrene and d6-butadiene deuterated monomers were supplied by Cortectnet (purity of 99%) and Eurisotop (purity of 99%), respectively. Before their use for copolymerization, the monomers were first dried over BuLi for d6-butadiene and over calcium hydride and dibutyl magnesium for d8-styrene and then distilled to get the purified monomers. Similarly to hydrogenated SBR copolymers in classical runs, the deuterated SBR copolymer samples were synthesized by anionic polymerization by the Michelin Company. The copolymerization was initiated by BuLi in cyclohexane at 50°C. Deuterated monomers were mixed in appropriate conditions to reach a microstructure as close as possible to the one of the hydrogenated chains used here (26%wt. of Styrene, 74%wt. of Butadiene; M n = 40.000 g/mol, M w =41.700 g/mol, PDI =1.05). Two samples of deuterated SBR copolymers were obtained with the same microstructure as the hydrogenated ones (26%wt. of Styrene, 74%wt. of Butadiene) and satisfying enough chain mass distribution Batch 1 : M n =49 700 g/mol ,M w =100 500 g/mol, PDI =2.0, Batch 2 :

M n =45 000 g/mol , M w =96.270 g/mol , PDI =2.1). The large polydispersities (PDI ≥ 2.0) obtained for the deuterated copolymers is due to the presence of butene derivatives impurities present in nonnegligible quantities in the supplied deuterated butadiene.

II. 2. Preparation of the PNCs

The matrix consists systematically in a mixture of chains from the hydrogenated and the deuterated batches, in the appropriate volume proportions to match the silica. Considering the neutron scattering length densities of the three components (ρ SiO2 =3.41 10 10 cm -2 , ρ h-SBR = 0.59 10 10 cm -2 and ρ d-SBR = 6.67 10 10 cm -2 ), the matrix composition is fixed to 54%v. h-SBR + 46%v. d-SBR as it theoretically matches silica. All samples were prepared by solvent casting route with Dimethylacetamide (DMAc) as solvent.

The filler is Ludox type colloidal silica supplied by Aldrich (LS-30); the size distribution of these nanoparticles was finely characterized by SANS and gives a mean particle radius R 0 =7.8nm with a lognormal distribution of 0.15.

The uncross-linked samples were synthesized with deuterated chains from Batch 1. The steps and conditions of this synthesis -Ludox transfer from water to DMAc, dissolution of matrix, addition of Ludox/DMAc suspension at 130°C, solvent casting under nitrogen flow at 130°C -were precisely explained in a previous paper [18] and rigorously reproduced here for uncross-linked samples.

Dispersing agents, coating (OCTEO) and coupling (Si69) additives are in a liquid form at room temperature. They are diluted at 7% v/v in methanol (MeOH), and this solution is then introduced in the Ludox-DMAc suspension in the nominal quantity of 1 or 4 triethoxysilane groups per nm² of silica for OCTEO and 1/2 or 1 triethoxysilane groups per nm² of silica for Si69. Quantities were only adapted to both the small amount of Batch 1 and the following SANS experiment conditions (beam size). Uncrosslinked samples consist in malleable bulks of about 50mg, which can be fixed in a calibrated cell.

Cross-linked samples were synthesized with deuterated chains from Batch 2. The modifications relative to uncross-linked samples are the following. Five minutes before starting the casting phase, a crosslinking mixture is added in the polymer/silica/DMAc mixture at 130°C: soluble sulfur as cross-linker (2%wt. with matrix) and CBS as vulcanization accelerator (2%wt. with matrix). These cross-linking agents are fine powder under normal conditions, and their solubility was previously checked in pure DMAc at 130°C. In the following phase, mixtures are poured into round Teflon molds (diameter =

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Small Angles X-ray Scattering SAXS and Ultra Small Angles X-rays Scattering USAXS

In SAXS experiments, we measure in filled systems the silica scattering due to the electronic contrast existing between SBR and silica: the scattering length densities are respectively ρ SBR = 8.41 10 10 cm -2 and ρ Silica =17.36 10 10 cm -2 , calculated from the chemical composition, giving the electronic contrast ∆ρ=8.95 10 10 cm -2 . Uncross-linked systems were measured in their glass cell, at SOLEIL (Saclay) on the beam-line SWING, with a single configuration defined by a wavelength of λ = 1.77 Å (E = 7keV) and a sample-to-detector distances of 6.5m. As the center of scattering was not coinciding with the one of the detector, this single configuration yield a remarkably large total Q-range from 1 10 -3 to 1.6 10 -1 Å - 1 , by merging two measurements with two different beam-stops (a small one and a large one). Standard data treatments include merging, subtraction and normalization. Cross-linked systems were measured at ESRF on the beam-line ID02, with a wavelength of 1Å (E=12,4keV) and two sample-to-detector distances of 1 and 10 m, yielding a total Q range from 1 10 -3 to 5 10 -1 Å -1 . On the same beam-line, the Bonse-Hart setup allowed to reach the lower Q value of 3 10 -4 Å -1 . SAXS measurements were performed exclusively at rest, after accurate measurement of the thickness of samples. Standard treatments of data include isotropic averaging of the 2D isotropic scattering patterns, merging and normalization. After measuring the scattering of the unfilled system, the pure silica scattering in a nanocomposite of the silica volume fraction φ SiO2 , is obtained by subtracting for each sample the matrix contribution, according to the following operation:

I SiO2 = I Sample -(1-φ SiO2 ) x I Matrix .

Transmission Electronic Microscopy TEM

For TEM characterization, samples were cut by ultracryomicrotomy (mechanical cut) at -80°C (lower than the glass transition temperature of SBR) at a desired thickness of about 50nm, then laid on a TEM grid before observation. Observations were realized using a Philips CM200 LaB6 (200kV) microscope in bright field mode.

III Results and Discussion

Chain Conformation

The scattering intensity of a single chain in a nanocomposite can be obtained directly by SANS using the zero average contrast (ZAC) method. This method was first used for a mix of normal, i.e., non deuterated (h) and deuterated (d) chain in a solvent: the total scattering intensity I(Q) can be expressed by the relation [41]:

I(Q) = (ρ d -ρ h )²φ h φ d νΦNP(Q)+( φ d ρ d+ φ h ρ h -ρ 0 )²[ νΦNP(Q)+VΦ²S(Q)] (1)
Where ρ d, ρ h and ρ 0 are respectively the scattering length density of deuterated chains, hydrogenated chains and solvent, φ h , φ d are the volume fractions of hydrogenated and of deuterated chains, ν is the volume of the monomer, Φ is the volume fraction of polymer, V is the global volume, P(Q) is the form factor of the chain, and S(Q) is the inter-chain structure factor. When the average contrast between polymer mixture and solvent is adjusted to zero, i.e.:

φ d ρ d+ φ h ρ h -ρ 0 = 0 (2)
The scattering intensity is only related to the form factor of one chain:

I(Q) = (ρ d -ρ h )²φ h φ d νΦNP(Q) (3)
In our case, the role of the solvent is played by the silica particles. The scattering length density has been determined by previous contrast variations experiments [42] in solvent and found to be equal to 3.41 10 10 cm -2 . For our system, the matching composition to satisfy equation ( 2) is achieved by mixing 54% v/v of hydrogenated SBR chains (SLD = 0.59 10 10 cm -2 ) and 46% v/v of deuterated SBR chains (SLD = 6.74 10 10 cm -2 ). When both hydrogenated and deuterated chains have equivalent molecular weights and when there is no interaction between the monomer, for ideals mixture, the form factor of a single chain corresponds to a Gaussian chain in a melt whose scattering function can be described by the Debye function [START_REF] Higgins | Polymers and neutron scattering[END_REF]:

P(Q) = 2/(Q²R g ²)²[e -Q²Rg² -1+Q²R g ²) (4)
Where R g is the radius of gyration of the chain. However mixtures of hydrogenated and deuterated chains are usually not ideal and one needs to introduce a monomeric interaction parameter (Flory-Huggins) χ. The equation ( 3) is extended to the random phase approximation (RPA) [START_REF] De Gennes | Scaling Concepts in Polymer Physics[END_REF]:

V ² 2 ) Q ( P N ² 1 ) Q ( P N ² 1 ) Q ( I 1 d d d d h h h h ρ ∆ χ - υ φ ρ ∆ + υ φ ρ ∆ = (5)
Where where d is the density in g/cm 3 and N A the Avogadro number. As the two monomers have the same structure, the numbers of parameters can be simplified according to R gd = R gh (M wd /M wh ) 1/2 . The Figure 1 presents the normalized SANS curve obtained for the pure h/d SBR uncross-linked mixture (batch 1) under the ZAC condition, without filler and additives. The curve is perfectly fitted with the RPA function that enables to extract the following parameters with an uncertainty equal to 5%: R gh = 7.00 nm and χ=6 10 -4 . The radius of gyration of the d chains can be deduced from the molecular masses ratio and is equal to R gd = 8.30 nm. We consider now the situation where we added increasing content of filler in the h/d polymer matrix first without dispersing agents. The SANS curves are presented in Figure 2 for 1, 10 and 15% vol. of silica. We can observe that up to a Q value of 0.01 Å -1 , all the curves superimpose perfectly with the pure h/d matrix. Below this Q value, one can see an upturn of the low Q scattering that will be discussed in details below. We performed the SANS analysis with the RPA function (full black lines) by fixing the χ parameter to the h/d blend value and fitted the R gh to the height of the Kratky plateau (only one fitting parameter). The resulting fitting R gh values are reported in the Table 1. We now consider the influence of the dispersing additive using the coating agent OCTEO that can covalently bind to the surface of the silica and is known to improve the compatibility with the polymer.

