Florian Bouville

Sylvain Deville

Dispersion of Boron Nitride Powders in Aqueous Suspensions with Cellulose

The dispersion of nitride and carbide ceramic particles in water is difficult, due to the absence of hydroxyl groups on their surface. Boron nitride is not an exception and despite its numerous applications, no effective dispersant has been identified so far. We demonstrate here the dispersion properties of two cellulose derivatives for hexagonal boron nitride powders, hydroxyl ethyl cellulose and methyl cellulose. The effect of particle sizes and cellulose concentration was investigated. The adsorption of cellulose onto the surface of the BN particles was confirmed by isotherm adsorption.

Zeta potential measurements shows a charge screening effect of the cellulose The suspensions obtained were highly loaded and stable versus pH, and thus could lead to homogeneous codispersion of BN particles and sintering additives.

I. Introduction

Hexagonal boron nitride (BN) exhibits a unique combination of high thermal conductivity, high resistance to thermal shock, high thermal/chemical stability, and low surface energy. These properties arise from its crystalline structure 1 : covalently bonded boron and nitrogen that forms layers that are weakly bonded by Van der Walls forces. The structure is similar to graphite but electrically insulating. Because of its low surface energy, BN is particularly useful in applications involving liquid metals handling where it acts as a barrier to prevent corrosion and contamination.

Such benefits are also particularly useful for high temperature crucibles [START_REF] Eichler | Boron nitride (BN) and BN composites for high-temperature applications[END_REF] . These two main applications usually rely on aqueous powder processing routes, to cast a preform or deposit a coating. The low sinterability of BN requires then the use of densification techniques such as hot isostatic pressing (HIP) in presence of sintering additives [START_REF] Ertug | Microstructural aspects of the hot-pressed hexagonal boron nitride ceramics with limited content of boron oxide[END_REF][START_REF] Lee | Thermal conductivity of hot-pressed BN ceramics[END_REF] .

Unfortunately, the low surface energy of BN is a major drawback for its use in aqueous media [START_REF] Crimp | Suspension properties of hexagonal BN powders : effect of pH and oxygen content[END_REF] , because it is not easily dispersed. This dispersion issue often results in heterogeneities in the green body, both in terms of particles packing density and sintering additives. An effective dispersant could allow the use of highly loaded suspensions with a homogenous mix of BN particles and sintering additives. The final ceramic parts could therefore present more homogenous microstructures and thus improved mechanical properties.

Cellulose derivatives are usually added in suspension as thickening agent, to prevent the settling of large particles. We demonstrate through zeta potential, viscosity, and adsorption measurements that cellulose are also efficient dispersants of BN particles in aqueous suspensions.

II. Experimental procedure

BN powders were provided by Saint-Gobain BN products (Très BN series). Those powders are mainly used in cosmetics products and thus present a relatively high purity and crystallinity, but also low boric acid content. The main characteristics of the different powders are given in the table 1. The granulometric distributions have been obtained by laser granulometry, which assumes a relative sphericity of the particles. The equivalent diameters measured are the diameter of the platelets, values confirmed by SEM observations. The thicknesses of the platelet have been evaluated by SEM observations. Two types of cellulose were tested: a hydroxyethyl cellulose HEC (Tylose H 4000 P2 from ShinEtsu, Mv = 720 000) and methyl cellulose MC (Methocel A4M from Dow Chemicals, Mw = 88 000). The celluloses present the same viscosity of 4 Pa.s at 2 wt.% in water but their structures are different, the MC is more hydrophile than the HEC. The suspensions were prepared by mixing the powder, the cellulose, and distilled water with a magnetic stirrer or a propeller mixer depending on their viscosity. Suspensions with hydroxyethyl cellulose were mixed for at least 4 hours to ensure total hydration before being used. Ultrasonic mixing was applied with a sonotrode (Digital Sonifier 250) with an applied energy around 150 W.h/kg. Adsorption isotherm measurements were conducted using a total organic carbon (TOC) analyzer (TOC VCSN, Shimadzu), which provides a quantitative measure of the non-adsorbed fraction of cellulose in solution. BN/Cellulose suspensions from 0 to 1 wt.% of HEC were prepared. After complete hydration and mixing, the suspensions were centrifuged at 7200 rpm for 1h and supernatant aliquots were taken to be analyzed. Zeta potentials were measured with a ZetaProbe after dispersion of the particles in distilled water by sonication. The pH was adjusted with 1 mol/L hydrochloric acid and sodium hydroxide. To calculate the zeta potential with this technique, the relative sphericity of the particles is assumed. This assumption is not valid with the anisotropic particles used here, so the measured values offer only a qualitative comparison of the zeta potential values between samples.Rheological measurements were performed in a concentric cylinder viscosimeter (Bohlin, Malvern, UK). The suspension was presheared at 100 s -1 during 30 s followed by 30 s at rest. Two procedures were followed: a shear rate ramp was applied from 0.05 s -1 to 600 s -1 or a constant shear rate for 1 min. A shear rate of 150 s -1 was chosen to compare the viscosity of the different suspensions. Most of the suspensions have reached a quasi-Newtonian behavior before this value, which ensure the stability of the measure.

III. Results and Discussion (1) Effect of ageing time

A 19 vol.% of BN-16 suspension, with HEC to powder ratio of 0.019, was mixed by a magnetic stirrer at 800 rpm and a part of the suspensions was taken at different ageing times to perform rheological measurements. The evolution of viscosity versus ageing time is plotted in figure 1. The viscosity decreased by a factor six just by maintaining the stirring for fifteen days. The final viscosity was even slightly below the viscosity of the solvent with the same amount of HEC (0.11 Pa.s for the slurry and 0.15 Pa.s for 1 wt.% HEC solution). The evolution could be explained by the adsorption of the cellulose on particle surface. This phenomenon could help the breakage of agglomerates, and thus could reduce the suspension viscosity.

(2) Effect of amount of cellulose Rheological characterization (fig. 2) has been conducted on 19 vol.% BN-8 and BN-1 suspensions with HEC to powder weight ratio ranging from 0 to 0.03. Concerning the BN-1 suspensions, only the HEC/BN weight ratios of 0.01, 0.02, and 0.03 were tested because the suspensions corresponding to the other ratios presented a viscosity too high to be assessed by the concentric cylinder rheological setup.

Repeated HEC additions minimized the viscosity of the suspensions for both particle sizes. After this minimum, the viscosity increases at different rates for the various particle sizes. The optimum weight ratio of HEC/BN is 0.0025 for 8 µm powder and 0.02 for 1 µm powder. The presence of a viscosity minimum is generally associated with a ratio where the dispersant molecules cover the maximum of particle surface without any excess in the suspension [START_REF] Briscoe | Optimising the dispersion on an alumina suspension using commercial polyvalent electrolyte dispersants[END_REF] . If we assume that HEC acts as a dispersant for BN, this particular ratio must be constant with the evolution of surface area. Suspension viscosity evolution versus HEC/powder ratio divided by powder surface area is shown in figure 2b. The minimum viscosity is reached at almost the same ratio for both powders, which is consistent with an optimum repartition of a dispersant on particle surface, independent of its size.

