Philippe Lacorre

Ania Selmi

Gwenaël Corbel

Brigitte Boulard

On the flexibility of the structural framework of cubic LAMOX compounds, in relationship with their anionic conduction properties

A mathematical analysis of the cubic crystal structure of fast oxide-ion conductor b-La 2 Mo 2 O 9 (and derived members of the LAMOX family) shows that its cationic sublattice can behave as a semi-rigid framework. Tilt/rotation of rigid [O1La 3 Mo] anti-tetrahedral units about their 3-fold axis can open up tunnels in the cationic framework, therefore favoring the mobility of O2 and O3 oxide ions located in these tunnels, as confirmed by molecular dynamics simulations. Such a process is likely to assist the anionic transport and explain the postulated transition from Arrhenius-type to VTF(Vogel-Tammann-Fulcher)-type behavior propounded to account for the peculiar conductivity curvature observed at high temperature in all the cubic LAMOX compounds. It also clarifies the correlated extra cell volume expansion observed at the same temperature in all these materials.

with the volume expansion generated by the doping. Correlations between ionic conduction and thermal (or chemical) expansion are well-known in oxide ion conductors. Some alternative explanations to free volume have been advanced to explain them, such as the dependence of both conduction and expansion on oxygen vacancy concentration, or the decreasing interaction between ions when the crystal expands (see for instance refs 9 and 10 and references therein). Among the studies focusing more specifically on free volume, depending on the structural type, correlations with ionic conduction have either been clearly evidenced (fluorite 11 or brownmillerite 12 types) or shown to be partly counterbalanced by conflicting effects (perovskite 13,14 or brownmillerite 15 types).

The purpose of this paper is to try to evidence such type of correlations between oxide-ion conduction and thermal expansion in the LAMOX family, and to interpret it in terms of the flexibility of its structural framework.

Experimental Section

Synthesis. Powder samples were prepared by conventional solidstate reaction of intimate mixtures of elementary oxides or carbonates. A preheating of several hours at 500 °C was applied for pre-reaction in order to avoid molybdenum oxide sublimation. One or several firings were performed at high temperature (generally between 900 and 1200 °C depending on composition, 2,16 with intermediate grindings) until a pure phase is obtained. Pellets with high relative density (>95%) were obtained from ball-milled powders (FRITSCH planetary micromill pulverizette apparatus, with 900 mg of sample in agate vials together with six 10 mm diameter agate balls, in ethanol; 4 alternations of 15 min milling sequences at 700 rpm, with a 15 min pause in between). After mixing with an organic bending agent, dried powders were first shaped in a uniaxial press (∼5 mm diameter and 3-5 mm thickness) and then pellets were isostatically pressed at 500 MPa in a Top Industrie room-temperature apparatus. The optimal sintering conditions were determined by dilatometry (see below).

X-ray Diffraction. X-ray powder thermodiffractograms were recorded in air from RT to 1060 °C on a θ/θ Bragg-Brentano Philips X'pert MPD PRO diffractometer (Cu KR 1+2 radiations) equipped with the X'celerator detector and an Anton Paar HTK1200 high-temperature heating chamber. Diffractograms were collected at each temperature for 184 min in the [15-130°] scattering angle range, with a 0.0167°step (heating rate 10 °C/min between each thermal step, with 20 min temperature stabilization before recording). Crystal cell parameters were determined by full pattern matching refinement of the powder XRD patterns in space group P2 1 3 using the Fullprof program. 17

Dilatometry. For dilatometric measurements, a vertical SET-ARAM TMA Setsys Evolution 16 dilatometer was used. Linear thermal expansion coefficients were measured on sintered pellets in a 20 mL/min air flow from RT to 1000 °C (5 °C/min heating rate).

Conductivity Measurements. Complex impedance measurements were performed on dense pellets, with thin platinum films (deposited by magnetron sputtering on both flat faces) as electrodes. The impedance spectra were recorded over the 32 MHz to 0.1 Hz frequency range (signal amplitude 50 mV) in dry air flow at high temperature using a Solartron SI 1260 frequency response analyzer combined with the Dielectric Interface (Solartron 1296). A 35 min thermalization time was applied before each measurement.

Molecular Dynamics. A series of molecular dynamics (MD) simulations were performed in order to test the effect of antitetrahedra rotation on oxygen mobility. The pair-potential U ij used in this study was the Busing approximation of Born-Mayer-Huggins form where r ij is the distance between ions, z i the effective valence of ion i, f o a constant equal to 6.947 × 10 -19 J Å -1 , c i and c j the parameters of the molecular force term, and a i and b i the repulsion term parameters. A list of the parameters used for the current simulation can be found in Table 1.

Conductivity and Crystal Cell Volume Anomalies in the LAMOX Family

A correlation between conductivity increase and volume expansion obviously exists in La 2 Mo 2 O 9 (see Figure 1, experimental data) since the R/β phase transition is accompanied by both an increase of anionic conductivity of about 1 order of magnitude 1 and a cell volume expansion of ca. 0.5%. 18 In the cubic members of the LAMOX family, of course no such abrupt change in cell volume is observed

U(r ij ) ) z i z j r ij + f o (b i + b j )exp ( a i + a j -r b i + b j ) -c i c j r 6
since no symmetry change occurs in the whole temperature range. However, a change in the thermal expansion coefficient of several cubic members of the LAMOX family (namely, V-, W-, and Bi-substituted compounds) above ca. 650 K has already been observed. 19 It was associated by the authors to either a disordering of oxygen sublattice or possibly a minor oxygen loss indicated by increasing of n-type electron transference number when the temperature increases. We think the second alternative is most unlikely, specially in the case of W-substituted compounds which are known to be more stable to reducibility than La 2 Mo 2 O 9 , 20 this last compound having a very low electronic transference number at 750 °C in air (around 10 -3 , from ref 4). Our interpretation is that such a change in volume expansion should rather be correlated with a change in oxygen transport mechanism from thermally activated (Arrhenius) at low T to thermally assisted (VTF) at high T, in other words with a transition from a static to a dynamic disorder of oxide ions.

As a matter of fact, such an increase in cell volume occurs at the same temperature as the conductivity increase (which can also be seen in oxygen tracer diffusion measurements 21 ). This is illustrated in Figure 1 with the experimental conductivity and volumetric curves of another cubic member of the LAMOX family, La 1.9 Sr 0.1 Mo 2 O 8.95 .

The previous assumption is in qualitative agreement with the correlation established in the earlier study on modified oxide glasses. 8 The advantage of a crystalline over an amorphous matrix is that the known crystal host framework can be directly analyzed in terms of the ability of its atomic arrangement to accommodate for such an extra volume expansion relative to its own natural thermal expansion. The next section is devoted to such an analysis. The conventional manner to represent an oxide structure, featuring the ligand coordination environment of cations, is not the most appropriate way to describe the structural arrangement of β-La 2 Mo 2 O 9 . This is due to the fact that the partial occupancy of most oxide ions sites does not allow an easy or convenient polyhedral representation, especially when our understanding of transport properties requests a differentiation between those oxide ions which are likely to migrate and those which belong to the host framework.

Flexibility of the β-La

Recently, we have proposed a new kind of description of the β-La 2 Mo 2 O 9 structural arrangement based on antipolyhedra. 16 As a matter of fact, fully occupied sites in the structure (La, Mo, and O1, in 4a positions of space group P2 1 3) form tetrahedral units around the O1 sites, hereafter called [O1La 3 Mo] anti-tetrahedra. These anti-tetrahedral units are supposed to be the most stable part of the structure and O1 the less mobile oxide ion since it is surrounded by four cations. In comparison, O2 and O3 are surrounded by three cations only 22 and are therefore less trapped than O1. Moreover, O2 and O3 sites are partially occupied and have larger and more anisotropic Debye-Waller thermal factor than O1. 18 Each [O1La 3 Mo] anti-tetrahedron shares its three La vertexes with two other anti-tetrahedra to form a 3D framework with formulation [OLaMo]. Such a framework delineates tunnels in which are located partially occupied O2 and O3 sites on 12b positions (see Figure 2). Incidentally, these two O2 and O3 sites are thought to be occupied by charge carriers, 21 thus providing an interesting and suggestive connection with transport properties. In this new description, the molybdate formula becomes [OLaMo] 2 O 7 0 with seven oxide ions (and a vacancy, according to the LPS concept 23 ) inside the voids of the [OLaMo] framework.

