Jesús E Vega-Castillo

Uday K Ravella

Gwenaël Corbel

Philippe Lacorre

Alberto Caneiro

Thermodynamic stability, structural and electrochemical characterization of mixed ionic and electronic conductor La 2 Mo 2 O 8,96

the generation of an electronic contribution arising from the partial reduction of molybdenum. This contribution is more pronounced when the temperature and the tungsten content y are low.

 They found that the electrical conductivity of these partial reduced samples varies depending on both the porosity of initial ceramics and the W content.

In this work, we aim to analyse the stability limits of pO 2 for LM at equilibrium conditions and temperatures close to those meant for IT-SOFC operation. We also study the ranges of stability of the phases present in the La 2 Mo 2 O 9-δ system at these temperatures.

Experimental Synthesis of LM powders and ceramics

La 2 Mo 2 O 9 raw powders were prepared by standard solid state reaction. Stoichiometric amounts of La 2 O 3 and MoO 3 were weighed and mixed in an agate mortar. La 2 O 3 was previously dehydrated and decarbonated at 1000 °C for 2 h. The mixed oxides were annealed at 500 °C for 12 h to prevent Mo sublimation and then fired at 900 °C for 12 h. Several grindings and heat treatments were necessary to obtain pure LM.

In order to prepare a dense LM pellet, 5 g of raw powders were ball-milled into an agate jar with 20 agate balls (10 mm in diameter) in ethanol for 15 min. Approximately 500 mg of milled powder were shaped into a cylindrical pellet by pressing uniaxially under 1 ton. The green pellet was sintered at 1200 °C for 2 h. The sintered pellet had 92% of the theoretical density value with thickness and diameter of 4.53 and 5.30 mm, respectively.

Equilibrium TGA and pO 2 control Equilibrium measurements at constant temperatures of 718 and 608 °C in a controlled atmosphere were carried out by using a Cahn 1000 thermobalance coupled to an electrochemical pO 2 control system. 27 Atmospheres with controlled pO 2 are set by flowing a carrier gas through an electrochemical oxygen pump and an electrochemical oxygen sensor. The pump enables to modify the pO 2 in the gas following Faraday's law while the sensor uses Nernst's law to measure the set pO 2 value. The symmetrical thermobalance allows both to detect when the steady state is reached and to measure the oxygen content of the sample.

Ar, CO 2 and Ar-H 2 were used as carrier gases to supply the oxygen pump. The highest pO 2 corresponded to that obtained in Ar ( pO 2 = 3 × 10 -6 atm, value due to residual oxygen in commercial Ar 5.0). Intermediate pO 2 range was set by electrochemical reduction of CO 2 giving CO-CO 2 atmospheres. The low pO 2 implemented range was obtained by oxidizing Ar-H 2 and generating Ar-H 2 -H 2 O. The gas flow rate for all equilibrium measurements was set to 6 L h -1 . Each equilibrium point was recorded after a constant TGA signal (mass variation lower than 1 μg) for at least 5 h.

Approximately 500 mg of LM raw powder were placed into an alumina crucible in the thermobalance. The sample was heated up in flowing Ar; once at the working temperature, the equilibrium was reached. The mass value obtained under Ar with pO 2 = 3 × 10 -6 atm was used as reference for stoichiometric La 2 Mo 2 O 9 (δ = 0). This reference was verified under pure oxygen ( pO 2 = 1 atm). The pO 2 value was then decreased by switching the atmosphere and the annealing maintained until a new equilibrium was detected. This procedure was repeated as the pO 2 was gradually decreased down to the minimum adjustable value with dry Ar-H 2 .

Structural characterization

The phase purity of powder and pellet samples after their annealing in the thermobalance and quenching under controlled oxygen partial pressure was checked by XRD at room temperature. Diffractograms were collected on a Philips PW1700 diffractometer with Cu(K α1 + K α2 ) radiation and a graphite monochromator and a PANalytical q/q Bragg-Brentano X'pert MPD PRO diffractometer Cu(K α1 + K α2 ) equipped with the X'Celerator multi-elements detector.

In situ XRD measurements under controlled oxygen partial pressure were carried out at different annealing temperatures by using an Anton-Paar HTK10 camera coupled to the electrochemical pO 2 control system described in the previous subsection. In this manner it is possible to record XRD patterns at high temperature under controlled pO 2 atmosphere.

Electrical characterization

LM and LM896 were characterized by EIS in different atmospheres using a symmetrical electrochemical cell. Pt electrodes were sputtered onto both flat sides of cylindrical LM pellet. The sample was placed in an alumina cell inside a cylindrical furnace, with Pt mesh as current collectors. This setup can also be connected to the electrochemical pO 2 control system.

EIS spectra were collected from 250 up to 608 °C by means of a frequency response analyser (FRA) coupled to an Autolab potentiostat, within a frequency range of 1 MHz to 0.1 Hz and voltage amplitude of 50 mV. The measured spectra were analyzed using Zview software. [START_REF] Associates | A Software Program for IES Measurements and Analysis[END_REF] A R||Cpe circuit was used to model the high frequency contribution (bulk) for spectra recorded in air and Ar (stoichiometric LM). For spectra recorded on partially reduced La 2 Mo 2 O 9-δ , 0 < δ ≤ 0.04, in Ar-H 2 -H 2 O atmosphere, the high frequency contribution was adjusted with only one resistance element. The total conductivity values σ for LM and La 2 Mo 2 O 9-δ were calculated from the resistance values R in the respective equivalent circuits, the measured thickness L and surface area A of the pellet as follows: σ = L(AR) -1 . For all spectra, the Pt electrode contribution was modelled with a short Warburg element in series with the high frequency circuit. decreased. The oxygen content remains as 9 as long as the pO 2 is above 10 -15 atm. It delineates the thermodynamic stability range for stoichiometric LM at 718 °C. As pO 2 is slightly decreased down to 10 -16 atm, an abrupt oxygen loss of δ = 0.43 is observed after five days. A further decrease of the oxygen partial pressure down to 10 -22 atm does not induce any change in the oxygen content of the sample. Therefore, the 9 -δ = 8.57 stoichiometry is thermodynamically stable at 718 °C within the range 10 -22 < pO 2 < 10 -15 atm.

Results and discussion

When decreasing pO 2 below 10 -22 atm a slow reduction takes place again and in this case a steady state, indicating thermodynamic equilibrium, is never properly reached. Instead, a pseudoplateau with a drift down is detected in the TGA signal vs. time at 9 -δ values lower than 7 (open triangles in Fig. 1). The presence of a linear drift that increases when pO 2 decreases suggests that the compound La 2 Mo 2 O 7-y has no fixed oxygen stoichiometry and it losses oxygen continuously with kinetics depending on the pO 2 .

In order to determine the nature of phases (or mixture of phases) corresponding to the found stoichiometries 9 -δ = 8.57 and 7y, two batches of LM raw powders were annealed at 718 °C in the TGA equipment in atmospheres with pO 2 values of 10 -20 and 10 -24 atm, respectively. Once the oxygen content of each batch reached constant 8.57 and 7y (with a drift as described in previous paragraph), respectively, the samples were quenched down to room temperature while maintaining the atmosphere. XRD patterns of quenched powders compared to those of initial LM are shown in Fig. In our equilibrium experiments, La 7 Mo 7 O 30 remained stable at 718 °C in atmospheres with pO 2 values within the range 10 -22 < pO 2 < 10 -15 atm (we have kept the treatment for even several days with no change observed in the mass signal). This indicates that La 7 Mo 7 O 30 is thermodynamically stable in the described conditions. The use of high gas flow rates in our equilibrium experiments enhances the kinetics of the reduction processes and avoids confusing meta-stable states with thermodynamically stable ones.

Regarding the observed amorphous compound (see Fig. 2(c)), in our previous work we had referred it to La 2 Mo 2 O 6.88 . 5 Nevertheless, subsequent experiments including the one here reported indicate an oxygen non-stoichiometric range for this amorphous phase, leading us to revise that chemical formula. We have opted to use La 2 Mo 2 O 7-y as approximated chemical formula for this In order to determine whether the La 7 Mo 7 O 30 phase exists at a lower temperature, an equilibrium experiment in controlled atmosphere at 608 °C was performed.

Fig. 3 shows the oxygen content 9 -δ in La 2 Mo 2 O 9-δ as a function of pO 2 at 608 °C. At this temperature the pO 2 stability limit for La 2 Mo 2 O 9 is 10 -22 atm. Below this value the sample undergoes a slight reduction down to an oxygen content of 8.96. The stoichiometry 8.96 remains unchanged within the range 10 -25 < pO 2 < 10 -23 atm. Below 10 -25 atm the sample reduces slowly down to amorphous La 2 Mo 2 O 7-y .

The reproducibility of La 2 Mo 2 O 8.96 (LM896) was checked by several oxidizing-reducing cycles between 9 -δ = 9 and 8.96. LM896 can also be recovered by oxidation from amorphous La 2 Mo 2 O 7-y . The reproducibility of LM896 upon reduction-oxidation confirms that the found pO 2 values corresponding to LM896 represent true thermodynamic equilibrium states. LM896 powders for XRD characterization were obtained from annealing LM at 608 °C in an atmosphere with a pO 2 of 10 -24 atm in the TGA setup until total stabilization at 9 -δ = 8.96 was reached. The reduction of 500 mg from LM to LM896 took 200 min. The homogeneously light gray LM896 powders were then quenched down to room temperature and analyzed by XRD; the results are shown in Fig. 4. The X-ray diffractograms of LM and LM896 are identical. Both pseudo-cubic (2 3 1) reflections (at 2θ around 47.5°) are compared in the inset, the notorious peak splitting characteristic of αLa 2 Mo 2 O 9 indicates that LM896 has also a monoclinic symmetry at room temperature. XRD patterns of LM and quenched LM896 were fitted using constant scale factors (Lebail) within FULLPROF software, [START_REF] Rodríguez-Carvajal | [END_REF] in monoclinic P2 1 space group ignoring weak (2 × 3 × 4) superstructure peaks. 30 The obtained cell volumes were 366.05(1) Å 3 The structure of αLa 2 Mo 2 O 9 has been described as one of the most complex structures reported to date. 31 The monoclinic cell has a number of formula units Z = 48, so, the chemical formula contained in the asymmetric cell can be written as La 96 Mo 96 O 432 . In the case of αLa 2 Mo 2 O 8.96 , 48 formula units would equal to La 96 Mo 96 O 430 . This corresponds to a loss of two O atoms per monoclinic cell, which is consistent with maintaining the monoclinic P2 1 symmetry of αLa 2 Mo 2 O 8.96 , with departure from La 2 Mo 2 O 9 of 2 equivalent O atoms related by the 2 1 screw axis.

