Jason Richard

Anthony Phimphachanh

Julien Schneider

Shyamapada Nandi

Eline Laurent

Patrick Lacroix-Desmazes

Philippe Trens

Sabine Devautour-Vinot

Nathalie Marcotte

Corine Gérardin
email: corine.gerardin@enscm.fr

An integrated process for structuring and functionalizing ordered mesoporous silica to achieve superprotonic conductivity

Polyacid-functionalized inorganic mesoporous materials have attracted considerable interest as catalysts, permselective molecular sieves or drug carriers. Despite their great interest, their synthesis into ordered mesostructures incorporating polyacids densely and homogeneously distributed in the mesopores, is a challenge. Moreover, their properties as conductors for energy applications remain completely unexplored. Here we report an efficient, one-shot environmentally friendly synthesis route to prepare ordered mesoporous silica functionalized with strong polyacids, which exhibits excellent proton conductivity. We used polyion electrostatic complex micelles as structure-directing, functionalizing and pore-generating agents to obtain a material of remarkable textural and functional quality. It presents large and ordered mesopores hosting monodisperse polyacid chains corresponding to a dense and homogeneous functionalization of 1.2 mmolSO 3 H.gSiO 2 -1 and a function density of 1 SO3H per nm 3 of mesopore volume. Overcoming the performance-limiting inhomogeneities, we designed a superprotonic conductor, while the high value of the conductivity, 0.024 S cm -1 at 363 K/95% relative humidity, was maintained at least 7 days.

Introduction

Ordered mesoporous silica (OMS) functionalized by polymers is an emerging class of hybrid materials that combines the singular textural characteristics of inorganic networks with the specific physico-chemical properties of polymers. This gives the materials unique properties that can be attractively triggered by environmental stimuli. [START_REF] Soler-Illia | Multifunctional Hybrids by Combining Ordered Mesoporous Materials and Macromolecular Building Blocks[END_REF][START_REF] Brilmayer | Recent Trends in Nanopore Polymer Functionalization[END_REF][START_REF] Ngo | One-Pot Synthesis of Organic Polymer Functionalized Mesoporous Silicas[END_REF] Among OMS, those functionalized with polyacids have proven to be particularly attractive for their acidic properties, complexation ability and pH-responsiveness, showing excellent performances in many applications such as heterogeneous catalysts, [START_REF] Li | An Efficient Solid Acid Catalyst: Poly-p-Styrenesulfonic Acid Supported on SBA-15 via Surface-Initiated ATRP[END_REF][START_REF] Martín | Acid Hybrid Catalysts from Poly(Styrenesulfonic Acid) Grafted onto Ultra-Large-Pore SBA-15 Silica Using Atom Transfer Radical Polymerization[END_REF][START_REF] Jiang | Catalytic Performance of Strong Acid Catalyst: Methyl Modified SBA-15 Loaded Perfluorinated Sulfonic Acid Obtained by the Waste Perfluorinated Sulfonic Acid Ion Exchange Membrane[END_REF][START_REF] Ma | SBA-15-Supported Poly(4-Styrenesulfonyl(Perfluorobutylsulfonyl)Imide) as Heterogeneous Brønsted Acid Catalyst for Synthesis of Diindolylmethane Derivatives[END_REF][START_REF] Alghamdi | Catalytic Performance of SBA-15-Supported Poly (Styrenesulfonic Acid) in the Esterification of Acetic Acid with n-Heptanol[END_REF] permselective mesoporous thin films [START_REF] Calvo | Facile Molecular Design of Hybrid Functional Assemblies with Controllable Transport Properties: Mesoporous Films Meet Polyelectrolyte Brushes[END_REF][START_REF] Tom | Fabrication and in Situ Functionalisation of Mesoporous Silica Films by the Physical Entrapment of Functional and Responsive Block Copolymer Structuring Agents[END_REF] and drug carriers, [START_REF] Yuan | Preparation of PH-Responsive Mesoporous Silica Nanoparticles and Their Application in Controlled Drug Delivery[END_REF][START_REF] Samart | Preparation of Poly Acrylic Acid Grafted-Mesoporous Silica as PH Responsive Releasing Material[END_REF][START_REF] Yang | Smart Nanoreactors for PH-Responsive Tumor Homing, Mitochondria-Targeting, and Enhanced Photodynamic-Immunotherapy of Cancer[END_REF] to name a few. Surprisingly, the proton conduction performance of OMS functionalized with strong polyacids has never been reported, despite the discovery of promising porous materials as proton conductors, e.g. porous sulfonated organic polymers [START_REF] Li | Simple and Universal Synthesis of Sulfonated Porous Organic Polymers with High Proton Conductivity[END_REF] and monosulfonated OMS. [START_REF] Mikhailenko | Solid Electrolyte Properties of Sulfonic Acid Functionalized Mesostructured Porous Silica[END_REF][START_REF] Marschall | Ordered Functionalized Silica Materials with High Proton Conductivity[END_REF][START_REF] Marschall | Nanoparticles of Mesoporous SO3H-Functionalized Si-MCM-41 with Superior Proton Conductivity[END_REF][START_REF] Bibent | Proton Conductivity versus Acidic Strength of One-Pot Synthesized Acid-Functionalized SBA-15 Mesoporous Silica[END_REF][START_REF] Fujita | Enhancement of Proton Transport by High Densification of Sulfonic Acid Groups in Highly Ordered Mesoporous Silica[END_REF] Obviously, the substitution of a monosulfonic acid function with a polysulfonic acid in the porous framework of a stable inorganic backbone is expected to increase the number of labile protons, resulting in a higher density of acid functions for a similar mesopore size. Therefore, the already promising proton conductivity (7.2 x 10 -3 S.cm -1 ) observed by Mikhailenko et al. [START_REF] Mikhailenko | Solid Electrolyte Properties of Sulfonic Acid Functionalized Mesostructured Porous Silica[END_REF] for monosulfonic acidfunctionalized OMS particles should be improved.

The lack of studies on materials functionalized with polysulfonic acids is due to the difficulty in achieving dense and homogeneous functionalization of the mesoporous silica as well as controlled polymer characteristics, including dispersity and block molecular weights. Most current polymer functionalization strategies rely on post-functionalization of mesopores using the "grafting to" [START_REF] Yuan | Preparation of PH-Responsive Mesoporous Silica Nanoparticles and Their Application in Controlled Drug Delivery[END_REF] or "grafting from" [START_REF] Li | An Efficient Solid Acid Catalyst: Poly-p-Styrenesulfonic Acid Supported on SBA-15 via Surface-Initiated ATRP[END_REF][START_REF] Martín | Acid Hybrid Catalysts from Poly(Styrenesulfonic Acid) Grafted onto Ultra-Large-Pore SBA-15 Silica Using Atom Transfer Radical Polymerization[END_REF] approaches. In these respective approaches, a polymer is either grafted onto the pore surface or grown by in situ polymerization from the activated surface. While the "grafting to" method results in inhomogeneous functionalization with presynthesized polymers of controlled molecular weight, the "grafting from" method produces polymers of polydisperse molecular weight, which can ultimately be densely integrated, but at the expense of the pore volume required for most of applications. [START_REF] Kruk | Surface-Initiated Controlled Radical Polymerization in Ordered Mesoporous Silicas[END_REF] Furthermore, regardless of the post-functionalization strategy, an ordered porous material must first be synthesized. This requires numerous steps involving the use of many solvents and reagents, high energy consumption to remove the pore-forming sacrificial agent directing the silica structure, which subsequently limits the large-scale development of such polymerfunctionalized OMS. Alternatively, one-pot synthesis strategies for functionalizing OMS with polyacids should be suitable methods to overcome those limitations, using a pre-synthesized polyacid directly introduced into the synthesis medium. In this context, perfluorobutylsulfonylimide-based alkoxysilanes were co-condensed with tetraethoxysilane (TEOS) under sol-gel conditions. [START_REF] Yang | Mesoporous Silica Supported Water-Stable Perfluorobutylsulfonylimide and Its Catalytic Applications in Esterification[END_REF] However, non-uniform agglomerated mesoporous particles with irregular porosity were obtained. Even the presence of a structure-directing agent in the synthesis medium did not improve the formation of an ordered silica mesophase. As reported, [START_REF] Paris | Silica Nanoparticles Decorated with Polymeric Sulfonic Acids Trigger Selective Oxidation of Benzylic Methylenes to Aldehydic and Ketonic Carbonyls[END_REF] silica nanoparticle agglomerates without structure or with missing porosity were obtained by mixing quaternary ammonium surfactants with perfluorosulfonic acid ionomers and TEOS.

These examples highlight the difficulty of precisely controlling the interactions between all the reactants in a one-pot synthesis, which is crucial for obtaining polyacid functionalized OMS with controlled structure and specific functionalities. Given these limitations, it is imperative to design new routes for direct functionalization of OMS with polyacids to improve the quality and efficiency of functionalization. This includes controlling the length, dispersity and spatial distribution of polymers, combined with reducing the environmental impact of the synthesis method.

In this work, we present a method for the direct and environmentally friendly functionalization of ordered mesoporous silica with strong polyacids of perfectly defined length, which are densely and homogeneously distributed in the mesopores. The strategy relies on the use of a novel structure-directing agent, namely polyion complex (PIC) assemblies, which incorporate desired functionalities that are masked during the sol-gel process, preventing disruption of the mesostructuring process. As reported, [START_REF] Houssein | Mesoporous Silica Templated by Polyion Complex Micelles: A Versatile Approach for Controlling the Mesostructure[END_REF][START_REF] Molina | PH-Mediated Control over the Mesostructure of Ordered Mesoporous Materials Templated by Polyion Complex Micelles[END_REF][START_REF] Baccile | Ecodesign of Ordered Mesoporous Materials Obtained with Switchable Micellar Assemblies[END_REF][START_REF] Richard | Dual Control of External Surface and Internal Pore Structure of Small Ordered Mesoporous Silica Particles Directed by Mixed Polyion Complex Micelles[END_REF] PIC micelles formed with a weak polyacidbased double-hydrophilic block copolymer (DHBC) and an oppositely charged polyelectrolyte direct the structure of PIC-silica materials with tunable textural and structural properties (2D hexagonal, lamellar, cubic), called MesoPIC materials. In most subsequent studies, the PIC assembly has been completely removed from the organic-inorganic hybrid material. In the present study, we seek to preserve the DHBC of the PIC assembly in the material to give acidic properties to the pores, using a strong polyacid-based DHBC containing a poly(styrene sulfonic acid) (PSS) block. The new PSS-functionalized OMS is hereafter referred to as MesoPIC-PSS.

The process operates under purely aqueous conditions and at room temperature, making it an environmentally friendly and energy-efficient synthesis procedure. The resulting MesoPIC-PSS material obtained has a very high volume density of 1 SO3H function per nm [START_REF] Ngo | One-Pot Synthesis of Organic Polymer Functionalized Mesoporous Silicas[END_REF] of mesopore volume; it also exhibits a superprotonic conductivity, i.e.  = 2.4 x 10 -2 S.cm -1 at 363 K / 95 % Relative Humidity (RH), as well as constant performances maintained for 7 days, proving its excellent hydrolytic and chemical stability. These results demonstrate for the first time the interest of mesoporous silica functionalized with polyacids as proton conductors.

