Khiem Trad

Aurore Castets

Alain Wattiaux

Claude Delmas

Ben Amara

Dany Carlier
email: dany.carlier@icmcb.cnrs.fr

Li0.75Mn1.50Fe1.75(PO4)3: First Alluaudite-type Iron Phosphate Containing only Li + as Alkaline Ions

)2(PO4)3 general formula with a = 12.002(9) Å, b = 12.509(9) Å, c = 6.404(7) Å, β = 115.07(7)° in the monoclinic C2/c space group. The 3D framework consists of infinite chains of edge-sharing M(2)2O10 dimers and M(1)O6 octahedra connected by phosphate tetrahedra leading to two sets of hexagonal tunnels. The Li + and Mn 2+ ions partially occupy one of them, the other tunnel being empty. The Mössbauer spectroscopy confirmed the occurrence of only Fe 3+ ions in octahedral environment. Electrochemical cycling tests in Li cells were performed using Li0.75Mn II 1.50Fe III 1.75(PO4)3 powder as electrode material. Only a small amount of Li + ions can be reversibility deintercalated/intercalated corresponding to a capacity as low as 30 mAh/g with a 3.2 V average voltage. Moreover, a strong polarization due to low electronic and ionic conductivities is observed. The presence of Mn 2+ ions in the same tunnel as Li + probably hinders a good Li + ionic diffusion.

Introduction:

Iron phosphates are among the most studied phosphate materials due to their complex and rich structural chemistry owing to the possible +2/+3 mixed valence of iron and its tendency to form various complex frameworks with the phosphate groups [START_REF] Moore | Crystal chemistry of the basic iron phosphates[END_REF]. The interest devoted to these materials has been accentuated since the discovery, by [START_REF] Padhi | Phospho-olivines as Positive-Electrode Materials for Rechargeable Lithium Batteries[END_REF] of LiFePO4 as a promising electrode material for Li-ion batteries [2]. Compared to the previously used LiCoO2 oxide, this material has the advantages of low cost, low toxicity, high thermal stability, high capacity (170 mAh.g -1 ), good cycling stability, and a flat discharge plateau (at 3.4 V).

Besides LiFePO4, other iron phosphates were tested as electrode materials for rechargeable batteries [3]. Among polyanionic positive electrode materials, the alluaudite-type compounds are attracting much attention for sodium and/or lithium batteries due to their interesting open framwork. Some phosphates phases were investigated as LixNa2-xFeMn2(PO4)3 [START_REF] Richardson | Phosphate-stabilized lithium intercalation compounds[END_REF] NaMnFe2(PO4)3 [5], LixNa(1-x)MnFe2(PO4)3 [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF], Li0.47Na0.2FePO4 [START_REF] Kim | LiFePO4 with an alluaudite crystal structure for lithiumion batteries[END_REF], Na2Fe1.96V0.96(PO4)3 [START_REF] Kim | Development of a new alluaudite-based cathode material with high power and long cyclability for application in Na ion batteries in real-life[END_REF] and Na1.702Fe3(PO4)3 [START_REF] Lui | Synthesis, Crystal Structure, and Electrochemical Properties of Alluaudite Na1.702Fe3(PO4)3 as a Sodium-Ion Battery Cathode[END_REF]. Recently the group of Yamada showed very high operating voltages for Na2+2xFe2-x(SO4)3 sulfates compounds in sodium batteries [START_REF] Barpanda | A 3.8-V earth-abundant sodium battery electrode[END_REF][START_REF] Oyama | Off-stoichiometry in alluaudite-type sodium iron sulfate Na2+2xFe2-x(SO4)3 as an advanced sodium battery cathode material[END_REF][START_REF] Oyama | Sodium intercalation mechanism of 3.8 V class alluaudite sodium iron sulfate[END_REF]. Subsequently, analogous alluaudite-type sulfates Mn-based [START_REF] Dwibedi | Na2.44Mn1.79(SO4)3: a new member of the alluaudite family of insertion compounds for sodium ion batteries[END_REF] and Co-based [START_REF] Dwibedi | Na2.32Co1.84(SO4)3 as a new member of the alluaudite family of high-voltage sodium battery cathodes[END_REF] were also studied as positive electrodes for Na-ion batteries. Very recently, a molybdate compound has also been reported: Na2.67Mn1.67(MoO4)3 as candidate for Na-ion batteries [START_REF] Gao | Mn1[END_REF]. Although the alluaudite family encloses a large number of isostructural phosphates and arsenates either natural or synthetic, no alluaudite type materials containing only Li + ions as alkaline ions have been reported so far to the best of our knowledge. During our investigation of the LixNa(1-x)MnFe2(PO4)3 solid solution, no pure phase with x higher than 0.75 could be prepared [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF]. K. Kang and co-workers, actually aimed to prepared an alluaudite-type LiFePO4, by ion-exchange from the analogous Naalluaudite, but finally synthetized Li0.47Na0.2FePO4 with remaining Na + in in the lattice acting as pillars. Stronger ion-exchange conditions resulted in the collapse of the structure [START_REF] Kim | LiFePO4 with an alluaudite crystal structure for lithiumion batteries[END_REF].

The alluaudite structure has been first reported by Moore who proposed the X(2)X( 1)M(1)M( 2)2(PO4)3 structural formula where X(2), X(1), M(1) and M(2) are cationic sites arranged in decreasing size [START_REF] Moore | Crystal chemistry of alluaudite structure type[END_REF]. Hatert et al., based on detailed structural refinements reported that X(1) has three distinct positions labeled A(1), A(1)' and A(1)", in the tunnel 1 and that X(2) has two distinct positions labeled A(2) and A(2)'in tunnel 2 [START_REF] Hatert | First experimental evidence of alluauditelike phosphates with high Li-content : the (Na1-xLix)MnFe2(PO4)3 series (x = 0 to 1)[END_REF]. The structural formula of Moore was than reformulated as A(2)A(2)'A(1)A(1)'A(1)"M(1)M( 2)2(PO4)3. The structure can be described as a 3D skeleton structure [M(1)M( 2)2(PO4)3] with two types of tunnels parallel to the c direction as shown in Figure 1a. All the existing alluaudite phases contain atoms only on A(1), A(2)', M(1) and M(2) sites [START_REF] Lee | Hydrothermal Synthesis and Structures of Na3In2 (PO4)3 and Na3In2 (AsO4)3: Synthetic Modifications of the Mineral Alluaudite[END_REF][START_REF] Antenucci | Combined X-ray Rietveld, infrared and Raman study of a new synthetic variety of alluaudite[END_REF][START_REF] Antenucci | Synthèse et cristallochimie de NaCaCdMg2(PO4)3, phosphate nouveau à structure alluaudite sans cation trivalent[END_REF][START_REF] Yakubovich | The crystal structure of a synthetic variety of alluaudite Na2(Fe3+0.5Fe2+0.5)2Fe2+[PO4]3[END_REF][START_REF] Corbin | Synthesis of Open-Framework Transition-Metal Phosphates Using Organometallic Precursors in Acidic Media[END_REF][START_REF] Korzenski | Hydrothermal synthesis, structure, and characterization of a mixed-valent iron (II/III) phosphate, NaFe3.67(PO4)3: a new variation of the alluaudite structure type[END_REF][START_REF] Chouaibi | Neutron diffraction, Mössbauer spectrum, and magnetic behavior of Ag2FeMn2(PO4)3 with alluaudite-like structure[END_REF][START_REF] Hatert | The crystal chemistry of lithium in the alluaudite structure: a study of the (Na1-xLix)CdIn2(PO4)3 solid solution (x= 0 to 1)[END_REF][START_REF] Hidouri | Characterization by X-ray diffraction, magnetic susceptibility and Mössbauer spectroscopy of a new alluaudite-like phosphate:: Na4CaFe4 (PO4) 6[END_REF][START_REF] Hidouri | Exploratory synthesis of new Iron (ii, iii)-containing polyanion framework solids of electrochemical and magnetic interest[END_REF] except for the following three phases: (i) Cu1.35Fe3(PO4)3 in which copper is distributed on the A(2), A(1)', A(1)" sites and iron is located at the M(1) and M(2) sites [START_REF] Warner | Synthesis and Structure of Cu1.35Fe3(PO4)3 and Cu2Mg3(PO4)3: New Mixed Valence Compounds of the Alluaudite Structure Type[END_REF], (ii) K0.53Mn2.37Fe1.24(PO4)3 in which potassium occupies the A(2)' site, manganese occupies A(1)" and M(1), it also shares the M(2) site with iron [START_REF] Hidouri | Mn2.37Fe1.24[END_REF] (iii) the solid solution LixNa(1-x)MnFe2(PO4)3 in which lithium and sodium are distributed over the A(1) and A(1)' sites while manganese and iron are distributed over the M(1) and M(2) sites [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF].

In this paper, we report the synthesis of a new lithium iron phosphate Li0.75Mn1.50Fe1.75(PO4)3 with an alluaudite-type structure prepared as single crystals by a flux method. The structural characterization was performed by single crystal X-ray diffraction. Then by classical solid state chemistry this material was obtained as a powder for further characterization (Mössbauer spectroscopy, 7 Li MAS NMR). The electrochemical properties of this material were also studied vs Li.

Experimental

Single crystals of the title phase were isolated during an attempt to grow LiMnFe2(PO4)3 in a flux of lithium dimolybdate Li2Mo2O7. A starting mixture of LiNO3, MnO, Fe(NO3)3.9H2O, (NH4)2HPO4 and MoO3 corresponding to a molar ratio of LiMnFe2(PO4)3 : Li2Mo2O7 = 2:1 was dissoved in nitric acid, then the resulting solution was evaporated by heating at 353 K. The obtained dry residue was ground in an agate mortar, and then progressively heated in an open platinum crucible up to 873 K to expel NH3 and CO2. The sample was then reground, melted for 1h at 1173 K and subsequently cooled slowly down to room temperature at 10°C.h -1 rate.

Brown and hexagonally shaped crystals were obtained by washing the whole product with hot water in order to dissolve the flux.

A single crystal of dimensions 0.06 × 0.07 × 0.16 mm was chosen for the structure determination. Data were collected with an Enraf-Nonius CAD4 diffractometer using graphite monochromated MoKα radiation (λ = 0.7107 Å). The unit cell parameters (monoclinic) and the orientation matrix were determined from a least squares fit of 25 intense reflections in the range 9.72° ≤ θ ≤ 14.74°. Among a total of 1269 unique reflections collected using the (ω-2θ) scan mode (2θmax = 29.96°), only 1180 were considered as observed according to the statistic criterion

( ) [ ] 2 2 4 o o F F σ >
. No detectable decay was observed during the data collection, according to the intensities of two standard reflections measured every two hours.

