Sana Ben Moussa

Afef Mehri

Michel Gruselle

Patricia Beaunier

Sana Ben Moussa

Guylène Costentin

Béchir Badraoui

Combined effect of magnesium and amino glutamic acid on the structure of hydroxyapatite prepared by hydrothermal method

Keywords: hydrothermal method, apatite surface, glutamic acid, hybrid compound

come

Combined effect of magnesium and amino glutamic acid

on the structure of hydroxyapatite prepared by hydrothermal method

Introduction

Among the minerals having an interest from an economic point of view, apatites, mostly hydroxy-(CaHAp) and fluoro-apatites (FAp), are of considerable interest in numerous research areas [START_REF] Elliot | Structure and Chemistry of the Apatites and Other Calcium Orthophosphates[END_REF][START_REF] Govindaraj | Carbon nanotubes/pectin/minerals substituted apatite nanocomposite depositions on anodized titanium for hard tissue implant: In vivo biological performance[END_REF][START_REF] Gomez-Morales | Progress on the preparation of nanocrystalline apatites and surface characterization : overview of fundamental and applied aspects[END_REF]. Apatites are used in several applications such as sorbents, catalysts and biomaterials [START_REF] Ben Moussa | Calcium, Barium and Strontium Apatites: A New Generation of Catalysts in the Biginelli Reaction[END_REF][START_REF] Gruselle | Apatites: a new family of catalysts in organic synthesis[END_REF].

Apatites are component of bones and teeth [START_REF] Wang | FT-IR and XRD study of bovine bone mineral and carbonated apatites with different carbonate levels[END_REF][START_REF] Kanwal | In-vitro apatite formation capacity of a bioactive glass -containing toothpaste[END_REF]. Apatites belong to the phosphate family of compounds, and are of general formula M 10 (PO 4 ) 6 Y 2 where M is a divalent cation: Ca, Sr, Ba, Pb…, and Y a hydroxyl (OH) or halide (F, Cl) [START_REF] Bulina | Fast synthesis of Lasubstituted apatite by the dry mechanochemical method and analysis of its structure[END_REF][START_REF] Lim | Enhancing osteoconductivity and biocompatibility of silver-substituted apatite in vivo through silicon cosubstitution[END_REF]. Apatites may contain several different substituents in their structure. This ability to trap substituents in their structures leads to the formation of total or *Manuscript Click here to view linked References partial solid solutions. The substitution processes are controlled by the crystallographic rules related mainly to: ionic radius, charge, electronegativity and polarisability [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF][START_REF] Ben Moussa | Synthesis, Rietveld refinements and electrical conductivity of news fluorobritholite based on lead Ca 7x Pb x La 3 (PO 4 ) 3 (SiO 4 ) 3 F 2 (0  x  2)[END_REF]. Previous works [START_REF] Kaur | Lanthanide (= Ce, Pr, Nd and Tb) Ions Substitution at Calcium Sites of Hydroxyl Apatite Nanoparticles as Fluorescent Bio Probes: Experimental and Density Functional Theory Study[END_REF][START_REF] Badraoui | Synthesis and characterization of Sr (10-x) Cd x (PO 4 ) 6 Y 2 (Y = OH and F): A comparison of apatites containing two divalent cations[END_REF][START_REF] Aissa | Synthesis, X-ray Structural Analysis and Spectroscopic Investigations (IR and 31 P MAS NMR) of Mixed Barium/Strontium Fluoroapatites[END_REF] have shown that steric hindrance related to cations bigger in size than calcium ions, plays an important role in the limitation of the cationic substitution processes. Magnesium is undoubtedly one of the most important bivalent ions associated to biological apatites [START_REF] Geng | Synthesis, characterization and the formation mechanism of magnesium-and strontium-substituted hydroxyapatite[END_REF][START_REF] Geng | Synthesis, characterization and biological evaluation of strontium/magnesium-co-substituted hydroxyapatite[END_REF]. It has been verified that in calcified tissues, the amount of magnesium associated to the apatitic phase is higher at the beginning of the calcification process and decreases on increasing calcification [START_REF] Gu | Influences of doping mesoporous magnesium silicate on water absorption, drug release, degradability, apatitemineralization and primary cells responses to calcium sulfate based bone cements[END_REF][START_REF] Bigi | Structural and chemical characterization of inorganic deposits in calcified human mitral valve[END_REF][START_REF] Burnell | Normal maturational changes in bone matrix, mineral, and crystal size in the rat[END_REF]. This interest is increasing taking into consideration the ability of such mixed apatites to lead to hybrid organic-inorganic materials by reaction with amino-acids [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF][START_REF] Yu | Synergistic antibacterial activity of multi components in lysozyme/chitosan/silver/hydroxyapatite hybrid coating[END_REF][START_REF] Hong | Fabrication, biological effects, and medical applications of calcium phosphate nanoceramics[END_REF][START_REF] Padilla | High Specific Surface Area in Nanometric Carbonated Hydroxyapatite[END_REF][START_REF] Sanchez-Salcedo | In vitro structural changes in porous HA/β-TCP scaffolds in simulated body fluid[END_REF][START_REF] Zhou | Nanoscale hydroxyapatite particles for bone tissue engineering[END_REF][START_REF] Carrodeguas | α-Tricalcium phosphate: synthesis, properties and biomedical applications[END_REF]. The expected benefit from the introduction of an amino-acid such as a glutamic acid in magnesium modified apatites is correlated with the ability of magnesium to coordinate amino-acids more strongly that calcium ions do and consequently to bind a greater proportion and more firmly proteins on their surfaces.

In this aim, we have carried out a structural, morphology and chemical investigation of the combined effect of magnesium and amino glutamic acid on hydroxyapatite structure. We have also studied the interaction between glutamic acid and the apatite surface. The results show that the acid forms a complex on the apatite surface. As glutamic acid functionalization should potentially modify the electrostatic interaction, the corresponding change in the surface charge of the powder was monitored by zeta-potential measurements and [H + ] consumed as a function of the pH value.

Experimental and methods

Synthesis

The mixed Mg/CaHAp of general formula: Ca (10-x) Mg x (PO 4 ) 6 (OH) 2 (x = 0, 0.5 and 1.0), named Ca (10-x) Mg x HAp, have been synthesized using the hydrothermal method [START_REF] Bigi | Magnesium influence on hydroxyapatite crystallization[END_REF]. A demineralised water solution (14 mL, 0.75 M) of a mixture of the two nitrates Ca(NO 3 ) 2 .4H 2 O and Mg(NO 3 ) 2 .6H 2 O in the desired proportions is added to a (NH 4 ) 2 HPO 4 water solution (25 mL, 0.25 M). The pH of the final solution is adjusted to 10 by adding a NH 4 OH solution (d = 0.89, Purity = 28%). The final solution is transferred to an autoclave. The mixture is maintained at 120°C for 12 hours. After filtration and washing using hot demineralised water, the mineral is dried at 120°C overnight. The hybrid materials were prepared according to the same experimental protocol, with addition of a quantity of organic reagent glutamic acid (GA) to the phosphate solution before pH adjustment [START_REF] Bigi | Microstructural investigation of hydroxyapatite-polyelectrolyte composites[END_REF]. The samples will be named as Ca (10-x) Mg x HAp-GA(n), where n is the value of the glutamic acid/(CaHAp) molar ratio (n= 10 and 20).

Powder characterization

N 2 adsorption-desorption isotherms were performed at 77 K using a Micromeritics ASAP 2000 instrument. The Brunauer-Emmett-Teller equation was used to calculate the specific surface area (S BET ). X-ray diffraction (XRD) analysis were carried out by means of a X'Pert Pro Panalytical X-pert diffractometer using Cu-Kα radiation (λ=1.5418 Å, with θ-θ geometry, equipped with an X'Celerator solid detector and a Ni filter). The 2θ range was from 20 to 70° with a step size Δ2θ=0.0167°. The experimental patterns were compared to standards compiled by the Joint Committee on Powder Diffraction and Standards (JCPDS cards) using the X'Pert High-Score Plus software [29]. The infrared (IR) adsorption analysis of the samples were obtained using a Spectrum Two 104462 IR spectrophotometer equipped with a diamond ATR setup in the range 4000-400 cm -1 . Nitrogen sorption isotherms for dried powders were recorded at 77K using a sorptometer EMS-53 and KELVIN 1040/1042 (Costech International). Points of Zero Charge (PZC) and Iso-Electric Point (IPE) of the samples are determined by zeta potential measurements, using a Malvern Nano ZS. Suspensions were prepared using NaCl (0.1M) as a background electrolyte with each powder using aqueous solutions, starting in an alkaline medium and stopping at pH = 4 under N 2 at 25°C [START_REF] Wu | Surface complexation of calcium minerals in aqueous solution[END_REF]. The titrations were carried out on suspensions of different samples of apatite obtained by adding 0.15 g of apatite to 30 mL of electrolyte (NaCl) and then 1.5 mL of 0.1M NaOH. The titrant used is 0.1 M hydrochloric acid prepared from 1M HCl, at the same ionic strength as the electrolyte by the addition of NaCl. The phosphorus and calcium contents were obtained by ICP-OES on a Horiba Jobin Yvon modele activa.

