Tobias Lange

Thibault Charpentier

Frédéric Gobeaux

Sophie Charton

Fabienne Testard

Antoine Thill
email: antoine.thill@cea.fr

The phosphonic acid moiety is commonly used as an anchoring group for the surface modification of imogolite. However, the impact of the reaction on its structure has never been clearly analyzed before.

We study the reaction of imogolite and decylphosphonic acid by combining infrared spectroscopy, X-ray scattering, scanning electron microscopy, transmission electron microscopy and solid-state nuclear magnetic resonance spectroscopy. Instead of a surface functionalization, we observe the formation of a lamellar phase interconnected with imogolite bundles. Although we find no evidence for grafted imogolite tubes, we observe the expected dispersion characteristics and stabilization of water in toluene emulsions described in literature. Based on the surface chemistry of imogolite we propose an explanation for the observed reactivity and link the structural features of the obtained composite material to its dispersibility in toluene and its observed properties at the toluene-water interface.

Partial Transformation of Imogolite by Decylphosphonic Acid Yields an Interface Active Composite Material Introduction

Surface modification is often a mandatory step to explore the properties and potential applications of inorganic nanoparticles. Different approaches and compounds were developed to meet the specific requirements in diverse domains (polymer reinforcement, nanophotonics, catalysis, batteries, nanomedicine, etc.). [START_REF] Kango | Surface modification of inorganic nanoparticles for development of organic-inorganic nanocomposites-A review[END_REF][START_REF] Xavier | Advances in optoplasmonic sensors -combining optical nano/microcavities and photonic crystals with plasmonic nanostructures and nanoparticles[END_REF][START_REF] Avvakumova | Biotechnological approaches toward nanoparticle biofunctionalization[END_REF] However, the potential influence of the used agent on the resulting particle shape, size and integrity keeps the topic challenging. [START_REF] Guerrero | Organically modified aluminas by grafting and sol-gel processes involving phosphonate derivatives[END_REF] This is especially relevant for the surface modification of delicate structures like the one of imogolite nanotubes.

Imogolite is a nanotubular clay, characterized by a diameter between 2 nm and 3 nm and a general formula unit (OH) 3 Al 2 O 3 Si(OH). [START_REF] Thill | Characterisation of Imogolite by Microscopic and Spectroscopic Methods[END_REF] The tube wall consists of a single gibbsite-like sheet, which is linked to isolated silanol groups at the internal surface. [START_REF] Cradwick | Imogolite, a Hydrated Aluminium Silicate of Tubular Structure[END_REF] The gibbsite-like external surface is covered by Al 2 (µ-OH)groups.

Various approaches and compounds have been studied to modify it, e.g. catechol [START_REF] Miura | Nanocomposites composed of poly(epsilon-caprolactone) and oligocaprolactone-modified imogolite utilizing biomimetic chelating method[END_REF] , phosphates [START_REF] Ma | Surface Modification of Individual Imogolite Nanotubes with Alkyl Phosphate from an Aqueous Solution[END_REF][START_REF] Ma | Preparation and properties of PVC/PMMA-g-imogolite nanohybrid via surface-initiated radical polymerization[END_REF][START_REF] Ma | Poly(methyl methacrylate) grafted imogolite nanotubes prepared through surface-initiated ARGET ATRP[END_REF][START_REF] Yamamoto | Preparation and properties of [poly(methyl methacrylate)/imogolite] hybrid via surface modification using phosphoric acid ester[END_REF] , silane [START_REF] Qi | Surface-modified imogolite by 3-APS-OsO 4 complex: Synthesis, characterization and its application in the dihydroxylation of olefins[END_REF] , pyrrole [START_REF] Lee | Conducting properties of polypyrrole coated imogolite[END_REF] , and stearic acid [START_REF] Yamamoto | Preparation of a novel (polymer/inorganic nanofiber) composite through surface modification of natural aluminosilicate nanofiber[END_REF] . However, most of the studies aim to graft octadecylphosphonic acid or other phosphonic acids onto imogolite. [START_REF] Yamamoto | Preparation of a novel (polymer/inorganic nanofiber) composite through surface modification of natural aluminosilicate nanofiber[END_REF][START_REF] Yamamoto | Surface modification of aluminosilicate nanofiber "imogolite[END_REF][START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF][START_REF] Yah | Molecular Aggregation States of Imogolite/P3HT Nanofiber Hybrid[END_REF][START_REF] Yah | Molecular Aggregation State and Electrical Properties of Terthiophenes/Imogolite Nanohybrids[END_REF][START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF][START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF][START_REF] Li | Dispersing surface-modified imogolite nanotubes in polar and non-polar solvents[END_REF][START_REF] Li | Synthesis of polyamide thin-film nanocomposite membranes using surface modified imogolite nanotubes[END_REF] Yamamoto et al. were the first to study the chemisorption of octadecylphosphonic acid (ODPA) on imogolite. [START_REF] Yamamoto | Preparation of a novel (polymer/inorganic nanofiber) composite through surface modification of natural aluminosilicate nanofiber[END_REF][START_REF] Yamamoto | Surface modification of aluminosilicate nanofiber "imogolite[END_REF] They concluded that ODPA adsorbs to the surface of imogolite. Park et al. modified imogolite with ODPA and tetradecylphosphonic acid (TDPA). [START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF] The product was dispersible in toluene and microscopy pictures showed regular patterns, which were attributed to homogenously grafted imogolite. Bac et al. studied the reaction between ODPA and germanium imogolite. [START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF] They concluded a successful grafting, although they observed the formation of a lamellar product in XRD. Picot et al. used ODPA modified imogolite to stabilize oil in water Pickering emulsions. [START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] All studies demonstrate the reactivity between imogolite and the phosphonic acid moiety. However, the evidence for the presence of grafted phosphonic acid on intact dispersed imogolite is ambiguous. The IR results only reveal the reactivity, but not the identity of the resulting phase. [START_REF] Yamamoto | Preparation of a novel (polymer/inorganic nanofiber) composite through surface modification of natural aluminosilicate nanofiber[END_REF][START_REF] Yamamoto | Surface modification of aluminosilicate nanofiber "imogolite[END_REF][START_REF] Yah | Molecular Aggregation States of Imogolite/P3HT Nanofiber Hybrid[END_REF][START_REF] Yah | Molecular Aggregation State and Electrical Properties of Terthiophenes/Imogolite Nanohybrids[END_REF][START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF][START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF][START_REF] Li | Dispersing surface-modified imogolite nanotubes in polar and non-polar solvents[END_REF][START_REF] Li | Synthesis of polyamide thin-film nanocomposite membranes using surface modified imogolite nanotubes[END_REF] XRD and SAXS identified changes in the organization of the product, but led to different interpretations. [START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF][START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF][START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] Finally, the investigations by microscopy do not have the required resolution to evidence isolated grafted nanotubes with clarity or lack complementary characterization. [START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF][START_REF] Yah | Molecular Aggregation States of Imogolite/P3HT Nanofiber Hybrid[END_REF][START_REF] Yah | Molecular Aggregation State and Electrical Properties of Terthiophenes/Imogolite Nanohybrids[END_REF] A possible alternative reaction path for imogolite is the formation of bulk aluminum phosphonates. It is already known for the intercalation of gibbsite and kaolinite with phenylphosphonic acid (PPA) [START_REF] Gardolinski | On the destruction of kaolinite and gibbsite by phenylphosphonic, phenylphosphinic and phenylarsonic acids: evidence for the formation of new Al compounds[END_REF] , the modification of aluminum oxide by ODPA [START_REF] Gao | Self-assembled monolayers of alkylphosphonic acids on metal oxides[END_REF] and PPA [START_REF] Guerrero | Organically modified aluminas by grafting and sol-gel processes involving phosphonate derivatives[END_REF] , and the modification of germanium imogolite by ODPA. [START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF] Nonetheless, it was shown that choosing favorable conditions prevents the bulk phosphonate formation and enables grafting of alumina by octylphosphonic acid. [START_REF] Lassiaz | Organo-lined alumina surface from covalent attachment of alkylphosphonate chains in aqueous solution[END_REF] To assess if an efficient surface modification of imogolite is achieved, a combination of structural and chemical characterizations is necessary. In the present study, we explore the fate of imogolite during the reaction with alkylphosphonic acid at ambient conditions. We combine Fourier Transform Infra-Red (FTIR) spectroscopy, Small Angle X-ray Scattering (SAXS), Scanning Electron Microscopy (SEM), Transmission Electron Microscopy (TEM) and multinuclear Magic-Angle Spinning Nuclear Magnetic Resonance (MAS NMR) to study the chemical and structural modifications of imogolite.

