Serra Gürcan

Nicolas Tsapis

Franceline Reynaud

Stéphanie Denis

Juliette Vergnaud

Özgen Özer

Elias Fattal
email: elias.fattal@universite-paris-saclay.fr

Combining dexamethasone and TNF-α siRNA within the same nanoparticles to enhance antiinflammatory effect

Keywords: corticosteroids, dexamethasone, siRNA, inflammation, macrophages

We propose to combine two therapeutic anti-inflammatory approaches with different mechanisms of action in a single drug delivery system consisting of cationic dexamethasone palmitate nanoparticles (CDXP-NP) associated with TNF-α siRNA. The CDXP-NPs are obtained by the solvent emulsion evaporation technique using dexamethasone palmitate, a prodrug of dexamethasone, associated with a cationic lipid, DOTAP. Their physicochemical properties as well as their ability to bind siRNA were evaluated through gel electrophoresis and siRNA binding quantification. SiRNA cellular uptake was assessed by flow cytometry and confocal microscopy on RAW264.7 macrophages. TNF-α inhibition was determined on LPS-activated RAW264.7 macrophages. Stable and monodisperse nanoparticles around 100 nm with a positive zeta potential (+59mV) were obtained with an encapsulation efficiency of the prodrug of 95%. A nitrogen/phosphate (N/P) ratio of 10 was selected that conferred the total binding of siRNA to the nanoparticles. Using these CDXP-siRNA-NPs, the siRNA was strongly internalized by RAW264.7 macrophage cells and localized within the cytoplasm. On the LPS-induced RAW264.7 macrophages, a larger inhibition of TNF-α was observed with CDXP-siRNA-NPs compared to CDXP-NPs alone. In conclusion, from these data, it is clear that a combination of DXP and TNF-α siRNA therapy could be a novel strategy and optimized alternative approach to cure inflammatory diseases.

Introduction

Inflammation is a biological response to a sudden injury, leading to classical symptoms such as swelling, redness, pain. Physiopathology involves a variety of related cells and cytokines and a set of changes in those triggers the occurrence of inflammation [START_REF] Chen | Inflammatory responses and inflammation-associated diseases in organs[END_REF]. Among all cytokines, tumor necrosis factor-alpha (TNF-α) plays a major role such as activating immune cells during inflammation [START_REF] Zelová | TNF-α signalling and inflammation: interactions between old acquaintances[END_REF]. TNF-α has been shown to play a pivotal role in orchestrating the cytokine cascade in many inflammatory diseases and for this reason, it has been proposed as a therapeutic target for many diseases including rheumatoid arthritis, Crohn's disease, atherosclerosis, psoriasis, sepsis, diabetes, and obesity [START_REF] Parameswaran | Tumor necrosis factor-α signaling in macrophages[END_REF].

Glucocorticoids (GCs) are steroid hormones, with anti-inflammatory and immunosuppressive properties, that are widely used for the treatment of inflammation and autoimmune diseases [START_REF] Chan | Use of glucocorticoids in the critical care setting: Science and clinical evidence[END_REF]. GCs have been applied for the therapeutic care of inflammatory and immune disorders such as rheumatoid arthritis, multiple sclerosis, inflammatory bowel disease, and nephrotic syndrome [START_REF] Hardy | Therapeutic glucocorticoids: mechanisms of actions in rheumatic diseases[END_REF][START_REF] Spies | Glucocorticoids[END_REF]. The recent pandemic of COVİD-19 has shown that a GC such as dexamethasone (DXM) was among the only drugs efficient in reducing the post-infection hyper inflammation resulting in a significant diminution of mortality [START_REF] Johnson | Dexamethasone in the management of covid -19[END_REF]. GCs administration is however limited to short-term treatment and low doses. Indeed, it has been shown that GCs can cause the emergence of serious side effects such as diabetes, high blood pressure, or Cushing syndrome [START_REF] Money | The Risks of Chronic Corticosteroid Exposure[END_REF][START_REF] Oray | Long-term side effects of glucocorticoids[END_REF]. Besides, GCs are hydrophobic molecules with relatively poor oral bioavailability. Furthermore, GCs systemic administration leads to broad distribution throughout the body that is responsible for the observed side effects. As a result, the overall therapeutic efficacy and patient compliance with GC-based treatments can be poor [START_REF] Onishi | Kinetic analysis of in vitro and in vivo release of prednisolone from the conjugate of glycol-chitosan and succinyl-prednisolone[END_REF].

To overcome issues related to GCs administration, one approach consists of increasing GCs accumulation within the targeted tissue using nanoparticulate drug delivery platforms. Regarding Dexamethasone (DXM), it has been already encapsulated into polymeric nanoparticles (NPs) [START_REF] Gomez-Gaete | Dexamethasone acetate encapsulation into Trojan particles[END_REF][START_REF] Gómez-Gaete | Encapsulation of dexamethasone into biodegradable polymeric nanoparticles[END_REF] with however a low encapsulation efficiency, a strong burst rlease and the presence of drug forming crystals. DXM has also been encapsulated into liposomes again with a weak encapsulation efficiency [START_REF] Koning | Targeting of angiogenic endothelial cells at sites of inflammation by dexamethasone phosphate-containing RGD peptide liposomes inhibits experimental arthritis[END_REF][START_REF] Rauchhaus | Separating therapeutic efficacy from glucocorticoid side-effects in rodent arthritis using novel, liposomal delivery of dexamethasone phosphate: long-term suppression of arthritis facilitates interval treatment[END_REF]. To solve problems related to GCs encapsulation and release, we have recently designed lipid nanoparticles to deliver dexamethasone palmitate (DXP) in a collagen-induced arthritis murine model (CIA) (Lorscheider et al., 2019b). The fatty acid derivative was synthesized and used to obtain, through controlled hydrolysis, a long-term effect. Pharmacokinetics parameters were evaluated after intravenous (IV) injection of dexamethasone palmitate nanoparticles (DXP-NP) to healthy mice. The release of DXM from DXP-NP in plasma was controlled for up to 18 hours compared with the free drug, which was rapidly eliminated from plasma after administration (Lorscheider et al., 2019a).

The physiopathology of inflammation is complex and involves a variety of related cells and cytokines.

Thus, it is difficult to achieve an ideal therapeutic effect with one single small molecule drug or one anti-cytokine therapy due to the large number of targets that one of these drugs should reach. It appears more effective to combine two therapeutic approaches with different mechanisms of action.

When multiple therapeutic agents target the same cellular pathway, they can cooperate to obtain a higher therapeutic effect and higher targeting selectivity. Based on these considerations, developing the co-delivery of GCs and siRNA together appears a simple, direct, and feasible method. As shown in the literature, a potent approach should be based on the co-administration of GCs with a companion nucleic acid such as TNF-α siRNA [START_REF] Wang | Targeting NF-kB signaling with polymeric hybrid micelles that co-deliver siRNA and dexamethasone for arthritis therapy[END_REF].

Although siRNA are effective, safe, and most importantly target-specific therapy, there are still barriers that must be taken into consideration for their delivery [START_REF] Singh | Lipoplex-based therapeutics for effective oligonucleotide delivery: a compendious review[END_REF]. SiRNA are unstable in biological fluids which makes them poorly available in their intact form at the target site [START_REF] Fattal | State of the art and perspectives for the delivery of antisense oligonucleotides and siRNA by polymeric nanocarriers[END_REF]. Moreover, unfavorable physicochemical properties of siRNA such as high molecular weight and negative charge prevent them to pass through the cell membrane. Even though after a successful entry into cells siRNA must be protected by delivery systems in terms of endosomal escape, lysosomal degradation, and provide siRNA release [START_REF] De Rosa | Long-term release and improved intracellular penetration of oligonucleotide-polyethylenimine complexes entrapped in biodegradable microspheres[END_REF][START_REF] De Rosa | Poly(lactide-coglycolide) microspheres for the controlled release of oligonucleotide/polyethylenimine complexes[END_REF][START_REF] Deng | Transdermal Delivery of siRNA through Microneedle Array[END_REF][START_REF] Lechanteur | Cationic Liposomes Carrying siRNA: Impact of Lipid Composition on Physicochemical Properties, Cytotoxicity and Endosomal Escape[END_REF][START_REF] Toub | Innovative nanotechnologies for the delivery of oligonucleotides and siRNA[END_REF].

It is therefore a challenge to deliver dexamethasone (hydrophobic) and siRNA (hydrophilic) within the same nanoparticle while keeping the latest intact and favoring its intracellular penetration. To address this challenge, we propose a novel type of nanoparticle which composition is based on the formulation of DXP-NP described by our group (Lorscheider et al., 2019a(Lorscheider et al., , 2019b). These nanoparticles have been covered by a cationic lipid able to bind a TNF-α targeted siRNA through electrostatic interactions [START_REF] Zhi | A review on cationic lipids with different linkers for gene delivery[END_REF]. Particle characterization as well as in vitro cell trafficking and TNF-α inhibition were explored offering interesting perspectives for this novel nanomedicine combination.

Materials and Methods

Materials

Dexamethasone palmitate (DXP) was purchased from Interchim (France). 1,2-dioleoyl-3 trimethylammonium-propane (chloride salt) (DOTAP) was obtained from Avanti Polar Lipids, Inc. N-(carbonyl-methoxypolyethylenglycol-2000)-1,2-distearoyl-sn-glycero-3-phosphoethanolamine, mPEG-2000-DSPE, Na-salt (DSPE-PEG) was provided by Lipoid., Dexamethasone disodium phosphate (DSP) was purchased from Fagron (The Netherlands). Milli-Q water was purified using a RIOS system from Merck-Millipore (France). High-performance Liquid chromatography (HPLC)-grade chloroform was obtained from Carlo Erba Reagents (France). HPLC-grade acetonitrile and dimethyl sulfoxide (DMSO) was purchased from VWR Chemicals (France). Duplex siRNA directed against TNF-α was provided by Integrated DNA Technologies Inc. (IDT, 100 Coralville, IA, USA) as lyophilized and purified duplexes. The TNF-α siRNA consisted of the following duplex sequences:

Sense 5'-GUCUCAGCCUCUUCUCAUUCCUGT-3', Antisense 3'-UACAGAGUCGGAGAAGAGUAAGGACGA-5', (Mw 16,807.3 Da).

A negative control sequence of siRNA (Ctr siRNA) was acquired from Eurogentec (Eurogentec, Angers, France). The exact sequence of Ctr siRNA was not disclosed by the supplier. Fluorescent label dye siRNA 6-FAM was obtained from Eurogentec (Eurogentec, Angers, France). Agarose Type-D5 DNA Grade was bought from Merck-Millipore (France). 3-(4,5-Dimethyl-2-thiazolyl)-2,5-diphenyl-2Htetrazolium bromide, MTT, Methylthiazolyldiphenyl-tetrazolium bromide was purchased from Sigma.

Fluorescent stain ethidium bromide solution was obtained from Bio-Rad laboratories. TBE Buffer (Tris-borate-EDTA) was purchased from Promega. Nuclease free water was acquired from Sigma, RNaseZap TM .

Preparation of cationic dexamethasone palmitate nanoparticles

Nanoparticles were prepared at five different lipid concentrations using a fixed prodrug concentration with the following DXP/DOTAP ratios: 5/2.5; 5/2; 5/1.5; 5/1; 5/0.5 in mg/mL using an emulsion solvent evaporation method derived from Lorscheider et al (Lorscheider et al., 2019a).

Briefly, 10 mL of Milli-Q water was pre-chilled at 4°C. DXP was dissolved with the desired amounts of DOTAP (25; 20; 15; 10 or 5 mg) and completed to 1 mL with chloroform. Then, the 10 mL water phase was placed on ice and the organic phase was injected using a solvent-compatible syringe. The mixture was then vortexed for 30 seconds to obtain a pre-emulsion. The pre-emulsion was then ultrasonicated at 40% amplitude (300W Branson Digital Sonifier) for 2 minutes by keeping it in the ice bath. The organic phase was evaporated under reduced pressure using a rotary evaporator for 1h.

Afterward, the suspension volume was completed to 10 mL with the chilled milli-Q water in a volumetric flask, and nanoparticles were stored under an inert atmosphere either at 4°C or room temperature (RT) protected from light.