We thus adopt, with this kind of representation, a symmetrical approach 24 to that commonly used in describing the structure/properties relationships in cation conductors: indeed, cationic anti-polyhedra for anionic conductors are equivalent to (or play the same role as) anionic polyhedra for cationic conductors. For instance, migrating oxide anions are likely to move, inside the [OLa 3 Mo] anti-tetrahedra-based framework in β-La 2 Mo 2 O 9 , the same way as lithium cations do inside the [TiO 6 ] octahedral-based framework in the LLTO perovskite structure.

Therefore, what we want to check now is whether antitetrahedra in cubic LAMOX can easily tilt and rotate the same way as octahedra can do in the perovskite structure.

Anti-tetrahedral [O1LaMo]

Host Matrix as a Semirigid Framework. As reported in the Introduction, lithium migration above room temperature in LLTO perovskites is thought to be assisted from perovskite cage to perovskite cage by octahedra tilting/rotation. In this respect the octahedral framework is considered to be semirigid, with antitetrahedral units (green), with O2 and O3 oxide ions sites imbedded in its tunnels. 16 Pink spheres ) Mo 6+ or substitute, blue spheres ) La 3+ or substitute, yellow and red transparent spheres ) O2 and O3 oxide ion sites, respectively.

Flexibility of Structural Framework of Cubic LAMOX Compounds

Inorganic Chemistry, Vol. 45, No. 2, 2006 629 rigid octahedral units allowed to rotate around their vertexes. The equivalent of the perovskite rigid octahedral unit in cubic LAMOX is the anti-tetrahedral [OLa 3 Mo] unit represented in Figure 3, whose geometry in this symmetry is fixed by the three interatomic distances La-La (d ll ), La-Mo (d lm ), and Mo-O1 (d mo ). In space group P2 1 3, all three atoms are located in 4a sites on a 3-fold axis (O1 and Mo on the same 3-fold axis), with only one atomic coordinate each (x ) y ) z). Indeed the interatomic distances can be expressed in terms of the atomic positions (hereafter called x l , x m , and x o for La, Mo, and O1, respectively) and of the a cubic cell parameter. The main idea is then to deduce from these relationships the expression of the atomic position parameters x l , x m , and x o as a function of the cell parameter a and of geometrical characteristics of the anti-tetrahedra d ll , d lm , and d mo . The core of our approach can be outlined as follows, with f( ), f′( ), f′′( ), g′( ), g′′( ), h′′( ), etc..., representing functions of ( ): leading finally to

The expressions of d ll , d lm , and d mo are quadratic forms of x l , x m , and x o . The atomic coordinates are therefore roots of the corresponding equations that can be successively inserted into the next equation in order to get a global solution to the system, provided that these roots exist in the functional definition ranges (positive discriminant). One ends up with the following set of expressions: with with Formula (1′) is valid, and therefore the cationic arrangement is self-consistent, only if ∆ 1 g 0, that is, from (1′′): d ll /a g [3/8] = 0.61237. According to (1′), the maximum possible value of x l is obtained for ∆ 1 ) 0, that is, x l max ) 7/8 ) 0.875. Formula (2′) is more complex since it relates the atomic positions of two different atoms. Of course, it requests that ∆ 2 g 0 which, through (2′′), delimits bounds on interatomic distances and/or lanthanum position. Defining precisely these bounds is of little interest here. Just notice that, for the current study range of relevance (see below), ∆ 2 is always positive definite. An illustration is given in Figure 4, which shows the evolution of x m as a function of d ll /a and d lm /a (when either the geometry of anti-tetrahedral units or the cell parameter are modified).

Finally, formula (3′) is a straightforward relationship which relates the O1 site position to the Mo site position through their interatomic distance.

The interesting point about the current analysis is that, given a reasonable set of La-La, La-Mo, and Mo-O1 interatomic distances and a cubic cell parameter, it is possible to find the La, Mo, and O1 atomic positions which satisfy this set of distances. In other words, given the geometry of a rigid [O1La 3 Mo] anti-tetrahedral unit, it is possible, for any value of the a cubic cell parameter (smaller than d ll / [3/8]), to find the atomic coordinates which satisfy the anti-tetrahedra geometry (see Figure 5). This is an important result since it shows that the LAMOX anti-framework can be, at least theoretically, flexible enough to accommodate, without any change in symmetry (the above calculations being carried out in the P2 1 3 space group), the tilt/rotation of rigid anti-tetrahedral units and concomitant cell volume change. The anti-framework can thus be folded or unfolded by just closing or opening up the O1-La-O1 interunits angle without modifying the geometry of the anti-tetrahedral units.

d ll ) g′(a,x l ) w x l ) f′(a,d ll )
(1) In these formulas δ is negative, δ ) 0 corresponding to maximal volume expansion (for x l ) 7/8 or d ll /a ) [3/8]). Figure 7 shows the calculated cell parameter evolution when the rotation angle varies, as deduced from formula (6).

d lm ) g′′(a,x l ,x m ) w x m ) h′′(a,x l ,d lm ) ) f′′(a,d ll ,d lm ) (2) d mo ) g′′′(a,x m ,x o ) w x o ) h′′′(a,x m ,d mo ) ) f′′′(a,d ll ,d lm ,d mo ) (3) (x l ,x m ,x o ) ) f(a,d ll ,d lm ,d mo ) (4) x l ) (7 -∆ 1 )/8 (1′) ∆ 1 ) 8(d ll /a) 2 -3 ( 1 ′′) x m ) (2 -∆ 2 -x l )/3 (2′) ∆ 2 ) -8x l 2 + 14x l + 3(d lm /a) 2 -13/2 (2′′) x o ) x m + d mo /(a 3) (3′)
4.3. Framework Flexibility and Ionic Conductivity. In an ionic conductor, the free volume is a volume available for ions to move. The larger this volume, the higher the ionic mobility. In the hypothesis of an Arrhenius/VTF type transition, it seems reasonable to think that the increase of mobility in the VTF regime is correlated to an expansion of free volume, and therefore in our case of the crystal cell parameter. Such an expansion and its correlation with the conductivity increase are clearly evidenced in the experimental data displayed in Figure 1. It is also observed in all other cubic members of the LAMOX family (see Figure 8) with more or less magnitude depending on the substituent, which also influences the Arrhenius/VTF transition temperature. 3 Two expansion regimes are observed: a regular one at low temperature, with constant expansion coefficient, and an extra expansion at higher temperature, correlated to the appearance of the VTF regime. From the very nature of the VTF mechanism, the host matrix is supposed to assist oxygen migration, and it is likely that the flexibility of the LAMOX framework is at the origin of the observed volume increase, through anti-tetrahedral rotation/vibration. The "empty" volume (cell volumeatomic volume) is relatively large in La 2 Mo 2 O 9 since it represents about 38% of the cell volume (as compared to 26% in a close packed arrangement). Any additional volume freed up by the anti-tetrahedra rotation is likely to favor the ionic mobility, or to be correlated to it. Figure 9 reports the experimental thermal expansion of the tan(δ) ) (8x l -7)/ 3 ( 5) In both compounds, an extra 0.5-1% volume expansion cannot be accounted for by the regular thermal expansion observed at low temperature. Our proposition is that this additional expansion originates from the rotation of anti-tetrahedral units, at the origin of the high mobility of O2/O3 oxide ions in the anti-framework tunnels. Such an interpretation is illustrated in Figure 10, which combines the experimental data of Figure 1 and our model calculation of Figure 7.

) -[8(d ll /a) 2 /3 -1] a ) d ll / [3(1 + tan 2 (δ))/8] (6)
A rotation of the order of 1°, over about 500 K, of [O1(La,Sr) 3 Mo] units around their 3-fold axis would account for the observed extra increase in cell volume.

In the next section, we will try to evidence a correlation between such a volume expansion through anti-tetrahedra rotation and an increase of oxide-ion conduction, using molecular dynamics simulations.