The thermal analysis of the carbon was carried out using a SETARAM SETSYS 1750. Heating was performed in a platinum crucible in air flow at a rate of 10°C /min up to 800°C. For transmission electron microscopy (TEM) investigations, samples were prepared by dispersing the powders in a slurry of dry ethanol, deposited on a copper grid covered with a carbon thin film. High-resolution transmission electron microscopy (HRTEM) observations were performed on a JEOL JEM 2010 transmission electron microscope equipped with a LaB 6 filament and operating at 200 kV. The images were collected with a 4008 X 2672 pixels CCD camera (Gatan Orius SC1000). Circular dichroism (CD) experiments were performed at solid state using 5 mg of powder dispersed in nujol between NaCl pellets [START_REF] Castiglioni | Experimental Aspects of Solid State Circular Dichroism[END_REF]. The measurements were performed by a TASCO J-815 spectropolarimeter. The scans were recorded from 190 to 300 nm wavelength with the following parameters: 0.5 data pitch, 2 nm bandwidth, 100 nm/min scanning speed, and are the result of 3 accumulations.

Results and discussion

Elemental analysis

The results of chemical analysis for mixed CaMgHAp with glutamic acid are reported in Table 1. The CaHAp sample shows a (Ca/P) molar ratio very close to the targeted stoichiometric value of 1.67. For the apatite series, the (Ca+Mg/P) ratio decreased from the starting CaHAp (1.68) to Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF] (1.55). The presence of the organic anion in the precipitated material is attested and quantified by the total carbon analysis. We can note the increase in carbon amount with the concentration of Mg, this great affinity of glutamic acid to mixed CaMgHAp could be explained by the high electronegativity of magnesium (χ Ca = 1, χ Mg = 1,31) [START_REF] Pauling | The Nature of the Chemical Bond[END_REF], in agreement with the results previously reported for CaCuHAp modified by polyaspartic acid [START_REF] Othmani | Surface modification of calcium-copper hydroxyapatites using polyaspartic acid[END_REF] and CaZnHAp modified by tartric acid [START_REF] Turki | Surface modification of zinccontaining hydroxyapatite by tartaric acid[END_REF]. The larger absorption of carbon in the CaMgHAp-GA would explain their loss of stoichiometry. This indicates that our samples are indeed hydroxyapatite-glutamic acid composites.

For the samples CaHAp, Ca 9.5 Mg 0.5 HAp and Ca 9 Mg 1 HAp the increase in carbon amount with the concentration of Mg is explained by the disorder induced by the magnesium substitution promoting the incorporation of carbonates.

Thermal analysis

The (TG) curves of Ca 9 Mg 1 HAp, Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF] samples are reported in figure 1. The thermal decomposition shows a first weight loss between 50°C and 200°C, assigned to the removal of physisorbed water. The second one, between 200°C and 500°C, corresponds to the elimination of the organic matter of glutamic acid. The weight loss associated with this second process allows evaluation of the relative amount of glutamic acid in the composite hybrids. The values obtained for carbon expressed as wt% of the solid product are reported in Table 2. The relative amount of glutamic acid increases with its increasing concentration in the reaction with CaHAp. This result is in agreement with the increase of the percentage of carbon determined by chemical analysis. Figure 2 reports the differential thermal analysis curves (DTA) of the samples. These curves display an unexpected endothermic effect associated with water desorption.

This effect was already reported for Mg modified hydroxyapatites [START_REF] Yasukawa | Preparation and characterization of magnesiumcalcium hydroxyapatites[END_REF][START_REF] Diallo-Garcia | Influence of Magnesium Substitution on the Basic Properties of Hydroxyapatites[END_REF]. It was assigned to the fact that heating in the presence of physisorbed water first induce a structuring effect. The latter may be related both to the surface relaxation at solid-water interface upon water release [START_REF] Ben Osman | Control of calcium accessibility over hydroxyapatite by post-precipitation steps: influence on the catalytic reactivity toward alcohols[END_REF] and to 6 the polarization process of OH groups from the columns known to occur at 200°C that initiates the proton mobility inside the columns [START_REF] Nakamura | Proton Transport Polarization and Depolarization of Hydroxyapatite Ceramics[END_REF]. An exothermal effect is observed in the temperature range 200-500°C with a peak top at 300°C for Ca9Mg1HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca9Mg1HAp-GA(20) samples.

This peak, which is absent in the DTA plot of non-modified Ca9Mg1HAp, corresponds to the combustion of the organic material. Furthermore, the intensity of these peaks increases with increasing grafted amount content. In fact, their presence confirms that the prepared samples correspond to hydroxyapatite-glutamic acid composites, similar to that previously found for hydroxyapatite modified with glycine and sarcosine acids [START_REF] Ben Moussa | Hybrid organic-inorganic materials based on hydroxyapatite structure[END_REF].

Table 2

Results of TG analysis of ungrafted and grafted apatites.

Samples

Infrared investigation

The IR spectra recorded with or without glutamic acid onto the apatite surface are illustrated in figure 4. The vibrations in the range 1800-1200 cm -1 are summarized in table 3. In particular, the bands observed at 1598, 1445 and 1261cm -1 (Fig. 4b,4c), which are not present in the spectrum of Ca 9 Mg 1 HAp (Fig. 4a) can be attributed to sorbed glutamic acid [START_REF] Navarrete | Vibrational study of aspartic acid and glutamic acid dipeptides[END_REF]. The vibration located at 1635 cm - 1 is attributed to adsorbed water [START_REF] Garcia-Ramos | The adsorption of acidic amino acids and homopolypeptides on hydroxyapatite[END_REF]. The higher water content in the presence of glutamic acid is consistent with the decrease in the ratio (Ca + Mg) / P. In order, to eliminate the hypothesis of a simple mechanical mixture between glutamic acid and apatite, further data have been recorded from such 20) sample. The FT-IR spectrum of this mixture, which is reported in Fig. 4d, displays a number of bands due to the two discrete components of the mixture, and indicates the absence of specific interaction between the carboxylic groups of the glutamic acid and the calcium ions of hydroxyapatite. The comparison with the spectra reported in Fig. 4b and4c, which show also a small shift of the carboxylic stretching band to lower wave numbers, in agreement with an increase of the C-O bond length can be attributed to those of the organic moieties grafted on calcium or magnesium atoms on the surface of the hydroxyapatite, in agreement with the results previously reported for CaHAp modified by amino-acids [START_REF] Ben Moussa | Hybrid organic-inorganic materials based on hydroxyapatite structure[END_REF][START_REF] Bachoua | Preparation and characterization of functionalized hybrid hydroxyapatite from phosphorite and its potential application to Pb 2+ remediation[END_REF].

Table 3

FTIR Spectral data (±5) of Ca 9 Mg 1 HAp-GA(10), Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], and pure glutamic acid.

Tentative assignments

Wave number (cm -1 )

Glutamic acid [START_REF] Navarrete | Vibrational study of aspartic acid and glutamic acid dipeptides[END_REF] Asym.: Asymmetrical, str.: stretching

X-ray analysis

The X-ray powder diffractograms for CaHAp and Ca 9 Mg 1 HAp synthesized with or without the presence of glutamic acid are shown in figure 5. In table 4, we reported the size of the apatite crystallites induced by magnesium and glutamic acid for the reflections (002) and (310). For all samples, we observe a unique apatitic phase belonging to the P6 3 /m space group (n° 9-432-ICDD-PDF). We could not prepare the grafted apatites Ca 8.5 Mg 1.5 HAp-GA(n), since all our preparations failed. Several authors have shown that the particular behavior of the Ca 8.5 Mg 1.5 HAp compound can therefore be explained by the fact that sample Ca 8.5 Mg 1.5 HAp is a pure non stoichiometric CaMgHAp which would turn into whitlockite under the effect of temperature and already made up of a mixture of phases CaMgHAp crystalline and amorphous whitlockite [Ca 3y Mg y (HPO 4 ) z (PO 4 ) 2-2z/3 ], which is crystallized under the effect of heat treatment at 900 °C [START_REF] Chaudhry | Synthesis and characterization of magnesium substituted calcium phosphate bioceramic nanoparticles made via continuous hydrothermal flow synthesis[END_REF][START_REF] Correia | Wet synthesis and characterization of modified hydroxyapatite powders[END_REF].

Broadening of the diffraction lines increases with the concentration of magnesium and glutamic acid. The crystallite sizes were calculated from the broadening of the (0 0 2) and (3 1 0) using the Scherrer equation [START_REF] Bigi | Strontium-substituted hydroxyapatite nanocrystals[END_REF]:

β 1/2 cosθ Kλ D hkl 
Where  is the diffraction angle,  the wavelength and K a constant depending on the crystal (chosen as 0.9 for apatite crystallites) and β ½ is the line width at full width at half maximum (FWHM), of a given reflection. The line broadening of the (0 0 2) and (3 1 0) reflections was used to evaluate the crystallite size along the c-axis and along a direction perpendicular to it. The crystallinity (Xc) is defined as the fraction of the crystalline apatite phase in the investigated volume of powdered sample.