Preparation of cationic dexamethasone nanoparticle-associated-TNFα siRNA

Cationic dexamethasone palmitate nanoparticles (CDXP-NP) were mixed with anionic siRNA varying the nitrogen/phosphate (N/P) ratio (the charge molar ratio between the cationic DOTAP and the anionic siRNA). They were prepared at 1, 2, 5, 10 and 20 N/P ratios. Desired volumes of CDXP-NP were placed into Eppendorf tubes then fixed 50 µL of 5 µM siRNA solution were added producing (CDXP-siRNA-NP). Subsequently, the solutions were vortexed for 30 seconds and incubated at RT for 30 minutes. All the experiments were carried out under a biosafety cabinet and at RNase free conditions. N/P Ratio was calculated according to the following formula, since DOTAP bears one charge and siRNA 53 charges:

Dynamic Light Scattering

Particle size distribution and zeta potential were determined by using a Zetasizer Nano-ZS from Malvern Instrument (UK), based on quasi-elastic light scattering. Measurements were performed in triplicate at an angle of 173° and a temperature of 25°C. Samples were diluted (1/10) in Milli-Q water for size measurements and in 1 mM NaCl for zeta potential measurements.

Agarose Gel Electrophoresis Assay

The binding affinity between anionic siRNA and cationic CDXP-NP was evaluated by the agarose gel electrophoresis method. To prepare 1% agarose gel, 1 g of agarose powder was dissolved in 100 mL of 1X TBE buffer. The mixture was boiled to obtain transparent agarose solution and was then left under a fume hood for 5 minutes. Later, 1.25 μL of 10 mg/mL ethidium bromide (EtBr) solution was added to the warm agarose solution and it was decanted to the tray smoothly without the formation of bubbles allowing to form a gel by cooling. After decanting, a comb was placed on the gel to obtain wells. Meanwhile, samples were prepared by using DNA loading dye/sample at 1:3 ratio and 5 µL from prepared samples were loaded into wells. DNA loading dye consisted of 0.25% bromophenol blue, 0.25% xylene cyanol FF, 30% glycerol in water Electrophoresis was run at 120 V for 30 minutes using Apelex PS 608 electrophoresis power supply. siRNA bands were visualized using Bio Rad® ChemiDoc gel imaging system under UV light.

SiRNA Quantification

Quantification siRNA associated to CDXP-NP was calculated from measuring the amount of unbound free siRNA for each N/P ratio (1, 2, 5, 10, and 20) when preparing CDXP-siRNA-NP. Samples were diluted until reaching a concentration of 1000 ng/mL siRNA and were ultra-centrifugated at -4°C, 40.000 rpm for 15 min. Quantification of free siRNA was performed on the supernatant.

QuantiFluor® RNA System kit containing RNA-binding dye (492nm excitation/540nm emission) was used and measurements were carried out using a spectrofluorimeter. A 1 mL RNA standard solution was prepared at 5 µg/mL using nuclease-free water (NFW) and diluted with NFW to obtain different samples which concentration ranges between 39 ng/mL to 2500 ng/mL. RNA-dye solution was diluted at a 1/200 ratio with 1X TE buffer in the dark and vortexed. One to one volume of RNA-dye solution was added to each RNA standard solution, to each sample, and as well as to the NFW as a negative control (Emission 0, Em 0 ). The calibration curve was obtained from emission values corresponding to each sample which was calculated using Em-Em 0. The equation of the calibration curve was the following Y = 0.3111X -20.82, R 2 = 0,9986. Sample values were interpolated to the calibration curve to calculate the quantity of free siRNA. Afterward, entrapment efficiency was determined by comparing entrapped/associated siRNA to the initial amount of siRNA in the preparation of CDXP-siRNA-NP. Results were expressed as a percentage.

Quantification of Dexamethasone palmitate

Quantification of DXP concentration in CDXP-siRNA-NP was calculated using HPLC with a UV detection. The analysis was carried out at 240 nm using a SymmetryShieldTM RP18 column (5 µm, 250×4.6 mm; Waters, France). The column temperature was maintained at RT. A gradient mobile phase system was used (Supplementary information Table S1). Eluent A was composed at 90:10 v/v with Milli-Q water: acetonitrile (ACN) while eluent B was composed of 100 % ACN. The mobile phase flow rate was set at 1mL/min with a 50 µL of injection volume and the run time was 40 min. The retention time of DXP was 22 min. The calibration curve was obtained from DXP in ACN between 62.5-1000 µg/mL and found linear, Y = 65145X + 429882, R 2 = 0,9993. DXP encapsulation efficiency was calculated as a percentage ratio between the actual encapsulated DXP amount compared to the initial DXP amount used. Drug loading of DXP and corresponding DXM was calculated as a percentage of actual encapsulated DXP amount compared to the total amount used in CDXP-siRNA-NP.

Transmission Electron Microscopy

Transmission electron microscopy (TEM) was performed at I2BC facility (CNRS, Gif-sur-Yvette, France). First, 400 mesh formvar-coated copper grids were plasma treated to render them hydrophilic. Then, 5 µL volumes of 1/5 diluted samples were placed onto formvar-coated copper grids for 2 minutes. Subsequently, the grids were gently dried and negative staining was performed by adding a drop of 2% w/w phosphotungstic acid (PTA) for 30 seconds. Excess solution was removed, and grids were left to dry before observation. The observations were carried out on a JEOL JEM-1400 microscope at an acceleration voltage of 120kV. Images were acquired using an Orius camera (Gatan Inc, USA).

Cell Culture

The murine macrophage cell line RAW 264.7 was purchased from American Type Culture Collection 165 (ATCC). The cell suspension was obtained by scraping the cells in 10 mL of fresh Dulbecco's modified Eagle's medium (DMEM, high glucose) (Sigma Aldrich, St. Louis MO, USA) supplemented with 10% fetal bovine serum (FBS), 10 000 unit/mL penicillin and 10 mg/mL streptomycin. For cell splitting, 1 mL cell suspension was added into 75 cm 2 flask and completed with 14 mL fresh full medium and incubated at 37°C and under 5% CO 2 -95% O 2 atmosphere. The cells were sub-cultured twice a week and used at 80% confluence between 3-12 passage numbers.

Cell Viability

RAW264.7 cell viability was evaluated as a function of CDXP-NP and CDXP-siRNA-NP concentration using the 3-[4,5-dimethylthiazol-2-yl]-3,5-diphenyl tetrazolium bromide (MTT) colorimetric assay.

Cells were seeded in 96-well plates at 8X10 3 cells/100 µL density per well using full medium and incubated 24h at 37°C. The next day, samples were prepared by dilution in FBS-free medium at equivalent DXP concentrations (2-124 µg/mL concentrations). Subsequently, the full medium was removed gently from the wells and replaced with 100 µL of samples. After 4h of incubation at 37°C, 100 µL medium supplemented with 20% FBS was added into wells, and incubation was maintained for an additional 20h. Then, 20 µL from 5 mg/mL MTT/Phosphate-buffered saline (PBS) solution was added to each well and incubated 2h at 37°C to allow the transformation of formazan crystals. The medium was then carefully discarded and 200 μL of DMSO was added to dissolve formazan crystals.

The 96-well plate was shaken for 5 minutes at 300 rpm then the absorbance was measured at 570 nm with a microplate reader. Cell viability was then calculated as a percentage using the formula below. All measurements were performed n=5, 3 times.

% cell viability = - - *100
Where Abs is the absorbance. The same protocol was used for the cytotoxicity investigation of dexamethasone sodium phosphate (DSP), DOTAP-only, free siRNA, DOTAP/siRNA at equivalent concentration either of DOTAP or siRNA or CDXP-NP.

Quantification of Secreted TNF-α

The anti-inflammatory efficacy of both the CDXP-NP and CDXP-siRNA-NP was evaluated based on the amount of secreted TNF-α levels after activating the RAW264.7 macrophage cells by lipopolysaccharide (LPS). Cells were seeded in 96-well plates at a density of 8x10 3 /100 µL per well using full medium and incubated 24h at 37°C. Samples were prepared by diluting in FBS-free medium at equivalent to 34 µg/mL DXP concentration. Also, samples were tested without using cells to check their potential interference with the MTT assay. MTT assay was performed at the end of the experiment to confirm that all experiments are carried out in subtoxic conditions. At the end of the 24h incubation, the full medium was kindly removed from wells and replaced with 90 µL of samples.

Additionally, two columns were replaced with 90 µL of FBS-free medium as further controls.

Incubation remained 4h at 37°C. Afterward, a 100 µL volume of 20% FBS contained medium was added on wells and incubation carried 20h more. LPS solution was prepared at 1 µg/mL in 20% FBS containing medium. 10 µL volume of LPS solution was added either to the samples or to one of the two columns as obtaining positive control (LPS-only) 3 hours before the ends of the incubation time (hour 21 st ). On the other hand, 10 µL volume from 20% FBS contained medium was added to the other one of the two columns to have negative control (No treatment). At the end of the experiment, supernatants were collected and stored at -20°C until performing TNF-α assay using a Cytometric Bead Array (CBA) mouse inflammation kit as described below. Meanwhile, the first 200 µL of the full medium then 20 µL MTT/PBS solution at 5 mg/mL concentration were added to cells and incubated 2 hours at 37°C. The same protocol was used for the anti-inflammatory activity investigation of DSP, DOTAP-only, free siRNA, DOTAP/siRNA, and CDXP-NP and CDXP-siRNA-NP and Ctr siRNA. For that reason, all samples were prepared in an amount equivalent or corresponding to the 34 µg/mL DXP contained in CDXP-NP or equivalent siRNA concentration.

CBA Assay

For the CBA test, BD Biosciences CBA mouse inflammation kit was used. First, all the beads were taken in an Eppendorf tube and diluted with Assay Diluent to obtain Top Standard (TS) solution as described in the CBA kit. Then, it was diluted 8 times with a 1:1 ratio using assay diluent in volumes of 300 µL each time. Finally, the assay diluent was taken into an Eppendorf tube of the same volume and incubated for 15 minutes at RT. Following this step, respectively at volumes of 50/5/5 µL from standard solution/TNF-cytokine solution/PE-detection reagent was taken in an Eppendorf tube. This process was repeated for each standard solution and one time for assay diluent to obtain a calibration curve (between 0-5000 pg/mL) then incubated at RT for 2 hours. Meanwhile, samples were defrosted then diluted at 1/10 ratio with the CBA wash buffer and vortexed at low speed. At volumes of 50/5/5 µL from diluted samples/TNF-cytokine solution/PE-detection reagent was taken in an Eppendorf tube, respectively. This process was repeated for each sample and incubated at RT for 2 hours. At the end of the incubation time, a volume of 1 mL of CBA wash buffer was added to either calibration samples or the samples and centrifuged at 200 g, 5 minutes. Following, supernatants were aspirated then a volume of 250 µL of CBA wash buffer was added to each sample to resuspend the pellets and vortexed before running the experiment. Samples were quantified with the BD Accuri C6 Cytometer. All measurements were performed in triplicate.

The cellular siRNA uptake and its intracellular distribution with both CDXP-siRNA-NP and free siRNA in RAW264.7 cells were assessed against various incubation times by flow cytometry and confocal microscopy. For flow cytometry, 2 mL of the cell suspension at a final concentration of 15x10 4 / mL was added to the 6-well plates then left to incubate at + 37°C for 24 hours. The next day, CDXP-siRNA-NP and free siRNA were prepared using 1% fluorescent siRNA. Subsequently, the samples were diluted to 2 mL volumes with FBS-free medium at a final DXP equivalent concentration of 34 µg/mL or the corresponding siRNA concentration. Meanwhile, after 24 hours of incubation, the old cell media was gently removed from the wells with a pipette. A volume of 1 mL was added to the wells for the samples with an incubation time of 1, 2, 3, and 4 hours. Also, 1 mL volume of cell suspension alone was added to the wells to obtain negative control columns versus incubation hours.

At the end of their incubation periods of 1, 2, 3, and 4h investigations were performed by flow cytometry. The following procedure was used to collect the cells: samples were first removed from the wells and replaced with 1 mL PBS and cells were scraped off and placed into an Eppendorf tube.

The scrapping procedure was repeated once. Cells in Eppendorf tubes were centrifuged at 400 g for 4 minutes and the supernatant was removed. 1 mL of PBS was added to the pellets for reconstitution.

Samples were vortexed for 30 seconds before being analyzed. Relative fluorescence intensity was measured against time with the help of Plot 7 region FL2-A / SSC-A filter in a flow cytometer. The analysis was carried out with 5×10 4 activity, 220 µL for 2 minutes, and in slow mode.