Experimental Section

Chemicals

Oligochitosan with a molar mass of 2500 g.mol -1 was purchased from Creative PegWorks® and used as received. A deacetylation degree of 97% was measured by 1 H NMR. The following global molar composition was determined by a combination of 1 H NMR analysis, TGA, elemental analysis, aqueous acid-base titration, conductivity and chloride concentration measurements: DeacetylatedUnit(0,97), AcetylatedUnit(0,03), C3H5O3(0,52), Cl -(0,48), H2O(0,85), where C3H5O3 is lactate counter-ion. A small amount of inorganic cations may be present but was not detected. Tetraethoxysilane (TEOS), 2-(Dodecylthiocarbonothioylthio)-2methylpropionic acid (DDMAT, 98%), sodium 4-vinylbenzenesulfonate (SSNa, ≥ 90% technical grade), 4,4′-Azobiscyanopentanoic acid (ACPA, Aldrich, 98%) were purchased from Sigma-Aldrich and used as received. Poly(ethylene glycol) methyl ether acrylate (APEO, Mn =480 g.mol -1 , 8.5 EO units on average) was purchased from Sigma-Aldrich; prior to use, APEO was stirred for 30 minutes with basic alumina and filtered to remove the inhibitor. The chemical formula of DDMAT, APEO, and SSNa are represented in Figure S1.

Synthesis of PAPEO-b-PSS

Synthesis of Poly(poly(ethylene glycol)-methyl ether acrylate) by aqueous RAFT

Polymerization of APEO

Synthesis of P(APEO)25-DDMAT was adapted from a previous work [START_REF] Ma | Synthesis of Hydrophilic/CO2-Philic Poly(Ethylene Oxide)b-Poly(1,1,2,2-Tetrahydroperfluorodecyl Acrylate) Block Copolymers via Controlled/Living Radical Polymerizations and Their Properties in Liquid and Supercritical CO2[END_REF] and performed as follows: DDMAT (0.55 g, 1.5 mmol, 1 eq), APEO (15 g, 31.3 mmol, 20.9 eq), ACPA (0.08 g, 0.3 mmol, 0.2 eq) and distilled water (12.79 g) were introduced in a round-bottom flask and stirred (solids content= 55 %). The mixture was degassed with nitrogen bubbling for 30 min under magnetic stirring. The mixture was then heated in an oil bath at 353 K for 4 h under nitrogen atmosphere. At the end of the polymerization, water was evaporated under reduced pressure. Degree of polymerization (DP) and conversion were determined by [START_REF] Soler-Illia | Multifunctional Hybrids by Combining Ordered Mesoporous Materials and Macromolecular Building Blocks[END_REF] H NMR (DPP(APEO) = 25.4, monomer conversion > 99.2%). NMR sample preparation: 0.5 mL of D2O was added to 0.1 mL of crude polymerization medium and quenched in liquid nitrogen. Determination of Mn, Mw and dispersity Đ was performed by size exclusion chromatography in THF (SEC-THF) with PMMA calibration (Mn = 8040 g.mol -1 , Mw = 9610 g.mol -1 and Đ = Mw/M = 1.19). The full characterization is given in Figures S2 andS3.

Synthesis of Poly(poly(ethylene glycol)-methyl ether acrylate)-b-poly(sodium 4vinylbenzenesulfonate) by aqueous reversible addition-fragmentation chain transfer (RAFT) polymerization

The synthesis of the DHBC was adapted from previous works [START_REF] Bathfield | Asymmetric Neutral, Cationic and Anionic PEO-Based Double-Hydrophilic Block Copolymers (DHBCs): Synthesis and Reversible Micellization Triggered by Temperature or PH[END_REF][START_REF] Molina | Drug-Polymer Electrostatic Complexes as New Structuring Agents for the Formation of Drug-Loaded Ordered Mesoporous Silica[END_REF][START_REF] Phimphachanh | Characterization of Diblock Copolymers by Capillary Electrophoresis: From Electrophoretic Mobility Distribution to Distribution of Composition[END_REF] and performed as follows: P(APEO)25-DDMAT (15.02 g, 1.22 mmol, 1 eq), sodium 4-vinylbenzenesulfonate (SSNa, 6.92 g, 30.3 mmol, 24.3 eq), ACPA (0.0702 g, 0.25 mmol, 0.2 eq) and distilled water (9.15 g) were introduced in a round bottom flask and stirred (solids content= 70 %). The mixture was degassed with nitrogen bubbling for 30 min under magnetic stirring. The reaction medium was then heated in an oil bath at 353 K for 4 hours under nitrogen atmosphere. The resulting double hydrophilic block copolymer (DHBC) was then lyophilized. Degree of polymerization (DP) and conversion were determined by [START_REF] Soler-Illia | Multifunctional Hybrids by Combining Ordered Mesoporous Materials and Macromolecular Building Blocks[END_REF] H NMR (DPP(SSNa) = 21.4, monomer conversion > 99.9 %, Mn, P(APEO)-b-P(SSNa) = 16 954 g.mol -1 , Figure S4). Due to the polyionic structure, no SEC analysis was realized on this DHBC but a capillary-electrophoresis (CE) experiment was performed (Figure S5), according to conditions previously reported, [START_REF] Phimphachanh | Characterization of Diblock Copolymers by Capillary Electrophoresis: From Electrophoretic Mobility Distribution to Distribution of Composition[END_REF] to check the presence of the covalent link between the two blocks in synthesized P(APEO)25-b-P(SSNa)21-DDMAT.

Synthesis of the polyacid-functionalized silica material MesoPIC-PSS

MesoPIC-PSS particles were prepared following a one-pot synthesis approach. The concentrations of oligochitosan and TEOS were fixed with respect to the number of styrene sulfonate (SS) and ethylene oxide (EO) units of the PAPEO-b-P(SSNa) copolymer: molar ratios were set at 1 nitrogen per SS (N/SS = 1) and 1 silicon per EO (Si/EO = 1). The final concentration of the reaction medium was set at 2.0 wt% of DHBC polymer. 100 mg of PAPEO-b-P(SSNa) was dissolved in a mixture of 4.04 g of deionized water and 27.2 mg of oligochitosan. The pH of the solution was then adjusted to 2 using H2SO4 (1 M). Into the obtained clear solution, 244 mg of TEOS was added and the solution was stirred at 303 K for 1h. After that, the pH was rapidly increased to 4.5 with an aqueous NaOH solution (2 M). The mixture was stirred slowly for 5 hours at 303 K. The precipitate formed was then recovered by centrifugation, and dispersed in 12 ml of aqueous solution at pH=4.5 for 20 min to remove excess polymer. The product was centrifuged, the supernatant was eliminated, and the solid was dispersed and stirred for 16h in 12 ml of an aqueous solution with 0.05M H2SO4 and 1M NaCl. The dispersion was centrifuged and the process was repeated a first time for 3h with 12ml of an aqueous solution with 0.5M H2SO4 and 1M NaCl, and a second time for 1h with 12ml of an aqueous solution with 0.5M H2SO4 but without NaCl. Finally, the solid was recovered by centrifugation and air-dried at 353 K for 16 hours; typically, 100 mg of dry powder was obtained.

Characterization

Structural characterizations TEM images were acquired on microtome cut samples using a JEOL 1200 EX II operating at 120 kV. Samples SEM images were obtained using a Hitachi S4800 FEG-HR operating at 5 kV. SAXS experiments were run using a homemade instrument with a copper X-ray tube =1.54 Å). Nitrogen sorption analyses were performed at 77K using a Micromeritics Tristar 3000 after degassing samples one night at 353 K under secondary vacuum. The specific surface area was calculated using the BET equation in the range 0.15 < p/p° < 0.35, volumes of micro-and mesopores were calculated using the alpha-S method, distributions of pore diameters were calculated on the adsorption branch of the sorption isotherm using Broekhoff and Boer equations.

Chemical characterization

TGA were performed using a PerkinElmer STA 6000 under air flow from 300 K to 1173 K with a 10 K.min -1 rate. The silica content SiO2 wt.% was determined as the residual final mass at 1173 K. Elemental analyses were performed on a Elementar Vario Micro Cube to determine C wt.%, S wt.% and N wt.%, respectively the mass content of carbon, sulfur and nitrogen. TGA-MS measurements were done with a Setaram STA-MS operating under air, from 303 K to 1173 K with a heating rate of 5 K.min -1 . 29 Si MAS NMR experiments were run on a Varian VNMRS 400 MHz at 293K with a speed rotor rotation of 5 kHz. Pulses of π/6 were repeated every 60 s. Octa(dimethylsiloxy)-octasilsesquioxane (Q8M8) was used as the chemical shift reference (δ = -12,6 ppm). The proton exchange capacity was determined by titration of the H + in solution after ionic exchange with Na + . Typically, 50 mg of MesoPIC material was added to 10 ml of an aqueous solution of sodium chloride (NaCl, 1M) and stirred for 24h at room temperature. The suspension was then filtered and the obtained solution was titrated with a sodium hydroxide aqueous solution (NaOH, 0.01M) using a pH-meter. DLS analyses of PIC micelles were performed using a Malvern 4800 spectrogoniometer device with a laser source of 50 mW operating at 582 nm. Hydrodynamic diameters were obtained from autocorrelation functions fitted with CONTIN algorithm. Suspensions were prepared in deionized water with a concentration of PAPEO-b-P(SSNa) of 2wt.% and with a molar ratio between oligochitosan units and SSNa units of 1. The pH was adjusted at 4.5 with NaOH 2M solution. Analyses were performed without filtration.

Water adsorption. Water adsorption was performed at 298 K using a home-made apparatus. [START_REF] Trens | Adsorption and Separation of Xylene Isomers Vapors onto the Chromium Terephthalate-Based Porous Material MIL-101(Cr): An Experimental and Computational Study[END_REF][START_REF] Fertier | Responsive Bridged Silsesquioxane[END_REF] In brief, this sorption device is based on manometric measurements of the equilibrium water pressure before and after each sorption step. The pressure is recorded using two capacitive pressure gauges (10 and 1000 Torr). The whole device is placed in a climatic chamber allowing for a thermal stability better than 0.1 K at constant temperature (here, 298 K). The sample cell can be disconnected from the sorption line, allowing for an activation of the sample before the sorption measurements. The sorption thermodynamic equilibrium is ensured for each sorption step by setting a time criterion at 300s, which means that the pressure must remain constant for 300 s before recording the equilibrium pressure.