HT-XRD on LM896 powders

In order to determine if LM896 undergoes a monoclinic-cubic transition, approximately 20 mg of LM raw powders were heated up to 608 °C inside the HT-XRD setup in He atmosphere. At this temperature, the atmosphere was switched to He-Ar-H 2 -H 2 O with a set pO 2 of 10 -24 atm. After 200 min of annealing, the stabilization of La 2 Mo 2 O 8.96 was assumed, based on the kinetics observed in TGA experiments described in previous subsection. Fig. 5 shows the XRD pattern collected at 608 °C on the LM896 phase (black line) in the vicinity of the most intense (1 2 0) peak of a beta-LM-type phase (gray line). A slight shift to higher 2θ values is visible indicating once again a contraction of the unit cell after reduction from LM to LM896. At 608 °C LM896 is in cubic symmetry (isostructural to βLa 2 Mo 2 O 9 ) as indicated by the absence of splitting in (2 3 1) reflection shown Aiming at determining the α ↔ β transition temperature for LM896, the (2 3 1) reflection was monitored at different temperatures by HT-XRD while cooling down from 608 °C. Fig. 6 shows how the monoclinic distortion shows up at 550 °C upon cooling, while on heating, both symmetries coexist at 560 °C. These values are identical to those observed for LM powders in air upon both cooling and heating in the same HTXRD setup.

Electrical characterization of La 2 Mo 2 O 8.96

As other authors have observed, the partial reduction of Mo in La 2 Mo 2 O 9-δ leads to an increase of electrical conductivity. 20,25,26 This behaviour would be expected for LM896 compound. Arrhenius plots of measured conductivity values for both LM (in air) and LM896 (in Ar-H 2 -H 2 O) are shown in Fig. 7. The curve corresponding to LM (squares) was recorded while heating up to 608 °C in air. At this point the atmosphere was switched by Ar-H 2 -H 2 O with a controlled pO 2 of 10 -24 atm. The total resistance of the sample was then monitored as a function of time (see inset in Fig. 7) and once a plateau was obtained, the stabilization of LM896 was assumed, after 40 h. The Arrhenius curve of LM896 shows a significant increase of conductivity values compared to those of LM, for instance at ∼500 °C the difference is about two orders of magnitude. The curve presents two visible linear zones with different slopes with a break at around 550 °C. From adjusting the linear zones with Arrhenius's law, the calculated activation energies are 0.4 and 0.72 eV for the low and high temperature zones, respectively. Keeping in mind that 550 °C is the measured β → α transition temperature for LM896 at cooling, the increase of activation energy can be related to the increase of ionic conductivity associated to this transition. The implemented technique does not allow to discriminate between electronic and ionic conductivity components. Nevertheless, the low activation energies for LM896 suggest predominant electronic conductivity.

Discussion on La 2 Mo 2 O 8.96 Some authors have suggested that LM exhibits both ionic and electronic conductivities while being partially reduced at temperatures close to 600 °C. 22,25 In fact, Jacquens et al. reduced a LM pellet in Ar-(10%)H 2 at 608 °C while monitoring its conductivity by EIS. Once they found stabilization of resistivity values, they characterized the conductivity of the reduced sample as a function of temperature while cooling down in the same atmosphere. 25 The results obtained by Jacquens et al. are identical to those reported in this work on La 2 Mo 2 O 8.96 electrical characterization. In the cited work, after EIS experiments, XRD identification showed the conservation of LAMOX structure so the authors concluded that LM can withstand partial reduction without decomposing.

We suspect that the results of Jacquens et al. correspond to MIEC La 2 Mo 2 O 8.96 and they were able to stabilize this compound in dry Ar-(10%)H 2 . This is in apparent contradiction with our equilibrium results, in which while decreasing the pO 2 down to 20 -26 atm, the compound amorphisizes. Theoretically, the oxygen activity in pure dry Ar-H 2 should tend to zero (log pO 2 → -∞). Nevertheless, in the practice, there is always residual oxygen in high purity gas mixtures and there are leakages in any gas circuit. The effect of incoming oxygen through leakages is more important when the flow rate of the gas source is low since the total flux is the arithmetical addition of all individual gas inputs. Jacquens et al. used a very low flow rate (0.6 L h -1 ) of dry Ar-(10%)H 2 so only a small amount of incoming air could have been enough to oxidize a part of H 2 setting an Ar-H 2 -H 2 O atmosphere. One of the aims of using high gas flow rate (6 L h -1 ) in our experiments is to minimize the effect of the leakages in controlled atmospheres. Another one is to minimize oxygen pressure gradient around the sample by flushing it out. The high oxide ion conductivity of βLa 2 Mo 2 O 9 has been attributed to its crystalline structure and explained with a lonepair substitution concept. 32 Since the cationic framework of the LAMOX structure is seemingly preserved after reduction, it suggests that oxide ions remain mobile in La 2 Mo 2 O 8.96 . On the other hand, the EIS measurements show considerably higher total conductivity values with low activation energies for LM896 compared with those of LM at temperatures below 608 °C. These facts lead us to conclude that the difference in conductivity values between LM and LM896 is due to the appearance of significant electronic conductivity in the latter compound. LM896 is an MIEC stable in reducing atmospheres. Thermal and chemical compatibilities of LM896 with LAMOX based electrolyte materials are presumed, with this novel MIEC being part of the LAMOX family as well.

The compensation of charges for La 2 Mo 2 O 8.96 , assuming that the valence state for La and O are 3+ and 2-respectively, yields a total valence state of +5.96 for Mo. The reduction of Mo 6+ to a Mo 6+ /(4%)Mo 5+ combination is therefore compensated by the generation of O vacancies. The reduction to LM896 takes place at 608 °C in cubic beta form; this structure has three oxygen sites and two of those, O2 and O3, are partially occupied. It is logical to suppose that oxygen leaves from O2 and/or O3 sites. In the alpha form there are 216 oxygen sites and the vacancies might have a statistical distribution in several sites. Nevertheless, the strict plateau reached at 9 -δ = 8.96 leads us to think that the vacancies should correspond to a full loss at one monoclinic site. Unfortunately, the information available from the present study is not enough to clarify the oxygen vacancies distribution in LM896, neither in alpha or beta symmetry.

The decrease of the unit cell volume indicates that the slight decrease in oxygen stoichiometry involves a larger reduction in volume than the expansion arising from the increase in ionic radius as the valence state of molybdenum is lowered (0.455 A for Mo 6+ and 0.497 A for Mo 5+ in 4.5 coordination). Note that in β-LM, taking into account the partial occupancy of O2 and O3 sites, the average number of oxygen atoms surrounding Mo is 4.5.

Conclusions

The thermodynamic pO 2 limits for LM at 718 and 608 °C were determined as 10 -15 and 10 -22 atm, respectively. The stability range for perovskite related compound La 7 Mo 7 O 30 at 718 °C was found between 10 -15 and 10 -22 atm of pO 2 . The novel compound La 2 Mo 2 O 8.96 was found at 608 °C within the range of 10 -23 < pO 2 < 10 -25 atm. This material is isostructural to α(monoclinic) La 2 Mo 2 O 9 below 560 °C and undergoes a transition to a β(cubic) LAMOX structure above this temperature. LM896 is a mixed conductor that exhibits an important electronic conductivity component added to the ionic conductivity values of LM oxide conductor.

Fig. 1

 1 Fig. 1 Oxygen content as a function of pO 2 in La 2 Mo 2 O 9-δ at 718 °C.

Fig. 2

 2 Fig. 2 XRD patterns of La 2 Mo 2 O 9-δ stabilized at 718 °C in different atmospheres and quenched down to room temperature with 9 -δ values of (a) 9, (b) 8.57 and (c) 7y.

  for LM and 365.71(1) Å 3 for LM896 showing a slight contraction after reduction. The respective cell parameters were a = 7.1414(1) Å, b = 7.1538(1) Å, c = 7.1653(1) Å and β = 89.530(1)°for LM and a = 7.1411(1) Å, b = 7.1520(1) Å, c = 7.1608(1) Å and β = 89.584(1)°for LM896.

Fig. 4

 4 Fig. 4 Room-temperature X-ray diffraction patterns of raw La 2 Mo 2 O 9 and quenched La 2 Mo 2 O 8.96 powder samples showing the superstructure peaks (*) and the splitting of the cubic (2 3 1) peak (inset), both characteristic of the monoclinic low temperature α form.

Fig. 3

 3 Fig. 3 Oxygen content as a function of pO 2 in La 2 Mo 2 O 9-δ at 608 °C.

Fig. 6

 6 Fig. 6 Cubic or pseudo-cubic (2 3 1) XRD reflection for LM896 at different temperatures: (a) while cooling down, (b) while heating.

Fig. 5

 5 Fig. 5 Comparison of XRD patterns at 608 °C of cubic LM and LM896: main, (1 2 0) reflection and in the inset, (2 3 1) reflection for LM896 at 608 °C. Fig. 7 Conductivity as a function of temperature for LM in air and LM896 in Ar-H 2 -H 2 O.

  Reduction of La 2 Mo 2 O 9 at 718 °C: stability of La 7 Mo 7 O 30 and amorphous La 2 Mo 2 O 7-y The oxygen content 9 -δ in La 2 Mo 2 O 9-δ as a function of decreasing pO 2 at 718 °C is shown in Fig. 1. As explained in Experimental section, the first measurement was performed in Ar (closed square) and afterwards the pO 2 was gradually This journal is © The Royal Society of Chemistry 2012

	Dalton Trans., 2012, 41, 7266-7271 | 7267

  2. La 2 Mo 2 O 8.57 has been confirmed as La 7 Mo 7 O 30 perovskite related phase, while La 2 Mo 2 O 7-y is amorphous.

Discussion on equilibrium pO 2 data at 718 °C Several authors have observed the formation of La 7 Mo 7 O 30 as intermediate product during partial reduction of LM in different reducing atmospheres.