The data were corrected for Lorentz and polarization effects, and for the absorption by ψ-Scan (Tmin,max: 0.57, 0.72).

A pure powder sample with a composition close to the Li0.75Mn1.50Fe1.75(PO4)3 single crystals was synthesized by the conventional solid state reaction starting from a stoichiometric mixture of LiNO3, MnO, Fe(NO3)3.9H2O and (NH4)2HPO4. After an initial treatment, similar to that undertaken for the synthesis of the single crystals until 873 K, the sample was subjected to final calcinations at 1113 K for 48 h with intervening grinding. A light brown powder was obtained by quenching in air. Its purity was checked from the examination of its X-ray powder diagram (Fig. 2) collected in the range 5°≤ 2θ ≤80° on a PANalytical diffractometer using CuKα radiation (λ = 1.5406 Å ). This pattern was refined by the Full profile matching, using the Fullprof program [START_REF] Rodriguez-Carvajal | Fullprof version 1.9c[END_REF]. The refined parameters a = 12.004(1) Å, b = 12.484(1) Å, c = 6.389(1)

Å and β = 115.00(1)° are in a good agreement with those obtained from single crystal X-ray data. The ICP measurements lead to the following composition: Li0.8(1)Mn1.4(1)Fe1.8(1)(PO4)3.

The particles size was analyzed by Scanning Electron Microscopy : the aggregates size ranges between 10 and 60 μm in diameter and consisted of 2 to 5 μm primary particles.

57 Fe Mössbauer measurements were performed using a constant acceleration Halder-type spectrometer operating in transmission geometry with a room temperature 57 Co source (Rh matrix). The velocity scale was calibrated according to the 57 Fe Mössbauer spectrum of a pure α-Fe 0 foil recorded at room temperature. The polycrystalline absorbers were prepared in order to contain about 10 mg/cm 2 of iron and thus, to avoid saturation effects. The refinement of Mössbauer hyperfine parameters (δ isomer shift, Δ quadrupole splitting, Γ signal linewidth and relative areas) was performed using both homemade programs and the WinNormos software.

Distributions of hyperfine parameters (quadrupole splitting) have been considered for overlapped components presenting apparent linewidth higher than 0.30 mm/s, following the method described by Hesse and Rübartsch [START_REF] Hesse | Model independent evaluation of overlapped Mössbauer spectra[END_REF].]. polypropylene foil were used as separators. The electrochemical tests were carried out in the galvanostatic mode with a Bio-logic VMP system using a C/50 cycling rate.

Results and discussion

Structural characterization

The structure of the single crystal of Li0.75Mn1.50Fe1.75(PO4)3 was solved by direct methods [START_REF] Altomare | Completion and refinement of crystal structures with SIR92[END_REF] and subsequently refined on F 2 by full matrix least squares techniques [START_REF] Sheldrick | SHELXL97, A Program for the Solution of the Crystal Structures[END_REF] using the C2/c space group. Crystallographic data and structural refinement conditions are summarized in Table I.

All of the atoms were refined with anisotropic thermal parameters leading to R1 = 0.028 and wR2 = 0.074 and to the atomic coordinates given in Table II. Relevant interatomic distances are listed in Table III x)MnFe2(PO4)3 [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF] and is in good agreement with the site stability study performed by DFT and DFT+U calculations on the LiMnFe2(PO4)3 phase [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF].

A polyhedral view of the structure as projected along the [0 0 1] direction is shown 3a). Thus, its environment could be considered as a square pyramid (5+1 coordination). A similar behavior was previously observed in our study on Na1-xLixMnFe2(PO4)3 [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF]. The average <M(1)-O> = 2.207(3) Å distance is comparable to that (2.241(4) Å) observed for the fully occupied M(1)

site by Mn 2+ in the alluaudite phosphate Ag2Mn2Fe(PO4)3 [START_REF] Chouaibi | Neutron diffraction, Mössbauer spectrum, and magnetic behavior of Ag2FeMn2(PO4)3 with alluaudite-like structure[END_REF].

As usually observed in the alluaudite-type phosphates, the two distinct PO4 tetrahedra are slightly distorted as indicated by the P-O distances between 1.522(2) and 1.552(2) Å, leading to an mean values of 1.541 Å for P(1) and 1.537 Å for P(2), close to that calculated by Baur for the anhydrous monophosphates [START_REF] Baur | The geometry of polyhedral distortions. Predictive relationships for the phosphate group[END_REF].

The A(1)' and A(1)" sites, located in the large tunnels of the alluaudite structure are partially occupied by Li + and Mn 2+ ions, respectively (Fig. 3b). These sites which reside in the tunnel at (½, 0, z) share one face; therefore, they cannot be occupied simultaneously due to their short separation of 1.265 Å. Their oxygen environments were determined using the procedure of Donnay and Allman [35] with the ionic radii of Shannon [36]. Accordingly, the environment of A(1)' consists of four oxygen atoms with A(1)'-O distances ranging from 1.85(3) to 2.11(4) Å close to those observed for the A(1)' site in the alluaudite phosphates Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4(from 1.81(5) to 2,10(1) Å). The environment of Mn 2+ ions within the A(1)" site consists of five O atoms with A(1)"-O distances ranging from 2.216(5) to 2.371(5) Å. This environment of A( 1)" agree with that reported in K0.53Mn2.37Fe1.24(PO4)3 in which Mn-O distances range from 2.253(4) to 2.400(4) Å.

Mössbauer study

The Mössbauer spectrum (Fig. 4) exhibits two resonance absorption lines with large line widths.

An initial refinement using Lorentzian profile lines indicated a single paramagnetic doublet assigned to a unique non symmetry equivalent iron site in accordance with the crystallographic study. The derived isomer shift and quadrupolar splitting are typical of Fe 3+ ions in a distorted octahedral field [START_REF] Menil | Systematic trends of the 57Fe Mössbauer isomer shifts in (FeOn) and (FeFn) polyhedra. Evidence of a new correlation between the isomer shift and the inductive effect of the competing bond T-X (→ Fe) (where X is O or F and T any element with a formal positive charge)[END_REF]. Even so, the obtained linewidth at half maximum was excessively large (Γ = 0.340(2) mm.s -1 ) and this result was taken as an indication of the presence of a quadrupolar splitting distribution which may be associated with a significant disorder resulting from the statistical distribution of Mn 2+ and Fe 3+ in the M(2) site. Moreover, as previously mentioned, these sites are strongly distorted (Table III and Fig. 3a) and they share also an oxygen with the A(1)' and A(1)" sites which can be occupied either by a Li + and a Mn 2+ . Accordingly, a second fit of the spectrum on the basis of a distribution of quadrupolar splitting was undertaken using a fixed linewidth of 0.20 mm.s -1 . The isomer shift δ = 0.425 mm.s -1 is close to those 0.431 and 0.433 mm.s -1 reported for octahedral Fe 3+ ions in Na2Ca0.5Fe2(PO4)3 and Na1.25Mg1.10Fe1.90(PO4)3, respectively [START_REF] Corbin | Synthesis of Open-Framework Transition-Metal Phosphates Using Organometallic Precursors in Acidic Media[END_REF][START_REF] Korzenski | Hydrothermal synthesis, structure, and characterization of a mixed-valent iron (II/III) phosphate, NaFe3.67(PO4)3: a new variation of the alluaudite structure type[END_REF]. The mean value of the quadrupolar splitting <∆*> = 0.62 (2) mm.s -1 indicates that the Fe 3+ site is strongly distorted in agreement with the deformation of the M(2)O6 octahedron determined by the structural refinement (Fig. 3a). The quadrupolar splitting values are widely distributed (0.20 < Δ < 1.00 mm/s) confirming the strong disorder in the M(2) environment mentioned above.

7 Li NMR study

The 7 Li MAS NMR spectrum (Fig. 5a) exhibits only one isotropic resonance located at +79 ppm. From the large shift observed, one can conclude that the Fermi contact interaction predominates: i.e, the shift is mainly due the presence of electronic spin density on the Li nucleus, typically observed for paramagnetic compounds [START_REF] Grey | NMR studies of cathode materials for lithium-ion rechargeable batteries[END_REF][START_REF] Carlier | Transferred Hyperfine Interaction between a Tetrahedral Transition Metal and Tetrahedral Lithium: Li6CoO4[END_REF]. However, one could have expected to observe several distinct Li signals or a distribution of signals, since the A'( 1) lithium site can by surrounded differently by Fe 3+ and Mn 2+ paramagnetic ions distributed over the M(1) and M(2) sites (Figure 5b for Li paramagnetic environment). However, these two ions exhibit the same electronic spin state (t2g 3 eg 2 ), and the occupation of M(1) and M(2) sites by Fe 3+ or Mn 2+ can lead to a similar spin transfer on adjacent Li nucleus thus leading to similar shifts. The Mn 2+ ions in A(1)'' sites could have led to additional different spin transfer its neighboring Li, but since the A(1)' and A(1)'' sites are too close to allow occupancy of adjacent sites, the Li ions in A(1)' sites have no neighboring Mn 2+ in A(1)''(Fig. 3b). The hypothesis of the observation of a single Li signal due to rapid exchange at the NMR timescale between the different Li sites was also considered but not further considered since the global lineshape (extension of the spinning side bands) remains large.

Electrochemical properties

The electrochemical properties of the Li0.75Mn II 1.50Fe III 1.75(PO4)3 phase were studied as positive electrode in lithium batteries using a slow C/50 rate, similar to the protocol used in Ref. [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF].

The corresponding galvanostatic cycling curve is shown in Fig. 6. Starting with a charge, a very small amount of Li + ions can be removed from the lattice, therefore associated to the oxidation of the Mn 2+ ions at rather high voltage. The 4.5 V limit voltage is rapidly reached due to a strong polarization of the cell. During the following discharge, lithium is reintercalated up to the composition Li1.15. Therefore Mn 3+ are reduced back to Mn 2+ and moreover some Fe 3+ ions are reduced to Fe 2+ . As a conclusion, a very limited amount of Li + ions can be reversibility deintercalated/intercalated between 1.5 and 4.5 V versus Li + /Li, involving the two Mn 3+ /Mn 2+ and Fe 3+ /Fe 2+ redox couples, what corresponds to a capacity as low as around 30 mAh/g. Similar low capacities were obtained for the most Li-rich compounds in the LixNa(1-x)MnFe2(PO4)3

series [START_REF] Trad | Structural Study of the Li0.5Na0.5MnFe2(PO4)3 and Li0.75Na0.25MnFe2(PO4)3, Alluaudite Phases and Their Electrochemical Properties As Positive Electrodes in Lithium Batteries[END_REF].