(3) Adsorption isotherm of cellulose

The adsorption isotherm for HEC on BN-8 is plotted in the figure 3. A nearly total adsorption on the surface (>90% for 10 mg/m²) is observed for a relative low amount of cellulose added (1g/L corresponding to a HEC/powder ratio of 0.0025). Then the adsorbed amount reached a plateau at 15 mg/m² when more cellulose is added. Those value are relatively high, compared to the typical value of well-known system such as poly(acrylic acid) (PAA) on oxides, that are around 1 mg/m² [START_REF] Cesarano | Stability of Aqueous alpha-Al2O3 Suspensions with Poly(methacrylic acid) Polyelectrolyte[END_REF][START_REF] Kirby | Poly(acrylic acid)-Poly(ethylene oxide) Comb Polymer Effects on BaTiO 3 Nanoparticle Suspension Stability[END_REF] . This quantity is usually related to the polymer conformation on particle surface, the more the polymer is stretched, the lower the concentration needed to entirely cover the surface. However, the cellulose used here presents chains nearly an order of magnitude longer (the molecular mass is MV = 720 000) than the usual dispersant used (MW around 10 000 for most of the commercially available PAA). This amount could come from the larger number of cellulose chains needed to cover the surface. The large molecular weight could also explain why a full coverage is not achieved. The cellulose derivatives are effectively adsorbed on boron nitride platelet and thus play the role of a steric dispersant.

(4) Effect of cellulose on zeta potential

Due to the low boric acid content (<0.1 wt.%), zeta potential of these BN particles is low, between -8 mv to -20 mV. Only the boric acid present on particle surface exhibit hydroxyl groups and thus can induce surfaces charges changes with the pH [START_REF] Lewis | Colloidal processing of ceramics[END_REF] . This behavior has also been reported by Crimp et al. [START_REF] Crimp | Suspension properties of hexagonal BN powders : effect of pH and oxygen content[END_REF] with two different powders of different boric acid content in aqueous suspension. The higher the surface charge, the higher the oxygen content. However, the addition of the HEC decreased the zeta potential from -20 mV to -4 mV at the solution pH and at pH 4. The decrease in zeta potential is usually associated with a more agglomerated state of particles, which induces higher suspension viscosity, but HEC addition lowered the viscosity (figure 4). The addition of this type of cellulose possibly masks the surface charge, thus lowering the zeta potential, and acts as a steric barrier to prevent agglomeration. This behavior of polysaccharides has been observed previously by Wang and al [START_REF] Wang | Mechanisms of ethyl(hydroxyethyl) cellulose-solid interaction: influence of hydrophobic modification[END_REF] with clay and ethyl (hydroxylethyl) cellulose. Clay particles have a flake morphology (a few microns in diameter and a hundred of nanometer thick) with a flat surface usually presenting a low negative surface charge (with a small numbers of hydroxyls groups), while the edges are positively charged (with hydroxyl group). Even if the interactions between cellulose and minerals are not well established yet, the frequently accepted explanation consists in adsorption due to a mix of hydrophobic forces on faces and hydrogen bonding on edges. Wang et al. [START_REF] Wang | Mechanisms of ethyl(hydroxyethyl) cellulose-solid interaction: influence of hydrophobic modification[END_REF] conclude that hydrogen bonding -thus interactions with edges-is predominant. With BN, edges should not play a key role in interaction due to a different particle size and aspect ratio. A similarity between clay faces and BN can be noticed because none of them present hydroxyl groups, and thus supposedly the main mechanism of cellulose adsorption on surface is hydrophobic forces. The zeta potential is lowered in neutral to acidic pH range, a range where cellulose is considered stable; a proper dispersion can theoretically be obtained within this range.

(5) Effect of cellulose type on dispersion

Another type of cellulose, methyl cellulose (MC), has been mixed with BN and the viscosity measured under the same conditions. MC was chosen to assess if the hydrophilicity influences the dispersion behavior. The resulting viscosity at different ratio and the data obtained with HEC are represented in figure 5.

An optimum viscosity is found for both MC and HEC at different cellulose/powder ratio (0.0025 and 0.005 for HEC and MC respectively). This suggests that the nature and/or amount of cellulose functional groups do not change its capacity to disperse the BN particles. Indeed, if the main mechanism of adsorption is hydrophobic bonding, the nature of cellulose does not have great impact as long as they have the same type of interaction with the solvent and particle, e.g. the same type of hydrophobic and hydrophilic functional groups. The amount of HEC required is slightly lower, possibly because of a larger molecular mass (Mw = 80 000 for MC and MV = 720 000 for HEC) that leaded to a maximum coverage of the surface with less molecules.

(6) Effect of solid loading

The effect of solid loading on the behavior of a suspension is a critical parameter in order to use the slurry in typical ceramic processing, like slip or tape casting. A high aspect ratio of particles is a drawback to obtain loaded suspension because of low packing ability due to a greater steric hindrance. This can be partially tackled if the particle-particle interactions are limited due to an optimum amount of dispersant. In order to as determine maximum solid loading of cellulosedispersed BN, the rheological behavior at different solid loading have been investigated (figure 6), using the dispersant /BN ratio obtained previously (figure 2).

A typical shear thinning behavior is obtained for the three different slurries, due to the breaking of particles arrangement and their alignment along flow direction as the shear gradient increases [START_REF] Barnes | An introduction to rheology[END_REF] .

Experimental data are plotted in figure 6 for the BN suspensions. For comparison a suspension of BN-8 without HEC at 17 vol.% was also investigated. The suspension without HEC presents a high shear-thinning behavior, its viscosity decreases by two orders of magnitude. The viscosity at low shear rate of the suspension with HEC is more than an order of magnitude lower than the viscosity of the suspension without HEC. The viscosities values of both suspensions get closer as a higher shear rate is applied, probably because the agglomerates are broken by the applied stress. The high viscosity at low shear rate makes it hard to cast the suspension in any mold. The dependencies of the relative viscosity (viscosity of suspension at high shear rates on viscosity of the medium [START_REF] Kirby | Poly(acrylic acid)-Poly(ethylene oxide) Comb Polymer Effects on BaTiO 3 Nanoparticle Suspension Stability[END_REF] ) values with the solid volume fraction φ are plotted in figure 7 with and without HEC. The relative viscosities without dispersant are higher at lower volume fractions (three times higher for a volume fraction of 0.3). The data with dispersant were fitted by a Krieger-Dougherty model [START_REF] Kirby | Poly(acrylic acid)-Poly(ethylene oxide) Comb Polymer Effects on BaTiO 3 Nanoparticle Suspension Stability[END_REF][START_REF] Lewis | Colloidal processing of ceramics[END_REF] (1). The high shearthinning behavior of the suspensions without cellulose additives makes it hard to precisely determine the viscosity minimum and thus to apply the same model on it. [] (1)

 𝑟𝑒𝑙 = (1 -   𝑚 ) - 𝑚
Equation ( 1) allows calculating m, the volume fraction of particle where the viscosity tends to the infinity. In this equation, s is the viscosity of the dispersing medium and [] is the intrinsic viscosity.

The intrisic viscosity and the maximum volume fraction are fonction of particle's shape. The value of

[] = 2.5 and m = 0.64 are obtained with perfectly spherical and monodispersed particle, but [] increases and m decreases as the particle aspect ratio increases. The value obtained with BN, [] = 3.1 and m = 0.36, are consistent with these evolutions and the values found in litterature with various materials [START_REF] Bergstrom | Shear thinning and shear thickening of concentrated ceramic suspensions[END_REF][START_REF] Mora | Dispersant-free colloidal fabrication of Bi2Sr2CaCu2O8 superconducting thick films[END_REF] . Barnes et al. [START_REF] Barnes | An introduction to rheology[END_REF] found an empirical formula for the intrinsic viscosity evolution with the aspect ratio of disks [] = 3 (axial ratio) /10 and leads, in this case, to an aspect ratio of approximately 10, which is consistent the aspect ratio of 16 for BN particles, considering the high polydispersity of these powders (cf. table 1). As described previously, the plate-like shape of these particles prevents a high solid loading suspension.

IV. Conclusion

Hexagonal boron nitride (BN) aqueous slurries can be dispersed using cellulose derivatives. The mechanism of the adsorption on particles surface is not explained in this study although experimental evidences point toward hydrophobic forces. Because cellulose is relatively stable versus pH, the dispersion can be achieved within a relatively large pH range. Obtaining a higher volume fraction could be achieved by adjusting more precisely the molecular weight.