Confocal Laser Scanning Microscopy (CLSM) images were obtained after 4h incubation. For that purpose, cells were seeded at 5x10 4 /300 µL per well using IBIDI 8-well plates then incubated 24h along at + 37°C. The next day, CDXP-siRNA-NP and free siRNA were prepared using 1% fluorescent siRNA in an FBS-free medium. Following the end of the incubation period, old cell media were removed from the plates, and samples in a volume of 140 µL were replaced and left for incubation at + 37°C for 4 hours. Samples were then imaged with an inverted STED-gated Leica TCS SP8 microscope (Leica, Germany) using an HC PL APO CS2 63x/1.40 oil immersion objective lens and under optimal incubation conditions provided by a heating stage. The instrument was equipped with a WLL (White Light Laser) laser. Red fluorescence emission was collected with a 560-760nm wide emission slit using a sequential mode. The pinhole was set at 1.0 Airy unit giving an optical slice thickness of 0.89µm. Twelve-bit numerical images were acquired using Leica SP8 LAS X software (Version 2.0.1; Leica, Germany). The same protocol was used for the fluorescent background investigation of cells alone (No Treatment).

3

Results and Discussion

Preparation of cationic dexamethasone palmitate nanoparticles

To produce CDXP-NP, various DOTAP concentrations with a fixed DXP content were used applying the solvent emulsion evaporation method as described (Lorscheider et al., 2019a). The lowest DOTAP concentration of 0.5 mg/mL led to nanoparticles with the highest hydrodynamic diameter (120 nm) but the lowest PdI (0.13) and highest zeta potential (70mV) (Figure 1). When DOTAP concentration increased, although the mean diameter was reduced, the polydispersity increased from 0.17 to 0.22 with increasing DOTAP concentration. As shown by Jensen et al. [START_REF] Jensen | Design of an inhalable dry powder formulation of DOTAP-modified PLGA nanoparticles loaded with siRNA[END_REF], DOTAP is thought to limit the enlargement of polymeric NPs due to its surfactant and condensation characteristics which affect the size. The zeta potential diminished to 57mV as DOTAP concentration increased (Figure 1). Statistical analysis was carried out using one-way ANOVA by Dunnett's multiple comparisons test and we concluded that NP with the least amount of DOTAP (0.5 mg/mL) had a significant difference in terms of hydrodynamic particle size (D H ) and zeta potential over all other CDXP-NP formulations. The zeta potential decrease can be explained by two hypotheses: i) as DOTAP concentration increases, the ionic strength increases and electrostatic screening becomes more important [START_REF] Israelachvili | Intermolecular and Surface Forces: Third Edition, Intermolecular and Surface Forces: Third Edition[END_REF] ii) for high concentrations, DOTAP may distribute within the nanoparticles and not only on their surface, therefore, hiding some charges. One can also imagine a combination of the two phenomena. Formulation stability was evaluated over 7 weeks at 4°C and room temperature (RT). At 4°C neither hydrodynamic diameter (102 ± 4 nm) and PdI (0.18 ± 0.02) nor zeta potential (59 ± 4 mV) changed significantly (Supporting information Figure S1). At RT, the hydrodynamic diameter started to increase moderately after 3 weeks of storage while neither the PdI nor the zeta potential was drastically modified (Supporting information figure S1). Storage at 4°C was therefore preferred to favor stability.

All these results suggest that DOTAP lipophilic chains interact strongly with the lipophilic dexamethasone palmitate [START_REF] Doi | Physicochemical properties of dexamethasone palmitate, a high fatty acid ester of an anti-inflammatory drug: Polymorphism and crystal structure[END_REF] and the cationic lipid is efficient to stabilize DXP into nanoparticles. Recent literature showed DXP could be successfully formulated with a high encapsulation efficiency (98% w/w) in anionic PEGylated NP using DSPE-PEG 2000 (Lorscheider et al., 2019b) as a stabilizer. These data support the idea that amphiphilic lipid molecules can coat the surface of lipophilic prodrug nanoparticles. This is not a peculiar property of lipid nanoparticles since DOTAP was also shown to cover the surface of poly(lactide-co-glycolide) (PLGA) nanoparticles either unloaded [START_REF] Jensen | Design of an inhalable dry powder formulation of DOTAP-modified PLGA nanoparticles loaded with siRNA[END_REF] or loaded with budesonide [START_REF] Leng | Engineering of budesonide-loaded lipidpolymer hybrid nanoparticles using a quality-by-design approach[END_REF].

Preparation of cationic dexamethasone nanoparticles associating TNF-α siRNA

From a very first fast screening, using gel electrophoresis, we looked at siRNA binding strength at two N/P ratio to select DOTAP concentrations to be used among the ones providing a low size and good monodispersity (1, 1.5, and 2 mg/mL). From all samples, at an N/P ratio of 10, siRNA migration was detected only for DOTAP concentration of 1 mg/mL but not for higher concentrations (supporting information Figure S2). Considering their good physical characteristics and their ability to tightly bind siRNA, DXP-DOTAP 5/1.5 mg/mL (CDXP-NP) were chosen as a good compromise for further studies.

The encapsulation efficiency and loading of DXP in CDXP-NP formulation were then quantified by HPLC. DXP encapsulation efficiency was found to be 95% w/w. The corresponding DXP loading and the corresponding active drug loading DXM was therefore 73% w/w leading to an equivalent loading of the parent drug DXM of 45% w/w. Our results confirmed DXP could be efficiently formulated in CDXP-NP using DOTAP as a stabilizer with similar high encapsulation efficiency as with PEGylated lipids in place of the cationic lipids (Lorscheider et al., 2019a).

Binding of siRNA to CDXP-NP was performed at various N/P ratios = 1, 2, 5, 10, and 20 and yielded complexes noted CDXP-siRNA-NP. At N/P=1, large and polydisperse aggregates were formed with a hydrodynamic diameter around 3000 nm, a PdI of 1, and almost a neutral zeta potential. The neutral zeta potential explains the observed aggregation (Figure 2 A). As the N/P ratio increases, the size of CDXP-SiRNA-NPdecreases from 180 ± 2 nm (n=3) for N/P=2 down to 90 ± 1 nm for N/P=20. PdI is reduced below 0.3 for all samples and even below 0.2 for N/P 10 and 20 (Figure 2A). The zeta potential becomes positive between 60 and 80 mV as soon as there is an excess of positive charges arising from CDXP-NPs (Figure 2B). The complexation efficiency of siRNA was then determined. At N/P=1, the percentage of complexed siRNA was around 68%. At higher N/P ratios, almost all the siRNA was complexed (95%) due to an excess of positive charges provided by CDXP-NPs (Figure 2C).

To assess the strength of complexation, gel electrophoresis was performed with CDXP-siRNA-NP and free siRNA. At N/P = 1, 2, and 5, the electric field led to some level of siRNA migration. Almost all siRNA migrated for N/P=1 and the amount migrating was reduced as N/P increased. For N/P 10 and 20, the siRNA migration could not be observed (Figure 2D). Since the strength of complexation may have a strong impact on the efficiency of siRNA delivery [START_REF] Singh | Lipoplex-based therapeutics for effective oligonucleotide delivery: a compendious review[END_REF], the formulation CDXP-SiRNA-NP at N/P 10 was selected for further experiments as a compromise between small size (around 100 nm), positive zeta potential (67 mV), complexation efficiency (95%), and siRNA amount.

The loading efficiency was of the same magnitude as PLGA cationic nanoparticles [START_REF] Hasan | Delivery of multiple siRNAs using lipid-coated PLGA nanoparticles for treatment of prostate cancer[END_REF], around a few percent.

TEM observations

TEM was performed to observe CDXP-NP and CDXP-siRNA-NP and compare the structures obtained to those formed by DOTAP and DOTAP-siRNA-lipoplexes (LPX). Indeed, the excess of DOTAP was not removed nor dosed due to the scarcity of methods to quantify it. CDXP-NP possess a spherical shape and some polydispersity can be observed (Figure 3A). DOTAP micelles were difficult to image properly and possess a more faceted morphology (Figure 3B). These types of structures are not visible on CDXP-NP images suggesting that all DOTAP is associated to the nanoparticles. When siRNA is complexed to CDXP-NP, spherical and non-spherical particles could be observed (Figure 3C). Similar morphologies were observed with DOTAP-siRNA-LPX. Sizes are in the same range as measured by dynamic light scattering (Figure 2A and Supporting information Table S2).

In-vitro studies

Before evaluating the inhibition of TNF-α, the cytotoxicity of all formulations was investigated by MTT assay on RAW264.7 macrophages. Since cationic nanoparticles may aggregate in serumsupplemented cell culture medium [START_REF] Doktorovova | Nanotoxicology applied to solid lipid nanoparticles and nanostructured lipid carriers -A systematic review of in vitro data[END_REF], formulations were first incubated in an FBS-free medium for 4h before adding FBS for the remaining time, preventing aggregation (data not shown). Interestingly, CDXP-NP and CDXP-siRNA-NP, above an equivalent DXP concentration of 34 µg/mL, have a very similar cytotoxicity profile (Figure 4). Other samples such as soluble dexamethasone (DSP), DOTAP, DOTAP-siRNA-LPX, and free siRNA did not induce a reduction of cell viability in the studied concentration range. In other words, while the components of nanoparticles and lipoplexes did not show a toxic effect alone, their combination leads to some toxicity (Figure 4).

Cytotoxicity does not arise from the cationic charges, as DOTAP and DOTAP-siRNA-LPX are also positively charged. Most probably, cytotoxicity arises from the better cellular entry of dexamethasone as a prodrug since it is formulated as cationic nanoparticles, whereas entrance is lower for soluble form DSP. Given the absence of cytotoxicity up to 34 µg/mL (eq. DXP concentration), the inhibition of TNF-α release was performed at this concentration. siRNA-NP as compared with those incubated with free siRNA or untreated cells (Figure 5).

Fluorescence in punctuated and mostly perinuclear.

The siRNA internalization kinetics was then assessed by flow cytometry. As shown in Figure 6, siRNA internalization was much more important after incubation with CDXP-siRNA-NP than with free siRNA, with stronger uptake as soon as after 1-hour incubation that kept increasing overtime until 4 hours (p < 0.05). After 4 hours incubation, uptake was found ~15 times higher with CDXP-siRNA-NP than with free siRNA. These data demonstrate cationic lipidic prodrug nanoparticles possess the same capacity as cationic polymers to induce a high siRNA uptake by RAW264.7 cells compared to free siRNA [START_REF] Bohr | Treatment of acute lung inflammation by pulmonary delivery of anti-TNF-α siRNA with PAMAM dendrimers in a murine model[END_REF][START_REF] Bohr | Anti-Inflammatory Effect of Anti-TNF-α SiRNA Cationic Phosphorus Dendrimer Nanocomplexes Administered Intranasally in a Murine Acute Lung Injury Model[END_REF]. Finally, TNF-α release inhibition was evaluated (Figure 7). There are three main groups for the different treatments. First of all, neither treatments with DOTAP, DOTAP-SiRNA-LPX nor free siRNA led to a reduction of TNF-α secretion as compared with LPS-activated cells. In the second group, treatments were significantly different compared to LPS-activated cells for DSP, CDXP-NP, and CDXP-CtrsiRNA-NP (p < 0.05), leading to a strong reduction of TNF-α secretion compared to untreated cells.

These effects were different than those induced by CDXP-siRNA-NP (p < 0.05) for which TNF-α secretion was even significantly greater. These results show that neither DOTAP-siRNA-LPX nor free siRNA can reduce TNF-α secretion at the siRNA concentration used (28 nM). DOTAP alone displays a poor efficiency of siRNA inhibition unless it is combined with helper lipids such as dioleylphosphatidylethanolamine (DOPE) [START_REF] Zhi | A review on cationic lipids with different linkers for gene delivery[END_REF]. However, in most cases where DOTAP was associated to polymer nanoparticles, no helper lipid was needed to obtain inhibition [START_REF] Hasan | Delivery of multiple siRNAs using lipid-coated PLGA nanoparticles for treatment of prostate cancer[END_REF].

One can notice that all treatments containing dexamethasone either soluble or as lipidic prodrug nanoparticles led to a reduction of TNF-α secretion although to a different extent. We can observe that CDXP-NP and CDXP-CtrsiRNA-NP were found equivalent to DSP probably due to the antiinflammatory effect of dexamethasone itself either in soluble form or as lipidic prodrug nanoparticles. A clear significant difference was found for CDXP-siRNA-NP compared to DSP, CDXP-NP and CDXP-CtrsiRNA-NP (p < 0.05). These results prove that the further reduction of TNF-α is sequence-specific and does not only arise from dexamethasone. Indeed, the combination of CDXP-NP and TNF-α siRNA leads to a synergistic effect in reducing TNF-α secretion.