An empirical relation between X c and β ½ was deduced, according to the following equation [START_REF] Ren | Characterization and structural analysis of zinc-substituted hydroxyapatites[END_REF]: X c = [K A / 1/2 ] 3 . Where K A is a constant set at 0.24 and β ½ is the FWHM of the (0 0 2) reflection, the 4. The crystallite size and the crystallinity decrease with increasing magnesium and amino acid concentration. It can also be deduced that the crystallites are of nanometric sizes and the decrease is more important in the (3 1 0) than in the (0 0 2) direction. Such observation was earlier reported for other organic moieties grafted onto apatite surfaces and can be explained by a better interaction of the glutamic acid with faces parallel to the c axis [START_REF] Boanini | Nanocomposites of hydroxyapatite with aspartic acid and glutamic acid and their interaction with osteoblast-like cells[END_REF][START_REF] Othmani | Surface modification of calcium-copper hydroxyapatites using polyaspartic acid[END_REF]. The individual effect of magnesium and glutamic acid on the crystallite size can be observed in figure 6. The crystallite sizes D (002) and D (310) decrease slowly with the concentration of magnesium whereas the addition of glutamic acid induces a bigger change in the crystallinity. It would appear that the glutamic acid is the component most responsible for the loss of crystallinity which could be explained by the presence of the groups COO -on the surface of materials. We do not observe a change in position of the peaks between CaHAp and Ca 9 Mg 1 HAp diffractograms, here the main phenomenon is the broadning of the peaks, which has already been observed during the substitution of calcium by another divalent ion. When glutamic acid is present, we observe no difference between Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA(20) diffractograms. In contrast, peak shifts are observed compared to the CaHAp and Ca 9 Mg 1 HAp one, in particular for the (211) and ( 222) peaks. This phenomenon that we have already described was attributed to a better interaction of the glutamic acid with the faces parallel to the axis c during the process of crystalline growth [START_REF] Ben Moussa | Hybrid organic-inorganic materials based on hydroxyapatite structure[END_REF].

TEM observations

Transmission electron microscopy (TEM) analysis micrographs of the samples are illustrated in figure 7A. From the photomicrographs, it can be seen that the size of precipitated apatite particles, prepared with and without glutamic acid, is on the nanometer scale. CaHAp is constituted of well dispersed plate-shaped crystals with an average size of about 40-150 nm long and about 30 nm wide.

A small addition of magnesium induces a decrease of the size: 30-60 nm long and about 15-20 nm wide for the Ca 9.5 Mg 0.5 HAp. The presence of magnesium and glutamic acid in the start solution completely modifies the aspect. We obtain large bundles of CaMgHAp-GA fibers (300 nm length /80 nm width). The HRTEM images (Fig. 7B) reveal that theses fibers are thin (15 nm wide) and stacked together to each other. The analysis of the FFT patterns show that the growth of the particles always occurs in the (002) direction, indicating that they are growing along the c axis direction such as the

[[D (310) ] Mgx, GA(n) / [D (310) ] CaHAp ]% 0 0,2 0,4 0,6 0,8 1

Functionalization of CaHAp with glutamic acid

Speciation of the interface apatite-solution

The influence of the glutamic acid amount on the textural properties of CaHAp was examined (Table 5). We obtained 43 m 2 /g for Ca 9 Mg 1 HAp and after functionalization 30 m 2 /g and 24 m 2 /g for Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], respectively. The decrease in the specific surface area of the apatite samples is related to the structural arrangement of the 2-aminopentanedioic acid on the surface of the solid. These results are in good agreement with those obtained by Oberto Da Silva et al. [START_REF] Silva | Hydroxyapatite organofunctionalized with salivating agents to heavy cation removal[END_REF]. The initial pH i value measured in aqueous solutions increase with the glutamic acid ratio. We recorded 8.9 and 9.3 for Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA(20) respectively against only 8.4 for Ca 9 Mg 1 HAp. The Zeta potential of different apatite samples is presented in figure 8. The values of Point of Zero Charge (PZC) and iso-electric point (IEP) were determined (Table 5). In a basic medium, the Ca 9 Mg 1 HAp sample is characterized by a highly negative zeta potential indicating a deficit of positive surface charge, a value lower than that of the apatites Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF].

This result indicates an increase in positive surface charge sites, on amino-acid grafted apatites [START_REF] Kollath | A Modular Approach To Study Protein Adsorption on Surface Modified Hydroxyapatite[END_REF].

Whereas the value of pH IEP is 7. The total number of protons consumed during titrations of different samples is determined using this equation:

                                   OH ) pH ) Ke ( Log ( 10 H pH 10 V blanc ) b V b C a V a C ( susp ) b V b C a V a C exp S

8(B)

). This quantity of protons (H + ) is deduced from the difference between the amount of (H + ) added to the suspension and the amount of free (H + ) in solution. This is calculated directly from the measured pH. The amount of the added protons is corrected by the amount of hydroxyl initially added to the suspension.

The proton quantity consumed by apatite samples [H + ] shows the same evolution: in an acid medium, [H + ] is between 9 and 12 μmol/m² (pH = 5) for Ca 9 Mg 1 HAp, Ca 9 Mg 1 HAp-GA [START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA [START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], respectively. Then, it decreases regularly and becomes negative in a basic medium with the protons consumed in the order of -8 and -10 μmol/m² at pH = 11.

Proposed glutamic acid sorption mechanism

The IR spectra provide data about the ionization state of the carboxylate groups grafted onto the apatite surface. Indeed, the presence of bands characteristic of -COO -groups and the absence of bands attributed to COOH groups (1700 cm -1 ) indicate the carboxylate form. This shows that the interaction is mainly due to the electrostatic interaction between -COO -groups of the glutamic acid and the calcium Ca 2+ /Mg 2+ ions of the hydroxyapatite. We cannot exclude also that interactions between COO -and surface POH groups are possible. The fixation is due to the simultaneous presence of -COO-/Ca 2+ / Mg 2+ electrostatic interactions and H-bonds between NH 3 + protons and surface oxygen atoms of the PO 4 group [START_REF] Almora-Barrios | Density functional theory study of the binding of glycine, proline, and hydroxyproline to the Hydroxyapatite (001) and (010) surfaces[END_REF][START_REF] Rimola | Ab initiomodelling of protein/biomaterial interactions: glycine adsorption at hydroxyapatite surfaces[END_REF].

The nature of interactions between the apatite surface and a glutamic acid depends on the pH value of the medium. Taking into consideration that the reaction is carried out at a pH above 9.5, we can consider that in aqueous solution glutamic acid exists as a carboxylate ion, the amino group being neutral [57]. Under the same conditions, the apatite surface is considered as negatively charged, therefore some authors consider that the electrostatic interactions between the surface and the aminoacid are very weak [START_REF] Palazzo | Amino acid synergetic effect on structure, morphology and surface properties of biomimetic apatite nanocrystals[END_REF][START_REF] Brown | An analysis of hydroxyapatite surface layer formation[END_REF]. In the present study, glutamic molecules are present in the form of carboxylate ions, which can lead to calcium complexes that participate in the formation of the crystalline edifice. The observation of the IR spectra, showing characteristic vibrations of carboxylate salts, lead us to conclude that the amino-acid in this carboxylate ionic form exchanges an hydroxyl ion.

Different model grafting mechanisms proposed based on the results obtained are shown in scheme 1.

Scheme 1. Formation of Ca carboxylate salt leading to the grafting of glutamic on the CaHAp surface.

Conclusion

In conclusion, we have successfully synthesized hydroxyapatite-glutamic composites of different glutamic acid content using the hydrothermal method. The presence of the glutamic acid and/or magnesium in the reaction solution does not change the apatite structure, but reduces the crystallinity and the crystallite sizes. According to IR spectroscopy, the new vibrations after adsorption can be attributed to those of the organic moieties grafted on calcium or magnesium atoms onto the surface of

FirstFig. 2 .

 2 Fig. 2. DTA plots; (a) Ca 9 Mg 1 HAp, (b) Ca 9 Mg 1 HAp-GA(10) and (c) Ca 9 Mg 1 HAp-GA(20).

Fig. 1 .

 1 Fig. 1. TG plots; (a) Ca 9 Mg 1 HAp, (b) Ca 9 Mg 1 HAp -GA(10) and (c) Ca 9 Mg 1 HAp-GA(20).

Fig. 3 .

 3 Fig. 3. (Left) Circular dichroism spectrum and (right) UV absorption curves of (a) glutamic acid and (b) Ca 9 Mg 1 HAp-GA(20).

9

 9 Mg 1 HAp and glutamic acid in the same relative amounts as those contained in the Ca 9 Mg 1 HAp-GA(

Fig. 4 .

 4 Fig. 4. FT-IR spectra of: (a) Ca 9 Mg 1 HAp, (b) Ca 9 Mg 1 HAp-GA(10), (c) Ca 9 Mg 1 HAp-GA(20), (d) Mixed Ca 9 Mg 1 HAp-glutamic acid and (e) glutamic acid.

  Ca 9 Mg 1 HAp-GA(10) Ca 9 Mg 1 HAp-GA(20) COO-asym.str.

CaHAp Ca 9

 9 Mg 1 HAp Ca 9 Mg 1 HAp-GA[START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] Ca 9 Mg 1 HAp-GA[START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF]

Fig. 5 .

 5 Fig. 5. X-ray diffractograms for CaHAp and Ca 9 Mg 1 HAp ungrafted and grafted.

Fig. 6 .

 6 Fig. 6. Effect of the Mg and GA concentration on the crystallite size D (002) (A) and D (310) (B) relative to the CaHAp values taken as references.

Fig. 7B .

 7B Fig. 7A. TEM images (Scale bars=50 nm) of CaHAp, Ca 9.5 Mg 0.5 HAp, Ca 9 Mg 1 HAp-GA(10) and Ca 9 Mg 1 HAp-GA(20).

  specific surface area (m 2 /g) exposed in the suspension, V is total volume of solution; C a and C b are the HCl and NaOH concentration used for titration. K e is the dissociation constant of water. ( H+ ) and ( OH-) are the coefficient of the dissociation activity of H + and OH -calculated by Debye-strength of solution (mol/L), A=0.507, B=0.328.10 -8 and  i 0 is the effective diameter (Fig.

Fig. 8 .