Materials and Methods

Synthesis of Imogolite

Imogolite was synthesized using an adapted protocol inspired from Farmer et al. [START_REF] Farmer | Synthesis of imogolite: a tubular aluminium silicate polymer[END_REF] Tetraethoxysilane was introduced under stirring into a 2 mM solution of AlCl 3 until a Si/Al ratio of 0.55 was reached. Under vigorous stirring, a 0.1 M NaOH solution was then added at a rate of 8 mmol•min -1 until an OH/Al ratio of 2 was reached. The suspension was then transferred to an oven, where it was kept at 90 °C for 5 days.

Ultrafiltration with a 10 kDa membrane (SpectrumLabs) was then used to concentrate the imogolite dispersion up to a final concentration of 10 g•L -1 . The concentrated dispersion was centrifuged for 1 hour at 10,000 rpm to remove any trace of gibbsite, and then dialyzed against MilliQ water until reaching a residual conductivity of less than 2 µS•cm -1 . The resulting transparent dispersion was stored in a plastic container. Chemicals were obtained from Sigma-Aldrich and used as received.

Modification of Imogolite

In order to study the surface modification of imogolite by decylphosphonic acid (DPA) we adapted the protocols described in literature. [START_REF] Yamamoto | Preparation and properties of [poly(methyl methacrylate)/imogolite] hybrid via surface modification using phosphoric acid ester[END_REF][START_REF] Yamamoto | Surface modification of aluminosilicate nanofiber "imogolite[END_REF][START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF][START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF][START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] The prepared samples are listed in Table 1. For samples A1-A4 and B, 10 mL of aqueous imogolite dispersion (5 g•L -1 ) were diluted with ethanol (CARLO ERBA Reagents, p.a.) under strong stirring. Decylphosphonic acid (DPA, Sigma-Aldrich, 97 %) was dissolved in ethanol and then added to the imogolite dispersion. The resulting mixture was stirred at room temperature and turned whitish during reaction. The dispersed precipitate was collected by centrifugation and washed by three dispersion -centrifugation steps in ethanol. The sediment was finally dried under ambient conditions. For sample A5, DPA was dissolved in an aqueous ethanol solution (66:34, v/v, EtOH:H 2 O) before adjusting the pH with 0.1 mol•L -1 NaOH (Sigma-Aldrich). The remaining procedure is unchanged.

We typically stopped the reaction after one day (samples A1-A5) to characterize the product. In order to obtain a complete reaction, one sample was kept stirring for 18 days (sample B). Sample A2 is a repetition of sample A1, to check the results repeatability for a given set of preparation conditions. Samples A3 and A4 have a smaller P: Al ratio compared to sample A1. Sample A5 was prepared in order to observe the influence of the initial acidity.Table 1 summarizes the different experimental conditions. Fourier Transform Infrared Spectroscopy FTIR Spectra were collected in transmission from 4000 cm -1 to 400 cm -1 with a resolution of 4 cm -1 (Vertex 70, Bruker). KBr pellets were prepared by compressing approximately 1 mg of sample dispersed in 100 mg of KBr.

Small Angle X-ray Scattering SAXS was acquired at 1.542 Å under vacuum with a Xeuss 2.0 apparatus (Xenocs). The intensity is plotted against the scattering vector q = 4π•sin(θ)•λ -1 , with λ being the wavelength and 2θ the scattering angle. The displayed q range from 1.0 nm -1 to 16.0 nm -1 was attained with a single sample to detector distance. The latter was calibrated with tetradecanol and the detector count was normalized by direct beam measurements. Samples were sealed in glass capillaries (diameter 1.5 mm, wall thickness 0.1 mm, WJM-Glas). Standard procedures were applied to subtract background scattering and to normalize the intensities. [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] The lamellar repeat distances were calculated by the position of their 0k0 progression, using

d = 2•π•q -1 .

Magic Angle Spinning Nuclear Magnetic Resonance Spectroscopy

MAS NMR data have been collected on a Bruker Avance II 500WB spectrometer operating at a magnetic field of 11.72 T using ZrO 2 rotors (outer diameter 4 mm) and Bruker HX and HXY CPMAS probes at a spinning frequency of 12500 Hz.

27 Al (I=5/2) MAS NMR spectra were acquired using a short single-pulse excitation of 1 μs (tip angle ~10°, to ensure quantitativeness of the spectra because of the non-homogeneous excitation of all sites with respect to their quadrupolar interaction strength) and a recycle delay of 1 s. Two-dimensional Multiple Quantum MAS (MQMAS, the triple quantum transition was used) were acquired using the three-pulse Zfilter sequence. [START_REF] Massiot | Twodimensional magic-angle spinning isotropic reconstruction sequences for quadrupolar nuclei[END_REF] 31 P (I=1/2) MAS NMR spectra were acquired using a presaturation (90°-tau) N=20 period followed by a recovery delay of 32 s to ensure quantitativeness of the spectra.

Additionally, [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al{ 31 P} Rotational Echo DOuble Resonance (REDOR) experiments were performed to investigate the Al-P spatial proximities. [START_REF] Fyfe | Solid-State Double-Resonance Nmr Experiments Involving Quadrupolar and Spin 1/2 Nuclei[END_REF] The data was processed using an in-house made code. [START_REF] Angeli | Boron Speciation in Soda-Lime Borosilicate Glasses Containing Zirconium[END_REF]

Dispersion of modified Imogolite, Emulsification and Interface Experiments

The modified imogolite was dispersed in toluene (CARLO ERBA, 99.8 %) directly after the washing steps (dispersion -centrifugation), and without prior drying. Emulsions were formed by mixing the dispersed modified imogolite suspension at 0.5 g•L -1 with water in 50/50 v/v proportion. Droplets of water were deposited on the interface between water and the above mentioned dispersion. The emulsions were observed on an Olympus IX81 microscope. High purity water (18.2 MΩ•cm) was used to create emulsions and liquid-liquid interfaces.

Scanning Electron Microscopy

The morphology of imogolite after the reaction with DPA was observed by scanning electron microscopy.

The dispersed product was deposited on a plasma cleaned Si surface and left to dry. Observations were conducted on a Hitachi SU-70 microscope operated at 5 kV.

(Cryo-)Transmission Electron Microscopy

The morphology of synthesized imogolite was observed with cryogenic transmission electron microscopy.

A drop of the solution was deposited on a copper grid covered with a holey carbon film (Quantifoil R2/2) that was previously treated by a plasma glow discharge. Excess of liquid on the grid was blotted out with filter paper and the grid was quickly plunged in liquid ethane to form a thin vitreous ice film. The whole process was performed using a Vitrobot apparatus (FEI Company). Observations were conducted at low temperature (-180 °C) on a JEOL 2010 FEG microscope operated at 200 kV. Images were recorded with a Gatan camera. To observe the morphology of the product dispersed in toluene, the dispersion was deposited on the copper grids and left to dry. The same camera and microscope were used at ambient temperature.

Results and Discussion

Transformation of imogolite by decylphosphonic acid

Imogolite

Imogolite was synthesized according to the protocol described in Materials and Methods. Cryo-TEM observations of the obtained aqueous dispersion (Figure 1a) evidence imogolite tubes in coexistence with globular allophane and proto-imogolite. The typical presence of other objects than imogolite reflects in the dampened SAXS curve (Figure 1b) and is in agreement with literature data on imogolite SAXS patterns. [START_REF] Du | Insights into the formation mechanism of imogolite from a full-range observation of its sol-gel growth[END_REF][START_REF] Amara | Tunable-Diameter, Metal Oxide Nanotubes for Trapping of Organic Molecules[END_REF][START_REF] Picot | Exploring hybrid imogolite nanotubes formation via Si/Al stoichiometry control[END_REF] Fitting the obtained curve with the scattering pattern of a hollow cylinder gives an outer tube radius of 2.8 nm, which is a typical value for synthetic dispersed imogolite.

The reactivity between the suspended particles and DPA is studied for the protocol described in material and methods. The products are characterized by FTIR, SAXS, multinuclear MAS NMR and electron microscopy to investigate the evolution with the reaction process. To identify the occurring transformations, all the prepared samples are systematically compared to pristine imogolite.