CONCLUSION

Results clearly demonstrate that it is possible to combine within the same nanoparticle two different antiinflammatory drugs, the first being a lipophilic prodrug of dexamethasone, the second a TNF-α siRNA. Both drugs were loaded to a high extent through the use of a cationic lipid that interacts strongly with the DXP-NP through the acyl chains and with siRNA through the cationic head groups.

The resulting nanoparticles called CDXP-siRNA-NP were stable and highly internalized in vitro by macrophages. As a result, a strong inhibition of TNF-α was observed related to the combined effect of dexamethasone and the siRNA. As a perspective, we believe that such small particles would extensively migrate to inflamed tissues in several inflammatory diseases.

Figure 1 :

 1 Figure 1: Effect of various DOTAP concentrations in terms of hydrodynamic diameter (D H ) and PdI (A) and zeta potential (B).

Figure 2 :

 2 Figure 2: Physico-chemical characterization of lipoplexes (CDXP-siRNA-NP) formed by the association of siRNA and CDXP-NP at different N/P ratios in terms of size and PdI (A), zeta potential (B), percentage of complexed siRNA (C). Agarose gel electrophoresis of free siRNA or CDXP-siRNA-NPs at different N/P ratios (D).

Figure 3 :

 3 Figure 3: TEM images of CDXP-NP (A), DOTAP micelles (B), CDXP-siRNA-NP (C) and DOTAP-siRNA-LPX (D). The scale bars represent 200 nm in all images

Figure 4 :

 4 Figure 4: RAW264.7 murine macrophages viability after treating with DSP solution (green circles), CDXP-NP (blue squares), CDXP-siRNA-NP (purple squares), DOTAP (white circles), DOTAP-siRNA-LPX (orange circles), and free siRNA (pink circles) for 24 h at different concentrations. Concentrations are expressed as equivalent DXP.

Figure 5 :

 5 Figure 5: CLSM images of cells after 4h incubation with either free siRNA or CDXP-SiRNA-NP. The scale bar represents 20 microns.

Figure 6 :

 6 Figure 6: siRNA uptake by RAW264.7 cells versus vary incubation time for CDXP-siRNA-NP (purple) or free siRNA (pink). Uptake was significantly different between groups at all times (p < 0.05). Data are presented as mean±SD (N=3).

Figure 7 :

 7 Figure 7: Quantification of TNF-α cytokine secretion as a function of treatment: No treatment (light grey), LPS (dark grey), DSP solution (green), CDXP-NP (blue), CDXP-siRNA-NP (purple), CDXP-CtrsiRNA-NP (dark blue), DOTAP (white), DOTAP-siRNA-LPX (orange), free siRNA (pink). siRNA concentration was 28 nM. Data represent mean±SD (N=3). Statistical analysis was performed using an unpaired nonparametric Mann-Whitney test between groups (*, p < 0.05).

  was supported by TÜBİTAK 2214/A International Research Fellowship Program for PhD students. This research received support from Laboratory of Excellence LERMIT thanks to an ANR grant (ANR-10-LABX-33) and from CNRS. The present work has benefited from Imagerie-Gif core facility supported by l'Agence Nationale de la Recherche (ANR-11-EQPX-0029/Morphoscope, ANR-10-INBS-04/FranceBioImaging ; ANR-11-IDEX-0003-02/ Saclay Plant Sciences) with the precious help of C. Boulogne and C. Gillet. The authors would like to thank V. Nicolas (MIPSIT, Université Paris-Saclay) for help with confocal microscopy.

Figure S2 :

 S2 Figure S2: Agarose gel electrophoresis of CDXP-siRNA-NP at different N/P ratios and different DOTAP concentrations

Table S1 :

 S1 Gradient mobile phase system

	Process	Time	Flow	rate	% Eluent A	% Eluent B
		(minute)	(mL/min)
	1.		1.00		65.0	35.0
	2.	5.00	1.00		65.0	35.0
	3.	7.00	1.00		10.0	90.0
	4.	30.00	1.00		10.0	90.0
	5.	34.00	1.00		65.0	35.0
	6.	40.00	1.00		65.0	35.0

Table S2 :

 S2 Hydrodynamic diameter D H , PdI and zeta potential of formulations imaged by TEM

		Z-Average ; PdI ; ZP
	CDXP-NP	97 nm ; 0,2 ; + 62 mV
	CDXP-SiRNA-NP N/P 10	104 nm ; 0,2 ; + 67 mV
	DOTAP	76 nm ; 0,19 ; + 58 mV
	DOTAP-siRNA-LPX N/P 10	85 nm ; 0,28 ; + 41 mV

Supporting Information

Combining dexamethasone and TNF-α siRNA within the same nanoparticles to enhance antiinflammatory effect Serra Gürcan 1,2, Nicolas Tsapis 1 , Franceline Reynaud 1,3 , Stéphanie Denis 1 , Juliette Vergnaud 1 Özgen Özer 2 and Elias Fattal 1,* 1 Université Paris-Saclay, CNRS, Institut Galien Paris-Saclay, 92296, Châtenay-Malabry, France.


domain: spi phys chim

text:
Salim Chaibi1
email: salim.chaibi@onera.fr

Christophe Bouvet

Frédéric Laurin

Johann Rannou

Julien Berthe

Fabrice Congourdeau

Dominique Martini

Experimental and Numerical Study of Damages Induced by Low-Velocity Impact on 4th Generation Composite Laminates

The use of composite fiber reinforced polymer (CFRP) structures in the aircraft industry needs to demonstrate that they satisfy the same safety requirements as for metal structures. Among these requirements, impact damage tolerance is a major sizing criterion for those structures. Impact induce complex out-of-plane loadings that results on large damage extensions. Experimental campaigns are realized at the industrial level to design composite structures but, it induces long and costly development for innovative solutions. Finite Element Model (FEM) is a good tool for sizing efficiently composite components even those exposed to impact. Unfortunately, numerical models developed by academics are too complex to be used in the industry. Besides, due to the highly non-linear problem, explicit solvers are used to avoid convergence issues, which require an expertise in the results verification. In this way, we present a damage model developed at the ONERA [1,2] to predict the progressive rupture of last generation composite laminates subjected to low-velocity impact (LVI)/low-energy impact (LEI). This model has been developed with a fair level of complexity in order to be used in aeronautical industries. This material has thick resin layers at each interface between each plies, which increase the toughness of the laminate. In order to understand damage mechanisms of this specific material when subjected to LVI/LEI, an investigation of impact damages has been carried out in this study. Since LVI/LEI do not enter in the scope of fast transient dynamics, we use an implicit integration scheme and the first results using the damage model are compared with the experimental data given by experimental testing.

INTRODUCTION

In aeronautics, impact events occurring during ground operations or manufacturing such as tool dropping can induce a 3D accidental loading state. Therefore, when structures made of composite laminates are subjected to LVI, they undergo important strength reduction in compression when subjected to out-of-plane loadings due to their poor performances. Consequently, residual performances can fall up to 50 % of initial strength and must be taken into account for sizing composite structures.

To ensure impact damage tolerance requirements, industrials have developed phenomenological models based on several and costly experimental testing as explained in [START_REF] Fualdes | Composite Airbus damage tolerance methodology[END_REF]. The main limitation is the validity framework of these models which are restricted to the impact configuration. Intending to assist industrials, academics have developed numerical models [START_REF] Bouvet | Low velocity impact modeling in composite laminates capturing permanent indentation[END_REF][START_REF] Baluch | An efficient numerical approach to the prediction of laminate tolerance to Barely Visible Impact Damage[END_REF] but, although these models provide quite good results, the possibility of using them by engineers in industries is still a challenge. Indeed, such numerical models are quite complex to be used in aeronautical industries. As nonlinearities are introduced such as contact, damage laws, non-linear geometrical formulations, it results on a highly non regular problem. Besides, in order to exempt from convergence problems, explicit solvers are used. The use of explicit solvers needs proficiency in the result analysis, on the top of that, the use of explicit solvers may be out of the habits of some industrials.

These last decades, the understanding of impact damages and especially damages due to delaminations has driven material manufacturers to develop materials in order to improve impact resistance. The enhancement of interfaces between each plies is one of the numerous examples. In our study, the material is a CFRP of last generation tougher than the oldest generation, one of the specificity of this material is that interfaces between each plies are thick. As reported in [START_REF] Aksoy | Interlaminar shear fracture of interleaved graphite/epoxy composites[END_REF] graphite/epoxy interleaved composites allow the increasing of the interlaminar Mode-II fracture toughness, G IIC , which is an important parameter in impact.

Thus, Dassault Aviation and ONERA have launched a 3 years project called MARCOS 2 with the aim to progress in virtual testing on composite materials for impact tests. A physically based damage model [START_REF] Germain | Evaluation des capacités prédictives et de la robustesse de modèles à bases physiques pour la prévision de la tenue des structures composites présentant des singularités géométriques[END_REF] developed initially for in-plane loadings will be enriched to out-of-plane loadings (impact). Furthermore, the validation of the model will be performed on a CFRP laminate made of unidirectional (UD) plies of new generation. The expertise of ONERA and Dassault Aviation in the use of implicit solvers has allowed choosing such a solver to treat the impact problem, and to run post-impact calculation, as compression after impact (CAI).

In this study, the preliminary experimental characterization of the material is presented. Damage investigations by means of 2D (micrographs) and 3D (ultrasonic scan, computed tomography) inspections are conducted after LVI/LEI tests. Then, damage laws are detailed for the intra-ply and inter-ply damages. In order to understand each damage mechanisms, simulation results using cohesive zone model (CZM) are firstly presented, then simulations using both intra-ply and CZM are performed. The interaction between intra-ply matrix cracks and delaminations is then discussed. Finally, a first comparison between experimental results and simulations is presented.

EXPERIMENTAL INVESTIGATION OF IMPACT DAMAGES

In this section, the experimental study of LVI/LEI on a new composite material is presented to :  (i) Identify the different kinds of impact damages which contribute to dissipate the energy such as (delaminations, matrix cracking and fiber failures).  (ii) Observe the damage locations through-the-thickness.  (iii) Quantify which damage mechanisms are leading the material degradation.

In order to perform such investigation, assessment of impact damages has been performed with ultrasonic control, optical micrographs, and X-ray tomography (CT).

Material

The specificity of this material is that matrix is an epoxy toughened resin. Studies of similar materials have been performed in [START_REF] Priasso | Modélisation multi-échelle de l'endommagement des composites stratifiés avec intercouches[END_REF][START_REF] Bull | Damage assessment of particle-toughened carbon fibre composites subjected to impact and compression-after-impact using 3D X-ray tomography[END_REF], it has been shown that they are tougher than classical CFRP laminates. In fact, in [START_REF] Bull | Damage assessment of particle-toughened carbon fibre composites subjected to impact and compression-after-impact using 3D X-ray tomography[END_REF] it is demonstrated that the mode-II fracture toughness G IIc is increased, it can be noted that G IIc is one of the most important parameter to simulate impact because delamination propagates mainly in mode II.

Materials are provided by our partner Dassault Aviation, 4 different laminates are tested and are summarized in the table I.

Experimental set-up

To understand if the damage process of this material is similar as for oldest materials such as T700GC/M21 in terms of classical impact damages (fiber breakages, delamination cracks, matrix cracks), experimental tests have been performed at ONERA. Post-impact inspections (ultrasonic scans, micrographs, X-ray tomography scans) on coupons have been carried out to assess impact damages and, to supply data for comparing with simulation results.

Coupon sizes are 150 mm x 100 mm as recommended by the standard ASTM D7136 [START_REF]Standard test method for measuring thedamage resistance of a fiber-reinforced polymer matrix composite to a drop-weight impact event[END_REF]. Coupons are simply supported on an impact window as shown in [START_REF]Standard test method for measuring thedamage resistance of a fiber-reinforced polymer matrix composite to a drop-weight impact event[END_REF], the clamping system recommended in the standard is not used in order to simplify the boundary conditions. The impactor diameter is a 16 mm diameter with a weight of 13.01 kg. An anti-rebound system is used to avoid a second impact. In figure 1, the experimental set-up is presented. Three impact energy levels are considered (11J, 21J, 35J) by modifying the impact velocity and therefore the initial height of the impactor. In terms of instrumentation, a cell load is located within the impactor, the velocity of the impactor is measured by a photoelectric cell and, displacement of the impactor is calculated by integrating twice the acceleration. 2 samples have been tested for each configuration in order to estimate the scattering, table II provides a summary of the experimental campaigns.