 8 Fig. 8. (A) Zeta potential IEP (in mV) as a function of pH spectra of the as-prepared Ca 9 Mg 1 HAp-GA(n) (n=10 and 20) and sample Ca 9 Mg 1 HAp. (B) Number of protons consumed by the surface for different samples reported in µmol/m 2 during their 2 hours immersion in a solution containing 0.1 M NaCl + HCl or NaOH as a function of pH of the as-prepared Ca 9 Mg 1 HAp-GA(n) (n=10 and 20) and sample Ca 9 Mg 1 HAp.

  the apatite. TEM images confirm the reduction of crystallite sizes and indicate the change in its morphology.

Table 1

 1 Chemical composition (% weight ±0.02) of grafted mixed CaMgHAp.

	Samples	%Ca	%Mg	%P	%C	(Ca + Mg)/P
	CaHAp	39.31	-	18.09	0.12	1.68
	Ca 9.5 Mg 0.5 HA	38.14	0.98	18.66	0.15	1.65
	p Ca 9 Mg 1 HAp	35.28	1.82	18.01	0.18	1.64
	CaHAp-GA(10)	39.05	-	18.15	0.72	1.66
	Ca 9.5 Mg 0.5 HAp-	37.94	1.08	18.63	0.87	1.64
	GA(10) Ca 9 Mg 1 HAp-	34.05	1.89	17.98	1.12	1.59
	GA(10) CaHAp-GA(20)	38.97	-	18.23	1.25	1.65
	Ca 9.5 Mg 0.5 HAp-	37.23	1.13	18.58	1.49	1.63
	GA(20) Ca 9 Mg 1 HAp-	32.51	2.03	17.88	2.58	1.55
	GA(20)

Table 4 :

 4 Evolution size of the apatite crystallites with magnesium and glutamic acid for the reflexion (002) and (310).

	Samples	β 1/2 (002)	D (002) (Å)	β 1/2 (310)	D (310) (Å)	Crystallinity (X C )
	CaHAp	0.191(1)	427	0.477(9)	177	1.981
	Ca 9.5 Mg 0.5 HAp	0.207(2)	394	0.640(1)	132	1.554
	Ca 9 Mg 1 HAp	0.264(2)	309	0.712(1)	119	0.749
	CaHAp-GA(10)	0.230(4)	354	0.710(4)	119	1.042
	Ca 9.5 Mg 0.5 HAp-GA(10) 0.283(4)	288	0.931(3)	91	0.607
	Ca 9 Mg 1 HAp-GA(10)	0.364(5)	224	1.286(3)	66	0.285
	CaHAp-GA(20)	0.233(1)	350	0.761(2)	111	1.029
	Ca 9.5 Mg 0.5 HAp-GA(20) 0.322(3)	253	1.084(1)	75	0.413
	Ca 9 Mg 1 HAp-GA(20)	0.389(2)	210	1.385(1)	61	0.234

Table 5

 5 Surface area and pH value of different apatite samples

	(c)			(d)
		002
		100
	Samples	S BET (m 2 /g) IEP	PZC	pH i in aqueous solution
	Ca 9 Mg 1 HAp	43	7.1	8.2	8.4
	Ca 9 Mg 1 Hap-GA(10)	30	8.2	8.6	8.9
	Ca 9 Mg 1 Hap-GA(20)	24	8.6	8.8	9.3
	PZC: point zero charge is determined by an acid-basic titration equilibrated at 16h
	IEP: Iso-electric point (mV) is determined by zeta-metry

  1, the PZC for the Ca 9 Mg 1 HAp sample is around 8.2. The same value is found in the literature[START_REF] Bell | The point of zero charge of hydroxyapatite and fluorapatite in aqueous solutions[END_REF][START_REF] Attia | The equilibrium composition of hydroxyapatite and fluoroapatitewater interfaces[END_REF][START_REF] Saleeb | Surface properties of alkaline earth apatites[END_REF][START_REF] Wu | Surface complexation of calcium minerals in aqueous solution[END_REF] (Figure8(B)). This result can be explained by the presence of initial charge and specific adsorption at the surface of apatite. After glutamic acid functionalization, the Zeta-potential curve is shifted towards alkaline pH values, with pH IEP of 8.2 and 8.6 for Ca 9 Mg 1 HAp-GA[START_REF] Hamad | Synthèse et étude physico-chimique de fluoroapatites mixtes à cations bivalents Pb-Cd, Pb-Sr et Sr-Cd[END_REF] and Ca 9 Mg 1 HAp-GA[START_REF] Li | Dopamine Modified Organic -Inorganic Hybrid Coating for Antimicrobial and Osteogenesis[END_REF], respectively. This result verified by measurement of PZC (from 8.6 to 8.8), confirms the functionalization of the surface of Ca 9 Mg 1 HAp by the glutamic acid.

Acknowledgments

This research was carried out with the financial support of the University of Monastir, (Tunisia); the University Pierre and Marie Curie and CNRS (France).


Benoit Voillot
email: voillot@lmt.ens-cachan.fr

Jean-Lou Lebrun

René Billardon

François Hild

J.-L Lebrun

F Hild Lmt

Validation of registration techniques applied to XRD signals for stress evaluations in titanium alloys

Keywords: DIC, in-situ test, integrated methods, stress analyses, XRD

To estimate stresses near specimen surfaces, X-ray diffraction (XRD) is applied to titanium alloys. Some of these alloys are difficult to study since they are composed of various phases of different proportions, shapes and scales. For millimetric probed volumes, such multi-phase microstructures induce shallow and noisy diffraction signals. Two peak registration techniques are introduced and validated thanks to tensile tests performed on two titanium alloy samples.

Introduction

High performance titanium alloys are used in aeronautical industries and especially large forgings (e.g., landing gears). These critical parts of an aircraft are submitted to various heat and mechanical treatments [START_REF] Hill | Engineering residual stress in aerospace forgings[END_REF]. One of the consequences are changes in microstructure, roughness and residual stresses [START_REF] Cox | The effect of finish milling on the surface integrity and surface microstructure in Ti-5Al-5[END_REF][START_REF] Aeby-Gautier | Isothermal α formation in β metastable titanium alloys[END_REF]. Each of these parameters has an impact on the global in-service mechanical behavior of the whole structure and in particular in fatigue [START_REF] Lütjering | Titanium[END_REF][START_REF] Guillemot | Prediction of the endurance limit taking account of the microgeometry after finishing milling[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF]. It is of high interest to be able to describe the effect of these treatments by several ways (e.g., microstructure studies, roughness measurements). Another way to quantify these treatments is via residual stress analyses. Various methods exist to estimate residual stresses. Most of them are destructive (e.g., incremental hole [START_REF]ASTM E[END_REF], contour methods [START_REF] Prime | Cross-sectional mapping of residual stresses by measuring the surface contour after a cut[END_REF]). To carry out analyses on in-service structures, nondestructive methods need to be performed.

One of the most popular nondestructive stress analysis techniques is X-ray diffraction (XRD) [START_REF] Reed Reed | The influence of surface residual stress on fatigue limit of titanium[END_REF][START_REF] Fitzpatrick | Determination of residual stresses by x-ray diffraction -issue 2[END_REF][START_REF] Freour | Determination of the macroscopic elastic constants of a phase embedded in a multiphase polycrystal -application to the beta-phase of Ti17 titanium based alloy[END_REF][START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF]. Various post-processing algorithms are used to evaluate stresses from XRD measurements [START_REF] Pfeiffer | Evaluation of thickness and residual stress of shallow surface regions from diffraction profiles[END_REF][START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. Among them the centered barycenter is common [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF]. It consists in determining positions of the diffraction peak by the sliding center of gravity of all the points in its neighborhood. This method is very powerful for well-defined peaks. However it is known that stress analyses are challenging for titanium alloys, especially multi-phase grades [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Suominen | Residual Stress Measurement of Ti-Metal Samples by Means of XRD with Ti and Cu Radiation[END_REF]. This difficulty comes from high levels of fluorescence and high noise to signal ratios. As a result, it is difficult to accurately locate diffraction peaks [START_REF] Withers | The precision of diffraction peak location[END_REF]. Other post-processing methods are currently used to register diffraction peaks. Peak to peak registrations are also utilized in some cases [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. A prerequisite is good similarity between peaks, which is not necessarily very suitable for two-phase materials or materials whose fluorescence varies with the angle of measurement. The latter phenomenon is observed in two-phase titanium alloys [START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Lütjering | Titanium[END_REF]. All these methods are implemented in most of commercial codes (e.g., Stressdiff [START_REF]Stressdiff[END_REF] or Leptos by Bruker [START_REF] By | [END_REF]). The last method used in existing commercial codes consists in modelling peaks with a mathematical function (e.g., Lorentz, Pearson VII, pseudo-Voigt or Gauss distributions [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]). This post-processing method (i.e., model with a known function) will be used to benchmark the integrated approach proposed herein.

Stress analyses are most of the time benchmarked with stress-free configurations (i.e., powder of the material [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Noyan | Residual Stresses: Measurements by Diffraction and Interpretation[END_REF]) or standard coupon [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF]. One of the issues with the former is finding a powder whose lattice structure is close to those of the material of interest. In some cases such as multi-phase materials, it is hard or impossible to get powder of the real material. An alternative route is to monitor a mechanical test with XRD means. Tools used for evaluating residual (i.e., ex-situ) stresses can also determine applied (i.e., in-situ) stresses [START_REF] Geandier | Elastic strain distribution in metallic filmpolymer substrate composites[END_REF] during mechanical tests.