A stable dispersion with a high solid loading offers interesting possibilities to obtain homogeneous green bodies with fewer defects, especially when associated with electrostatic dispersion of sintering additives. An alternative to make BN ceramic can now be considered: the use of thermal gel casting. Indeed cellulose ethers form a gel when heated above a certain temperature and thus could directly be used for ceramic shaping.

FiguresFig. 1 .

 1 Figures

Fig. 2 .

 2 Fig. 2. (a) Effect of HEC/Powder weight ratio on different sizes of boron nitride particles. (b) Effect on

Figure 3 .

 3 Figure 3. Adsorption isotherm of HEC on BN-8 surface.

Fig. 4 .

 4 Fig. 4. Effect of HEC on Zeta potential BN particles.

Fig. 5 .

 5 Fig. 5. Effect of cellulose type on suspension viscosity.

Fig. 6 .

 6 Fig. 6. Rheological behavior of BN-8 with and without HEC suspension at different solid loadings.

Fig. 7 .

 7 Fig. 7. Minimum viscosity versus particle volume fraction with and without HEC. The solid line is the

Acknowledgements

We acknowledge the financial support of the ANRT (Association Nationale Recherche Technologie) and Saint-Gobain through a CIFRE fellowship, convention #808/2010.


Grégoire P Millet

Franck Brocherie

Raphael Faiss

Olivier Girard

Clarification on altitude training

We congratulate our colleagues for their careful reading of our work in their Hot Topic Review 'Does 'altitude training' increase exercise performance in elite athletes?' (Lundby & Robach 2016) but strongly believe that they have a selective appreciation of the available literature that requires clarification.

Living High Training low (LHTL)

 detailing the controlled training and environmental parameters along with the high level of reproducibility of the duplicate Hb mass method used.

Repeated Sprint Training in Hypoxia (RSH)

The authors question how RSH might lead to "performance gains" as high as 55%. Precisely, improved repeated-sprint performance post-RSH training in the quoted study (Faiss et al. 2014) was highlighted by longer exercise duration (or a larger number of completed sprints) before reaching exhaustion (carefully defined as 70% of peak power). While such performance improvement remains very specific (test protocol, participants background, training content), the reader needs to be cognizant that even a small change in an individual's power produces a large change in time to exhaustion [START_REF] Allen | Age of Peak Competitive Performance of Elite Athletes: A Systematic Review[END_REF]. Such specific performance enhancement is in fact discussed carefully in our articles as to avoid any over-interpretation of our data [START_REF] Faiss | Response[END_REF]. [START_REF] Montero | Enhanced Performance after Repeated Sprint Training in Hypoxia: False or Reality[END_REF] criticize the "criteria for fatigue" we have used and based on fancied ex nihilo calculations, claim that the outcomes we have reported result from "differentiated criteria" between intervention groups. We contend that this has no foundation and have carefully replied to this criticism, explaining how inaccurate their reasoning is [START_REF] Faiss | Response[END_REF]. Curiously, in their first RSH study [START_REF] Montero | Repeated Sprint Training in Hypoxia Versus Normoxia Does Not Improve Performance: A Double-Blind and Cross-Over Study[END_REF], the authors have used the same criteria. How coherent is this?

A careful reading of the available literature would mention the very likely greater gains in a specific repeated agility test (cumulated times) after RSH compared to similar training near sea-level (Brocherie et al., 2015a). Additionally, in [START_REF] Brocherie | Live High-Train Low and High" Hypoxic Training Improves Team-Sport Performance[END_REF], the LHTL+RSH group had a twice larger RSA enhancement than the LHTL + RSN group and maintained this positive outcome for three weeks.

We also disagree with Lundby and Robach's interpretation of the study by [START_REF] Gatterer | Shuttle-run sprint training in hypoxia for youth elite soccer players: a pilot study[END_REF] who actually reported a training × group interaction on the fatigue slope calculated during repeated sprint exercise. This in fact confirms a better "resistance to fatigue" and Gatterer et al. concluded "this type of hypoxic training led to larger improvements in RSA when compared to normoxia training". That is the opposite of Lundby and Robach's conclusions.

The authors then cite their recent study [START_REF] Montero | Repeated Sprint Training in Hypoxia Versus Normoxia Does Not Improve Performance: A Double-Blind and Cross-Over Study[END_REF] where 12 different tests in 3 days post-intervention were conducted. We [START_REF] Girard | Repeated sprint training in hypoxia versus normoxia does not improve performance due to methodological shortcomings?[END_REF] pointed out many methodological flaws in this study, questioning how individuals were able to maximally perform so many exhaustive exercise tests in such a small time period without any confounding deleterious influence of residual fatigue, likely to induce negative pacing strategies e.g. no control of the training intensity or of the pacing during RSA test. We concluded that many methodological shortcomings may explain why no additional effect of RSH versus RSN was observed.

A rare point agreed with the authors is that the readers/practitioners should critically assess the strength and weaknesses of the different studies and of the different sides of this debate.

Finally, we predict that the authors may change their mind in the future regarding RSH as they did regarding the efficiency of LHTL. A few years ago with the method they use for measuring Hb mass , a larger than 10% increase in Hb mass was reported for 250 h of exposure at 2500-3000 m, corresponding to an increase > 4% per 100 h in elite endurance athletes [START_REF] Brugniaux | Eighteen days of "living high, training low" stimulate erythropoiesis and enhance aerobic performance in elite middle-distance runners[END_REF]. One of the authors then stated "LHTL improves VO 2 max and associated power output. This improvement represents a marked increase, especially for elite athletes". Whereas recently [START_REF] Robach | The role of haemoglobin mass on VO(2)max following normobaric 'live high-train low' in endurancetrained athletes[END_REF] they stated "the positive effects of LHTL on oxygen transport appear to be negligible among elite cyclists who already possess very high aerobic capacities conferred by high Hb mass and VO 2 max".


G Boil

AND S. V Ũ NGO . C N Raymond

ON THE STABILITY OF THE SCHWARTZ CLASS UNDER THE MAGNETIC SCHRÖDINGER FLOW

We prove that the Schwartz class is stable under the magnetic Schrödinger flow when the magnetic 2-form is non-degenerate and does not oscillate too much at infinity.

1. Introduction 1.1. Motivation and context. This paper is devoted to describing the solutions to the magnetic Schrödinger equation. Let B be a smooth and closed 2-form on R d . Let A : R d → R d be a 1-form (identified with a vector field) such that dA = B. The magnetic Schrödinger operator is the essentially self-adjoint differential operator

L h = (-ih∇ -A) 2 = d j=1 L 2 j ,
where h > 0 and, for all j ∈ {1, . . . , j}, L j = -ih∂ j -A j . Its domain is given by

Dom(L h ) = {ψ ∈ L 2 (R d ) : (-ih∇ -A)ψ ∈ L 2 (R d ) , (-ih∇ -A) 2 ψ ∈ L 2 (R d )} = {ψ ∈ L 2 (R d ) : (-ih∇ -A) 2 ψ ∈ L 2 (R d )} .
The time dependent magnetic Schrödinger equation is given by

-ih∂ t ψ = L h ψ , ψ(0) = ψ 0 ∈ Dom(L h ) . (1.1)
By Stone's theorem, this Cauchy problem admits a unique solution, evolving in the domain of L h , and it is given by ∀t ∈ R , ψ(t) = e itL h h ψ 0 . By unitarity of the flow, we have

∀t ∈ R , ψ(t) = ψ 0 , L h ψ(t) = L h ψ 0 ,
where • denotes the usual norm on L 2 (R d ). This norm controls the rough phase space localization of the quantum state ψ(t); a natural question is to know to which extent a strong phase space localization of ψ 0 is preserved by the flow. More precisely, this paper was inspired by the following rather naive question. Is it true that