The very high polarization observed shows that the electronic and/or ionic transfers are difficult in this structure which contains unconnected chains either for electron and ionic transfers.

Moreover, within the M(2)-M(1) chains, the alternation of Mn and Fe hinders a good hopping phenomenon due to the large difference in their redox couple. Concerning the Li + ionic diffusion, it may be difficult as Li ions are present only in the tunnel 1 (A(1)' site) isolated from each other and a significant amount of Mn 2+ ions (0.13) in the A(1)'' also in tunnel 1 probably hinder the diffusion (Fig. 3b). This explains why the reversible capacity is so small and the polarization so large. In the case of LiFePO4 the presence of few percent of iron in the tunnels (few percent of antisite defect) limits strongly the electrochemical performances [START_REF] Liu | Effects of antisite defects on Li diffusion in LiFePO4 revealed by Li isotope exchange[END_REF]. The average voltage around ~ 3.2 V that is lower than the one observed for the olivine LiFePO4 (~ 3.5 V [3]) for a same main active redox couple Fe III /Fe II and the polarization observed is really strong due to bad electronic and ionic conductivities and will prevent the material to be further used as positive electrode on Li-ion batteries.

Conclusion

Li0.75Mn1.50Fe1.75(PO4)3 has been prepared and exhibits an alluaudite structure consisted by chains of edge-sharing M(2)2O10 [M(2) = 0.25 Mn + 1.75 Fe] and MnO6 octahedra that run along the [1 0 1 ] direction. Such chains are connected by means of the PO4 tetrahedra and the resulting 3D framework delimits two sets of hexagonal tunnels, the first at (0, 0, z) is empty and the second (½, 0, z) is partially occupied by the Li + and Mn 2+ ions. Mössbauer spectroscopy gave accurate information about the distribution of iron and its local environment. The presence, in this phase, of large tunnels partially occupied by Li + ions, shows that it is of interest to know whether it can be a candidate as a positive-electrode material for lithium batteries.

Electrochemical cycling tests using Li0.75Mn1.50Fe1.75(PO4)3 as positive electrode in Li batteries were performed: the strong polarization due to poor electronic and ionic conductivities will prevent the application of the material in Li-ion batteries.

O( 21)

O(22) O (23)
O

A(1)'

A a U(eq) is defined as one-third of the trace of the orthogonalized Uij tensor.

Table III. Selected interatomic distances (Å) for Li3/4Mn3/2Fe7/4(PO4)3 (Li0.75Mn1.50Fe1.75(PO4)3).

  . Further details of the crystal structure investigation may be obtained from the Fachinformationszentrum Karlsruhe, D-76344 Eggenstein-Leopoldshafen (Germany), on quoting the depository number CSD-434283. The crystal data for Li0.75Mn1.50Fe1.75(PO4)3 (i.e: Li3/4Mn3/2Fe7/4(PO4)3) evidences its belonging to the alluaudite compounds characterized by the A(2)A(2)'A(1)A(1)'A(1)"M(1)M(2)2(PO4)3 general formula [17]. The site occupancies indicate the following cationic distribution: 0.75 Li in A(1)' and 0.25 Mn in A(1)"; 1 Mn on M(1) and [0.25 Mn + 1.75 Fe] in M(2). The presence of lithium in the A(1)' site has already been observed in the solid solution LixNa(1-

Figure 1a .

 1a Figure 1a. The structural units are M(2)2O10 dimers of edge-sharing M(2)O6 octahedra, M(1)O6

Figure 1 :

 1 Figure 1: (a) Perspective view of the structure: projected along the [0 0 1] direction. b) Chain formed by the M(1)O6 and M(2)O6 octahedra.

Figure 2 :

 2 Figure 2: Observed (red) and calculated (black) powder X-ray diagrams for Li0.75Mn1.50Fe1.75(PO4)3 and their difference curve (blue line).

Figure 3 :

 3 Figure 3: a) The M(1) and M(2) site environments versus Oxygens (in red) with the M(1)-O and M(2)-O distances given in Å. b) Perspective view of the tunnel 1 where are localised the Li + ions.

Figure 4 :

 4 Figure 4: The room temperature 57 Fe Mössbauer spectrum for Li0.75Mn1.50Fe1.75(PO4)3 together with the fitting results.

Figure 5 :

 5 Figure 5: a) 7 Li MAS NMR spectrum (vr = 30 KHz) of the Li0.75Mn1.50Fe1.75(PO4)3 recorded at 38 MHz with a Hahn Echo sequence.* = Spinning Side Bands. b) Li environment versus paramagnetic Fe 3+ and Mn 2+ ions located in the M(1) and/or M(2) sites.

Figure 6 :

 6 Figure 6: Electrochemical cycling curve of Li0.75Mn1.50Fe1.75(PO4)3 prepared by solid state reaction used as positive electrode in a Li cell (C/50 rate at room temperature).

  thin sheet and then was cut into a disk (14 mm diameter) and Swagelok ® type cells were assembled. Metallic lithium was used as negative electrode and a 1 M LiPF6 solution in a mixture of propylene carbonate (PC), ethylene carbonate (EC) and dimethyl carbonate (DMC) (1:1:3)) was used as electrolyte. Two Celgard ® 2400 separators and a Viledon ®

7

Li MAS NMR spectrum was recorded at 38 MHz respectively using a Bruker Avance solid state spectrometer with a Wide Bore Bruker magnet activated to 2.35 T. The spinning speed was varied up to 30 kHz using a standard 2.5 mm Bruker MAS probe. Single pulse and Hahn echo type experiments at 30 kHz (one rotor period as echo delay) were used with a 1.2 μs 90° pulse. The spectra were referenced to a 1 M aqueous solution of LiCl. The recycle time used for

7

Li NMR, D 0 = 1 s, is long enough to avoid T 1 saturation effects for the studied nuclei in a paramagnetic phase.

The positive electrode was made of 80% active material, 15% carbon black conductive additive and 5% PTFE binder. The mixture was homogenized by ball milling for 15 min then dried under vacuum at 100°C overnight. Into an argon filled glove box, the mixture was finally crushed into a

Table I .

 I Crystal data and structure refinement parameters for Li3/4Mn3/2Fe7/4(PO4)3 (Li0.75Mn1.50Fe1.75(PO4)3)

			Crystal data
	Formula unit	Li3/4Mn3/2Fe7/4(PO4)3 (Li0.75Mn1.50Fe1.75(PO4)3)
	Crystal system	Monoclinic
	Space Group	C 2/c
	a /Å		12.002(5)
	b /Å		12.509(9)
	c /Å		6.404(7)
	β /°		115.07(4)
	3 V /Å Z D cal/ g.cm	-3	870.9(12) 4 3.587
		Intensity measurements
	Crystal dimensions/mm	0.07 × 0.07 × 0.14
	Apparatus	CAD4 (Enraf-Nonius)
	Wavelength	λ(MoKα) = 0.71073 Å
	Monochromator	Graphite
	µ /mm -1		5.631
	Scan mode	ω/2θ
	Scan speed	variable
	2θmax /°		60
	Background measuring time	tmax/2
	Unique reflections, Rint	1269, 0.031
	Retained reflections [Fo 2 > 4 σ(Fo 2 )]	1180
	Indices		-16 ≤ h ≤15, -1 ≤ k ≤ 17, -1 ≤ l ≤ 8
	F(000)		905
		Structure solution and refinement
	Intensities correction	Lorentz-polarization
	Resolution method	Direct
	Absorption correction	Ψ-scan
	Tmax/min		0.67, 0.53
	Agreement factors	R1 = 0.026, wR2 = 0.068 , S = 1.127
			[Fo 2 > 4 σ(Fo 2 )]
	Number of parameters	102
	Extinction parameter (∆ρ)max, min /e. Å -3	0.0065(5) 0.602, -1.065
	Weighting scheme	W = 1/[σ 2 (Fo 2 ) + (0.0324P) 2 + 4.121P] where
			P = [max(Fo 2 , 0) + 2Fc 2 ]/3

Table II .

 II Atomic coordinates and equivalent isotropic displacement parameters in Å

						2 (ESDs
	in parentheses).
	Atom	Site	Wyckoff Occupation	x/a	y/b	z/c	U( eq)	a
	Li
	Mn(1)
	Mn
	(Mn, Fe)
	P(1)
	O(11)
	O(12)
	P


Florent Goujon

Alain Dequidt

Aziz Ghoufi

Patrice Malfreyt
email: patrice.malfreyt@uca.fr

How Does the

published or not. The documents may come

Introduction

The coarse-grained (CG) models have been widely applied in molecular simulations of biomolecules, 1 polymers 2-4 and surfactants. [START_REF] Ndao | Coarse-Grained Modeling of the Oil-Water-Surfactant Interface Through the Local Definition of the Pressure Tensor and Interfacial Tension[END_REF] In the CG model, the interactions are simplified with a CG particle corresponding to many atoms or molecules. These models are designed to model systems on time scales over hundred of nanoseconds that are not accessible by standard molecular simulations. Among the various CG force fields, the MARTINI force field [START_REF] Marrink | Coarse Grained Model for Semiquantitative Lipid Simulations[END_REF][START_REF] Marrink | The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations[END_REF][START_REF] Monticelli | The MARTINI Coarse-Grained Force Field: Extension to Proteins[END_REF][9][10][11][12] developed by Marrink et al. is by far the most commonly used. This model can be implemented in Monte Carlo (MC) and Molecular Dynamics (MD) simulation methods. Other CG potentials [START_REF] Espanol | Perspective: Dissipative Particle Dynamics[END_REF] are used with mesoscopic methods such as dissipative particle dynamics (DPD). [13][14][15][16] There is an increasing need to use these CG models to calculate the interfacial tension of LV and LL interfaces. CG simulations are also important in the industry and in detergency issues because they can be applied to multi-component mixtures. [START_REF] Ndao | Coarse-Grained Modeling of the Oil-Water-Surfactant Interface Through the Local Definition of the Pressure Tensor and Interfacial Tension[END_REF] A key-property in the simulations of these interfacial systems is the interfacial tension. A recent review 17 shows that some CG force fields present significant deviations of the calculated interfacial tension from experiments. 18 The first simulations of the LV interface of water with the MARTINI CG model 7 produced surface tensions of 45 and 30 mN m 1 at 298 K obtained for small and big system sizes, respectively. For the electrostatic version of the water MARTINI model, 10 the simulated surface tension is found to be 30 mN m 1 while the corresponding experimental value [START_REF] Jasper | The Surface Tension of Pure Liquid Compounds[END_REF][START_REF] Vasquez | Surface Tension of Alcohol + Water from 20 to 50 C[END_REF] is 72 mN m 1 . The MARTINI model produces more quantitative predictions in the case of the LV interface of alkanes. [START_REF] Marrink | The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations[END_REF] The surface tension of water 16 has also been calculated from the MDPD method. 15 In this work, a top-down approach has been carried out to develop of the potential parameters.