FTIR

The FTIR spectrum of imogolite has three main contributions (Figure 2). First, the hydroxyl stretching (3510 cm -1 ) and bending vibrations (1636 cm -1 ) originating mainly from internal water. [START_REF] Liao | Self-supporting thin films of imogolite and imogolite-like nanotubes for infrared spectroscopy[END_REF] Secondly, the Al-O vibrations at 692 cm -1 , 596 cm -1 , 564 cm -1 , 503 cm -1 and 426 cm -1 , which are characteristic for the imogolite local structure (Figure 2b). [START_REF] Thill | Characterisation of Imogolite by Microscopic and Spectroscopic Methods[END_REF] Lastly, the doublet appearing at 988 cm -1 /941 cm -1 , which is related to the asymmetric stretching mode of the isolated SiO 4 tetrahedra and is typical for the tubular shape of imogolite. [START_REF] Cradwick | Imogolite, a Hydrated Aluminium Silicate of Tubular Structure[END_REF][START_REF] Du | Insights into the formation mechanism of imogolite from a full-range observation of its sol-gel growth[END_REF][START_REF] Wilson | Tetrahedral rehydration during imogolite formation[END_REF][START_REF] Creton | Molecular Dynamics Study of Hydrated Imogolite. 1. Vibrational Dynamics of the Nanotube[END_REF][START_REF] Lee | Thermodynamic Control of Diameter-Modulated Aluminosilicate Nanotubes[END_REF] After reaction with DPA, all the characteristic regions of imogolite are retrieved in the FTIR spectra of samples A1-A5. Only the SiO 4 doublet shifts for each of these samples to 994 cm -1 -992 cm -1 / 938 cm -1 -935 cm -1 . For sample B the SiO 4 doublet is not observable, the appearance of the characteristic Al-O region (426 to 692 cm -1 ) changes, and both the intensity and shape of the hydroxyl bands are different.

The appearance of CH 2 and CH 3 vibrations between 2960 cm -1 and 2850 cm -1 and at 1471 cm -1 indicate the presence of alkyl chains in all samples (Figure 2c). Comparing the CH 2 stretching bands of the products to those of DPA reveals no shift for sample B, while a qualitative shift towards higher wavenumbers is observed for samples A1-A5. This shift of the vibration bands for samples A1-A5 indicates slightly less ordered chains than in sample B, for which a similar degree of order as in the all-trans DPA is evidenced. [START_REF] Peng | Lamellar Mesostructured Aluminophosphates: Intercalation of n-Alkylamines into Layered Aluminophosphate by Ultrasonic Method[END_REF] Furthermore, a shouldered peak appears at 1080 cm -1 for samples A1-A5. The shoulder turns into several sharp bands for sample B. This new contribution indicates the formation of P-O-Al bonds in all samples and therefore a successful reaction between imogolite and DPA. Interesting to note is the disappearance of the P=O signal of DPA at 1232 cm -1 after reaction. A possible explanation could be a tridentate binding mode in the formed product. [START_REF] Szillies | Formation and stability of organic acid monolayers on magnesium alloy AZ31: The role of alkyl chain length and head group chemistry[END_REF][START_REF] Yah | Selective modification of halloysite lumen with octadecylphosphonic acid: new inorganic tubular micelle[END_REF] Overall, the spectra of samples A1-A5 show a high content of imogolite and are in good agreement with the findings of Yamamoto et al. for the reaction of ODPA with imogolite, and Bac et al. for the reaction between ODPA and germanium imogolite. [START_REF] Yamamoto | Preparation of a novel (polymer/inorganic nanofiber) composite through surface modification of natural aluminosilicate nanofiber[END_REF][START_REF] Yamamoto | Surface modification of aluminosilicate nanofiber "imogolite[END_REF][START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF] The spectrum of sample B suggests a rather low amount of residual imogolite due to the modification of the typical hydroxyl, Al-O and Si-O bands. It compares well with the spectrum of ODPA-modified imogolite presented by Picot et al. [START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] The presence of several sharp P-O-Al peaks was also evidenced for aluminum organophosphonates and might indicate the presence of such a phase. [START_REF] Chaplais | Novel aluminium phenyl, benzyl, and bromobenzylphosphonates: structural characterisation and hydration-dehydration reactions[END_REF] Although IR demonstrates that a reaction has occurred between imogolite and DPA, it does not provide sufficient information to assess the integrity of the nanotubes after the reaction.

SAXS

The x-ray scattering pattern of imogolite shows a broad peak at 2.76 nm -1 , which is typical of disordered aggregated bundles in the dry powder (Figure 3). Significant modifications occur after reaction with DPA.

For samples A1-A5, three peaks at 2.26 nm -1 , 4.51 nm -1 and 6.71 nm -1 superimpose the scattering curve of imogolite. Additional peaks are present at higher q values but not clearly visible due to noise. The exact position of these reflections varies slightly with the reaction conditions. Sample B shows peaks at 2.11 nm -1 , 4.26 nm -1 , 6.35 nm -1 , 8.49 nm -1 , 9.31 nm -1 , 10.58 nm -1 , 12.73 nm -1 , 13.65 nm -1 and 14.80 nm -1 , while the contribution of imogolite is hardly noticeable. Excluding the peaks at 9.31 nm -1 and 13.65 nm -1 , all the peaks in sample B are consistent with the 0k0 progression of a lamellar phase. For samples A1 and A4, this is not so evident on the first glance, since the first peak of imogolite bundles seems to shift towards smaller q values. However, the resulting increase in d spacing is not consistent with a DPA grafted tube, since the difference to the radius of an imogolite tube is only 0.6 nm. For sample B, the lamellar peaks are shifted to smaller angles and the reflections are better defined. This effect is consistent with larger lamellar domains and a higher order of organization of the alkyl chain, as observed in IR. The inter-lamellar spacing is 2.8 nm for samples A1-A5 and 2.9 nm for sample B. It is close to the values reported for lamellar phases obtained with other decylphosphonate metal oxide phases such as aluminium decylphosphate (3 nm), calcium decylphosphate (3.1 nm) and calcium hydroxyapatite modified by decylphosphate (3 nm). [START_REF] Tanaka | Modification of amorphous aluminum phosphate with alkyl phosphates[END_REF][START_REF] Tanaka | Formation and structure of calcium alkyl phosphates[END_REF][START_REF] Tanaka | Modification of Calcium Hydroxyapatite Using Alkyl Phosphates[END_REF] Bac et al. already observed the formation of a lamellar phase following the reaction between germanium imogolite and ODPA. [START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF] They observed a peak in XRD at 1.65 nm, which most probably represents the 030 peak of aluminum octadecylphosphonate with a corresponding lamellar repeat distance of 5.0 nm.

Picot et al. found peaks at approximately q = 1.3 nm -1 , 2.6 nm -1 , 3.9 nm -1 in SAXS and interpreted them as the k0 progression of a 2D hexagonal phase. [START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] Calculating the distance for a lamellar 0k0 progression from their values yields 4.8 nm, which is in good agreement with aluminum octadecylphosphonate.

Direct comparison with the XRD data on the reaction between ODPA and imogolite of Park et al. is difficult, since the used wavelength is missing and the data is noisy. [START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF] Nonetheless, the authors present scanning tunneling microscopy pictures where they observe a pattern with a repeat distance of d=5.1 nm for ODPA and 4.7 nm for TDPA after reaction with imogolite. These distances correlate to the chain length of the used phosphonic acid, but the presence of imogolite tubes in the structure is not unambiguously proven.

The d spacings of all these compounds appear to increase linearly with the carbon number of the alkylphosphonic acid (Figure 4), which indicates that the length of the alkyl chain is the main factor influencing the thickness, regardless of the nature of the inorganic part. Even for short chain phosphonates, the trend holds true. The only deviation is observed with the microscopy data reported by Park et al. for TDPA. The variation in d spacing for the reported octadecylphosphonates could be associated to different crystal structures as reported for phenylphosphonates. For aluminum phenylphosphonate the structure indeed depends on the synthesis condition. [START_REF] Cabeza | Aluminum Phenylphosphonates: A Fertile Family of Compounds[END_REF] From SAXS, we can conclude that the reaction of imogolite with alkylphosphonic acid yields a lamellar phase for all experimental conditions (Table 1) and that no evidence of grafted imogolite is observed.

MAS NMR

Although MAS NMR is a useful tool for the characterization of imogolite, no study of the reaction between imogolite and phosphonic acids has been reported so far. [START_REF] Barron | Detection of imogolite in soils using solid state 29Si NMR[END_REF][START_REF] Yucelen | Formation of single-walled aluminosilicate nanotubes from molecular precursors and curved nanoscale intermediates[END_REF][START_REF] Zanzottera | Physico-Chemical Properties of Imogolite Nanotubes Functionalized on Both External and Internal Surfaces[END_REF] Since no relevant differences were observed between the samples A1-A5, the NMR investigation was restricted to the unreacted imogolite (reference), sample A1 and sample B.