ψ 0 ∈ S (R d ) =⇒ ∀t ∈ R , ψ(t) ∈ S (R d ) ? (1.2)
If so, what kind of explicit control do we have in terms of the Schwartz semi-norms? These questions are motivated by the recent investigation of the propagation of coherent states by the magnetic Hamiltonian flow in two dimensions (see the Ph. D. thesis of the first author [START_REF] Boil | Propagation quantique en temps longs d'états cohérents dans un champ magnétique fort[END_REF]). The present paper gives a positive answer to (1.2). Our explicit estimates of the Schwartz semi-norms (in terms of the semiclassical parameter h), combined with the use of the Birkhoff normal form from [START_REF] Raymond | Geometry and spectrum in 2D magnetic wells[END_REF], turn out to be the key ingredients in the study by [START_REF] Boil | Long time dynamics of coherent states in strong magnetic fields[END_REF] of the propagation of coherent states up to times of order h -N , for all N ∈ N. This gives a quantum analog to the low energy (say of order ε) classical propagation for times of order ε -∞ (see [START_REF] Raymond | Geometry and spectrum in 2D magnetic wells[END_REF]Theorem 1.2]). Taking into account the analysis of [START_REF] Helffer | Magnetic wells in dimension three[END_REF], one can even hope to extend these results to three dimensions where the classical dynamics has a more complex behavior.

Independently of this motivation, the answer to (1.2) has an interest of its own, especially because it lives at the confluence of two closely related domains: hypoellipticity and semiclassical analysis with magnetic fields. On these vast subjects, the literature is enormous, and we only refer to [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF][START_REF] Helffer | Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs[END_REF][START_REF] Helffer | Caractérisation du spectre essentiel de l'opérateur de Schrödinger avec un champ magnétique[END_REF][START_REF] Helffer | Décroissance à l'infini des fonctions propres de l'opérateur de Schrödinger avec champ électromagnétique polynomial[END_REF][START_REF] Tataru | On the Fefferman-Phong inequality and related problems[END_REF]16,[START_REF] Fournais | Spectral methods in surface superconductivity[END_REF][START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]. In this paper, we will use many classical ideas from these two contexts, and provide an elementary and self-contained presentation.

1.2. Main results. Let us now describe our assumptions and results.

Let P be the class defined by

P = {ψ ∈ C ∞ (R d ) : ∀α ∈ N d , ∃(C, m) ∈ R + × R + , ∀x ∈ R d , |∂ α ψ| C x m } .
The following assumption will hold throughout the paper, where we identify B with its antisymmetric matrix obtained in the usual basis (dx j ∧ dx k ; j < k).

Assumption 1.1. We assume that i. A belongs to P (in particular B ∈ P), ii. there exists b 0 > 0 such that, for all x ∈ R d ,

Tr + B(x) b 0 ,
where Tr + B(x) denotes the sum of the moduli of the eigenvalues with positive imaginary part of the matrix B(x), iii. for all α ∈ N d , there exists C > 0 such that, for all x ∈ R d , ∂ α B(x)

C B(x) , where • denotes a norm on the space of matrices. Assumption 1.1 is stronger than really necessary as we can see in our proofs. In this context, we will use the following lemma (see [START_REF] Helffer | Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells[END_REF]Theorem 2.2]). Lemma 1.2. We have

inf sp(L h ) = h inf x∈R d Tr + B(x) + o(h) .
In particular, there exist C > 0 and h 0 > 0, such that, for all h ∈ (0, h 0 ), L h is invertible and

L -1 h Ch -1 .
In the following, we will always assume that h is small enough and such that L h is invertible.

Definition 1.3. For all n ∈ N, we let

Dom(L n h ) = {ψ ∈ L 2 (R d ) : ∀ ∈ {1, . . . , n} : L -1 h ψ ∈ Dom(L h )} .
The operator L n h is defined by induction by

∀ψ ∈ Dom(L n h ) , L n h ψ = L h (L n-1 h ψ) .
The operator (Dom(L n h ), L n h ) is self-adjoint and invertible. The following theorem proves some magnetic elliptic estimates, showing that iterations of the magnetic laplacian L h control iterations of the magnetic derivatives (L j ) 1 j d . This will be an important tool on the proof of the main result of the paper.

Theorem 1.4. Let Assumption 1.1 hold. Let n ∈ N. There exist h 0 > 0 and C > 0 such that, for all h ∈ (0, h 0 ), and all ψ ∈ Dom(L n h ),

σ∈A(2n) L σ ψ Ch -3n/2 L n h ψ , (1.3)
where, for k ∈ N, A(k) = ∪ k p=0 {1, . . . , d} {1,...,p} , and for p ∈ N, for σ ∈ {1, . . . , d} {1,...,p} , L σ := L σ(1) . . . L σ(p) , with the convention L ∅ = Id.

In the case where A is bounded, Theorem 1.3 is closely related to [15, Theorem 3], which deals with the context of general Gårding inequalities. Definition 1.5. For all k ∈ N and all ψ ∈ S (R d ), we let

p k (ψ) = max (α,β)∈N 2d |α|+|β| k x α ∂ β ψ ∞ .
We can now state the main result of this paper.

Theorem 1.6. Let Assumption 1.1 hold. For all t ∈ R, we have

e itL h h S (R d ) ⊂ S (R d ) .
More precisely, for all M ∈ N * , for all k ∈ N, there exist h 0 > 0, C > 0, N ∈ N * and K ∈ N, such that, for all h ∈ (0, h 0 ), and for all ψ 0 ∈ S (R d ), and all t ∈ [0, h -M ],

p k (e itL h h ψ 0 ) Ch -N p K (ψ 0 ) .
Theorem 1.6 is related to the (pseudo-differential) analysis in [16, Section 7]. In this work, under the assumption that the derivatives of order two or higher of the symbol of the propagator should be bounded, a parametrix of the evolution operator was constructed. Closely related is also the paper [START_REF] Robert | Propagation of coherent states in quantum mechanics and applications[END_REF]Corollary 2.11], based on the analysis of coherent states, where the derivatives of order three or higher of the symbol have to be bounded. Our approach here is more directly related to the structure of the magnetic Laplacian, and is reminiscent of the analysis in [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF] of the ellipticity of certain algebras of non-commuting vector fields. 1.3. Organisation of the proofs. In Section 2, we prove Theorem 1.4 by using a regularization argument involving exponentially weighted estimates and commutator estimates. In Section 3, we apply our magnetic elliptic estimates to prove Theorem 1.6.

Magnetic elliptic estimates

This section is devoted to the proof of Theorem 1.4.

2.1. Density argument. Let us explain here why it is sufficient to prove Theorem 1.4 for ψ ∈ S (R d ).

Let us consider

f ∈ L 2 (R d ). There is a unique ψ ∈ Dom(L n h ) such that L n h ψ = f . Consider f k ∈ C ∞ 0 (R d ) converging to f in L 2 (R d ) and consider the unique ψ k ∈ Dom(L n h ) such that L n h ψ k = f k . Note that, by continuity of (L n h ) -1 , ψ k converges to ψ in L 2 (R d ).
Lemma 2.1. We let

H ∞ exp (R d ) = {ψ ∈ L 2 (R d ) : ∀α ∈ N d , ∃β > 0 : e β x ∂ α ψ ∈ L 2 (R d )} . Consider f ∈ H ∞ exp (R d ). Then, the unique solution u ∈ Dom(L h ) to L h u = f satisfies u ∈ H ∞ exp (R d ).
Proof. The proof follows from the classical Agmon estimates (see [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF][START_REF] Helffer | Semi-classical analysis for the Schrödinger operator and applications[END_REF]). Consider β > 0 such that e β x f ∈ L 2 (R d ). Let ε > 0 and Φ ε = β min( x , ε -1 ). We have the Agmon formula (see [START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]Section 4.2]):

L h u, e 2Φε u = R d |(-ih∇ -A)e Φε u| 2 dx -h 2 ∇Φ ε e Φε u 2 .
In particular,

R d |(-ih∇ -A)e Φε u| 2 dx -β 2 h 2 e Φε u 2
e Φε f e Φε u .