Molecular Dynamics Simulations

To check the incidence of such a tilt/rotation of antitetrahedra units on the migration of O2 and O3 oxygen sites located in the tunnels of the anti-structure, a series of molecular dynamics simulations were carried out with different rotation angles of these units. The pair-potential function used in this study is given in the Experimental Section. The simulation parameters for O 2-ions and effective charge for all ions were taken from ref 26. The parameters for Mo 6+ and La 3+ ions were determined so as to minimize atomic displacement from the atomic positions determined on a La 2 MoO 6 crystal. 27 This set of parameters, as listed in Table 1, has been shown to reproduce accurately the first coordination sphere of La and Mo in La 2 Mo 3 O 12 , 28 La 2 -Mo 4 O 15 , 29 and β-La 2 Mo 2 O 9 at 943 K. 18 The pair radial distribution functions, involving O2 and O3 oxide ions only, in a 64 cells sample of β-La 2 Mo 2 O 9 after relaxation at 943 K (see below) are shown in Figure 11 (for full coordination sphere of cations, add one O1 for Mo 6+ and three O1 for La 3+ ). They are consistent with the crystal structure. 18 A cubic MD simulation box consisting of 4a × 4a × 4a crystallographic unit cells was chosen. The d ll , d lm , and d mo interatomic distances of β-La 2 Mo 2 O 9 at 943 K were taken as reference. 18 A series of rotation angles were chosen, spanned around the value measured in β-La 2 Mo 2 O 9 at this temperature (δ ) -5.85°). The corresponding crystal cell parameters a and La, Mo, and O 1 atomic coordinates, as deduced from formulas ( 6), (1′), (2′), and (3′), respectively, were kept fixed during each simulation run (see Table 2). The O2 and O3 oxygen ions where chosen randomly to match the occupation ratio, but a constraint was used to exclude small O-O distances. However, short O-O (i.e., 1.2-1.7 Å) could not be avoided. The box thus contained 256 Mo, 256 La, 256 O1, 600 O2, and 296 O3 ions, periodic boundary conditions were employed, and the time step was equal to 2 × 10 -15 s. The initial velocity of O2 and O3 ions were sampled from a Maxwell-Boltzmann distribution at the temperature specified for the simulation, with Mo, La, and O1 being kept fixed. The system was relaxed at 943 K for 25000 time steps and then heated at 1500 K in one step and then relaxed again for 80000 more time steps. The 1500 K temperature was selected instead of 943 K in order to obtain accurate displacement.

The bulk oxygen tracer diffusion coefficient D tracer was computed via the time-dependent mean square displacement 30 denoted by s 2 (t):

where r b n (t) is the position vector of the nth oxygen ion (O2 or O3) at the time t; C is a constant. D tracer is taken as the asymptotic value of D(t) with a time calculation of 160 ps (i.e., 80000 time steps).

The dependence of the diffusion coefficient on the antitetrahedra tilt/rotation angle is plotted in Figure 12. A significant increase, of about 1 order of magnitude in a few degrees of rotation, is observed between -10°and -5°, that is to say just below the tilting angle measured at 670 °C in β-La 2 Mo 2 O 9 (δ ) -5.85°). These simulations therefore show that an opening up of the anti-structure tunnels through antitetrahedra rotation would be responsible for an increase in the O2 and O3 oxygen mobility, thus confirming the selfconsistency of our model.

We also attempted to observe individual oxygen jumps during the simulation. Since trajectory analysis requires long displacements, we had to use a smaller box, i.e., a 3a × 3a × 3a crystallographic unit cell (108 Mo, 108 La, 108 O1, 253 O2, and 125 O3 ions) in order to reduce time calculations. Figure 13 shows 3D plots of the trajectory of some O2 or O3 oxygen recorded at 1500 K in the most favorable geometry (δ ) 0°), as well as the related atomic coordinates as a function of time. One clearly sees discrete jumps, which could be responsible for the oxygen diffusion in La 2 Mo 2 O 9 . The mean jump length involving O2/O3 ions is 2.8 Å, close to the mean O-O distance obtained on the simulated radial distribution function, and to the nearest oxide ion neighbors distance measured from the short-range-order modulation of the neutron diffraction background. 31,16 Note also that the jumps detected on O2 and O3 oxide ions #699 and #402 are time-correlated at around 330 ps, with ion O2 occupying after its jump the previous position of ion O3 just before its jump.

(30) Sangster, M. J. L.; Dixon, M. AdV. Phys. 1976, 25, 247-342. (31)

s 2 (t) ) 1 N ∑ 1 N 〈[ r b n (t o + t) -r b n (t o )] 2 〉 ) C + 6D(t) × t

Flexibility of Structural Framework of Cubic LAMOX Compounds

Inorganic Chemistry, Vol. 45, No. 2, 2006 633

Discussion

The flexibility of polyhedral networks, and in particular tetrahedral networks, has been extensively studied by various techniques, including simulation and geometrical analysis (see for instance refs 32 and references therein). The basic concept of all these studies is that of the "rigid-unit mode" (abbreviated as RUM) 33 model, which assumes that there exist, in these networks, rigid polyhedral units whose low-energy vibration modes ensure structural flexibility. The most famous and studied applications of such a concept concern silicates or alumino-silicates(phosphates) with corner-sharing rigid (Si,Al,P)O 4 tetrahedra. As a matter of fact, in SiO 4/2 networks, the energy cost for a distortion of the 2-fold coordination environment of oxide ions is probably smaller than that for a distortion of the 4-fold coordination polyhedra of silicon.

Our approach in this paper is to invert the pattern and use OM 4 anti-tetrahedral units as the building rigid blocks of the RUM model. This is a risky assumption since the coordination sphere of cations is usually larger, in term of ligand numbers, than the coordination sphere of anions, so that a distortion of the latter would remain less energy costly than a distortion of the former, as in the previous silicate cases. Indeed, if one takes into account partial occupancies of oxide ions, the average coordination numbers of Mo 6+ and La 3+ in β-La 2 Mo 2 O 9 are 4.5 and 10, respectively, 18 as compared to 4 for O1 anion. However, as mentioned earlier and due to the fast oxide-ion properties of this molybdate, O2 and O3 are rather mobile ligands around molybdenum and lanthanum, themselves elements which tolerate a high flexibility in their coordination spheres. If one assumes that fully occupied O1 is the most tightly bound ligand, therefore creating the most rigid links, Mo 6+ and La 3+ have only one and three such rigid bounds in their surroundings, respectively, 18 whereas O1 has four. And on average, one therefore ends up with twice more strong bounds around O1 than around cations, a reversed situation when compared to that of silicates. Even if at first sight it might appear astonishing that the flexibility of the β-La 2 Mo 2 O 9 framework could rely upon the displacement of such heavy atoms as La or Mo, a similar case has already been referred to in the literature. In zirconium tungstate ZrW 2 O 8 , a compound known for its negative thermal expansion properties, it has been shown that the Zr-O-W linkage remains rather stiff with temperature (with no important static 34 or dynamic 35 opening up of the interatomic angle) and that the low-energy vibrational modes involve heavy Zr and W atoms. That a comparable type of low-energy mode involving rigid La-O-(Mo,B) links could occur in La 2 (Mo,B) 2 O 9 is therefore not so surprising.

On the other hand, even with a certain stiffness of La-O1-(Mo,B) links in anti-tetrahedral units, the assumption that these units keep exactly the same geometry upon heating, even with some isotropic thermal expansion, might not be very realistic. As a matter of fact, we have already shown that anti-tetrahedra can distort upon chemical substitution, 3 and it would not be so surprising if chemical bonds had different thermal expansion depending on the nature of the involved cation. Such additional effect, not taken into account in our model, could for instance be at the origin of differences in oxygen tracer diffusion coefficients, which appear to be somewhat smaller in our MD simulations than the ones measured on real samples. 21 Nevertheless, we think that our proposition, even crude as it is, gives a plausible explanation for the main structural/transport anomalies observed in these compounds and should be tested further.