The obtained [START_REF] Du | Insights into the formation mechanism of imogolite from a full-range observation of its sol-gel growth[END_REF] Pristine imogolite has a single peak at 8 ppm in [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al MAS NMR (Figure 5b), which is assigned to octahedral coordinated aluminum and clearly confirms the imogolite local structure. [START_REF] Barron | Detection of imogolite in soils using solid state 29Si NMR[END_REF][START_REF] Yucelen | Formation of single-walled aluminosilicate nanotubes from molecular precursors and curved nanoscale intermediates[END_REF] Three peaks are visible in the [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al MAS NMR spectra for samples A1 and B. The peak centers and the corresponding relative area were obtained by deconvolution of the [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al MAS NMR spectra (Figure S3). The signals of sample A1 are centered at -5 ppm (24%), 8 ppm (69 %) and 47 ppm (7 %). The peak at -5 ppm exhibits a noticeable broadness. For sample B the signals are located at -13 ppm (69 %), 8 ppm (28 %) and 46 ppm (3%).

By collecting [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al { 31 P} REDOR curves (Figure 5c), the spatial proximity of P to Al atoms was probed. The Al REDOR signal at -5 ppm and 47 ppm in sample A1 rapidly increases with time suggesting that Al atoms are in close vicinity to P atoms. This is not the case for the slowly increasing signal at 8 ppm, indicating that this site is able to interact with phosphorous, but does not directly coordinate to it. For sample B the Al atoms resonating at -13 ppm are close to P atoms, while the 8 ppm peak is not coordinated by P. Low peak intensity made it impossible to collect the data for the signal at 46 ppm and turned the data for the 8 ppm peak noisy.

In [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al MQMAS NMR (Figure 6) imogolite shows a NMR shift distribution mainly due to quadrupolar interactions. The MQMAS experiments for samples A1 and B confirm the presence of three sites. For sample A1 the signal at 8 ppm has a similar NMR shift distribution as unmodified imogolite, while the two new peaks at -5 ppm and 47 ppm display mainly a distribution in the isotropic shift (see dashed lines in Figure 6). This distribution indicates a lower degree of crystallinity in the new environments. For sample B the signal at -13 ppm has a more pronounced quadrupolar distribution than the signal at -5 ppm in sample A1 and is therefore more crystalline.

All samples show a signal at 8 ppm, hinting on the presence of unreacted imogolite tubes. [START_REF] Fyfe | Solid-State Double-Resonance Nmr Experiments Involving Quadrupolar and Spin 1/2 Nuclei[END_REF] Si MAS NMR also confirms the presence of unreacted imogolite (Figure S2b). Due to the vicinity of P and Al, the peaks at -5 ppm and -13 ppm can be assigned to AlO 6 octahedrons with P in their second coordination shell, and the peaks at 47 ppm and 46 ppm to Q 4 (4P) AlO 4 -tetrahedra. [START_REF] Müller | Solid-state aluminium-27 nuclear magnetic resonance chemical shift and quadrupole coupling data for condensed AlO4tetrahedra[END_REF] Similar shifted signals were reported for AlO 4 41, 45 and AlO 6 50 coordination in aluminum phenylphosphonates. From this result and the previous characterization it can be concluded that sample B mainly consists of rather crystalline aluminum decylphosphonate. This is supported by the sharp peaks in the [START_REF] Du | Insights into the formation mechanism of imogolite from a full-range observation of its sol-gel growth[END_REF] P spectrum, which were identified as a sign for crystalline aluminum alkylphosphonate. [START_REF] Gao | Self-assembled monolayers of alkylphosphonic acids on metal oxides[END_REF][START_REF] Lassiaz | Organo-lined alumina surface from covalent attachment of alkylphosphonate chains in aqueous solution[END_REF][START_REF] Lassiaz | Modification of silica by an organic monolayer in aqueous medium using octylphosphonic acid and aluminium species[END_REF] Sample A1 still has a high content of imogolite and the new octahedral Al environment is less well defined. Furthermore, the [START_REF] Du | Insights into the formation mechanism of imogolite from a full-range observation of its sol-gel growth[END_REF] P spectrum shows a broad peak, which was identified by Lassiaz et al. as an indicator for a successful grafting reaction on aluminum oxide. [START_REF] Lassiaz | Organo-lined alumina surface from covalent attachment of alkylphosphonate chains in aqueous solution[END_REF][START_REF] Lassiaz | Modification of silica by an organic monolayer in aqueous medium using octylphosphonic acid and aluminium species[END_REF] However, in the present case the unresolved 31 P peak could also reflect the distributed chemical environments in a not exclusively crystalline phosphonate.

SEM and TEM

SEM of sample A5 (Figure 7a, b) shows aggregates with zones of needle-shape and platelet-like morphologies. Both types of morphology are connected. These aggregates appear as dark domains in TEM (Figure 7c). In their periphery, assemblies of aggregated nanotubes are visible. In between the network of imogolite bundles, small domains are present. Depending on their orientation in space, the aggregates show a lamellar organization. We did not find any structures hinting on grafted imogolite, i.e. imogolite surrounded by a corona of DPA.

Degradation versus surface modification

The obtained results challenge the commonly used scheme of grafted imogolite, where the phosphonic acid reacts with OH groups on the external wall of imogolite and forms covalent bonds by condensation. 18- 19, 21-22 IR, [START_REF] Du | Insights into the formation mechanism of imogolite from a full-range observation of its sol-gel growth[END_REF] P MAS NMR and [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al { [START_REF] Du | Insights into the formation mechanism of imogolite from a full-range observation of its sol-gel growth[END_REF] P} REDOR experiments give evidence for the formation of P-O-Al bonds.

However, the found chemical shift in [START_REF] Lindner | Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter 1st[END_REF] Al MAS NMR and the SAXS results are in favor of the formation of a lamellar aluminum decylphosphonate structure. SEM and TEM support these findings by showing a mixture of lamellar phase and imogolite tube bundles, interconnected with each other. Overall, the results do not support a grafting reaction of DPA on imogolite but give evidence for the formation of a composite material.

The basis for this reactivity is the surface chemistry of imogolite, which is similar to gibbsite. The tube wall is covered in bridging Al 2 (µ-OH) groups that correspond to the crystal face of gibbsite, while the terminal points have Al-OH / Al-OH 2 + groups, similar to the crystal edges of gibbsite. The difference between those two sites is their affinity to participate in adsorption. The single coordinated hydroxyls are active, while the bridging hydroxyls are inert for adsorption processes at low adsorbate concentrations. [START_REF] Parfitt | ADSORPTION ON HYDROUS OXIDES: II. OXALATE[END_REF][START_REF] Mcbride | Chemisorption of catechol on gibbsite, boehmite, and noncrystalline alumina surfaces[END_REF] For phosphate, gibbsite starts to form aluminum phosphate phases at acidic pH that deposit on the faces of gibbsite. [START_REF] Van Emmerik | 31P solidstate nuclear magnetic resonance study of the sorption of phosphate onto gibbsite and kaolinite[END_REF] It was suggested that the complexation of the gibbsite edge is an intermediate in this process. 54- 55 Based on the structural similarity of metal phosphates and phosphonates, [START_REF] Cao | Layered metal phosphates and phosphonates: from crystals to monolayers[END_REF] we presume a similar reactivity for the sorption process of phosphonic acid on imogolite as for phosphate on gibbsite. At the studied conditions DPA does not adsorb or react with the bridging Al 2 (µ-OH) groups of imogolite. It mainly complexes Al-OH and Al-OH 2 + groups and degrades imogolite tubes by forming lamellar aluminum decylphosphonate. Since those groups are mainly present at the terminal points of the tubes, the degradation is slow and it explains the high amount of residual tubes evidenced for samples A1-A5.