Either the testing machine is put within the goniometer [START_REF] Geandier | Development of a synchrotron biaxial tensile device for in-situ characterization of thin films mechanical response[END_REF] or the goniometer is mounted inside the testing machine [START_REF] Rekik | Dispositif de mesure du comportement magnéto-mécanique d'un alliage de fer-silicium sous chargement mécanique multiaxial[END_REF]. This methodology will be used to validate the stress estimates reported herein. Further, it will also be combined with 2D digital image correlation [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF] to monitor surface displacements. Such combination has been used to validate XRD measurements and determine stress/strain curves in biaxial experiments on thin films [START_REF] Djaziri | Combined synchrotron x-ray and image-correlation analyses of biaxially deformed w/cu nanocomposite thin films on kapton[END_REF][START_REF] Djaziri | Investigation of the elastic-plastic transition of nanostructured thin film under controlled biaxial deformation[END_REF].

To address the issue of high noise to signal ratio associated with titanium alloys, it is proposed to require the peak shifts to be expressed in terms of the quantities of interest, namely, the sought elastic strains (or stresses). When a peak registration procedure is used it corresponds to an integrated approach as used in digital image correlation [START_REF] Hild | Digital image correlation: From measurement to identification of elastic propertiesa review[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF][START_REF] Réthoré | A fully integrated noise robust strategy for the identification of constitutive laws from digital images[END_REF], stereocorrelation [START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from digital images using {3D} kinematics[END_REF][START_REF] Beaubier | CAD-based calibration of a 3D-DIC system: Principle and application on test and industrial parts[END_REF][START_REF] Dufour | CAD-based displacement measurements. Principle and first validations[END_REF][START_REF] Dufour | Shape, Displacement and Mechanical Properties from Isogeometric Multiview Stereocorrelation[END_REF] and digital volume correlation [START_REF] Hild | Toward 4d mechanical correlation[END_REF]. The outline of the paper is as follows. First, the studied alloys and the experimental setup are presented. Then the two registration techniques used herein are introduced. Last, the results obtained on the two titanium alloys are analyzed and validated.

Studied alloys

Two two-phase titanium alloys are studied herein. First, Ti64 alloy is selected for benchmark purposes because it is well-known and its microstructure is favorable for XRD measurements [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Suominen | Residual Stress Measurement of Ti-Metal Samples by Means of XRD with Ti and Cu Radiation[END_REF]. Second, Ti5553 is a two-phase material used by Safran Landing Systems for landing gears [START_REF] Boyer | The use of beta titanium alloys in the aerospace industry[END_REF].

The composition of Ti64 is given in Table 1. It is composed of 6 wt% of Aluminum and 4 wt% of Vanadium. It is a quasi pure α-phase (i.e., 95 wt% of α-phase and only 5 wt% of β-phase). The α-phase is hexagonal close packed (HCP) and the β-phase is body centered cubic (BCC). Table 1: Chemical composition of Ti64

Alloying elements of Ti5553 are mostly composed of 5 wt% of Aluminum, 5 wt% of Vanadium, 5 wt% of Molybdenum, and 3 wt% of Chromium (Table 2). It is a quasi β-metastable material.

It contains 60 wt% of α-phase and 40 wt% of β-phase. Such metastable alloys are of interest for aeronautical applications thanks to their high specific strength. Only the α-phase will be considered in XRD measurements. The same lattice family (i.e., {213}-planes) will be studied.

The proportion of β-phase in Ti64 can be neglected. This is not the case for Ti5553 alloys for which the stress evaluation will be incomplete. Figure 1 shows micrographs of both alloys. For Ti64 the α-grains are observed as well as the former β-grains. For Ti5553 the matrix of the primary β-phase appears in gray and contains primary and secondary α-nodules (a few µm in diameter) and lamellae (at sub-µm scale) in black [START_REF]Standard Test Method for Determining Average Grain Size[END_REF]. The coexistence of these two phases is an issue for XRD analyses [START_REF] Lu | Handbook of measurement of residual stresses[END_REF][START_REF] Voillot | Evaluation of residual stresses due to mechanical treatment of Ti5553 alloy via XRD[END_REF]. Figure 3 shows that the size of β-grains is millimetric. This dimension is comparable to the extent of the volume probed by the X-ray beam (i.e., 1 mm in diameter for a depth of 5 µm, see Section 3). For one XRD analysis, only few grains of the β-phase are analyzed. It results that the β-phase lattices of the probed volume are not always in Bragg's conditions. Consequently, the β-phase is not adequate for stress analyses. This EBSD picture also reveals that there is no significant texture. The orientations of β-grains are not linked to one another. The α-phase made of nodules and lamellae do not show any preferential orientation with the β-grain in which they are nested [START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. This is of prime importance for XRD measurements since a random orientation of α-grains in the interaction volume ensures that Bragg's condition will be satisfied for the α-phase in any probed direction. A complete stress estimation is possible with a lab goniometer [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF] for this phase. 3 and4). Consequently, it is expected that the peak positions are more difficult to determine with Ti5553 than Ti64 alloys, which will presumably have an impact on stress evaluations [START_REF] Withers | The precision of diffraction peak location[END_REF][START_REF] Voillot | Evaluation des contraintes residuelles induites par traitements mecaniques dans un alliage de titane bi-phase par trois methodes de depouillement differentes[END_REF]. The aim of the following analyses is to assess the feasibility of XRD analyses under such challenging conditions. 3 Experimental setup

The experimental setup consists of a biaxial testing machine [START_REF] Bertin | Integrated digital image correlation applied to elasto-plastic identification in a biaxial experiment[END_REF] mounted in the X-ray goniometer (Figure 6). The advantage of such testing machine, which can also be used in an SEM chamber, is that two actuators are used to load the sample. In the chosen control mode, the center of the sample is motionless. It was checked (a posteriori) with DIC analyses that the maximum motion was less than 10 µm, which is negligible with respect to the probed surface (i.e., 1 mm 2 ). The testing machine is moved and relocated into the goniometer at each step of loading in order to acquire images for DIC purposes. The goniometer used in the present work follows the χ-method [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF], see Figure 7. The radius of the goniometer is 150 mm. It is equipped with a mobile head composed of the X-Ray source (X) and a single linear position sensor (LPS) that turns about a motionless coupon to be analyzed. The LPS sensor gives a 1D measurement of diffraction intensity. S1, S2, S3 are the basis axes corresponding to the analyzed sample S. Here as the coupon is motionless during all the measurements, the basis corresponds to the goniometer reference axes. 2θ is the diffraction angle between the incident and diffracted beams. For diffraction in titanium alloys, a copper source was selected. The use of a collimator gives a probed volume of the order of 1 mm 2 × 5 µm. A Nickel (K β ) filter is added in front of the LPS sensor to only analyze the K α rays of the copper source. The duration of X-Ray exposure (i.e., 3 min per diffractogram), intensity and voltage (i.e., 20 mA and 40 kV) are a compromise between duration of measurement and diffraction peak quality in the current conditions (Figure 5). To have access to various zones on the diffraction sphere, rotations are made possible in the Eulerian cradle of the goniometer as illustrated in Figure 8. Along the S3 axis (i.e., normal to the sample surface and the vertical axis of the goniometer) various angles φ are reached. They correspond to the main direction of measurement. The rotation along the S1 axis allows various angles χ or ψ to be probed. For one single stress analysis, thirteen diffractograms at various angular positions ψ ranging from -50 • and 50 • are selected [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. It is worth noting that the goniometer does not allow for oscillations during measurements. When the moving head is turning within the goniometer, the center of the surface area of the volume probed by the X-ray beam is moving by less than 0.1 mm. This observation ensures that the same volume of material is always impacted by X-rays at each angle of measurement.

To check the errors due to the use of the goniometer, an analysis on a stress-free powder made of pure titanium was carried out before and after each measurement campaign. All evaluated stress components varied about 0 MPa with a standard deviation of 10 MPa. Further, a height change of 50 µm of the coupon led to an error in stress of about 4 MPa. This stability of the beam interaction with the probed surface and the verification of stress-free evaluations with a Ti powder are indications of good working conditions for stress analyses.

In order to validate the stress analyses performed on titanium alloys, in particular the new integrated method, dog-bone samples (Figure 9) were machined. The coupon has been designed to ensure a controllable and homogenous stress and strain area in the central area of the coupon where X-ray analyses are performed. The aim is to carry out in-situ tensile tests. It is an alternative way to benchmark stress analyses via XRD. Usual validations are carried out with standard coupons [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF] or powders assumed to be stress-free. For pure titanium, powders exist and were proven to yield good results for the goniometer used herein [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. However, the lattice parameters of pure titanium are slightly different for Ti5553 or Ti64 alloys. The position and quality of peaks also vary (Figure 5). Consequently, having reliable results on titanium powders does not necessarily ensure trustworthy estimates for titanium alloys. The Ti64 grade will be the reference material since it is more suitable to XRD analyses. The Ti5553 alloy will be subsequently studied once the registration procedures have been validated. In-situ uniaxial tensile tests are carried out under a load control mode. The direction φ is chosen to coincide with the longitudinal direction S1 of the coupon (Figures 7 and9). To measure the total strains on the sample surface, 2D DIC will be used. The sample surface is coated with black and white paints in order to create a speckle pattern (Figure 9(b)). To avoid any bias, the analyzed XRD zone is not covered. When the targeted load level is reached, the sample and testing machine are moved out of the goniometer to acquire pictures. A telecentric lens is mounted on the digital camera to minimize as much as possible the spurious effects associated with out of plane motions. The actuators of the testing machine induce stress variations less than 16 MPa (standard deviation) for the investigated stress range. An isostatic set-up ensures the testing machine to be positioned very precisely in the goniometer. The angular standard uncertainty in relocation is less than 2 • , which corresponds to a small error in final stress estimations. The spatial repositioning error is less than 0.1 mm, a value ten times smaller than the probed surface diameter.