(2.1)

On the other hand, by Lemma 1.2, we get, for some c > 0,

(hc -β 2 h 2 ) e Φε u 2 R d |(-ih∇ -A)e Φε u| 2 dx -β 2 h 2 e Φε u 2 .
Choosing β small enough, we get, for some C(h) > 0 independent of ε, e Φε u C(h) e Φε f . Then, we take the limit ε → 0 and apply Fatou's Lemma to find

e β x u C(h) e β x f . (2.2)
Coming back to (2.1), and replacing β by β < β, we get

R d |e Φε (-ih∇ -A -ih∇Φ ε )u| 2 dx -β2 h 2 e Φε u 2 e Φε f e Φε u .
Thus,

h 2 2 R d |e Φε ∇u| 2 dx e Φε f e Φε u + β2 h 2 e Φε u 2 + 2 (ih∇Φ ε + A)e Φε u 2
e Φε f e Φε u + β2 h 2 e Φε u 2 + 4 ∇Φ ε e Φε u 2 + 4 Ae Φε u 2 .

Using that A ∈ P, (2.2), and Fatou's Lemma, we get e β x ∇u ∈ L 2 (R d ). Considering the equation L h u = f , we get, in the sense of tempered distributions,

-h 2 ∆u = f -|A| 2 u -ih(∇ • A)u -2ih(A • ∇)u .
(2.3)

Noticing that we have just controlled the terms of order at most one, we deduce that, for some β > 0,

e β x ∆u ∈ L 2 (R d ) ,
and also

∆ e β x u ∈ L 2 (R d ) .
By Fourier transform, we get

e β x u ∈ H 2 (R d ) ,
which implies that, for all α ∈ N d with |α| 2,

e β x ∂ α u ∈ L 2 (R d ) .
The higher order derivatives can be controlled by induction (taking successive derivatives of (2.3)).

Remark 2.2. By the Sobolev embeddings, we have

H ∞ exp (R d ) ⊂ S (R d ). By Lemma 2.1, we see that ψ k ∈ S (R d ). Assume that (1.3) holds for any ψ ∈ S (R d ), with ψ = ψ k -ψ , for all (k, ) ∈ N 2 , this shows that, for all σ ∈ A(2n), (L σ ψ n ) n∈N is a Cauchy sequence in L 2 (R d ). Thus, in the sense of distributions, L σ ψ ∈ L 2 (R d ).
It remains to use again (1.3) with ψ = ψ k and to take the limit k → +∞.

Preliminary lemmas.

Lemma 2.3. For all ψ ∈ Dom(L h ), we have

(-ih∇ -A)ψ 2 1 2 ψ 2 L h
where ψ 2 L h = ψ 2 + L h ψ 2 . Proof. We recall that, by definition of the domain, for all ψ ∈ Dom(L h ),

L h ψ, ψ = (-ih∇ -A)ψ 2 = d j=1 L j ψ 2 , so that d j=1 L j ψ 2 ψ L h ψ .
Lemma 2.4. There exist C > 0 and h 0 > 0 such that, for all ψ ∈ S (R d ) and all h ∈ (0, h 0 ),

B ψ + B 1 2 (-ih∇ -A)ψ Ch -1 ψ L h . (2.4)
Moreover, for all ψ ∈ Dom(L h ), we have

Bψ ∈ L 2 (R d ) and |B| 1 2 (-ih∇ -A)ψ ∈ L 2 (R d ) ,
and (2.4) holds for ψ ∈ Dom(L h ).

Proof. By integration by parts and using Assumption 1.1,

R d B |(-ih∇ -A)ψ| 2 dx = B (-ih∇ -A)ψ, (-ih∇ -A)ψ L h ψ, B ψ + C B ψ (-ih∇ -A)ψ C B ψ ψ L h .
(2.5)

Then, we have

R d |hB| 2 |ψ| 2 dx = (k, )∈{1,...,d} 2 R d |hB k, | 2 |ψ| 2 dx
and we write

(k, )∈{1,...,d} 2 R d |hB k, | 2 |ψ| 2 dx = (k, )∈{1,...,d} 2 | [L k , L ]ψ, hB k, ψ | Ch B ψ (-ih∇ -A)ψ + Ch R d B |(-ih∇ -A)ψ| 2 dx ,
where we used an integration by parts and Assumption 1.1. By (2.5), it follows

R d |B| 2 |ψ| 2 dx Ch -1 B ψ ψ L h
and then B ψ Ch -1 ψ L h . Using again (2.5), the conclusion follows.

2.3.

Case n = 1. The estimate of Theorem 1.4 is obvious when n = 0. Let us consider the case when n = 1 to explain the principle producing these estimates.

Lemma 2.5. There exist C > 0, h 0 > 0 such that, for all ψ ∈ S (R d ) and all h ∈ (0, h 0 ),

L 2 1 ψ + L 2 2 ψ + L 1 L 2 ψ + L 2 L 1 ψ Ch -1 L h ψ .
Proof. Let us consider ψ ∈ S (R d ) and let

L h ψ = f .
Consider j ∈ {1, . . . , d}. We have

L h (L j ψ) = L j f + [L h , L j ]ψ , and
(-ih∇ -A)(L j ψ) 2 = L j f, L j ψ + [L h , L j ]ψ, L j ψ , We have [L h , L j ] = d k=1 [L 2 k , L j ] = d k=1 [L k , L j ]L k + L k [L k , L j ] .
Thus,

| [L h , L j ]ψ, L j ψ | C Bψ (-ih∇ -A)ψ + C R d |B||(-ih∇ -A)ψ| 2 dx .
We have, for all ε ∈ (0, 1),

| L j f, L j ψ | ε L 2 j ψ 2 + C f 2 , and then (-ih∇ -A)(L j ψ) 2 C f 2 + C Bψ (-ih∇ -A)ψ + C R d |B||(-ih∇ -A)ψ| 2 dx . With Lemma 2.3, noting that ψ 2 L h C(1 + h -2 ) f 2 , we find (-ih∇ -A)(L j ψ) 2 Ch -3 f 2 .
2.4. Induction. Let n ∈ N * . Let us assume that, for all k ∈ {1, . . . , n}, the ellipticity property (1.3) is true. Let us consider f, ψ ∈ S (R d ) such that

L h ψ = f .
Consider σ ∈ A(2n). Since the functions are in the Schwartz class, all the following computations are justified.

We have

L h L σ ψ = L σ f + [L h , L σ ]ψ ,
and then

L h L σ ψ, L σ ψ = L σ f, L σ ψ + [L h , L σ ]ψ, L σ ψ .
(2.6) By using the Cauchy-Schwarz inequality and the induction assumption, we have for all ε ∈ (0, 1),

| L σ f, L σ ψ | Ch -3n L n h f L n h ψ Ch -3n L n+1 h ψ 2 + Ch -3n L n h ψ 2 , so that, by Lemma 1.2, | L σ f, L σ ψ | Ch -(3n+2) L n+1 h ψ 2 .

Let us now deal with

[L h , L σ ]ψ, L σ ψ . The commutator [L h , L σ ]
is the sum of various terms. Each of them is the composition of at most 2n -2 of the L j and with exactly one of the B k, . By commuting the B k, to put it on the left, and using Assumption 1.1, we get

[L h , L σ ]ψ C τ ∈A(2n-2) B L τ ψ .
By applying Lemma 2.4, and then the induction assumption, we get

[L h , L σ ]ψ Ch -1 τ ∈A(2n-2) L τ ψ L h Ch -(3n-1)/2 L n h ψ .