Dispersion of modified Imogolite, Emulsification and Interface Experiments

In the majority of the published studies the dispersibility of modified imogolite was taken as a proof of concept for the successful surface modification of imogolite by phosphonic acid. [START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF][START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF][START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF][START_REF] Li | Dispersing surface-modified imogolite nanotubes in polar and non-polar solvents[END_REF] Although our results

give no hint for surface modified imogolite tubes, the DPA modified imogolite is dispersible in toluene. It gives an almost clear liquid, which is slightly scattering light. Mixing this dispersion with water stabilizes a water in oil emulsion (Figure 8a, b and Figure S4). In contrast, bare imogolite gives an oil in water emulsion, as was already observed for hexane. [START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] With the composite material, the droplets do not coalesce and can undergo deformations. This is also observed at the macroscopic scale, where an interface between the dispersed composite material and water is able to stabilize large aqueous droplets against coalescence (Figure 8c). This very specific behavior is related to the DPA modified imogolite and cannot be obtained with the interface between toluene and a bare imogolite dispersion. The ability to stabilize develops with contact time between the two phases and is generally not instantaneously present. The droplets are stable up to one year (experimental time), but collapse if the toluene evaporates. Illumination of the sample unveils the otherwise almost invisible surface precipitate (Figure 8), that forms after the contact between the dispersion and water, but does not exist in the toluene dispersion itself. The stabilized droplets can also be moved on the interface (see SI). We retrieve characteristics of "liquid marbles" for an oil/water system. Liquid marbles are defined for air/liquid systems as stabilized liquid droplets covered with hydrophobic particles. Thanks to the nanoparticles, the droplets are able to maintain their shapes even after deformation by external forces. [START_REF] Abe | Reversible Shape Transformation of Ultrathin Polydopamine-Stabilized Droplet[END_REF][START_REF] Aussillous | Liquid marbles[END_REF][START_REF] Liu | One-Step Synthesis of Multifunctional Zinc-Iron-Oxide Hybrid Carbon Nanowires by Chemical Fusion for Supercapacitors and Interfacial Water Marbles[END_REF] Up to now, the dispersibility of alkylphosphonic acid modified imogolite in organic solvents was explained by a successful grafting reaction, and the resulting grafted imogolite was sketched as a tube with a brushlike corona of grafting agent. [START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF][START_REF] Yah | Molecular Aggregation States of Imogolite/P3HT Nanofiber Hybrid[END_REF][START_REF] Yah | Molecular Aggregation State and Electrical Properties of Terthiophenes/Imogolite Nanohybrids[END_REF][START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF][START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] The difference in hydrophilicity between the internal and the external surfaces of such a grafted tube was used to explain the formation of Pickering emulsions by ODPA modified imogolite. [START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] However, the structure of dried modified imogolite (Figure7) suggests that the mode of stabilization has to be different.

Presuming a similar structure in its dispersed state as in its dried state (Figure 7), the imogolite-lamellar phase network would exhibit zones where either hydrophobic (lamellar phase) or hydrophilic areas (residual imogolite) dominate. While the hydrophobic areas promote the dispersion in toluene, the hydrophilic areas and the nanostructuration of the composite material could be responsible for its surface activity at the toluene/water interface. Further investigations are needed to elucidate the real stabilization mechanism for this system, but are out of the scope of this study.

Conclusion

We characterized the reaction between imogolite and DPA by several complementary methods, in particular a complete NMR study. Instead of a grafting reaction, we found the partial transformation of imogolite into a lamellar aluminum phosphonate. The resulting composite material consists of interconnected imogolite tube bundles and lamellar phase. The specific nanostructure of this material enables the dispersion in organic solvent and gives rise to interfacial properties leading to the stabilization of water droplets in toluene. Our results demonstrate that DPA (or phosphonic acids in general) are not appropriate reagents to modify the surface of imogolite nanotubes. We believe that the observed process might give insights into an approach to create hierarchical structures with novel properties from colloidal particles under mild conditions. [START_REF] Tanaka | Modification of amorphous aluminum phosphate with alkyl phosphates[END_REF] , Lassiaz et al. [START_REF] Lassiaz | Organo-lined alumina surface from covalent attachment of alkylphosphonate chains in aqueous solution[END_REF] , Picot et al. [START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] , Gao et al. [START_REF] Gao | Self-assembled monolayers of alkylphosphonic acids on metal oxides[END_REF] , Park et al. [START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF] , Bac et al. [START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF] and our data. Scattering patterns and conditions for aluminum octadecyl-and butylphosphonate can be found in supplementary material (Figure S6).

  P MAS NMR spectra (Figure5a) show no remaining DPA at 31 ppm (FigureS2) for the different products. Sample A1 has a single broad peak from 0 ppm to 35 ppm, while sample B shows pronounced peaks at 2 ppm, 9.5 ppm and 13.5 ppm. The broad peak in sample A1 evidences the formation of P-O-Al bonds with distributed chemical environments. The latter develops into a more ordered structure with prolonged reaction time, as observed for sample B.

Figure 1 .

 1 Figure 1. (a) Cryo-TEM picture of imogolite. Globular structures are allophane and proto-imogolite. (b) SAXS pattern of dispersed imogolite and a fit by a core-shell cylinder (see SI), yielding an outer diameter of 2.8 nm.(c) Inset of the imogolite structure, grey tetrahedra are centerd by silicon, blue octahedra are centerd by aluminum, red dots are oxygen and hydrogens are omitted.

Figure 2 .

 2 Figure 2. (a) FT-IR spectra of DPA(red), of imogolite (blue), samples A1 -A5 (orange) and sample B (green), normalized to maximum absorbance.

Figure 3 .

 3 Figure 3. Powder X-ray scattering patterns of imogolite (blue), samples A1 -A5 (orange) and sample B (green).

Figure 4 .

 4 Figure 4. Repeat distance d in relation to the used alkylphosphonic acid chain length in several phosphonate phases by Tanaka et al.[START_REF] Tanaka | Modification of amorphous aluminum phosphate with alkyl phosphates[END_REF] , Lassiaz et al.[START_REF] Lassiaz | Organo-lined alumina surface from covalent attachment of alkylphosphonate chains in aqueous solution[END_REF] , Picot et al.[START_REF] Picot | Behaviour of hybrid inside/out Janus nanotubes at an oil/water interface. A route to self-assembled nanofluidics?[END_REF] , Gao et al.[START_REF] Gao | Self-assembled monolayers of alkylphosphonic acids on metal oxides[END_REF] , Park et al.[START_REF] Park | Two-dimensional alignment of imogolite on a solid surface[END_REF] , Bac et al.[START_REF] Bac | Surface-modified aluminogermanate nanotube by OPA: Synthesis and characterization[END_REF]

Figure 5 .

 5 Figure 5. (a) 31 P MAS NMR spectra of sample A1 (orange) and sample B (green). (b) 27 Al MAS NMR spectra of imogolite (blue), sample A1 (orange) and sample B (green), normalized to highest peak intensity. (c) 27 Al { 31 P} REDOR signals of the three sites found in sample A1 (left) and sample B (right). Lines between data points are for clarity.

Figure 6 .Figure 7 .

 67 Figure 6. 27 Al MQMAS (triple quantum) spectra of (a) imogolite (blue), (b) sample A1 (orange) and (c) sample B (green). Dashed lines represent the directions of broadening induced by the distribution of isotropic chemical shift ( ) and second-order quadrupolar induced shift (). 𝛿 𝑖𝑠𝑜 𝛿[START_REF] Xavier | Advances in optoplasmonic sensors -combining optical nano/microcavities and photonic crystals with plasmonic nanostructures and nanoparticles[END_REF] 𝑄

Figure 8 .

 8 Figure 8. (a), (b) Water in oil emulsion, stabilized by sample A5 in toluene. (c) A 20 µL water droplet stabilized by modified imogolite at the toluene/water interface. (d) Stabilized droplet, illuminated from below.

Table 1 :

 1 Summary of the investigated reaction conditions, giving the stirring time, the molar ratio between

	phosphorous and aluminum, the volumetric solvent ratio (ethanol : water), and the concentration in DPA
	for each sample.
	Sample	Time,	P : Al Solvent ratio	C PA ,
		days		EtOH : H 2 O	mmol•L -1
	A1	1	5	95 : 5	5
	A2	1	5	95 : 5	5
	A3	1	3	95 : 5	3
	A4	1	1	90 : 10	1
	A5 *	1	5	62.5 : 37.5	5

* pH adjusted to 5

Acknowledgements

We want to thank M. Moskura for her help with MAS NMR and P. Picot for the helpful discussions. FEGSEM & EDX instrumentation was facilitated by the Institut des Matériaux de Paris Centre (IMPC FR2482) and was funded by Sorbonne Université, CNRS and by the C'Nano projects of the Région Ile-de-France. (Cryo-) TEM observations were made thanks to "Investissements d'Avenir" LabEx PALM (ANR-10-LABX-0039-PALM). This work was financed by the Amont-Aval CEA program.


Cécile Le Guern

Béatrice Bechet

Chloé Besnard

Sophie Bretesché

Nicolas Briant

Ghozlane Fleury-Bahi

Liliane Jean-Soro

Joel Knoery

Alexandra Lepinay

Gilles F Montavon

Béatrice

POLLUSOLS: diffuse contamination from land to sea

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Human activities (transport, industry, agriculture,...) are sources of many contaminants that can be disseminated into the environment through various ways, impacting soils and sediments. Small quantities of any compound dispersed over a large area may amount to "a nonpoint source pollution", with a persistence of several decades.

POLLUSOLS is a research program focusing on the issues of nonpoint source pollution on the land-to-sea continuum. It aims at a) structuring research teams from different fields (biology, sociology, chemistry, psychology, physics, economics,...), b) improving the understanding of the pollution cycle, and c) proposing relevant tools for managing soils and sediments impacted by diffuse pollution.