It is also probable that the observed lattice expansion does not result from a smooth regular opening of the anti-structural framework only, but that it takes place through a libration process with dynamical breathing of the framework. Such a libration would thus assist oxygen migration, in somewhat the same way as in LLTO perovskites, octahedral librations (which allow for the opening/closing of the inter-cage bottleneck), do for lithium migration.

Conclusion

In this paper, we have studied through a mathematical analysis the ability of the β-La 2 Mo 2 O 9 cationic sublattice to distort as a semirigid framework of tilting [OLa 3 Mo] antitetrahedral units. It is theoretically possible in a surprisingly large rotational domain (although atomic repulsion certainly limits it within a narrower range). It has been proposed that the extra volume expansion, observed at the postulated Arrhenius/VTF transition, originates from such a tilting mechanism. This hypothesis has been tested using molecular dynamics simulations, which confirmed that such an opening of anti-tetrahedral framework induces an increase of oxideion diffusion in the tunnels of the structure. However, the magnitude of the structural distortion is expected to be very small (around 1°over a 500 K range 36 ), which should make it difficult to be detected, specially as it is most certainly correlated to a dynamic redistribution of O2/O3 ion sites. Further studies using different techniques will be devoted to its detection.

Finally, the current study has shown that the way we are looking at structures and describing them is not neutral since it has an incidence on our understanding of their properties. As a matter of fact, this analysis would not have been possible if an alternative description of the structural arrangement of β-La 2 Mo 2 O 9 had not been striven for, inspired by the pioneering work by O'Keeffe and Hyde 20 years ago. 24

Figure 1 .

 1 Figure 1. Comparison between measured conductivity increase (top) and measured volumic expansion (bottom) as a function of temperature for La2-Mo2O9 (dotted lines) and La1.9Sr0.1Mo2O8.95 (full lines). The gray vertical bars indicate the monoclinic/cubic R/β phase transition in La2Mo2O9 at high temperature and the postulated Arrhenius/VTF transition in cubic La1.9-Sr0.1Mo2O8.95 at lower temperature.

Figure 2 .

 2 Figure 2. Representation of the β-La2Mo2O9 structural type in terms of a [OLaMo] framework built up from La corner-sharing [O1La3Mo]antitetrahedral units (green), with O2 and O3 oxide ions sites imbedded in its tunnels.16 Pink spheres ) Mo 6+ or substitute, blue spheres ) La 3+ or substitute, yellow and red transparent spheres ) O2 and O3 oxide ion sites, respectively.

Figure 3 .

 3 Figure 3. [O1La3Mo] anti-tetrahedral basic units, with the interatomic distances considered in the text.

Figure 4 .

 4 Figure 4. Molybdenum atomic coordinate xm as a function of dll/a and dlm/a in a semirigid [OLaMo] framework.

Figure 6

 6 Figure6shows, from two different standpoints, the unfolding process. Along the tunnels, one can clearly see that the anti-tetrahedra tilting opens up the channels, thus releasing extra free volume for mobile ions. The view along the 3-fold axis (main diagonal axis of the cubic cell) is also interesting because it makes clear the true nature of the antitetrahedra movement during the folding/unfolding process: it consists of a rotation around the 3-fold axes([111] directions) of the structure, passing through the [O1-Mo] axis of the anti-tetrahedra units. The rotation angle δ can be calculated from the (x,y) La atomic coordinates in a hexagonal reference frame, obtained by applying the rhombohedral-to-hexagonal transformation matrix to the La coordinates in cubic reference (see ref 25). It gives a simple relationship between δ and x l or d ll /a:

(

  25) Hahn, Th., Ed. International Tables for Crystallography, Vol. A, SpaceGroup Symmetry, 4th ed.; Kluwer Academic Publishers: Norwell, MA, 1995; p 78. Namely, the matrix is applied to the (-1 /2 + xl, 3 /2 -xl, 1 -xl) lanthanum coordinates.

Figure 5 .

 5 Figure 5. Position of the atoms of rigid [O1La3Mo] anti-tetrahedral units in the crystal cell as a function of the cubic cell parameter (from interatomic distances measured in β-La2Mo2O9 at 670 °C: 18 dll ) 4.454 Å, dlm ) 3.703 Å, dmo ) 1.839 Å).

Figure 6 .

 6 Figure 6. Folded (left, exaggerated) and unfolded (right) anti-framework as viewed along two directions of the β-La2Mo2O9 type structure: along the tunnels parallel to the cubic cell axis (upper) and along the 3-fold axis (lower). The rotation of [O1La3Mo] anti-tetrahedral units is outlined.

Figure 7 .

 7 Figure 7. Calculated cubic cell parameter as a function of the δ rotation angle of rigid [O1La3Mo] anti-tetrahedral units (with imposed geometry dll ) 4.454 Å, dlm ) 3.703 Å, dmo ) 1.839 Å). The dashed lines indicate the cell parameter and rotation angle of β-La2Mo2O9 at 670 °C (from crystal structure data, ref 18) and their approximation in a pseudo-cubic cell at room temperature.

Figure 8 .

 8 Figure 8. Departure from linearity at high temperature in the thermal expansion (bottom) and the Arrhenius plot of conductivity (top) of a series of LAMOX compounds (whose cationic stoichiometry is indicated). The two temperature scales coincide around 1000/T(K) ∼ 1.4, T ∼ 440 °C.

Figure 9 .

 9 Figure 9. Experimental LAMOX cell volume expansion with temperature: (a) raw data with emphasis on extra expansion relative to regular thermal expansion (La2Mo2O9 in black and La1.9Sr0.1Mo2O8.95 in gray); (b) relative extra volume expansion in excess to regular thermal expansion (black circles for La2Mo2O9, open circles for La1.9Sr0.1Mo2O8.95).

Figure 10 .

 10 Figure 10. Thermal expansion data of Figure 9 replotted as a function of the δ rotation angle (in insert) deduced from analytical formula (5), for La2Mo2O9 (black circles) and La1.9Sr0.1Mo2O8.95 (open circles).

Figure 11 .

 11 Figure 11. Radial distribution function gij(r) and coordination number distribution nij(r) versus the interionic distance r in the cation-anion (Mo-O2,3 and La-O2,3) and anion-anion (O2,3-O2,3) pairs, after molecular dynamics relaxation at 943 K. For full cationic coordination numbers, one and three O1 extra neighbors should be added to Mo and La polyhedra, respectively. Vertical lines correspond to the crystal structure interatomic distances (from ref 18).

Figure 12 .

 12 Figure 12. O2+O3 diffusion coefficient at various anti-tetrahedra rotation angles δ, from molecular dynamics simulations at 1500 K. The reported values correspond to the average over 10 runs of 160 ps duration each. The solid dot corresponds to the structural framework arrangement of β-La2-Mo2O9 at 943 K. 18

Figure 13 .

 13 Figure13. From the molecular dynamics simulation of a sample with fully opened framework (δ ) 0°) at 1500 K, the trajectory and atomic coordinates versus time of some oxide ions indexed as #402 (O3), #672 (O2), and #699 (O3). NB: O2 and O3 are labeled according to their original positioning, but subsequent migrations prevent any real differentiated allocation (localizations in black in the 3D trajectory charts).

Table 1 .

 1 Potential Parameter Set Used for Molecular Dynamics Simulations

	parameters

a Taken from ref 26.

  Goutenoire, F.; Isnard, O.; Retoux, R.; Lacorre P. Chem. Mater. 2000, 12, 2575-2580.

Table 2 .

 2 Crystallographic Cell Parameter a and Corresponding Atom

	Coordinates (Calculated from Formulas (1′), (2′), (3′), and (6)) Used in
	the Molecular Dynamics Simulations
	δ (deg)	a (Å)	x(La)	x(Mo)	x(O1)
	0	7.273352	0.875000	0.163496	0.309474
	-5	7.245674	0.856058	0.169002	0.315537
	-5.85	7.23511 a	0.8528 a	0.1697 a	0.3165 a
	-7.5	7.211127	0.846496	0.171172	0.318409
	-10	7.162853	0.836824	0.172958	0.321188
	-15	7.025518	0.816987	0.175372	0.326500
	-20	6.834715	0.796198	0.176189	0.331536

a From ref 18.