Stress extraction

When a material is loaded or has residual stresses, crystal lattices deform. XRD procedures measure the variations of inter-reticular distances by analyzing diffraction peaks in (poly)crystalline materials [START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF][START_REF] Noyan | Residual Stresses: Measurements by Diffraction and Interpretation[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Lu | Handbook of measurement of residual stresses[END_REF]]. In the case of titanium alloys analyzed by a copper source, the peak commonly used to carry out stress analyses diffracts at an angle ≈ 140 • [10] and corresponds to the {213} family of the α-phase, which can be found in both Ti64 and Ti5553 alloys [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Boyer | The use of beta titanium alloys in the aerospace industry[END_REF][START_REF] Aeby-Gautier | Isothermal α formation in β metastable titanium alloys[END_REF]. The inter-reticular distance for this angle is given by the dimension of the HCP crystal structure where the edges of the hexagon are a = 0.295 nm and the height of the lattice is c = 0.468 nm [START_REF] Leyens | Titanium and titanium alloys: fundamentals and applications[END_REF]. The link between diffraction peak angle and inter-reticular distances is given by Bragg's law

2d {213} sin θ = λ KαCu (1)
where d {213} is the inter-reticular distance for {213} planes of the α-phase, θ the diffraction angle, and λ KαCu the wavelength of the X-ray beam (e.g., copper source with λ KαCu = 0.154 nm). Peak registrations presented in the following do not dissociate K α1 and K α2 rays. Elastic stresses near the sample surface in α-phases are evaluated via XRD by estimating peak shifts [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF].

From channels to stresses

Figure 10 describes the whole procedure used from the acquisition of diffractograms by the goniometer equipped with a linear position sensor (LPS) that gives an information expressed in channels x to stress estimations. and angular position depends on the geometry of the goniometer (Figure 11)

2θ(x) = 2θ ref + tan -1 L x -x ref x max (2)
with

x ref = x max + 1 2 (3)
where 2θ ref is the angle corresponding to the central channel of the linear sensor (this angle is equal to 140 • thanks to the use of Ti powder). The ratio /L is a geometric parameter of the goniometer (i.e., it depends on the length of the LPS sensor, = 50 mm, and the distance between the analyzed surface and the sensor, L = 150 mm). The angles φ and ψ determine the location of the goniometer during one single measurement with φ the main measurement direction and ψ the incidence angle of the X-ray source and sensor (see Figures 7 and8). the interreticular distance for the undeformed configuration, and 2θ 0 the corresponding diffraction angle.

To get an estimate of the stress tensor in one direction, several measurements (i.e., angular positions ψ) are needed for each point of analysis [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF]. Inter-reticular distances d φψ and corresponding angles θ φψ depend on the main measurement direction φ but also on the incidence angle of the X-ray source ψ. As measurements are carried out very close to the sample surface, the normal stress component perpendicular to the surface (along the axis S3, see Figure 7) will be assumed to vanish (σ 33 = 0), in order to be consistent with a traction-free surface. The relationship between stress and elastic strain components is then given by

˜ φψ = - 1 2 S {213} 2 σ φ sin 2 ψ - 1 2 S {213} 2 τ φ sin 2ψ + ˜ ψ=0 (5)
where ˜ φψ = ln(sin θ φψ ), S are invariant for any probed angle for the α-phase. It is worth noting that with the present setting, the knowledge of θ 0 is not needed when using ˜ φψ instead of φψ . However, to probe the consistency of the results, it will be checked a posteriori when analyzing tensile tests (in particular, the value of θ 0 ).

In the present case, the X-Ray elasticity parameters

S 1 = - ν α E α and 1 2 S 2 = 1 + ν α E α depend on
the Poisson's ratio ν α and Young's modulus E α of the α-phase (i.e., 1 2 S 2 = 11.9 × 10 -6 MPa -1

and S 1 = -2.64 × 10 -6 MPa -1 [START_REF] Fréour | Influence of a two-phase microstructure on XEC and XRD stress analysis[END_REF][START_REF] Bruno | Surface and Bulk Residual Stress in Ti6Al4V Welded Aerospace Tanks[END_REF]. These values correspond to E α = 109 GPa and ν α = 0.3.

The stress tensor components σ φ and τ φ are determined by least squares minimization min

σ φ ,τ φ ,˜ ψ=0 ψ [˜ φψ + 1 2 S 2 σ φ sin 2 ψ + 1 2 S 2 τ φ sin 2ψ -˜ ψ=0 ] 2 (6)
in addition to the composite strain ˜ ψ=0 .

Registration of diffraction peaks

The previous section has shown that the sought stresses can be related to the strains (Equation ( 5))

or equivalently to the channel position of the diffraction peak (Equations ( 2)-( 4)). In both cases, the location of the peaks θ φψ has to be determined. One of the standard approaches consists in registering the measured diffractogram with an a priori chosen function 0 ≤ g(x) ≤ 1 whose maximum location is known (i.e., g(x = x 0 ) = 1). The registration procedure consists in finding the location x 0 of the peak in addition to other parameters needed to model the measured signal to minimize the sum of squared differences min

x 0 (ψ),ξ 0 ,∆I,I b ,D x η 2 (x; x 0 (ψ), ξ 0 , ∆I, I b , D)

with

η(x; x 0 (ψ), ξ 0 , ∆I, I b , D) = f (x; ψ) -I b -Dx ∆I -g x -x 0 (ψ) ξ 0 ( 8
)
where f is raw signal measured by XRD for the angular position ψ, x the channel position on the LPS sensor, ξ 0 the width of the function that is proportional to the full width at half maximum (FWHM), and ∆I = I max -I bgl the peak intensity above the background line (bgl).

In the present case, the background intensity is modelled with a linear function I bgl (x) = I b + Dx.

Consequently, there are five unknowns to be determined, namely, x 0 , ξ 0 and I max that characterize the diffraction peak, in addition to I b and D that describe the background line (Figure 12). Each diffractogram is analyzed independently (i.e., for each considered angle ψ) with a registration technique that can be referred to as diffraction signal correlation or DSC (i.e., it is the one dimensional version of (2D) digital image correlation or (3D) digital volume correlation [START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF][START_REF] Hild | Digital Image Correlation[END_REF]. A Gauss-Newton algorithm is implemented to minimize the sum of squared differences [START_REF]ASTM E[END_REF] for each considered angle ψ. The covariance matrix associated with the evaluated parameters, which are gathered in the column vector {p} = {x 0 , ξ 0 , ∆I, I b , D} † , reads [START_REF] Hild | Digital Image Correlation[END_REF] [

Cov p ] = γ 2 ∆I 2 [M] -1 ( 9
)
where γ is the standard deviation of the acquisition noise that is assumed to white and Gaussian, and [M] the Hessian used in the minimization scheme

[M] = [m] † [m] (10)
with

[m] =     ∂η ∂x 0 (x) ∂η ∂ξ 0 (x) ∂η ∂∆I (x) ∂η ∂I b (x) ∂η ∂D (x) . . . . . . . . . . . . . . .     (11)
This covariance matrix allows the resolution 1 of the registration technique to be assessed [START_REF] Hild | Digital Image Correlation[END_REF]. The standard resolution of the peak position is then equal to the diagonal term of [Cov p ] corresponding to x 0 , provided the other parameters do not change.

The influence of the peak position uncertainty on stress resolutions is easily computed since the stress extraction is a linear least squares problem in terms of ˜ φψ (see Equation ( 6)). The covariance matrix of the stress extraction technique reads

[Cov σ ] = [C] -1 [c] † [Cov ][c][C] -1 (12)
with

[C] = [c] † [c] (13)
and

[c] =     1 2 S 2 sin 2 ψ 1 2 S 2 sin 2ψ 1 . . . . . . . . .     (14)
1 The resolution of a measuring system is the "smallest change in a quantity being measured that causes a perceptible change in the corresponding indication" [START_REF]International Vocabulary of Metrology -Basic and General Concepts and Associated Terms, VIM. International Organization for Standardization[END_REF].

Commercial softwares (e.g., Stressdiff [START_REF]Stressdiff[END_REF] or Leptos [START_REF] By | [END_REF]) use peak positions issued from such minimization to end up with so-called sin 2 ψ methods [START_REF]Non-destructive Testing, Test Method for Residual Stress analysis by X-ray Diffraction[END_REF][START_REF] Cullity | Elements of x-ray diffraction[END_REF][START_REF] Lütjering | Titanium[END_REF]. This procedure will serve as benchmark for the validation of the following integrated procedure.