J A 2   -  B 2   -  C 2   -  21 J A 2     B 2     C 2     35 J A 2     B 2     C 2    
Results from impact experiments and investigation of impact damages

GLOBAL RESPONSE

After the impact, the following data results are available.  Load over time  Displacement of the impactor over time These informations have been plotted in figure 2 to evaluate the global response of the material and to observe non-linearities occurring during the impact event. In this paper, only the results obtained on quasi-isotropic laminate (A) are presented.

First of all, it can be seen that scattering is very low especially for the two first levels of impact energy where curves are overlapping.

A first non-linearity is observed around 55% of the maximum load for all energy levels in figure 2a and 2b in black line. This nonlinearity has been observed also in [START_REF]Standard test method for measuring thedamage resistance of a fiber-reinforced polymer matrix composite to a drop-weight impact event[END_REF] for the T700GC/M21. According to [START_REF] Liv | Experimental study into compression after impact strength of laminates with conventional and nonconventional ply orientations[END_REF], this nonlinearity could be due to plastic deformation of particles during delamination. Therefore, delamination could be responsible of this non-linearity. Furthermore, it has been observed on slice tomography that fiber breakages in compression occur since the first impact energy level and affect the first three plies at the impacted face. Fiber breakages in compression could also explain the loss of stiffness occurring around 55% of the maximum load observed in figure 2a and2b.

A 37 % global stiffness decrease appears at 100% of the maximum load for the highest energy level (35 J), it seems that the laminate has reached it maximum loadbearing capability. This load drop is due to fiber breakages in tension of the plies located at the opposite of the impacted face.

In figure 2b, a typical load/displacement response is presented. During the loading, curves are superimposed, for the 35 J, it can be noted that after the load drop considerable oscillations can be seen until 90% of the maximum displacement where the unloading stage is quite feeble. The 35 J curve exhibits a different behavior from the other energy levels, which could be explained by large damage extends. Dissipated energies are calculated from figure 2b for the 3 levels of impact energies. Each energy level dissipates gradually (2.4 J, 4.6 J, and 35 J).

In figure 2c, the deformation energy is plotted, it can be seen that all impact energies haven't been recovered at the end of impact due to dissipated damages. Moreover, at 35 J, almost the energy has been dissipated in damages confirming the presence of higher dissipated mechanisms such as fiber failures.

In figure 2d the residual dent is plotted in function of the energy level, the higher is the energy, the deeper is the residual dent. The figure 2d shows that the evolution of the residual dent is not linear and an important difference can be relieved at 35 J. It could be explained by the increase of fiber breakages in compression at the impacted face which influences the residual dent. As no scattering has been noticed for residual dent, only the first residual dents of first samples have been plotted. ASSESSMENT OF DAMAGES Cut at 0° and 90° have been performed in order to quantify impact damages. Then, micrographs have been performed using scanning electron microscopy (SEM) at ONERA on quasi-isotropic (A) samples impacted at 21 J. In Figure 3a, we find classical damage mechanisms in composite laminate such as transverse cracks and delaminations at the interface between the 2 plies. Moreover, the strong interaction between matrix cracks due to out of plane shear stresses and delaminations is observed with the presence of fragments. In figure 3b, fiber breakages in compression, well known as fibers kinking are observed. The figure 3b illustrates the interaction between each damages and their complexity. It allows the observation and the comprehension of impact damages but the scenario is still unknown.

Besides, another intra-ply damage mechanism not much known is observed in figure 3b. This damage has already been mentioned in [START_REF] Bull | Damage assessment of particle-toughened carbon fibre composites subjected to impact and compression-after-impact using 3D X-ray tomography[END_REF] for this specific material and, could be explained by the high toughness of interphase between plies, resulting on easier delamination propagation within the plies and not between them. Ultrasonic scans (US) with 1 mm resolution and X-ray tomography with 23μm resolution allow us to assess the damaged zones through the thickness. In figure 4a, the projected damaged area of delamination obtained by ultrasonic inspection is presented, all metric values are normalized using the highest length (at 35 J) obtained by C-scan. Damaged area grows for the 2 first levels of energy and at 35 J, damaged area grows longitudinally especially for the lower interface in red [0°/45°] due to large delamination cracks interacting with splitting cracks on the last ply. Besides, a -45° oriented crack appears on the impacted surface since the first level of energy (11 J) and is growing with energy levels. C-Scan provides an useful data in impact which is the projected damaged area. The projected damaged area is growing with impact energy (values are normalized using the highest damaged area). Unfortunately, inspection by US is insufficient to understand precisely the damage mechanisms inside the laminate, therefore complementary X-ray tomography have been performed.

In figure 5b, a slice of CT-scan from sample impacted at 35 J shows the damage through the thickness, each damage is easily distinguished and allow a better understanding of damage location rather than C-scan. X-ray tomography allows us identifying each kind of damage mechanisms.

In figure 4b, damage segmentation through-the-thickness obtained with computed tomography (CT) scan is presented. From figure 5a, it can be observed that the cracks at -45° observed in C-scan are fiber breakages in compression located at the impacted face. These fibers kinking occur since the first level of energy (11J) and through the 3 first plies. It is due to high compression stress localized close to the impact zone. Concerning the damage extends in figure 4b, despite the fact that the damage patterns obtained with the 2 methods are quite similar, the damaged area obtained by X-ray is always slightly lower than that obtained by C-scan. Same assessment has been reported in [START_REF] Léonard | The quantification of impact damage distribution in composite laminates by analysis of X-ray computed tomograms[END_REF] and can be explained by the lower resolution of X-ray than C-Scan. For the 2 first levels of energy, no damage is observed under the impact due to compressive pressure that delays the onset of cracks, as observed in [START_REF] Bull | Damage assessment of particle-toughened carbon fibre composites subjected to impact and compression-after-impact using 3D X-ray tomography[END_REF]. This is discussed in the next section.

FINITE ELEMENT MODELING APPROACH

In this section, the damage model is firstly detailed with a special emphasize on the intra-ply damage law, then the inter-ply damage model is discussed with the introduction of a new onset criterion that takes into account shear reinforcement in presence of out-of-plane compression. Finally, the results of simulations are analyzed and a first simulation/experiment comparison is performed.

General principle

A multiscale hybrid approach for damage and failure predictions has been proposed in the WWFE-III [START_REF] Laurin | A multiscale hybrid approach for damage and final failure predictions of composite structures[END_REF]. During this exercise, the damage model has been evaluated on different test cases for different composite laminate materials. Although this model has predicted accurately the failure in several test cases, it is difficult to use such a complex damage model in industries. Therefore, during the MARCOS project, a simplified model has been proposed [START_REF] Germain | Evaluation des capacités prédictives et de la robustesse de modèles à bases physiques pour la prévision de la tenue des structures composites présentant des singularités géométriques[END_REF] with some inheritances from WWFE-III.

The main described sources of non-linearity are summarized in equation 1.

σ = C ̃∶ (ε -ε nl -ε ve -ε th ) (1)
With σ is the Cauchy stress ε is the total strain C ̃ is the effective stiffness due to damage with C ̃= S ̃-1 ε nl is the nonlinear elastic strain ε ve is the viscous strain ε th is the thermal strain A DAMAGE MODEL BASED ON OBSERVABLE VARIABLES As shown in figure 6, when a transverse crack reaches the interface between 2 adjacent plies, local-delamination appears at crack-tip where the stress gradient is very high. This local-delamination induces the saturation of transverse cracks.

It is important to take into account of matrix cracking because it induces load transfer to the neighborhood and stress concentration which are important to predict ply failure. Therefore, as proposed in [START_REF] Huchette | Sur la complémentarité des approches expérimentales et numériques pour la modélisation des mécanismes d'endommagement des composites stratifiés[END_REF], damage variables are introduced using observable parameters called 𝜌̅ and 𝜇̅ as shown in equation 2, which are respectively the normalized matrix crack density and the delamination ratio, where L being the length of the interface and h the thickness of the ply. A damage model can be decomposed in three main parts (i) onset of damage, (ii) evolution laws and (iii) effect of damages. These parts are described in the following.

{ 𝜌̅ = 𝜌 * ℎ 𝜇̅ = 𝜇 𝐿 (2)

DAMAGE ONSET

To create a crack in a UD ply, it is assumed that only 𝜏 12 , 𝜎 22 and 𝜏 23 stresses expressed in the material axis can contribute. Therefore, these 3 stresses are taken into account in the calculation of the driving force Y reported in equation 3.

Y

From the equation, it can be relieved that:  The contribution of shear in the driving force Y is led by 2 parameters 𝑎 24 and 𝑎 26 depending on the shear strength (𝑆 12 𝑅 , 𝑆 23 𝑅 ) and on the initial compliance tensor 𝑆 0 .  Y varies not only with stresses but also with effective compliance which increases with damage.  Positive part of the stress tensor 𝜎 + is used. It allows predicting the apparent reinforcement of the strength obtained for combined shear/compression loading. It is assumed that the onset of transverse cracking is reached when the driving force threshold 𝑌 0 detailed in equation ( 5) is achieved.

Y 0 = 1 2 (Y T 2 S 0 22 ) (5)
The threshold is identified using the in-situ transverse strength 𝑌 𝑇 . Due to the strongly dependency of the damage threshold to the ply thickness, the in-situ strength is expressed using 2 complementary criteria (a stress criterion and an energetic criterion) that have to be fulfilled for damage onset. The in-situ strength 𝑌 𝑇 is determined based on the work of [START_REF] Dvorak | Analysis of Progressive Matrix Cracking In Composite Laminates II. First Ply Failure[END_REF]. In order to obtain accurate onset of damage for shear loadings the (𝑎 24 , 𝑎 26 ) parameters are considered and depend on the shear strength. The in-plane strength 𝑆 12 𝑅 is calculated using [START_REF] Camanho | Prediction of in situ strengths and matrix cracking in composites under transverse tension and in-plane shear[END_REF] methodology. As no Interlaminar Shear Strength (ILSS) tests haven't been yet performed, the resistance 𝑆 23 𝑅 is used from T700GC/M21.

DAMAGE EVOLUTION LAW

Once the driving force Y becomes higher than the threshold 𝑌 0 , damage is initiated, and evolves thanks to a damage evolution law defined in equation 6, proposed firstly in [START_REF] Laurin | A multiscale hybrid approach for damage and final failure predictions of composite structures[END_REF].

 A strong coupling between 𝜌̅ and 𝜇̅ has been introduced, as noticed in [START_REF] Huchette | Sur la complémentarité des approches expérimentales et numériques pour la modélisation des mécanismes d'endommagement des composites stratifiés[END_REF], local delamination tends to slow down the evolution of transverse cracks.

 𝛼 𝐼 , 𝑛 are material parameters which control the damage kinetics of transverse cracking, 𝑎 ℎ and 𝑏 ℎ being also material parameters that control local delamination kinetics.  It can be seen that for thicker plies, transverse cracks and local-delaminations will be more important due to the introduction of the ply thickness h in the equation 6.

{ ρ ̅ = (1 -μ ̅)ℎ𝛼 𝐼 〈Y -Y 0 〉 𝑛 + μ ̅ = 〈a h ρ ̅ 2 + b h hρ ̅〉 + (6)

DAMAGE EFFECT

These two damage variables affect the effective stiffness matrix as exposed in equation 7. The effective compliance is calculated by adding to the initial elastic compliance S 0 , fourth order tensors to express damage effects in the different directions of the material. The effect tensors H X (with x ∈ (a, b, c)) are diagonal tensors, the different parameters ahead of tensor effects are identified numerically. Indeed, stiffness reduction is determined numerically thanks to finite element simulations on a representative elementary volume (REV) with a discrete crack as proposed in [START_REF] Laurin | A multiscale hybrid approach for damage and final failure predictions of composite structures[END_REF][START_REF] Germain | Evaluation des capacités prédictives et de la robustesse de modèles à bases physiques pour la prévision de la tenue des structures composites présentant des singularités géométriques[END_REF][START_REF] Huchette | Sur la complémentarité des approches expérimentales et numériques pour la modélisation des mécanismes d'endommagement des composites stratifiés[END_REF]. The effect tensors are therefore determined by homogenization knowing only the 3D elastic properties of a composite material.