Acknowledgment. B. Boulard is grateful to J. Kieffer and C.A. Angell (Arizona State University) for providing the Molecular Dynamics program.

Supporting Information Available: Animated movie of the unfolding-folding process described in the text, whose Figure 6 gives snapshots. This material is available free of charge via the Internet at http://pubs.acs.org. (36) Although comparable in magnitude, the first-order transition in La2-Mo 2O9 cannot be explained the same way since it results from an order-disorder type transition between phases with different symmetries. 18,37 The physical origin of the transition is however comparable to that observed in cubic LAMOX compounds, except that in this last case the random cationic distribution prevents any oxide-ion/vacancy ordering (and therefore any symmetry breaking). ( 37


Stéphanie Lambour

Françoise Mechin

Jean-Pierre Pascault

A NEW PROCESS FOR CROSSLINKING THERMOPLASTIC POLYURETHANES USING RUBBER INDUSTRY TECHNIQUES

Keywords: Thermoplastic Polyurethanes (TPU), processing, vulcanization, allophanates, thermomechanical behavior

New polyurethane materials were obtained after chemical modification of a commercial thermoplastic polyurethane (TPU) using diisocyanates and diols or diamines. The reaction can be easily processed by the well-known calendering technique often used by the rubber industry. It leads to a remarkable improvement of the thermomechanical properties of the material, due to the formation of chemical crosslinks between the TPU chains ("vulcanization"). As this new processing consists only of an intimate mixing of the starting TPU with solid reagents, followed by molding and curing stages, it allows the use of any and possibly highly reactive solid monomer (especially diamines) which could not otherwise be considered except with constraining techniques like RIM; in the same way the incorporation of organic or inorganic fillers also becomes possible.

INTRODUCTION

Polyurethane elastomers with alternating sequences of hard and soft segments make up an important class of industrial materials [START_REF] Saunders | High polymers -Polyurethanes Chemistry and Technology, Part I[END_REF]. They show excellent mechanical properties such as high tensile strength and abrasion resistance. According to the monomer reactivity, especially that of the diisocyanate and chain extender, and depending on the kind of application, different types of synthesis can be applied, starting with simple solvent-based coatings to cast-molding, reactive extrusion and finally RIM technology for the most reactive systems [START_REF] Macosko | Fundamentals of Reaction Injection Molding[END_REF].

We propose here another way to work on polyurethanes with a technique deriving from rubber industry [START_REF] Schroeter | [END_REF]. Instead of mixing the classical reactants (polyol [polyester or polyether glycol], diisocyanate and chain extender [diol or diamine]), the new initial reactive blend consists of a preformed thermoplastic polyurethane, TPU, of reasonable molar mass (quite amorphous, or with a low melting temperature, T m -typically 40°C) and of a diisocyanate for example. A chain extender in the solid state can also be easily added to this formulation. In this process, all solid compounds are mixed in open mills or calendered at room temperature or just below the melting temperature of each component. This constitutes the main advantage of this process, because chemical reactions occur during press curing only at high temperature. Therefore, the use of monomers with high reactivities becomes possible. Another important aspect is that this technique allows the addition of large quantities of fillers, either mineral or organic, reactive or not, a step which is usually difficult in classical processes like cast molding.

The resulting polymer is a thermoset with a particular, slightly crosslinked structure involving allophanate or biuret bonds. In classical polyurethane synthesis, these chemical links are usually formed when using excess isocyanate with respect to hydroxyl or amine groups (4). Other side reactions (isocyanate dimerization and mainly trimerization) can also occur and change the structure and the properties of the final product (5,6).

As mentioned before, the final polymer is completely insoluble and has an elastomeric behavior. Its properties are nearly equivalent to those of classical linear segmented polyurethanes with a low glass transition, a high melting point and a high storage modulus in relation with the type and proportion of hard segments formed. The stress-strain properties are also rather improved by the additional covalent bonds.

In the present work, we attempt to explain the chemical structure of such systems (TPU/diisocyanate/chain extender, possibly plus a filler), with the help of dynamic mechanical analysis to understand the final network behavior. The advantages of the process were illustrated by the preparation of materials modified with highly reactive monomers and/or fillers.

EXPERIMENTAL Materials

All the reagents used in this work are displayed in Figure 1. A commercial thermoplastic polyurethane (TPU) was used as precursor. Millathane ® E49 (TSE industries) is a semi-crystalline polymer obtained from a polyetherpolyol ( n M between 1000 and 3000 g/mol), extended with toluene than a diol because of its higher reactivity.

Polyamide 11 (Rilsan ® , AtoFina) was chosen as organic filler for its high melting point, around 190°C, which can confer better thermal properties to the polymer.

Sample preparation and analytical techniques

Polyurethane precursors and other monomers were mixed by calendering at 80°C. Pellets or sheets of thermoplastic polyurethane were in a highly viscous liquid state during this step, with a high shear level between the rolls. The solid compounds were intimately mixed with this liquid PU. Reactions took place during press curing only at high temperature, between 120 and 150°C, above all melting temperatures, but below the splitting temperature of di-TDI.

Size exclusion chromatography (SEC) was performed in THF using a Waters chromatograph equipped with 3 microstyragel columns (Millipore HR1, HR2 and HR3) and double detection (differential refractometry and UV absorption). When necessary, a small amount of LiCl was added in order to destroy hydrogen bonds (i.e. principally for E49 TPU). DSC thermograms of polyurethane rubbers were obtained by means of a METTLER TA3000 calorimeter, at a heating rate of 7.5 °C/min from -150 to 400°C under inert atmosphere.

The mechanical properties of these polymers depend on their complex crosslinked structure. Dynamic mechanical properties were measured using a Rheometric RSA II dynamic viscoelasticimeter with a frequency of 1Hz from -120 to 220°C (3°C/min).

Stress-strain properties were evaluated by using a tensile tester at a crosshead speed of 50mm/min with dumbbell type pieces at room temperature.

RESULTS AND DISCUSSION

Two-component systems (TPU/diisocyanate)

All formulations contained a thermoplastic polyurethane, a diisocyanate, a chain extender (diol or diamine) and occasionally a filler. The TPU precursor has mostly hydroxyl and amine reactive end groups. The main possible reactions between each reagent are given below (Equations 1 to 5).

Chain extension with amine groups:

R 1 -NCO + (TPU)-NH 2 R 1 -NH-C-NH-(TPU) O (1)
Exchange reaction with isocyanate groups:

R 1 -NCO + (TPU)-NH-C-O-R R 1 -NH-C-O-R + (TPU)-NCO O O (2)
Exchange reaction with alcohol groups:

R 2 -OH + (TPU)-NH-C-O-R R-OH + (TPU)-NH-C-O-R 2 O O (3)
Urethane formation:

R 1 -NCO + R 2 -OH R 1 -NH-C-O-R 2 O ( 4
)
Allophanate formation:

R 1 -NCO + (TPU)-NH-C-O-R (TPU)-N-C-O-R C=O NH R 1 O O (5)
Chain extension (Eq. 1) can occur when PU chains have amine functional end groups. Exchange reactions (Eq. 2 & 3), also known as "transurethanization" (8)[START_REF] Eisenbach | Advances in Elastomer and Rubber Elasticity[END_REF][START_REF] Koberstein | [END_REF](11)(12) can be observed at high temperature as a consequence of urethane scission (Eq. 4), since urethane formation is thermally reversible above ~ 150°C.

Finally, Eq. 5 describes the main reaction involved in this process with excess NCO groups, i.e. the formation of allophanate bonds and thus of a crosslinked ("vulcanized") polyurethane.

Simple formulations were obtained at first from two-component systems, TPU and diisocyanate. The proportions used are reported in Table 1.

The degree of crosslinking can be characterized by checking the insolubility of the samples in tetrahydrofuran (THF) or dimethylformamide (DMF), if necessary with added LiCl. Beyond a critical diisocyanate proportion, there is no soluble fraction in these formulations. In contrast, when the diisocyanate proportion is lower than this critical value, a small soluble fraction of unlinked TPU chains is observed. In the case of E49 this value falls between 7 and 8 phr;

this is illustrated in Figure 2 where the size exclusion chromatograms of the THFsoluble fractions of the starting TPU and of the materials obtained with 7 or 8 phr of di-TDI are displayed. No soluble polymer chain can be detected for 8 phr, whereas the use of only 7 phr leads to the existence of a soluble fraction with a molar mass higher than the starting TPU (presumably extended or branched TPU molecules).