Even though the calculation of the surface tension from two-phase atomistic simulations of a slab geometry is now under control, it has been at the heart of several debates over the last 40 years. 17, [START_REF] Liu | Phase Separation of Lennard-Jones Systems: A Film in Equilibrium with Vapor[END_REF][START_REF] Lee | Surface Structure and Surface Tension: Perturbation Theory and Monte Carlo Calculation[END_REF][START_REF] Chapela | Computer-Simulation of Gas-Liquid Surface[END_REF][START_REF] Rao | Surface-Structure of a Liquid-Film[END_REF][START_REF] Chapela | Computer-Simulation of a Gas-Liquid Surface[END_REF][START_REF] Salomons | Atomistic Simulation of Liquid-Vapour Coexistence:Binary Mixtures[END_REF][START_REF] Blokhuis | Tail Corrections to the Surface Tension of a Lennard-Jones Liquid-Vapour Interface[END_REF][START_REF] Guo | Long Range Corrections to Thermodynamic Properties of Inhomogeneous Systems with Planar Interfaces[END_REF][START_REF] Goujon | Direct Monte Carlo Simulations of the Equilibrium Properties of n-Pentane Liquid-Vapor Interface[END_REF][START_REF] Goujon | Dissipative Particle Dynamics Simulations in the Grand Canonical Ensemble: Applications to Polymer Brushes[END_REF][START_REF] Holcomb | A Critical Study of the Simulation of the Liquid-Vapour Interface of a Lennard-Jones Fluid[END_REF][START_REF] Janeček | Interfacial Properties of Cyclic Hydrocarbons: A Monte Carlo Study[END_REF][START_REF] Ibergay | Molecular Simulations of the n-Alkane Liquid-Vapor Interface: Interfacial Properties and Their Long Range Corrections[END_REF][START_REF] Shen | Comparative Study of the E↵ect of Tail Corrections on Surface Tension Determined by Molecular Simulation[END_REF][START_REF] Biscay | Calculation of the Surface Tension from Monte Carlo Simulations: Does the Model Impact on the Finite-Size E↵ects?[END_REF][START_REF] Ferrando | A Transferable Force Field to Predict Phase Equilibria and Surface Tension of Ethers and Glycol Ethers[END_REF][START_REF] Werth | The Inflence of the Liquid Slab Thickness on the Planar Vapor-Liquid Interfacial Tension[END_REF][START_REF] Míguez | Influence of the Long-Range Corrections on the Interfacial Properties of Molecular Models using Monte Carlo Simulation[END_REF] Actually, these slab simulations gave rise to several debates to important methodological problems such as the range of the interatomic interactions, [START_REF] Holcomb | A Critical Study of the Simulation of the Liquid-Vapour Interface of a Lennard-Jones Fluid[END_REF][START_REF] Biscay | Calculation of the Surface Tension from Monte Carlo Simulations: Does the Model Impact on the Finite-Size E↵ects?[END_REF][START_REF] Trokhymchuk | Computer Simulations of Liquid/Vapor Interface in Lennard-Jones Fluids: Some Questions and Answers[END_REF][START_REF] Lopez-Lemus | Thermodynamic and Transport Properties of Simple Fluids using Lattice Sums: Bulk Phases and Liquid-Vapour Interface[END_REF][START_REF] Grosfils | Dependence of the Liquid-Vapor Surface Tension on the Range of Interaction: A Test of the Law of Corresponding States[END_REF] the truncation of the potential, [START_REF] Ibergay | Molecular Simulations of the n-Alkane Liquid-Vapor Interface: Interfacial Properties and Their Long Range Corrections[END_REF][START_REF] Trokhymchuk | Computer Simulations of Liquid/Vapor Interface in Lennard-Jones Fluids: Some Questions and Answers[END_REF] the mechanical and thermodynamic definitions used for the calculation of the surface tension [START_REF] Ibergay | Molecular Simulations of the n-Alkane Liquid-Vapor Interface: Interfacial Properties and Their Long Range Corrections[END_REF][START_REF] Gloor | Test-Area Simulation Method for the Direct Determination of the Interfacial Tension of Systems with Continuous or Discontinuous Potentials[END_REF][START_REF] Ghoufi | Entropy and Enthalpy Calculations from Perturbation and Integration Thermodynamics Methods using Molecular Dynamics Simulations: Applications to the Calculation of Hydration and Association Thermodynamic Properties[END_REF][START_REF] Ghoufi | Multiple Histogram Reweighting Method for the Surface Tension Calculation[END_REF] and the long-range corrections to surface tension [START_REF] Salomons | Atomistic Simulation of Liquid-Vapour Coexistence:Binary Mixtures[END_REF][START_REF] Blokhuis | Tail Corrections to the Surface Tension of a Lennard-Jones Liquid-Vapour Interface[END_REF][START_REF] Guo | Long Range Corrections to Thermodynamic Properties of Inhomogeneous Systems with Planar Interfaces[END_REF][START_REF] Goujon | Direct Monte Carlo Simulations of the Equilibrium Properties of n-Pentane Liquid-Vapor Interface[END_REF][START_REF] Goujon | Dissipative Particle Dynamics Simulations in the Grand Canonical Ensemble: Applications to Polymer Brushes[END_REF][START_REF] Janeček | Interfacial Properties of Cyclic Hydrocarbons: A Monte Carlo Study[END_REF][START_REF] Ibergay | Molecular Simulations of the n-Alkane Liquid-Vapor Interface: Interfacial Properties and Their Long Range Corrections[END_REF][START_REF] Shen | Comparative Study of the E↵ect of Tail Corrections on Surface Tension Determined by Molecular Simulation[END_REF] and to the configurational energy. [START_REF] Guo | Long Range Corrections to Thermodynamic Properties of Inhomogeneous Systems with Planar Interfaces[END_REF][START_REF] Janeček | Interfacial Properties of Cyclic Hydrocarbons: A Monte Carlo Study[END_REF][START_REF] Shen | Comparative Study of the E↵ect of Tail Corrections on Surface Tension Determined by Molecular Simulation[END_REF][START_REF] Míguez | Influence of the Long-Range Corrections on the Interfacial Properties of Molecular Models using Monte Carlo Simulation[END_REF][START_REF] Ghoufi | Multiple Histogram Reweighting Method for the Surface Tension Calculation[END_REF][START_REF] Guo | On the Long-Range Corrections to Computer Simulation Results for the Lennard-Jones Vapor-Liquid Interface[END_REF] An other input parameter that may impact the value of the simulated surface tension is the size of the interfacial area. A number of works have reported the dependence of the surface tension of the Lennard-Jones (LJ) fluid on the surface area. [START_REF] Biscay | Calculation of the Surface Tension from Monte Carlo Simulations: Does the Model Impact on the Finite-Size E↵ects?[END_REF][START_REF] Orea | Oscillatory Surface Tension due to Finite-Size E↵ects[END_REF][START_REF] Gonzalez-Melchor | Stress Anisotropy Induced by Periodic Boundary Conditions[END_REF][START_REF] Velazquez | Finite-Size E↵ects in Dissipative Particle Dynamics Simulations[END_REF][START_REF] Errington | Calculation of Surface Tension via Area Sampling[END_REF] showing the ratio of the surface tension to the surface tension 1 calculated at the largest surface area.

Nowadays the CG models are extensively used to model large and complex interfacial systems where the surface tension is at the origin of the transport of matter through the interface. As a consequence, it is important to investigate the interface area dependence on surface tension to avoid spurious computational e↵ects. To do so, we propose in a first step to study the impact of the surface area on the interfacial tension calculated by the MARTINI Model. We will investigate the LV interface of water and n-octane. We will complete by the LL alkane-water interface. In the CG MARTINI model, the interactions are calculated by both the truncated and shifted LJ and Coulombic potentials. 18 In the case of CG interactions treated with softer potentials, we will extent the study of the size-e↵ects to a mesoscopic simulation method (dissipative particle dynamics, DPD) by using di↵erent shapes of CG potentials. In a first step, we plan to study the size-e↵ects for generic DPD models for the water LV and oil-water LL interfaces. In these cases, we will use the standard conservative interactions of the DPD model and its multibody version. 15,16,[START_REF] Ghoufi | Coarse Grained Simulations of the Electrolytes at the Water/Air Interface from Many Body Dissipative Particle Dynamics[END_REF][START_REF] Ghoufi | Recent Advances in Many Body Dissipative Particles Dynamics Simulations of Liquid-Vapor Interfaces[END_REF] In a second step, we will investigate the size-dependence of the interfacial tension of the LV interface of the n-pentane by using a realistic DPD potential. [START_REF] Canchaya | Development of DPD Coarse-Grained Models: from Bulk to Interfacial Properties[END_REF] Since the MARTINI and the di↵erent DPD models are the most commonly used for the mesoscale modeling of complex systems, the results concerning the size-dependence of the interfacial tension of coarse-grained interfaces will be very useful for the future simulations in this field.

The paper is organized as follows. Section 2 describes briefly the di↵erent potential models, the methods of simulations and the computational details. In Section 3 we discuss the di↵erent finite-size e↵ects of the MARTINI model for di↵erent interfaces. We complete by examining the surface area dependence of the interfacial tension for di↵erent CG DPD models. Section 4 contains our conclusions and recommendations.

Model and simulation methodology 2.1 MARTINI model

In the LV interface of the n-octane, the molecule 6 is described by two beads of type C 1 .

The bonding energy between two beads a and b is calculated by using a harmonic potential

U bond (r ab ) U bond (r ab ) = 1 2 k bond (r ab r 0 ) 2 (1)
where r ab is the distance between beads a and b and r 0 the equilibrium distance. The van der Waals diameter of the CG beads is fixed to 0.47 nm and ✏ is scaled to 75% of the original value of C 1 . The values of k bond , r 0 , and ✏ are listed in Table 1.

The parameters of the polarizable version of the water MARTINI force field 10 are listed in Table 1 et can be found in Ref. 10. In the case of the modeling of the n-octane-water LL interface, the cross-interactions between water (W) and alkane (C1) beads are given in Table 1.