The partners apply a common methodology on different types of experimental sites reflecting environmental and societal issues on the Loire Estuary: vineyards, urban areas (former landfills, allotment gardens, and road environment), former uranium mines, and the estuary. The methodology considers three main connected steps: sources of contamination (origin and characterization), transfers (mobility in the environment, evolution of use) as well as impacts and heritage (including management options and patrimonial considerations).

We focus on classical and emerging contaminants: metals that are toxic at low doses like lead and mercury, oligo-elements such as copper and zinc that become toxic at high doses, emerging contaminants such as PGE (platinum-group elements) or pharmaceutics, and radionuclides including uranium and its decay chains. In addition to site-scale approaches, Cu contamination is studied at the regional scale. Indeed, the content of Cu has doubled within the last 30 years in the mussels of the Loire estuary, the receptacle of all water-borne contamination.

The multidisciplinary and even interdisciplinary approaches developed within the POLLUSOLS program will be illustrated (case of urban sites and former mines). The partners are looking forward to European collaboration. They can offer their "in the field" experience around various sites and watersheds.

  and emerging contaminants ThS 3d Policy strategies of environmental concerns of emerging contaminations POLLUSOLS: diffuse contamination from land to sea C. Le Guern (BRGM-IRSTV), B. Béchet (IFSTTAR-IRSTV), C. Besnard (OSUNA-IRSTV), S. Bretesché (LEMNA-IMT Atlantique), N. Briant (IFREMER), G. Fleury (U-Nantes, IRSTV), L. Jean-Soro (IFSTTAR-IRSTV), J. Knoery (IFREMER), A. Lépinay (OSUNA-IRSTV), G. Montavon (Subatech, IMT Atlantique), O. Navarro (U-Nantes, IRSTV), B. Plottu (AgroCampus Ouest, IRSTV), M. Tendero (AgroCampus Ouest, IRSTV), T. Lebeau (OSUNA, U-Nantes, IRSTV)


Franck Ramaharo
email: franck.ramaharo@gmail.com

A generating polynomial for the two-bridge knot with Conway's notation C(n, r)

Keywords: generating polynomial, shadow diagram, Kauffman state

We construct an integer polynomial whose coefficients enumerate the Kauffman states of the two-bridge knot with Conway's notation C(n, r).

Introduction

A state of a knot shadow diagram is a choice of splitting its crossings [2, Section 1]. There are two ways of splitting a crossing:

(A) =⇒ , (B) =⇒ .
By state of a crossing we understand either of the split of type (A) or (B). An example for the figure-eight knot is shown in Figure 1.

Let K be a knot diagram. If m denotes the initial number of crossings, then the final states form a collection of 2 m diagrams of nonintersecting curves. We can enumerate those states with respect to the number of their components -called circles -by introducing the sum

K(x) := S x |S| , (1)
where S browses the collection of states, and |S| gives the number of circles in S. Here, K(x) is an integer polynomial which we referred to as generating polynomial [START_REF] Ramaharo | Enumerating the states of the twist knot[END_REF][START_REF] Ramaharo | Statistics on some classes of knot shadows[END_REF] (in fact, it is a simplified formulation of what Kauffman calls "state polynomial" [2, Section 1-2] or Figure 1: The states of a crossing.

"bracket polynomial" [START_REF] Louis | State models and the Jones polynomial[END_REF]). For instance, if K is the figure-eight knot diagram, then we have K(x) = 5x + 8x 2 + 3x 3 (the states are illustrated in Figure 2). In this note, we establish the generating polynomial for the two-bridge knot with Conway's notation C(n, r) [START_REF] Lafferty | The three-variable bracket polynomial for reduced, alternating links[END_REF][START_REF] Overduin | The three-variable bracket polynomial for two-bridge knots[END_REF]. We refer to the associated knot diagram as B n,r , where n and r denote the number of half-twists. For example, the figure-eight knot has Conway's notation C(2, 2). Owing to the property of the shadow diagram which we draw on the sphere [START_REF] Denton | Shadow movies not arising from knots[END_REF], we can continuously deform the diagram B n,r into B r,n without altering the crossings configuration. We let B n,r B r,n express such transformation see Figure 3 (a) . Besides, we let B n,0 and B n,∞ denote the diagrams in Figure 3 (b) and (c), respectively. Here, "0" and "∞" are symbolic notations -borrowed from tangle theory [2, p. 88] -that express the absence of half-twists. If r = ∞ and n ≥ 1, then B n,∞ represents the diagram of a (2, n)-torus knot B n-1,1 . Correspondingly, we let B 0,r and B ∞,r denote the diagrams pictured in Figure 3 (d) and (e), respectively.

Generating polynomial

Let K, K and be knot diagrams, where is the trivial knot, and let # and denote the connected sum and the disjoint union, respectively. The generating polynomial defined in (1) verifies the following basic properties:

(i) (x) = x; (ii) K K (x) = K(x)K (x); (iii) K#K (x) = 1 x K(x)K (x). Furthermore, if K K , then K(x) = K (x) [6].
Lemma 1. The generating polynomial for the knots B n,0 and B n,∞ are given by The key ingredient for establishing (2) and (3) consists of the states of specific crossings which produce the recurrences respectively, with initial values B 0,0 (x) = x and B 0,∞ (x) = x 2 [START_REF] Ramaharo | Enumerating the states of the twist knot[END_REF]. Note that the lemma still holds if we replace index n by r.

B n,0 (x) = x(x + 1) n (2) and B n,∞ (x) = (x + 1) n + x 2 -1. (3) 2
B n,0 (x) = ( B n-1,0 ) (x) + B n-1,0 (x) and B n,∞ (x) = B n-1,0 (x) + B n-1,∞ (x), (a) B n,r B r,n . (b) B n,0 . (c) B n,∞ (2, n)-torus knot . (d) B 0,r . (e) B ∞,r (
Proposition 2. The generating polynomial for the two-bridge knot B n,r is given by the recurrence

B n,r (x) = B n-1,r (x) + (x + 1) n-1 B ∞,r (x), (4)
and has the following closed form:

B n,r (x) = (x + 1) r + x 2 -1 x (x + 1) n + x 2 -1 (x + 1) r -1 x . (5)
Proof. By Figure 4 we have where the last relation follows from property (iii). Solving the recurrence for n yields

B n,r (x) = B n-1,r (x) + B n-1,0 #B ∞,r (x) = B n-1,r (x) + (x + 1) n-1 B ∞,r (x),
B n,r (x) = B 0,r (x) + B ∞,r (x) (x + 1) n -1 x .
We conclude by the closed forms in Lemma 2.

Remark 3. We can write

B n,0 (x) = x 2 α n (x) + x (6) and B n,∞ (x) = xα n (x) + x 2 , (7)
where α n (x) := (x + 1) n -1 x , so that identity (5) becomes

B n,r (x) x 2 α n (x) + x + x 2 α r (x) + xα n (x)α r (x) . (8)
Since the coefficients of α n (x) are all nonnegative, it is clear, by [START_REF] Ramaharo | Enumerating the states of the twist knot[END_REF], that the polynomial x 2 α n (x) counts the states of B n,0 that have at least 2 circles. This is illustrated in Figure 5 (a). Likewise, we have an interpretation of (7) in Figure 5 (b). In Figure 5 and 6, the dashed diagrams represent all possible disjoint union of -1 circles ( = n or r, depending on the context), counted by α (x) and eventually empty.

Therefore, for n, r / ∈ {0, ∞}, identity (8) means that we can classify the states into 4 subset as shown in Figure 6. In these illustrations, there are 2 n -1 and 2 r -1 states of (a) and (b) kind, respectively, and

n 1 × r 1 + 1 one-component states of (c) and (d) kind.
The remaining states are of (c) kind, bringing the total number of states to 2 n+r . where δ 1,k is the Kronecker symbol. By (1), we recognize b(n, r; k) as the cardinal of the set {|S| = k : S is a state B n,r }, i.e., the number of states having k circles. In this section, the coefficients b(n, r; k) are tabulated for some values of n, r and k. We give as well the corresponding A-numbers in the On-Line Encyclopedia of Integer Sequences [START_REF] Neil | The On-Line Encyclopedia of Integer Sequences[END_REF].

Particular values

k) = n + r k + 1 + n k -1 + r k -1 - n k + 1 - r k + 1 -δ 1,k , (a)
• b(n, 0; k) = x k x(x + 1) n , essentially giving entries in Pascal's triangle A007318 (see Table 1).

n \ k 0 1 2 3 4 5 6 7 8 • b(n, 2; k) = x k + 2)(x + 1) n + x 2 -1 (x + 2) , giving triangle in A300454 (see Table 3). Table 3: Values of b(n, 2; k) for 0 ≤ n ≤ 7 and 0 ≤ k ≤ 8.