Thus, we deduce

| L h L σ ψ, L σ ψ | Ch -3n-1/2 L n+1 h ψ 2
. This shows that, for all γ ∈ A(2n + 1),

L γ ψ Ch -(3n/2+1) L n+1 h ψ . (2.7)
Now, we want to get the control for γ ∈ A(2n + 2). Let σ ∈ A(2n + 1). We consider again (2.6). By integration by parts, we can write

L σ f, L σ ψ = L σf, L σψ ,
with σ ∈ A(2n) and σ ∈ A(2n + 2). Thus, by Cauchy-Schwarz, and the induction assumption, for all ε > 0, there exists C > 0 such that

| L σ f, L σ ψ | ε L σψ 2 + Ch -3n/2 L n+1 h ψ 2 . (2.8)
As previously, we have

[L h , L σ ]ψ C τ ∈A(2n-1)
B L τ ψ .

We use Lemma 2.4 and (2.7) to find

[L h , L σ ]ψ Ch -1 τ ∈A(2n-1) L τ ψ L h Ch -(3n/2+2) L n+1 h ψ .
Then, with Cauchy-Schwarz and (2.7), we get

| [L h , L σ ]ψ, L σ ψ | Ch -(3n+3) L n+1 h ψ 2 . (2.9) From (2.6), (2.8), and (2.9)
, summing over σ ∈ A(2n + 1), and choosing ε small enough, we get (1.3) with n replaced by n + 1. This achieves the proof of Theorem 1.4 when ψ ∈ S (R d ) and it remains to use the discussion of Section 2.1.

Application to the evolution problem

We can now prove Theorem 1.6. Let ψ 0 ∈ S (R d ). We denote by ψ(•) the solution to the Schrödinger equation (1.1).

Notation. For κ, λ ∈ N, we define Π κ,λ the set of the operators P that are composition of operators taken among (L j ) 1 j d and and (x k ) 1 k d with κ occurences of x and λ occurences of L. We also set Π =

(κ,λ)∈N 2 Π κ,λ .
The aim of this section is to prove the following proposition. Proposition 3.1. Let (κ, λ) ∈ N 2 and consider P ∈ Π κ,λ . There exist h 0 > 0, C 0 and N ∈ N such that, for all t 0, and all ψ 0 ∈ S (R d ), for λ 2n λ + 1,

P ψ(t) Ch -N (1 + t κ ) |α| κ:|α|+ν κ+n L ν h x α ψ 0 .
This proposition implies the control of the Schwartz semi-norms and achieves the proof of Theorem 1.6. Indeed, from Sobolev embeddings, for all k ∈ N, there exists K ∈ N such that for all f ∈ S (R d ),

p k (f ) max |α|,|β| K x α ∂ β f .
Using now that ∂ j = (-ih) -1 (L j + A j ), and A ∈ P, there exists C > 0 and N ∈ N such that, for all f ∈ L 2 (R d ), if P f < +∞ for all P ∈ Π, then x α ∂ β f ∈ L 2 (R d ) for all α, β ∈ N d , and

x α ∂ β f Ch -N P ∈R P f ,
where R is a finite part of Π. Then, Proposition 3.1 implies, for N , κ and n large enough,

x α ∂ β ψ(t) Ch -N P ∈R P ψ(t) Ch -N (1 + t κ ) |γ| κ,|ν| n L ν h x γ ψ 0 .
Finally, we conclude by

L ν h x α ψ 0 C (1 + x 2 ) m L ν h x α ψ 0 ∞ Cp K (ψ 0 ) , for m, K large enough.
3.1. Case when κ = 0. For all t 0, we have, by definition,

ψ(t) = e it h L h ψ 0 .
For all ∈ N, we get

L h ψ(t) = e it h L h L h ψ 0 .
Applying Theorem 1.4, this establishes the estimate of Proposition 3.1 when κ = 0.

3.2.

Case when κ = 1. Before starting the induction procedure, let us understand first the mechanism with only one occurence of x. Let j ∈ {1, . . . , d}. We have

-ih∂ t (x j ψ) = L h (x j ψ) + [x j , L h ]ψ .
Note that [x j , L h ] = 2hL j . With the Duhamel formula, we have, for all t 0,

x j ψ(t) = e it h L h (x j ψ 0 ) + t 0 e i(t-s) h L h 2L j ψ(s)ds . (3.1)
With Lemma 2.3, we get

x j ψ(t) x j ψ 0 + 2 t 0 L j ψ(s) ds x j ψ 0 + √ 2 t 0 ψ(s) L h ds .
Since the evolution is unitary, we get, for all t 0,

x j ψ(t) x j ψ 0 + 2 t 0 L j ψ(s) ds x j ψ 0 + t √ 2 ψ 0 L h .
More generally, with (3.1), we have, for all ∈ N,

L h x j ψ(t) = e it h L h L h (x j ψ 0 ) + t 0 e i(t-s) h L h 2L h L j ψ(s)ds . so that L h x j ψ(t) L h (x j ψ 0 ) + 2 t 0 L h L j ψ(s) ds L h (x j ψ 0 ) + Ch -N t 0 L +1 h ψ(s) ds L h (x j ψ 0 ) + Cth -N L +1 h ψ 0 .
It remains to apply Theorem 1.4 and to commute the x j with the L k .

3.3. Induction. Let us now end the proof of Proposition 3.1 by induction. We set the following two induction assumptions. For κ ∈ N, let

Q κ : ∀n ∈ N, ∀α ∈ N d , |α| = κ, ∃N ∈ N s.t. L n h x α ψ(t) L n h x α ψ 0 + C |β| κ-1:|β|+ν κ+n L ν h x β ψ 0 h -N (1 + t κ ) ;
and

P κ : ∀P ∈ Π κ,λ , P ψ(t) C |α| κ:|α|+ν κ+n L ν h x α ψ 0 h -N (1+t κ ), for λ 2n λ+1 .
We have proved propositions P 0 , Q 0 and Q 1 . We assume now that for a given κ ∈ N * , for any k κ, Q k and P k hold, and we prove P κ+1 and Q κ+1 . We begin with Q κ+1 . Let α ∈ N d , with |α| = κ + 1 and n ∈ N. We have As L n h P 1 is a sum of elements in Π κ,(2n+1) , we can apply P κ , and integrating in time and using the unitariness of e i t h L h , we find, for some integer N ∈ N

-
L n h x α ψ(t) L n h x α ψ 0 + Ch -N |β| κ:|β|+ν κ+n+1 L ν h x β ψ 0 (1 + t κ+1 )
that proves Q κ+1 . It remains to prove P κ+1 . We consider so some P ∈ Π κ+1,λ , for a given λ ∈ N. Then, because of the commutation relation of the (x k ) 1 k d with the (L j ) 1 j d , that is [x k , L j ] = -ihδ k,j , there is α ∈ N d , |α| = κ + 1 such that P = P 2 + P 3 x α with P 2 a sum of elements in Π κ,λ-1 and P 3 ∈ Π 0,λ . So P ψ(t) P 2 ψ(t) + P 3 ψ(t) .