18,19 Indeed, Goutenoire was able to synthesize a large amount of La 7 Mo 7 O 30 in order to determine its crystal structure from neutron diffraction analysis by reducing LM in N 2 -(6%)H 2 at 760 °C. 23 Nevertheless, they clarify that if the atmosphere and temperature are maintained the reduction continues until amorphization of the compound. In other words, La 7 Mo 7 O 30 is a meta-stable phase at 760 °C in N 2 -(6%)H 2 atmosphere. We think that the reproducible transient stabilization of La 7 Mo 7 O 30 from reduction of LM in N 2 -(6%)H 2 at 760 °C is due to kinetic effects related to the gas flow rate (not specified in the cited work) and also to the used gas mixture. In fact, we have been able to reproduce Goutenoire's experiment under the same conditions obtaining the TGA plateau of La 7 Mo 7 O 30 . Nevertheless when we have repeated the experiment under the same conditions but replacing N 2 -(6%)H 2 by Ar-(5%)H 2 , this plateau was not reached and the reduction continued directly until the amorphization of the sample.

Downloaded by UNIVERSITE DU MAINE-SCD on 31 August 2012 Published on 05 April 2012 on http://pubs.rsc.org | doi:10.1039/C2DT30261F View Online

Acknowledgements

This work was financially supported by CNEA (Comisión Nacional de Energía Atómica), CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas), UNCuyo (Universidad Nacional de Cuyo) and ECOS-SUD program, project A07E03.


Bholanath Pahari

Noureddine Mhadhbi

Gwenaël Corbel

Philippe Lacorre

Jens Dittmer

Analysis of the local structure of phosphorus-substituted LAMOX oxide ion conductors

extremely broad signals due to quadrupolar coupling. This coupling reflects the electric field gradient (EFG) around the lanthanum cation, a property that is sensitive to the local ground state electronic environment.

Experimental Synthesis

Compositions of the series La 2 Mo 2-y P y O 9-y/2 were prepared by a conventional solid state reaction method from a stoichiometric mixture of La 2 O 3 , (NH 4 ) 2 HPO 4 and MoO 3 powders. Prior to use, La 2 O 3 was calcined in air for 12 h at 1000 °C in order to remove adsorbed water and carbon dioxide. The well mixed powders were calcined in an alumina crucible at 500 °C for 12 h. The desired high-purity samples have been obtained after several intermediate grindings and firings in the temperature range 900-1175 °C (heating and cooling rates of 5 °C min -1 ).

For the purpose of comparison, the monazite LaPO 4 was prepared by heating a stoichiometric mixture of La 2 O 3 and (NH 4 ) H 2 PO 4 at 500 °C for 12 hours (heating and cooling rates of 2 °C min -1 ) and then at 1300 °C for 12 hours (heating and cooling rates of 5 °C min -1 ).

X-ray powder diffraction

Phase purity and identification as well as crystal cell parameters were determined by X-ray powder diffraction at room temperature on a θ/θ Bragg-Brentano Philips X'pert MPD PRO diffractometer (Cu Kα 1+2 radiations) equipped with the X'celerator detector. For cell parameter refinements, the diffractograms were collected over 300 min in the scattering angle range [9-130°], with a 0.0084°step, and the program FullProf 22 was used in the Le Bail's fit mode. 23 To detect any trace of LaPO 4 as impurity, scans were collected during 150 min in the scattering angle range [27.2-29.2°], with a 0.0084°step. The integrated intensity of the diffraction peak at 28.64°was determined using the fit profile option of the software X'Pert HighScore Plus, version 2.0a (PANanalytical B.V., Almelo, Netherlands).

Solid-state NMR

4 mm magic angle spinning (MAS) NMR rotors were filled with equal amounts of La 2 Mo 2-y P y O 9-y/2 for a series of samples with different concentrations y/2 of P. 31 P and 139 La NMR experiments were performed on a Bruker Avance III 300 spectrometer operating at a magnetic field of 7.04 T. 139 La NMR spectra were recorded under static conditions by sweeping the carrier stepwise over the spectral range and acquiring a signal with 1024 scans at each carrier position. For each scan, 10 echoes were generated by multiple refocusing in the QCPMG fashion in intervals of 150 μs. 24 The rf strength of the pulses was 93 kHz (p 90 = 2.7 μs), the recycle delay was 0.2 s. A 50 mM aqueous solution of LaCl 3 was used as frequency reference. The echoes were coadded, a Gaussian window of 200 kHz was applied, and the time signal was Fourier transformed. For each partial experiment, the intensity was determined as the y-value at the respective carrier frequency. The final spectra consist of the ensemble of these intensities as a function of the carrier frequency. The spectra assembled this way were analyzed by means of the software dmfit 25 and a self-written analysis tool. The effect of a Czjzek distribution 26 was simulated by the "Czjzek simple" model 27 of dmfit, in which the distributions of chemical shift and quadrupolar coupling are not coupled. For reasons of computing time the chemical shift distribution was adjusted after fitting. All 31 P NMR experiments were performed under MAS at either 3, 5 or 7.5 kHz using a 4 mm dual band probehead. 31 P MAS NMR spectra acquired with a spinning rate of 30 kHz did not show a better resolution than those acquired with 7.5 kHz (indicating that the line width is determined by site inhomogeneity rather than chemical shift anisotropy). 31 P chemical shifts are referenced to 85 mol% H 3 PO 4 in water. For quantitative 1D spectra 128 scans were acquired with a relaxation delay of 10 min. The set of spectra was deconvoluted in a conjoined fashion by a selfwritten Matlab script. A DQ-SQ experiment was applied to the composition y = 0.04 with a POST-C7 28 recoupling time of 1.14 ms for each direction, 64 increments with 8 scans each, and a relaxation delay of 2 min. For a quantitative analysis, they were applied for a series of samples with 16 increments and between 16 and 176 repetitions. A POST-C7 DQ-SQ experiment with 2.85 ms recoupling in each direction was combined with a anteceding saturation recovery unit in order to measure longitudinal relaxation times of a DQ-filtered signal. For the sample y = 0.04, the experiment was done in the 2D fashion with between 16 and 64 repetitions. For spin-counting experiments, 29 multiplequantum coherences were excited during a mixing period of 2.0 ms by the application of BABA recoupling 30 and subsequently reconverted to longitudinal magnetization. The MAS frequency was 3 kHz. In order to separate the multiple-quantum coherences, we have used the phase-incrementation scheme of Shykind et al. 31 with 64 increments. Overall, 200 repetitions have been accumulated for y = 0.04 and 32 for LaPO 4 ; the interscan delay was 40 s. All NMR experiments have in addition been applied to LaPO 4 for sake of comparison.

Results

Existence and extension of solid solution, phase transition

Several compounds of the solid solution La 2 Mo 2-y P y O 9-y/2 with compositions y = 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 and 0.10 were prepared according to the procedure described in the Experimental section, and analyzed by X-ray powder diffraction (300 min, [9-130°] range, 0.0084°step). All compounds up to y = 0.05 appear to be pure in these patterns, thus attesting the existence of a solid solution. Samples with compositions y = 0.06 to 0.10 show the presence of LaPO 4 as impurity phase.

As the phosphorus content y increases, the diffraction patterns reveal a phase transition from a monoclinic α type (y = 0.01 and 0.02) to a cubic β type (y = 0.03, 0.04 and 0.05) LAMOX phase (Fig. 1A, 47.5°). This is interesting as there is an analogy to the transition to the high temperature, high conductivity cubic β phase of the parent compound La 2 Mo 2 O 9 .

As a further test of the solid solution domain, the evolution of the crystal cell volume as a function of the P content is plotted in Fig. 1B. Since phosphorus has a smaller radius than molybdenum (0.17 Å and 0.41 Å, respectively, for CN = 4), 32 a decrease of the cell volume was expected up to y = 0.05; then, in This journal is © The Royal Society of Chemistry 2012 Dalton Trans., 2012, 41, 5696-5703 | 5697 the ideal case, a constant volume. Fig. 1B shows an overall decrease of the volume with y, but this decrease is hardly more distinct than the experimental error of each point. The limiting factor for the precision appears to be the mechanics of the diffractometer.

Therefore the major (120) peak of LaPO 4 at 28.64°(PDF no. 01-084-0600) [START_REF]The International Centre for Diffraction Data (ICDD), 12[END_REF] was recorded with high accuracy during very long scans (150 min scans, [27.2-29.2°] range, 0.0084°step). These extended scans revealed that the y = 0.05 composition also contains a small amount of LaPO 4 precipitate while for the y = 0.04 composition no signal was detected. Fig. 1C displays the evolution of the LaPO 4 amount as a function of y, the straight line corresponding to a linear regression of the points for which a non-zero intensity value was measured. It shows that the solid solution limit is just below y = 0.05. 139 La NMR 139 La (I = 7/2) exhibits a sizeable quadrupolar coupling constant which usually leads to extremely broad signals that exceed by far the band width of the excitation pulses and the reception circuit. Spectra of La 2 Mo 2-y P y O 9-y/2 with y = 0.00-0.06 (in increments of 0.01), 0.08, and 0.10 were therefore recorded by sweeping the carrier frequency stepwise through the whole frequency range. Each sampled point displays the intensity at the carrier frequency of the respective spectrum. In order to enhance the sensitivity, the time signal was sampled by means of the QCPMG scheme. 24 Fig. 2A shows 139 La spectra of La 2 Mo 2-y P y O 9-y/2 with y = 0.00-0.05 and 0.10, and of the monazite LaPO 4 for comparison. The monazite spectrum shows a broad central transition with distinct features of the second-order term of the quadrupolar coupling. They can be described by the corresponding standard model function with an enormous coupling constant ν Q = 3.4 MHz, an asymmetry parameter η = 0.68 and a Gaussian line broadening of 64 kHz. The signals of La 2 Mo 2-y P y O 9-y/2 are featureless, asymmetric, and have an unusual sharp summit. A tentative modeling with a crystal powder quadrupolar coupling model signal as for LaPO 4 is not completely poor, but does not match these particular properties, and moreover yields implausible results as a quadrupolar coupling of only ν Q = 0.4 MHz for the parent compound, almost ten times smaller than that of LaPO 4 and also much smaller than other 139 La coupling constants that can be found in the literature (see, for example, ref. 34). The coordination environment of La in LAMOX, 3,4 however, does not make it likely that its electric field gradient has high cubic symmetry. As the line shape is rather untypical for polycrystalline materials, we try to model the spectra with a Czjzek distribution 26 of the quadrupolar coupling, which is normally used for amorphous structures such as glasses. We focus here on the region dominated by the central transition. The best fits are achieved with a critical exponent d = 3, reflecting that the distribution of the quadrupolar coupling obviously is not isotropic. The quadrupolar coupling constants at the peak of the distribution (Fig. 2B) are of the same order of magnitude as that of This journal is © The Royal Society of Chemistry 2012 (6 to 12) numbers of oxygen neighbours are thus so different that they lead to a broad distribution of the electric field gradient (EFG), for which a conventional crystal model (which would assume no distribution) is not appropriate. The fit however improved when broadening the lines with a broad chemical shift distribution: a width of 30 kHz, or a distribution of ±350 ppm around an average chemical shift of about +700 ppm, was appropriate for all La 2 Mo 2-y P y O 9-y/2 spectra. This width appears relatively large and apparently compensates for the residual imperfection of the model.