Depending on the sorption process (monolayer sorption, capillary condensation, ...), equilibrium times are different and they can reach up to 600 min for a single step. [START_REF] Trung | Hydrocarbon Adsorption in the Isostructural Metal Organic Frameworks MIL-53(Cr) and MIL-47(V)[END_REF] Conductivity. The impedance measurements were carried out using a Solartron analytical Modulab XM MTS over a frequency range from 1Hz to 10 6 Hz with an ac applied voltage of 0.020 V. The powder sample was placed between two gold electrodes in a parallel plate capacitor configuration with an annular Teflon spacer to provide insulation, allowing the use of the "two-probe" method for the electrical measurements. The Relative Humidity (RH) and Temperature (T) were monitored by a 'Bench-Top type climatic chamber' from ESPEC Corporation. It is composed of a small liquid water "bath" located in the mixing plenum; as the chamber air is drawn into the plenum, it passes the bath, picking up vapor submitted to the sample. Impedance was recorded with varying RH from 35 to 95 %, while T was fixed in the [298 K -363 K] range. The sample was equilibrated for at least 12 hours, before changing the RH/T conditions.

The resistance R of the solid was determined from the analysis of the Nyquist plots (-Z" = f(Z')), i.e. from the real-axis intercept of the signal and/or the fitting procedure with equivalent circuits using the ZView Software. In the latter case, the best fit was obtained considering the following equivalent circuit model: where R represents the bulk resistance, CPE1 and CPE2 refer to the constant phase elements for the non-ideal capacitance behaviors of the bulk solid and the electrodes, respectively. The conductivity values were obtained using σ=1/R×l/S, where l and S are the sample thickness and surface, respectively. As shown in the impedance measurements section of SI, both approaches converge towards comparable values of , supporting that both methods can be equally considered. The conductivity values are given with an uncertainty of  = 0.01. The activation energy associated with the proton transport process was deduced from the linear fitting of ln (σT) versus 1/T, according to the Arrhenius equation σ(T) = σ0/T exp(-E/kT),

where k is the Boltzmann constant.

Results and discussion

Scheme 1 describes the general concept of the functional material synthesis. First, a mesostructured PIC-silica hybrid phase is synthesized by macroscopic precipitation at room temperature in water at pH = 4.5 due to a cooperative organization between a P(APEO)25-b-P(SSNa)21 DHBC of well-defined block molecular weights synthesized by controlled radical polymerization (see Characterization of the DHBC section and Materials and Methods section in SI), a hydrosoluble renewable biomass oligochitosan and silicic species obtained by prehydrolysing TEOS used as a silica source. Formation of the hybrid phase results from electrostatic complexation between the SO3 -functions of the DHBC and the NH3 + functions of oligochitosan, together with hydrogen bonding between the ethylene oxide units of the DHBC neutral blocks and the silanol groups of the silicic species. Upon dispersion of the hybrid material in an aqueous solution of hydrochloric acid (pH = 1) and sodium chloride (1 mol.L -1 ), the electrostatic interactions between the polyions are disrupted, which ensures the dissociation of PIC micelles. The objective is threefold: (i) elute the oligochitosan for potential reuse (recycling of the micellization auxiliary), [START_REF] Baccile | Ecodesign of Ordered Mesoporous Materials Obtained with Switchable Micellar Assemblies[END_REF] (ii) generate mesoporosity in the material, (iii) decorate the mesopore walls with accessible sulfonic acid-rich polyions. The PAPEO chains of the copolymer remain anchored to the silica walls via hydrogen bonds, allowing dense and homogeneous functionalization of the mesopores with the PSS polyacid chains. Scheme 1: Illustration of the eco-designed pathway to form large-pore ordered silica functionalized with polyacid chains, called MesoPIC-PSS

The PSS-functionalized silica material (MesoPIC-PSS) exhibits a long-range ordered 2D hexagonal structure. TEM images of the particles sectioned by ultra-microtome cutting reveal the ordered hexagonal network of the cylindrical mesopores (Figure 1a, supplementary images on Figure S6). The SAXS pattern shows four well-defined peaks, indicating the longrange order of the P6mm mesostructure with a d0 spacing of 14.9 nm (Figure 1b). SEM images reveal the long-range ordered internal structure in micrometer-sized particles (Figures 1c andS7).

The chemical composition of MesoPIC-PSS was determined by combining thermogravimetric analysis (TGA) and elemental analysis (EA). According to TGA, the residual mass recorded at T = 1173 K corresponds to 47 wt.% of silica. [START_REF] Molina | Drug-Polymer Electrostatic Complexes as New Structuring Agents for the Formation of Drug-Loaded Ordered Mesoporous Silica[END_REF] Si MAS NMR analysis (Figure 1d) reveals that 51 %, 46 % and 3 % of Q4, Q3 and Q2 species respectively, constitute the silica framework, resulting in an overall formula of SiO1.74(OH)0.52. It should be noted that a silanol-rich MesoPIC-PSS was successfully synthesized in this way in conjunction with the absence of any calcination step during the elaboration process. Most importantly, the nitrogen content measured from the EA (< 0.1 wt. %N, Table S1) highlights a complete removal of oligochitosan from the elution process, ensuring the complete decomplexation of sulfonic acid functions. According to EA, the MesoPIC-PSS material contains a large amount of PAPEO-b-PSS in the mesopores (886 mg.g -1 SiO2, Table 1), which corresponds to 99% of the DHBC polymers initially present in the hybrid MesoPIC material. This key feature ensures an intrinsically dense and homogeneous spatial distribution of the polyacid chains in the mesopores considering that: (i) the PAPEO-b-PSS polymers associated with oligochitosan are the pore structuring agents, and (ii) they remain in the silica-based material during the PIC disassembly process. The density of sulfonic acid functions was calculated to be 1.16 mmol.g SiO2 -1 or 0.54 mmol.g material -1

, which is within the range of those reported in the literature for mesoporous silica particles functionalized with mono-or polysulfonic acid. Table S2 lists the relevant sulfonic acidfunctionalized OMS in particle or thin film form, including their density of sulfonic acid functions as well as their textural properties. Among them, sulfonic acid-functionalized mesoporous silica thin films synthesized by Fujita et al. exhibit the highest density of sulfonic acid functions. [START_REF] Fujita | Enhancement of Proton Transport by High Densification of Sulfonic Acid Groups in Highly Ordered Mesoporous Silica[END_REF] The chemical composition of the copolymer in the material was preserved, as indicated by the similar value of the C/S mass ratio in the final material (9.8) compared to that of the raw PAPEO-b-PSS copolymer (10.5). Most importantly, the volume density of SO3H functions in the mesopores reaches 1 function per nm 3 of mesopore volume, which is the highest value reported to date for mesoporous particles (calculations are provided in ESI together with comparisons to values for other materials in the literature, Table S2). In addition, the acidic protons in MesoPIC-PSS are highly accessible: 80% of them were exchanged in aqueous solution, giving an exchange capacity of 0.44 mmol H + .g material -1

. This result contrasts with a previous report of a PSS-functionalized SBA-15, in which only 14% of the acidic functions were accessible to ion exchange due to the hydrophobicity of the material. [START_REF] Martín | Acid Hybrid Catalysts from Poly(Styrenesulfonic Acid) Grafted onto Ultra-Large-Pore SBA-15 Silica Using Atom Transfer Radical Polymerization[END_REF] In our case, the excellent accessibility may be the consequence of (i) the high hydrophilicity of MesoPIC-PSS due to the numerous silanol and SO3H functions, and (ii) the homogeneous distribution of the polymer chains within the pores. Therefore, the present strategy provides mesoporous materials with large pores, high pore volume and very high volume density of accessible and strong acid functions in the mesopores.

Table 1: Textural properties and chemical composition of MesoPIC-PSS.

Vmicro / cm 3 .g -1 Vmeso / cm 3 .g -1 a dmeso / nm b d0 / nm c twall /nm d SBET / m².g -1 mgDBHC .gSiO2 -1 e

mmol-SO3H

.gSiO2 -1 e 0.00 0.24 10.0 14.9 The dynamic thermal stability of MesoPIC-PSS was evaluated under air by TGA coupled with mass spectrometry (TGA-MS, Figure S8). The PAPEO-b-PSS copolymer decomposes from 475 K to 950 K in the material, generating CO2, H2O and SO2. The thermal decomposition of PAPEOb-PSS occurs in the same temperature range (Figure S9), indicating that the thermal stability of the copolymer is not affected by the mesoporous structure, probably due to the highly porous structure and large diameter of the mesopores.

The accessibility to these large pores from the outer surface of the particles is clearly evident in the SEM images in Figure 1c. The sharp symmetric hysteresis loop of the type IV nitrogen sorption isotherm (Figure 2a) further validates the accessibility and homogeneity of the mesoporosity in the particles. Capillary condensation of nitrogen occurs in MesoPIC-PSS in the same manner as in the case of polymer-free OMS (calcined material), suggesting that nitrogen sorption is not affected by the presence of polymers in the mesopores (Table 1). The pore size distribution indicates monodisperse mesopores of 10.0 nm in diameter. A consistent diameter (10.5 nm) was estimated by analyzing the TEM images (Figure S10), from which the pores are distinctly delineated from the silica walls, confirming the fact that nitrogen sorption analysis allows access to the mesopore diameter as if no polymers were present. Given the presence of many functional polymers in the mesopores and the large thickness of the silica walls (7 nm, Table 1), the mesopore volume (Vmeso) of 0.24 cm 3 . g material -1

can be described as really high.

No microporosity could be detected by t-plot in contrast to SBA-15-type materials, in which micropores are generated by removal of the micelle PEO corona by calcination. [START_REF] Kruk | Characterization of the Porous Structure of SBA-15[END_REF] In the present approach, the stable anchoring of the comb-type PAPEO blocks in the silica walls and the absence of any calcination step avoid microporosity formation.

