D Zheng

P Roumanille

P Hermet

M Cambon

J Haines

O Cambon

Enhancement of the piezoelectric effect in Fesubstituted GaAsO 4 : a combined XRD, Raman spectroscopy and first principles study

GaAsO4 is the highest performance piezoelectric material of the M III X V O4 alpha-quartz type group.

First principles based calculations of the pure FeAsO4 show that the piezoelectric properties are strongly improved by replacing Ga 3+ by a chemical element of the d-bloc like Fe 3+ . For GaAsO4, our calculations give, d11 =7.08 and d14 =5.76 pC/N, while these values are significantly higher in FeAsO4 to reach d11 =18.77 and d14 = 13.78 pC/N; about a factor of three greater than in GaAsO4 due to the increase of the elastic compliances of FeAsO4 with respect to GaAsO4. These results show that the electronic configuration and particularly the involvement of the d-orbitals in the M-O bonds in the M III O4 tetrahedron could be at the origin of this increasing.

By considering a linear dependence of d11 or d14 between the parent compounds, GaAsO4 and FeAsO4, we get the following equations for the Ga(1-x)FexAsO4 solid solution: d11 [pC/N] = 11.689 xFe + 7.082 and d14 [pC/N] = 8.021 xFe + 5.762. These predictions motivated the first synthesis of mixed Ga(1-x)FexAsO4 solid solutions with x= 0,09; 0,18 and 0,45. The hydrated compounds, Ga1-xFexAsO4,2H2O, were obtained by hydrothermal synthesis and the alpha-quartz phases Ga1-xFexAsO4 were crystallized with a thermal treatment in an oven at 600°C for 24h. XRD patterns show that the cell parameters are in accordance with a Vegard's law for the orthorhombic hydrated phases (Pbca space group) and for the trigonal dehydrated alpha-quartz phases (P3121 space group). Raman spectroscopy data obtained for the hydrated phases confirm the results of the literature on FeAsO4•2H2O, GaAsO4•2H2O and other arsenate analogues as well as their substituted analogs on the M and X sites. For the dehydrated alpha-quartz phases, all the main peaks observed in Raman spectra have been observed and the assigned by analyzing the experimental spectra measured in solid solutions and the calculated spectra in the parent end-

member compounds. DFT calculations were used to identify the Raman vibrational modes. As in the case of GaAsO4, no libration mode of oxygen atoms is present for FeAsO4 that could lead to a very high thermal stability. These results prove the interest of -quartz type FexGa1-xAsO4 solid solutions.

INTRODUCTION

Stability is one of the most important properties of piezoelectric materials from the point of view of current research and future applications 1,2 . Physical properties of piezoelectric materials, such as dielectric, piezoelectric and electromechanical coupling coefficient etc., are often restricted by the thermal stability 3,4 . Starting from the commonly used piezoelectric material, α-quartz itself, and GeO2 to members of the α-Quartz-type family compounds with the general formula MXO4 (M= Al, Ga, Fe, B, and X=P and As), research on their crystal structures, crystal growth conditions and piezoelectric properties has been carried out over the past decades 5, 6 . However, there are still some existing barriers limiting their piezoelectric performances and potential applications at high temperature 7,8 . This has motivated the study of MXO4 compounds. The α-quartz homeotypes MXO4 belong to the P3121 or P3221 space groups. Their piezoelectric properties and thermal stability are closely related to the structural distortion of the α-phase with respect to the β-phase, and this structural distortion can be described by the intertetrahedral bridging angle  and the tetrahedral tilt angle δ. Indeed, with the recent advances in research, some important empirical structure-property relationships for α-quartz homeotypes have been established between thermal stability, piezoelectric, physical properties and the distortion with respect to the β-quartz structure 9 . The piezoelectric coupling coefficient of these α-quartz homeotypes materials was found to be a linear function as the structural distortion with respect to β-quartz structure and it increases with the structural distortion. For example, SiO2 (θ=144.8°, δ=16.3°) and AlPO4 (θ=142.8°, δ=17.6°), have phase transition temperatures of 573 and 586 °C respectively 10,11 , along with a higher degree of structural distortion (the bridging angle θ reduces while tilt angle δ increases) and their electromechanical coupling coefficient k increases from 8% to 11%. Similarly, FePO4 (θ=137.8°, δ=21.5°) and GaPO4 (θ=134.4°, δ=23.3°), which have a similar structure distortion, could have a similar piezoelectric coupling coefficient at room temperature of about k=16%. These two materials also have a good thermal stability up to temperatures of 707 and 927 °C respectively 12- 14 . It is also reasonable to predicted that GaAsO4, which possess the most distorted crystal structure (θ=129.6°, δ=26.9°), should exhibit the highest thermal stability (up to 1300 K) and the highest piezoelectric coupling coefficient (about 22%) 15 . Based on the above results, this paper describes the investigation of the most promising piezoelectric material by use of the chemical doping method, as chemical doping is a powerful tool to study physical and chemical properties of materials. It can bring about changes in conductivity, bond lengths, bond angles, lattice distortions and lattice parameters of the target materials depending upon the synthesis conditions. Accordingly, we used hydrothermal methods for preparing the Ga1-xFexAsO4 series using iron lying in the same period of the periodic table of elements with a slightly larger radius than that of gallium to systematically dope or substitute gallium in the crystal structure of GaAsO4. In fact, increasing the average ion size of M site cations of MXO4 is one of the main ways to modify the structural distortion. Thus, we can investigate the important relationship between the crystal structural distortion and the chemical composition of the solid solution obtained in the process of materials synthesis and then integrate the design and synthesis of such tunable structures with adjustable physical properties in target α-quartz homeotype piezoelectric materials. All these adjustable intrinsic properties of piezoelectric materials will provide excellent performance for future applications in high-temperature environments and in micro-electronic components requiring high frequencies and consequently miniaturization. Even in the field of second harmonic generation (SHG) these materials may be of great interest due to their expected nonlinear optical (NLO) properties [START_REF] Hermet | FULLPROF, a Rietveld refinement and pattern matching analysis program[END_REF] .

EXPERIMENTAL AND METHODS SECTION

Sample Preparation.