Upon substitution of Mo with P, the peak quadrupolar coupling initially increases with y. There is a significant jump from y = 0.02 to 0.03, the point of phase transition from a monoclinic to cubic phase. The general shape however is preserved. Afterwards the peak coupling constant does not change any more (within the range of experimental uncertainty). The increase of the peak coupling constant with the phase transition is on the first glance counterintuitive, as one would expect that the quadrupolar coupling decreases with increasing crystal symmetry (compare for example ZrO 2 in ref. 35), but the local symmetry is not necessarily correlated. The preservation of the amorphouslike shape of the signal rather shows that a local disorder remains and that the crystal structure represents, as suggested by Evans et al. for the high temperature β phase of the parent compound, a dynamic average. 4 The increase in the peak quadrupolar coupling will then be due to a higher average local symmetry distortion around the La nuclei, as it can be due to the higher number of possibilities to create an asymmetric La environment by oxygen vacancies of the O2 and O3 site.

Quantitative 31 P NMR Each 31 P single pulse spectrum of the series y = 0.01 to 0.04 shows only one asymmetric peak, which has a maximum intensity that shifts with y from about -1 to -4.5 ppm (Fig. 3). For deconvolution, the spectra were normalized to the P concentration (weighted by 1/y) in order to avoid an under-representation of the lower y spectra in the error sum. One spectrum alone would not allow a detailed deconvolution, however we make use of the surplus of information given by the series of spectra that should have most parameters in common. We therefore perform in a first approach a curve fitting for all spectra together, assuming that peak positions and shapes are the same and that only the intensities of the peaks vary (Table 1, Fig. 4). These constraints reduce the number of parameters for a three site Lorentzian/Gaussian model from 48 (4 spectra × 3 peaks × 4 parameters, i.e., position, width, Gaussian percentage, intensity) These constraints not only ensure a consistent result, but also allow and even require the refinement by proceeding from a twoto a three-site model: a two site model with 16 parameters yields, as may be expected by inspection, an upfield site whose population is increasing with y and a downfield site decreasing with y (Fig. 3). Each spectrum alone could be well described by a two-site model, but the constraints in a joint fit lead to an inappropriate model function for either low or high P concentration. Changing from a two-site to a three-site model leads to satisfactory fits for all spectra, and the error sum improves by 22%.

Fig. 4A shows the three relative peak intensities as a function of y, representing three populations. At low P concentrations, the spectra can be described by two peaks at 0.05 ppm (signal P1) and -2.83 ppm (signal P2). There is a discontinuity between y = 0.02 and y = 0.03; P1, contributing about 35% at low P concentration, vanishes beyond this point. This transition in the peak intensities coincides with the observed structural phase transition from the α to the β phase between y = 0.02 and 0.03. Under this observation the fitting approach is not fully appropriate, as we give the constraint of same peak positions and shapes for the α and the β phase. The observation that one site is vanishing while the other takes all the population may be induced by the above fitting approach. As a consequence, we adjust the fitting model by separating the four spectra into two groups, below (y = 0.01-0.02) and above the phase transition (y = 0.03-0.04). As in each spectrum in the joint three-site approach only one or two of the three sites had a significant population, we reduce the number of sites to two in the separated approach (right column in Table 1). The error sum improves slightly by 5%, with 20 rather than 21 parameters. Signals P1 and P2 now unify to a common peak P1/2 in the second group of spectra (Fig. 4B), representing a more homogeneous group of sites. However, at the same time a signal at -4.60 ppm (signal P3) arises, increasing with the P concentration.

We have acquired 31 P NMR spectra of several reference compounds, the most relevant being the impurity LaPO 4 found in the La 2 Mo 2-y P y O 9-y/2 series for y = 0.05 or higher. LaPO 4 gives a line at -4.56 ppm, similar to signal P3, indicating that P3 not only has 4-fold coordination but also a very similar or identical chemical environment as LaPO 4 . In fact, at a concentration y = 0.10 this signal becomes the dominating contribution. The straightforward interpretation is thus precipitation of LaPO 4 from y = 0.04, which is however in contradiction to the outcome of the X-ray diffraction study.

In order to test whether the site heterogeneity is due to the presence of hydroxide ions in the coordination sphere of the phosphorus ion in the LAMOX phase, spectra of NH 4 H 2 PO 4 and (NH 4 ) 2 HPO 4 were acquired, whose signals however did not show any similarity (0.88 and 1.24 ppm).

Exchange experiments

The 31 P nuclei in the compounds La 2 Mo 2-y P y O 9-y/2 have long relaxation times T 1 in the order of some minutes, which are favourable for chemical exchange experiments to study the ionic motions in solids. When an O 2-ion moves to or away from a P site, the coordination of the respective P sites and the corresponding chemical shift should change. If this chemical exchange is faster than the NMR time scale, the signal frequencies are averaged. However, if it is slower, 2D exchange experiments can show correlations between P ions with different coordinations by cross peaks in the 2D spectrum. The exchange experiments were performed with the samples with y = 0.03 and 0.04 with mixing times of 1 ms to 60 s. However, all the experiments did not show any exchange cross peaks between P1/2 and P3, neither at room temperature nor at higher temperature (350 K).

P-P coupling

In order to understand the nature of P3 as well as the lack of exchange with P1/2, a 2D double-quantum-single-quantum (DQ-SQ) experiment has been conducted by means of POST-C7 28 recoupling for the composition y = 0.04 (Fig. 5A). The spectrum shows one cross peak at -4.6 ppm SQ resonance (P3) correlated to its own DQ resonance position (about -9.2 ppm). In contrast, cross peaks between P3 and P1/2 or P1/2 and itself cannot be observed. The existence of a cross-peak indicates that P3-type of P atoms are relatively close to each other, giving rise to dipolar connectivity. A P-P distance of about 4.5 Å, the average distance between two neighbouring Mo/P sites, causes a dipolar coupling of about 200 Hz, just sufficient for a transfer with an adequate recoupling sequence. P1/2 type P atoms however must be farther apart from each other and from P3 because the respective dipolar coupling is too weak to excite DQ coherences. The existence of cross peaks thus indicates a clustering of P3-type phosphorus on neighbouring Mo/P sites. Fig. 5B shows the intensity of the cross peak as a function of y (it should therefore be correlated to the behaviour of the P3 signal in Fig. 4B). After sufficiently long acquisition, a significant DQ-SQ signal was observed already at y = 0.03, and for y = 0.02 there appears to be a signal just above noise level. In order to resolve the apparent contradiction with the diffraction results and to study the P3 signal in more detail, we have measured the longitudinal relaxation times of the P3 signal in y = 0.04 and of LaPO 4 . The separation from the other signals was achieved by a DQ-filter. As the saturation recovery curves in Fig. 5C show, the relaxation of the P3 signal in y = 0.04 is significantly faster than that of LaPO 4 . The times are T 1 = 78 s for y = 0.04 and 186 s for LaPO 4 .

Excitation of multiple quantum 31 P coherences

In principle, already in a homogeneous distribution of P there is a certain probability that two P atoms replace two Mo on neighbouring sites. If this had been the reason for the observed cross peaks, one should see DQ-SQ cross peaks also for the site P1/2. As a further test, we tried to excite higher order multiplequantum coherences. These spin counting experiments, usually applied for 1 H with their strong couplings, are not very sensitive for 31 P. For LaPO 4 up to 5-quantum coherences could be excited (data not shown). Fig. 5D shows a multiple-quantum diagram of the P3 signal of the compound with y = 0.04, acquired by means of BABA recoupling. 29,30 Coherences of 1, 2 and maximal 3 quanta could be excited with an intensity that is significantly above noise, while corresponding coherences for P1/2 were absent except for the single-quantum signal (not shown). Note that the zero-quantum peaks can contain baseline artefacts and are therefore omitted in the graph. The low statistical probability in connection with the low efficiency of 3-quantum excitation make it unlikely that the 3-quantum signal in Fig. 5D, about one third of the 2-quantum signal, stems from phosphorus atoms in a homogeneous distribution on Mo sites. It therefore confirms that there are clusters of at least three P3 phosphorus atoms on neighbouring Mo sites in the y = 0.04 P-substituted LAMOX. Note that this does not mean by any means a limitation to three; this number is only the experimental limit of multiple-quantum excitation.

Discussion

The results of the different 31 P NMR experiments, in particular the multiple-quantum excitation of P3 and the lack of an exchange peak demand a principle distinction between the character of the P3 site on the one hand and the P1 and P2 sites on the other hand. The P1 and P2 phosphorus are distributed randomly over the Mo lattice positions. At higher P concentrations (β phase), they can be described by one model signal, while below the phase transition concentration (α phase) two are necessary. Line widths and spectral overlap in combination with signal-to-noise ratio, however, do not allow an interpretation in terms of exactly two distinct phosphorus sites. In contrast, a model of two peaks is sufficient to describe a potential multitude of sites, for which more detailed information is not obtainable. In fact, the LAMOX α phase contains 48 sites divided into three groups. The phase transition from α to β phase is reflected in both 139 La and 31 P NMR spectra. Evans et al. have proposed the hypothesis that the β phase of the parent compound is a dynamic average of the α phase. 4 This would match with the transition to only one average 31 P NMR site at the y = 0.03 P-substituted β phase. Earlier crystal structure analysis 3,4 of the La 2 Mo 2 O 9 parent compound in its β phase showed that two of the three oxygen sites (O2 and O3) are partially occupied and one (O1) is fully occupied. In the α-La 2 Mo 2 O 9 phase, the Mo atoms were found in three different basic coordination types of 4, 5 and 6. Under normal conditions, phosphorus in phosphate compounds can however only be surrounded by four oxygens due to its small size. On the other hand one can argue that in LAMOX compounds the structure and distances are dominated by the Mo properties due to the low concentration of the substituting element, and that therefore P could take the coordination of the atom it replaces. The phosphorus chemical shifts in P-substituted LAMOX only show relatively little dispersion when compared to the overall chemical shift range of 31 P. This rather indicates that phosphorus remains in its preferred 4-fold coordination at all concentrations. For comparison, we refer to the behaviour of yttrium in YSZ, where 89 Y NMR shows a shift from 8-to 7-and 6-fold coordination, significant however only beyond much higher concentrations of 20% (7-fold) and 40% (6-fold) Y 2 O 3 in zirconia. 36 The possibility to excite multiple-quantum coherences of P3 and the lack of exchange peaks between P3 and P1/2 indicate the formation of clusters in which Mo is substituted by P. The P3 signal is similar to that of LaPO 4 . The NMR data alone could thus be interpreted in terms of a precipitation of LaPO 4 grains. However, saturation observable by X-ray diffraction begins only at about y = 0.048 (Fig. 1C) as compared to the observation of a DQ-SQ signal already at y = 0.03 or even 0.02 (Fig. 5B). The different longitudinal relaxation behaviour (Fig. 5C) indicatesif one presumes a similar chemical environmenta different mobility of the phosphorus in P3 compared to LaPO 4 . It makes it unlikely that P3 stems from real LaPO 4 grains. A solution to this apparent contradiction could be a low dimensional (non-3D) clustering of P atoms on Mo sites. In the first article on LAMOX oxide ion conductors, 1 a comparison was made between the cationic arrangement of the β-La 2 Mo 2 O 9 -type structure and that of the LaPO 4 monazite structure: each can be considered as the periodic micro-twinning of the other, through cationic planes parallel to a LAMOX crystal cell face. One can therefore consider that, if P cations tend to cluster in a LAMOX-type structure, they will preferentially occupy such a plane and induce a twinning corresponding to a LaPO 4 -type slab between two LAMOX type grains (see Fig. 6). This type of clustering has a 2D dimensionality. Above such a substitution rate of 2 mol% (i.e. 1.9 mol% P randomly distributed in LAMOX grains and 0.1 mol% P clustered in twinning planes), the clustering loses its 2D character, which results in 3D LaPO 4 crystallites as detected by X-ray diffraction.