In summary, the MesoPIC process can produce porous silica-based solids with both high porosity and a high degree of functionalization. On one hand, the pore volume, which results from the elution of the oligochitosan, is related to the amount of oligochitosan initially incorporated into the material. On the other hand, the degree of functionalization of the MesoPIC-PSS material is related to the amount of block copolymer incorporated in the material. Knowing that the core of the PIC assembly results from the charge compensation between the PSS and the oligochitosan, this suggests that the pore volume and the degree of PSS functionalization should both concomitantly increase when the length of the PSS block is increased. This represents a major advantage over the former grafting post-functionalization approaches, in which an increase in polymer block length or degree of grafting inherently leads to a decrease in pore volume. Water sorption analyses were performed to evaluate the water sorption capacity and affinity of the functional surfaces for water, which can then be related to proton conductivity properties. The water vapor sorption isotherm recorded at 298 K is shown in Figure 2b. The maximum water uptake at RH = 97% reaches 308 mgH2O.g material -1

. The linear shape at low relative pressure is characteristic of a mixture of type IV and V sorption isotherms, [START_REF] Sing | Reporting Physisorption Data for Gas/Solid Systems with Special Reference to the Determination of Surface Area and Porosity (Recommendations 1984)[END_REF] as previously reported for bare and sulfonic functionalized mesoporous silica. [START_REF] Maaz | Systematic Investigation of the Pore Structure and Surface Properties of SBA-15 by Water Vapor Physisorption[END_REF][START_REF] Timm | A Novel and Versatile Grafting Procedure: Toward the Highest Possible Sulfonation Degree of Mesoporous Silica[END_REF] Maaz et al.

proposed using the water pore filling volume ratio at p/p° = 0.1 as an indicator of the hydrophilicity of the material, which has been shown to increase with silanol density and microporosity. [START_REF] Maaz | Systematic Investigation of the Pore Structure and Surface Properties of SBA-15 by Water Vapor Physisorption[END_REF] For MesoPIC-PSS, the water pore filling volume ratio at p/p° = 0.1 was calculated to be 0.06, which is one of the highest values of all materials reported by Maaz et al. [START_REF] Maaz | Systematic Investigation of the Pore Structure and Surface Properties of SBA-15 by Water Vapor Physisorption[END_REF] Since no microporosity was found for MesoPIC-PSS, the strong hydrophilicity is due either to the presence of the many silanols revealed by [START_REF] Molina | Drug-Polymer Electrostatic Complexes as New Structuring Agents for the Formation of Drug-Loaded Ordered Mesoporous Silica[END_REF] Si MAS NMR (see above), the large number of sulfonic acid functions or most likely a combination of both. A comparison of the water and nitrogen sorption isotherms provides useful insights. A clear difference in the isotherms appears at intermediate RH levels. The high uptake in this region can been attributed to a capillary condensation process in the case of nitrogen adsorption; however, in the case of water adsorption, the curve is not as steep in the region of intermediate RH levels. In addition, water uptake increases gradually up to saturation. This could be related to the adsorption of water molecules onto the many sulfonic acid groups of the polymeric chains, as usually observed for Proton Exchange Membranes (PEM). [START_REF] Legras | Sorption and Diffusion Behaviors of Water in Nafion 117 Membranes with Different Counter Ions[END_REF] For example, perfluorosulfonic acid membranes with hydrophobic and hydrophilic domains show no capillary condensation of water, which first condenses on the sulfonic acid groups before percolation. [START_REF] Legras | Sorption and Diffusion Behaviors of Water in Nafion 117 Membranes with Different Counter Ions[END_REF][START_REF] Tsonos | Water Sorption and Dielectric Relaxation Spectroscopy Studies in Hydrated Nafion (-SO3K) Membranes[END_REF][START_REF] Wu | Differences in Water Sorption and Proton Conductivity between Nafion and SPEEK[END_REF] In contrast, nitrogen, which has no specific affinity for the polymer, condenses as it does in the pores of bare mesoporous silica. This would explain the larger volume of water adsorbed (0.31 cm 3 .g -1 )

compared to nitrogen (0.24 cm 3 .g -1 ). Since MesoPIC-PSS has both a hydrophilic silica surface and hydrophilic PSS blocks, one would expect water molecules to adsorb onto both types of sites and condense via a combination of the mechanisms reported for bare mesoporous silica and PEM. The structuration of the polymer chains in the mesopores could also play a role in the differences between nitrogen and water adsorption isotherms. While at 77K these chains are rigid and cover the surface of the mesopore walls, at 298 K, the polymer chains are flexible and can adopt a brush-like conformation. In the first case, the mesoporosity is fully accessible to nitrogen and capillary condensation can therefore occur according to the Kelvin's law. In the second case, the presence of polymer chains in the mesopores prevents the formation of a homogeneous meniscus of water in the mesopores, which is a prerequisite for homogeneous capillary condensation. Therefore, the mesopores can be progressively filled with water as the relative pressure increases. Because of the inflection of the water adsorption isotherm noticeable at 60% RH, capillary condensation probably occurs between 60% RH and 95% RH. Still, further experiments or simulations are needed to confirm these interpretations.

Given the thermal stability of the inorganic framework, the well-defined mesoporous structure, the dense functionalization and the numerous silanol functions, MesoPIC-PSS appears to be a promising candidate for proton conductivity applications. [START_REF] Noureddine | Tailoring the Hydrophilic/Lipophilic Balance of Clickable Mesoporous Organosilicas by the Copper-Catalyzed Azide-Alkyne Cycloaddition Click-Functionalization[END_REF][START_REF] Trens | Evolution of Surface Composition, Porosity, and Surface Area of Glass Fibers in a Moist Atmosphere[END_REF] AC impedance spectroscopy was used to explore its proton conductivity at 95% Relative Humidity for temperatures ranging from 298 K to 363 K. The Nyquist plots are shown in Figure 3a. They correspond to very low impedances and are dominated by the capacitive tail due to the ionicblocking electrodes. The total impedance was extrapolated from the linear region to the real axis. The corresponding conductivity values increase from 2.5 x 10 -3 S.cm -1 to 2.4 x 10 -2 S.cm -1 upon heating from 298 K to 363 K, illustrating the superprotonic conductivity behavior of MesoPIC-PSS (σ > 10 -4 S.cm -1 ). [START_REF] Colomban | Proton Conductors: Solids, Membranes and Gels[END_REF] Interestingly, the performances of MesoPIC-PSS exceed that of the best sulfonic acid based OMS particles (𝛔 𝟑𝟔𝟑𝐊,𝟏𝟎𝟎%𝐑𝐇 𝐇 + = 𝟏. 𝟏𝟎 -𝟐 𝐒. 𝐜𝐦 -𝟏 ), while exhibiting a larger pore diameter (Table S3). Furthermore, the outstanding performance recorded at 363 K and 95% RH is maintained over at least one week (Figure 3b), highlighting the long-term hydrothermal stability of the silica at the T/RH operating conductivity conditions. This is further highlighted by the comparable TEM images of the material collected before and after the impedance measurements (Figure S11). These results confirm the robustness and high hydrothermal stability of MesoPIC-PSS, which can be attributed to the thick walls of the material and the absence of any microporosity.

The activation energy for proton transfer in MesoPIC-PSS was deduced from an Arrhenius plot (Figure 3c). It reaches 0.28 eV, which is consistent with a Grotthuss transfer mechanism (E < 0.40 eV). [START_REF] Grotthuss | Sur La Décomposition de l'eau et Des Corps Qu'elle Tient En Dissolution à l'aide de l'électricité Galvanique[END_REF] It is assumed that protons are transferred from -SO3H groups to water molecules interacting through hydrogen-bonds, creating a pathway for proton migration into the silica porosity. To verify that water assists the proton transfer, the humidity dependence of the conductivity for MesoPIC-PSS was studied at 298 K and 363 K (see Figures S12-S14 and Table S4). The significant increase in conductivity with increasing humidity, from 40 % to 95 % of RH, confirms that H2O molecules are involved in the proton migration mechanism, and this trend is valid regardless of the temperature. Propitiously, the dependence of conductivity with relative humidity perfectly follows the water uptake profile obtained from the manometric adsorption isotherm (Figure 3d). This excellent correlation confirms that the water-mediated proton conductivity of MesoPIC-PSS is proportional to the concentration of adsorbed water molecules inside the porosity, which are assumed to act as charge carriers through the path they create via the H-bond network. [START_REF] Nandi | Revisiting the Water Sorption Isotherm of MOF Using Electrical Measurements[END_REF] Noteworthy, conductivity measurements recorded at 303 K as a function of RH perfectly illustrates the acid character of the protogenic groups of the MesoPIC-PSS. This also provides evidence that the proton conduction process occurs through the intrinsic bulk solid, i.e. through the water-filled pores of the silica particles, and not through the grain boundaries or interparticle domains. [START_REF] Mileo | Proton-Conducting Phenolate-Based Zr Metal-Organic Framework: A Joint Experimental-Modeling Investigation[END_REF][START_REF] Mileo | Highly Efficient Proton Conduction in a Three-Dimensional Titanium Hydrogen Phosphate[END_REF][START_REF] Donnadio | A Layered Mixed Zirconium Phosphate/Phosphonate with Exposed Carboxylic and Phosphonic Groups: X-Ray Powder Structure and Proton Conductivity Properties[END_REF][START_REF] Levenson | Particle Size Dependence of Proton Conduction in a Cationic Lanthanum Phosphonate MOF[END_REF]

Conclusions

An environmentally friendly integrated strategy has been developed to prepare ordered mesoporous silica functionalized with strong polyacid chains that are densely and homogeneously distributed in the mesopores, allowing the preparation of superprotonic conductors. The synthesis process takes place in an aqueous solvent at room temperature and relies on the use of complex micelles of oppositely charged hydrophilic polymers. A functional polyacid DHBC complexed with an oligoamine (oligochitosan) was used to direct the structure of silica into an ordered 2D hexagonal mesostructure with large mesopores of 10 nm diameter.

The oligoamine was then released into water by changing the pH and ionic strength, generating mesoporosity and revealing the acidic functions. More than 99% of poly(styrene sulfonic acid) chains remained anchored in the material during the PIC dissociation step, demonstrating the high efficiency of the process resulting in an inherently dense and homogeneous distribution of functional polymers along the mesopores. The PSS chains have a known and well-defined degree of polymerization of 21, which leads to a large amount of acid functions in mesopores: 1.16 mmol per gram of silica, which corresponds to the highest volume density reported so far of 1 SO3H function per nm 3 of mesopore volume. The success of this direct and integrated strategy can be attributed to (i) the use of PIC micelles as structuring, pore-forming and functionalizing agents, and (ii) the selectivity of the elution process that allows the removal of the oligoamine while keeping the totality of the functional copolymers in the mesopores. Moreover, overcoming performance-limiting inhomogeneities, we designed a superprotonic conductor: the sulfonic acid functionalized MesoPIC-PSS material exhibits superprotonic conductivity, up to 0.024 S cm -1 at 95% RH and 363 K, with a Grotthus-like mechanism. This performance was extremely stable for at least seven days, due to the high hydrothermal stability of the mesoporous hybrid material characterized by thick silica walls, large pores and no microporosity.

These results demonstrate for the first time the major interest of stable large pore mesoporous materials functionalized by dense and homogeneous brushes of strong polyacids as solid-state proton electrolytes. Moreover, this strategy could be adapted to design Li-or Na-ion conducting materials. Furthermore, the versatility of the process can be easily exploited to control the ordered silica structure and to tune the pore properties of functionalized hybrid materials through the reactivity of PIC micelles to pH, concentration or ionic strength. [START_REF] Houssein | Mesoporous Silica Templated by Polyion Complex Micelles: A Versatile Approach for Controlling the Mesostructure[END_REF][START_REF] Molina | PH-Mediated Control over the Mesostructure of Ordered Mesoporous Materials Templated by Polyion Complex Micelles[END_REF] Overall, our synthesis strategy paves the way for the large-scale sustainable development of well-defined polymer-functionalized mesoporous materials.