S ̃= S 0 + ρ ̅H a + μ ̅ 1-μ ̅ H b + ρ ̅ 2 1-μ ̅ H c (7)

Delamination modeling

To model delamination, a cohesive zone model has been considered. A damage model is associated with interface elements with initial zero-thickness introduced between plies with different orientations. Such a damage model can be decomposed in two parts, (i) the onset of delamination and (ii) the delamination propagation. An initial elastic stiffness is associated with the interface element fixed here at 10 6 MPa/ mm, as it is widely used in the literature and allows a good compromise between convergence and spurious softening of the structure.

ONSET CRITERION 2 criteria have been considered for the onset of delamination :

 Quadratic criterion based on Yeh and Stratton criterion

( 〈σ 33 〉 + Z t ) 2 + ( τ 13 S c ) 2 + ( τ 23 S c ) 2 = 1 (8)
Where 〈-〉 + is the Macauley bracket, 𝑍 𝑡 is the out-of-plane tensile strength and 𝑆 𝑐 is the out-of-plane shear strength. It is assumed that the out-of-plane shear strength associated to τ 13 is equal to that associated with τ 23 .  Quadratic criterion with a strength reinforcement under combined shear/outof-plane compression [START_REF] Vandellos | Développement d'une stratégie de modélisation du délaminage dans les structures composites stratifiées[END_REF] ( σ 33

+ Z t ) 2 + ( τ 13 + S cr ) 2 + ( τ 23 + S cr ) 2 = 1 (9)
Where σ + corresponding again to the positive part of the stress. For the first classical criterion, it is assumed that the onset of delamination due to shear is insensitive to the level of out-of-plane compression (σ 33 < 0). But experimentally, it has been reported in [START_REF] Catalanotti | The effect of through-thickness compressive stress on mode II interlaminar fracture toughness[END_REF][START_REF] Li | Predicting the effect of through-thickness compressive stress on delamination using interface elements[END_REF] that through-thickness compressive stress delays the delamination onset and increases the mode-II interlaminar fracture toughness G IIc . During an impact, the presence of a high gradient of through-thickness compressive stress under the impactor has led us to consider the second onset criterion. It has been developed in [START_REF] Vandellos | Développement d'une stratégie de modélisation du délaminage dans les structures composites stratifiées[END_REF] for bolted joint problem with pre-loading but hasn't been tested for impact test.

In figure 7, the two onset criteria are compared. Using the first criterion, the onset of delamination doesn't evolve with through-thickness compression σ 33 . Whereas, for the second criterion the failure onset evolves with through-thickness compression. Benzeggagh-Kenane criterion [START_REF] Benzeggagh | Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed mode bending apparatus[END_REF] is used to model delamination propagation. The criterion reported in equation 10, is formulated with interlaminar fracture toughness. Double cantilever beam (DCB) and mixed-mode bending (MMB) have been performed for this material in [START_REF] Germain | Evaluation des capacités prédictives et de la robustesse de modèles à bases physiques pour la prévision de la tenue des structures composites présentant des singularités géométriques[END_REF] to identify interface properties. 4-point end notch flexure (ENF) should be performed soon to consolidate the identification of the mode II fracture toughness.

𝐺 𝑐 = 𝐺 𝐼𝑐 + (𝐺 𝐼𝑐 -𝐺 𝐼𝐼𝑐 ) ( 𝐺 𝐼𝐼𝑐 𝐺 𝐼𝑐 + 𝐺 𝐼𝐼𝑐 + 𝐺 𝐼𝐼𝐼𝑐 ) 𝜂 (10)

Finite element model

The proposed finite element model simulates the impact experiments on 150 mm x 100 mm laminated samples. The sample is simply supported on an impact window of 125 mm x 75 mm. The sample is impacted by a 13.01 kg impactor, with an initial velocity of 1.8 m. s -1 corresponding to a 21 J impact. Each ply of the composite laminate is meshed using a solid element as presented in figure 8. Interface elements are introduced to model delamination between each ply. The damage model is associated with each ply. The impactor and the impact window are meshed using solid elements, the behavior of the impactor and the impact window are assumed isotropic linear elastic using the properties E=200 GPa and ν=0.3. Contact without friction is used with a penalty contact law for the impactor/laminate and laminate/impact window. The assumption for small displacement is not relevant, geometric nonlinearity is thus taken into account.

Despite all the non-linearities and convergence issues, a dynamic implicit scheme (HILBER-HUGHES-TAYLOR method) is used. The FE simulations have been performed using Zset code, co-developed by ONERA and Ecole des Mines.

Analysis and discussion of calculation results

5 simulations summarized in table III, are compared. Firstly, the effect of the two onset criterion on delamination is analyzed in the simulations 1 and 2. Simulations 1 and 2 were run with CZM using interface elements between each ply and without intra-ply damage law. The simulation 3 was run with only intra-ply damage law for each ply in order to study the effect of matrix cracking and to visualize the shape of matrix cracks. Then, the simulations 4 and 5 were carried out using intra-ply model and CZM to study how damages (inter-ply/intra-ply) interact. Finally, a comparison of simulations 4 and 5 are made with the experimental results in terms of the global response and projected delaminated area.

INFLUENCE OF ONSET CRITERION ON DELAMINATION

In figure 4b, X-ray tomography have relieved that for the impact energy at 11J and 21J, no damages are localized under the impact. In fact, under combined shear/out-of-plane compression, shear reinforcement is observed experimentally. Therefore, the second onset criterion has been used in simulation 2, in order to study the influence of the onset criterion on damage extend. By using the second onset criterion, the onset of damage under the contact zone (i.e. where a high through the thickness compressive stress gradient is present) is delayed due to the reinforcement of the material in presence of out-of-plane compression.

In figure 9, projected delaminated area is presented between simulation 1 and 2 for the front (impact side) and rear views. In this figure, all damaged Gauss points (d >0, where d being the damaged variable) are shown, the figure is observed as a C-scan in depth with 0 being the impacted face and 1 being the non-impacted face for the front view.

Delamination patterns are differents between simulations 1 and 2. Thus, by taking into account the shear reinforcement, a significant change on the delamination extend is observed. In figure 9, the simulation 2 using the reinforcement criterion is getting closer from the experimental assessment than simulation 1, indeed on the rear view, the first interfaces are more damaged as seen in the tomography. Moreover, as shown in figure 10, the simulation 2 allows a delamination distributed more regularly for each interface. By comparing the simulation 3 with the simulations 4 and 5, the introduction of interface elements with CZM changes in a meaningful way the matrix crack shapes. As delamination cracks propagate, the initial laminate flips through sub-laminates shifting the neutral axis, resulting in different matrix crack patterns.

In figure 12, the effect of intra-ply damage on the delamination extend is shown. By comparing, for fixed onset criteria, the simulation with CZM and elastic behavior and simulation with intra-ply damage + CZM, it can be relieved that the matrix cracks have an influence on delamination extend. The influence of intra-ply damage on delamination is higher using the reinforced onset delamination criterion, especially at the rear side. In fact, transfers loading of damaged gauss points due to matrix cracks effects on the neighborhood, induce modifications in delamination onset. A first comparison is illustrated in figure 13 between the simulations 4 and 5 and the experimental results at 21 J. Values are normalized using the experimental results. For the macroscopic behavior, it can be seen that the global impact response is almost the same for the 2 simulations even if delamination patterns are quite different. The global behavior of the laminate is well reproduced until 55% of the load and then simulations are getting stiffer. It could be explained by the fact that fibers kinking under the impactor are not yet taken into account in the current model, resulting on underestimated dissipated energies as seen in figure 13. A damage model for fiber failure in tension and compression will be developed soon.

In figure 14, projected delaminated area for the front and the rear views are shown. The extent of damages is higher in simulations forecasting than obtained by Xray tomography. As explained previously, fibers damage are not taken into account actually, therefore, the two only dissipative mechanisms are matrix cracking and delamination which will surestimate the dissipated energy by those mechanisms. Besides, the value of G IIc has been fitted using the mixed mode bending curves for a 100 % mixity. For this kind of material, the impact toughness is linked to the increase of G IIc . 4 points-ENF test will be performed soon to estimate rigorously G IIc value. For the rear view, the simulation 5 (using the reinforcement criterion) seems to show a better agreement with the experiment than the simulation 4.

CONCLUSIONS Conclusions and outlooks

A first study has been presented on LVI/LEI for a carbon/epoxy composite interleaved laminate. The data obtained from inspections allowed us understanding the damage pattern obtained after the LVI/LEI in a new generation of UD ply material. However, questions remain. All of inspections have been performed after impact event, therefore, the evolution of damages hasn't been monitored during loading. To validate the damage model, it seems relevant to understand how damage kinetics evolves and how the different damage mechanisms interact with each other. To do so, modifications of our impact experimental device have been performed. It will allow to monitor damage in real-time by adding high-speed infrared and optical cameras for digital image correlation in order to establish precisely the damage scenario.

Numerically, an intra-ply damage model based on observable variables has been introduced in order to simulate LVI/LEI on an industrial CFRP laminate. This model has been identified and validated previously in [START_REF] Germain | Evaluation des capacités prédictives et de la robustesse de modèles à bases physiques pour la prévision de la tenue des structures composites présentant des singularités géométriques[END_REF] for in-plane loadings and will be enhanced for out-of-plane loadings. Impact simulations have been performed using the in-house Z-set code. Despite the complexity of the problem, simulations have been carried out with a dynamic implicit scheme and first comparisons with experimental data have been performed. Results have shown that the current limitation of the numerical model is the non-introduction of a fiber damage law to simulate the progressive but violent degradation of mechanical properties of fiber breakage. Therefore, the numerical model will be improved taking into account the fiber breakage in tension and compression in a near future.

Figure 1

 1 Figure 1 Experimental set-up available at ONERA

Figure 2

 2 Figure 2 Effects of the impact energy on global responses of quasi-isotropic samples (A)

Figure 3

 3 Figure 3 Damage patterns observed in SEM for 90° cut on quasi-isotropic (A) sample impacted at 21 J

Figure 4 (

 4 Figure 4 (a) Projected damaged area for the 3 energy levels applied on quasi-isotropic samples (A) obtained by C-scan in (a) and CT in (b)

Figure 6

 6 Figure 6 Damage variables observable from SEM micrograph on quasi-isotropic (A) sample impacted at 21 J

Figure 7

 7 Figure 7 Comparison of Yeh and Stratton and the reinforcement criterion

Figure 8 FEM

 8 Figure 8 FEM of the impact experiment

Figure 9 2 Figure 10

 9210 Figure 9 Comparison of the delamination extends in the simulations 1 and 2

Figure 11 , 4 and 5 Figure 12

 11512 Figure 11 Longitudinal cut under the impact showing matrix cracks and delamination shape for the simulations 3, 4 and 5

Figure 13

 13 Figure 13 Macroscopic comparison of load-displacement curves between tests and simulations on (A) laminate for 21 J impact energy

TABLE I

 I

	SUM UP OF THE COMPOSITE LAMINATES TESTED
	Laminates	Numbers of plies Lay-ups (0°/45°/-45°/90°)
	(A) Quasi-isotropic	16	4/4/4/4
	(B) 0° oriented	20	10/4/4/2
	(C) 0°/90° oriented	20	8/2/2/8

TABLE II . SUMMARY OF THE EXPERIMENTAL CAMPAIGNS

 II

	Experiment		Inspection
	Energies Laminates Samples Ultrasonic	CT	Micrographs Measure of the
	scans	scans	residual dent
	11

TABLE III .

 III FINITE ELEMENT SIMULATIONS SUMMARY


Bin Feng
email: binfengcq@163.com

Jie Wu
email: jie.wu@math.cnrs.fr

THE ARCSINE LAW ON DIVISORS IN ARITHMETIC PROGRESSIONS MODULAR PRIME POWERS

Keywords: 2010 Mathematics Subject Classification. 11N37 Selberg-Delange method, arcsine law, arithmetic progressions

Let x → ∞ be a parameter. In 2016, Feng proved that Deshouillers-Dress-Tenenbaum's arcsine law on divisors of the integers less than x also holds in arithmetic progressions for non Siegel 'exceptional' modulus q exp{( 1 4 -ε)(log 2 x) 2 }, where ε is an arbitrarily small positive number. In this paper, we shall show that in the case of prime-power modulus (q := p with p a fixed odd prime and ∈ N) the arcsine law on divisors holds in arithmetic progressions for q x 15/52-ε .