φ d,h , N d,h , ν d,

Uncross-linked systems

Q (Å -1 ) Q (Å -
The different series containing respectively 1 and 4 equivalent (1 and 4 TEOS group per nm² of silica) of coating agent have been probed with SANS for 1, 10 and 15% vol. of silica filler (Figure 3). We previously double-checked that the additives do not contribute to the SANS scattering by comparing the pure h/d blend with the same blend with the same amount of OCTEO or Si69. No specific scattering coming from the additive can be detected with SANS.

Q (Å -1 ) Q (Å -1 )
I.Q² (cm -1 As for the case of silica without additive, the scatterings follow the pure blend up to 0.01 Å -1 and also reveal an upturn for the low Q region. This effect is especially pronounced when increasing the content of coating agent (O4) for high filler content 10 and 15% vol. of silica. This is illustrated by a decrease of the agreement between the experimental data and the RPA calculation, which is particularly visible in the Kratky representation. The full line is the best fitted obtained with the RPA function. The curves have been shifted for clarity.

A similar trend is observed with ½ and 1 equivalent of the coupling agent Si69 that can be covalently bonded to the particles and the polymer chains. Up to 0.01 Å -1 , the scattering again follows the blend 1: R gh values deduced from the RPA analysis by fixing the χ to the initial blend value 6.10 -4 . The uncertainty is ± 5%.

Q (Å -1 ) Q (Å -

Cross-linked systems

The Figure 5 presents the SANS curves obtained with the cross-linked samples (batch 2). We can notice that crosslinking modified the h/d blend with the appearance of a peak visible in the Kratky representation. This peak is related to an intra-chain correlation induced by the crosslinking process.

The mean distance between two junctions in the Gaussian pathway is reduced with a resulting modification of the chain form factor. Such a form factor has already been computed by Bastide et al. [START_REF] Bastide | [END_REF]. We choose not to use this formula for the present study that is beyond the scope of the paper. In addition, RPA is still valuable in the Guinier regime to capture the R g values and it variations as function of filler content, additive and elongation rates. We get the following parameters with a larger uncertainty (10%) than for uncross-linked systems (2%): R gh =6.80 nm, R gd =10.78 nm and χ=0. The value for the h chains is a little bit smaller than the one obtained for uncross-linked chains as expected from the intra-chains correlation effects. For the same reason, we can see a drop of the Flory parameter.

Faraday Discussions Accepted Manuscript

One gets a higher value for the d chains due to the larger dispersion of the molecular masses for the batch 2. We applied the same analysis in presence of coating and coupling agent for 1% vol. and 5 % vol. of silica. The best fitted values of the R gh are reported in Table 2. The uncertainty is ±(nm) 10%. We applied the RPA analysis on elongated samples the same way as the non-elongated ones. The variations of the R gh as a function of the elongation rate are reported in Table 2. Under elongation, the deformation is lost at large angle (loss of affinity), the chains adopted their initial configuration for

Q (Å -1 ) Q (Å -1 ) I/ϕ polymer (cm -1 ) Q (Å -1 ) Q (Å -

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small distances, typically for distances smaller than the mean distance between two cross-linking junctions.

Nanoparticles Assembly

We performed SAXS experiments to evaluate the NP dispersion inside the polymer blend. For x-rays radiation the contrast is mainly between silica particles and the polymer matrix (there is no contribution of the labelling h/d chains). The experiments have been made on the same samples prepared for SANS.

The SAXS curves are presented in Figure 7 for the different conditions (no additive, coating and coupling agent) as function of the silica loading. For comparison, we reported the spherical form factor of the primary particles (full black line) previously determined [18].

I/φ SiO2 (cm -1 ) I/φ SiO2 (cm -1 ) I/φ SiO2 (cm -1 ) I/φ SiO2 (cm -1 ) I/φ SiO2 (cm -1 ) Q (Å -1 ) Q (Å -1 ) Q (Å -1 ) Q (Å -1 ) Q (Å -1 )
No calculated using a spherical function convoluted with a log-normal distribution with R 0 =7.8nm and σ=0.15. The bottom/right plot illustrates the effect of cross-linking on the NP dispersion. The curves have been normalized by the silica volume fraction.

We can observe a systematic increase of the low Q scattering suggesting that the primary particles are aggregated inside the polymer matrix and are forming clusters of NPs. The slope of the decrease of the intensity in the intermediate Q range, close to 3, indicates the formation of compact aggregates. We can also observe a diminution of the low Q scattering level when increasing the particle volume fraction coming from an increase of the interaction between the clusters arising when the clusters are forming a continuous connected network as a result of a percolation process. The absence of plateau prevents the determination of a characteristic size of the clusters. The formation of cluster seems to be systematic and not controlled by the nature and the quantity of the additives. That can be due to the low molecular weight of the matrix used (40k) that gives rise to a very low viscosity that cannot balanced the cluster formation during the solvent casting process. To evaluate the influence of the crosslinking on the NP assembly, we compared the SAXS curves for all the samples uncross-linked and cross-linked. One example is given in Figure 7 (bottom/right). We can see that the scattering superimposed perfectly illustrating that cross-linking do not modify the NP assembly inside the polymer matrix. We completed the analysis with TEM pictures presented in Figure 8. TEM pictures confirmed the formation of dense and compact aggregates that are interacting at shorter distance when increasing the silica volume fraction. We can see clearly the transition from individual clusters to the formation of a continuous network visible for 10% vol. of silica. To evaluate the size of the clusters, we calculated the radially averaged autocorrelation function C(r) using the image processing software ImageJ [46]. C(r) basically calculates the two-point correlation of the pixels of an image, after applying a binary filter, as a function of distance and gives information about the typical feature size in an image. The crossing line with zero gives the cluster sizes as illustrated in Figure 8 which are respectively 170 nm and 115 nm for 1 and 10% vol. of silica which is consistent with the filler network formation.

IV Discussion

As previously shown through the different results, we can have access to the radius of gyration of the SBR chains for uncross-linked and cross-linked filled nanocomposites at rest and under uniaxial elongation. We can manage the NP assembly from individual clusters to a continuous filler network with the NP loading, coating and coupling additives. We successfully used the RPA function to analyze the SANS data with most of the time a consistent agreement between the model and the experimental points for both uncrosslinked and crosslinked systems. However, one can observe for some situations a discrepancy especially in Figures 3 and4. The most probable explanation of this discrepancy is an unperfect matching of the silica.

S(Q) S(Q) S(Q) S(Q) S(Q) S(Q) Q (Å -1 ) Q (Å -1 ) Q (Å -1 ) Q (Å -1 ) Q (Å -1 ) Q (Å -1 ) S(Q) Q (Å -1 ) S(Q) Q (Å -1 )
O This can be illustrated with the following idea: for un-perfect matching, the SANS curve will be modified by a contribution of the NP structure factor. We thus divide the SANS scattering intensity by the chain form factor, calculated with the RPA function, to extract an "apparent" structure factor. We compared this apparent structure factor deduced from SANS with the "effective" NP structure factor that can be calculated by dividing the SAXS intensity by the well-known NP form factor. The structure factors have been determined from intensities in absolute unit and normalized by the x-rays and neutron contrast terms. We report two examples of such comparison in Figure 10 for the two series containing 1 (top) and 4 equivalent (bottom) of coating agent. For complete silica matching, the SANS signal remain below the SAXS as can be seen for the O 1 series. However, for uncompleted silica matching, the SANS becomes equal or even larger than the SAXS scattering. In addition, we can see the appearance of a peak (in the inset) that corresponds exactly to the NP-NP interactions, NP in close contact inside the cluster that is visible on both apparent and effective structure factors. There is always a low Q contribution on the SANS scattering, beyond the size of the polymer chain, which is enhanced for unperfect matching of the filler. The typical size of this contribution can be evaluated approximatively using a simple Guinier function that gives R g =50nm and 40nm for 1% vol. and 10% vol. of silica. One can evaluate the corresponding sphere equivalent size using the relation R g ²=3/5R² and we obtain typical diameter of 130 nm and 105 nm for respectively 1% and 10% vol. of silica. These values are similar whatever the additives used. The interesting feature here is that these values are close to the silica cluster size determined from the TEM analysis in Figure 9. That means that we can see the large cluster with SANS even if the contrast matching condition is reached. This effect can be due to specific random adsorption of h or d chains at the surface of the particles during the sample processing. A recent study has estimated an average grafting density of 0.01 chains/nm² corresponding to adsorbed bound layer in similar NP-polymer systems [47]. This amount, around a 1-2 nm adsorbed monolayer, is sufficient to modify the mean scattering neutron contrast of the cluster that makes them visible even under the ZAC condition. An alternative explanation should be that NP induced the formation of h/d phase separation domains of typical sizes close to the silica cluster size (R g ∼ 30-50 nm). A rapid simulation shows us that even a very low amount of the domains, typically 0.01% in volume fraction, is enough to create the low Q additional scattering. However, due to the limited number of data in the low Q regime (5-6 points), this is difficult to going further in the quantitative analysis and to conclude clearly between the different explanations. One way to check this idea deeply should be to anneal the sample for a longer time at high temperature to see whether the low Q contribution is decreasing due to chains desorption or not. Here we demonstrated that whatever the matching condition (perfect or not), the low Q contribution, certainly related to the silica cluster, can be treated separately from the determination of the radius of gyration of the polymer chains. As a result, we can follow the evolution of the normalized radius of gyration of the uncross-linked h chains from the values reported in Table 1. The normalized R gh , the ratio between the filled and unfilled values, are reported in Figure 11a. This representation show clearly that the polymer chain conformation at rest is not modified within a range of ±5% whatever the degree of confinement by the filler (managed by the NP loading) and whatever the degree of interaction between the NP and the matrix (the nature and amount of additives). This is the first conclusion of the present study which is line with several previous conclusions on model systems [27,28,29,31,32,33].