Referring to the above equations, one can reasonably assume that the isocyanate functions react at first with all available OH or NH 2 groups; this can possibly lead to some chain extension. Afterwards, allophanate formation is enabled, as far as enough isocyanate groups are still available; but also exchange reactions can become non-negligible.

The effect on dynamic mechanical properties can be seen in Figure 3a and 3b, which show the temperature dependence of the storage modulus and of the loss factor of Millathane E49 TPU and of three formulations containing different di-TDI ratios. The starting TPU shows at first a main relaxation at -70°C, typical from polyoxytetramethylene soft segments, with some crystallization of these soft segments around -40°C, followed by their melting around 10°C. The three modified materials show the same main relaxation around -70°C, with a sharp tanδ peak at the same temperature for all the samples. The rubbery modulus holds constant till 150°C, where a very important drop is encountered. This phenomenon can be attributed to the partial or total cleavage of the uretdione ring, allowed by the simultaneous melting of di-TDI-based crosslinks. Pure di-TDI dissociates above 150°C (7), whereas the splitting temperature can be higher for model diurethanes based on this molecule (13), but also sometimes lower in the presence of reactive hydroxyl groups (14). Apart from this, the fact that the sample obtained with 7 phr di-TDI was not completely crosslinked has no visible influence on this part of the curve.

All the materials are crosslinked and the rubbery plateau modulus increases as the number of crosslinks increases. More and more urethane groups are changed into allophanate links, which results in higher crosslink densities together with shorter chains between crosslinks.

In the case of the lowest diisocyanate proportion (7 phr), a shoulder is observed on the storage modulus curve after the main transition, just like for pure TPU. This can be related to some long parts of linear polyurethane chains, the amount of diisocyanate being too low to strongly crosslink the polymer and to prevent molecular motions for the soft segments. Their crystallization is thus enabled, in contrast to highly crosslinked networks where most soft segments have both ends tethered by allophanate crosslinks.

Three-component systems: use of a chain extender

The properties of the above materials can be further adjusted by using greater amounts of diisocyanate and by compensating for part of the isocyanate excess with a short diol or diamine (chain extender).

The networks can be depicted as in Figure 4 and 5. First, if we imagine the simple reaction between diisocyanate and polyurethane, a thermoset polymer is obtained with very few and short di-TDI segments between crosslinks (Figure 4).

When the chain extender is added, all previous reactions (Eq. 1 to 5) can take place and lead to a different network with more distributed (-di-TDI-HQEE-) hard segments (Figure 5). Actually, HS are inserted in the "soft" network as a result of exchange reactions, and allophanate crosslinks can finally also be formed on these hard segments.

This structure can also be explained by the proportions used in the formulations. For example for 100g of TPU, 30 phr of di-TDI and 12.54 phr of HQEE are added. 8 phr of di-TDI represents the minimal proportion to obtain a completely crosslinked material. Then, the excess isocyanate functions have multiple choices to react as shown in Eq. 1 to 5. Exchange reactions are possible, and hydroxyl functions from HQEE or urethane groups from the TPU are also potentially reactive.

Figure 6 shows typical viscoelastic spectra of three formulations containing two different chain extenders, HQEE and MCDEA, and also another diisocyanate without uretdione ring, PPDI. Different amounts of hard segments were generated in these materials; their overall compositions are indicated in Table 2 and correspond almost exactly to the same TPU/diisocyanate/chain extender molar ratios. The formulation TPU/di-TDI/HQEE has 30 wt% HS (calculated by the relative amount of rigid monomers, di-TDI + HQEE, in the overall material) and a rubbery modulus (100°C) equal to 120 MPa while HS proportion in TPU/di-TDI/MCDEA is slightly higher, 35wt% and is associated with an increased rubbery modulus, 180 MPa. This value thus strongly depends on the HS content, as it is well-known for classical linear segmented polyurethanes, polyureas and polyurethane-ureas (15)(16)(17)(18)(19). It can also explain why the rubbery modulus found for the material based on PPDI is so low (36 MPa), since this sample only contains 20 wt% HS. Moreover, in the polymer based on MCDEA, the hard segments are partially crystalline and their melting point is so high that only a tiny proportion of uretdione rings can split at their usual degradation temperature; this can be a reason why, in contrast to HQEE, no sharp collapse of the modulus is observed, but rather a smooth decrease. In the last polymer with PPDI as diisocyanate, the sudden high temperature drop disappears, and is replaced by the simple melting of [PPDI/HQEE] hard segments.

For this kind of material, we noticed that the stress at break, σ break , was higher than for classical thermoplastic polyurethanes, but in return strain at break, ε break , was limited by additional covalent bonds. In the literature, 30 to 40 MPa are usually found for σ break , and ε break usually lies between 350 and 450 %.

Here, formulations like TPU/di-TDI/HQEE or TPU/di-TDI/MCDEA show 45 and 48

MPa for σ break , and 420 and 370 % for ε break , respectively.

This technique of combined calendering and high temperature press curing thus allows the use of highly reactive monomers, usually impossible to process by classical casting. Other diisocyanates can easily substitute for di-TDI, as long as they are solids at the processing temperature. In the same way, any chain extender can be introduced to control the properties of the final network.

Effect of an organic filler

Using the same process, formulations can be modified with quite high amounts of organic or mineral fillers. In this work, a linear polymer with good thermal properties (melting temperature: 190°C) was added to TPU/di-TDI/HQEE. 20 wt% TPU were replaced with polyamide 11 which represents 14 wt% of the whole formulation (see table 3). All the precursors were calendered at 80°C and then press-cured for 4h30 at 130°C as for previous formulations.

Figure 7 depicts the temperature dependence of storage modulus and loss factor of the filled formulation TPU/di-TDI/HQEE/PA11. The relaxation attributed to the amorphous phase of PA11 is shown in this figure by a broad tanδ peak between 30 and 130°C. High temperature flow begins after 190°C, the melting temperature of PA11; which is soon followed by the melting of the polyurethane [di-TDI/HQEE] hard segments, as can be seen in the DSC curves depicted in Figure 8. The presence of these two peaks suggests that PA11 was only dispersed as a classical filler and did not react with the other components of the network. However the cure was entirely carried out below its melting temperature, and a postcure above this temperature (195°C) induces a total disappearance of these melting peaks. At such a high temperature, new reactions and rearrangements may affect the TPU chains (12,20), probably after HS melting; also the urethane bond reversibility can no longer be neglected. In the case of di-TDI they can further be accompanied by the release of new isocyanate functions from uretdione rings. Some reactions between amide groups and isocyanates have been described in the literature, and are said to evolve CO 2 [START_REF] Saunders | High polymers -Polyurethanes Chemistry and Technology, Part I[END_REF][START_REF] Parodi | Isocyanate-derived polymers[END_REF]. In our case the material indeed gave rise to foam (density: 0.87 to 0.90 g.cm -3 , instead of 1.13 g.cm -3 for the pure TPU/di-TDI/HQEE material).

In Figure 8, two glass transitions are also visible on the first curve that can be associated with the polyoxytetramethylene soft phase and the amorphous part of PA11, respectively. After postcuring the first one remains unchanged, while that of PA11 tends to disappear; a slight phenomenon is nevertheless visible around 110°C. Consequently the reactions during postcuring must rather imply TPU hard segments and PA11.

CONCLUSIONS

Polyurethane elastomers synthesized classically by cast molding, extrusion or RIM technology can also be easily processed and afterwards cured by a technique well-known in the rubber field.

Curing the polyurethane chain precursors with a diisocyanate results in the formation of allophanate or biuret bonds in the polymer material. Many reactions take place between the different species and depend on the isocyanate content and processing temperature. The network properties are easily modified by the addition of a chain extender, with the creation of hard segments and exchange reactions which allow to insert them between the soft segments. The thermomechanical properties are typical of segmented polyurethanes, with a low glass transition temperature (well below 0°C), a relatively high melting point depending on the type and proportion of hard segments, and a high rubbery plateau modulus (around or above 100 MPa).