Introduction.

For each positive integer n, denote by τ (n) the number of divisors of n and define the random variable D n to take the value (log d)/ log n, as d runs through the set of the divisors of n, with the uniform probability 1/τ (n). The distribution function F n of D n is given by (1.1) F n (t) := Prob(D n t) = 1 τ (n) d|n, d n t 1 (0 t 1).

Deshouillers, Dress and Tenenbaum ([4] or [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF]Theorem II.6.7]) proved that the Cesàro means of F n converges uniformly to the arcsine law. More precisely, the asymptotic formula

(1.2) 1 x n x F n (t) = 2 π arcsin √ t + O 1 √ log x
holds uniformly for x 2 and 0 t 1 and the error term in (1.2) is optimal. Various variants of (1.2) have been investigated by different authors. In particular, Cui & Wu [START_REF] Cui | The Selberg-Delange method in short intervals with an application[END_REF] and Cui, Lü & Wu [START_REF] Cui | The Selberg-Delange method in short intervals with some applications[END_REF] considered generalisation of (1.2) to the short interval case; and Feng & Wu [START_REF] Feng | Beta law on divisors of integers representable as sum of two squares[END_REF] showed that the average distribution of divisors over integers representable as sum of two squares converges to the beta law. Based on Cui-Wu's method [START_REF] Cui | The Selberg-Delange method in short intervals with an application[END_REF], Feng [START_REF] Feng | On the arcsine law on divisors in arithmetic progressions[END_REF] studied analogue of (1.2) for arithmetic progressions. His result can be stated as follows: Let a and q be integer such that (a, q) = 1, and suppose that q is not a Siegel 'exceptional' modulus. Then for any ε ∈ (0, 1 4 ) we have

(1.3) 1 (x/q) n x n≡a(mod q) F n (t) = 2 π arcsin √ t + O ε e √ log q √ log x
uniformly for 0 t 1, x 2 and q exp{( 1 4 -ε)(log 2 x) 2 }, where log 2 := log log.

The aim of this paper is to improve the result above in the case of prime power modulus. Our result is as follows.

Theorem 1. Let q := p with p an odd prime and ∈ N. Then for any ε > 0, we have

(1.4) 1 (x/q) n x n≡a(mod q) F n (t) = 2 π arcsin √ t + O p,ε 1 √ log x
uniformly for 0 t 1, x 2, q x 15/52-ε and a ∈ Z * such that (a, q) = 1, where the implied constant depends on p and ε at most.

Our improvement is double. Firstly, with q = p any Siegel zero must occur for L(s, χ) where χ is a real character modulo p. Since the implied constant in Theorem 1 is allowed to depend on p, there is no Siegel zero for the modulus q = p . These considerations allow to remove the assumption of Siegel zero in Feng's result for q = p with an implied constant in the error term depending on p. Alternatively, this follows from Feng's result and Corollary 3.4 of the Banks and Shparlinski paper [START_REF] Banks | Boun ds on short character sums and L-functions for characters with a smooth modulus[END_REF] (cf. Lemma 2.3 below). Secondly the domain of q is extended significantly.

Preliminary

Our first lemma is an effective Perron formula (cf. [10, Corollary II.2.2.1]). Lemma 2.1. Let F (s) := ∞ n=1 a n n -s be a Dirichlet series with finite abscissa of absolute convergence σ a . Suppose that there exist some real number α > 0 and a non-decreasing function B(n) such that:

(a) ∞ n=1 |a n |n -ς (ς -σ a ) -α (ς > σ a ), (b) |a n | B(n) (n 1).
Then for x 2, T 2, σ σ a and κ := σ a -σ + 1/ log x, we have

n x a n n s = 1 2πi κ-iT κ+iT F (s + w)x w dw w + O x σa-σ (log x) α T + B(2x) x σ 1 + x log T T .
Lemma 2.2. Let q > 2 be an integer.

(i) If χ is a Dirichlet character modulo q, then we have

L(σ + iτ, χ) q 1-σ (|τ | + 1) 1/6 log(|τ | + 1). (ii) If χ is a non principal Dirichlet character modulo q, then for 0 < ε < 1 2 , ε σ 1, |τ | + 2 T , we have L(σ + iτ, χ) ε (q 1/2 T ) 1-σ+ε .
Proof. See [9, p.485, Theorem 1] and [START_REF] Tenenbaum | Théorie analytique et probabiliste des nombres: 307 exercices corrigés[END_REF]Exercise 241].

The next lemma is due to Banks-Shparlinski [1, Corollary 3.4.], which will play a key role in the proof of Theorem 1.

Lemma 2.3. Let q = p with p an odd prime and ∈ N. For each constant A > 0, there is a constant c 0 = c 0 (A, p) > 0 depending only on A and p such that for any character χ modulo q, the Dirichlet L-function has no zero in the region

(2.1) σ > 1 - c 0 (log q) 2/3 (log 2 q) 1/3 and |τ | q A .
The following lemma is a key for the proof of Theorem 1.

Lemma 2.4. Let q := p with p a prime and ∈ N and let χ 0 be the principal character to the modulus q. Then we have

(2.2) n x χ 0 (n) τ (nd) = hx √ π log x g(d) + O (3/4) ω(d) log x
uniformly for x 2, 1 d x and 1, where the implied constant is absolute, ω(d) is the number of all distinct prime factors of d,

(2.3) h := 1 -p -1 (p,p)=1 1 -p -1 log(1 -p -1 ) -p -1
and

(2.4) g(d) := p α d ∞ j=0 (χ 0 (p)p -1 ) j j + α + 1 -χ 0 (p)p -1 log(1 -χ 0 (p)p -1 ) • Proof.
As usual, denote by v p (n) the p-adic valuation of n. By using the formula

(2.5) τ (dn) = p (v p (n) + v p (d) + 1),
we write for e s > 1

f d (s, χ 0 ) := ∞ n=1 χ 0 (n) τ (dn) n -s = p ∞ j=0 (χ 0 (p)p -s ) j j + v p (d) + 1 = (p,d)=1 ∞ j=0 (χ 0 (p)p -s ) j j + 1 × p α d ∞ j=0 (χ 0 (p)p -s ) j j + α + 1 = p ∞ j=0 (χ 0 (p)p -s ) j j + 1 × p α d ∞ j=0 (χ 0 (p)p -s ) j j + α + 1 ∞ j=0 (χ 0 (p)p -s ) j j + 1 -1 = L(s, χ 0 ) 1/2 G d (s, χ 0 ), (2.6)
where

G d (s, χ 0 ) := p ∞ j=0 (χ 0 (p)p -s ) j j + 1 1 -χ 0 (p)/p s p α d ∞ j=0 (χ 0 (p)p -s ) j j + α + 1 ∞ j=0 (χ 0 (p)p -s ) j j + 1 -1
is a Dirichlet series that converges absolutely for e s > 1 2 .

We easily see that

p α d ∞ j=0 (χ 0 (p)p -s ) j j + α + 1 ∞ j=0 (χ 0 (p)p -s ) j j + 1 -1 = 1 α + 1 + O 1 √ p .
for e s 1 2 , where the implied constant is absolute. This implies that for any ε > 0,

(2.7) G d (s, χ 0 ) p α d 1 α + 1 + O 1 √ p C ε 3 4 ω(d)
for e s 1 2 + ε, where C ε > 0 is a constant depending on ε only. We can apply Lemma 2.1 with the choice of parameters

σ a = 1, B(n) = 1, α = 1 2 and σ = 0 to write n x χ 0 (n) τ (nd) = 1 2πi b+iT b-iT f d (s, χ 0 ) x s s ds + O ε x log x T ,
where b = 1 + 2/ log x and 100 T x such that ζ(σ + iT ) = 0 for 0 < σ < 1. Let M T be the boundary of the modified rectangle with vertices ( 1 2 + ε) ± iT and b ± iT as follows:

• ε > 0 is a small constant chosen such that ζ( 1 2 + ε + iγ) = 0 for |γ| < T ; • the zeros of ζ(s) of the form ρ = β + iγ with β > 1
2 + ε and |γ| < T are avoided by the horizontal cut drawn from the critical line inside this rectangle to ρ = β + iγ;

• the pole of ζ(s) at the points s = 1 is avoided by the truncated Hanke contour Γ (its upper part is made up of an arc surrounding the point s = 1 with radius r := 1/ log x and a line segment joining 1 -r to ( 1 2 + ε).

Γρ Γ L 1 L 2 L 4 L 3 b = 1 + 2 log x τ O 1 2 + ε 1 Figure 1 -Contour M T
Clearly the function f d (s, χ 0 ) is analytic inside M T . By the residue theorem, we can write (2.8)

n x χ 0 (n) τ (nd) = I + 1 2πi I 1 + • • • + I 4 + β> 1 2 +ε, |γ|<T I ρ + O ε x log x T ,
where

I := 1 2πi Γ f d (s, χ 0 ) x s s ds, I ρ := Γρ f d (s, χ 0 ) x s s ds, I j := L j f d (s, χ 0 ) x s s ds. A. Evaluation of I. Let 0 < c < 1 10 be a small constant. Since G d (s, χ 0 )((s -1)ζ(s)) 1/2 (1 -p -s ) 1/2 is holomorphic and O((3/4) ω(d)
) in the disc |s -1| c thanks to (2.7), the Cauchy formula allows us to write

G d (s, χ 0 )((s -1)ζ(s)) 1/2 (1 -p -s ) 1/2 = G d (1, χ 0 )(1 -p -1 ) 1/2 + O((3/4) ω(d) |s -1|) for |s -1| 1 2 c. In view of L(s, χ 0 ) = ζ(s)(1 -p -s ) and G d (1, χ 0 )(1 -p -1 ) 1/2 = hg(d), it follows that f d (s, χ 0 ) = hg(d)(s -1) -1/2 + O((3/4) ω(d) |s -1| 1/2 ) for |s -1| 1 2 c. So we have (2.9) I = hg(d)M (x) + O((3/4) ω(d) E 0 (x)),
where

M (x) := 1 2πi Γ (s -1) -1/2 x s ds, E 0 (x) := Γ |(s -1) 1/2 x s ||ds|.
Firstly we evaluate M (x). By using [10, Corollary II.5.2.1], we have

(2.10) M (x) := x √ log x 1 Γ( 1 2 ) + O x -c/2 .
Next we deduce that

E 0 (x) 1-1/ log x 1/2+ε (1 -σ) 1/2 x σ dσ + x (log x) 3/2 x (log x) 3/2 ∞ 1 t 1/2 e -t dt + 1 x (log x) 3/2 • (2.11)
Inserting (2.10) and (2.11) into (2.9) and noticing that Γ( 12 ) = √ π, we find that (2.12)

I = x √ π log x hg(d) + O ε (3/4) ω(d) log x .
B. Estimations of I 1 and I 2 .

It is well known that (cf. [10, Corollary II.3.5.2])

(2.13) |ζ(σ + iτ )| |τ | (1-σ)/3 log |τ | ( 1 2 σ 1 + log -1 |τ |, |τ | 3).
Noticing that q := p , it follows that

(2.14) L(s, χ 0 ) = ζ(s)(1 -p -s ) |τ | (1-σ)/3 log |τ | for 1 2 σ 1 + log -1 (|τ | + 3)
|I 1 | + |I 2 | ε (3/4) ω(d) 1+2/ log x 1/2+ε T (1-σ)/6 (log T ) x σ T dσ ε (3/4) ω(d) x T log T.
C. Estimations of I 3 and I 4 .