Faraday Discussions Accepted Manuscript

The cross-linking of the samples does not modify the NP assembly and the chain conformation. Then one can directly compare the pure chain deformation with the one in presence of filler, coating (Figure 11b) and coupling agent (Figure 11c). We also computed the theoretical phantom network model deformation [48] for a functionality f=4 for comparison. On can see that the pure polymer is less deformed compared to the theoretical prediction especially in parallel direction. This can be due to the chain relaxation during the deformation that can occur for low cross-linking, namely only 3 crosslinking junction par chains. Surprisingly this effect is reduced in presence of filler for which the chain deformation is closer to the theoretical prediction in parallel direction. Silica cluster can then act as additional cross-linking junctions reducing the chain relaxation and then enhancing the chain deformation. Such conclusions are of a great interest for further understanding of the macroscopic mechanical properties of these materials.

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V Summary and Conclusion

We report a complete study about the interplay between polymer chain conformations with NP assembly in model industrial SBR/silica nanocomposites. By mixing d and h chains, we applied the ZAC method with SANS that enables to extract the polymer chain form factor while matching the silica filler contribution. SAXS/TEM permits to characterize the NP assembly as function of NP loading and dispersing agent, coating and coupling additives. NPs are forming individual clusters at low loading that can connect to form a continuous network at high silica loading. Using RPA analysis, we show that the radius of gyration of the polymer chain remains unchanged -compared to the pure polymer -within a range of ±5% whatever the crosslinking, the degree of confinement induced by NP loading and the degree of interaction between NP and polymer balanced by the nature (coating or coupling) and the amount of the dispersing additives. The extra scattering at low Q, that do not influenced the R g determination, can be attributed to NP cluster whose neutron contrast is enhanced from ZAC condition by trapped h/d chains due to randomly chain adsorption at the surface of the NPs. Under stretching, NP acts as additional cross-linked junction preventing the chains relaxations that are deformed of a larger amplitude than the pure polymer.

  Published on 10 September 2015. Downloaded by University of Pennsylvania Libraries on 12/10/2015 12:59:40. View Article Online DOI: 10.1039/C5FD00130G50mm, thickness = 10mm) and let cast in an oven at constant temperature T Cast = 130 °C under nitrogen flow for 4 hours. During this phase both solvent casting and crosslinking are taking place. This yields dry films of about 1mm of thickness, which do not creep at rest and under moderate traction.Small Angles Neutron Scattering SANSThe composition of the matrix in all samples (54%v. H-SBR + 46%v. D-SBR) is supposed to reach the Zero Average Contrast condition in SANS experiments. In this case, the scattering is directly proportional to the form factor of a single chain. SANS measurements in ZAC condition were performed on two spectrometers. Uncross-linked systems were measured at rest at the Laboratoire Léon Brillouin (LLB Saclay) on the spectrometer PAXE, by varying the wavelength λ and the sample-todetector distance D. The three configurations defined by λ = 4Å / D = 1m, λ = 6Å / D = 5m and λ = 17Å / D = 5m, cover a total Q-range from 3 10 -3 to 5 10 -1 Å -1 . Samples were measured in calibrated quartz cells. The 2D patterns were reduced to 1D spectra I(Q) after radial averaging around the center of scattering. Standard corrections by sample thickness, neutron beam transmission, empty cell signal subtraction, detector efficiency, subtraction of incoherent scattering were applied to get the scattered intensities on absolute scale. Cross-linked systems were measured at the Institue Laue Langevin (ILL Grenoble) on the spectrometer D11. The three configurations defined by λ = 8Å / D = 1,2m, λ = 8Å / D = 5m and λ = 8Å / D = 20m cover a total Q-range from 3 10 -3 to 4 10 -1 Å -1 . Samples were fixed in a home-made traction device allowing to perform measurements at fixed traction rate ζ=L/L 0 , ranging from 1 (sample at rest) to 1.8. The resulting 2D SANS patterns were anisotropic and radially averaged in 30° degree sectors along the parallel and the perpendicular direction of stretching.

  h , P d,h (Q) are the volume fraction, the number of monomeric unit per chain, the volume of monomeric unit and the form factor of the chains respectively for the hydrogenated and for the deuterated SBR polymers. V is the average monomer volume = φ h ν h + φ d ν d . The molecular weight determination with SEC experiments permit to extract the N d,h ν d,h product like M wd,h = dN A N d,h ν d,h

Figure 1 :

 1 Figure 1: SANS curve the pure h/d SBR chain mixtures at the ZAC composition in log-log

Faraday

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Figure 2 :

 2 Figure 2: SANS curves for the silica filled h/d SBR chain mixtures at the ZAC composition without

Figure 3 :

 3 Figure 3: SANS curves for the silica filled h/d SBR chain mixtures at the ZAC composition with (top)

Figure 4 :

 4 Figure 4: SANS curves for the silica filled h/d SBR chain mixtures at the ZAC composition with (top)

Figure 5 :

 5 Figure 5: (top) SANS curves for the pure h/d SBR chain cross-linked (CL) mixtures at the ZAC

Figure 6 :

 6 Figure 6: SANS for cross-linked stretched samples at different elongation ratio ζ=1.3 (brown), 1.5

Figure 7 :

 7 Figure 7: SAXS curves for the different conditions (top) no additive and coating agent (O 1 and O 4 )

Figure 8 :

 8 Figure 8: TEM pictures for h/d blends filled with (a) no additive at 10% vol. of silica, (b) coating agent

Figure 9 :

 9 Figure 9: c(r) on TEM pictures for (red) coating agent O 1 at 10% vol. of silica and (blue) coupling

Figure 10 :

 10 Figure 10: Comparison between effective structure factor S(Q) deduced from SAXS and apparent

Figure 11 :

 11 Figure 11: Reduced radius of gyration of h chains (a) uncross-linked samples as function of the silica

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Page 10 of 29 Faraday Discussions Faraday Discussions Accepted Manuscript Published

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	I/ϕ polymer (cm -1 )	No additive	I.Q² (cm -1 .Å -2 )	1 ) 1% vol. SiO 2 10% vol. SiO 2 15% vol. SiO 2	Faraday Discussions Accepted Manuscript

  Published on 10 September 2015. Downloaded by University of Pennsylvania Libraries on 12/10/2015 12:59:40. but an upturn appears at low Q that disturbs the agreement between the experimental data and the RPA calculation. We performed the RPA analysis by fixing the Flory parameter to 6 10 -4 and fitted the R g of the h chains. Results are reported on the table 1 for the different experimental conditions.

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	I/ϕ polymer (cm -1 ) Coating agent O 1 No additive Coating agent O 4 Coupling agent Si 1/2 Coupling agent Si 1 Table	I.Q² (cm -1 .Å -2 ) 1% vol. SiO 2 Coupling agent Si 1 Pure h/d blend R gh (nm) R gh (nm) 7.00 7.00 7.00 7.05 7.00 7.25 7.00 6.95 7.00 6.75	1 ) 1% vol. SiO 2 10% vol. SiO 2 15% vol. SiO 2 1% vol. SiO 2 10% vol. SiO 2 15% vol. SiO 2 10% vol. SiO 2 R gh (nm) 7.00 6.60 7.00 6.85 6.70	15% vol. SiO 2 R gh (nm) 6.65 6.80 6.75 6.90 6.65	Faraday Discussions Accepted Manuscript

signal

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Table 2 :

 2 Published on 10 September 2015. Downloaded by University of Pennsylvania Libraries on 12/10/2015 12:59:40.Beyond the range of the error bars, one can see that at rest for ζ=1, the R g values of the h chain in presence of filler and additives do not seem to be different to the uncross-linked situation. That suggests that cross-linking does not modify significantly the polymer chain conformation in nanocomposites. We stretched the samples for different elongation rates ζ=1.3, 1.5 and 1.8 and compared the pure chain deformation with the one in presence of silica (1% and 5% Vol.) and 1 equivalent of coating and coupling agents. We show only two cases, the pure h/d and the coating O 1 at 5% vol. of silica. The results are reported in Figure6for two specific averaging directions, parallel (left) and perpendicular (right) to the stretching direction in log-log and Kratky representations. R gh values deduced from the RPA analysis by fixing the χ to the initial blend value equal to 0.