Powders of the Ga1-xFexAsO4 (x =0.11, 0.18, 0.60) solid solution series were synthesized by hydrothermal methods and followed by a subsequent heat treatment process. 7.7g (0.075mol) GaOOH and 40g (0.1mol) Fe(NO3)3,9H2O were separately dissolved in the arsenic acid solution (25ml H3AsO4 + 25ml H2O). Next, when the two solutions were clarified, they were mixed them under continuous magnetic stirring. Finally, this mixture was transferred in a PTFE-lined autoclave, maintained at 170°C for 3 days and then gradually heated at about 5 degrees per day up to 245°C.

After rapid cooling, the solid precipitate was filtered, washed several times with distilled water and absolute ethanol and dried at 60 °C yielding the hydrated phases. The anhydrous α-quartz-type phases were obtained by heating the above powders in an oven at 600°C for 24h.

SEM AND EDX

The synthesized products were characterized using a scanning electron microscope (SEM) equipped with an Energy Dispersive X-Ray (EDX) analyzer to determine the chemical composition of the sample. A Quanta 200 field emission gun (FEI) equipped with an SDD diode was used as detector (Oxford INCA). Spectra were measured under a low vacuum at 15 kV. At least three different areas were checked for each sample. In this paper, the iron content x of three compositions studied was 0.11, 0.18 and 0.60 respectively. The sample Ga0.40Fe0.60AsO4 and its hydrate are presented as examples with their EDS and SEM results in Figure S1 and Tables S1 (Supporting Information).

Powder XRD

The samples were characterized by X-ray diffraction (XRD) on a PanAnalytical X'Pert diffractometer equipped with an X'Celerator detector using Cu-K radiation (K = 1.5405 Å, K = 1.5444 Å). To eliminate the fluorescence due to the excitation of the iron in samples, a graphite monochromator was placed between the sample and the detector. The powder samples, which had been ground and passed through a 20-µm sieve, were placed in a rotating sample holder.

The angular range investigated was as follows: 2θ = 7°-100°. Acquisition times were approximately 12 hours with a 0.008° step size. Le Bail fits and Rietveld refinements were performed with the FULLPROF program 17 . Quantitative analysis by two-phase Rietveld refinement indicated that the true compositions of the solid solutions are (x=0.09, 0.18, 0.45) with the remaining iron being found in a secondary pure iron end member phase (Details in SI).

Raman spectroscopy

Raman measurements were performed at room temperature on a Horiba Jobin-Yvon Labram Aramis Raman spectrometer equipped with a blue diode laser (λ = 473 nm). The powder was placed on a glass plate and the laser power was ~2 mW. The spectrometer was calibrated using samples of Si and GaAsO4. The Ga1-xFexAsO4 Raman lines were fitted using pseudo-Voigt functions. The spectral resolution is about 1cm -1 .

First principles based calculations

Calculations for GaAsO4 and FeAsO4 were performed within the density functional theory (DFT) framework as implemented in the ABINIT package [START_REF] Gonze | [END_REF] . The all-electron potentials are replaced by norm-conserving pseudopotentials generated according to the Troullier-Martins scheme. Ga (4s 2 , 3d 10 , 4p 1 ), Fe (3d 6 , 4s 2 ), As (4s 2 , 4p 3 ) and O (2s 2 , 2p 4 )-electrons are considered as valence states. The electronic wave functions are expanded in plane-waves up to a kinetic energy cutoff of 55 Ha. Lattice parameters and atomic positions were fully relaxed using a Broyden-Fletcher-Goldfarb-Shanno algorithm until the maximum stresses and residual forces were less than 7×10 -4 GPa and 6×10 -6 Ha/Bohr, respectively. Born effective charges, dielectric tensors, piezoelectric tensors and dynamical matrix (yielding the phonon frequencies and eigenvectors) were obtained within a variational approach to density functional perturbation theory whereas the Raman susceptibilities were calculated from a nonlinear response formalism taking advantage of the 2n+1 theorem 19 . GaAsO4 was studied using the generalized gradient approximation (GGA), according to the Perdew, Burke and Ernzerhof (PBE) parametrization 20 , and a 10×10×6 mesh of special k-points 21 . This exchange-correlation functional gave an excellent prediction with the experimental data for the study of crystallographic structure 22 , lattice dynamics and piezoelectric properties in non-magnetic -quartz-type M III X V O4 compounds 6 with M = B, Al, Ga and X = P, As. In the case FeAsO4, the primitive unit-cell has been doubled along the c-axis to impose an antiferromagnetic order. We found a magnetic moment on iron of 4.15 , which is in good agreement with the reported value 23 of 4.53  measured at 5 K for FePO4. These values are consistent with a high spin 3d 5 Fe 3+ ion (5.92 ) when extrapolated the experimental magnetic moment of FePO4 at 0K. We used the local density approximation (LDA) according to the Perdew and Wang (PW) parametrization 24 and a 6×6×2 mesh of special k-points 21 . This change of functional with respect to GaAsO4 is supported by a better description of the FeAsO4 crystal structure that is the only experimental data known for this compound. Indeed, the LDA/PW relaxed lattice parameters of FeAsO4 (a = 5.050 and c = 11.626 Å) are underestimated by less than 2% with respect to the experimental ones 25 (a = 5.154 and c = 11.470 Å). In contrast, the GGA/PBE relaxed ones (a = 5.277 and c = 12.084 Å) are overestimated by more than 5 % along c.

Considering the strong correlation between the crystal structure and the piezoelectric efficiency in -quartz-type M III X V O4 compounds 6 , prediction of the piezoelectric constants in FeAsO4 could be less reliable at the GGA/PBE level. The piezoelectric constants of Ga1-xFexAsO4 solid solutions have been estimated supposing a Vegard's law between the two parent compounds. The orthorhombic lattice parameters a, b and c were calculated using Le Bail fits over the range 10°< 2θ < 90°. The values of a, b and c are plotted as a function of the Fe concentration x in the Ga1-xFexAsO4,2H2O solid solution as shown in Figure 2(a). Among them, the data of the end members GaAsO4,2H2O and FeAsO4,2H2O come from the literature for comparison 26,27 . Actually, the lattice parameters almost showed a linear change generally in line with Vegard's law for the present solid solution series. The small negative deviations from the Vegard's law are extremely small for x<0.25, while discernibly positive for x>0.25, resulting in the slight "S" shape of the unit-cell parameter-composition curves. Similar behavior occurs for the dependence of unit cell volumes V as a function of the Fe concentration in the solid solution as presented in Figure 2(b).