TEM images of the y = 0.05 sample, recorded on a 200 kV side-entry JEOL 2010 electron microscope, show that the grains are formed of domains with size ranging from 5 to 100 nm. The domains, well crystallized, are slightly disoriented with respect to each other. The reason for this domain formation, not observed in any other cubic substituted LAMOX phase, might

Conclusions

LAMOX with molybdenum substitution by phosphorus is a representative of cationic substitution systems where saturation is already reached at low concentrations. It is a favorable example for NMR studies of the structural differences induced by a low amount of substituent because of the relatively high sensitivity of 31 P in NMR and the high sensitivity of the large quadrupolar coupling of 139 La towards changes in the environment. Substitutions between 0.5 and 5% are a challenge not only for NMR; also X-ray diffraction is in some respects at the limit of precision, in particular with respect to the small changes of the unit cell volume parameter. Our study shows how observations with one technique are detectable in the other one: while the phase transition from α to β crystallographic phase is of course very easily detectable and identifiable by X-ray diffraction, it is a secondary effectbut visiblein 31 P NMR, and it is reflected as a sudden increase in the mean quadrupolar coupling of 139 La. The multitude of crystallographic La sites as well as the statistics in occupation of the oxygen sites leads to 139 La signals that rather resemble those of amorphous rather than crystalline phases; the quadrupolar couplings apparently follow a relatively broad distribution. Well comparableand controversialis the determination of the saturation limit of phosphorus: while diffraction does not show a typical signal of LaPO 4 precipitates below y = 0.05, 31 P NMR shows a LaPO 4 -like signal and moreover couplings which identify neighbouring P atoms already at y = 0.03. 31 P relaxation however indicates that the environment around the respective P atom is not completely identical with LaPO 4 . We try to explain the apparent contradiction by a hypothetical model that contains two-dimensional LaPO 4 areas as interfaces of crystal twinning, representing the beginning of precipitation which only at higher concentrations becomes three-dimensional and thus detectable by diffraction.

LaPO 4 .

 4 The model function represents quite well the special characteristics of the LAMOX 139 La signal. The environments of the 48 crystallographically different La sites with their different

Fig. 2 (

 2 Fig. 2 (A) 139 La NMR spectra of La 2 Mo 2-y P y O 9-y/2 for y = 0.00 to 0.05 and 0.10 and the central transition of LaPO 4 . Black line: fitted model spectra. (B) Peak quadrupolar coupling constant as a function of the phosphorus content. Fig. 1 (A) X-ray diffraction patterns of the La 2 Mo 2-y P y O 9-y/2 series. Stars indicate superstructure peaks of the monoclinic cell. The excerpt at 46.5-48.5°shows a phase transition from a monoclinic (α) phase to a cubic (β) phase. (B) Unit cell volume as a function of P concentration. (C) Intensity of the (120) diffraction peak of LaPO 4 (2θ = 28.6°, not visible in Fig. 1A) as a function of P concentration. The line corresponds to a linear regression fit of the data points representative of compositions where a impurity peak is observed (y ≥ 0.05).

Fig. 3

 3 Fig. 3 31 P NMR spectra of La 2 Mo 2-y P y O 9-y/2 with y = 0.01 to 0.04, normalized to the P content (blue line). Deconvolution by joint fits of the two groups y = 0.01-0.02 and 0.03-0.04, respectively, with two signals each: black dotted lines: separate partial model signals, black solid lines: sum.

Fig. 4

 4 Fig. 4 Normalized populations of the three phosphorus sites as a function of the phosphorus content y. (A) Joint fit of all four 31 P spectra with a three-site model. (B) Joint fits of the two groups y = 0.01-0.02 and y = 0.03-0.04, respectively, with a two site model each.

Table 1

 1 Common curve fit parameters in two different joint fit approaches. The intensities are shown in Fig. 4. The first approach is a three site model for all four spectra together, the second joins the spectra in the α and in the β phase, respectively, in a two-site model. Errors are estimated to be in the order of 0.1 ppm for chemical shifts and line widths, 20 percent points for the Gaussian fraction, and 10% for the intensities Joint fit of 3 sites model 2 groups, 2 sites model y = 0.01-0.04 y = 0.01-0.02 (α) y = 0.03-0.

Fig. 5 (

 5 Fig. 5 (A) 31 P DQ-SQ MAS spectrum of La 2 Mo 2-y P y O 9-y/2 with y = 0.04. On top 1D signal (black) with deconvolution (violet, blue) (B) Intensity of the DQ-SQ peak (2P3 × P3) as a function of y. (C) Saturation recovery of the DQ-filtered P3 signal of y = 0.04 and of LaPO 4 . Relaxation times, y = 0.04: T 1 = 78 ± 20 s; LaPO 4 : T 1 = 186 ± 10 s. (D) Multiple-quantum excitation (spin counting) diagram.

  This journal is © The Royal Society of Chemistry 2012 Dalton Trans., 2012, 41, 5696-5703 | 5701 be related to distortions caused by the formation of LaPO 4 -type planes of a limited extension.

Fig. 6

 6 Fig. 6 Hypothetical model of La 2 Mo 2 O 9 /LaPO 4 /La 2 Mo 2 O 9 twinning consistent with both NMR and X-ray diffraction results. Compare with the LAMOX and LaPO 4 structures in ref. 1. Only the cations are shown, oxygens are omitted in order to simplify the figure. The twinning between two La 2 Mo 2 O 9 -type domains creates a 2D LaPO 4 -type slab.

  5699to 21 (3 peaks × 3 common parameters, i.e. position, width, Gaussian percentage, plus the intensities of 3 peaks × 4 spectra).

							04 (β)
	Signal	P1	P2	P3	P1	P2	P1/2	P3
	Chem. shift [ppm]	0.05	-2.81	-4.59	-0.39	-3.41	-2.78	-4.60
	Line width [ppm]	4.87	6.84	2.28	5.59	7.26	6.74	2.10
	% Gaussian	0.06	0.73	0.62	0.41	0.80	0.72	0.70

This journal is © The Royal Society of Chemistry 2012 Dalton Trans., 2012, 41, 5696-5703 |

Downloaded by UNIVERSITE DU MAINE-SCD on 10 July 2012 Published on 09 March 2012 on http://pubs.rsc.org | doi:10.1039/C2DT12389D View Online

Acknowledgements

The authors would like to thank Dr Monique Body and Prof. Alain Bulou for valuable discussions and Dr Stéphanie Kodjikian for collecting TEM images. The financial support of the Region Pays de la Loire is gratefully acknowledged (Convention No. 2007-11860).

Notes and references


Sami Ben

Elhaj Salah

Azdine Nait-Ali
email: azdine.nait-ali@ensma.fr

Effect of aggregation phase on the elastic properties of a heterogeneous material

Keywords: Heterogeneous material, Inclusion, Covariogram, Equivalent morphology concept, Elastic properties, Homogenization, Tomography

In this study, the technique of homogenization and morphological analysis is used in order to quantify the relationship between two morphologies of voided material: two types of structures are considered. Each one is characterized by a scale which was observed. Structure 1 is an elastic matrix containing a random distribution of spherical inclusions corresponds to a smallest scale (microscopic scale). Structure 2 is a composite material consisting of spherical inclusions distributed randomly in an elastic matrix corresponds to a largest scale (mesoscopic scale). A well-known mathematical morphology technique is used in this investigation. The parameters of mathematical morphology, such as covariance, are determined and presented the relationship between the parameters of the two structures. All the obtained results suggest us to define a new concept called "the equivalent morphology". This concept is to replace the first composite containing inclusions with radius R1 by other fictif material containing inclusions with radius R2 that was expressed as a function of the parameters derived from the analysis of the initial morphology. These two composites 1 and its equivalent morphology 2 give the same effective elastic properties, but not necessarily with the same size of the deterministic representative volume element, DREV.

Introduction

Sectors were developed, like aeronautics or space require the development of more efficient materials. Composites, with their various mechanical properties, are materials that improve the quality of the face material to some use (lightness, stiffness to a force, etc.). However, their morphological characteristics and mechanical behavior attached to each phase are difficult to be modelized in this class of materials.