	1 )

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	Additive Coupling agent Si 1/2	Coating agent O 1 Coupling agent Si 1	0.01 0.1 1 10 100 1000 10 4 10 5 10 6	Coating agent O 4 0.01 0.1 I/φ SiO2 (cm -1 ) 0.001 Q (Å -1 ) Coupling agent Si 1 Uncross-linked 1% vol. SiO 2 Cross-linked 1% vol. SiO 2	Faraday Discussions Accepted Manuscript

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	1 1% vol. SiO 2 S (q) effective S (q) apparent O 4 1% vol. SiO 2	O 1 10% vol. SiO 2 O 4 10% vol. SiO 2	O 1 15% vol. SiO 2 O 4 15% vol. SiO 2	Faraday Discussions Accepted Manuscript

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	0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2	0	0.05 Reduced R gh (a)	0.1	0.15 No additives Coating O1 Coating O4 Coupling Si1/2 Coupling Si1 ϕ SiO2	0.2	0.6 0.8 1 1.2 1.4 1.6 1.8	1	1.2 pure h/d para pure h/d perp O1 1% vol. para 1.4 O1 1% vol. perp O1 5% vol. para O1 5% vol. perp phantom network para phantom network perp Reduced R gh (b)	1.6	1.8	ζ	2	0.6 0.8 1 1.2 1.4 1.6 1.8	1	1.2 pure h/d para pure h/d perp Si1 1% vol. para 1.4 Si1 1% vol. perp Si1 5% vol. para Si1 5% vol. perp phantom network para phantom network perp Reduced R gh (c)	1.6	1.8	ζ	2	Faraday Discussions Accepted Manuscript

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Acknowledgments

We thank G. Baeza, A.-C. Genix and J. Oberdisse (LCC, Montpellier) for fruitful discussions. We acknowledge S. K. Kumar (Columbia University) for critical review of this work.

http://rsc.li/fd-upcoming-meetings

Faraday Discussions Accepted Manuscript


Minh Ha-Duong

Affordable and reliable power for all in Vietnam

progress report

Ensure access to affordable, reliable, sustainable and modern energy for all

Our goal is to report how Vietnam is achieving

Outline

• Energy poverty indicators

• Access to grid and electricity use statistics

• Affordability statistics

Energy poverty is

• Not having access to enough energy to meet basic needs

• Spending more than 10 % of income on energy bill • Fuel costs above median, and the bill sinks you below the poverty line

Ferme du moyen âge à Saint Julien aux Bois, France. © la Mairie

Subjectively defined indicator

• Did you suffer from cold last winter?

• In the last month, did your consumption of electricity met the needs of your household?

Data, method and tools

Reference : General statistical office ( 2015 Descriptive data analysis using Python, Pandas, NumPy and Matplotlib.

The households survey details

• Population : Socio-demographics of all household members

• Housing : characteristics, equipments, ownership...

Access and usage

« From the early days [...], when the power system had just been unified,through to today, when more than 95 percent of all households have access to electricity, there is a compelling story.

It is not only a story of policies and institutions, or of financing and tariffs. It is also a story of Vietnam's growth as a nation and as a society built on the unifying force of electrification. » Trunh Hai (2011) in World Bank (2011) In 2014, about 50 % of the households used less than 100 kWh of electricity in the month.

Hoang

In 2014, about 50 % of the households used less than 100 kWh of electricity in the month. In 2014, about 50 % of the households paid less than 2.1% of income for electricity.

In 2014, about 50 % of the households paid less than 2.1% of income for electricity. • In 2010, one out of four households in Vietnam declared that their electricity use was insufficient to meet their needs. That ratio dropped under 3 % in 2014.

Half of the unsatisfied used less than 22 kwh per month

• In 2014, half of the households in Vietnam who declared insufficient electricity used less than 22 kWh per month.

• In 2014, among households using less than 22 kWh per month, only one out of six declared that their needs were not met.

Conclusion SDG7 level 1 complete

• Rural electrification : poster success

• Power supply quality : recently improved

• Transition to clean electricity : who will pay ?

)

  The 1/4/2014 Viet Nam Intercensal Population and Housing Survey: Major Findings. Vietnam: Ministry of Planning and Investment.

  The share of households not relying on the national grid for lightning decreased from 2010 to 2014 in most provinces.

	« Vietnam has gone through a rapid increase in electrification since 1990 where electrification 50 kWh monthly electricity budget
	levels jumped from a pre-policy reform rate of less than 50% in the late 1980s-early 1990s to 1 kWh Charge phone everyday
	77% by 2001 and 96% by 2009. 2 kWh Boil 5 teapots everyday	Basics,
	The Electrification Programme driven by the increased access for 82 million people between 17 kWh Fan, 8 hours per day Vietnamese government has resulted in add up to 7 kWh Lighting, 4 lamps 4 hour per day 27 kWh
	1976 and 2009. 23 kWh Rice cooker OR Television
	One million people primarily in the northern
	mountainous regions of Vietnam are currently without access to electricity. » 50 kWh Fridge Do not fit into
	60 kWh A/C, 2 hours per day	50 kWh budget
	Doan Van Binh, IES (2010)

State and People, Central and Local, Working Together: The Vietnam Rural Electrification Experience

• In 2014, 97.7 % of households in Vietnam used grid electricity for lighting.• In 2014, out of four Vietnamese households, one used less than 50 kWh per month, and another between 50 kWh and 100 kWh.

Affordability Preventing Power Tariffs From Stoking Inflation

Dr. Nguyen Minh Phong, The Saigon Times, Feb 25, 2010

Vietnam to raise power prices from July 1 after inflation slows.


Sylvain Harquel
email: sylvain.harquel@univ-grenoble-alpes.fr.

Lysianne Beynel

Nathalie Guyader

Christian Marendaz

Olivier David

Alan Chauvin

CortExTool: a toolbox for processing motor cortical excitability measurements by transcranial magnetic stimulation

Keywords: EMG processing toolbox, TMS, Cortical excitability, MEP, CSP

Background

Transcranial magnetic stimulation (TMS) is a noninvasive cortical stimulation technique, which is now widely used in both fundamental and clinical research [START_REF] Rossi | Safety, ethical considerations, and application guidelines for the use of transcranial magnetic stimulation in clinical practice and research[END_REF][START_REF] Sandrini | The use of transcranial magnetic stimulation in cognitive neuroscience: A new synthesis of methodological issues[END_REF][START_REF] Lefaucheur | Evidence-based guidelines on the therapeutic use of repetitive transcranial magnetic stimulation (rTMS)[END_REF]. Assessing cortical excitability (CE) on the motor cortex, by estimating the motor threshold (MT), is a mandatory step prior to any TMS experiment. Stimulation intensities are always expressed as a percentage of this threshold, in order to conform to safety guidelines and to normalise the stimulation power between subjects [START_REF] Rossi | Safety, ethical considerations, and application guidelines for the use of transcranial magnetic stimulation in clinical practice and research[END_REF][START_REF] Herbsman | Motor threshold in transcranial magnetic stimulation: The impact of white matter fiber orientation and skull-to-cortex distance[END_REF][START_REF] Lefaucheur | Recommandations françaises sur l'utilisation de la stimulation magnétique transcrânienne répétitive (rTMS) : règles de sécurité et indications thérapeutiques[END_REF]. Motor CE measurement is based on the feature analysis of motor evoked potentials (MEPs), such as amplitude and latency, recorded on the electromyogram (EMG) of the targeted muscle [START_REF] Wassermann | Variation in the response to transcranial magnetic brain stimulation in the general population[END_REF][START_REF] Säisänen | Motor potentials evoked by navigated transcranial magnetic stimulation in healthy subjects[END_REF][START_REF] Littmann | Variability of motor cortical excitability using a novel mapping procedure[END_REF]. CE can also be used as a biomarker of various cognitive states [START_REF] Beynel | Age-related changes in intracortical inhibition are mental-cognitive state-dependent[END_REF] and psychiatric disorders [START_REF] Malsert | Saccadic performance and cortical excitability as trait-markers and state-markers in rapid cycling bipolar disorder: a two-case follow-up study[END_REF][START_REF] Radhu | A meta-analysis of cortical inhibition and excitability using transcranial magnetic stimulation in psychiatric disorders[END_REF], when completed with the evaluation of cortical inhibitory and excitatory circuits through paired pulse stimulations [START_REF] Roshan | Two phases of short-interval intracortical inhibition[END_REF][START_REF] Stagg | Relationship between physiological measures of excitability and levels of glutamate and GABA in the human motor cortex[END_REF] and/or with the measurement of the cortical silent period (CSP) [START_REF] Fuhr | Spinal motor neuron excitability during the silent period after cortical stimulation[END_REF].

Major technical progress has been made in the practice of TMS over the last decade [START_REF] Gugino | Transcranial magnetic stimulation coregistered with MRI: a comparison of a guided versus blind stimulation technique and its effect on evoked compound muscle action potentials[END_REF][START_REF] Kantelhardt | Robot-assisted image-guided transcranial magnetic stimulation for somatotopic mapping of the motor cortex: a clinical pilot study[END_REF][START_REF] Finke | Brain-mapping using robotized TMS[END_REF][START_REF] Grau | Conscious brain-to-brain communication in humans using non-invasive technologies[END_REF]. However, there is still a lack of practical and easy-to-use software solution regarding off-line processing of EMG signals resulting from CE measurements. We therefore developed a new signal processing toolbox CortExTool, dedicated to CE measurement. This toolbox allows automatic processing of MEP data in order to quantify standard CE features in a fast and user-friendly way.