Table of Contents

Figure 1 :

 1 Figure 1: a) TEM image, b) SAXS pattern, c) SEM image, and d) 29 Si one-pulse MAS NMR spectrum of MesoPIC-PSS. Q2, Q3 and Q4 refer to Si(OH)2(OSi)2, Si(OH)(OSi)3 and Si(OSi)4 species respectively.

16 a

 16 Total pore volume at p/p° = 0.975 b mean mesopore diameter at adsorption c lattice parameter d thickness of silica walls e Determined by elemental analysis and TGA.

Figure 2 :

 2 Figure 2: MesoPIC-PSS sorption isotherms of a) nitrogen measured at 77K and related pore size distribution, and b) water measured at 298 K.

Figure 3 :

 3 Figure 3: a) Nyquist plots of the impedance for MesoPIC-PSS recorded under 95% RH for temperature varying from 298 K to 363 K. b) Long term stability of the conductivity of MesoPIC-PSS at 363 K/95% RH. The dashed curve is a guide for the eye. c) Arrhenius type plot of the conductivity for MesoPIC-PSS under 95 % RH. The dashed line corresponds to the linear least-square fit. d) Evolution of the water mass uptake (black circle) and the conductivity (red squares) for MesoPIC-PSS versus the water relative pressure at 298 K upon adsorption.

Acknowledgments

The authors thank Thomas Cacciaguerra for TEM analysis at the University of Montpellier and Didier Cot for SEM analysis at the European Institute of Membranes; they also acknowledge Philippe Gaveau and Emmanuel Fernandez for NMR experiments, Philippe Dieudonné for SAXS analysis and Amine Geneste for TG-MS analysis (University of Montpellier). The electrical measurements were performed with the support of the Balard Plateforme d'Analyses et de Caractérisation (PAC Balard). This work was supported by the ANR MESOPIC project (ANR-15-CE07-0005) and the ANR PASSCATA project (ANR-16-CE07-0029-02).

Supporting Information description

Supporting information includes: characterization of the DHBC by [START_REF] Soler-Illia | Multifunctional Hybrids by Combining Ordered Mesoporous Materials and Macromolecular Building Blocks[END_REF]


R Allais
email: roman.allais@centrale-marseille.fr

A Capart

A Da

O Boiron

Intervertebral disc mechanical response to axial rotation: the consequences of a mobile centre of rotation on the anisotropic component

Keywords: Intervertebral disc, Degeneration, Anisotropy, Hyperelasticity, Instantaneous centre of rotation

Introduction

The intervertebral discs (IVDs) are essential to the mobility of the spine as they ensure the mechanical stresses transmission between vertebral bodies and play an important role in shock absorption. In everyday life, the discs are mechanically solicited in compression, flexion (either in the sagittal or transverse plane) and in axial rotation (Figure 1). For these last two movements, the literature highlights complex kinematics that cannot be reduced to a pure rotation but rather to a set of compound movements and, more importantly, to non-static instantaneous centres of rotation (ICR). In our ongoing works, we are investigating the biomechanical impacts of the ICR mobility and we hypothesize it would yield lower average stress. As these organs are heterogeneous, we detail in this study the behaviour of the various constitutive parts of the IVDs. This work sheds light on the consequences of the ICR mobility on the collagen fibres.

Methods

Model

We used a 3-dimensional finite element model of a porcine IVD (Figure 1) in which a poroelastic biphasic swelling formulation [START_REF] Mow | Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments[END_REF] for the IVD mechanical behaviour is coupled with a nutrients transport model. The two phases are incompressible, the extracellular matrix (ECM) is assumed to be hyperelastic and a fibre reinforced model is used to account for the anisotropy of the collagen network in the annulus fibrosus. The mesh chosen after a mesh convergence analysis was made up of 56 568 elements and 342 699 degrees of freedom. A preliminary stationary simulation was run to put the IVD at an unloaded physiological state and initialize the osmotic pressure and the nutrients concentration fields. We define two fibre families characterized by their mean orientation (e1 and e2) with respect to the transverse plane. Two orientation invariants, I4 and I6, are computed using the family directions and fed in a ρmodel as defined by [START_REF] Gasser | Hyperelastic modelling of arterial layers with distributed collagen fibre orientations[END_REF]. Hence our medium is defined by a strain energy density which can be split into three parts: a volumetric component (Wvolumetric), an isochoric isotropic component (Wisotropic) and an isochoric anisotropic component (Waniso1+Waniso2), as expressed below.

Kinematics

As reported in several studies, axial rotation of the spine is a compound motion. We chose the range of motion of [START_REF] Pearcy | Axial rotation and lateral bending in the normal lumbar spine measured by three-dimensional radiography[END_REF] for a L4/L5 spinal unit and we modelled the motion as a 2-second prescribed displacement of the superior endplate, the inferior endplate being rigidly fixed and the lateral face free to swell. As the location of the ICR is a controversial topic, we implemented two different definitions reported in the literature: a static centre of rotation at the IVD barycentre and a mobile one, moving posteriorly in the opposite side of the rotation as described in [START_REF] Schmidt | The relation between the instantaneous center of rotation and facet joint forces-a finite element analysis[END_REF]. All the numerical simulations were performed on COMSOL Multiphysics 5.5 on a workstation with an Intel(R) Xeon(R) W-11955M CPU and 64GB of RAM.

Results and discussion

The aims of this work were to analyse the consequences of a mobile ICR on the stress tensors associated with the strain energy densities and to quantify the contribution of the anisotropic component of the solid phase during axial rotation. Our results show that the assumption made on the ICR kinematics considerably influences the fibres solicitation. A static ICR at the IVD barycentre creates higher stress (up to 0.21MPa) in the right side of the IVD for a left axial rotation while the simulation with a moving ICR gives higher stress (up to 0.40MPa) in the left posterior side, as illustrated on Figure 2 (a,b). These results are of particular interest as they can bring new insights on the origin of some injuries related to the IVD.

Regarding the contribution of the anisotropic component with respect to the isotropic one, Figure 2 (c,d) displays the ratio of the first anisotropic eigenvalue (i.e. max (𝑆𝑝{𝑃𝐾 + 𝑃𝐾 })) over the first isotropic eigenvalue (max (𝑆𝑝 𝑃𝐾

)). Although the anisotropic stress is up to two times higher when considering a mobile ICR, its proportion with respect to the isotropic stress stays similar, with the maximal values reached in the inner annulus. It indicates that in the areas of higher deformation, most of the stress is due to the anisotropic component of the solid phase.

Conclusions

ICR mobility is known to increase along with IVD degeneration [START_REF] Gertzbein | Centrode patterns and segmental instability in degenerative disc disease[END_REF]. Our results can help to precise this relationship as it appears that even though a mobile ICR might globally reduce joint reaction forces [START_REF] Senteler | Sensitivity of intervertebral joint forces to center of rotation location and trends along its migration path[END_REF], it also tends to locally put more stress on the fibres which could subsequently fuel the degeneration process or ease the apparition of acute injuries. Future work could focus on how physical therapy or muscle reinforcement might slow down the degeneration by restraining the ICR mobility and reducing the amount of stress put on the fibres.

Figure 1 :

 1 Figure 1: 3D finite element model of an IVD. X axis points anteriorly, y points left and z points superiorly.

Figure 2 :

 2 Figure 2: (a, b) First anisotropic eigenvalue (in Pa) in the median transverse plane. (c, d) Ratio of the first anisotropic eigenvalue over the first isotropic eigenvalue in the median transverse plane.

Acknowledgements

This work is supported by the Emplois Jeunes Doctorants 2021 grant of the Région Provence Alpes Côte d'Azur.


Dominic Schröder

Thomas Leweke

Ralf Hörnschemeyer

Eike Stumpf

VOLUMETRIC VELOCITY MEASUREMENTS OF A HELICAL VORTEX PAIR

Keywords: vortex flow, vortex instability, volumetric velocimetry, helical vortex system, rotor wake

This paper presents the results of an experimental study fn a helical vortex pair which is influenced by centrifugal instability. Volumetric velocity measurements using particle tracking methods are performed in a water channel for a one-bladed rotor in hovering conditions. The vortex pair is generated by a specific tip design which provokes the single tip vortex to split up into two separate vortices. The objective of this modification is to alter the characteristic vortex parameters by means of mutual interactions to minimise the negative effects associated with blade-vortex interactions. Previous studies showed that for selected vortex systems an immediate rapid core growth for both vortices is detected. The reason for this conduct was found in the occurence of centrifugal instabilities affecting both vortices. These instabilities cause a rapid disturbance of the vortex pair resulting in the observed significant core growth. To reveal the three-dimensional characteristics of the instability, volumetric measurements are performed. The results are compared to stereo particle image velocimetry data for validation purposes. Due to the unsteady behaviour, instantaneous snapshots are investigated to detect the secondary vortices caused by the instability. A statistical evaluation shows that the presence of those structures is pronounced in the early wake age, corresponding to the strongest impact of the instability.