RESULTS AND DISCUSSION

Hydrated Phases a. X-ray Powder diffraction

It is evident that the lattice parameters and unit cell volume of the synthesized phases are found to vary continuously as a function of composition between GaAsO4,2H2O and FeAsO4,2H2O end members. Because the non-bonding radius of Fe 3+ (1.68 Å) is slightly larger than that of Ga 3+ (1.63 Å), so it can reasonably deduced that the Fe-As distance should also be greater than that of Ga-As when Fe 3+ ions are used to substitute Ga 3+ in the GaAsO4,2H2O structure. The lattice parameters and cell volume correspondingly increase with x giving rise to lattice expansion similar to previous reports on -quartz-type homeotype phosphate solid solutions [28][29][30][31] .

Raman Spectroscopy

The hydrated phases Ga1-xFexAsO4,2H2O (x =0.09, 0.18, 0.45) were studied by Raman spectroscopy at room temperature. Generally, metal arsenates M III AsO4,2H2O (M III : Ga or Fe) belong to the orthorhombic space group Pbca (= D2h). The AsO4 3-groups occupy the C1 site symmetry in the crystal and the corresponding factor group analysis theoretically yields 9A modes, altogether 36 Raman active modes classified as Γ(AsO4 3-) = 9(Ag(Ra) + B1g(Ra) + B2g(Ra) + Au(Silent) + B1u(IR) +B2u(IR) + B3u(IR)). According to previous related research work 32,33 , the arsenate bending and stretching vibrations modes of the gallium(III) arsenate dihydrate (GaAsO4,2H2O) and iron(III) arsenate dihydrate (FeAsO4,2H2O, scorodite) can all be distinctively observed in the Raman spectrum between 327 and 484 cm -1 and the region from 800 to 900 cm -1 , respectively. However, the position and intensity of their Raman lines also exhibit obvious differences in these two regions. For GaAsO4,2H2O, the strongest intense band occurring in the Raman spectrum at 894 cm -1 assigned to the ν1(A1) totally symmetric stretch and the bands at 861, 837, and 826 cm -1 ascribed to the ν3(F2) antisymmetric stretch. However, for FeAsO4,2H2O, Significant changes in intensity are observed for the ν1 (A1) and ν3 (F2) arsenate modes between 800 and 900 cm -1 region. It appears that with the increase of iron content, the Raman line around 890 cm -1 gradually weakened and was no longer the most intense band; at the same time, the Raman line intensity near 810 cm -1 increased and became the strongest peak. If we compare the Raman line in this region with their parent compound (GaAsO4,2H2O) and the end member (FeAsO4,2H2O), our iron-doped products Ga1-xFexAsO4,2H2O (x =0.09, 0.18, 0.45) perfectly demonstrated intermediate cases concerning the peak intensities of the Raman active stretching modes observed between 800 and 900 cm -1 . This explains very well when the gallium in the crystal structure of GaAsO4,2H2O is gradually replaced by iron. In the Raman spectrum the intensity of the original ν1 (A1) (totally symmetric stretch vibration) peak gradually decreases, while the intensity of ν3 (F2) (antisymmetric stretch vibration) peak increases. All this will cause the original ν1 (A1) of GaAsO4,2H2O to weaken and be replaced by the antisymmetric stretching vibration of the FeAsO4,2H2O, and the original ν3 (F2) of the GaAsO4,2H2O is replaced by the strongest peak, that is the symmetric stretch vibration of FeAsO4,2H2O. This also in line with the point of view of Nakamoto et al. [START_REF] Nakamoto | Infrared and Raman spectra of inorganic and coordination compounds. Part A: theory and applications in inorganic chemistry[END_REF] and Gomez et al. 33 , that the symmetric stretching vibrations, which is a breathing type of mode, are always stronger than the antisymetric vibrations in terms of Raman intensity.

The ν1 (A1) (totally symmetric stretch vibration) peak should thus exhibit the strongest Raman intensity with respect to the ν3 (F2) (antisymmetric stretch vibration) peaks. This fact has been confirmed by many articles on the Raman spectroscopy of FeAsO4,2H2O, GaAsO4,2H2O and other arsenate analogues, as well as their substituted analogs on the M and X sites, which behave as those in the present study.

External and Lattice Modes

Correspondingly, for the bending modes, from the Table 1 we can see that the ν4 (F2) antisymetric mode of GaAsO4,2H2O and FeAsO4,2H2O are located at 460, 433 cm -1 and 483, 450, 420 cm -1 according to the references, while the ν2 (F2) vibrations are located at 394,346, 327 cm -1 and 380, 340 cm -1 , respectively. We can also observe the Raman spectra of all the above arsenate bending ν4 (F2) and ν2 (F2) modes as well as the external and lattice modes for our synthetic iron-doped dihydrate compound Ga1-xFexAsO4,2H2O (x =0.09, 0.18, 0.45) are located just between their parent compound (GaAsO4,2H2O) and the end-member compound (FeAsO4,2H2O), which is showing a perfect evolution of the trend. The detailed stretching, bending vibration modes and lattice modes of the arsenate in different regions of the Raman spectrum for synthetic dihydrate compounds Ga1-xFexAsO4,2H2O (x =0.09, 0.18, 0.45) are provided in Figures S3 andS4 (Supporting Information).