Toolbox description and functionalities

Overview

CortExTool is a free toolbox (under the GNU General Public License v3) implemented in Matlab (The MathWorks Inc., United States). It is based on a simple graphical user interface showing all the toolbox functionalities in a single window (Figure 1A). Every signal processing function (see 2.3, 2.4) is easily accessible via push buttons, lists, check boxes, and menu entries. Functions can be applied either on experimental conditions or on individual trials. Whenever necessary, appropriate data or results are automatically displayed. Results can be obtained both at individual and group level.

Data are processed as follows. The EMG signals are first loaded into the toolbox, then preprocessed using band-pass filters, artefact removal, and MEP/CSP automatic detection. The signal features can finally be viewed and processed in several ways, and/or be exported for further statistical analysis.

Input data

For now, CortExTool is compatible with data coming from Keypoint (Natus Medical Inc., United States), Spike2 (Cambridge Electronic Design Limited, England ) EMG recording systems, and any data exported in European Data Format (EDF and EDF+). Data has to be arranged within one folder per unique subject, containing one subfolder per condition or experimental block (see details in the user-guide documentation).

Preprocessing

Filtering

Data are automatically filtered when loaded in the toolbox using a 5-600 Hz Butterworth band-pass filter, encompassing the frequency range of the EMG activity [START_REF] Komi | EMG frequency spectrum, muscle structure, and fatigue during dynamic contractions in man[END_REF][START_REF] De Luca | Filtering the surface EMG signal: Movement artifact and baseline noise contamination[END_REF].

Automatic artefacts detection

Since the muscular contraction level of the pre stimulation period is highly correlated with some MEP features such as amplitude [START_REF] Darling | Variability of motor potentials evoked by transcranial magnetic stimulation depends on muscle activation[END_REF], it has to be controlled in protocol involving resting condition. Any trial showing a pre-stimulus contraction should then be considered as artefacts. Automatic artefact detection is based on the analysis of the pre-stimulation period, if available in the data. The root mean square power (RMS) of the EMG signal RM S b is computed in the time window preceding the stimulation onset. Any trial whose corresponding RM S b exceeds a certain threshold is then considered as an artefact. This threshold is tunable (500µV by default) and its optimal value depends on the recorded muscle and the EMG system recording parameters used (gain, filters, etc.). Trials can also be manually marked as artefact. Users can finally choose to discard any trial marked as artefact from further analysis.

MEP detection

MEPs are automatically detected and labelled using four markers: beginning (t start ), end (t end ), and local extrema t 1 and t 2 of the MEP (Figure 1B). A cross-correlation function s f * u between the filtered signal s f (t) and an EMG template u(t) representing a typical MEP (see below) is computed. Local maxima m 1:N and their corresponding time indices t m 1:N are then extracted from the cross-correlation function s f * u. Each maximum represents a possible location of the MEP in the EMG signal. Maxima are then sorted in descending order and sequentially processed according to the following algorithm. For the i th maximum:

1. List first local minima and maxima indexes t 1 and t 2 of s(t) surrounding m i 2. Test whether

• |s(t 1 )-s(t 2 )| > A min , • std(s(t mi ± w)) > θ std , • |t 1 -t 2 | ∈ [w ± kw],
with w being the time interval between the corresponding maxima and minima of u, std being the standard deviation operator, and A min corresponding to the minimum amplitude of the MEP (tunable, 50µV by default). θ std and k are the minimum standard deviation and sensibility parameters respectively. These parameters prevent false alarm by checking the width and the surrounding EMG activity level of the detected MEP. Their default values are 20µV and 1(AU ) respectively.

3. If one of those tests is false, iterate i = i + 1 and return to step 1, otherwise stop and set MEP peaks at t 1 and t 2 respectively.

Temporal boundaries of MEP are then found using the calculation of the surrounding standard deviation. Rise and fall of this value (found by thresholding of the first derivative) correspond to the beginning and the end of the muscle contraction.

By default, the toolbox uses a template of the first dorsal interosseous muscle (FDI). This EMG template was generated by averaging 1959 baseline MEPs of the FDI muscle coming from 49 different subjects of our own TMS experiments database (grey insert in Figure 1A -see 3). The algorithm can also handle other templates coming from any other muscles. A simple tool is provided within the toolbox to build other templates from recorded data. As a consequence, the toolbox can be turned into a detector of any particular signal, as long as it can be defined by a stereotypical shape.

CSP detection

CSPs are automatically delimited using two markers: beginning (t start ) and end (t end ) of the induced muscular atonia (Figure 1B). The automated CSP detection algorithm is based on the thresholding of the first derivative of the epoched signal. The CSP is then defined as the duration between t start and t end , for which the first derivative is under this threshold (see [START_REF] Julkunen | Feasibility of automated analysis and inter-examiner variability of cortical silent period induced by transcranial magnetic stimulation[END_REF] for the detailed routine). The threshold value is 22.5µV /ms by default, as proposed by [START_REF] Julkunen | Feasibility of automated analysis and inter-examiner variability of cortical silent period induced by transcranial magnetic stimulation[END_REF], but remains adjustable by users.

Analysis function EMG features

Five features can be extracted from analysed EMGs. The means and standard deviations of the selected features can then be plotted across conditions in the visualisation window (Figure 1C). The features are:

• latency (in ms), expressing the delay between the stimulation onset and t 1 for MEP or t start for CSP,

• duration (in ms) being the time between t start and t end ,

• amplitude (in µV ) representing the peak to peak amplitude of the signal at t 1 and t 2 ,

• mean RMS (in µV ) calculated as the RMS averaged between markers t start and t end ,

• template likeness (TL) index (in %), defined as: with u being the MEP template, s ′ the re-sampled MEP signal so that the time interval between t 1 and t 2 (see 2.3) equals the one of u, and * the normalized cross-correlation product.

T L = max(s(t min ± w 2 ) ′ * u) × 100 (1)
T L indicates the shape similarity between the analysed MEP and the template. Since the toolbox allows users to generate their own EMG template (see 2.3), this indicator can be turned into a likeness index of any other MEP or signal shapes. For example, this could be useful while trying to detect signal irregularities in a patients vs controls paradigm. Since the surface EMG results from the summation of motor unit action potentials (MUAPs), their synchronization generates high amplitude EMGs shaped like MUAPs [START_REF] Yao | Motor-unit synchronization increases EMG amplitude and decreases force steadiness of simulated contractions[END_REF]. If the template's waveform stays closed to underlying MUAP shapes, as for our default template [START_REF] Kleine | Magnetic stimulation-induced modulations of motor unit firings extracted from multi-channel surface EMG[END_REF][START_REF] Zhou | A study of surface motor unit action potentials in first dorsal interosseus (FDI) muscle[END_REF][START_REF] Luca | Decomposition of surface EMG signals from cyclic dynamic contractions[END_REF], see 2.3 and Figure 2d), T L will be a direct indicator of the MUAPs synchronisation level.

Stimulus-Response curve

Stimulus-Response (S-R) curves [START_REF] Ridding | Stimulus/response curves as a method of measuring motor cortical excitability in man[END_REF]) can be visualized (Figure 1C) once having defined the stimulation powers used to obtain the selected MEPs. A non-linear fitting function (nlinfit matlab function) then estimates the sigmoid curve linking all the MEP amplitudes, using the following parametric function:

f SR (t) = |P 1 | + P 4 1 + e (-P2 * (t-P3))
(2)

where P 1 , P 2 , P 3 and P 4 are the four estimated parameters.

Mapping

The toolbox offers the possibility to plot data coming from motor mapping protocols. The spatial pattern of a particular feature can thus be studied, its mean being plotted as a function of the stimulation site coordinates. After having entered the size of the stimulation grid, a 2D image showing the mean selected features according to each spatial node is displayed (Figure 1C, bottom right). Motor maps can be smoothed using a tunable bi-cubic 2D interpolation. The center of gravity (COG) of the map is calculated and displayed through a black cross.

Group level

Analysis can be conducted on the data of a group of subjects already processed by CortExTool (Figure 1D). Each of the five EMG features (see 2.4) can be analysed across conditions and subjects. Users can define a baseline condition in order to normalize values.

Output data

All the EMG markers are recorded in Matlab file format in the corresponding analysis folder. Extracted features of each trial can also be exported in spreadsheets for complementary statistical analysis using external softwares. At any time, figures displayed in the toolbox can easily be exported.

TMS methods

All the EMGs used for toolbox tests or illustration purposes in this manuscript come from our local database, recorded from healthy volunteers. None had histories of psychiatric illness or neurological disorders, and all were free of any medical treatment likely to modulate cortical excitability. All of them gave a written informed consent. The ethical committee of Grenoble University Hospital (ID RCB: 2012-A00316-37/1 and 2013-A01734-41) approved the studies covering these data.

The data were all recorded at IRMaGE Neurophysiology facility (Grenoble, France) following the same methodology. Electrodes were placed over the subject's first dorsal interosseous muscle in a belly-tendon montage. The EMG signal was amplified (1-10 K), low-pass filtered (6 kHz), and sampled at 12 kHz using a Dantec Keypoint portable system (Natus Medical Inc., United States). TMS was delivered using the MC-B65 or the cool B65-RO butterfly coils plugged in a dual-pulse stimulator (MagProX100 MagOption, MagVenture, Danemark ). The TMS coils were respectively handled by hand or by a TMS-robot (Axilum France). The TMS coil position was continually tracked using a neuronavigation device (TMS Navigator, Localite, Germany). The "hotspot" was defined as the optimal coil position to elicit the greatest motor evoked potential (MEP) in the contralateral muscle. Resting motor threshold (rMT) was then defined using the "threshold hunting" method [START_REF] Bibliography Awiszus | TMS and threshold hunting[END_REF] as the minimal stimulation intensity to evoke a 50µV MEP in fifty percent of trials. MEPs were finally recorded while subjects were at rest following various experimental conditions (depending on the study) including baseline, paired-pulse protocols, CSP measurements, and motor mapping procedures.