0 0 1 1 0 1 1 2 0 1 2 1 3 0 1 3 3 1 4 0 1 4
• b(n, n; k) = x k (x + 1) 2n + x 2 -1 2(x + 1) n -1)
x , giving triangle in A321127

(see Table 4).

• b(n, r; 1) = nr + 1, giving A077028, and displayed as square array in Table 5.

In Kauffman's language, b(n, r; 1) is, for a fixed choice of star region, the number of ways of placing state markers at the crossings of the diagram B n,r , i.e., of the forms The process is illustrated in Figure 7 for the figure-eight knot.

• b(n, r; 2) = n r 2 + 1 +r n 2 + 1 , giving square array in A300401 (see Table 6).

We paid a special attention to the case k = 2 because, surprisingly, columns b(n, 1; 2) n and b(n, 2; 2) n match sequences A000124 and A014206, respectively [START_REF] Ramaharo | Enumerating the states of the twist knot[END_REF]. The former gives the maximal number of regions into which the plane is divided by n lines, and the latter the maximal number of regions into which the plane is divided by (n + 1) circles.

• b n, r; d(n, r) = leading coefficient of B n,r (x), giving square array in A321125 (see Table 7). Here, d(n, r) = max(n + 1, r + 1, n + r -1) denotes the degree of B n,r (x), and gives entries in A321126. We have Table 8 giving the numbers d(n, r) for 0 ≤ n ≤ 7 and 0 ≤ r ≤ 7. n \ r 0 1 2 3 4 5 6 7 0 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 3 2 2 2 2 1 3 1 1 2 1 1 1 1 1 4 1 1 2 1 1 1 1 1 5 1 1 2 1 1 1 1 1 6 1 1 2 1 1 1 1 1 7 1 1 2 1 1 1 1 1

Table 7: Leading coefficients of B n,r (x) for 0 ≤ n ≤ 7 and 0 ≤ r ≤ 7.

We have the following properties:

  (a) The states of the figure-eight knot following the initial "A" split. (b) The states of the figure-eight knot following the initial "B" split.

Figure 2 :

 2 Figure 2: The states of the figure-eight knot.

  2, r)-torus knot .

Figure 3 :

 3 Figure 3: Two-bridge knots with Conway's notation C(n, r).

Figure 4 :

 4 Figure 4: The splits at a crossing allow us to capture B n-1,r , B n-1,0 and B ∞,r .

  r; k)x k := B n,r (x), or b(n, r; k) := x k B n,r (x). Then b(n, r;

  Illustration of x 2 α n (x) + x. (b) Illustration of xα n (x) + x 2 .

Figure 5 :

 5 Figure 5: Illustrations of B n,0 (x) and B n,∞ (x) as functions of α n (x) .

Figure 6 :

 6 Figure 6: The states of B n,r : states in (a) are counted by x 2 α n (x), those in (b) by x 2 α r (x), those in (c) by xα n (x)α r (x), and state in (d) is simply counted by x.

n \ k 0 1

 1

Figure 7 :

 7 Figure7: Illustration of b(2, 2; 1): mark two adjacent regions by stars ( * ), then assign a state marker at each crossing so that no region of B 2,2 contains more than one state marker, and regions with stars do not have any.

-

  d(n, r) = d(r, n); if r = 0, then d(n, r) = n + 1; if r = ∞,then sequence d(n, r) n begins: 2, 2, 2, 3, 4, 5, 6, 7, 8, . . . (A233583 with

Table 1 :

 1 = x k (x + 1) n+1 + x 2 -1 , generating a subtriangle in A300453 (see Table2).

		6 4 1
	5	0 1 5 10 10 5 1
	6	0 1 6 15 20 15 6 1
	7	0 1 7 21 35 35 21 7 1

Values of b(n, 0; k) for 0 ≤ n ≤ 7 and 0 ≤ k ≤ 8. • b(n, 1; k)

Table 2 :

 2 Values of b(n, 1; k) for 0 ≤ n ≤ 7 and 0 ≤ k ≤ 8.

Table 4 :

 4 Values of b(n, n; k) for 0 ≤ n ≤ 7 and 0 ≤ k ≤ 13.

	,	,	,	,

Table 5 :

 5 Values of b(n, r; 1) for 0 ≤ n ≤ 7 and 0 ≤ r ≤ 7.

	so that the resulting states are "Jordan trails" [2, Section 1-2]. Note that a state
	marker is interpreted as an instruction to split a crossing as shown below:
	=⇒	and	=⇒	.

Table 6 :

 6 2, 2; 1): mark two adjacent regions by stars ( * ), then assign a state marker at each crossing so that no region of B 2,2 contains more than one state marker, and regions with stars do not have any. Values of b(n, r; 2) for 0 ≤ n ≤ 7 and 0 ≤ r ≤ 7.

	n \ r 0 1 2 3	4	5	6	7
	0	0 1 2 3	4	5	6	7
	1	1 2 4 7 11 16 22
	2	2 4 8 14 22 32 44 58
	3	3 7 14 24 37 53 72 94
	4	4 11 22 37 56 79 106 137
	5	5 16 32 53 79 110 146 187
	6	6 22 44 72 106 146 192 244
	7	7 29 58 94 137 187 244 308

n \ r 0 1 2 3 4 5 [START_REF] Ramaharo | Enumerating the states of the twist knot[END_REF] Diagramatically, we give the corresponding illustration for some values of n and r in Figure 8. Remarquable values in Table 7 correspond to knots B 1,1 ("Hopf link", see Figure 9), B 2,2 (figure-eight knot, see Figure 1, 2) and B n,2 ("twist knot" [START_REF] Ramaharo | Enumerating the states of the twist knot[END_REF]) for n ≥ 3. The latter case can be observed from identity [START_REF] Neil | The On-Line Encyclopedia of Integer Sequences[END_REF] for which the leading coefficient is larger than 1 when n + 1 = n + r -1 is satisfied. Also, considere the identity below:

We notice that the leading coefficient is deduced from the contribution of the summands B n,0 (x) and B n,∞ (x) [START_REF] Ramaharo | Enumerating the states of the twist knot[END_REF].

10


Anke M Brock
email: anke.brock@enac.fr

Jessica Cauchard
email: jcauchard@acm.org

Markus Funk
email: funk@tk.tu-darmstadt.de

Jérémie Garcia
email: jeremie.garcia@enac.fr

Mohamed Khamis
email: mohamed.khamis@glasgow.ac.uk

Matjaž Kljun
email: matjaz.kljun@upr.si

Figure 2

Keywords: Display drones [13] CCS CONCEPTS, Human-centered computing → Human computer interaction (HCI); Human-Drone Interaction, Unmanned Aerial Vehicles (UAV), Flying Robots, Quadcopters, Human-Robot Interaction

Commercial drones have recently been developed to encompass use cases beyond aerial photography and videography. Researchers have explored wider applications of drones including using drones as social companions, as key components in virtual environments, as assistive tools for people with disabilities, and even as sport companions. However the uptake of research in Human-Drone Interaction (HDI) also brought forth a plethora of challenges that are unique to this platform. While drones were

INTRODUCTION

In the last years, the adoption rate of personal and commercial drones has exponentially increased. This rapid growth is both exciting and frightening. On the one hand, drones are presenting new opportunities, with applications ranging from entertainment (Figures 2 and3) to delivery, supporting people with special needs [START_REF] Soto | DroneNavigator: Using Leashed and Free-Floating Quadcopters to Navigate Visually Impaired Travelers[END_REF] (Figure 4), sports [START_REF] Floyd | Jogging with a Quadcopter[END_REF], pedestrian guidance [START_REF] Colley | Investigating Drone Motion As Pedestrian Guidance[END_REF], agriculture, and even search-and-rescue. On the other hand, many negative issues are arising around the adoption of drones and what it means to have them in our environments. The success and acceptability of this technology will depend on how well it can be used (Figure 1), integrated, and made acceptable to people. Many of these challenges fall in the realm of research in human-computer and human-robot interaction.

In this multidisciplinary workshop, we propose to bring together students, experienced researchers, and practitioners from diverse areas. We expect this workshop will raise interest to researchers from fields including, interaction techniques, autonomous vehicles, robotics, ubicomp, design, and cognitive science. In particular, we will foster discussions around interaction techniques, design, feedback strategies, automation, control, trust, as well as privacy and ethical issues in HDI research. These topics are relevant to ensure the successful integration of the technology.

Through two hands-on sessions and group discussions with experts from both academia and industry, our goal is to address key questions around the design and methodologies, and bring a community together to develop new approaches for future human-drone interaction research. While, HRI research has been growing over the years, only few papers are presented at the CHI conferences. This workshop will give the opportunity for researchers in the HRI field to discover CHI and participate in our community. This workshop will lead to a special edition of a journal, to help bridge the gap between the HCI and HRI communities in terms of methodologies for human-drone interaction research.