As before, (2.6) and (2.14) allow us to deduce

|I 3 | + |I 4 | ε (3/4) ω(d) T 1 (|τ | + 1) 1/12 log(|τ | + 1) x 1/2+ε |( 1 2 + ε) + iτ )| dτ ε (3/4) ω(d) x 1/2+ε T 1 (τ + 1) -1+1/12 dτ ε (3/4) ω(d) x 1/2+ε T 1/12 .
(2.16)

D. Estimation of I ρ .
With the help of (2.14) and (2.7), we can derive that for s = σ + iγ with (2.17)

I ρ ε (3/4) ω(d) β 1/2+ε |γ| (1-σ)/6 (log |γ|) 1/2 x σ |σ + iγ| dσ.
Denote (|τ | + 3) (c > 0 absolute constant). Summing (2.17) over |γ| < T and interchanging the summations and noticing that β < 1 -σ(T 1 ) (the Korobov-Vinogradov zero free region), we have

β> 1 2 +ε, |γ|<T |I ρ | (3/4) ω(d) (log T ) max T 1 T β> 1 2 +ε, T 1 /2<|γ|<T 1 |I ρ | ε (3/4) ω(d) (log T ) max T 1 T 1-σ(T 1 ) 1/2+ε T (1-σ)/6 1 • x σ T 1 • N (σ, T 1 ) dσ.
According to [START_REF] Huxley | The difference between consecutive primes[END_REF], it is well known that

(2.18) N (σ, T ) T (12/5)(1-σ) (log T ) 44
for 1 2 + ε σ 1, and T 2. Thus

β> 1 2 +ε, |γ|<T |I ρ | (3/4) ω(d) (log T ) 45 max T 1 T 1-σ(T 1 ) 1/2+ε T (1-σ)/6 1 x σ T 1 T (12/5)(1-σ) 1 dσ x(log T ) 45 max T 1 T 1-σ(T 1 ) 1/2+ε T 17/30 1 x 1-σ dσ x(log T ) 45 max T 1 T T 17/30 1 x σ(T 1 )
x(log T ) 45 T 17/30 x σ(T ) .

(

χ 0 (n) τ (nd) = x √ π log x hg(d) + O ε (3/4) ω(d) log x + O ε (R d,T (x)),
where

R d,T (x) := 3 4 ω(d) x T log T + x 1/2+ε T 1/12 + x(log T ) 45 T 17/30 x σ(T ) + x log x T •
Taking T = x and ε = 10 -3 and noticing that ω(d) (log x)/ log 2 x for d x, it is easy to verify that R d,T (x)

(3/4) ω(d) x/(log x) 3/2 for d x. This completes the proof.

Lemma 2.5. Under the notation in Lemma 2.4, we have

(2.20) h d x χ 0 (d)g(d) = (ϕ(q)/q)x √ π log x 1 + O 1 log x ,
where the implied constant is absolute.

Proof. According to (2.4), it is easy to see that g(d) is a multiplicative function and

(2.21)

g(p ν ) = j 0 (χ 0 (p)p -1 ) j j + ν + 1 k 0 (χ 0 (p)p -1 ) k k + 1 -1 = -χ 0 (p)p -1 log(1 -χ 0 (p)p -1 ) j 0 (χ 0 (p)p -1 ) j j + ν + 1 • For σ > 1, we can write n 1 χ 0 (n)g(n)n -s = L(s, χ 0 ) 1/2 n 1 β(n)n -s = ζ(s) 1/2 (1 -p -s ) 1/2 n 1 β(n)n -s ,
where β(n) is a multiplicative function determined by

(2.22) ν 1 β(p ν )ξ ν = (1 -χ 0 (p)ξ) 1/2 ν 0 χ 0 (p)g(p ν )ξ ν (|ξ| < 1).
Since |g(p ν )| 1, the right-hand side is holomorphic for |ξ| < 1 and we have β(p ν )

3 2 ν (ν = 1, 2, . . .). In addition, β(p) = χ 0 (p)(g(p) -1/2) = O(1/p). These imply the absolute convergence of β(n)n -s for σ > 1 2 and β(n)n -s ε 1 for σ 1 2 + ε. Applying Theorem II. 5.3 of [10], we have n x χ 0 (n)g(n) = x √ log x λ 0 ( 1 2 ) + O 1 log x ,
where we have

λ 0 ( 1 2 ) := (1 -p -1 ) 1/2 Γ( 1 2 ) p (1 -χ 0 (p)p -1 ) 1/2 ν 0 χ 0 (p) ν g(p ν ) p ν ,
thanks to (2.21) and (2.22). In view of (2.21), it follows, with the notation ξ = χ 0 (p)p -1 ,

ν 0 χ 0 (p) ν g(p ν ) p ν (1 -χ 0 (p)p -1 ) = j 0 ξ j j + 1 -1 (1 -ξ) ν 0 j 0 ξ j+ν j + ν + 1 = j 0 ξ j j + 1 -1 (1 -ξ) k 0 ξ k = -χ 0 (p)p -1 log(1 -χ 0 (p)p -1 ) • Thus λ 0 ( 1 2 ) = (1 -p -1 ) 1/2 √ π p (1 -χ 0 (p)p -1 ) -1/2 -χ 0 (p)p -1 log(1 -χ 0 (p)p -1 )
and hλ 0 ( 1 2 ) = (1 -p -1 )/ √ π = (ϕ(q)/q)/ √ π, which concludes the proof of (2.20).

Lemma 2.6. Let q = p with p an odd prime and ∈ N. For any ε > 0, there is a positive constant c 1 (ε) > 0 depending on ε such that we have

(2.23) χ =χ 0 χ(a)χ(d) n x χ(n) τ (nd) xe -c 1 (ε)(log x) 1/3 (log 2 x) -1/3
uniformly for d 1, x 2, q x 15/52-ε and a ∈ Z * such that (a, q) = 1.

Proof. Since the proof is rather close to that of Lemma 2.4, we only mentionne the principal points. As before, by (2.5), we can write for σ := e s > 1

(2.24)

f d (s, χ) := ∞ n=1 χ(n)τ (dn) -1 n -s = L(s, χ) 1/2 G d (s, χ),
where

G d (s, χ) := p ∞ j=0 (χ(p)p -s ) j j + 1 (1 -χ(p)p -s ) 1/2 p α d ∞ j=0 (χ(p)p -s ) j j + α + 1 ∞ j=0 (χ(p)p -s ) j j + 1 -1
is a Dirichlet series that converges absolutely for σ > 1 2 and verifies

|G d (s, χ)| C ε ( 3 4 ) ω(d) for σ 1 2 + ε and d 1,
where ε is an arbitrarily small positive constant and C ε > 0 is a constant depending only on ε.

We apply Lemma 2.1 with

σ a = 1, B(n) = 1, α = 1 2 and σ = 0 to write n x χ(n) τ (nd) = 1 2πi b+iT b-iT f d (s, χ) x s s ds + O x log x T ,
where b = 1 + 2/ log x and 100 T x such that L(σ + iT, χ) = 0 for 0 < σ < 1. Let M T be the boundary of the modified rectangle with vertices ( 1 2 + ε) ± iT and b ± iT as follows:

• ε > 0 is a small constant chosen such that L( 1 2 + ε + iγ, χ) = 0 for |γ| < T ; • the zeros of L(s, χ) of the form ρ = β + iγ with β > 1 2 and |γ| < T are avoided by the horizontal cut drawn from the critical line inside this rectangle to ρ = β + iγ.

Clearly the function f d (s, χ) is analytic inside M T . By the Cauchy residue theorem, we can write (2.25)

n x χ(n) τ (nd) = I 1 + • • • + I 4 + β> 1 2 +ε, |γ|<T I ρ + O x log x T ,
where

I j := 1 2πi L j f d (s, χ) x s s ds, I ρ := 1 2πi Γρ f d (s, χ)
x s s ds and L j and Γ ρ are as in Figure 1.

A. Estimations of I 1 and I 2 .

In view of (2.24) and Lemma 2.2, we have (2.26)

|I 1 | + |I 2 | 1+2/ log x 1/2+ε (q 1/2 T ) 1 2 (1-σ)+ε • x σ T dσ x T 1+2/ log x 1/2+ε q 1/4 T 1/2 x 1-σ dσ x T • B.
|I ρ | (log T ) max T 1 T 1-σ(T 1 ;q) 1/2+ε q 1 2 (1-σ) T 1/12+ε 1 x σ T 1 N (σ, T 1 , χ) dσ.
where σ(τ ; q) := C log -2/3 (q|τ | + 3q) log

-1/3 2 (q|τ | + 3q) (C = C(p)
is a positive constant depending on p) and we have used Lemma 2.3.

It is well-known that (cf. [START_REF] Montgomery | Topics in multiplicative number theory[END_REF]Theorem 12.1] and [START_REF] Huxley | The difference between consecutive primes[END_REF])

N (σ, T, q) := χ (mod q) N (σ, T, χ) (qT )

Thus

(2.29)

χ =χ 0 β> 1 2 +ε, |γ|<T |I ρ | log 10 (qT ) max T 1 T 1-σ(T 1 ;q) 1/2+ε q 1-σ 2 T 1/12+ε 1 x σ T 1 (qT 1 ) 12 5 (1-σ) dσ
x log 10 (qT ) max

T 1 T 1-σ(T 1 ;q) 1/2+ε
q 87/30 T 17/30 1

x 1-σ dσ x log 10 (qT ) max

T 1 T
q 87/30 T 17/30 1

x σ(T 1 ;q)

x log 10 (qT ) q 87/30 T 17/30 x σ(T ;q)

. provided q 87/30 T 17/30 x. Inserting (2.26), (2.27) and (2.29) into (2.25), we find that

χ =χ 0 χ(a)χ(d) n x χ(n) τ (nd)
qx log x T + x 1/2 q 5/4 T 1/12+ε + x log 10 (qT ) q 87/30 T 17/30 x σ(T ;q) (x -13 q 104 ) 1/17+ε + (x 33 q 42 ) 1/51+ε + x(log x) 10 x -εσ(T ;q)/195 thanks to the choice of T = (x 30(1-ε) q -87 ) 1/17 . This implies the required result.

Proof of Theorem 1

Firstly we write (3.1) S(x, t; q, a) := 1 (x/q) n x n≡a(mod q)

F n (t).

In view of the symmetry of the divisors of n about √ n, it follows that

F n (t) = Prob(D n 1 -t) = 1 -Prob(D n < 1 -t) = 1 -F n (1 -t) + O(τ (n) -1 ).
Summing over n x with n ≡ a (mod q), we have S(x, t; q, a) + S(x, 1 -t; q, a) = 1 (x/q) n x n≡a(mod q)

{1 + O(τ (n) -1 )} = 1 + O 1 √ log x
uniformly for x 3, q x 15/52-ε and a ∈ Z * such that (a, q) = 1, where we have used the orthogonality and Lemmas 2.4 and 2.6 with d = 1 to deduce that S(x, t; q, a) = q xϕ(q) n x χ(mod q) χ(a)χ(n) τ (n)

1 (x/q) n x n≡a(mod q) 1 τ (n) = q xϕ(q) χ(mod q) χ(a) n x χ(n) τ (n) (q/ϕ(q)) e c 1 (ε)(log x) 1/3 (log 2 x) -1/3
d|n, d n t 1 (n = dm) = q xϕ(q) d x t χ(mod q) χ(a)χ(d) uniformly for x 3, q x 15/52-ε and a ∈ Z * such that (a, q) = 1. Inserting (3.3)-(3.6) into (3.2), we find that S(x, t; q, a) = 2 π arcsin √ t + O p,ε 1 √ log x uniformly for 0 t 1 2 , x 3, q x 15/52-ε and a ∈ Z * such that (a, q) = 1.

  by N (α, T ) the number of zeros of ζ(s) in the region e s α and | m s| T and define σ(τ ) := c log -2/3 (|τ | + 3) log -1/3 2

1 - 3 d x t χ =χ 0 χ

 130 S 2 + S 3 -S 4 ,whereS 1 := d x t χ 0 (a)χ 0 (d) m x/d χ 0 (m) τ (md) , S 2 := d x t χ 0 (a)χ 0 (d)

  and |τ | 3. From (2.6), (2.7) and (2.14), we derive that

	(2.15)

  Estimations of I 3 and I 4 .

	By (2.24) and Lemma 2.2, we have
	(2.27)	|I 3 | + |I 4 |	1	T	q 1/4 (|τ | + 1) 1/12	x 1/2+ε 2 + ε) + iτ )| |( 1	dτ
			x 1/2+ε q 1/4 T 1/12 .
	C. Estimation of I ρ .
	With the help of (2.24) and Lemma 2.5, we have
	(2.28)	I ρ	β 1/2+ε	q	1-σ 2 |γ| 1/12+ε	x σ |σ + iγ|	dσ.
	Denote by N (σ, T, χ) the number of zeros of L(s, χ) in the region e s σ and | m s| T .
	Summing (2.28) over |γ| < T and interchanging the summations, we have
	β> 1 2 +ε, |γ|<T

(1-σ) log 9 (qT ).

Acknowledgements. This paper was written when the first author visited LAMA 8050 de l'Université Paris-Est Créteil during the academic year 2019-2020. He would like to thank the institute for the pleasant working conditions. This work is partially supported by NSF of Chongqing (Nos. cstc2018jcyjAX0540 and cstc2019jcyj-msxm1651).