Evaluation of MEP detection and Template likeliness calculation

We evaluated the performance of our automated MEP detection algorithm and the TL index calculation on simulated data. First, 3000 EMG signals were randomly generated using six different SNR ratios between the MEP template u and a white noise colored in the frequency range of typical EMG signals (50 -400Hz). Time onset and amplitude of the MEPs were identical for all the generated EMGs. Secondly, 5000 random MEP shapes were computed using a 4-Gaussian mixture model. Several parameters of the model were constrained in order to simulate realistic MEPs.

Figure 2a shows the results of MEP detection on noisy EMG data. The number of correct amplitude and latency estimations increases quickly with the SNR ratio, exceeding 50% between 10 and 15 dB, and 90 % between 20 and 25 dB. Our algorithm has a good performance, since typical recorded MEPs present SNR ratios high above 20-25 dB. The results of TL estimation on simulated MEPs are shown in Figure 2b. As expected, MEPs get a higher TL score when their shape is closest to the template's one. Having more than 2 Gaussians with a non-zero amplitude is penalizing (T L < 80%), whereas MEPs showing the typical asymmetry between the first and second peaks get high values (T L > 90%). Figure 2c shows two examples of real MEPs presenting TL indexes of 61 and 99%.

Conclusion

CortExTool is an efficient and easy-to-use tool to process and visualize EMG data coming from CE measurements by TMS. It offers users both fully automated and tunable processing pipelines, allowing the detection of MEPs and CSPs, the extraction of their features, and the display of results on both individual and group levels.

The toolbox is freely available on the corresponding author webpage. User-guides and regular updates are available on the corresponding author webpage. Ongoing developments of the toolbox include the handling of more input formats, the integration of statistical tests, and the software migration to a fully open-source programming language.

Conflict of interest

All authors declare no conflict of interest. All authors have approved the final article.

Figure 1 :

 1 Figure 1: CortExTool General User Interface (GUI) and its main functionalities. A -CortexTool GUI. Bottom right grey insert: default FDI template u(t). B -Preprocessing panel: results of automatic MEP (top) and CSP (bottom) detection.Markers tstart and t end are displayed in green, while t 1 and t 2 are coloured in red and blue respectively (see 2.3). C -Results panel, from top left to bottom right: means of the selected feature (here, TL index) across conditions, S-R curve, and surface map. D -Group level panel, from top to bottom: GUI, plot of peak-to-peak amplitude means in percentage of the defined baseline across conditions.

Figure 2 :

 2 Figure 2: Results of automated MEP detection and evaluation of Template likeliness. A.1 -Percentage of trials correctly estimated in terms of amplitude and latency across SNR values. A.2 -Examples of simulated MEPs for SNRs of 5, 15, and 25 dB. B -Gaussian mixture models sorted by TL. Individual boxes show examples of simulated data for TL indexes of 30, 60, and 95%. C -Examples of real MEPs showing TL indexes of 61 and 99%. D -FDI MEP template u(t) used by default.

  7. Authors contributionS.H. designed the study, wrote the ethical protocol, recorded TMS data, developed and tested the toolbox, wrote the article. L.B. wrote the ethical protocol, recorded TMS data, tested the toolbox. N.G. recorded TMS data, tested the toolbox. C.M. designed the study, wrote the article. O.D. designed the study, wrote the article. A.C. designed the study, recorded TMS data, tested the toolbox, wrote the article.

Acknowledgements

This work was partly funded by the French program "Investissement d'Avenir" run by the 'Agence Nationale pour la Recherche' ; grant 'Infrastructure d'avenir en Biologie Santé -ANR-11-INBS-0006.


Laetitia Giraldi

Pierre Lissy

Clément Moreau

Jean-Baptiste Pomet

Controllability of a bent 3-link magnetic microswimmer

In this paper, we focus on a variant of a 3link magnetic microswimmer which consists of three rigid magnetized segments connected by two torsional springs. In particular, we assume that one of the springs is twisted so that the swimmer is not aligned at rest. By acting on it with an external magnetic field, the swimmer twists and moves through the surrounding fluid. By considering the external magnetic field as a control function, we state a local partial controllability result around the equilibrium states. Then, we propose a constructive method to find the magnetic field that allows the swimmer to move along a prescribed trajectory. Finally, we show numerical simulations in which the swimmer moves along a prescribed path.

I. INTRODUCTION

At a microscopic scale, swimming in water or another similar fluid is a very different matter from the macroscopic one. Indeed, micro-swimmers face a very small Reynolds number (around 10 -6 ), which means that the intensity of inertial forces is negligible towards those of viscous ones. Due to promising perspectives of medical micro-robots performing delicate tasks inside the human body, interest in the study of micro-swimmers has been recently growing.

The shapes and propulsion techniques of these new robots could be inspired by biology, since micro-organisms such as sperm cells or bacterias developed efficient ways to move through a surrounding fluid (see [START_REF] Peyer | Bio-inspired magnetic swimming microrobots for biomedical applications[END_REF]). One direction of research is to use chemical reactions inside the micro-robot to drive it (see [START_REF] Mirkovic | Fuel for thought: chemically powered nanomotors out-swim nature's flagellated bacteria[END_REF]). Another technique consists in using an external magnetic field to drive a magnetized swimmer (see [START_REF] Gao | Cargo-towing fuel-free magnetic nanoswimmers for targeted drug delivery[END_REF], [START_REF] Dreyfus | Microscopic artificial swimmers[END_REF], [START_REF] Ghosh | Controlled propulsion of artificial magnetic nanostructured propellers[END_REF]).

In this paper, we focus on this type of propulsion, applied on a simple model of micro-swimmer consisting on three magnetized segments linked by elastic joints. Since the swimmer is supposed to be small, the hydrodynamic interaction between the swimmer and the fluid can be modeled by the local drag approximation of Resistive Force Theory introduced in [START_REF] Gray | The propulsion of sea-urchin spermatozoa[END_REF]. Such models, with different numbers of segments, have been studied for instance in [START_REF] Gutman | Simple model of a planar undulating magnetic microswimmer[END_REF] and [START_REF] Alouges | Selfpropulsion of slender micro-swimmers by curvature control: N-link swimmers[END_REF], in which the authors show that sinusoidal magnetic fields allow the swimmer to move forward in a prescribed direction.

In [START_REF] Giraldi | Local controllability of the two-link magneto-elastic micro-swimmer[END_REF], the authors show a local controllability result for the 2-segment model around its straight position. In this paper, we focus on a 3-segment magnetized micro-swimmer, under the assumption that it is not aligned at its equilibrium. By considering the external magnetic field as a control function, we study how to control the position of the swimmer without prescribing any constraints on the orientation and shape of the swimmer, i.e., we state a local partial controllability result. Then, we develop a constructive method to find a magnetic field such that the robot can move along prescribed paths.

The paper is organized as follows. In Section II, we detail the dynamics of the model and state the main result of local partial controllability for the bent swimmer. In Section III, we describe a practical method that explicitly compute the magnetic field to make the swimmer follow some prescribed trajectory, as soon as the swimmer does not go through its aligned position. Using the latter procedure, we give in Section IV some numerical simulations with leads to control the swimmer along some prescribed trajectories. Finally, Section V is dedicated to some perspectives of this work.

II. MICROSWIMMER MODEL AND CONTROLLABILITY ISSUES A. Formulation of the Problem

We follow the notations, assumption and modelisation introduced in [START_REF]Can magnetic multilayers propel artificial microswimmers mimicking sperm cells?[END_REF] and [START_REF] Giraldi | Local controllability of the two-link magneto-elastic micro-swimmer[END_REF]. In the present paper, we focus on a micro-swimmer consisting on 3 rigid magnetized segments connected by two torsional springs with stiffness κ, subject to an external uniform in space magnetic field H. The 3 segments, called S 1 ,S 2 and S 3 , have same length , same hydrodynamic drag coefficients ξ and η, and respective magnetic moments M 1 , M 2 and M 3 . The choice of the numerical values for the parameters will be detailed later in Section IV (see Table I). The swimmer can move in the 2d-plane defined by the vectors e x and e y . Let us define e z = e x × e y . Let x = (x, y) be the coordinates of the end of S 1 , we call θ the angle between (Ox) and S 1 , and α 1 and α 2 the angles between S 1 and S 2 and between S 2 and S 3 . The swimmer is then completely described by 5 variables (x, y, θ, α 1 , α 2 ) where the pair (x, y) represents the position of the swimmer, θ its orientation and the pair (α 1 , α 2 ) its shape. Let us also define the moving frames associated to S i for i = 1, 2, 3 as (e i, , e i,⊥ ). All the geometrical parameters are gathered in Figure 1).