Given the fact that the peak positions are parameterized in terms of the sought stresses and composite strain (i.e., x 0 = x 0 (σ φ , τ φ , ˜ ψ=0 ), see Equations ( 2) and ( 4)), the previous minimization can be performed over the whole set of diffractograms min

σ φ ,τ φ ,˜ ψ=0 ,ξ 0 ,∆ I ,I b ,D ψ x η 2 I (x, ψ; σ φ , τ φ , ˜ ψ=0 , ξ 0 , ∆ I , I b , D) (15)
with

η I (x, ψ; σ φ , τ φ , ˜ ψ=0 , ξ 0 , ∆ I , I b , D) = f (x, ψ) -I b -Dx ∆I -g x -x 0 (σ φ , τ φ , ˜ ψ=0 ) ξ 0 (16)
This approach corresponds to integrated DSC for which the sought quantities are directly determined from the registration procedure and do not need an additional minimization step. A Gauss-Newton scheme is also implemented and the initial guess of the sought parameters comes from a first non integrated analysis. The covariance matrix associated with the evaluated parameters, which are gathered in the column vector {p I } = {σ φ , τ φ , ˜ ψ=0 , ξ 0 , ∆ I , I b , D} † , reads

[Cov p I ] = γ 2 ∆I 2 [M I ] -1 (17)
where [M I ] is the approximate Hessian used in the global minimization scheme

[M I ] = [m I ] † [m I ] (18)
with

[m I ] =     ∂η I ∂σ φ (x, ψ) ∂η I ∂τ φ (x, ψ) ∂η I ∂˜ ψ=0 (x, ψ) ∂η I ∂ξ 0 (x, ψ) ∂η I ∂∆I (x, ψ) ∂η I ∂I b (x, ψ) ∂η I ∂D (x, ψ) . . . . . . . . . . . . . . . . . . . . .     (19)
Tests have been made by selecting various distributions (i.e., Gauss, Pearson VII, Lorentz).

They give the same types of results without any significant improvement for any of them in terms of registration residuals. In the following, a Gaussian distribution was selected

g ≡ g G (x) = exp - x -x 0 ξ 0 2 (20)
A preliminary step is performed to calibrate the LPS sensor in terms of local offset corrections.

The offset of the detector is determined by measuring the intensity during diffraction on a material that does not diffract at an angle close to that of {213} planes of the α-phase. Amorphous glass was chosen. The diffractogram on glass is obtained by a measurement lasting at least half an hour in order to get enough signal. After rescaling the offset is subtracted to each diffractogram used in stress analyses. Figure 13 shows that most of the spacial variations are erased and that the diffraction peak appears more clearly.

Fig. 13: Subtraction of detector offset

For the sake of convenience, the residuals that will be reported hereafter will refer to the raw acquisitions, namely,

ρ(x; ψ) = f (x; ψ) -∆Ig x -x 0 (ψ) ξ 0 + I b + Dx (21)
and are therefore expressed in counts. Any deviation from random noise will be an indication of model error (i.e., associated with the choice of g, and elasticity). The final outputs of the two registration algorithms are σ φ , τ φ , and ˜ ψ=0 = ln(sin θ 0 ) -S 1 tr(σ).

5 Results of stress analyses

Validation of the registration techniques

Six loading steps were applied (Figure 14) on the Ti64 sample ranging from 0 to 660 MPa. For such type of alloy, the maximum stress level is lower than the yield stress ≈ 900 MPa [START_REF] Lütjering | Titanium[END_REF][START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. The applied stress corresponds to the applied load divided by the cross-sectional area of the sample ligament. Interestingly, the integrated DSC residuals (i.e., 39 counts), which are expected to be higher since less degrees of freedom are available, remain close to the non integrated approach. The similarity of residual levels between integrated and non integrated approaches validates the assumption of elasticity in the probed volume. These two results also show that the registration was successful via DSC and integrated DSC. Stress analyses can also be performed via integrated DIC [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF]. The displacement field is first measured with FE-based DIC in which the kinematic field is parameterized with nodal displacements associated with an unstructured mesh (see Figure 16) made of three-noded triangles (with linear interpolation of the shape functions). Even though the central part of the sample was not speckled, the DIC code could converge (i.e., helped by the convergence in the speckled area) and yields results within the XRD measured area. From these measurements, only those corresponding to the two extreme transverse rows of elements are considered and they are prescribed as Dirichlet boundary conditions to an elastic FE analysis under plane stress hypothesis, which allows the stress field to be evaluated everywhere. Macroscopic elasticity constants used are the Young's modulus which is equal to 109 GPa and the Poisson ratio which is equal to 0.3. Figure 16 shows the longitudinal stress field for each loading step. In all the zone probed by XRD, the stress is virtually uniform and the corresponding standard deviation is less than 10 MPa for the highest load level. From this analysis, the mean longitudinal stress is reported and will be compared to the other stress evaluations.

Fig. 16: Longitudinal stress (expressed in MPa) field evaluated via integrated DIC at the last loading step. Online version: history of the six analyzed steps (Figure 14)

In Figure 17, the comparison between various stress analyses is shown. In the present plot, the reference (i.e., horizontal axis) is the applied stress. Integrated DIC and DIC give the same stress estimation and correspond to the mean longitudinal stress averaged over the XRD zone.

These levels are very close to the applied stress. For DIC, the measured strains are assumed to be elastic and the corresponding stresses are evaluated, as would be performed by XRD analyses (i.e., without any equilibrium requirement), by using isotropic elasticity under plane stress

The two DSC approaches yield results that are in very good agreement with each other and with the evaluations based on integrated DIC and load measurements.

Interestingly, the stress estimations based on both DIC results for which the strains are averaged over the zone of interest of XRD analyses are consistent with each other. This observation proves that the assumption of linear elasticity is satisfied. The fact that DSC results are consistent with IDSC, integrated DIC evaluations and applied stress estimates also validates the choice of X-Ray elasticity constants for the XRD measurements in Ti64 alloy. Figure 18 shows the error quantifications for DSC and integrated DSC with respect to the applied stress. The stress resolutions of both DSC methods are also shown. They are based on the effect of acquisition noise only (see Equations ( 12) and ( 12)). The standard resolution of the two DSC techniques is very low and of the order of 3 MPa (i.e., 2.7 MPa for integrated DSC and 3.3 MPa for regular DSC). Thanks to integration, the standard resolution is decreased by 20 %. The root mean square error between the applied stress and DSC or IDSC is respectively equal to 37 MPa and 41 MPa. This level corresponds to an upper bound given the fact that load fluctuations occur (i.e., inducing stress variations of the order of 16 MPa in the present case). The standard applied stress uncertainty is equal to 16 MPa

The last output of the DSC codes is the composite strain ˜ ψ=0 = ln(sin θ 0 ) -S 1 tr(σ). Given the fact that the trace of the stress tensor is equal to the longitudinal stress in a uniaxial tensile test, it is possible to evaluate 2θ 0 , which is the diffraction angle for zero stress of {213} planes for the α-phase. The mean value is 2θ 0 = 140.16 • , which is close to the expected value for pure titanium (i.e., 140 • ), see Figure 20. The corresponding standard deviation is equal to 0.019 • for DSC and 0.018 • for IDSC, which is 5 % lower than DSC. With Bragg's law, it is concluded that this angular uncertainty divided by tan(θ 0 ) is an estimate of the longitudinal strain uncertainty (i.e., ≈ 1.2 × 10 -4 for both methods), and a longitudinal stress uncertainty of ≈ 13 MPa. This level is of the same order as the stress fluctuations induced by the tensile stage. Thanks to the reported consistencies of stress estimations both DSC and IDSC algorithms are now considered as validated and will be used to study the Ti5553 grade.

Application to Ti5553 alloy

For the specimen made of Ti5553 alloy, 9 loading steps and one globally unloaded step were applied and analyzed via DIC and DSC (Figure 21). The lowest stress level is 0 MPa, and the highest is 1100 MPa (the yield stress for this alloy is of the order of 1250 MPa [START_REF] Martin | Simulation numerique multi-echelles du comportement mecanique des alliages de titane betametastable Ti5553 et Ti17[END_REF]). However during electropolishing, the geometry of the sample was roughened and the thickness of the coupon has been reduced from 0.5 mm to 0.45 mm and some imperfections on the edges were created. As a result, plasticity may occur during the test especially at high load levels. Further, the estimation of the applied stress is more delicate and thus will not be reported. The so-called DIC stresses are preferred. The occurrence of plasticity is confirmed in Figure 22 in which the stresses are evaluated with the DIC strain fields, which are assumed to be elastic. For high load levels, very high (and non physical) stress levels are observed in one area of the sample, which was not probed via XRD.

It is worth noting that the last step corresponds to total unloading, which does not result in vanishing strains and stresses, thereby confirming the presence of plastic strains in the lower part of the coupon. However, as will be illustrated in Figure 25, in the area impacted by X-rays it is believed the local stress state remains elastic and homogeneous during the whole experiment.

The standard stress uncertainty corresponding to the probed area is less than 30 MPa via DIC estimations.