In the following, only a voided material with random distribution of spherical inclusions in an elastic matrix under a displacement are considered. Thus, in this context, much progress has been made in characterizing the microstructure of statistically homogeneous two-phase random media, via covariance and integral range(IR). This is the case of Torquato [START_REF] Torquato | Morphology and effective properties of disordered heterogeneous media[END_REF], Jeulin and Ostoja-Starzewski [START_REF] Jeulin | Mechanics of random and multiscale microstructures[END_REF], for two-phase heterogeneous media, [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF] [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF] for Voronic mosaics, [START_REF] Kanit | Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry[END_REF] [START_REF] Kanit | Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry[END_REF] for ice cream, Willot and Jeulin [START_REF] Willot | Elastic behavior of composites containing boolean random sets of inhomogeneities[END_REF] for porous materials, Jean et al. [START_REF] Jean | A multiscale microstructure model of carbon black distribution in rubber[END_REF] for elastomer materials, Jeulin [START_REF] Jeulin | Morphology and effective properties of multi-scale random sets: A review[END_REF] for 3D composite materials and Painbon et al. [START_REF] Paiboon | Numerical analysis of effective elastic properties of geomaterials containing voids using 3d random fields and finite elements[END_REF] for two-phase heterogeneous media with overlapping and non-overlapping spherical particles. We studied voided structures by measurements on 3D images [START_REF] Delarue | 3d morphological analysis of composite materials with aggregates of spherical inclusions[END_REF], such as the covariogram [START_REF] Jeulin | Random texture models for material structures[END_REF] and [START_REF] Azdine | Catching the time evolution of microstructure morphology from dynamic covariograms[END_REF] which allows us to have maximum of microstructural information for the materials studied and permits us to have a good morphological description. Many morphological information can be determined from this statistical analysis method as the volume fraction, the isotropy or the ergodicity of the distribution of inclusion. The aim of this paper is to expand the concept of the equivalent morphology, thus, by using the data derived from this statistical analysis and the analytical estimation for the effective elastic properties. Hashin and Shtrikman [START_REF] Hashin | A variational approach to the theory of the effective magnetic permeability of multiphase materials[END_REF] - [START_REF] Hashin | The elastic moduli of heterogeneous materials[END_REF] proposed a more efficient framework and several homogenization techniques for two-phase materials named Hashin Shtrickman Bounds or known as up-scaling. It is based on solving a problem of inclusions embedded in a finite homogeneous matrix. We can define an equivalent heterogeneous composite made of two phase as well. Then, an optimization by an ellipsoidal model will be proposed. Finally, numerical and analytical results will be compared with the experimental for an EPDM under a diffusion of hydrogen, but in this case we will only focus on the mechanical approach and not on the phenomeno of diffusion . Indeed, under a decompression, the damage on these materials involves the nucleation and growth of spherical cavities.

Morphological characterization

We introduce the basic elements retained to characterize the morphology of random microstructures. The morphological tools for these aspects are the covariance.

covariance

The covariance is very useful characteristics for the description of the size, shape and spacial distribution of a given particle. Covariance C(x, x+h) [START_REF] George | Les variables regionalisées et leur estimation[END_REF]- [START_REF] Matheron | Random sets and integral geometry[END_REF] and [START_REF] Serra | Image analysis and mathematical morphology london[END_REF] is defined by the probability P for two points, separated by the vector h, to belong to the same stationary random set B. We have:

C(x, x + h) = P {x ∈ B, x + h ∈ B} (1)
See [START_REF] Azdine | Catching the time evolution of microstructure morphology from dynamic covariograms[END_REF] that the covariogram can be used in dynamic case. Such a function corresponds to the probability to intercept other particles at a distance h from a reference particle.

More particular properties of the covariogram are listed blow.

(P1) : It can be noticed first that the volume fraction of set B is directly given by covariogram for h=0: C(h)=p.

(P2) : If the events x {x ∈ B and x+h ∈ B} become independent as the magnitude of h tends to the higher values, moreover, C(h) converges to an asymptotic theoretical value equal to p 2 , then the probability distribution is ergodic, meaning that the inclusions distribution statistic is homogeneous, so whatever the draw, the macroscopic behavior will be the same and the occurrence probability is the same at any point of the volume.

(P3) : If the horizontal asymptote values is the same for all direction, the microstructure is isotropic at the macroscopic scale. But values can be different depending on the direction. In this case, the material is anisotropic.

(P4) : The first intersection between C(h) and the asymptote corresponds to the correlation length in the z direction, for example Dc(z). It is also called Integral Range by some authors [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF]. It is defined by:

Dc(z) = min h∈R * {Cz(h) -p 2 = 0} (2)
This distance corresponds to the maximal distance of statistical influence of the inclusion phase B. It provides information about the minimal size of the domain over which the volume is statistically representative. If the heterogeneous medium contains clusters of inclusions, Dc(.) provides an estimate of the statistical average size of the cluster in each direction. If Dc(.) is constant regardless of the direction, the material is isotropic at this scale [3]

Equivalent morphology concept

At a microscopic scale, the characteristic variables which are derived during the analysis of the initial morphology, allow us to define the concept of equivalent morphology. This concept represents a composite with inclusions corresponds to a mesoscopic scale, statistically equivalent to the original structure. It means replacing each cluster by one effective inclusion whose parameters are given in terms of the different distances from where any phase affects the neighborhood. It is called correlation length. In our case, it should be noted that the equivalent material will follow the same law of distribution of clusters in the original material. Thus, we are able to consider the assumption of ergodicity. Then, an optimization by an ellipsoid is realized in order that the mechanical fields around these iclusters are similar to the inclusions presented in the original material. The spferical and ellipsoid models (original and equivalent morphologies ) are given in Fig. 1 and2. Many different data can be estimated from this statistical analysis method through the covariogram for the initial images. In this part, only the correlation lengths Dc(.) are considered. In Fig. 3, it seems that we increased gradually the number of the clusters in our matrix in order to come closer to reality and have a significant results in the equivalent morphology. After the presentation of the basic morphological parameters, these available notions are used to characterize the geometry and the morphology of random composite. The objective of this paragraph is to find a relationship between the morphological parameters of initial and equivalent morphology. This relationship is used to define the concept of equivalent morphology in the two types of model: spherical and ellipsoidal. They are mentioned in the following equations:

spherical model : Requ= < Dc > 6 ( 3
)
ellipsoidal model :

     Rmax = M ax{Dc(.)} 2 Rmin = M in{Dc(.)} 2 (4)
The obtained results, based on laws of mathematical morphology that the error between the two microstructures examined on a macroscopic scale (Asymptote) is insignificant and does not overpass 2 percent. However, we noted that there is a slight increase on a local scale (Correlation lengths) because the inclusions are empty. Also, it is due to the morphology of local field. Thus it appears that there is a small oscillation because the clusters are the same in the original material with meadows of rotation. Using this condition, the equivalent morphology concept can be proposed for the voided materials. Figures 1 and2 describe schematically this concept that consists to replace the structure containing clusters by structure juste containing inclusions. These two structures give the same elastic properties but not necessary the same DREV.

Material, Microstructure and conditions

Loading condition

The boundary condition is a main challenge in modeling a structure by finite elements and it is the application of a force and/or constraint. For the particular model, two types of boundary condition were applied. The first one is that the model is fixed in the bottom surface and subjected to a displacement in top one.

Morphology and Meshes

In this section we presented our 3D model in the context of the linear elasticity. It is a two-phase heterogeneous material with an elastic matrix containing a random distribution of spherical inclusions. There is a distance between each two neighbor's center inclusions that avoids the connection between them. We compared two types of morphologies: The intial composite and the equivalent one. The finite element mesh, related to our model, is obtained by using the tetrahedral as type of elements with the free technique. All our composites are characterized by a volume of fractions of the inclusions phases.

Ovrage stress, strain and displacement

The first step in a numerical study is to create a geometry that represents the microstructure of the studied material. This geometry is a numerical representation of real microstructures. The finite element solver ABAQUS with its integrated interface of visualization and modeling (ABAQUS / CAE) is used to make our 3D model. Calculations were done using the finite elements code "Foxtrot" that was developed in the DPMM team and finally for the post-processing and visualization of our simulations, we had used Paraview with a VTK file as an imput.

In the present work, the definition of overall mechanical quantities usually defined as averages of microscopic and mesoscopic ones, must be explicited, see [START_REF] Vincent | Porous materials with two populations of voids under internal pressure: I. instantaneous constitutive relations[END_REF]. Let V denotes the total volume. The porous material is subjected to a displacement. In order to compare the two composite 1 and 2, it is classical to define the overall stress and strain as the volume averages of the corresponding local stress and strain fields over a volume V so we can write :

< σ >= 1 | V | V σ dx , < ε >= 1 | V | V ε dx (5)
An example of the obtained numerical results is presented respectively in Fig. 5, 6 and 7. These figures show that the mechanical fields in the two morphologies are nearby. Then, based on the obtained numerical results, a comparison has been done in order to quantify the difference. According to the results in Fig. 8, it seems that they are adequate in the macroscopic scale even the error does overpass 4 percent.

Homogenization

In this part, several analytical homogenization techniques are used to estimate the effective linear elastic properties (ELEP), of three dimensional (3D) microstructures consisting of spherical inclusions distributed randomly in an elastic matrix.

Scales and structures

The present work with primary and secondary mophologies involves two different scales. The smallest (or microscopic) corresponds to composite 1 and the seconde one is the largest (or mesoscopic). The up-scaling procedure is able to modelize a heterogeneous material and estimates the linear elastic properties of voided material subjected to displacement on its top surface

Identification with Hashin-Shtrikman bounds

In the early age, effective properties in porous material are estimated by using an analytical technique of homogenous. Thus, they were derived from Finite Element Calculations. In this context, Hashin and Shtrikman [START_REF] Hashin | A variational approach to the theory of the effective magnetic permeability of multiphase materials[END_REF] suggested a more useful framework for two phase materials named HS bounds. This technique aims to explain and solve the problem of a spherical inclusions randomly distributed in an elastic matrix. Compressibility and shearing ratios are associated to E and ν , so we can write:

K = E 3(1 -2ν) (6)
µ = E 2(1 + 2ν) (7)
Then, based on the homogenization process and using all the relations that are given bellow, we focus on estimating the values of young's modulus of equivalent inclusion. In this context we use the methode of optimisation based on the function of minimisation. It allows us to find the minimum of unconstrained multivariable function using derivative-free method. Fig. 9 shows the procedure of the up-scaling technique.

1 k HS + k * = cm k m + k * + cf k f + k * with k * = µ 0 (8) 1 µ HS + µ * = cm µ m + µ * + cf µ m + µ * with µ * = k 0 µ * k 0 + 2µ 0 (9)
and there are two types of bounds for this technique:

• HS Up-Bound: k 0 =k m et µ 0 = µ m • HS Down-Bound: k 0 =k f et µ 0 =µ f
With the results obtained by numerical simulation coupled with a process of homogenization, this paragraph is devoted to make a comparison in the mechanical field as displacement, stress and strain between the two composites under a displacement in the linear problem. Fig. 9 shows an example of the results after a numerical simulation in Finite Element Code. Some results of this comparison are given in Fig. 10. It seems that the results are suitable in macroscopic scale. Yet there was a small increase in stress level compared to strain and displacement. In other words, in this case we also decreased the properties' gradient. Thus the error in each mechanical field is constant because the clusters are distant and do not interact. Also, the same motif was repeated.