Let us describe briefly the forces applied to the robot. 1) Elasticity: The torsional springs which connect the swimmer segments exert a torque T el proportional to the shape angles α 1 and α 2 . Thus, the torque T el 2 exerted on S 2 is given by T el 2 = κα 1 e z and the torque T el 3 exerted on S 3 is given by T el 3 = κ(α 2 -α 0 )e z , where with α 0 ∈ (-π, π). Here, the specificity of the swimmer we study is that the spring that relies S 2 and S 3 is at rest when α 2 = α 0 . Hence, if α = 0, the springs tend to get the swimmer back to a bent shape, in which S 1 , S 2 are aligned and the angle between S 2 and S 3 is equal to α 0 (see Figure 2). 2) Hydrodynamics: Since the swimmer is assumed to be immersed in a fluid, hydrodynamic forces and torques derive from their interaction. According to the Resistive Force Theory (see [START_REF] Gray | The propulsion of sea-urchin spermatozoa[END_REF]), we assume that the drag force per unit length intensity is proportional to the velocity and to the hydrodynamics coefficients ξ and η. Let x s be a point on one of the segments S i . Its velocity u i (s) is given in the moving frame (e i, , e i,⊥ ) by u i (x s ) = u i, e i, + u i,⊥ e i,⊥ . The drag force exerted on this point is then given by

f i (x s ) = -ξ i u i, e i, -η i u i,⊥ e i,⊥ .
Let us integrate to obtain the total force F h i exerted on S i :

F h i = Si f i (x s )dx s .
Moreover, given a point x 0 , the drag torque for S i with respect to x 0 takes the form

T h i,x0 = Si (x s -x 0 ) × f i (x s )dx s .
Hydrodynamic drag effects are resistant : they oppose to the swimmer's movement. Then, without a magnetic field, the swimmer tends to its equilibrium bent shape.

3) Magnetism: We assume that we apply a uniform timevarying external magnetic field H(t) in the fluid around the swimmer. Here, this magnetic field is assumed to be the control function and in the following, we express H(t) such that the robot can move along a prescribed trajectory. We choose to decompose H in the moving frame associated to S 1 : H(t) = e 1, + H ⊥ e 1,⊥ . The magnetic field exerts a torque T m i on S i which is proportional to its magnetization coefficient M i :

T m i = M i e i, × H.

4) Dynamics equations:

The swimmer is considered sufficiently small to be at low Reynolds number regime, so that inertia may be neglected (see [START_REF] Purcell | Life at low Reynolds number[END_REF] for further considerations on low Reynolds number swimming). We apply Newton's second law to the system {S 1 + S 2 + S 3 } : the total force applied to the system is zero, and so is the total torque with respect to x. Same holds for the subsystems {S 2 + S 3 } and {S 3 }, with torques computed respectively with respect to S 2 end and S 3 end. It gives the following system of equations : (1) This system gives five scalar equations by projecting the first line on (Ox) and (Oy) and the last three on (Oz). After computing the different contributions, the system takes the form

           F h 1 +F h 2 +F h 3 = 0 T h 1,x +T h 2,x +T h 3,x +T m 1 +T m 2 +T m 3 = 0 T h 2,x 2 +T h 3,x 2 +T m 2 +T m 3 + T el 2 = 0 T h 3
M (α 1 , α 2 )R -θ Ż = Y, (2)
with

Z = x y θ α 1 α 2 T , R θ =   r -θ = cos θ sin θ -sin θ cos θ 0 0 I 3   and Y =           0 0 H (M2 sin α1+M3 sin (α1+α2)) -H ⊥ (M1+M2 cos α1+M3 cos (α1+α2)) -κα1+H (M2 sin α1+M3 sin (α1+α2)) -H ⊥ (M2 cos α1+M3 cos (α1+α2)) -κ(α2-α0)+H M3 sin (α1+α2)-H ⊥ M3 cos (α1+α2)          
.

M is a matrix that depends only on α 1 and α 2 .

Remark 1: Up to a rotation matrix that can be eliminated by a changing of basis, the dynamics only depends on the shape state variables α 1 and α 2 : the problem is invariant by any translation or rotation.

Remark 2: If the magnetic field is supposed to be zero, one can see that the equilibrium points are of the form (x, y, θ, 0, α 0 ) with (x, y, θ) ∈ R 3 .

Straightforward computations show that the determinant of M remains negative for all (α 1 , α 2 ), so M is invertible and we can rewrite the system (2) as a nonlinear control system given by

R -θ Ż = F 0 + H (t)F 1 + H ⊥ (t)F 2 , (3)
where F 0 ,F 1 and F 2 are combinations of the third, fourth and fifth columns of M -1 , denoted respectively in what follows by X 3 ,X 4 and X 5 :

F 0 = -κ(α 1 X 4 + (α 2 -α 0 )X 5 ); F 1 = (M 2 sin α 1 + M 3 sin (α 1 + α 2 ))(X 3 + X 4 ) +M 3 sin (α 1 + α 2 )X 5 ; F 2 = -M 1 X 3 -(M 2 cos α 1 + M 3 cos (α 1 + α 2 ))(X 3 + X 4 ) -M 3 cos (α 1 + α 2 )X 5 .

B. Partial Controllability and Small-Time Local Controllability

Let us remind that our aim is not to control either only the position of the swimmer or both its position and its orientation, without taking care of its shape. This type of problem is a partial controllability problem, or Π p -controllability problem (see [START_REF] Duprez | Contrôlabilité de quelques systèmes gouvernés par des équations paraboliques[END_REF]). Let us define Π p as the projection operator given by

Π p : R p × R n-p → R p (y 1 , y 2 ) → y 1 ,
where n is the state space dimension for the considered control system and 1 ≤ p ≤ n. Let us define the Π pcontrollability and state the classical Kalman condition for linear systems. Definition 1: Let (S) be the linear system of ordinary differential equations

(S) : ẏ = Ay + Bu in [0, T ] y(0) = y 0 , with y 0 ∈ R n , A ∈ M n (R), B ∈ M n,m (R), and u ∈ L 2 ([0, T ], R m ) the control. (S) is Π p -controllable at time T
if for all y 0 ∈ R n and y T ∈ R p , there exists u such that the solution of (S) verifies

Π p y(T ; y 0 , u) = y T . Theorem 1: The system (S) is Π p -controllable at time T if and only if Ker(K T Π T p ) = {0},
where K is the Kalman matrix given by K = B AB A2 B . . . A n-1 B . In other terms, (S) is Π p -controllable if and only if the submatrix of K consisting of the p first rows of K is of maximal rank p.

For nonlinear systems, global controllability results such as above are often hard to obtain. Hence, we aim for local results, such as small-time local partial controllability (abbreviated as STLPC). Let (N L) be the linear system of ordinary differential equations

(N L) : ( ẏ1 , ẏ2 ) = f (y 1 , y 2 , u),
where

y 1 ∈ L 2 ([0, T ], R p ), u ∈ L 2 ([0, T ], R n-p ), u ∈ L 2 ([0, T ], R m ) and f : R p × R n-p × R m verifies the condition of the global Cauchy-Lipschitz Theorem.
Definition 2: Let (y e 1 , y e 2 , u e ) ∈ R p × R n-p × R m be an equilibrium of a control system (NL). The control system (NL) is small-time locally partially controllable at (y e 1 , y e 2 , u e ) with respect to y 1 if for every > 0, there exists a real number η > 0 such that, for every (y 0 1 , y 0 2 , y f 1 ) ∈ B η (y e 1 )×B η (y e 2 )×B η (y e 1 ), there exists u ∈ L 2 ([0, ] → R m ) such that (y 1 , y 2 ) verifies system (NL),

(i) ∀t ∈ [0, ], |u(t) -u e | ≤ ; (ii) y 1 ( ) = y f 1 .
An immediate application of the inverse mapping theorem enables us to obtain the following usual sufficient condition for the STLPC around an equilibrium for a nonlinear system.

Theorem 2: The nonlinear control system (NL) is STLPC at an equilibrium if its linearized control system around this equilibrium is Π p -controllable for some time T > 0.

This last theorem justifies that we study the linearized system of (3) to get partial controllability around the equilibrium.

C. Local partial controllability result

In the following, we prove that the position of the 3link magnetic swimmer can be partially controlled by the external magnetic fields when the bent swimmer is close to its equilibrium. The main result states as follows.

Theorem 3: If α 0 = 0, then system (3) is STPLC with respect to (x, y) around any equilibrium point.