Fig. 22: Longitudinal stress (expressed in MPa) field evaluated via DIC (assuming elasticity) at the last loading step (Figure 21). Online version: history of the ten analyzed steps

Figure 23 shows root mean square residuals for both DSC approached. In the present case, the two residuals are virtually coincident for all investigated stress levels. This remarkable agreement between the two DSC techniques enables the hypothesis of elasticity to be fully validated in the present case. This observation is consistent with the fact that the α-phase is finely distributed within the probed volume (i.e., of the order of 1 mm 2 × 5 µm, see Figure 2). When compared to Ti64, the mean residuals of DSC and IDSC are lower (e.g., 25 counts instead of 39 counts for IDSC in Ti64). However, the signal levels were also significantly lower (Figure 5). Due to the coexistence of two phases in the studied alloy (Figures 3 and4), the results will first focus on elastic strains. Figure 24 displays the estimations of elastic strain by various techniques, namely integrated DIC in addition to DSC and integrated DSC. In the present case, the abscissa corresponds to the mean longitudinal strain in the XRD zone measured by DIC. The fact that DIC and integrated DIC results coincide proves that the hypothesis of elasticity was fulfilled during all the experiment and that plasticity was confined in an area that has not impacted the zone probed by XRD (as expected from Figure 22). This effect cannot be attributed to plasticity since linear relationships between DIC and integrated DIC results would not be observed. The explanation comes from the fact that DIC analyses are performed at the macroscale (i.e., the two-phase alloy) whereas DSC analyses only consider the α-phase. In the present case, it is shown that the mean elastic strains evaluated in the α-phase are about 73 % those of the alloy (i.e., at the macroscopic level). This difference is due to the presence of two phases in Ti5553 whose elastic properties are different [START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF][START_REF] Fréour | Influence of a two-phase microstructure on XEC and XRD stress analysis[END_REF][START_REF] Sylvain Fréour | Determining ti-17 β-phase singlecrystal elasticity constants through x-ray diffraction and inverse scale transition model[END_REF][START_REF] Martin | Simulation numerique multi-echelles du comportement mecanique des alliages de titane betametastable Ti5553 et Ti17[END_REF][START_REF] Herbig | D short fatigue crack investigation in beta titanium alloys using phase and diffraction contrast tomography[END_REF][START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. Contrary to Ti64, which is mostly composed of α-phase, the presence of 40 wt% β-phase induces a deviation for the determination of macroscopic stresses using X-Ray elasticity constants of pure titanium.

The deviation is estimated at 27 % in the present case.

Figure 25 shows the stress estimates for Ti5553. In the present case the DIC stresses (i.e., the macroscopic stresses) are the reference. Independent tensile tests on the same alloy provided a Young's modulus of 115 GPa, and Poisson's ratio equal to 0.35. DIC and integrated DIC stresses are close, which is consistent with the observations of Figure 24. The longitudinal stresses obtained with both DSC approaches are also in good agreement. By using the same elastic parameters for the α-phase as those considered for Ti64, the mean stress in the α-phase amounts to about 69 % of the macroscopic stress. This stress ratio is again due to the difference in elastic properties of α-and β-phases in this alloy [START_REF] Hauk | Structural and Residual Stress Analysis by Non Destructive Methods: Evaluation, Application, Assessment[END_REF][START_REF] Fréour | Influence of a two-phase microstructure on XEC and XRD stress analysis[END_REF][START_REF] Sylvain Fréour | Determining ti-17 β-phase singlecrystal elasticity constants through x-ray diffraction and inverse scale transition model[END_REF][START_REF] Martin | Simulation numerique multi-echelles du comportement mecanique des alliages de titane betametastable Ti5553 et Ti17[END_REF][START_REF] Herbig | D short fatigue crack investigation in beta titanium alloys using phase and diffraction contrast tomography[END_REF][START_REF] Duval | Mechanical Properties and Strain Mechanisms Analysis in Ti5553 Titanium Alloy[END_REF]. The standard DIC stress uncertainty is equal to 30 MPa

The shear stresses are again very small (Figure 26). The mean level is 13.5 MPa for IDSC, and 15 MPa for DSC. Therefore, there is a small bias in the present case. It may be due to small misalignments of the goniometer (whose standard uncertainty was estimated to be 2 • ). From this information, the stress state is subsequently determined. The second route consists in merging both steps into a single analysis. It is referred to as integrated diffraction signal correlation (i.e., IDSC). It is shown that the latter leads to lower stress resolutions when compared to the former.

For Ti64 a very good agreement is observed between the stresses estimated with both DSC techniques, integrated DIC and regular DIC. For Ti5553, it is shown that the elastic strains in the α-phase are about 73 % of the macroscopic strain, which is due to the difference in elastic properties of the two phases. Further, the fact that the two DSC techniques yield virtually identical results validates the hypothesis of elasticity at the level of the α-phase. For both materials, it is observed that the diffraction angle for the unstressed state is very close to 140 • (i.e., the reference level for pure titanium).

Stress resolutions and uncertainties have been systematically analyzed for both alloys. In particular, it is shown that the uncertainty associated with the knowledge of the unstressed configuration has a limited impact on the overall levels. With the implemented registration techniques there is only a 20 % increase of overall stress uncertainties for Ti5553 with respect to Ti64 even though the peak height has been decreased by a factor of 4. This result validates the two registration techniques and shows their robustness even for Ti5553.

Having validated the present registration techniques, they can now be used in other configurations to study, for instance, the surface integrity [START_REF] Lütjering | Titanium[END_REF][START_REF] Guillemot | Prediction of the endurance limit taking account of the microgeometry after finishing milling[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF] associated with different milling conditions of Ti5553 [START_REF] Cox | The effect of finish milling on the surface integrity and surface microstructure in Ti-5Al-5[END_REF]. They may also be used in benchmarks with other materials and procedures [START_REF] Lefebvre | External reference samples for residual stress analysis by x-ray diffraction[END_REF][START_REF] Suominen | Residual Stress Measurement of Ti-Metal Samples by Means of XRD with Ti and Cu Radiation[END_REF].

Fig. 1 :Figure 2

 12 Fig. 1: Micrographs of Ti64 (a) and Ti5553 (b) alloys

Fig. 2 :

 2 Fig. 2: (a) EBSD orientation map of Ti64 sample (α-indexation). The black box is enlarged in sub-figure (b)

Fig. 3 : 4 In Figure 4 ,

 344 Fig. 3: (a) EBSD orientation map of Ti5553 sample (β-indexation). The black rectangle is enlarged in (b). The boxed detail is shown in Figure 4

Fig. 4 :Figure 5

 45 Fig. 4: EBSD orientation map of Ti5553 sample (zoom of Figure 3(b)). (a) α-indexation, and (b) β-indexation

Fig. 5 :

 5 Fig. 5: Diffractograms of the two studied alloys and of pure titanium powder (the same dynamic range is used for all plots)

Fig. 6 :

 6 Fig. 6: In-situ tensile test enabling for XRD stress analyses

Fig. 7 :

 7 Fig. 7: Sketch of the linear position sensor (LPS), X-ray source (X), analyzed sample S in the goniometer (after Ref. [12])

Fig. 8 :

 8 Fig. 8: Sketch of a rotation of ψ = -50 • around χ axis and locations reached thanks to the Eulerian cradle of χ-method goniometer (after Ref. [12])

Fig. 9 :

 9 Fig. 9: Coupon for in-situ tests. (a) Dog-bone geometry (ligament width: 6 mm, thickness: 0.5 mm, radius of the hour-glass shape: 65 mm). (b) Speckle pattern used for DIC procedures with an unpainted area to allow for XRD measurements. The area probed by the X-ray beam is depicted in orange.

Fig. 10 :

 10 Fig. 10: General procedure used in stress analyses

Fig. 11 :

 11 Fig. 11: Linear position sensor (LPS) and goniometer geometry

2

 2 are the X-Ray elasticity constants depending on the diffracting planes, and ˜ ψ=0 = ln(sin θ 0 ) -S {213} 1 tr(σ) will be referred to as composite strain. The normal and shear stresses are denoted by σ φ ≡ σ φφ and τ φ ≡ σ φ3 , respectively. In the present case elastic isotropy and homogeneity are assumed since anisotropy in {213} planes of α lattice is relatively small, so that S

Fig. 12 :

 12 Fig. 12: Peak model and notations

Fig. 14 :

 14 Fig. 14: Loading history during the in-situ tensile test on Ti64 alloy. The standard stress uncertainty is equal to 16 MPa

Fig. 15 :

 15 Fig. 15: Root mean square residual for Ti64 alloy for the two DSC analyses

Fig. 17 :

 17 Fig. 17: Longitudinal stress σ φ in Ti64 alloy coupon measured with four different evaluation techniques. The error bars depict the root mean square difference between DSC or IDSC estimates and the applied stress. The standard applied stress uncertainty is equal to 16 MPa

Fig. 18 :Fig. 19 :

 1819 Fig. 18: Stress resolutions and errors for Ti64 alloy coupon. The standard applied stress uncertainty is equal to 16 MPa

Fig. 20 :

 20 Fig. 20: Estimation of 2θ 0 for Ti64 alloy from the analysis of the composite strain . The standard applied stress uncertainty is equal to 16 MPa

Fig. 21 :

 21 Fig. 21: Loading history during the in-situ tensile test on Ti5553 alloy. The standard DIC stress uncertainty is equal to 30 MPa

Fig. 23 :

 23 Fig. 23: Root mean square residual for Ti5553 alloy for the two DSC analyses. The standard DIC stress uncertainty is equal to 30 MPa

Fig. 24 :

 24 Fig. 24: Longitudinal elastic strains for the 10 loading steps with the four techniques

Fig. 25 :

 25 Fig. 25: Longitudinal stresses σ φ in Ti5553 alloy evaluated via different techniques. The error bars depict the room mean square difference between DSC or IDSC estimates and the DIC stresses.

Fig. 26 :Fig. 27 :Fig. 28 :

 262728 Fig. 26: Shear stress τ φ in Ti5553 alloy coupon evaluated with DSC and IDSC. The error bars depict the room mean square difference between DSC or IDSC estimates and a vanishing shear stress. The standard DIC stress uncertainty is equal to 30 MPa

Table 2 :

 2 Chemical composition of Ti5553

Acknowledgements This work was funded by Safran Landing Systems. The authors acknowledge Pierre Mella for providing Ti64 alloy and useful discussions on XRD analyses. The authors also thank Adam Cox for providing Ti5553 alloy samples and Thierry Bergey for electropolishing them.