Teste with EPDM Material

Tomograph and meshing of 3D images

This part is devoted to apply all our numerical simulations coupled with laws of mathematical morphology to a real elastomer EPDM subjected to a diffusion of hydrogen, used in the new energy. Thus, One of the best solutions available today to observe changes in the microstructure due to damage is High Resolution X-ray Computed Tomography (HRXCT) [START_REF] Orlov | Simulation of damage percolation within aluminum alloy sheet[END_REF] [START_REF] Buffiere | In situ experiments with x ray tomography: an attractive tool for experimental mechanics[END_REF]. It is not the purpose of the present paper to describe the tomography technique in detail. Many preceding papers have been published and can be referred to (see, for instance, the review in [START_REF] Baruchel | Advances in synchrotron radiation microtomography[END_REF] and [START_REF] Salvo | 3d imaging in material science: Application of x-ray tomography[END_REF]) illustrate the principle of X-ray tomograph. Therefore, the mesh in initial structure is generated using the 3D pictures derived by this technique that converts the voxel in hexahedral elements, but in the case of the equivalent morphology, we have studied the convergence of mesh and using the required mesh by the tetrahedral elements.

Results and discussion

The obtained set of numerical results coupled with laws of mathematical morphology is used to quantify the difference between two different random structures of porous material. Two cases of real images will be analyzed and discussed.

The main objective of this work is to make a morphological simplification which allows us to replace one microstructure via another, named here equivalent morphology. Also it ensures shorter calculation times. Table 1 shows an example of two different 3D images of real materials treated and it is clear that the error in the stress field did not exceed 10 percent. Concerning the time calculations, we are able to optimize approximately 40 times.

Conclusions

The main objective of this study is to quantify the difference between the two studied morphologies. The obtained results allow us to replace a composite containing clusters by other fictive composite containing inclusions that their radius was expressed as a function of the parameter (correlation lengths) derived from the analysis of the initial morphology. Thus a new concept is proposed and called "the concept of equivalent morphology". It was noted that these two morphologies have the same elastic properties, but not necessarily they have the same size of the deterministic representative volume element, DRVE.

Homogenization was considered the second main objective in this paper. In reality, it can currently consist of a step for some analysis of global behavior. Particularly, it is sensitive to the local responses of heterogeneous materials. As an example, the initiating of the damage by microcracks or composites microcavitation. Hence this work are also focuses on the prediction of the effective elastic properties of heterogeneous materials using a conventional analytical method of homogenizing called Hashin-Shtrikman Bounds. An extension of this study to other application (microstructures type, with pores having either different shape and containing an anisotropic material) will be discussed in the following works.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Concept of equivalent morphology: spherical model

Fig. 3 .

 3 Fig. 3. Variation of raduis of the inclusion in the equivalent morphology

Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 4567 Fig. 4. Initial morphology: (a): displacement, (b): stress, and (c): strain

Fig. 8 .

 8 Fig. 8. Process of homogenization

Fig. 9 .

 9 Fig. 9. Error between initial and equivalent morphology investigated in the macroscopic scale

Table 1 .

 1 results of calculations

		Error σ in % Time ratio
	Example
	1	9,18	44
	2	9,73	42


Jean-Luc Koning

Guillaume François

Yves Demazeau

J.-L Koning

Formalization and Pre-Validation for Interaction Protocols in Multiagent Systems

In this article, we take up validation techniques used so far in the domain of communication protocols in networks and adapt a similar method for the validation of interaction protocols in multiagent systems. First of all, we rely on Petri nets in order to model such protocols. Second, we move to a protocol validation stage through the analysis of a Petri net model. We also give a very brief presentation of a software tool that enabled us on one hand to run the analysis, and on the other hand to simulate the unfolding of typical scenarios for sequences of exchanged messages. Finally, we extract the global protocol properties and infer consequences on its prevalidity. The phrase pre-validity is used in this paper instead of validity since one is not interested in proving a complete validity of the negotiation protocol but rather a partial one, i.e., checking the satisfaction of some elementary properties.

Introduction

Multiagent Systems (MAS) are concerned with coordinating behavior among a collection of autonomous intelligent agents. Such software agents are designed to reconcile their own interests with the constraints implied by the other agents. The main justifications for making use of a multiagent model comes from its distribution, openness, flexibility and robustness features [START_REF] Th | A negotiation framework for antonomous agents[END_REF].

In this setting, a negotiation process translates into interactions among agents. Interaction is more than an exchange of messages. Issues associated with it, are: models of agents (beliefs, goals, representation and reasoning), interaction protocols (an interaction regime that guides the agents) and interaction languages (languages that introduce standard message types that all agents interpret identically). One way to structure message passing during agent communication is through protocols.

The design of interaction protocols for multiagent systems has already been tackled in the literature like in [START_REF] Burmeister | Generic, configurable, cooperation protocols for multiagent systems[END_REF]. These references usually discuss low level communication issues or tools for designing interaction protocols. In the remainder of this article, we will take up validation techniques used so far in the domain of communication protocols in networks [START_REF] Diaz | Modelling and analysis of communication and cooperation protocols using petri net based models[END_REF] and prototyping of distributed systems [START_REF] Estrailler | Applying petri net theory to the modelling analysis and prototyping of distributed systems[END_REF]. We will adapt a similar method for the pre-validation of interaction protocols in multiagent systems. The phrase pre-validity is used in this paper instead of validity since one is not interested in proving a complete validity of the negotiation protocol but rather a partial one, i.e., checking the satisfaction of some elementary properties.

First of all, we will rely on Petri nets in order to model such protocols. For lack of space we will briefly introduce related work both in the domain of communication networks and the domain of multiagent systems.

Second, we will move to a protocol validation stage through the analysis of a Petri net model. We will also give a very brief presentation of a software tool that enabled us on one hand to run the analysis, and on the other hand to simulate the unfolding of typical scenarios for sequences of exchanged messages.

Finally, we will extract the global protocol properties and we will infer consequences on its (pre-)validity.

Petri Nets and their Use in DAI

Modeling an interaction between two agents of a MAS comes down to building a per-agent Petri sub-net as well as a subnet for the virtual medium, i.e., modeling the overall service that enable to rout messages between agents.

In [START_REF] Ferber | Les systèmes multi-agents, vers une intelligence collective[END_REF], Ferber discusses a Petri net modeling of an interaction between two agents. It shows how agent A can ask agent B to provide a service. There are at least two reasons for criticizing that approach.

• The protocol that is described only deals with two agents that cannot play a symmetric role. Their statute is different; one requests and the other answers the request. • The exchanged messages modeled by tokens in the Petri net represent hypotheses or proposals. When a hypothesis is modified by an agent the original one is lost even though its processing by the agents is not over. This stems from the fact that there is a given number of tokens that circulate in the net: one per agent which conveys its internal state and one and only one for the message that is being processed. As a consequence, this Petri net does not model a simultaneous processing of several proposals derived from one initial hypothesis.

A real interaction protocol provides a detailed description of the possible interactions between n (more than two) agents. But there is no clear approach in the DAI literature for modeling interactions that take place between more than two agents at one time. This is probably due to the fact that interaction between agents in DAI are rather grounded on Speech Acts theory than on distributed systems, which constitutes a serious bias in our point of view. As an example, the cooperative learning protocol given in [START_REF] Singh | Adaptation based on cooperative learning in multi-agent systems[END_REF] only deals with interactions among two agents.

Other uses of Petri nets in Distributed Artificial Intelligence include planning. In [START_REF] El-Fallah-Seghrouchni | A framework for agent coordination oriented distributed planning and resource sharing managment[END_REF] a formalization of concurrent plans is proposed based on recursive Petri nets which is a suitable formalism for distributed planning. They also tackle there the management of potentially shared and conflicting resources through a recursive model.

Petri Nets For Representing Protocols

Modeling interactions between agents in a MAS comes down to building a Petri sub-net per agent and a Petri sub-net that represents the virtual medium. That is what we address in this section.

Models of Virtual Mediums Linking Communicating Entities

Let us first model a perfect virtual medium linking n communicating entities. In a perfect virtual medium the possible loss of messages is not modeled and the transmission capacity is infinite. Such medium must offer the n entities the capability to send and receive several sorts of messages. Figure 1 In a same medium linking n entities and capable of exchanging m different sorts of messages, the number of links would be n • (n -1) • m. Clearly this modeling technique is not suitable for a great many entities. The virtual medium's complexity should not depend on the number of communicating entities or at least not be proportional to its square. One could also rely on the bus communication principle, where there is a bus type line of transmission for any kind of message. Figure 2 shows a bus type virtual medium that links three communication entities and enables the exchange of two types of messages.

Box (i, j) carries out the following functions:

• collect j-type messages sent by entity i and propagate them on line j in both directions, i.e., toward box (i -1, j) and box (i + 1, j).

Entity (i-1) Entity (i) Entity (i+1) Box (i+1,j) Box (i-1,j-1) Box (i,j-1) Box (i+1,j-1)
Box (i-1,j) Line for message (j) Line for message (j-1) • transmit j-type messages coming from box (i -1, j) to box (i + 1, j) and vice versa, while duplicating them for entity i.

Box (i,j)

Figure 3 details box (i, j)

.

If [cond.] message (j) send E1 Entity (i) E2 E3 E4 Receive message (j) Box (i+1,j) Box (i,j) Box (i-1,j)
are the entity's E1, E2, E3 and E4 internal states Such a virtual medium liking n entities that is capable of exchanging m different sorts of messages, consists of m lines like those shown in figure 2. Hence, the medium's complexity does not depend on the number of communicating entities. The drawback with such a medium is that receivers of a message do not get it simultaneously. However, this is no problem when modeling asynchronous interactions.

As a summary, there are two techniques for modeling communication mediums that link n entities and support m different sorts of messages. One provides a loosely structured model whose complexity is O(m • n 2 ) links. The other provides a structured one whose complexity is O(m) links (or O(m • n) boxes).

Petri Net Models for Assistant Agents

In a Petri sub-net that models a communicating entity [START_REF] Ferber | Les systèmes multi-agents, vers une intelligence collective[END_REF], places represent internal states of those entities, transitions correspond to either the synchronizing of the reception of messages or actions agents perform, e.g., sending a message as a speech act.