In this way, the design and the preparation of a broad range of new materials become possible. The calendering technique allows the incorporation of highly reactive monomers (especially amines, which usually confer better thermomechanical properties to the material but are very difficult or impossible to process other than by RIM) and of high amounts of mineral or organic fillers.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 1 :Figure 2 :

 1234567812 Figure 1: Reagents used in this study

Figure 3 :

 3 Figure 3: Viscoelastic spectra of the starting TPU (a) ( , ) and of three formulations TPU/di-TDI with different weight percent of diisocyanate (b) ( , : 7phr; , ∆: 8phr; , : 16phr; dark symbols: storage modulus; clear symbols: loss factor)

Figure 4 :Figure 5 :Figure 6 :Figure 8 :

 4568 Figure 4: Proposed structure for the networks based on TPU and di-TDI

Table 1 : Composition of the two-component systems (TPU/di-TDI)

 1

	TPU (g)	di-TDI (g); (mol)
	100	7; 2.01.10 -2
	100	8; 2.30.10 -2
	100	16; 4.60.10 -2

Table 2 : Composition of the three-component systems (TPU/diisocyanate/chain extender)

 2

	Formulation	TPU (g)	diisocyanate	chain extender	hard segment
			(g); (mol)	(g); (mol)	(wt%)
	TPU/di-TDI/HQEE	100.00	30.00; 8.62.10 -2 12.54; 6.33.10 -2	30
	TPU/di-TDI/MCDEA 100.00	30.00; 8.62.10 -2 24.02; 6.32.10 -2	35
	TPU/PPDI/HQEE	100.00	13.42; 8.39.10 -2 12.26; 6.19.10 -2	20

Table 3 : Composition of the three-component system (TPU/di-TDI/HQEE), pure or modified by PA11

 3

	Formulation	TPU (g)	di-TDI (g); (mol)	HQEE (g); (mol)	PA11 (g)
	TPU/di-TDI/HQEE	100.00	30.00; 8.62.10 -2 12.54; 6.33.10 -2	0.00
	TPU/PA11/di-TDI/HQEE	80.00	30.00; 8.62.10 -2 12.54; 6.33.10 -2	20.00

ACKNOWLEDGEMENTS

The authors wish to express their thanks to SAMI (Société d'Applications de Moulages Industriels, Malesherbes, France) for financial support and fruitful discussions.


Nawal Eldrogi

Benoît Larroque
email: benoit.larroque@univ-pau.fr

Valérie Bolliet

Franck Luthon

Vision par ordinateur pour suivi automatique de civelles en bassin Computer vision for automatic detection and tracking of glass-eels

Keywords: Poisson, Soustraction de fond, Étiquetage en composantes connexes, Classification K-moyennes Elver, Background subtraction, connected component analysis, K-means, tracking

Le suivi d'organismes vivants permet de comprendre les dangers qui les guettent et qui pourraient un jour avoir des effets négatifs sur l'homme. Des laboratoires de l'Université de Pau et des Pays de l'Adour (UPPA) collaborent sur le thème de la gestion des poissons migrateurs du bassin de l'Adour. Une des actions de recherche concerne la vision par ordinateur pour le suivi automatique et l'acquisition de connaissances sur le comportement migratoire d'une espèce menacée, les alevins d'anguilles. Pour cela, quelques civelles sont marquées, puis introduites dans un milieu expérimental. Pendant toute la durée de l'observation, une grande base de données de séquences vidéo est collectée. Dans cet article, les premières étapes d'un algorithme permettant de détecter et de suivre les civelles dans des séquences vidéos sont développés. La méthode de soustraction de fond a été utilisée pour détecter les civelles, ensuite, une opération d'étiquetage des composantes connexes est appliquée pour extraire toutes les informations sur les civelles. Le suivi de civelle par l'utilisation du filtrage de Kalman est actuellement à l'étude.

Introduction

Le suivi automatique d'objet en mouvement est un domaine de recherche actif en vision par ordinateur. C'est un processus important pour de nombreuses applications telles que la reconnaissance d'objet, le traitement d'images médicales, l'analyse de comportement humain ou animal, la robotique et la surveillance automatique. En biologie, le suivi du déplacement d'animaux pose parfois des problèmes techniques, liés aux caractéristiques des espèces et des stades de développement. Les biologistes utilisent des séquences vidéo pour identifier et quantifier les poissons afin d'étudier leur comportement et leurs schémas de migration. Généralement, ces vidéos sont analysées manuellement par des experts. Cependant, l'analyse manuelle est un processus fastidieux qui prend plusieurs heures pour analyser une vidéo. L'automatisation, au moins partielle, de ce processus permettra de gagner beaucoup de temps dans l'analyse des vidéos. La majorité des recherches utilise la vision par ordinateur pour détecter et suivre des poissons pour caractériser leur comportement et leur trajectoire [START_REF] Niu | Survey of Fish Behavior Analysis by Computer Vision[END_REF]. Les méthodes les plus traditionnelles pour détecter et suivre les poissons automatiquement sont la détection des contours [START_REF] Shrivakshan | A comparison of various edge detection techniques used in image processing[END_REF] ainsi que le modèle de mélange gaussien [START_REF] Zivkovic | Improved adaptive Gaussian mixture model for background subtraction, Pattern Recognition[END_REF]. Les travaux de [START_REF] Sharma | Fish Counting from Underwater Video Sequences by Using Color and Texture[END_REF] utilisent le filtre de Canny pour détecter les poissons ainsi qu'une analyse par blob de la couleur et de la texture pour automatiser le traitement des vidéos. Dans [START_REF] Spampinato | Tracking and Counting Fish in Low Quality Unconstrained Underwater Videos[END_REF], le suivi est effectué par l'algorithme CamShift, un système de vision par ordinateur capable d'analyser des vidéos sous-marines pour détecter, suivre et compter les poissons. Des travaux récents [START_REF] Levy | Automated Analysis of Marine Video With Limited Data[END_REF] montrent comment l'apprentissage par transfert peut être utilisé pour obtenir des résultats de grande qualité, à la fois pour la détection et pour le suivi. FIGURE 1 -Marquages VIE sur les civelles [START_REF] Delcourt | Video tracking in the extreme : A new possibility for tracking nocturnal underwater transparent animals with fluorescent elastomer tags[END_REF].

Pour suivre les civelles, chaque individu est marqué par VIE Tag (Visible Implant Elastomer, voir Fig. 1) [START_REF] Bolliet | Individual patterns of rhythmic swimming activity in Anguilla anguilla glass eels synchronised to water current reversal[END_REF]. Ce marquage consiste à implanter sous la peau une pointe d'élastomère de couleur, visible sous UV. Le suivi des individus se fait sur des enregistrements vidéo [START_REF] Delcourt | Video tracking in the extreme : A new possibility for tracking nocturnal underwater transparent animals with fluorescent elastomer tags[END_REF], mais il représente un travail fastidieux, car il n'est actuellement pas automatisé. Les paramètres qui intéressent les biologistes sont principalement le sens de déplacement des civelles (dans le sens du courant ou à contre-courant) et leur vitesse. Toute mesure permettant d'évaluer la dépense énergétique est également recherchée, car les civelles jeûnent durant la migration, et leur statut énergétique pourrait jouer un rôle important dans le potentiel migratoire. Dans notre article, un système pour détecter automatiquement les civellesest présenté. Les étapes de ce système de traitement informatisé sont les suivantes :

• Détecter les civelles par la méthode de soustraction de fond suivie par une opération d'étiquetage des composantes connexes. • Construire une table d'étiquetage qui contient toutes les informations sur le poisson détecté (mesure géométrique, couleur du marquage,...) • Générer un suivi des civelles par des boîtes englobantes et l'utilisation du filtrage de Kalman. L'article est structuré de la façon suivante : en premier lieu le contexte expérimental est décrit puis la section 3 présente les principes généraux pour le suivi. Enfin la section 4 décrit notre approche et les résultats obtenus.