Then, applying P κ , we get, for some integer N ∈ N P 2 ψ(t) Ch

 L n x α ψ 0 +

 0 -N (1 + t κ ) |β| κ:|β|+ν κ+n L ν h x β ψ 0 (3.2)and applying Theorem 1.4 along with Q κ+1 , for some integer N ∈ N, and for λ 2n λ + 1, we haveP 3 x α ψ(t) Ch -n L n x α ψ(t) Ch -N (1 + t κ+1 )  (3.2) and (3.3), we find, for some integer N ,P ψ(t) Ch -N (1 + t κ+1 ) |β| κ+1:|β|+ν κ+n+1 L ν h x β ψ 0that proves P κ+1 and achieves the proof of Proposition 3.1.

  ih∂ t L n h x α ψ(t) = L n+1 Then, noting that [x α , L h ] = hP 1 with P 1 a sum of elements in Π κ,1 , we get from the Duhamel formula

	L n h x α ψ(t) = e

h x α ψ(t) + L n h [x α , L h ]ψ(t) . it h L h L n h x α ψ 0 + i t 0 e i t-s h L h L n h P 1 ψ(s)ds .

Acknowledgments. The authors are grateful to Karel Pravda-Starov for sharing useful references.

[16] D. Tataru. Phase space transforms and microlocal analysis. In Phase space analysis of partial differential equations. Vol. II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, pages 505-524. Scuola Norm. Sup., Pisa, 2004. E-mail address: gregory.boil@univ-rennes1.fr E-mail address: nicolas.raymond@univ-rennes1.fr E-mail address: san.vu-ngoc@univ-rennes1.fr IRMAR, Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes cedex, France


Marc Crampon
email: m.crampon@brgm.fr

Jennifer Hellal

Caroline Michel

Christophe Mouvet

Guillaume Wille

Patrick Ollivier

Crampon Marc

Hellal Jennifer

Michel Caroline

Mouvet Christophe

Wille Guillaume

Ollivier Patrick

Impact of the presence of a natural biofilm on mobility and reactivity towards tetrachloroethylene (PCE

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Biofilms were grown on sand using environmental groundwater samples as inoculum in nitrate reducing conditions. For mobility assays, suspensions of nZVI were injected into the columns and the outlet Fe concentrations monitored. Biofilm-nZVI interactions were characterized using SEM/STEM observations of sand after the NP breakthrough. Biofilms were further characterized using molecular approaches. For reactivity experiments, three columns were used to study the impact of the presence of a biofilm and of nZVI on the dechlorination of PCE in the nZVI reactive zone.

The predicted travel distances of nZVI are found to be 1.5 to 25 m for a 10 m d -1 flow. The presence of biofilm in the column decreased the total porosity of column from around 30% to around 15%. Though the recoveries of nZVI at the column outlet in the presence or absence of biofilm were similar, the analysis of the sand suggested nZVI-biofilm interactions (correlation TOC vs Fe concentrations, presence of aggregates). These interactions are confirmed by the SEM/STEM observations. Concerning reactivity experiments, it appeared that the dechlorination of PCE was slightly higher in the absence of biofilm (65% of PCE transformed into Cl -vs 60% in the presence of biofilm), but anaerobic corrosion, characterized by evaluating the formation of H2 within the porous media, was greatly diminished in the presence of biofilm, suggesting (i) interactions between nZVI and biofilm, and (ii) a better longevity of nZVI in the presence of biofilm, and thus a better reactivity towards contaminant over time.

It appears finally that reactive NP, even if they can represent a source of emerging contamination, are very useful for in situ groundwater treatment. However, the presence of biofilm in the porous media could modify the mobility and reactivity of nZVI and should be considered when designing a nanoremediation treatment.

Aknowledgements:

  We gratefully acknowledge the financial support provided to the PIVOTS project by the Région Centre -Val de Loire (ARD 2020 program and CPER 2015 -2020) and the French Ministry of Higher Education and Research (CPER 2015 -2020 and public service subsidy to BRGM).


M.-C Druguet

V Morel

J Annaloro

Computations of a Shock Layer Flow with a Vibrational-Specific Kinetics Model

The present work shows the ability of Navier-Stokes codes to handle detailed chemical kinetics models to compute the reactive and vibrational non-equilibrium gases in shock layers and the recent development done in our laboratories towards this goal. The detailed model used in the present study is a simplified version of a collisional-radiative model for N 2 , that has been implemented step-by-step in a Navier-Stokes code in order to show the effect of each group of chemical and vibrational processes. The processes taken into account are the vibration-vibration exchanges and the vibration-translation exchanges through molecular or atomic impact, leading to dissociation or not. The question of how the vibrational levels reach the Boltzmann distribution is also assessed. Comparisons of results obtained with global and detailed chemical kinetics models are performed.

Introduction

If simulating non-equilibrium gas flows in shock layers has been widely done so far with global models for the chemical reactions and the vibrational non-equilibrium relaxation, it is new to compute such flow fields with detailed chemical kinetics models. This is now possible as the CPU and memory resources of the computers allow running CFD codes with detailed models. State-specific vibrational models are efficient to model both the chemical reactions and the vibrational non-equilibrium at once, without making strong assumptions (like Boltzmann distribution) as it is the case with multi-temperature models for vibrational nonequilibrium modeling.

A major step has been done recently by IUSTI and CORIA in simulating the reactions between chemical species on their vibrational excited states (called pseudo species) and the evolution of these pseudo species in the shock layer surrounding a sphere flying into a nitrogen gas at a speed of 10.6 km/s, modeled by means of a simplified version of CoRaM-N 2 , a collisional-radiative (CR) model developed by CORIA [START_REF] Annaloro | Collisional-radiative modeling behind shock waves in nitrogen[END_REF]. This simplified model was recently implemented in the CFD code PINENS developed at IUSTI [START_REF] Sanders | Multidimensional Dissipation for Upwind Schemes: Stability and Applications to Gas Dynamics[END_REF][START_REF] Druguet | Effect of Numerics on Navier-Stokes Computations of Steady Hypersonic Double-Cone Flows[END_REF], in which the shock resolution is done with a shock-capturing method, an approximate Riemann solver in a finitevolume discretization of the Navier-Stokes equations.

State-to-State Modeling

Chemical Species and Vibrational Energy States

The vibrationally specific model subject of the present work is a simplified version of the collisional-radiative model CoRaM-N 2 . It takes into account heavy, neutral particles only, as the CFD code in which it is implemented is not yet able to take into account electrons and ions. This model considers each of the 68 vibrational states of N 2 (v = 0 67) on the ground electronic state as pointed out by Armenise et al. [START_REF] Esposito | N-N 2 state-to-state vibrational-relaxation and dissociation rates based on quasiclaisscal calculations[END_REF], as well as N ground electronic state, that represent the 69 pseudo species whose population density evolutions are to be computed with a Navier-Stokes code.

Collisional Elementary Processes

According to the CR model CoRaM-N 2 that is used as the basis to derive the present vibrational-state specific model, the vibrational excitation of molecules N 2 takes place through vibration-vibration processes under molecular impact (VV-m, 4489 processes), through vibration-translation processes under molecular impact (VT-m, 67 processes), or through vibration-translation processes under atomic impact (VT-a, 2125 processes). The data used for the rate coefficients for these processes were calculated by Esposito et al. [START_REF] Esposito | N-N 2 state-to-state vibrational-relaxation and dissociation rates based on quasiclaisscal calculations[END_REF] and Armenise et al. [START_REF] Armenise | Dissociation-recombination models in hypersonic boundary layer flows[END_REF]. Once vibrationnally excited, the molecules N 2 may dissociate during collisions. For the dissociation of N 2 through vibration-vibration processes under molecular impact (VV-m-D, 67 processes), through vibration-translation processes under molecular impact (VT-m-D, 1 process), or through vibration-translation process under atomic impact (VT-a-D, 68 processes), the data of Esposito et al. [START_REF] Esposito | N-N 2 state-to-state vibrational-relaxation and dissociation rates based on quasiclaisscal calculations[END_REF] and of Armenise et al. [START_REF] Armenise | Dissociation-recombination models in hypersonic boundary layer flows[END_REF] are also used. The total number of processes is 6817 forward reactions. The backward collisional elementary processes are calculated from the forward rate coefficient and the corresponding equilibrium constant using the detailed balance principle. Thus, the set represents a total of about 13,600 forward and backward elementary processes between 69 pseudo species.