Dehydrated

Phases with Alpha-Quartz Structure

XRD data

Fig. 4 XRD patterns of the as prepared Ga1-xFexAsO4 (see text for meaning of the asterisks)

Heating the above hydrated phases Ga1-xFexAsO4,2H2O in an oven at 600°C for 24h leads to the loss of their crystalline water yielding the corresponding dehydrated α-quartz-type trigonal (P3121 space group) phases Ga1-xFexAsO4 (x=0.09, 0.18, 0.45). The X-ray diffraction patterns of these dehydrated phases and the anhydrous GaAsO4 for comparison are presented in Figure 4. From above XRD pattern, we can see that the diffraction patterns are very similar to that of the trigonal (P3121) GaAsO4 end member. However, there are several additional Bragg reflections (marked with asterisk in figure 4) in the diffraction patterns for iron contents of x= 0.09 and x=0. 45, respectively, which correspond to reflections of monoclinic FeAsO4 (See SI). This may suggest that the monoclinic FeAsO4 phase found by XRD analysis is mainly derived from the dehydration of the corresponding hydrated phase, which loses its structural water during the higher temperature treatment and finally leads to a nucleation and growth decomposition mechanism 31,32 . At the same time, it is evident that all diffraction peaks shift to lower 2θ values with increasing x in Ga1- The detailed effects of this incorporation of iron on the unit-cell parameters of the Ga1-xFexAsO4 trigonal (P3121) solid solutions are shown in Fig. 5. Among them, the data of the end members GaAsO4 and FeAsO4 come respectively from the literature 25 and theoretical calculations for comparison. Firstly, it can be clearly seen from Fig. 5 (a) that the lattice parameter c/a ratio of the solid solutions decreases monotonously with x, while the behavior of the unit cell volumes is opposite, increasing with x (Fig. 5 (b)). This means that the lattice of Ga1-xFexAsO4 expands as Fe 3+ with a larger ionic radius substitutes for Ga 3+ in GaAsO4, but the increase of the a parameter value is larger than that of c parameter. However, the unit cell volume does not vary linearly with the composition x between the GaAsO4 and FeAsO4 end members nor does the lattice parameter c/a ratio. In general, the values of c/a lie a little slightly above that expected by linear interpolation between limiting values for GaAsO4 and FeAsO4, while the volumes display a fluctuation with changes in iron content. Secondly, in detail, the deviation from the Vegard's law are extremely small for the trigonal (P3121) solid solutions with x=0.09 and 0.18, but for the sample with x = 0.45, the deviation is a little larger. However, when the GaAsO4 and FeAsO4 end members are excluded, the unit cell volume of Ga1-xFexAsO4 (x=0.09, 0.18, 0.45) exhibits an approximate linear increase as a function of composition x (Fig. 5 (b)). Upon incorporation of the larger Fe 3+ cations on Ga 3+ sites in the Ga1-xFexAsO4 under the current reaction conditions, the increasing of the M-As non-bonding distance leads to a larger unit cell volume and further to a decrease of degree of structural distortion, which corresponds to an increase in the inter-tetrahedral bridging angle θ and a decrease in the tetrahedral tilt angle δ with x. This approximate linear evolution relationship of the unit cell volume as a function of x (Fig. 5 (b)) is especially useful to estimate the iron content in the Ga1-xFexAsO4 solid solutions.

Raman Spectroscopy

According to group theory predicts, Ga1-xFexAsO4 belongs to space group trigonal P3121 or P3221 (Z = 3) and will exhibit 54 modes of vibration: Γ= 8A1 + 10A2 + 18E Among them 8A1 + 17E are predicted to be Raman active. The resulting Raman measurements of the Ga1-xFexAsO4 samples along with the spectra of undoped compound GaAsO4 for comparison are shown in Figure 6.

Fig. 6 Raman spectra of Ga1-xFexAsO4 solid solutions as a function of iron content

Although the profiles of these spectra are very similar, there are still a few significant changes in the position and the relative intensity of several peaks as the iron content increases. In order to better understand these subtle changes, we divide the spectra into three wavenumber ranges: (i) A series of weaker lines below 400 cm -1 , (ii) a very intense line along with several weak lines between 400 and 600 cm -1 , and (iii) several strong lines with significant changes in intensity and line width within the 800-1000 cm -1 range. According to previous reports [START_REF] Bhalerao | [END_REF]40 , for GaAsO4, similar to other related α-quartz-type materials, such as FePO4 and GaPO4, the A1 mode near 260 cm -1 involving oxygen displacements is not a libration, but is a more complex mode relate to bending of the various inter-and intratetrahedral angles caused by different components of O atoms displacements in different planes in the structure 41 . The Raman modes (cm -1 ) of α-quartz-type Ga1-xFexAsO4 solid solutions as a function of composition are listed in Table 2.

Table 2. Main frequencies (in cm -1 ) observed in Raman spectra of α-quartz-type Ga1-xFexAsO4 solid solutions.

Proposed Assignment x = 0 Calc.

Exp.

x = 0.09 Exp.

x = 0.18 Exp.

x = 0.45 % Exp.

x Most of Raman lines exhibit a progressive change in wavenumber as the iron content increases.

Figure 6 shows the evolution of the position of the Raman lines. The Raman shifts of the A1 mode centered at 260 cm -1 and bending modes at 425 cm -1 have a slight downshift with increasing x, which is related to the weakening of the M-O bonds.

The DFT calculations demonstrated the absence of pure libration of the oxygen atoms in GaAsO4

16 and FePO4 39 . The calculated low frequency A1 mode at 226 cm -1 for GaAsO4 and 193cm -1 for FePO4 involving the simultaneous displacement of the cations and the oxygen atoms. This mode exhibits strong damping prior to the - quartz transition in FePO4 linked to a high degree of dynamic disorder. In the case of FeAsO4 (Figure 7), the pure libration mode of the oxygen atoms is also absent.

Fig 7.

Calculated A1 vibrational modes of FePO4, FeAsO4, and GaAsO4 at 0 K. Large purple, medium yellow, and small red spheres represent (Fe, Ga), (P, As) and O atoms, respectively.