Contexte expérimental

Pour réaliser cette étude, un aquarium en forme d'ovale, de longueur 1, 50m et de largeur 50cm, est séparé par une paroi rigide en son centre [START_REF] Bolliet | Individual patterns of rhythmic swimming activity in Anguilla anguilla glass eels synchronised to water current reversal[END_REF] (Fig. 2). FIGURE 2 -Schéma de l'aquarium [START_REF] Célia | Impact du méthylmercure sur le potentiel de migration, le statut énergétique et les mécanismes de détoxication chez la civelle d'anguille européenne (Anguilla anguilla) : Etudes in situ et expérimentale[END_REF].

Deux pompes, situées aux extrémités opposées de l'aquarium, permettent d'alterner les courants d'eau toutes les 6,2 heures afin de reproduire le rythme des marées. La température de l'eau est maintenue à 11±0.5 • C, et une couche de gravier couvre le fond de l'aquarium. La vitesse du courant d'eau est en moyenne de 11 cm/sec. Une moitié des civelles est marquée avec des combinaisons de couleurs (système de marqueur VIE).

Processus général de suivi

De nombreuses approches de suivi d'objets ont été proposées. Dans cette section, les différentes méthodes et procédures de suivi décrites dans [START_REF] Parekh | A survey on object detection and tracking methods[END_REF] sont présentées.

FIGURE 5 -Processus général de suivi

Comme le montre la Fig. 5, les systèmes de suivi se composent généralement des étapes suivantes :

Détection d'objet

Méthode de Soustraction de fond La soustraction de fond est largement utilisée pour détecter des objets en mouvement dans des séquences vidéo. Cette méthode a été utilisée et a fait ses preuves dans de nombreuses études [START_REF] Sobral | A comprehensive review of background subtraction algorithms evaluated with synthetic and real videos[END_REF]. Elle est considérée comme la méthode la plus simple qui donne des détails complets sur l'objet par rapport aux autres méthodes de détection de mouvement comme la méthode de différenciation d'images ou de flux optique. Comme l'établit l'équation 1 la soustraction de fond consiste à soustraire l'image actuelle de l'image de référence (appelée aussi image de fond).

|I actuelle -I ref erence | > seuil (1)
Un seuillage de cette différence donne ensuite un masque avec les objets du premier plan. L'image de référence est une image de la scène statique sans objet mobile. Création de l'image référence L'image de référence contient les parties statiques de la scène mais elle n'est pas toujours disponible au début de la séquence vidéo, il faut d'abord la construire puis la réactualiser au cours du temps afin qu'elle s'adapte aux évolutions de la scène.

La méthode basique est de faire un moyennage des images de la séquence vidéo par l'équation 2.

I ref (x, y) = 1 N N k=1 I(x, y, k) (2)
Avec I(x, y, k) la valeur du pixel (x, y) de la k-ème image, et N le nombre d'images dans la séquence vidéo.

La classification

Il existe deux approches principales pour la classification : la classification supervisée et la classification nonsupervisée. L'approche supervisée se fait par une étape d'apprentissage permettant d'extraire l'information pertinente de grands ensembles de données contrairement à la classification non-supervisée qui se fait principalement par séparation automatique des nuages de points dans un espace sans le besoin de fournir des données d'apprentissage. Parmi les algorithmes de classification non supervisés, on peut citer la méthode des K-moyennes comme un outil efficace et rapide car il n'est pas nécessaire de faire l'apprentissage sur l'ensemble des données. Algorithme K-moyennes C'est l'un des plus simples algorithmes d'apprentissage non supervisés, son principe est de classer les points ou les objets en K groupes (souvent appelés clusters) et de minimiser la variance intra-cluster [START_REF] Dhanachandra | Image segmentation using K-means clustering algorithm and subtractive clustering algorithm[END_REF]. Les K clusters sont produits de façon à minimiser la fonction objective décrite dans l'équation 3.

E = K k=1 xj ∈S k (x j -g k ) 2 (3)
où : S k (pour k = 1, 2, ..., K) sont les K groupes et g k leur centre de gravité. En effet, cette méthode, initialise k groupes qui correspondent au nombre de groupes attendus par l'utilisateur. Ensuite, elle calcule le centre de gravité de chaque ensemble et construit une nouvelle partition en associant chaque point avec le centre de gravité le plus proche. Une alternance a lieu entre le calcul des centres de gravité des nouveaux clusters et l'appariement des points avec le centre de gravité le plus proche jusqu'à convergence. Cet algorithme est très populaire car extrêmement rapide en pratique. En effet, le nombre d'itérations est typiquement inférieur au nombre de points.

Suivi d'objet

De nombreux algorithmes de suivi ont été proposés dans la littérature [START_REF] Balaji | A survey on moving object tracking using image processing[END_REF]. Certaines méthodes sont dédiées au suivi d'objets spécifiques [START_REF] Qian | Realtime and robust hand tracking from depth[END_REF]

Étape 2 : La classification K-moyennes

Ensuite, à partir de la table d'étiquetage l'algorithme des K-moyennes a été appliqué comme décrit dans la section (3.2) sur la composante Hue pour obtenir la plage de chaque couleur de marquage.

Étape 3 : Suivi individuel des civelles

Dans l'étape de suivi, il est nécessaire d'identifier les civelles en premier lieu (marquage unique ou double marquage). Ce traitement peut être effectué en utilisant une boîte englobante (Fig. 12) autour de chaque marquage dont les dimensions prennent en compte les paramètres géométriques des civelles afin d'identifier si un autre marquage à proximité appartient à la même civelle. La Fig. 13 présente les premiers résultats issus de la mise en correspondance des doubles marquages par boîte englobante. Afin de pouvoir suivre et identifier chaque civelle pour chacune des vidéos il est possible d'utiliser les boîtes englobantes avec une approche similaire à celle décrite en Fig. 12. Dans le cas où plusieurs civelles se croisent il est nécessaire de prendre en compte les évènements passés. En effet, lors de cette étape et pour prendre une décision à chaque image il est utile de connaître la situation dans les images précédentes. L'utilisation du filtre de Kalman permettra de pré-

FIGURE 3 -FIGURE 4 -

 34 FIGURE 3 -Extrait d'une image issue d'une vidéo de nuit et comportant 4 civelles avec double marquage

  et d'autres méthodes sont conçues d'une manière plus générique[START_REF] Zhang | Pyramid-based visual tracking using sparsity represented mean transform[END_REF]. Les travaux menés actuellement sont liés à l'utilisation du filtre de Kalman qui permet d'estimer les paramètres dynamiques des objets. Ce filtre peut être utilisé pour obtenir la meilleure estimation de la trajectoire réelle et de diminuer les erreurs pouvant être dues au changement d'éclairage, au changement de fond et à l'apparition des autres objets en mouvement (croisement de civelles sur une image).4 Méthodologie de suivi des pibales et résultats obtenus4.1 Étape 1 : La détectionLa détection de la civelle se fait par deux modules. Le premier est basé sur la soustraction de fond suivi par une opération morphologique et le deuxième est d'effectuer l'analyse des composantes connexes sur l'image résultante afin d'extraire ses informations géométriques (Fig.6).

FIGURE 6 -FIGURE 7 -

 67 FIGURE 6 -Séquence de détection

FIGURE 8 -FIGURE 9 -

 89 FIGURE 8 -Algorithme K-moyennes

FIGURE 10 -FIGURE 11 -

 1011 FIGURE 10 -Système de carte couleur HSV.

FIGURE 12 -FIGURE 13 -

 1213 FIGURE 12 -Taille de la fenêtre de suivi

TABLE 1 -

 1 Extrait d'une table d'étiquetage la couleur de chaque objet. Pour cela, la procédure suivante est appliquée :• Transformer chaque image RVB en une image HSV en raison de l'indépendance entre les couleurs dans cet espace perceptif.

• Récupérer sa composante de teinte (Hue) en se basant sur la couleur du pixel du centre de gravité de coordonnées (C x , C y ) de chaque objet.

• Ajouter les trois composantes d'image HSV (Hue, Saturation, Valeur) à la table d'étiquetage (voir tableau 1).