The molecular dynamics (MD) simulations were performed with a modified version of the DL POLY code. [START_REF] Todorov | 3: New Dimensions in Molecular Dynamics Simulations via Massive Parallelism[END_REF] The changes consist of implementing the shift of the forces from r s = 9 Å to r c = 12 Å. The procedure is described in Ref. 18. The simulations of the LV interface were carried out in the constant-NVT statistical ensemble with the Nose-Hoover 54 algorithm using a thermostat relaxation time of 0.1 ps. For the LL interface, the simulations were performed in the constant Np N AT statistical ensemble with the thermostat-barostat Nose-Hoover algorithm. [START_REF] Nosé | Constant Pressure Molecular Dynamics for Molecular Systems[END_REF] The thermostat and barostat relaxation times were fixed to 0.1 and 5.0 ps, respectively. The normal component of the pressure tensor p N was maintained to 0.1 MPa and T = 298 K. The integration timestep was fixed to 10 fs.

The di↵erent bulk liquid phases of octane and water were equilibrated separately during 2 ⇥ 10 7 steps of simulations (10 7 NVT steps followed by 10 7 NpT steps). For the LV simulation, L z is elongated to form the vapor phase. For LL simulations, the bulk phases of the two liquids were joined to form the initial configuration. The equilibration phase consisted of 10 7 steps and the calculation of the average properties was carried out over 10 7 additional steps during the acquisition phase. The simulation time of the acquisition phase is then 100 ns.

DPD method 2.2.1 Generic DPD model

In the DPD method, the total force f i sums three forces: a conservative f C ij force, a random f R ij force and a dissipative f D ij force. The conservative repulsive force

f C ij is f C ij = 8 > < > : a ij ! C (r ij ) rij (r ij < r c ) 0 ( r ij r c ) (2)
where a ij is the repulsion energy between beads i and j, r ij is the interbead distance and rij is the corresponding unit vector. r c is the cuto↵ radius. In our DPD and MMC simulations, the particle mass, the temperature and interaction range were chosen as units of mass, energy and length. The weight function

! C (r ij ) is defined to 1 r ij /r c for r ij  r c
and vanishes for r ij r c . The expressions of the di↵erent forces can be found in Refs 30,51.

The LL interface of the AB system was simulated by placing an equal number particles of A and B in the simulation box. The L z dimension was fixed to 60 in reduced DPD units corresponding to a reduced number density of 3.0. L x and L y were changed from 2.5 to 20.0 in reduced DPD units leading to a variation of a number of particles from 1122 to 72000. The repulsion a AA , a BB and a AB parameters were fixed to 78.0, 78.0 and 105.0, respectively. The equilibration phase was composed of 10 6 steps and the averages were performed over 5⇥10 6 steps during the production phase. The thermodynamic properties were calculated every 5 steps leading to the calculation of profiles of pressure components and surface tension over 10 6 configurations. The z-dimension corresponds the dimension normal to the interface.

Generic MMC model

To model a LV interface, the conservative repulsive potential ! C (r ij ) must be modified in order to consider the local density. In the MDPD approach, the conservative force 15,16,[START_REF] Ghoufi | Coarse Grained Simulations of the Electrolytes at the Water/Air Interface from Many Body Dissipative Particle Dynamics[END_REF][START_REF] Ghoufi | Recent Advances in Many Body Dissipative Particles Dynamics Simulations of Liquid-Vapor Interfaces[END_REF] becomes dependent on the local particle density as

f C ij = A! C (r ij )e ij + B [ ⇢i + ⇢j ] ! d (r ij )e ij (3)
where the first term represents an attractive interaction (A < 0) and the second many- Monte Carlo (MMC) simulations were carried out using the potential defined by Eq.(4).

U C was calculated from the integration of Eq.( 3) where u i is the energy of particle i and A corresponds to a ij in Eq.( 2).

U C = N X i u C i = N X i ✓ ⇡r 3 c 30 A X j6 =i ! ⇢ (r ij , r c ) + ⇡r 4 d 30  X j6 =i b ij ! ⇢ (r ij , r d ) ⇣ ! ⇢ (r ij , r d ) + ⇢ 0 i + ⇢ 0 j ⌘ 2 ◆ (4)
In Eq.( 4) ⇢

⇢ 0 i = X k6 =i,k6 =j  1 r ik r d 2 , ⇢ 0 j = X k6 =j,k6 =i  1 r jk r d 2 (5)
If we assume that b ij = B we obtain the simplified relation

U C = N X i u C i = N X i ✓ ⇡r 3 c 30 A X j6 =i ! ⇢ (r ij , R = r c ) + ⇡r 4 d 30 B  X j6 =i ! ⇢ (r ij , R = r d ) 2 ◆ (6)
The systems were equilibrated during 25 ⇥ 10 6 steps and the average thermodynamic properties were averaged over 25 ⇥ 10 6 additional steps in the constant-NVT ensemble.

The simulation boxes were orthorhombic boxes of dimensions and the coexisting liquid density of water at 298 K.

L x L y L z . The

Realistic DPD model

For the LV interface of the n-pentane, the pentane is described by one bead. In this methodology, the potential ! C (r ij ) was replaced by a tabulated potential which has been developed by using a new bayesian strategy. [START_REF] Canchaya | Development of DPD Coarse-Grained Models: from Bulk to Interfacial Properties[END_REF][START_REF] Dequidt | Bayesian Parametrization of Coarse-Grain Dissipative Dynamics Models[END_REF] The shape of this potential is given for completeness later in the paper in Figure 6a. This CG potential has been used in constant-NVT Monte Carlo simulations with a cuto↵ radius of 20 Å at two temperatures 300 K and 400 K. 5

Results and discussions

As discussed in the introduction of this paper, the truncated Lennard-Jones (LJ) potential is associated with a strong dependence of the surface tension on the surface area. [START_REF] Oliveira | Test-Area Surface Tension Calculation of the Graphene-Methane Interface: Fluctuations and Commensurability[END_REF] Depending on the value of the surface area, the surface tension can vary by ± 20% compared to the value of surface tension ( 1 ) which is defined as the surface tension calculated with the largest surface area (see Figure 1). It means that along this curve, we can observe significant deviations up to 40% between consecutive values of surface tensions. In the case of LJ fluids, it is possible to avoid the size-e↵ects dependence 17 of the surface tension by choosing a surface area of 11 ⇥ 11 2 . We propose now to investigate the surface area dependence of the surface tension of the MARTINI model for di↵erent LV and LL interfaces.

Figure 2a shows the evolution of the surface tension of the LV interface of the n-octane at 300 K. The surface tension is calculated from the integral of the profile along the direction normal to the interface of the di↵erence between the normal and tangential components of the pressure tensor. [START_REF] Kirkwood | The Statistical Mechanical Theory of Surface Tension[END_REF][START_REF] Irving | The Statistical Mechanical Theory of Transport Processes .IV. The Equations of Hydrodynamics[END_REF][START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF] For this system, only the repulsion-dispersion interactions are modeled by the use of LJ potential. After a small oscillation with an amplitude of 3 mN m 1 that represents a variation of about 15% of the value obtained at the largest interfacial area, the surface tension stabilizes around 19. We now investigate the interfacial tension of an oil-water system where the interactions are calculated using a conservative CG model 1 2 (1 r/r c ) 2 potential which is continuous at the cuto↵ radius r c . The simulations were performed using the DPD method. Figure 3a shows the interfacial tension as a function of the interfacial area in reduced DPD units.

For a surface area smaller than 5 ⇥ 5, the surface tension shows small oscillations of around 2% and the convergence is obtained from L ⇤ x = 10. Figure 3b shows the profiles of the normal and tangential components of the pressure tensor for two interfacial areas. These profiles represent the total pressure components which sum the kinetic and configurational contributions. As expected from the mechanical equilibrium, the normal and tangential components are equal in the two bulk phases with a negative peak of the tangential part in 9 the interfacial region. Part c) of Figure 3 displays the profiles of the di↵erences between the normal and tangential components of the pressure tensor at di↵erent areas. On the right axis of Figure 3, the integral of these di↵erences are also shown for completeness. The profiles show the same features independently of the surface area :

p ⇤ N (z ⇤ ) p ⇤ T (z ⇤
) is zero in the bulk liquid phases and present two symmetric peaks at the interfaces. The analysis of ⇤ (z ⇤ ) confirms that the two-phase configurations respect the mechanical equilibrium expected for a planar LL interface. Figure 3c also establishes that the di↵erences in the value of the surface tension cannot be attributed to a lack of convergence of the simulations. By focusing on a specific interface, Figure 3d shows that the profile of p ⇤ N (z ⇤ ) p ⇤ T (z ⇤ ) is subject to small oscillations at the interface for small interfacial areas. The amplitude of these oscillations decreases with increasing surface areas. For larger surface areas, the profiles do not show any oscillations in the interfacial region whereas their shapes in the bulk phases are much more defined leading to a decrease of the anisotropy of the pressure components.

Figure 4 shows the surface tension of water calculated with a density-dependent CG potential at di↵erent surface areas. The parameters of this potential have been optimized through a top-down a procedure to reproduce the surface tension of water calculated by using atomistic models. 16 Simple relationships have been established to link the atomistic and mesoscopic length and time scales and operational parameters have been obtained with the MDPD method to reproduce the surface tension, coexisting densities of water at di↵erent temperatures. The degree of coarse-graining was fixed to 4 indicating that a water bead corresponds to 4 water molecules. We propose in Figure 4 to investigate the dependence of the surface tension of water on interfacial areas changing from 40 Å to 120 Å. To make a link with reduced units, 16 the reduced L ⇤

x should be divided by the cuto↵ r c = 8.52 Å indicating that L ⇤ x changes from 4.7 to 14. This curve shows an oscillatory behavior but with a very weak dependence of on L x . No plateau is clearly identifiable on this curve. The maximum and minimum values of are 73 and 67 mN m 1 , respectively. These deviations correspond to variations of ± 4% compared to the value of 70 mN m 1 over a large range of L x of 80 Å. The observed variations are within the statistical fluctuations of the calculation. Let us return to the use of the standard DPD method. We replace the usual conservative potential by a realistic potential model developed from atomistic simulations. The procedure is thoroughly described in Ref. 56 and its application for the calculation of the surface tension of pentane in Ref. 52. In a first step, we study the e↵ects of L x on the surface tension of the n-pentane at two temperatures. Figure 5a shows that decreases slightly with L x . Actually, decreases from 14.4 to 14.1 mN m 1 with small oscillations leading to a variation of 3% along the range of L x values. As shown in Figure 5b for T = 400K, we observe a monotonic decrease of of 0.45 mN m 1 from 40 to 70 Å. For larger interfacial areas, we observe a plateau extending over 30 Å indicating that the surface tension becomes independent of the interfacial area.