Piezoelectric properties

In the case of GaAsO4 and FeAsO4 α-quartz-type compounds, the piezoelectric-stress tensor has only two independent elements, namely e11 = -e12 = -e26 and e14 = -e25. Their magnitudes, given in Table 3, show that the substitution of Ga by Fe-atoms has only a small effect on the overall piezoelectric-stress components: e11 ~ 0.215 and e14 ~ 0.145 C/m 2 . Their decomposition into a clamped-ion and an internal-strain contributions, such as 6 , e1= e1 cl + e1 str where 1 or 4 in Voigt notation, show a larger effect between these two compounds. Indeed, the magnitude (in absolute value) of the clamped-ion and internal-strain contributions are significantly higher in FeAsO4 than in GaAsO4 both for e11 and e14. The higher electronic polarizability in FeAsO4 associated with the d-orbitals is responsible of the increase of the clamped-ion contributions while the lower frequencies observed in FeAsO4 explain the higher internal-strain contributions as e str ∝ j -2 where j is the frequency of the normal mode j. The decrease in frequencies observed in FeAsO4 could be attributed to the decrease of the Fe-O interatomic force constants as the Fe-O distances (~1.90 Å) are longer than the Ga-O distances (~1.84 Å). For both compounds, the overall piezoelectric-stress values are dominated by the internal-strain contribution and the clamped-ion contribution decreases the magnitude of e11 or e14 by its negative sign. Thus, the increase of the e str contribution in FeAsO4 is pretty much offset by the increase of the e cl contribution, leading to an overall value of e11 and e14 similar to those predicted for GaAsO4 (Table 3). In contrast to α-quartz where the e14 value is very small 42 The piezoelectric-strain constants, d11 and d14, are also of primary importance for piezoelectric devices as they are related to the athermal AT-cut, where the shift of the resonance frequency is minimal as a function of temperature. They are defined by: d11 = e11(S11 -S12) + e14 S14 and d14 = 2 e11 S14 + e14 S44, where S is the compliance tensor defined under the condition of zero electric field (see Table 4). For GaAsO4, our calculations give d11 =7.08 and d14 =5.76 pC/N, while these values significantly increase in FeAsO4 to reach d11 =18.77 and d14 = 13.78 pC/N; about three times higher than in GaAsO4. Thus, the highest piezoelectric-strain values predicted in FeAsO4 are related to the increase of the elastic compliances with respect to GaAsO4 as the piezoelectric-stress values are similar for the two compounds. This softening of the network in FeAsO4 is associated with less rigid tetrahedra where the Fe-O and Fe-As distances are longer than Ga-O and Ga-As distances in the tetrahedra of GaAsO4. Similar to Vegard's law observed for the lattice parameters (Fig. 5), we can estimate the piezoelectric-strain values of the Ga1-xFexAsO4 solid solutions considering a linear dependence of d11 or d14 between the parent compounds, GaAsO4 and FeAsO4.

Within this assumption, we get the following equations: d11 [pC/N] = 11.689 xFe + 7.082 and d14

[pC/N] = 8.021 xFe + 5.762. Thus, the d11 value could reach 8.1, 9.2 and 12.3 pC/N for an iron concentration of 9, 18 and 45 %, respectively. For these concentrations, the d14 values could reach 6.4, 7.2 and 9.4 pC/N, respectively. The comparison of these values with other -quartz-type M III X V O4 compounds 6 shows that, even with 1 %at. of iron, the d11 estimation of Ga1-xFexAsO4 solid solutions are much higher than that reported in AlPO4 and GaPO4 (see Table 5). In the case of GaAsO4, the Ga1-xFexAsO4 solid solutions begin really interesting from 20 % iron doping (d11 calc = 9.42 pC/N) and can reach d11 calc = 18.77 pC/N for the pure FeAsO4 end member. Similar observations can be seen about the d14 piezoelectric value. Note that powder of FeAsO4 has been already synthesized which is the first step for a future growth of a single crystal.

All these calculated values of the piezoelectric constants will have to be confirmed by experimental piezoelectric measurements which are only possible on millimeter sized single crystals whose growth requires several years of technical development to get crystals of sufficient quality for measuring the true values of the piezoelectric constants. Moreover, these results show the interest to substitute M III p-elements like Ga 3+ by d-elements like Fe 3+ to form chemical bonds with dorbitals leading to decreasing the elastic constants values and increasing the d11 piezoelectric constant. Indeed, with the same M V p-element (P or As), the d11 is about 2.5 times higher when Ga 3+ is replaced by Fe 3+ . This d-hybridation could be one of the keys to increase the piezoelectric performance of the materials.

CONCLUSIONS

In this work, the orthorhombic di-hydrate phases Ga1-xFexAsO4•2H2O and the dehydrated αquartz-type trigonal Ga1-xFexAsO4 solid solution series were synthesized by hydrothermal methods with a subsequent heating treatment. All the samples were studied as a function of composition by X-ray diffraction and Raman spectroscopy. The results show that all the members of the solid solution series were found to exhibit some very interesting gradual evolution in terms of lattice parameters as a function of the iron content in the corresponding structure.

From DFT calculations, it was shown that pure libration mode of oxygen atoms is absent in FeAsO4. This suggests that the - quartz transition should be absent as in the case of GaAsO4.

Piezoelectric properties of FeAsO4 were predicted by DFT calculations. The piezoelectric-strain constants are 18.77 and 13.78 pC/N for d11 and d14 respectively. These values are the highest values presently obtained in the -quartz type family making the iron-substituted FexGa1-xAsO4 of great interest for potential applications.

Fig. 1 Figure 1

 11 Fig. 1 XRD patterns of the as prepared Ga1-xFexAsO4,2H2O

Fig. 2

 2 Fig. 2 Experimentally determined (a) lattice parameters and (b) cell volume V as a function of

Fig. 3

 3 Fig. 3 Raman spectra of dihydrate compounds Ga1-xFexAsO4,2H2O as a function of iron content

  xFexAsO4, indicating the lattice expansion of the GaAsO4 unit cell caused by incorporation of iron on gallium sites.

Fig. 5

 5 Fig. 5 c/a ratio (a) and unit cell volume (b) as a function of iron content in Ga1-xFexAsO4.For pure GaAsO4, the data come from ref 24 and for pure FeAsO4 the data are the theoretical values calculated in this paper.

Table 1 .