Introduction

The flows encountered in applications involving rotating blades such as e.g. helicopters or wind turbines are characterised by unsteady and three-dimensional flow phenomena. The superposition of concentrated tip vortices, the shear layers shed from the trailing edges of the rotor blades and the influence of the external flow create a highly complex flow field, which strongly affects the rotor aerodynamics. In contrast to the straight vortices emerging behind a plane, the superposition of rotational motion and the freestream velocity causes the tip vortices in the rotor wake to form a helical geometry while being convected downstream. Additional to their decreasing effect on the rotor performance due to the induced drag, these vortices can lead to detrimental fluid-structure interactions. With regard to helicopters, blade-vortex interactions (BVI) causing unwanted vibrations and an impulsive acoustic noise emission pose a limit to the flight envelope of the helicopter [START_REF] Yu | Rotor blade-vortex interaction noise[END_REF][START_REF] George | Helicopter Noise: State-of-the-Art[END_REF]. The vehemence of the interactions is mostly affected by the strength of the involved vortex, quantified by the vorticity gradients (concentrated vortices with small core radii posses high gradients in the core regions) and the bound circulation [START_REF] Schmitz | Helicopter impulsive noise: Theoretical and experimental status[END_REF][START_REF] Hardin | Concepts for reduction of blade/vortex interaction noise[END_REF]. In wind farms, the concentrated vortices resulting from the turbine upstream have a severe impact on the fatigue characteristics and the general performance [START_REF] Sørensen | Instability of helical tip vortices in rotor wakes[END_REF][START_REF] Vermeer | Wind turbine wake aerodynamics[END_REF]. Destabilising the wake would provide a possibility to reduce the aforementioned negative aspects for a variety of applications. Consequently, methods to influence the relevant tip vortex quantities and wake stability characteristics have been the topic of several studies over the past years, e.g. [START_REF] Quaranta | Local and global pairing instabilities of two interlaced helical vortices[END_REF][START_REF] Felli | Mechanisms of evolution of the propeller wake in the transition and far fields[END_REF][START_REF] Sørensen | Multiple helical modes of vortex breakdown[END_REF]. Active systems involve higher harmonic blade control mechanisms, active trailing edge flaps and active twist mechanisms [START_REF] Bauknecht | Blade tip vortex measurements on actively twisted rotor blades[END_REF]. Passive systems mainly consist of blade tip geometry modifications to affect the initial vortex system emerging from the trailing edge by splitting up the single concentrated tip vortex into separated vortices. In viscous flow, a vortex pair of two like-signed vortices eventually merge into one single vortex. The outcome yields a significant change of the characteristic parameters comparing the initial two vortices and the final merged vortex. The general vortex structure is spread out, resulting in an increased core radius and a decreased peak vorticity in the vortex centre [START_REF] Meunier | Physics of vortex merging[END_REF]. A detailed description of the two-dimensional merging procedure is given in [START_REF] Melander | Symmetric vortex merger in two dimensions: causes and conditions[END_REF][START_REF] Cerretelli | The physical mechanism for vortex merging[END_REF]. The presence of mutual instability phenenomena, which affect the vortex pair, accelerates the onset of a merging process and further enhances the beneficial effects, such as the widening of the vortex core [START_REF] Meunier | Three-dimensional instability during vortex merging[END_REF][START_REF] Meunier | Physics of vortex merging[END_REF]. Several of the proposed tip designs aim at the generation of more than one vortex and take advantage of their mutual interactions to provoke instability and merging phenomena in the rotor wake, see e.g. [START_REF] Brocklehurst | A review of helicopter rotor blade tip shapes[END_REF] for a review of helicopter blade tip shapes. For wind turbines, different blade tip and winglet designs with similar techniques to modify the vortex system are part of current research, an example is given in the study of Ostovan et al. [START_REF] Ostovan | Experimental Investigation of the Effects of Winglets on the Tip Vortex Behavior of a Model Horizontal Axis Wind Turbine Using Particle Image Velocimetry[END_REF]. The results show great potential for both applications, however, many aspects of the underlying physical mechanims are not fully understood yet. New insights in the flow phenomena and the mutual interactions involved in such closely spaced helical vortex systems are the objectives of the French-German research project TWIN-HELIX, which combines theoretical and experimental methods. Comprehensive experimental studies using planar and volumetric velocimetry measurements are carried out to investigate the interaction of two helical vortices. A specific tip design involving a parametric fin mounted on the pressure side is used to split up the single concentrated tip vortex into a vortex pair.

The presented work is the extension of previous experimental studies including stereoscopic particle image velocimetry (SPIV) and dye visualisations, which revealed that an unexpected rapid core growth of both vortices occurs for certain fin configurations which ultimately results into a merged vortex with strongly widened core. [START_REF] Schröder | Experiments on helical vortex pairs in the wake of a rotor[END_REF][START_REF] Schröder | Instability and merging of a helical vortex pair in the wake of a rotor[END_REF]. The occurence of those rapid groth rates prior to the onset of merging gave a hint to the presence of instability phenomena in the vortex system. Observations within the conducted visualisations revealed strongly disturbed vortex cores immediately behind the trailing edges confirming the aforementioned statement. Similar to straight vortex pairs (see [START_REF] Leweke | Dynamics and Instabilities of Vortex Pairs[END_REF]), the instability mechanisms occurring in helical vortex pairs are divided into two groups, depending on the perturbation wavelength compared to the vortex core diameter. Possible candidates for the investigated vortex systems are the general long-wave instability [START_REF] Okulov | On the stability of multiple helical vortices[END_REF][START_REF] Quaranta | Long-wave instability of a helical vortex[END_REF][START_REF] Quaranta | Local and global pairing instabilities of two interlaced helical vortices[END_REF] and the short-wave elliptic and curvature instability [START_REF] Blanco-Rodríguez | Elliptic instability of a curved Batchelor vortex[END_REF][START_REF] Blanco-Rodríguez | Curvature instability of a curved Batchelor vortex[END_REF][START_REF] Hattori | Short-wave stability of a helical vortex tube: the effect of torsion on the curvature instability[END_REF][START_REF] Leweke | Long-and short-wave instabilities in helical vortices[END_REF]. However, the time scales of the observed phenomena causing the rapid core growth were incompatible with the expected long-and short-wave instability mechanisms [START_REF] Leweke | Long-and short-wave instabilities in helical vortices[END_REF]. An explanation of the behaviour was found in the initial velocity profiles, which showed that both vortices were unstable with respect to centrifugal instability [START_REF] Bayly | Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows[END_REF][START_REF] Billant | A unified criterion for the centrifugal instabilities of vortices and swirling jets[END_REF]. This instability mechanism arises in vortices when the absolute value of the circulation declines with growing distance from the vortex center, i. e. caused by an accumulation of opposite-signed vorticity surrounding the vortex core [START_REF] Billant | Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities[END_REF]. Following the method proposed by Bayly [START_REF] Bayly | Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows[END_REF], the viscous growth rate of the perturbations and the corresponding most amplified wavelength were obtained, which agreed well with the calculated core growth rates and the observed perturbation scales in the visualisations. The presence of the centrifugal instability showed great potential with respect to the desired effect of the modification, therefore its origin and development requires further investigations. The previous experimental studies contained only planar velocimetry measurements, unable to reveal the threedimensional character of the perturbations resulting from the instability effects. Within this contribution, results from volumetric velocity measurements in the rotor wake using particle tracking methods are presented. The initial interaction process and development of the vortex pair influenced by the instability is investigated. To validate the volumetric data, a comparison to previous SPIV measurement campaigns is performed. The instability effect is observed in the formation of secondary structures between the vortices, affecting both vortex cores. Due to their unsteady behaviour, these structures are merely spotted in selected volumetric snapshots, which required the evaluation of all instantaneaous recordings. A qualitative analysis of the secondary vortices is given, Global coordinate system for the rotor geometry (free stream is in z-direction) and measurement volume dimensions.

revealing that their presence is pronounced in early wake age. Within this rotor phase interval, a significant increase of both primary vortex core radii is found. These initial effects contribute to a subsequent accelarated growth prior to the onset of the vortex merging.

Experimental Methods

Experiments were carried out in the large recirculating water channel of the Institute of Aerospace Systems. Its test section has dimensions of 1.5 m (width) × 1 m (height) × 6.5 m (length) and glass walls suitable for optical measurement techniques. Due to the reduced tangential velocities in the core regions for equal Reynolds numbers, the quantative analysis of vortical flows in water provides obvious advantages in comparison to air. Since the balance between centrifugal and pressure forces on seeding particles in the vortex cores is much easier to obtain, the recorded particle images are less effected by the well-known seeding void in the vortex core regions, ensuring a valid determination of the vortex core parameters.

A one-bladed rotor of constant chord length (NACA 0012 profile, radius R = 240 mm, chord c = 80 mm) equipped with a modified tip geometry was analysed in hover conditions. The blade is slightly twisted and has a rectangular planform. The simple rectangular blade geometry was chosen to focus on the basic mechanisms of generating two tip vortices, a detailed describtion of the design proces is found in [START_REF] Schröder | Experimental investigation of a rotor blade tip vortex pair[END_REF]. A perpendicular fin mounted on the pressure side in the vicinity of the tip generates the second vortex. The fin has an untwisted NACA 0012 cross section and its geometry is defined by the radial position d, height h, chord c f and angle of attack a f . The rotor rotates at a constant rotation rate of 1 Hz, resulting in a Reynolds number on the blade tip of Re = c(2π f )/ν = 120000. To prevent the flow from recirculating around the rotor disc, a small free-stream velocity in the water channel is required to ensure stationary flow conditions. Volumetric particle tracking measurements (PTV) using the "Shake-The-Box" (STB) technology introduced by Schanz et al. [START_REF] Schanz | Shake-The-Box: Lagrangian particle tracking at high particle image densities[END_REF] were carried out to acquire volumetric flow fields in the rotor wake. Figure 1(a) shows a schematic of the experimental set-up, a system consisting of four cameras (Imager sCMOS, 2048 × 2048 px resolution) was used to record the particle images, synchronized triggering is ensured by an optical encoder connected to the rotor shaft. Cubic polyamide particles of mean diameter 50 µm and density 1.016 g/cm 3 were used to seed the flow. To prevent aberrations due to the change of fluids in the optical path, water-filled prisms were used as suggested by Prasad & Jensen [START_REF] Prasad | Scheimpflug stereocamera for particle image velocimetry in liquid flows[END_REF]. Adapters fulfilling the Scheimpflug criterion were mounted in front of the cameras to ensure a uniformly focussed image. The measurement volume was illuminated by a double-pulsed laser (Quantel Twins Ultra) providing a pulse energy of 120 mJ at an effective wavelength of 532 nm. The dimensions of the measurement volume and the coordinate system used for the subsequent evaluation are displayed in Fig. 1(b), the center of the measurement volume was positioned directly below the axis of the rotor shaft. In order to track the evolution of the vortices despite the limited dimensions of the measurement region, linear axes are installed to alter the rotor position in relation to the fixed measurement volume. The previous SPIV studies revealed that the instabilities and the resulting rapid core growth rates occur primarily at early wake ages (rotor angles ϕ = 10 • -180 • ), therefore it was decided to apply the volumetric analysis within this phase interval. Each measurement volume has a thickness t = 30 mm, which covers approximately a rotor phase of ϕ vol = 7 • . In total, 26 measurement volumes (300 frames per volume) up to a rotor phase ϕ max = 185 • were recorded in the double-frame/single-pulse acquisition mode. A small overlap of the volumes was established to prevent data loss due to noisy data at the volume edges. The volumes contained 3 × 10 4 ± 20% valid particles. Due to the small free-stream velocity in the water channel, a constant seeding density was difficult to sustain for all volumes which explains the deviations in the particle number. A two-sided two-level target was used to calibrate the camera system. The results were refined by the volume self-calibration introduced by Wieneke [START_REF] Wieneke | Volume self-calibration for 3D particle image velocimetry[END_REF] and a calibration of the optical tranfer function established by Schanz et al. [START_REF] Schanz | Non-uniform optical transfer functions in particle imaging: calibration and application to tomographic reconstruction[END_REF]. The commercial PIV/PTV software DaVis 10.0.5 was used for image acquisition, calibration process and subsequent processing of the particle images.

a)

b)

ϕ = 18 • ϕ = 90 • ϕ = 162 • Tip V. Fin V.