The p N (z) p T (z), shown in Figure 5c profiles confirms that the mechanical equilibrium of the planar LV is verified for the two interfacial areas. Interestingly, for the smallest L x value of 40 Å, we observe the profile which is much noisier in the bulk phase leads to a larger surface tension. However, this di↵erence in the surface tension does not come from the contribution of the bulk phase but rather to the interface region on the liquid side as shown in the profile of (z). The dependence of the surface tension on the longitudinal dimension L z is shown in Figure 5d. The dependence of on L z is clearly weak from 125 Å to 225 Å with an increase of only 0.2 mN m 1 . From L z = 250 Å, we do not observe any monotonic increase of . The main conclusions we can draw with the di↵erent DPD methods, is that they lead to relatively small dependencies of the surface tension on box dimensions.

The di↵erences observed between the di↵erent values are on the order of magnitude of the statistical fluctuations.

We plan now to investigate the e↵ect of the truncation on the = f (L x ) dependence. We do so on using a realistic potential of the pentane developed with a cuto↵ r c = 20 Å. With r c = 20 Å, the potential decreases smoothly to zero with no discontinuity in the potential as shown in Figure 6a. We take the route of truncating this potential at a smaller value of r c = 15 Å introducing then a discontinuity at this point. We represent in Figure 6b, the evolution of the ratio of to 1 at 400 K where 1 means the surface tension calculated with the largest box dimension L x . In both cases, the surface tension decreases with L x but the magnitude of the decrease varies from 30% to 10% as r c is changing from 20 Å to 15 Å. The introduction of the discontinuity in the potential at 15 Å increases significantly the size-e↵ects on the surface tension. This result is in line with observations made with the LJ potential modified with a cubic spline function. [START_REF] Biscay | Calculation of the Surface Tension from Monte Carlo Simulations: Does the Model Impact on the Finite-Size E↵ects?[END_REF] We complete this study by representing on Figure 7 the ratio / 1 for the LV interface of di↵erent CG models as a function of the reduced box dimension L ⇤ x . The reduced value of

L ⇤
x is obtained by dividing the real value by for the LJ and MARTINI models, by r c = 8.52 Å for the water MDPD model. 16 For the pentane DPD model, is defined by the position of the well of the CG potential represented in Figure 6a. Interestingly, we observe that the LJ potential give rise to large amplitudes of oscillations with a plateau obtained with three successive oscillations. The water MDPD model is also subject to significant oscillations that extend over a wide range of L ⇤ x . Indeed, these oscillations are not yet stabilized at 22 even if their amplitude are relatively smaller than those of the LJ fluid. When the MARTINI model is used with only the dispersion-repulsion interactions, e.g in the case of the octane molecule, only one oscillation is observed at 6 . For higher values of the box dimension, there is no more dependence of the surface tension on the surface area. For the mesoscopic simulation methods, the dependence is much less marked. For the water MDPD model, there are small oscillations but we can consider that reduced interfacial areas of 13⇥13 are acceptable to avoid any size-e↵ects in the calculation of the surface tension. Finally, in the case of the use realistic DPD model, we observe rather a small decrease of the surface tension with the interfacial area. We conclude that a reduced interfacial area of 10⇥10 is reasonable to avoid any dependence of the surface tension.

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Since there is a growing demand in the use of CG models for the calculation of the interfacial tension of complex systems and geometries, it was useful to investigate the dependence of the interfacial tension on the surface area for these models. In the case of atomistic models, it is now well-established that the surface tension of Lennard-Jones fluids exhibits an oscillatory behavior as function of the surface area. Relatively large surface area of 11 ⇥ 11 2 are required to provide an accurate calculation of the surface tension by using the LJ potential.

We have performed a large number of simulations with di↵erent CG models, namely, the MARTINI force field, the standard DPD model and realistic DPD and MDPD models. We have also investigated two types of interfaces : the LV and LL interfaces. Firstly, the size-e↵ects are much less strong with the CG models compared to those observed in atomistic simulations. For the MARTINI model, we observe a weak oscillatory e↵ect with small fluctuations for the LV and LL interfaces. This weak dependence is also observed in the LL interface with the standard DPD model. In the case of the MDPD model for water, the LV surface tension exhibits also a weak oscillatory behavior with small amplitudes. When we use a realistic model for the DPD, we observe that the slight oscillatory behavior observed at 300 K is replaced by a monotonic decrease of at T= 400 K. We show here that the truncation impacts the dependence of the surface tension on the surface area as already underlined for atomistic models.

At the end, we give some recommendations to avoid any size-e↵ects of these models by plotting the surface tension as a function of the reduced interfacial area. The di↵erences that appear on the interfacial area dependence with the MARTINI model find their origin in the use of the electrostatic interactions. Figure 7: Ratio of the surface tension to the value 1 calculated at the largest L x box dimensions for the LV interfaces of di↵erent systems and models as indicated in the legend. The horizontal axis is represented in reduced units of L x . For the LJ and MARTINI models, the reduced L ⇤

x is equal to L x / . For the water MDPD model, the reduced box dimension is L x /r c where r c is 8.52 Å (see Ref. 16). For the n-pentane DPD model, L x is divided by where represents the position of the well of the potential (see Figure 6a).

Figure 1

 1 illustrates this dependence for the LJ fluid by

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  body term a repulsive interaction (B > 0). The expressions of the weight functions, the values of the A and B parameters, the values of r c and r d are given in Ref. 16. Multibody

0 i and ⇢ 0 j are defined as 6 Page 6 of 36 ACS

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  interfacial area is defined by A = L x L y where L x = L y . The MMC simulations used a simulation cell constituted by a number of molecules changing from 1000 to 30000 water beads. In this CG model, a bead corresponds to 4 water molecules. This choice implies that the reduced DPD unit of r c corresponds to 8.52 Å. The L x and L y dimensions were changed from 36 Å to 111 Å. The values of the interaction parameters a ij = A and b ij = B are 50 and 25, respectively. These values have been developed in order to reproduce the surface tension 16

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 36 Paragon Plus EnvironmentJournal of Chemical Theory and Computation ⇥10 5 cycles of equilibration were performed to stabilize the interface and additional 2 ⇥ 10 6 cycles were carried out to average the coexisting densities and surface tensions. When L x was changed from 40 Å to 100 Å the total number of beads increased from 1665 to 104411 and L z was fixed to 500 Å. To study the impact of the change in L z between 150 and 400 Å, L x and L y were fixed to 60 Å.

5 mN m 1 .

 1 Any size-e↵ects are detected from a surface area of 35 ⇥ 35 Å2 . The comparison with the corresponding LJ truncated potential shows that the surface tension becomes independent of the interfacial area at smaller values for the MARTINI model (8.5 ⇥ 8.5 2 where = 4.7 Å). When electrostatic interactions are Page 8 of 36 ACS Paragon Plus Environment Journal of Chemical Theory and Computation included for the modeling of the LV of water for example (see Figure2b), we observe that the surface tension is subject to small oscillations of 2 mN m 1 . The last oscillation is much less marked and extends over 40 Å. The largest interfacial area simulated here is 100 ⇥ 100 Å (21 ⇥ 21 2 ). If we expect a calculation of the surface tension with an uncertainty of 6%, we can accept an interfacial area of 60 ⇥ 60 Å2 . To complete the study with the CG MARTINI force field, we represent in Figure 2c the area dependence of the surface tension in the case of the LL interface of octane-water. The amplitude of the oscillations is again around 2 mN m 1 which is in the order of magnitude of the statistical fluctuations. This represents a change in of 5% compared to 1 . A good convergence of the surface tension with the interfacial area is obtained from a surface of 70 ⇥ 70 Å2 . For these three typical interfaces using the CG MARTINI model, we can conclude that the interfacial tension is much less sensitive to size-e↵ects by comparison to the truncated LJ potential. This weak dependence of the surface tension calculated with the MARTINI model can be explained by the fact that this force field uses a truncated and shifted potential avoiding any discontinuity at the cuto↵ radius. We have already underlined this point in Ref. 35 where the LJ truncated potential, modified by a spline polynomial, removes the discontinuity in the force and energy equations and reduces significantly the anisotropy of the pressure components.

Figure 1 : 22 PageFigure 2 :Figure 3 :

 12223 Figure 1: Ratio of the surface tension to the value of 1 for di↵erent L x box dimensions. 1 is calculated with the interfacial area A = L x L y = 46 ⇥ 46 Å2 . The data can be found in Ref. 35 and correspond to the values of methane at T = 120 K. The cuto↵ was fixed to r c = 2.5 .

Figure 4 :Figure 5 :Figure 6 : 27 Page

 45627 Figure 4: Values of surface tension (mN m 1 ) of water calculated with MDPD simulation as a function of the box dimension along the x-dimension.

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Table 1 :

 1 The surface tension is much less dependent on the surface area with the DPD model even if the water MDPD model is much sensitive to Lopez, C. A.; Rzepiela, A. J.; de Vries, A. H.; Dijkhuizen, L.; Hünenberger, P. H.; Marrink, S. J. Martini Coarse-Grained Force Field: Extension to Carbohydrates. J. Dynamics Simulations of the Wetting Properties of Graphitic Surfaces with Non-Ionic, Long-Chain, and T-Shaped Surfactants. J. Chem. Phys. 2012, 137, 094904. (12) Marrink, S. J.; Tieleman, D. P. Perspective on the Martini Model. Chem. Soc. Rev. Phys. Rev. E 2011, 83, 051601. (17) Ghoufi, A.; Malfreyt, P.; Tildesley, D. J. Computer Modelling of the Surface Tension of the Gas-Liquid and Liquid-Liquid Interface. Chem. Soc. Rev. 2016, 45, 1387-1409. Lennard-Jones parameters and electrostatic charges for di↵erent molecules described through the MARTINI force field.

	size-e↵ects.
	13

(13) Hoogerbrugge, P. J.; Koelman, J. M. V. A. Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics. Europhys. Lett. 1992, 19, 155-160. (14) Groot, R. D.; Warren, P. B. Dissipative Particle Dynamics: Bridging the Gap Between Atomistic and Mesoscopic Simulation. J. Chem. Phys. 1997, 107, 4423-4435. (15) Warren, P. B. Vapor-Liquid Coexistence in Many-Body Dissipative Particle Dynamics. Phys. Rev. E 2003, 68, 066702. (16) Ghoufi, A.; Malfreyt, P. Mesoscale Modeling of the Water Liquid-Vapor Interface: A Surface Tension Calculation. ? J. Chem. Theory. Comput. 2015, 11, 3818-3828.