 1 Raman spectral of the arsenate vibrations modes for Ga1-xFexAsO4,2H2O

	x=0	x=0.09	x=0.18	x=0.45	x=1	Band
	(ref.33 )				(ref.34 )	assignment
						(AsO4 3-)
	894	895	892	888		ν1(A1)
		812	810	802	800
	861,837,826 863,849	860, 846	888,860,810 880,870, 830	ν3(F2)
	460,433	460, 431	451,431	485,451,424	483,450,420	ν4(F2)
				,
	394,346,327 393, 347,	392,346,337	385,340,	380,340	ν2(F2)
		337,328	,325,
	290,273,240,	290,274,	289, 272,	316,295,	315,292,
	222,189	240,222,	261,	280,253,235	280,250,235,
		190	239,222,	,216,180	215, 181
			206, 188

Table 3 .

 3 Calculated independent piezoelectric-stress constants (C/m 2 ) in GaAsO4 and FeAsO4.

	Compound	e11 Clamped-ion Internal-strain	Total	e14 Clamped-ion Internal-strain Total
	GaAsO4	-0.20	0.42	0.22	-0.13	0.29	0.16
	FeAsO4	-0.33	0.54	0.21	-0.19	0.32	0.13

(-0.04 C/m 2 ), we highlight that this value is non negligible (~0.15 C/m 2 ) in the case of FeAsO4 or GaAsO4.

Table 4 .

 4 Calculated independent compliance constants defined under the condition of zero electric field (x10 -2 GPa -1 ).

		GaAsO4	FeAsO4
	S11	2.379	5.577
	S12	-0.738	-3.132
	S13	-0.403	-0.563
	S14	0.008	0.340
	S33	1.323	1.775
	S44	3.607	9.353
	S66	6.234	17.419

Table 5 .

 5 Experimental and calculated e11 and d11 piezoelectric constants (SiO2 is reference material).

to the ν1(A1) totally symmetric stretch and the bands at 880, 870, and 830 cm -1 to the ν3(F2) antisymmetric stretch. Savage et al.35 and Myneni et al.36 have different opinions on this assignment, they proposed that the most intense band at about 800 cm -1 should be an antisymmetric stretching mode associated with oxygen bridging between As and Fe (As-O-Fe). A shoulder at 831 cm -1 was assigned to the As-O-Fe symmetrical stretching mode and the weaker band at 893 cm -1 to a non-bridging oxygen that may be associated with crystal faces or defects.Actually, such problems with the mode assignments are understandable because of the changes in position and intensity may result from the interaction or mixing of vibration modes, especially when their energy regions are similar (e.g. ν1 (A1) and ν3 (F2) arsenate modes) depending on the orientation of the crystal with respect to the polarization and wavelength of the laser37 . The present Raman spectroscopy study is a good reflection of this situation and illustrates these changes very well.

Institut Charles Gerhardt Montpellier, UMR-CNRS-UM-ENSCM 5253, Université de Montpellier, Place E. Bataillon,

34095, Montpellier, Cedex 5, France.3 Institut de Recherche Technologique Antoine de St Exupéry, 3 rue Tarfaya, CS 34436, 31405 Toulouse cedex 4, France.

P. Labeguerie, M. Harb, I. Baraille et al., Phys. Rev. B 81, 045107:1-045107:9 (2010).

ACKNOWLEDGMENT

This work was financially supported by national foundation for study abroad of China and the Natural Science Foundation of Shaanxi Province (No. 2015JM2065).

Funding Sources

Any funds used to support the research of the manuscript should be placed here (per journal style).

Notes

Any additional relevant notes should be placed here.

ASSOCIATED CONTENT

Supporting Information. Supplementary data are given in the Supporting Information file.

Table S1 andFig

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. These authors contributed equally.


Gabriel Kerbart

Charles Manière

Christelle Harnois

Sylvain Marinel

Master Sintering Curve with dissimilar grain growth trajectories: A case study on MgAl 2 O 4

Keywords: Sintering, Master Sintering Curve, Grain growth, Transparent Ceramics, MgAl 2 O 4, Modeling Graphical Relative density Conventional MSC assumption: Unique trajectory

Sintering is a key step in the processing of high performance ceramics. Both the density and the grain size play a crucial role on the ceramic sintering kinetics and the final material properties. The master sintering curve (MSC) is a well-known tool for exploring sintering models kinetics. However, the conventional MSC theory assumes a unique sintering trajectory, while our study on MgAl 2 O 4 spinel shows dissimilar growth response. Park's MSC theory has been applied and compared with the conventional MSC approach for obtaining the activation energy with and without dissimilar grain growth trajectories.