Results and Discussion

The previous measurement campaigns included a parameter study to detect fin configurations, which create suitable vortex systems for the desired mutual interaction process in the wake, see [START_REF] Schröder | Experiments on helical vortex pairs in the wake of a rotor[END_REF]. Based on the previous results, the configuration which showed the most significant growth rates for both distinct vortices prior to the subsequent vortex merging was identified and chosen for the volumetric analysis. The geometric specifications defining the fin geometry are: d/c = 0.2, height h/c = 0.3, chord c f /c = 0.8. This configuration produced a vortex system with a slighty stronger fin vortex (Γ f /Γ t ≈ 1.15), which ultimately merged into a single vortex with a core growth up to a factor 5 after one rotor revolution in comparison to an unmodified blade. For all recorded volumes, the freestream velocity was set to u ∞ = 11.4 cm/s, resulting in a tip speed ratio λ = (2π f R)/u ∞ = 13. Figure 2(a) shows a qualitative impression of the wake generated by the modified blade. To create this representation, the location of each measurement volume in relation to the blade was tracked during the recordings, giving the oppurtunity to align them with respect to the vortex axis detected in the previous volumes. Two distinct vortices emerging from the blade are clearly visible, illustrated by the isosurfaces of the normalised azimuthal vorticity component. The tip vortex is coloured in blue, the fin vortex in red. Phase-averaged data is used to create the isosurfaces displayed in Fig. 2(a). No evident three-dimensional disturbances surrounding the vortices or any wavy perturbations of the cores are visible. These findings confirm the expected unsteady behaviour of the instability, indicating that the evaluation of instantaneous snapshots is required to discover the related effects. The general impact of the instability presence on the wake evolution is exposed by the fieldcuts given in Fig. 2(b), their position in relation to the blade is marked by the green slices in Fig. 2(a). The cuts show contours of the phase-averaged azimuthal vorticity component, demonstrating the development of the vortex system. The roll-up process for both vortices is completed at early wake stages, at a rotor phase of ϕ = 18 • two concentrated co-rotating vortices have formed. While moving downstream, the vortices start to rotate around each other, combined with a decrease of core separation and a slight deformation of the vortex structure. Looking at the fields at ϕ = 90 • and ϕ = 162 • , it is evident, that both the core radiii of both vortices are strongly widened and the peak vorticities in the centers significantly decreased, a consequence of the instability and the ongoing merging process.

Comparison between SPIV and Shake-The-Box data

For validation purposes, the obtained volumetric STB results are compared to the data of a previous SPIV campaign. The equipment, rotor model and the flow conditions were identical for both measurements. The SPIV data was recorded up to a rotor phase of ϕ max,SPIV = 540 • in steps of ∆ϕ SPIV = 18 • . Data processing was performed with a final correlation window size of 48 × 48 px, with a 75 % overlap. This spatial resolution provided a sufficient amount of 8-10 vectors within the radii of the smallest vortex cores measured in these experiments. In comparison, the particle fields and corresponding Langrangian particle tracks calculated from the STB data allowed a reconstruction to a regular Cartesian grid with a voxel size of 96 × 96 × 96 px with a 75 % overlap, yielding a coarser resolution in comparison. The time-varying seeding density and the low free-stream velocity prevented a finer spatial resolution. With regard to the rotor phase discretization, the STB volumes covered a rotor phase of ϕ vol = 7 • , which yields a total of 38 Cartesian field cuts, resulting in a much finer angular resolution of ∆ϕ STB ≈ 0.19 • . The following comparison of the calculated vortex data is based on the phase-averaged velocity fields/volumes. A conditional averaging procedure to minimize the effec of vortex wandering on the vortex core profiles (swirl velocities, radius), compare [START_REF] Devenport | The structure and development of a wing-tip vortex[END_REF][START_REF] Van Der Wall | Analysis methodology for 3C-PIV data of rotary wing vortices[END_REF], was neglected in the present analysis since it is expected, that the influence of the secondary vortex is dominant in comparison to the impact of vortex wandering.

To extract the relevant parameters characterising the tip vortices, the measured veloctity fields for both measurements have been analysed with the same routines. The vortex centers have been detected utilising the λ 2 -criterion introduced by Jeong&Hussain [START_REF] Jeong | On the identification of a vortex[END_REF]. Due to the sufficient seeding density within the core regions, no further filtering techniques as e.g. convolution filters [START_REF] Van Der Wall | Analysis methodology for 3C-PIV data of rotary wing vortices[END_REF] were needed for the centre identification. A projection of the velocities on a local polar grid (ρ, ξ ) and subsequent averaging the azimuthal component u ξ for each radius ρ is used to obtain the swirl velocity profiles of a detected vortex. Figure 3(a) shows an example of the deduced swirl profiles for both data sets, the shown profile is extracted from the tip vortex at a rotor phase of ϕ = 18 • (compare the vorticity contours in Fig. 2(b). Despite the coarser resolution of the STB data, both profiles are in good agreement with each other. The core radii a max , defined as the distance from the vortex centre to the point of maximum swirl velocity, are almost identical. It was expected, that the peak velocities calculated from the STB measurements would yield a reduced value due to the smearing effect of the coarser resolution, which would result in greater differences of the extracted core radii.

The second quantity describing the vortices is the circulation and the correspponding ratio within the vortex pair. To determine the value for each vortex, the vorticity field is separated into two distinct polygons, based on the previously detected vortex centers (a detailed description of the procedure is given in [START_REF] Schröder | Experimental investigation of a rotor blade tip vortex pair[END_REF]). For both regions, the edges of the polygon define the border of the line integral to determine the circulation. An example of a radial circulation profile for SPIV and STB data is presented in Fig. 3(b). It is again extracted from the tip vortex at a rotor phase of ϕ = 18 • . Similar to the radial swirl profile, the STB data matches the SPIV results quite well. Both profiles show a slight decrease from the maximum value at ρ/c ≈ 0.1, indicating the presence of counter-rotating vorticity regions in the vicinity of the vortex core, which is the criterion for centrifugal instability. Taking a look at the corresponding contour in Fig. 2(b), it is indeed visible, that both vortices are sourrounded by vorticity patches of opposite sign. The effect of the instability on the growth rates of the vortex core radii is clearly visible in the evolution of the core radii, presented in Fig. 4(a). In this diagramm, the equivalent Gaussian core radii a = a max /1.12 is used to allow comparability with theoretical analysis, see [START_REF] Schröder | Experiments on helical vortex pairs in the wake of a rotor[END_REF] for more details. Both fin and tip vortex cores grow significantly before the merging process begins. The SPIV measurements showed that the excessive growth rates for both vortices occur immediately after emerging from the trailing edge, up to vortex ages of ϕ ≈ 160 -180 • , compare the highlighted region in the smaller diagramm. During the period ϕ ≈ 180 -270 • , the ongoing merging process hinders a valid extraction of the core radii until the final vortex is formed. It was therefore decided to perform the volumetric measurements within half a rotor revolution to cover the growth phase, which is mainly affected by the instability. Again, both measurements are in good agreement, the data of the STB measurements is extracted for each fieldcut of the volumes, providing a much finer angular resolution of the core growth. In Fig. 4(b), the circulation development during the early wake stage is displayed. As expected, the circulation for both vortices remain nearly constant during the growth phase. The slight decrease at phases ϕ ≈ 120 -150 • for the tip vortex (blue) and the corresponding total value (black) in the system is related to a slightly inaccurate position of the rotor in relation to the measurement plane/volume, which caused a partly cut-off tip vortex and consequently a) It should be mentioned that the presence of the fin increases the averaged total circulation in comparison to an unmodified blade (not shown here), indicating that the general circulation generated by the blade is enhanced as well [START_REF] Schröder | Experimental investigation of a rotor blade tip vortex pair[END_REF]. The averaged values for the STB measurements are modestly higher than the corresponding SPIV results. With regard to the merging procedure, the distance between both vortices is of interest, since the relation between the Gaussian core radii a and the vortex distance b is an indicator for the onset of merging [START_REF] Meunier | Physics of vortex merging[END_REF]. Figure 5(a) displays the development of the distance based on the determined center positions. Up to a rotor phase of ϕ = 60 • , the separation increases, afterwards both vortices move towards each other. The critical ratio characterising the onset of merging is given by a/b = 0.24 [START_REF] Meunier | Physics of vortex merging[END_REF]. Figure 5(b) reveals, that this ratio is exceeded at ϕ = 90 • . For subsequent wake stages, the core growth is expected to be driven by a combination of instability and merging effects. Prior to the beginning of merging, the centrifugal instability is considered as the main contributor. With regard to the evolution of a/b, it is evident, that the core growth rates compensate for the expanding vortex distance, yielding a continous increase. Despite the lower spatial resolution, it can be stated that the STB data is in surprisingly good agreement with the SPIV data, providing an accurate determination of the relevant vortex parameters for this study. The notably increased angular resolution allowed a much finer observation of the parameter evolution in comparison to previous studies. Still, higher seeding densities and consequently more determined particle tracks should be achieved for future measurement campaigns.

Analysis of secondary vortex structures

To get a better understanding of the phenomena caused by the instability, the volumetric data in the early wake stages is investigated in detail. In Fig. 6(a), the position of the measurement volume in relation to the rotor and the local coordinate system used for the following volume analysis is shown. Due to the small coverage of ϕ vol = 7 • and the negligible curvature of the vortices within the volume, no transformation into the rotor hub cylinder coordinates is performed. The results presented in 3.1 were determined using phase-averaged data. However, it is expected, that any occuring unsteady effects related to the instability are smoothed out due to the averaging process, which leaves the phase-averaged volumes unsuitable to detect any secondary structures. The confirmation of this assumption is given in Fig. 6(b), which displays the vortical structure within the phase-averaged volume at a rotor phase ϕ = 24 • , the isosurfaces represent a positive threshold of the Q-criterion [START_REF] Hunt | Eddies, streams, and convergence zones in turbulent flows[END_REF].

The tip and fin vortices are clearly visible, their core structure appears relatively undisturbed and no secondary structures are detected in their vicinity. With regard to the immidiate core growth and the appearance of counter-rotating vorticity close to both vortices, compare Fig. 8(b), the influence of the instability should already be present at this wake age. Due to the lack of time-resolved data, single snapshots are used for further examination. The theoretical analysis yields, that the instability causes a wavy perturbation of the vortex cores, the most amplified wavelength can be determined from the velocity fields with the methods described in [START_REF] Bayly | Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows[END_REF]. This deformation of the internal core structure causes the generation of secondary structures near the vortex cores. It should be stated, that the centrifugal instability is no mutual instability phenomenon. Nevertheless, in the presented case, both vortices fulfill the criterion of a decrease within the radial circulation profile with increasing distance from the vortex center. Consequently, the instability should affect both vortices. phase-averaged data, a secondary vortex between both primary vortices is indeed detected. Both vortex cores appear disturbed, but the identification of a dominant perturbation wavelength is unfeasible. Unfortunately, it is not possible to track the further evolution of the secondary structure due to its unsteady behaviour. To differentiate between primary vortices and secondary structures, a method introduced by Wolf et al. [START_REF] Wolf | Experimental study of secondary vortex structures in a rotor wake[END_REF] is used for further evaluation. In their study, they gave the experimental proove for the existence of similar secondary structures entangling the primary tip vortices in a hovering rotor wake, which were previously regarded as a numerical phenomenon. Wolf et al. showed that the generation of those structures has a physical background and is not purely associated to numerical issues. The following methodology is based on their work. The vorticity unit vector is used to decompose the scalar Q-criterion into a Cartesian framework [START_REF] Gibeau | Secondary instabilities in the wake of an elongated two-dimensional body with a blunt trailing edge[END_REF]. Within the local coordinate system introduced in Fig. 6(a), it is given by:

ω n =   ω n,x ω n,y ω n,t   = ∇ × v |∇ × v| . (1)
The primary vortices are mainly rotating around the t-axis, so the corresponding part of the normalised vorticity vector is used to identify tip and fin vortices with the following criterion:

Q tip = -ω n,t • Q; ω n,t < 0 0; ω n,t ≥ 0 . (2)
To account for the negative rotational sense of the primary vortices, the negative sign is inserted. The secondary structures are identified using the y-component of the normalised vorticity vector, since it is expected, that they mainly form in y-direction. The Q-values are multiplied with the projection of ω n on the xy-plane, which is given by:

ω n,xy = ω 2 n,x + ω 2 n,y . (3)
The criterion to discover the secondary structures yields:

Q sec = ω n,xy • Q; ω n,y = 0 0; ω n,y = 0 . (4)
A differentiation between the rotational direction of the secondary structures, as done in [START_REF] Wolf | Experimental study of secondary vortex structures in a rotor wake[END_REF] is neglected in this analysis. It should be stated that only positive tresholds for the Q-values are considered, since only the rotational parts of the flow are of interest here. Due to the sufficient seeding density within the vortex cores, no further filtering procedure is needed. Applying the decomposition to the snapshot displayed in Fig. 7(a) yields the result presented in Fig. 7(b). The primary vortices (coloured in green) are clearly distinguished from the secondary structure (red). Both isosurfaces are created with the equal threshold for Q tip and Q sec , respectively. The decomposition is applied to all snapshots, which were recorded for the different rotor phases, further examples for ϕ = 24 • are given in Fig. 7(c-d). As expected, the manifestation of the secondary structures and their position along the primary vortices' axes is an unsteady phenomenon. Consequently, they are not detected in every snapshot, compare Fig. 7(d), despite a visible deformation of the vortex cores. Determining the characteristic parameters of the secondary structures is impeded by the limited spatial resolution and the influence of the adjacent primary vortices. However, the qualitative impressions given in Fig. 7 suggest that strength and core size of the secondary vortices is similar to the primary ones.

A direct comparison to the results from Wolf et. al [START_REF] Wolf | Experimental study of secondary vortex structures in a rotor wake[END_REF] is problematic considering the completely different blade geometries and flow conditions, still it appears, that the presence of the centrifugal instability and the enhanced perturbations of the vortex cores provoke the generation of more pronounced secondary vortices. A topview of the decomposed isosurfaces, given in Fig. 8, accentuates the geometry of the secondary vortices. It is characterised by an S-shaped layout, entangling both primary vortices. The observed shape is similar to the structures detected between two tip vortices in the experiments conducted by Wolf et al. [START_REF] Wolf | Experimental study of secondary vortex structures in a rotor wake[END_REF]. Previously conducted visualizations, see [START_REF] Schröder | Experiments on helical vortex pairs in the wake of a rotor[END_REF] gave the impression that the vortex system is disturbed almost immediately after emerging from the trailing edge, resulting in the prompt core growth deduced from the phase-averaged data in Fig. 4(a). In accordance with those findings, the secondary vortices are detected within the volume containing the youngest vortices (ϕ = 17 • ). With regard to the findings shown in Fig. 5, it is likely that the impact of the instability and consequently the presence of secondary structures is pronounced up to a rotor phase of ϕ ≈ 60 • . Due to the lack of temporaly resolved data, a statistical approach was used to evaluate the strength of secondary vortices for each measured volume. The aforementioned decomposition of the Q-criterion was performed for all 300 snapshots recorded for each rotor phase and the results were subsequently averaged. To account for the varying position with regard to the t-axis of the secondary structures, an additional averaging was performed in t-direction over the entire volume depth. The resulting isocontours of Q tip and Q sec are shown in Fig. 9 for different rotor phases. The primary vortices identified by Q tip are marked in green, the averaged secondary structures are depicted with the red colour scale. A quantitative evaluation is restrained due to the unsteady characteristics of the secondary vortices within each volume. Still, a qualitative insight on the evolution of the entire vortex system is given by the deduced contours. The formation of the vortex braids start in the vicinity of the tip vortex. With proceeding phase angle, the structure of the secondary vortices expands towards the fin vortex. At a rotor phase of ϕ = 45 • , the braid nearly entwines both primary vortices. In terms of the strength of the tip vortices, a decrease of Q tip is observed for increasing phase angles, in agreement with the development of the azimuthal vorticity component shown in Fig. 2(b). The statistical results confirm that the influence of the secondary structures is especially high within the early wake stages. With widening primary core radii and reduced maximum Q tip values, the contours start fading out (ϕ > 45 • ). Since the actual amount of structures detected during the averaging process was not tracked, it is not clear if the strength or the total count of the secondary vortices is decreased for higher phase angles. Still, the obtained results are in good agreement with the conduct found for the phase-averaged data in 3.1. Their presence appears to be directly connected with the core growth, causing the concentrated vorticity distribution of the primary vortices to be widened. Nevertheless, time resolved experimental data of the formation and subsequent evolution of the secondary structures is still missing. This issue will be adressed in subsequent measurement campaigns which will involve high-speed volumetric measurements. In addition, the characteristic parameters of the secondary structures are of special interest. In a follow-up study to the work of Wolf et al. [START_REF] Wolf | Experimental study of secondary vortex structures in a rotor wake[END_REF], Schwarz et al. provided a comprehensive analysis of the secondary structures and their evolution in the wake of a two-bladed rotor in hovering conditions, combining experimental and numerical methods [START_REF] Schwarz | Development of secondary vortex structures in rotor wakes[END_REF]. Higher spatial resolution allowed the extraction of the secondary vortex parameters, consequently an increased spatial resolution should also be targeted for future experiments within the presented project.

ϕ = 24 • ϕ = 31 • ϕ = 38 • ϕ = 45 • ϕ = 59 • ϕ = 73 •

Conclusions

Within this contribution, we have presented the outcome of an experimental study containing volumetric velocity measurements of a helical vortex pair influenced by centrifugal instabilities. The vortex pair was generated by a one-bladed rotor in hover conditions, which was investigated in a water channel. A parametric fin is used to generated the secondary vortex, its geometric specifications have a direct impact on the emerging vortex system, compare [START_REF] Schröder | Experiments on helical vortex pairs in the wake of a rotor[END_REF]. The main objective of the proposed tip design is a modifaction of the rotor wake properties in order to mitigate the negative effects associated with blade-vortex interactions. Previous experimental studies showed, that the desired goal can indeed be achieved through mutual interactions and an eventual vortex merging for the generated tip vortex pair. For certain fin configurations, an imminent rapid growth of both vortex core radii before the subsequent vortex merging process was found, hinting at the presence of an instability phenomenon. It was identified as the centrifugal instability, which affects both vortices and provokes the observed growth rates. To gain new insights in the occuring flow phenomena, volumetric velocity measurements using the "Shake-The-Box" technology were performed for one chosen configuration in the early rotor wake. A comparison with previously recorded planar PIV data showed good agreement between the phase-averaged data of both measurements with regard to the extracted characteristic vortex parameters. The instability effect in terms of any occuring secondary flow structures was not detected in the volumetric phase-averaged data. Instantenous snapshots were analysed to reveal the presence of secondary vortices entangling both tip and fin vortex, comparable to the structures found in rotor wakes within the study of Wolf et al. [START_REF] Wolf | Experimental study of secondary vortex structures in a rotor wake[END_REF]. A qualititave statistical evaluation revealed that the secondary strucures are pronounced in the early wake stages, which is in accordance with the core evolution obtained from the phase-averaged data. Their presence is directly connected to the core growth and the decreasing vorticity peaks of the primary vortices. A limited spatial resolution impeded the determination of valid vortex parameters for the secondary vortices, an improvement will be targeted in future measurements. In addition, time-resolved volumetric STB is planned to gain new insights in the formation and development of the structures. With regard to the occurence of the centrifugal instability, its physical origin still remains unclear. To address this issue, a new rotor geometry providing a different radial circulation distribution will be investigated in the future.

Figure 1 -

 1 Figure 1 -(a) Schematic of the experimental setup for the particle tracking velocimetry measurements. (b)

Figure 2 -

 2 Figure 2 -(a) Volumetric isosurfaces of the normalised azimuthal vorticity component up to a rotor phase of ϕ = 185 • for the modified blade (green slices indicate the position of the fieldcuts shown in (b)). (b) Phase-averaged vorticity contours at ϕ = 18 • , ϕ = 90 • and ϕ = 162 • , black line represents the trailing edge of the blade.

Figure 3 -

 3 Figure 3 -(a) Radial swirl velocity and (b) radial circulation profiles for the tip vortex at a rotor phase of ϕ = 18 • . Profiles extracted from both STB and SPIV data.

Figure 4 -Figure 5 -

 45 Figure 4 -(a) Gaussian core radii evolution of both vortices in the region of rapid growth (marked with the black elipse within the full SPIV diagram (a 1 )) for STB and corresponding SPIV data. (b) Circulation values obtained for the distinct vortices and the total field (black).

Figure 7 (

 7 Figure 7(a) shows an instantaneous snapshot of the same measurement volume shown in Figure 6(b), the isosurfaces correspond to the equal threshold of the Q-criterion. In contrast to the

Figure 6 -

 6 Figure 6 -(a) Sketch of the STB measurement volume in relation to the rotor blade and local coordinate system used for subsequent analysis. (b) Isosurface of the Q-criterion at a rotor phase ϕ = 24 • for phase-averaged data.

Figure 7 -

 7 Figure 7 -(a) Instantaneous isosurfaces of the Q-criterion at a rotor phase ϕ = 24 • . (b) Decomposition of the instantaneous Q-criterion into Q tip (green) and Q sec (red). (c-d) Additional examples for instantaneous isosurfaces using the decomposed criteria.

Figure 8 -

 8 Figure 8 -(a-b) Top view of the decomposed criteria shown in Fig. 7(b-c).

Figure 9 -

 9 Figure 9 -Phase-averaged volumes showing the development of the primary vortices Q tip (green) and the secondary vortex structures Q sec (red) for different phase angles ϕ (volumes are averaged in t-direction).

Acknowledgements

This work is part of the German-French project TWIN-HELIX, supported by the Deutsche Forschungsgemeinschaft (grant no. 391677260) and the French Agence Nationale de la Recherche (grant no. ANR-17-CE06-0018).

Contact Author Email Address

The contact author email address is: dominic.schroeder@ilr.rwth-aachen.de

Copyright Statement

The authors confirm that they, and/or their company or organization, hold copyright on all of the original material included in this paper. The authors also confirm that they have obtained permission, from the copyright holder of any third party material included in this paper, to publish it as part of their paper. The authors confirm that they give permission, or have obtained permission from the copyright holder of this paper, for the publication and distribution of this paper as part of the ICAS proceedings or as individual off-prints from the proceedings.