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Philippe Stempflé
email: philippe.stempfle@enit.fr

Olivier Pantalé
email: olivier.pantale@enit.fr

Richard Kouitat Njiwa

Marthe Rousseau
email: rousseam@gmx.netlopez@mnhn.fr

Evelyne Lopez

Xavier Bourrat
email: xavier.bourrat@univ-orleans.fr

Friction-induced sheet nacre fracture: effects of nano-shocks on cracks location

Keywords: sheet nacre, friction, nanowear, nanoindentation, numerical simulation, dynamic impact test

Nacre (the pearly internal layer of molluscan shells) is an attractive nanocomposite displaying high mechanical properties, low density and a good biocompatibility with human bones. It is currently studied for both the prosthesis design and the creation of new organic/inorganic hybrid materials by mimicking biomineralization processes. These exceptional mechanical properties are ascribed to its highly ordered layered 'bricks & mortar' microstructure and more particularly to the energy absorption ability of the mortar during crack propagation. However, this ability appears to be drastically reduced in presence of nano-shocks generated during friction by the dynamic sollicitations. This paper compares two Finite Element simulations -a quasi-static compression test and a dynamic impact test -in order to consider the fracture mechanisms induced by friction. It reveals that cracks migrate from the mortar to the bricks, involving in the latter case, the formation of wear nano-debris. These numerical results are confronted with experimental results during friction.

Introduction

Nacre (mother-of-pearl) is a natural nanocomposite studied for the design of new organic/inorganic hybrid materials by mimicking biomineralization processes [1][START_REF] Vincent | Biomimetics, Biologically Inspired Technologies[END_REF][START_REF] Carlson | Biomimetics, Biologically Inspired Technologies[END_REF]. This biocomposite is generated at ambient temperature and pressure [1] and displays an exceptional high strength [START_REF] Currey | [END_REF]5], stiffness [6,7] and toughness [8] compared to others biological ceramic composites [9]. These exceptional mechanical properties are generally ascribed to its peculiar microstructure [10,11] described as a 'bricks and mortar' arrangement [12] (Figure 1). In this configuration, the bricks (97% in weight) refer to flat crystals of calcium carbonate (CaCO 3 ) -in the crystalline form of aragonite -with thickness less than 500 nm ; and the mortar (3% in weight) is an 'intercrystalline' thin network (about 40 nm) of a biological organic adhesive -mainly composed of β-chitin and silk-fibroin-like proteins [1,7,9]. Besides, the platelet is itself a nanograin-reinforced organic matrix composite (Fig. 1c) because each aragonite platelet is constituted of CaCO 3 nanosized crystals [13,14] surrounded by an 'intracrystalline' organic phase organised as a foam with very thin walls and closed porosity [15][16]. In this paper what is usually described as 'interlamellar' and 'intercrystalline' organic matrix will be call 'intercrystalline' and opposed to 'intracrystalline'. The deformation and fracture of sheet nacre have already been studied by many authors using various quasi-static mechanical tests: tension [6], three-and four-point bending [17,18], shear [18,19] and microindentation [20]. Although their results reveal the existence of different fracture mechanisms depending on the mechanical stresses (sliding of platelets, formation of organic ligaments between platelets [21], crack deflections, crack blunting, crack branching, crack bridging, micro-crack formation, platelets pull-out, diversion); they necessarily involve the 'intercrystalline' organic matrix surrounding the aragonite platelets. Besides, the cracks remain confined inside the organic matrix -or at the interface with the organomineral platelets -and never pass through the platelet. Hence, platelets are never directly cut in quasi-static mechanical tests.

Recently [22] it has been revealed that the fracture of sheet nacre under friction occurs mainly inside of the organomineral platelets involving the formation of nano-debris (size: 20-35nm) even though the stress level required to fracture the platelets is greater than the one for the matrix [8]. In fact, the main difference between the tribological test and the quasi-static tests (tension, compression, bending…) is the presence of repeated nanoshocks induced by dynamic solicitations.

The aim of this work is then to highlight the effects of these friction-induced nano-shocks on the location of the fracture by using numerical simulations. Two Finite Element models -devoted respectively to quasi-static and dynamic loads -are considered. The mechanical properties of each of sheet nacre's components -i.e organomineral platelets and 'intercrystalline' organic phase -are assessed either experimentally or by a numerical identification from a structural model. For this purpose, AFM image analysis will be a suitable tool for characterizing and modelling the multiscale structure of nacre.

Experimental

Samples

Samples are made of sheet nacre extracted from giant oyster Pinctada maxima [22]. They are polished more or less parallel to the aragonite platelets (Ra = 14.5 ± 0.6 nm). The average size of the initial nanograins as determined by AFM image analysis (cf. section 2.4) is 38 ± 21nm. The thickness of the 'intracrystalline' matrix is about 4 nm.

Nanoindentation

The mechanical properties of the samples are obtained with a NHT nanoindenter manufactured by CSM Instruments. It is composed of two elements: an instrumented nanoindenter and an optical microscope (enlargements, ×50 and ×1000). These elements are linked with an electromechanical positioning system (X,Y), which allows a relocation of the sample between the indenter and the microscope within a precision of 0.5µm. The vertical displacement and loading resolutions are, respectively, 0.03 nm and 1 µN. The compliance of the apparatus is 0.25 nm.mN -1 . A 5 µm radius spherical diamond indenter is used (E i = 1141 GPa, ν i = 0.07). A minimum of 30 indentations is carried out for each test.

Tribological tests

We used a pin-on-disc tribotester manufactured by CSM Instruments. The tests are carried out at ambient air and room temperature in dry conditions by repeated friction of a 3.5 mm square shaped pin of nacre against the surface of a polished disc of nacre (∅ 44 mm). The normal load varies from 1 N to 6 N corresponding to a mean contact pressure of 0.1 MPa to 0.5 MPa. The speed and the length of sliding are respectively 10 mm.s -1 and 100 m.

AFM and image analysis

Topography is assessed by using an Atomic Force Microscope Dimension 3000 connected to an electronic controller: Nanoscope IIIa manufactured by Digital Instruments (USA). Its spatial and vertical resolutions are lower than 1 nm and the field depth in-between 100 nm and 100 µm. The micrographs were achieved in high resolution (512 × 512 pixels) by using an intermittent contact mode (Tapping Mode™), which minimises the interactions between the probe and the surface during the acquisition and largely enhances the resolution compared to the contact-mode [23]. A Phase Detection Imaging (PDI) provides the phase contrast maps which improve the detection of the particles edges during the images analysis [24]. The silicon nitride probe is displaying a tip rounding lower than 10 nm. The work frequency, the stiffness and the cantilever amplitude are respectively: 270 kHz, 42 Nm -1 and 25nm. According to the size of the images (between 0.25 µm 2 and 25 µm 2 ), the scanning rate varies from 1 µms -1 to 2.4 µms -1 .

The size, the shape and the volume fraction of biocrystals nanograins were determined, from the phase contrast maps, with a specific algorithm integrated in the SPM data analysis software Gwyddion [25]. Analysis was made from cumulated measurements of at least 5,000 particles (images of 0.25 -1µm 2 ).

Results and Discussion

Tribological tests

Figure 2(a) contains the plots of the friction force vs. sliding distance for different normal loads. The high disturbance of the curves is due to local nano-shocks resulting from the "natural" structure of sheet nacre (Figure 1). A smoothed version of the above curves is presented in Figure 2(b). They reveal the existence of a run-in period, probably associated with the geometric adaptation of the surfaces. The deduced friction coefficient is rather high -i.e about 0.45.

Figure 3 shows a typical AFM view of the friction track after 100 m sliding. It can be observed that the worn surface is strongly degraded by parallel cracks aligned with the structure of sheet nacre. The morphology of these parallel cracks reveals two sorts of typical edge lines:

• On the one hand (Figure 4), the crack is cut clear. Then, the damage mechanism is clearly a brittle fracture around the aragonite platelets which involves the fracturing inside the 'intercrystalline' organic matrix along the interface between adjacent platelets.

• On the other hand (Figure 5), the crack edges are rough and strongly disturbed.

The damage is a kind of crumbling of the aragonite platelets. In this case, the fracture is supposed to occur inside the platelets itself and involves the fracturing of the 'intracrystalline' organic matrix. Although frequently reported in the case of quasi-static mechanical tests [26][27][28][29][30][31][32][33][34], the first mechanism is rarely observed after friction because it is generally followed by the platelet fragmentation [22]. Thus, this second mechanism (Figure 5) appears to be the main damage mechanism observed after sliding.

A Finite Element Analysis should enable to understand this particular friction-induced fracture mechanism. The required mechanical parameters of nacre's components -i.e the organomineral platelet and the "intercrystalline" organic phase -are determined by nanoindentation.

Nanoindentation of organomineral platelets

Figure 6 shows the load-depth indentation curves obtained with a 5 µm radius indenter for 2 mN maximal load. Within this range of deformation a purely elastic behaviour of platelets may be assumed. As the curves are slightly disturbed, the standard analysis methods [35,36] are not really suitable. Hence, a Hertzian contact analytical solution is fitted onto the experimental curves in order to determine the Young's modulus (Figure 7). The average value identified from the whole tests is : E = 62.49 ± 11.26 GPa. 8), the dissipative component corresponds to a plastic deformation involving piling-up phenomena. Hence, the previous Hertzian model enables to extract the limit of the elastic domain -i.e the compressive yield stress (about σ y = 2.7 ± 0.4 GPa).

Figure 6 Experimental load-depth curves performed on aragonite platelets at low loads

Figure 8 Experimental load-depth curves performed on aragonite platelets at high load. Inset spherical imprints after tests

When the normal load is pushed up to 20 mN, the penetration depth becomes higher than 300 nm and the contribution of the "intercrystalline" matrix is clearly observed. The Young's modulus decreases from 62.5 to 54.4 GPa, corresponding to the Young's modulus of the nanocomposite -i.e., platelets + "intercrystalline" organic phase (E = 54.42 ± 3.27 GPa).

Beyond 20 mN, the mechanical properties stay constant (about 54.5 GPa).

Properties of the "intercrystalline" organic matrix

Knowing the mechanical properties of the organomineral platelets (62.5 GPa) and the nanocomposite (54.4 GPa), the mechanical properties of the "intercrystalline" organic phase could be drawn from a structural model. According to Rousseau et al. [15,16], the Pinctada maxima sheet nacre does not present mineral bridges between the platelets. The "intercrystalline" matrix is then continuous. Hence, it can be considered in a first approximation (Figure 9) that the Young's modulus -assessed at high loads spherical nanoindentation -corresponds in fact to the transversal Young's modulus (E T ) of a stacking of layers made up of pure matrix and reinforced layers (mineral and organic).