In our framework [START_REF] Demazeau | From interactions to collective behaviour in agent-based systems[END_REF], the agents' internal states and the actions the agents can accomplish are gathered in a state transition graph that describe their behavior. While Ferber's approach focuses on internal states we focus on behavioral states. See [START_REF] Koning | An approach for designing negotiation protocols in a multi-agent system[END_REF] for a detailed approach on the designing of interaction protocols in MAS. Figure 4 exemplifies a state transition graph for a simple negotiation protocol.

In this graph arrows go from the sending agent's state prior to sending a message and end on a state the addressed agent(s) reach after receiving the message. In its initial ("Init") state an agent can send a proposal ("propose") which makes the receiving agent enter state "Opinion?". When an agent receives an modification (respectively acceptance) of proposal from another agent that is in state "Opinion" (respectively "Agreement?"), then it enters state "Opinion?" (respectively "Agreement?") too. In the "Opinion?" state an agent can also "refuse" a proposal which makes the receiving agent go to state "Agreement?". When the agents come to an agreement or want to abort the current negotiation they can stop negotiating ("report end") which makes the receiving agent enter state "End". Let us see now the Petri sub-net model for agents whose behavior corresponds to the use of such a negotiation protocol.

Places are as follows:

• For internal states: P:init, P:opinion?, and P:end. We have merged states "Init" in state "Agreement?". • For the receiving and storing of messages: P:S-propose, P:S-accept, P:S-refuse, P:S-modify, and P:S-report-end.

Transitions are:

• For the sending messages: T:propose, T:accept, T:refuse, T:modify, and T:report-end. • For the reading of a received message: T:R-propose, T:R-accept, T:R-refuse, T:R-modify, and T:R-report-end.

An agent is in state s if the place that corresponds to this state holds a mark (token). This mark that is unique to each agent will be called a state-mark. It can only go to places denoting the agent's internal state.

The other kind of mark that is present denotes the messages sent. They are only located in places where messages can be received. Hence, in such a place there is as many marks as messages of that type received and not yet read by the agent.

At this point, places, transitions and marks are identified. A Petri sub-net modeling the agent can be built. Places and transitions need to be linked by an arc so that a change of internal state caused by various actions matches the description of an agent's behavior (figure 1). Figure 5 shows the corresponding Petri sub-net. There must be one token in the P:init agent's place-the state mark. This indicates the agent is in its initial state. From there, the agent is able to choose which action to perform and which transition to fire. This current Petri sub-net does not specify the way the transition to fire is chosen. As a matter of fact, interaction protocols do not provide this kind of information since this is the concern of the agent's decision. It falls within the application domain.

Complete Negotiation Protocol Model

A global Petri net can now be built in order to model our negotiation protocol. Petri sub-nets that correspond to agents have been defined. The virtual medium still needs to be chosen. Modeling the protocol and performing validation tests on a real application (with more than two agents) does not depend on the type of virtual medium. When there are very few communicating entities the loosely structured model whose complexity is O(m • n 2 ) links is bearable (see figure 1). On the other hand, for simulation reasons, when agents are complex and numerous there is a need for the simplest possible medium (i.e., bus type medium).

For space reason, no example of a final Petri net is shown here. But let us just mention that with a perfect virtual medium (the loosely structured one) places for getting and storing messages prior to their reading and storing by the agent, like P:S-propose, P:S-modify, etc. (see figure 5) belong in fact to the medium. Indeed, as shown in figure 1, the medium contains one place per agent and per type of message received. With a bus type medium, those places also belong to the medium and are duplicated in each of its boxes (see figure 2 and3).

What is the theoretical significance of this complete negotiation protocol modeling? A protocol is a language L whose vocabulary V is the set of possible messages. V * denotes the set of V 's elements combinations, i.e., the set of all message sequences. The language L associated with the protocol is a sub-set of V * . Given these definitions, we are building an automata A1 by means of a Petri net, and whose language L(A) is such that L(A) ⊆ L. In fact L(A) is the restriction of L to a special case where n communicating entities are concerned. However, L(A) can be viewed as a good approximation for L. The protocol validation can thus be carried out with the n communicating entities.

Protocol Pre-Validation

Theoretical Approach

Performing a protocol pre-validation boils down to conveying expected services and then check whether there is adequacy between those services and the ones the protocol really supplies. Needless to mention those services correspond to a verification of general properties that should characterize any protocol. Those properties have nothing to do with the application's semantic. We will call these elementary services.

Figure 6 explains the theoretical approach for verifying protocol properties. The first step is to go through a formal modeling of the protocol given in figure 4. This has been done in the previous section. Let us now convey the expected elementary services as far as our negotiation protocol. This amounts to check whether it satisfies general properties necessary to any protocol [START_REF] Diaz | Modelling and analysis of communication and cooperation protocols using petri net based models[END_REF][START_REF] Courtiat | Petri nets are good for protocols[END_REF].

• The first property one may wish is that the global protocol model, represented as a Petri net, be bounded. Indeed, a not bounded model means the protocol leads to a process with an infinite number of states. • The second property is that the global protocol model be quasi-livable. If such a property is satisfied, all the model's transitions are fireable starting from the initial marking. This ensures all protocol's messages are used.

• Thirdly, it is important to know whether the protocol leads to deadlocks and if so what they are. Obviously in case of a negotiation protocol, all markings associated with a model's deadlock 4 (or sink state) must contain a token in each of the agents' "End" state.

• The last property one may wish to see satisfied is that the protocol's model must enable ending with the desired sink state situations. In other words, the negotiation process infered by the protocol must end. A number of other properties can be subjected to verification. However, these are specific to negotiation.

Having discussed the theoretical issues of a protocol's prevalidation principles, we will now detail more computational aspects

Use of a Simulation/Validation Tool

Editing the Petri nets that model negotiation protocols and analyzing certain general properties related to the net's structure have been carried out by means of a software tool [11]. Among other things, this tool allows for a graphical representation of Petri nets, their simulation, and the search for properties. In order to check on these properties, this software conducts an analysis of the net by means of its states; the state of a Petri net being given by its marking. At first, the software sets up the graph of reachable markings, i.e., the list of states the net can reach and the sequences of fireable transitions to go from one state to another.

The graph of markings may be infinite and therefore cannot be built. It is thus necessary to build the coverability graph which is some condensed version of the graph of markings. The purpose of such a graph is to obtain a sub-set of states that gives enough information on the set of reachable states, especially when it is infinite [START_REF] David | Petri nets and Grafcet: tools for modelling discrete event systems[END_REF].

Subsequently, seeking properties consists in examining the coverability graph according to a procedure specific to each of the properties. 4 A deadlock is a marking such that no transition is enabled.

Properties and Validity

Properties of the Petri net that model our negotiation protocol are given in the appendix of this article. In this section we discuss the properties found and infer the elementary services actually provided by the protocol.

1. The net is bounded. This implies that (1) this Petri net leads to a finite number of states. In other words, the negotiation process thus modeled has a finite number of states. Furthermore, there is no infinite sequence of transitions in this net. This does not help us identify the final states of these sequences. On the other hand, it implies that (2) the negotiation process does not enter cyclic or infinite sequences but always reach a terminal state. 2. In the final Petri net that model our negotiation protocol with three agents, we have made several hypotheses in order to restrict the number of reachable states (for a simulation purpose): agent 1 is the only one to send a proposal (agents 2 and 3 cannot "propose": T:propose-2, T:propose-3), agents 2 and 3 can only send modifications (agent 1 do not "modify": T:modify-1), and agent 1 is the only agent allowed to end the negotiation process (agents 2 and 3 do not "report end": T:report-end-1, T:report-end-2). Consequently, agent 1 does not receive and therefore read any "propose" message (T:R-propose). Indeed, messages are not sent to the sending agent even when in broadcast mode. Thus, the only non quasi-live transitions are those six transitions. In particular, to each forbidden transition in an agent there corresponds an equivalent transition in a neighbor agent that is not forbidden and quasi-live. Had we not precluded some transitions on purpose, all would have been quasi-live. Thus, (3) the protocol leads to negotiation processes in which no message is unused. 3. The expression of the terminal states with the coverability graph shows that the terminal markings have a common feature: none have a mark in place P:init-x and all have a mark in place P:end-x. In other words, the three agents reach their internal state "end" when the net is in a terminal state. Thus, (4) all the net's terminal states correspond to the end of the negotiation process. 4. Places P:init-x, P:opinion?-x et P:end-x are in mutual exclusion (i.e, it is impossible to have more than one mark in those places) and actually, there is always one mark in either of these places. Thus, (5) at one time, agents (modeled in figure 5) have a unique internal state, and are never in a state different from "init", "opinion?" and "end".

All four expected elementary services (given in section 4.1) are achieved: conclusion (1) implies service (a) is achieved, conclusion (2) and (4) implies services (c) and (d) are achieved, and conclusion (3) implies service (b) is achieved.

Concluding Remarks and Future Work

In this article we described an approach that formalizes and partially validates interaction protocols for multiagents systems once protocols have been designed. This means that one needs first to identify the agents' behavior and define the state transition graph of the interaction protocol. Once this is done, we have seen 1. how to build a Petri sub-net model for the agents, 2. which virtual medium to chose in order to link the agents, 3. how to build a complete Petri net for modeling the entire protocol, and 4. what it takes to seek such a Petri net's properties, by means of a software tool capable of simulations.

Usually the issue of interaction in MAS is almost exclusively tackled from a Speech Acts theory perspective. But the approach we have taken in this paper stems rather from the

Figure 2 .

 2 Figure 2. Bus type virtual medium.

Figure 3 .

 3 Figure 3. Inside of box (i, j).

Figure 4 .

 4 Figure 4. State transition diagram of a negotiation protocol.

Figure 5 .

 5 Figure 5. Petri sub-net modeling an agent that makes use of the negotiation protocol seen in figure 4.

Figure 6 .

 6 Figure 6. Principle of protocol pre-validation

LEIBNIZ-INPG, Grenoble, France

CELAR, Issy-les-Moulineaux, France

LEIBNIZ-CNRS, Grenoble, France

Distributed Systems' domain. Very few attempts have been investigated in this way (let us mention [START_REF] Vernadat | Modélisation par réseau de petri d'un langage d'acteurs[END_REF] though). We hope this paper will participate in offsetting this general tendency.

As far as future development, several domains need further work. One deals with the nature of properties to be verified. We have only touched on rather general properties that one wants to find in any interaction protocol. Those properties do not depend on the application. We are interested now in expressing and verifying semantic properties that are proper to the protocol given an application.

Appendix

Here is the file produced by PAPetry software for the seeking of properties in the Petri net modeling our interaction protocol.