ORGANIZERS

Our interdisciplinary organization team consists of 6 diverse international researchers from France, Israel, Germany, UK, and Slovenia. In addition, an industry partner from Balmore Group's Aerial Services division1 confirmed their participation in the workshop.

Anke Brock is an Assistant Professor at ENAC in Toulouse, France. She is interested in building drone-based navigation systems [START_REF] Anke | FlyMap: Interacting with Maps Projected from a Drone[END_REF], and making human-drone interaction accessible to people with impairments (Fig. 4).

Jessica Cauchard is an Assistant Professor at the Interdisciplinary Center (IDC) Herzliya, Israel where she heads the Ubiquitous Computing Lab. Jessica is interested in developing natural interfaces for human-drone interaction [START_REF] Cauchard | Drone & Me: An Exploration into Natural Human-drone Interaction[END_REF][START_REF] Rebecca Cauchard | Emotion Encoding in Human-Drone Interaction[END_REF][START_REF] Landay | Drone & Wo: Cultural Influences on Human-Drone Interaction Techniques[END_REF].

Markus Funk is a senior researcher (area head) at the Telecooperation Lab of the TU Darmstadt, Germany. Markus is interested in human-drone interaction and flying user interfaces [START_REF] Funk | Human-drone interaction: let's get ready for flying user interfaces![END_REF][START_REF] Knierim | Quadcopter-projected in-situ navigation cues for improved location awareness[END_REF].

Jérémie Garcia is an Assistant Professor at ENAC in Toulouse, France. He is interested in designing interactions to author and operate automated systems such as drones or air traffic control [START_REF] Conversy | Vizir: A Domain-Specific Graphical Language for Authoring and Operating Airport Automations[END_REF].

Mohamed Khamis is a Lecturer (Assistant Professor) at the University of Glasgow in the UK. His research is at the intersection of Ubiquitous Computing and User Privacy. He is interested in understanding the privacy implications of drones, as well as developing novel input methods for human-drone interaction [START_REF] Khamis | GazeDrone: Mobile Eye-Based Interaction in Public Space Without Augmenting the User[END_REF].

Matjaž Kljun is an Assistant Professor at University of Primorska (UP) where he co-directs HICUP lab (Humans Interacting with Computers at UP). Within the general HCI scope, he is also interested on how drones will become our companions [START_REF] Kljun | StreetGamez: A Moving Projector Platform for Projected Street Games[END_REF].

WEBSITE

The website will promote the workshop and serve as a platform to present the findings and foster the growing community of HDI researchers. It is available at: http://hdi.famnit.upr.si/.

PRE-WORKSHOP PLANS

Prior to the workshop, we will open a web submission form, issue a call for participation, advertise the workshop, select position papers and contact authors with relevant information.

The call for participation will be broadly distributed to the different research communities addressing the subject of the workshop. This includes posting to mailing lists (e.g., chi-announcements), on social medias (e.g., Facebook and Twitter), and using the organizers' personal networks to contact leading researchers with relevant contributions in the area.

Potential participants will be asked to submit a 2 to 6-page position paper in the CHI Extended Abstract format by February 12th, 2019, along with three topics which they would like to discuss during the workshop, and with their preference of presentation format (poster or oral presentation). Papers may be submitted with organizers as authors or co-authors. We will encourage authors to consider broader implications of HDI and HCI, and where appropriate, implications for related fields such as robotics. Participants will also be encouraged to bring in new perspectives. We will seek submissions from HCI and HRI researchers already familiar with mainstream HCI, as well as cutting edge drone and robot researchers who are less familiar with HCI, but who believe their work may have wider implications for interactivity. We will also encourage submissions from researchers in any area of HCI with insights relevant to HDI.

An Easychair account will be used for submitting the position papers 2 . Each paper will be reviewed by at least two of the workshop organizers. All organizers will take part in reviewing. The organizers will select around 12 position papers for presentation at the workshop, and authors will be informed of acceptance on or before March 1st, 2019. Papers will be selected based on quality and relevance to the workshop topic and goals. Accepted position papers will be uploaded to the workshop website.

Participants will be asked to read all of the accepted position papers in advance of the day so that they may come to the workshop primed with relevant questions and discussion points.

WORKSHOP STRUCTURE

The timeline for the workshop is shown in Table 1. First, the organizers will introduce themselves and talk briefly about the aim and scope of the workshop. Participants will then introduce themselves and their research interests in 30 seconds each.

Individual Presentations:

The morning will start with individual presentations of the position papers in one of two proposed formats: Short oral presentations (5 min) and poster presentations. We will invite pioneering researchers in the field of human-drone interaction to give a short keynote, with at least one researcher coming from academia, Prof. Roel Vertegaal from Queens University, Canada, and we are aiming to invite one from industry. This session will last up to 2 hours.

Group Activity 1: After the coffee break, participants will be divided into groups. Drones of different shapes and sizes will be brought in to stimulate the discussions. Approval to bring drones to the venue has been obtained from the CHI 2019 general chair Prof. Stephen Brewster. Participants will be encouraged to engage with the drone on their table. Following the observation and engagement stages, participants will discuss in their group the current and future research problems in Human-Drone Interaction. They will be asked to prioritize the research problems according to their significance and the amount of efforts required for each problem to be solved. The results of this activity will be used as basis for the second activity. Group Activity 2: Participants will be split into groups. Each group will focus on one of the themes identified in the first group activity and will discuss the upcoming steps to address this research problem. Each group will then be invited to present their outcome.

We will conclude by collecting feedback about the workshop and discussing the next steps the HDI community should undertake. We will encourage participants to start collaborating on the discussed themes and potentially submit new research work to the special journal issue.

POST-WORKSHOP PLANS

Interacting with drones is a topic which has recently sparked much interest in the HCI community. Yet the HDI community is sparse and this workshop will give an opportunity for researchers in the field to meet and discuss the future of the field. We expect this workshop to influence future research in this area and help establish well-needed methodologies for HCI research with drones.

After the workshop, the organizers will prepare a report to be submitted to ACM Interactions Magazine. All accepted submissions will be published on the workshop website and on HAL (https: //hal.archives-ouvertes.fr/browse/collection) which is indexed by Google scholar. Selected position papers' authors will be invited to submit for review longer versions of their submissions, including new material, together with a collaboratively produced paper. We plan to hold an open call in a special issue of a journal. The editors-in-chief of ACM THRI have been contacted and we are in the process of discussing the details of a human-drone interaction special issue to be published early 2020.

CALL FOR PARTICIPATION

Drones are becoming increasingly ubiquitous to our environments. Many applications have appeared such as using drones for delivery, security, search-and-rescue, or even as companions. Such applications led to new challenges in the design and development of drone interfaces. In this First International Workshop on Human-Drone Interaction (HDI), through hands-on sessions and group discussions with experts, we provide this emerging field a platform to bring together researchers and practitioners. We seek to structure previous research efforts, identify new research directions, and define methodologies for this novel research area.

We seek high quality contributions that explore the advances and challenges in HDI, and that suggest new ways of interacting with drones. Submissions are invited in, but not limited to, the following topics: The proceedings of the workshop will be published on the website and through HAL https://hal. archives-ouvertes.fr/, which is indexed by Google scholar. Selected authors will be invited to submit to a special issue in the THRI journal. Further information can be found on: http://hdi.famnit.upr.si At least one author must attend the workshop. All participants must register for the workshop and at least one day of the conference.

Figure 3 :

 3 Figure 3: Interactive games with drones [10].

Figure 4 :

 4 Figure 4: Making HDI accessible to people with impairments.

•

  Novel applications • Novel interaction techniques • Manual control to autonomous applications • Flying user interfaces • First-person view flying • Creating everyday interaction with drones • Increasing safety and security • Communicating drone interactivity • Accessibility Authors are invited to submit a 2 to 6-page position paper following the CHI Extended Abstracts format, on or before February 12th, 2019 at https://easychair.org/conferences/?conf=hdi2019.

Table 1 :

 1 Timeline

	08:30-08:45 or 14.30-14.45 Opening
	08:45-09:00 or 14.45-15.00 30-second introductions
	09:00-11:00 or 15.00-17.00 Presentations and posters
	11:00-11:30 or 17.00-17.30 Coffee break
	11.30-12.15 or 17:30-18:15 Group activity 1
	12.15-13.30 or 18.15-19.30 Group activity 2
	13.30-14.00 or 19.30-20.00 Summary and Closing
	Lunch or Dinner

https://balmoremedia.co.uk/

https://easychair.org/conferences/?conf= hdi2019