This transversal Young's modulus is given by the relation [START_REF] Gay | Hermes Lavoisier[END_REF]:

1 1 1 1 1 - ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - - = m f f m f T E V V E E E (1)
where E f , and E m are respectively the Young's modulus of the platelets and the "intercrystalline" organic matrix. V f is the volume fraction of the platelets, considering nacre as a composite of platelets and "intercrystalline" organic matrix. This relation provides a good approximation of the Young's modulus of the matrix (E m ) as a function of E f and E T , determined by nanoindentation respectively using low and high loads.

Figure 9 Illustration of the model used for identifying elastic properties of the 'intercrystalline' matrix (dark grey) vs. those of the platelet (light grey)

Results show that the "intercrystalline" organic matrix Young's modulus is ten times lower than that of the aragonite platelets (Table 1).

Table 1

Identified Young's modulus of the "intercrystalline" organic matrix

E T (GPa) E f (GPa) V f E m (GPa)
54.424 62.489 0.97 6.308

Knowing the mechanical properties of the different components of sheet nacre, a Finite Element analysis is now used in order to understand the specific fracture processes induced by friction. The main difference between the tribological and quasi-static tests is the presence of repeated nano-shocks induced by dynamic solicitations. Therefore, two numerical models will be compared in order to reveal the effects of these nano-shocks on the fracture location.

Numerical Simulation

Numerical models used for FEM simulations

The first numerical model simulates a quasi-static compression test of a sheet nacre specimen while the second one simulates the effect of a rigid impactor on the sample. The damage mechanism associated with both tests is analysed and correlated to the results of tribological tests. The Abaqus Explicit FEM code [START_REF] Hibbitt | [END_REF] is used to solve both problems. The choice of an explicit code is linked to the presence of severe nonlinearities induced both by the large deformations and the behaviour laws used.

Figure10 shows the main characteristics of both models:

• The inclination angle of 8° between the impactor trajectory and the platelets has been evaluated using the AFM facility on the specimens used for the tribological tests (Figure 3).

• In both numerical simulations, the rigid impactor is moved with a prescribed velocity of 10 mm.s -1 -corresponding to the relative tribological test velocity.

• Contact between the impactor and the sheet nacre specimen is assumed to be frictionless and without any influence on the results because contacting surfaces exhibit only a normal relative displacement during simulation.

• 6800 and 4100 elements are used in both models respectively. The "intercrystalline" organic matrix is meshed using two elements of 12.5 nm thickness.

• All displacements have a prescribed value of zero for the left and bottom edges in both models.

• From one model to the other one, only the boundary conditions linked to the rigid impactor vary as it will be detailed further.

• To simulate the fracture inside both materials, a ductile damage law based on the maximal equivalent plastic strain value (ε max ) is used.

• This damage law is coupled with the erosion algorithm of the Abaqus Explicit code: when the ductile damage criterion is reached, the corresponding element is considered instantly completely damaged, and is removed from the FE model.

Only the initiation part of this damage law is used in the simulations.

Figure 10 Geometrical description of the models used for both numerical simulations

The maximum strain value (ε max ) of the "intercrystalline" organic phase (10%) is extracted from the work of Ji and Gao [10].

Concerning the organomineral platelet, the plastic deformation is assumed to be controlled by the "intracrystalline" organic matrix surrounding the CaCO 3 nanograins (see Figure 11). Therefore, the maximum strain value of the platelet is estimated from the volume fraction of the organic matrix, within the platelet, measured by AFM image analysis -about 12% in 2D corresponding to 4.2% in 3D. This value is close to those obtained by biological extractions [START_REF] Bourrat | In line at CrystEngComm[END_REF]. Therefore, the previously unknown maximum strain value ε max of the platelets has been estimated to be about 12% of the one of the "intercrystalline" organic matrix (ε max = 10%) because both presented models are bidimensionnal ones.

Material mechanical properties used in both simulations are compiled in Table 2.

Table 2

Identified mechanical properties (a) from Ji and Gao [10])

E (GPa) σ y (GPa) ν ε max (%) [A] 62.5 2.5 0.2 1 [M] 6.3 0.116 0.3 10 a)

Quasi-static compression test

The initial contact length between the aragonite platelet and the rigid impactor is 100 nm i.e. 20% of the total platelet height. The contact zone contains two elements. An initial velocity equal to the impactor velocity is applied to all nodes of the contacted platelet in order to suppress the shock component at the beginning of the simulation. The velocity direction is parallel to the aragonite platelets arrangement.

Figure 12 shows the von Mises contourplot for a total displacement d = 30 nm of the rigid impactor. It can be observed that the "intercrystalline" organic matrix is completely fractured while the platelet remains intact after deformation and becomes free. This is not surprising because the mechanical properties (Table 2), and more precisely the yield stress σ y of the platelet [A] are higher than that of the biological "intercrystalline" organic matrix [M]. The same results have been obtained in traction. Hence, results from quasi-static tests, lead to the conclusion that high shear region and crack propagation may only be located inside of the organic matrix as already reported by both the experimental [4-9, 11, 17-20, 26-33] and other numerical works [10,32,[START_REF] Wu | [END_REF][41][42].

Note that, the explicit integration algorithm allows us to obtain the complete fracture of the organic matrix without rigid body problems. Fracture lines within nacre are clearly represented by the use of the erosion algorithm coupled with the damage law.

Dynamic impact test

The initial contact length between the aragonite platelet and the impactor is now 50 nm. The mesh has been refined locally in order to keep at least two elements in the contact zone. The retained approach is similar to the one proposed by Pantalé et al. [43] for the numerical simulation of a machining process.

Figure 14 shows the von Mises stress contourplot for a total horizontal displacement d = 125 nm of the impactor after the first contact with the platelet. Numerical results show a concentrated region of high stresses in the vicinity of the contact zone between the impactor and the platelet as reported in [43]. As a main consequence of the dynamic effects, and the presence of repeated shocks, the fracture zone illustrated in Figure 14 is totally different from the one reported in Figure 12. Major part of the fracture is now located inside of the aragonite platelet, in accordance with experimental observations using tribological tests. Stress distribution, reported in Figure 14, shows that during the impact, the ratio between the equivalent stress and the yield stress ( y σ σ

), inside of the biological "intercrystalline" organic matrix, is lower than the one inside of the platelet. Therefore, fracture propagation inside of the "intercrystalline" organic matrix is limited and platelet debonding does not occur as in the previous simulation.

Figure 14 von Mises contourplot for the impact test for a total displacement d = 125 nm (for colours see online version)

Different simulations have shown that the ratio between the numbers of fractured elements inside of the platelet and inside of the organic matrix increases with decreasing height of the contact zone -i.e when the shearing depth decreases. This usually occurs in tribological tests.

Figure 15 shows the time-history evolution of the von Mises stress for three selected elements A, B and C above the trajectory of the impactor during the simulation. These three points correspond respectively to the second, the sixth and the tenth element removed during the simulation. The analysis of the time-history plot in Figure 15 shows that the equivalent von Mises stresses for the three selected points increases linearly for a displacement up-to d = 11.5 nmi.e the instant when the damage criterion is reached in the first element. Then, as the first element is removed from the computation, the stress level vanishes until the impactor reaches the second element (element A). Then, the stress level increases in element A until the damage criterion is reached and the element is removed. The process is repeated during all the simulation, as represented in Figure 15.

Figure 15 Time-history analysis for the impact test (for colours see online version)

Thus, stress peaks (Figure 15) enable to satisfactorily explain the formation of nanometric wear debris -observed by AFM on the friction track (Figure 16) -which have a size very close to the initial nanograins constituting the platelets -about 40 nm (Figure 11). Hence, friction-induced nano-shocks seem to be responsible of possible nanoscale wear mechanism.

Conclusion

In order to consider the fracture mechanisms induced by friction and more particularly the location of the fracture inside the nanocomposite sheet nacre, this paper compares two Finite Element models: a quasi-static compression test and a dynamic impact test. Results reveal the following:

• in quasi-static mechanical tests (traction, compression, bending, shearing), high shear and crack propagation may only be located inside the 'intercrystalline' organic matrix • in impact tests, repeated shocks induced by a dynamic effect modify the location of fracture: cracks are observed inside the mineral platelets instead of the 'intercrystalline' matrix.

Since tribological tests also involve nano-shocks, these results rationalize the formation of wear nano-debris generated by the fracture of the mineral platelets. Thus, in dry friction, the wear behaviour is not only controlled by the 'intercrystalline' but also by the 'intracrystalline' organic phase surrounding the CaCO 3 nanograins.

Figure 1

 1 Figure 1 (a) Schematic cross section of the friction surface. Multiscale structure of sheet nacre; (b) aragonite platelets are surrounded by an 'intercrystalline' organic matrix and (c) each platelet is constituted by nanosized CaCO3 crystals (or nanograins) surrounded by an 'intracrystalline' organic phase

Figure 2 Figure 3 Figure 4 Figure 5

 2345 Figure 2 Variations of the friction force with the sliding length for various applied loads: (a) raw data and (b) smoothed with a 10 points adjacent averaging algorithm (for colours see online version)

Figure 7

 7 Figure 7 Typical load-depth curve and numerical Hertzian model allowing the identification of elastic mechanical properties of aragonite platelets

Figure 8

 8 Figure 8 shows the load-depth curves obtained for 10 mN maximal load. At this load, a dissipative component is clearly observed under the load-depth curves. As shown in the

Figure 11

 11 Figure 11 Typical AFM views of the polished surface of an aragonite platelet observed in tapping-mode topographic (a) and phase contrast maps (b) revealing the nanosized grains of CaCO3 surrounded by the 'intracrystalline' organic matrix (for colours see online version)

Figure 12 Figure 13

 1213 Figure 12 Von Mises contourplot for the quasi-static compression test at the end of the fracture process (for colours see online version)Figure13reports the time-history evolution of the von Mises stresses at four selected points within the "intercrystalline" organic matrix during the simulation. The von Mises stress level increases from zero up-to the yield stress σ y for all selected points. When the damage criterion is reached, due to accumulation of plastic deformation, the element is deleted from the computation process and the stress vanishes. During the simulation, the number of viable elements surrounding the platelet decreases as the stress inside of the organic matrix increases. Oscillations observed on the time-history curves of the points farther from the contact zone are purely numerical and due to the removing of elements algorithm. Crack initiates at point A and propagates through points B, C and D respectively.

Figure 13

 13 Figure 13 Time-history analysis for the quasi-static compression test (for colours see online version)

Figure 16

 16 Figure 16 Typical view (1 × 1 μm) of the wear nano-debris observed on the friction track: (a) height and (b) phase (for colours see online version)

Acknowledgments

Authors wish to thank C. Poigneau from SIERA SA for having supplied them with the nacre samples.