Proof. We only need to prove the result for the particular equilibrium point O = (0, 0, 0, 0, α 0 ). Indeed, according to Remark 1, solutions of (3) are invariant under the transformations

x y , θ, α 1 , α 2 , H , H ⊥ → Rθ x + x y + ȳ , θ + θ, α 1 , α 2 , H , H ⊥ ,
so if the result holds for O, it may be carried to an arbitrary equilibrium point. Following Theorem 2, we look at the linearized system around the equilibrium point O, which is given by

Ż = AZ + BH, (4)
where A is the Jacobian of Z → F 0 (Z) at O and the two columns of B are F 1 (O) and F 2 (O). Since we are interested in controlling on the position (x, y), we are looking for the Π 2 partial controllability of the swimmer, hence, according to Theorem 1, we have to check that the first two rows of the Kalman matrix

K = B AB A 2 B A 3 B A 4 B
give a matrix of rank 2. If we look at the 2 × 2 submatrix given by 1 the first two entries of the first columns of B and AB, a tedious but straightforward computation enables us to get the determinant of this matrix :

108M 2 3 κ(-9ηξ(19η+54ξ) cos α0-2Ξ(η+2ξ)) sin 3 (α0) L 7 η 2 (η 2 +34ηξ+28ξ 2 -(η 2 -11ηξ+28ξ 2 ) cos(2α0)) 2 , (5)
where

Ξ = (η 2 + 19ηξ + 7ξ 2 -(η 2 -8ηξ + 7ξ 2 ) cos(2α 0 )).
Along with the hypothesis α 0 = 0, the straightforward inequalities

η 2 + 19ηξ + 7ξ 2 > (η 2 -8ηξ + 7ξ 2 ) cos(2α 0 )
and

η 2 + 34ηξ + 28ξ 2 > (η 2 -11ηξ + 28ξ 2 ) cos(2α 0 )
show that the numerator and denominator of ( 5) are respectively the opposite sign of α 0 and positive. Therefore, the determinant is nonzero and its associated submatrix has rank 2. According to Theorem 1, the system (4) is partially controllable. We conclude by applying Theorem 2. Remark 3: None of the above applies for a non-bent 3link swimmer, i.e. if α 0 = 0. Indeed, the proof does not work in this case because the numerator of the determinant ( 5) is zero. Furthermore, a straightforward computation yields

A =       0 0 0 0 0 0 0 0 * * 0 0 0 * * 0 0 0 * * 0 0 0 * *       and B =       0 0 0 * 0 * 0 * 0 *      
, where stars stand for possibly nonzero entries, hence the first row of the Kalman matrix is zero, the linearized control system is not controllable, and Theorem 2 does not apply.

We do not know whether ( 3) is STPLC around its equilibrium if α 0 = 0, but non controllability of the linearized system in this case indicates that the non-bent 3-link swimmer is harder to control. It is why we focus here on the case of bend swimmer.

III. EXPLICIT PARTIAL CONTROL

In this Section, we describe a method to make the swimmer's position (x, y) follow an arbitrary trajectory, while its orientation and its shape are not prescribed, and we explain the theoretical difficulties that arise.

Let us focus on the two first lines of the system (3):

r -θ ẋ ẏ = F 0x + H F 1x + H ⊥ F 2x F 0y + H F 1y + H ⊥ F 2y . (6)
We denote by D(α 1 , α 2 ) the determinant of the 2 × 2 matrix F 1x F 2x F 1y F 2y ; it depends only on the state variables α 1 , α 2 .

It is clear that D vanishes at straight positions (D(0, 0) = 0). Restricting D to the open square K = (-π, π) × (-π, π) (values of (α 1 , α 2 ) outside K are not physical: the segments would then overlap), and using the values of the parameters used for our numerical simulations (see Table I), the plot in Figure 3 shows that it vanishes only at (0, 0). Let T > 0. Let f and g be two functions of class C 1 on [0, T ]. We require that the swimmer follows exactly the trajectory parameterized by f and g, i.e.

∀t ∈ [0, T ], x(t) y(t) = f (t) g(t) . ( 7
)
The problem is to find the control functions H , H ⊥ that achieve this goal. Differentiating (7) and using (6), we get

F 0x + H F 1x + H ⊥ F 2x F 0y + H F 1y + H ⊥ F 2y = r θ f (t) g (t)
.

Hence, at each time t, H ⊥ and H must solve the 2 × 2 linear system of equations It has a unique solution if the determinant D(α 1 (t), α 2 (t)) does not vanish.

F 1x F 2x F 1y F 2y H H ⊥ = -F 0x -F 0y + r θ f (t) g (t) . (8)
The functions H and H ⊥ solving the system (8) depend on f , g and the state variables θ, α 1 , α 2 , x and y. Following [START_REF] Isidori | Nonlinear control systems[END_REF]Chapter 7], this can be re-formulated in terms of "relative degree" and non-interactive control. Considering the control system (3) with two outputs f = x and g = y, and inputs H , H ⊥ , it has vector relative degree {1, 1} when D = 0 and the above mentioned expression of H , H ⊥ as functions of f , g , θ, α 1 , α 2 , x, y is a feedback transformation that solves the "Noninteracting Control Problem" [11, Section 7.3], i.e. it produces a control system with new controls f and g where the control f acts on the output x only and the control g acts on the output y only, as can be seen by computing ẋ = f and ẋ = g . Clearly, for any functions f (t), g(t), if such a feedback transformation exists and if f (t), g (t) are taken to be the time derivatives of f (t), g(t) and if x(0) = f (0) and y(0) = g(0), then (6) holds. Note that, because of the intermediary state feedback, the expression of H , H ⊥ as a function of time is obtained implicitly after solving the closed-loop ODE.

This method fails if the swimmer has to go through the straight configuration, where one cannot invert [START_REF] Giraldi | Local controllability of the two-link magneto-elastic micro-swimmer[END_REF]. Since we cannot guarantee in general that the straight confuguration will not be encountered, all we can do is try numerically, see next section where we see that both may happen: either the method works until the end of the trajectory or the alignment occurs and the controls blow up. This does not contradict Theorem 3 that is local and only concerned with short trajectories; Theorem 3 is however stronger since it applies even if D vanishes at the equilibrium (see the footnote on the thuird page).

IV. NUMERICAL SIMULATIONS

In this part, we show numerical result obtained by using the approach given in the previous Section, in the case where D vanishes only at α 1 = α 2 = 0. Table I gives the actual values for the parameters used in this Section (the values have been chosen according to [START_REF] Alouges | Selfpropulsion of slender micro-swimmers by curvature control: N-link swimmers[END_REF]). We have used Matlab and more particularly the function ode23t to integrate the system (3).

The routine takes functions f and g as an input, and returns the trajectory of the robot, including its orientation and shape, and the required controls H and H ⊥ . Figure 4 shows how it can follow a straight line. When the swimmer is closed to the aligned position, according to the system (8), the magnetic field goes to infinity, then the simulation stops and we cannot follow the entire prescribed trajectory. It is the case in Figure 5 and Figure 6.

Figure 7 shows a small circle trajectory that the swimmer is able to follow. Figure 8 presents its orientation, its shape and the external magnetic fields along this experiment. Finally, Figure 9 shows a more complex trajectory that leads to follow a path while remaining close to a global direction. Figure 10 shows the associated controls and angular variables during it.

V. CONCLUSION AND PERSECTIVES

Here, we proved local partial controllability of the "bent" swimmer. The same result for the non bent 3-link swimmer is still an open question, the present work indeed stemmed out from trying to prove STPLC via the return method of Coron [START_REF] Coron | Control and nonlinearity[END_REF]Chapter 6], rather than Theorem 2 (linear test).

Trying to go beyond local for the bent swimmer, we described a method to drive the position of the swimmer along a given trajectory. It fails if the swimmer passes through the straight shape, and we cannot ensure that this will not occur. In our numerical experiments, we observe that the method applies to some trajectories but that in other cases one has to go through the straight shape and the controls blow up, evidencing that this straight shape still represents a serious barrier to maneuverability, even though bending the swimmer provides linear controllability at the equilibrium.

Another perspective, under our investigation, is to conduct further numerical study, and add energetic aspects, in order to find a magnetic field which allows the swimmer to move close to a prescribed path by minimizing the kinetic energy of fluid-swimmer system (see [START_REF] Lighthill | Flagellar hydrodynamics[END_REF]).

Fig. 1 .

 1 Fig. 1. Model used for the 3-link microswimmer.

Fig. 2 .

 2 Fig. 2. The bent swimmer at its equilibrium.

Fig. 3 .

 3 Fig. 3.Aspect of D(α 1 , α 2 ) with numerical values from TableI. It vanishes only at (0, 0).

Fig. 4 .

 4 Fig. 4. Snapshots of the swimmer (in black) following a straight line (in blue). The blue plain line indicates the path already done, and the blue dotted line indicates the remaining path. The scale is in micrometers.

Fig. 5 .

 5 Fig. 5. An example of a bad case in which the swimmer is aligning.

Fig. 6 .

 6 Fig. 6. State variables and controls along the straight line. At the end of the time, we can see that α 1 and α 2 are going to zero and that H goes to infinity.

Fig. 7 .

 7 Fig. 7. Snapshots of the swimmer (in black) following a circle (in blue) and going back to its initial position. The blue plain line indicates the path already done, and the blue dotted line indicates the remaining path.

Fig. 8 .

 8 Fig. 8. State variables and controls along the circle trajectory.

Fig. 9 .

 9 Fig. 9. Snapshots of the swimmer (in black) following a complex trajectory (in blue). The blue plain line indicates the path already done, and the blue dotted line indicates the remaining path. The red line indicates the global direction that we want to follow.

Fig. 10 .

 10 Fig. 10. Angles and controls with respect the time (in sec.) during the trajectory described in Fig. 9. The middle plot shows the components H and H ⊥ of the control, whereas the bottom plot shows the Hx and Hy components, in the reference basis.

TABLE I NUMERICAL

 I VALUES USED FOR THE PARAMETERS

It would be more natural to chose the two columns of B. Indeed, for almost all values of the parameters, the two first lines of B compose an invertible

× 2 matrix (its determinant is called D(0, α 0 ) in Section III). For a precise set of parameters, however, this matrix has rank 1 only and one has to choose the first columns of B and AB instead; since this choice works in general, it is the one we make in the proof.