I-Introduction

A well-known tool of ceramist to study the sintering is the Master Sintering Curve concept (MSC) initially developed by Su and Johnson [START_REF] Su | Master Sintering Curve: A Practical Approach to Sintering[END_REF]. This tool is used to determine the activation energy of a solid state sintering model under the assumption of a single dominant diffusion mechanism [START_REF] Bordia | Current understanding and future research directions at the onset of the next century of sintering science and technology[END_REF][START_REF] Hansen | Combined-Stage Sintering Model[END_REF][START_REF] Coble | Sintering Crystalline Solids. I. Intermediate and Final State Diffusion Models[END_REF]. MSC theory implies the integration of the sintering model with, on one hand, the diffusion Arrhenius temperature dependence with time and, on the other hand, all others sintering terms, grain size, porosity function, relative density and constants. For the latter, the grain size (G) which is typically time and temperature dependent is assumed to depend only on the porosity (θ). Consequently, the so-called "sintering trajectory" G(θ) is assumed to be unique for all heating rates, which is the conventional MSC concept. This assumption allows an easy identification of the activation energy but prevents from any sintering optimization. The final stage effects on MSC were detected and studied in various ceramics and metals [START_REF] Robertson | Refinement of Master Densification Curves for Sintering of Titanium[END_REF][START_REF] Robertson | Some Effects of Particle Size on the Sintering of Titanium and a Master Sintering Curve Model[END_REF][START_REF] Song | Two-Stage Master Sintering Curve Approach to Sintering Kinetics of Undoped and Al2O3-Doped 8 Mol% Yttria-Stabilized Cubic Zirconia[END_REF]. For instance, a sintering optimization method like the "two stages sintering" is precisely based on dissimilar sintering trajectories [START_REF] Chen | Sintering dense nanocrystalline ceramics without final-stage grain growth[END_REF]. Park et al [START_REF] Park | Master Sintering Curve Formulated from Constitutive Models[END_REF] generalized the concept of MSC including both densification and grain growth evolution. This generalized conceptualization of MSC allows a better understanding of the sintering trajectory dissimilarities which are the basis of sintering optimizations. In the literature, the final stage phenomenon is also considered via transition of mechanism and/or activation energy in the MSC [START_REF] Pouchly | Two-stage master sintering curve applied to two-step sintering of oxide ceramics[END_REF]. The transition may be used to correct the MSC sintering response of two-step sintering [START_REF] Pouchly | Two-stage master sintering curve applied to two-step sintering of oxide ceramics[END_REF]. We will show in this study that both grain growth and mechanism transition may be considered. This paper focuses on MgAl 2 O 4 spinel. This material attracts much interest from the scientific community to industries for transparent polycrystalline ceramics applications like laser host, jewelry, spacecraft windows, IR windows and dome for military applications [START_REF] Wang | Transparent ceramics: Processing, materials and applications[END_REF]. The isotropic structure of this material is a great advantage for producing transparent ceramics as it is not sensitive to the detrimental grain size effect on the optical transmission coefficient (due to the birefringence phenomenon) [START_REF] Petit | Drying step optimization to obtain large-size transparent magnesiumaluminate spinel samples[END_REF][START_REF] Goldstein | Transparent polycrystalline MgAl2O4 spinel with submicron grains, by low temperature sintering[END_REF]. However, to ensure high mechanical properties and the absence of intra-grain pores, a careful control of grain growth is required [START_REF] Bonnefont | Fine-grained transparent MgAl2O4 spinel obtained by spark plasma sintering of commercially available nanopowders[END_REF]. For MgAl 2 O 4 spinel, we have previously shown that the grain growth law changes from surface diffusion for high porosity content, to lattice diffusion for low porosity content [START_REF] Kerbart | Predicting final stage sintering grain growth affected by porosity[END_REF]. The resulting grain growth model, inspired by Olevsky et al. [START_REF] Wang | Transparent ceramics: Processing, materials and applications[END_REF] and Zhao and Harmer [START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: III, Model Experiments[END_REF], therefore includes a transition mechanism with a porosity dependence. This advanced model allows reporting dissimilar sintering trajectories if needed. In this paper, the MSC tool is used via the conventional method (unique sintering trajectory) and with the Park's equations, which may involve different sintering trajectories. The impact of the grain growth during the final stage of the sintering process will be discussed accordingly to these two distinct models.

II-Theory and calculation

The MSC theory predicts a unique curve plotting the relative density vs the following term:

Θ = ∫ 1 T exp ( -Q RT ) dt t 0 (1)
with, T the absolute temperature, t the time, R the gas constant and Q the activation energy. For dissimilar grain growth trajectories, Park's model [START_REF] Park | Master Sintering Curve Formulated from Constitutive Models[END_REF] gives:

Θ = ∫ ( G 0 G ) w 1 T exp ( -Q RT ) dt t 0 (2)
with, G 0 the initial grain size and G the actual grain size, w the sintering mechanism exponent of 4 and 3 for grain boundary and lattice diffusion respectively. In this study, we assume grain boundary diffusion dominates.

The latter MSC approach considers the independent grain growth influence assuming no pore trapped gas influence on the final stage (high solubility of gas in solid phase or grain boundaries). The use of Park's model implies the knowledge of grain size throughout the whole sintering process.

Olevsky's grain growth model [START_REF] Olevsky | Fundamental Aspects of Spark Plasma Sintering: II. Finite Element Analysis of Scalability[END_REF] was chosen for MgAl 2 O 4 to predict the grain growth affected by porosity:

Ġ = 𝐾(𝑇) 𝐺 𝑝 ( 𝜃 𝑐 𝜃 + 𝜃 𝑐 ) 𝑛 (3)
with p, n mechanism exponents, 𝜃 the porosity, 𝜃 𝑐 the critical porosity and 𝐾(𝑇) an Arrhenius term.

In a previous study on the same powder [START_REF] Kerbart | Predicting final stage sintering grain growth affected by porosity[END_REF], the grain growth was determined experimentally focusing on the transition region from the onset of grain growth at low porosity to the grain growth behavior at high densification. We evidence a complex transition mechanism from high porosity to full densification requiring a change of the exponent p and the porosity function of Olevsky's model. This advanced grain growth model is of particular interest to use with MSC as a careful modeling of grain growth is required in the whole porosity range.

III -Experiment and method

The samples of this study were produced by cold isostatic pressing (CIP) at 300 MPa of a commercial Baikowski S30CR spinel powder. To avoid disturbance from binder or anisotropic sintering shrinkage, the green specimens were prepared without binder or previous uniaxial pressing (UP). This powder has a purity over 99%, a specific surface area between 25 and 28 m²/g (BET) and a d50 between 0.15 and 0.3 µm. The samples were sintered up to 1773 K with three different heating ramps of 1, 2 and 5 K/min.

For this purpose, we used a dilatometer (setsys 16/18, SETARAM, France) to record the vertical displacement of an alumina rod in contact with the surface of the samples. The final density of these samples was determined by Archimedes's method. Assuming the shrinkage is isotropic, we can calculate the density versus time during the overall sintering cycle (see Fig. 1). As expected, the density increases with lower heating rates. This sintering curves (Fig. 1) recorded on materials prepared via pure CIP without binder are very closed to those reported by Yalamaç results (UP + CIP) [START_REF] Yamalaç | Sintering, Co-sintering and Microstructure Control of Oxide based Materials : Zirconia, Alumina, Spinel, Alumina-Zirconia and Spinel-Alumina[END_REF]. Compared to our results, and using UP+CIP, Talimian et al [START_REF] Talimian | Impact of high energy ball milling on densification behaviour of magnesium aluminate spinel evaluated by master sintering curve and constant rate of heating approach[END_REF] found shrinkage curves slightly shifted towards lower temperature, whereas Benameur et al [START_REF] Benameur | Sintering Analysis of a Fine-Grained Alumina-Magnesia Spinel Powder[END_REF] reported shrinkage curves slightly shifted towards higher temperature (slip casting). These literature data show the slight impact of the powder shaping method on the sintering temperature. The shrinkage curves data are sufficient when using the Su and Johnson conventional MSC method. However, the use of Park's MSC model requires the knowledge of the grain size dependence with temperature/time. The method for this purpose is detailed hereafter.

IV -Results and discussion

Grain growth modeling

The modeling of grain growth, during sintering, is based on the Olevsky's equation (Eq.3). This model allows predicting the grain growth for the different heating rates, which is needed for the MSC Park's model.

We show on Fig. 2. (a) the results of the grain growth modeling applied to the three dilatometric ramping cycles. As expected, lower heating rates led to bigger grain size. The development of this model (from a previous study [START_REF] Kerbart | Predicting final stage sintering grain growth affected by porosity[END_REF]) is based on the analysis of twelve samples, sintered at three different temperatures (1673 K, 1773 K, and 1873 K) for four different dwell times (0h, 1h, 2h, 4h). The determination of the grain size and the porosity of these samples allows for the determination of the Eq.3 model parameters. As a result, the grain size dependence with density can be modeled as shown on figure 2 (b). The model can be applied to MSC cycles using the dilatometric data for the porosity function of Eq.3

Master Sintering Curve with and without grain growth

We show on Fig. 3. both MSC, realized respectively with Su and Johnson model (Fig. 3. a), and Park's (Fig. 3. b). On each graph, it is shown the relative density versus the logarithm of the Θ function (see Eq. 1. and Eq. 2.) while in the insert is shown the mean of residual squares versus the tested activation energy. The use of Su and Johnson and Park's MSC models has conducted to a close estimation of the activation energy of 450 and 485 kJ/mol, respectively. The two MSC have relatively close activation energies with a difference of 35 kJ/mol. Nevertheless, an apparent difference between these two MSC models is the slightly higher mean of residual squares for Park's model. This explains the weaker superposition of the curve for Park's approach.

The only difference between the two models is the dissimilar grain growth trajectory which is taken into account for Park's model. As shown on Fig. 2. grain growth starts at 1600 K for each sample, and this temperature corresponds to different densities due to the different heating rates. Therefore, the sintering trajectories cannot be considered unique in the range of (65 -90 %) of density, as clearly shown figure 2(b). There is a noticeable effect when the relative density is above 80 %, on the figure 3 (b), it can be seen that the superposition of the different curves is much better, with a very low mean of residual squares. This relative density range corresponds to the domain where grain growth is taking place (see fig. 2 b). If we split Park's MSC analysis in two zones "with/without grain growth" in Fig. 3. c and Fig. 3. d, a better superposition is observed with different activation energies (400 kJ/mole and 700 kJ/mole at low and high densities respectively). This may indicate a change in the densification mechanism when grain growth starts. This two zones MSC study maybe be useful for comparing the different activation energies for each sintering stages [START_REF] Manière | Modeling zirconia sintering trajectory for obtaining translucent submicronic ceramics for dental implant applications[END_REF].

In the literature, the MSC activation energies for the same powder grade are ranging from 750 to 950 kJ/mole for UP + CIP [START_REF] Yamalaç | Sintering, Co-sintering and Microstructure Control of Oxide based Materials : Zirconia, Alumina, Spinel, Alumina-Zirconia and Spinel-Alumina[END_REF] and slip casting [START_REF] Benameur | Sintering Analysis of a Fine-Grained Alumina-Magnesia Spinel Powder[END_REF] samples respectively. Our values are closer to the 530 kJ/mole obtained by Talimian et al [START_REF] Talimian | Impact of high energy ball milling on densification behaviour of magnesium aluminate spinel evaluated by master sintering curve and constant rate of heating approach[END_REF] on ground powder. Our green specimen preparation without binder, and shaped by pure CIP, seems to mimic the behavior of the ground powder prepared by Talimian et al.

V-Conclusion

Final stage sintering grain growth is a complex phenomenon influenced by the porosity and transition mechanism. Gain growth extends the diffusion distance and then decrease the sintering kinetics at the final stage. Controlling the densification/grain growth competition at the final stage is one of the key issues for obtaining high-performance ceramics and the sintering modeling can help predict such phenomena. The MSC is a method typically employed to predict the sintering model parameters such as the activation energy. However, the conventional method assumes a unique sintering trajectory instead of considering the final stage grain growth effect as a temperature-dependent separated variable. Based on Park's approach, the MSC has been investigated taking into account the final stage independent grain growth. In a previous study, the spinel grain growth behavior has been identified in a large temperature range and for the same preparation/sintering conditions. This has allowed us to test Park's MSC approach without having to conduct an extensive study to determine the grain growth of each curves. This study gives the following conclusions.

1

The sintering trajectories obtained from the densification curves and the grain growth model is not converged into a single curve as supposed by conventional MSC approach but shows noticeable dissimilarities.

Figure 1

 1 Figure 1 Relative density versus temperature for each heating ramp.

Figure 2 MSC

 2 Figure 2 MSC Grain size evolution during sintering and corresponding sintering trajectory curves.

Figure 3 a

 3 Figure 3 a) Su and Johnson model for spinel sintering. b) Park model for spinel sintering. c) Park model at low density. d) Park model at high density.

Fig. 1 .Fig. 2 .

 12 Figure CaptionsFig. 1. Relative density versus temperature for each heating ramp.Fig. 2. MSC Grain size evolution during sintering and corresponding sintering trajectory curves. Fig. 3. a) Su and Johnson model for spinel sintering. b) Park model for spinel sintering. c) Park model at low density. d) Park model at high density.

K/min 2 K/min 1 K/min

Acknowledgement

The authors would like to thank Baikowski for the powder, and the French ministry of research for the funding of the thesis. The authors are also thankful to Jerome Lecourt and Christelle Bilot for their precious help in conducting the experiments.

2

The sintering trajectory dissimilarity implies a higher value of activation energy compared to the conventional MSC. [START_REF] Hansen | Combined-Stage Sintering Model[END_REF] Park's MSC was also investigated via the two zones MSC consisting of estimating separately the activation energy in the initial/intermediate stage and in the final stage. Significant differences in the activation energy confirm the lowering of the sintering kinetics by grain growth. 4

The preparation of green specimens by CIP without binder seems to favor lower activation energy compared to the literature results.