Results and Analysis

Results obtained with Navier-Stokes simulations performed with the PINENS code are the evolution of N 2 and N concentrations throughout the axi-symmetric shock layer past a 1meter radius sphere flying at 10.6 km/s, for the 68 vibrational states of N 2 and the fundamental electronic state of N.

VV-m : 4489 Processes

The first step in analyzing the effect of each group of processes on the flow field (temperature, species densities, ...) consists in considering the processes for the vibrationvibration exchanges by molecular impact (VV-m). Surprisingly, introducing the reactions corresponding to these processes does not produce any visible effect on the temperature nor on the species profiles in the shock layer. The shock position, the temperature, and the total density evolutions throughout the shock layer are the same when VV-m processes are taken into account as when no chemical processes are taken into account. Therefore, these processes are not taken into account in the following computations.

VT-m : 67 Processes

The second step in analyzing the effect of a group of processes on the flow field and on the vibrational-level population densities consists in taking into account the vibrationtranslation exchanges by molecular impact (VT-m). Interestingly, the temperature profiles along the stagnation line (Fig. 1, left), are very similar to the results that are obtained when no chemical reactions are taken into account in the gas flow: the temperature is maximum in the entire shock layer, with a maximum reached near the stagnation area of the flow, right before the boundary layer. This means that the VT-m processes do not take much thermal energy from the gas. The evolutions of the mass fractions of each vibrational level of N 2 (from v = 0 to 67) along the stagnation line throughout the shock layer, including the boundary layer, are also shown in Fig. 1, left. The fundamental vibrational level of N 2 (v=0) is depopulated across the shock wave to the profit of the excited states v=1 to 67. As this is expected, the lowest levels are the most populated levels. The behavior of the first 10 vibrational levels right behind the shock wave is interesting: their population increases in the shock front, then decreases only 1 mm behind the shock front, to the profit of the excitation of higher vibrational states. The addition of the vibration-translation process by molecular impact leading to the dissociation (VT-m-D) leads to reduce the shock layer width from approximately 13 cm down to 10.5 cm (Fig. 1, right, compared to Fig. 1, left). The introduction of this single process into the computations also changes the temperature profile across the shock layer: the temperature is no longer maximum at 53,000 K throughout the shock layer but peaks at 46,000 K right behind the shock front then decreases. This new temperature profile is similar to the temperature profiles usually observed when chemical reactions are taken into account through global models, and is typical of shock layer where dissociation processes take place.

Regarding the species profiles within the shock layer, the atomic nitrogen is now the major species present in the shock layer. All the vibrational levels of N 2 have been depopulated, especially the highest levels -because of the dissociation of N 2 from the highest vibrational levels -to the profit of the atomic species. This means that the VT-exchange by molecular impact leading to dissociation (process VT-m-D) is a very efficient process among the many reactions of the vibrationally specific model implemented in the CFD code. This reaction of dissociation of N 2 (v) also absorbs a large amount of thermal energy : the temperature that was approximately equal to 53000 K in the shock layer 2 mm from the stagnation point (see Fig. 1, left) is now equal to 22000 K at the same location (Fig. 1, right). This shows that the VV processes (leading to dissociation or not) are not efficient in the present test-case.

VT-m + VT-m-D + VV-m-D + VT-a : 2260 Processes

The introduction of the 2126 vibration-translation processes by atomic impact (VT-a) has a large effect on the flow field (Fig. 2, left, compared to Fig. 1, right): the reactions VT-a, like the process VT-m-D, lead to decreasing the shock layer width (from 10.5 cm to 8.5 cm approximately) and the temperature peak (from 46,000 K down to 38,000 K). Regarding the population densities in the shock layer, it is clear that the highest vibrational levels of N 2 are even more depopulated to the profit of atomic nitrogen production than they were before. The curve of N population density (Fig. 2, left) shows that there is more N production in the shock layer when processes VT-a are considered, although there is no additional reaction of dissociation (only the VT-m-D reaction). Adding the 68 reactions corresponding to the vibration-translation processes by atomic impact leading to dissociation (VT-a-D) results in a complete dissociation of N 2 (v) (see Fig. 2, right). The flow field is strongly affected by the introduction of these additional reactions: again, the shock standoff distance is decreased (from 8.5 cm to 6.5 cm) and the temperature peak is decreased from 38,000 K to 18,000 K. Regarding the species densities, if the curves show that the various vibrational levels are all populated through the shock front, they are depopulated right behind the shock wave. This means that the additional reactions corresponding to the VT processes by atomic impact leading to dissociation (VT-a-D) are very efficient by transferring translational energy of atoms to vibrational energy of N 2 (v) in order to dissociate the molecules, whatever their vibrational level. Interestingly, the temperature profile along the stagnation line slightly increases until the boundary layer where the temperature drops to reach to wall temperature. Comparisons between results obtained with detailed models and results obtained with global models show how the global models perform compared to the detailed models in the prediction of the chemical species and vibrational states evolution past a shock wave (see Fig. 4). A reasonably good agreement is observed.

Conclusion

This work shows the ability of Navier-Stokes codes to handle detailed chemical models to compute the reactive and vibrational non-equilibrium gases in shock layers. The detailed model used in the present study has been implemented in the Navier-Stokes code step-by-step in order to show the effect of each group of chemical and vibrational processes in a given order. It was shown that the vibration-vibration exchanges do not have any visible effect on the present flow field, while the vibration-translation exchanges through molecular or atomic impact, leading to dissociation or not, are very efficient. Another main result is that the Boltzmann distribution of the vibrational levels is not reached until a certain distance behind the shock front. Finally comparisons between results obtained with global and detailed chemical kinetics models show good agreement. Future work is oriented towards the implementation of detailed models for Earth and Mars atmospheres into the PINENS code in order to simulate the non-equilibrium and reactive gas throughout an axi-symmetric shock layer with detailed models and compare the results with those obtained with global models classically used in such simulations.

Fig. 1

 1 Fig. 1 Temperature and mass fractions of N 2 (v=0,67) and N along the stagnation line. Effect of the VT excitation by molecular impact. Left figure : processes VT-m are included in the computations ; Right figure : processes VT-m + VT-m-D are included

3. 4

 4 VT-m + VT-m-D + VV-m-D : 135 Processes When the processes VV-m-D are added to the processes VT-m + VT-m-D in the simulations, the results are the same as when only the processes VT-m + VT-m-D are taken into account.

Fig. 2

 2 Fig. 2 Temperature and mass fractions of N 2 (v=0,67) and N along the stagnation line. Effect of the VT excitation by atomic impact. Left figure : processes VT-a are included in the computations in addition to the processes presented before ; Right figure : processes VT-a + VT-a-D are included in addition to the other processes 3.6 VT-m + VT-m-D + VV-m-D + VT-a + VT-a-D : 2328 Processes

3. 7

 7 Is Boltzmann distribution reached in the shock layer ? In order to show if the distribution of the vibrational level populations follows a Boltzmann distribution, Boltzmann plots of the N 2 vibrational levels at different positions behind the shock front (marked 01 to 12) are presented in Figs 3, left and right. The Boltzmann distribution is reached when the curves representing the level populations (or mass fractions) -expressed in a logarithmic scale -versus the energies of the levels -expressed in a linear scale -are linear. The Boltzmann diagrams plotted in Figs. 3, left, show that the Boltzmann distribution is not reached for the case (VT-m) + (VT-m-D) + (VV-m-D). On the other hand, the Boltzmann distribution is reached after a distance of 5 mm behind the shock front in the case (VT-m) (not shown), 8.5 mm behind the shock front for the case (VT-m) + (VT-m-D) + (VV-m-D) (not shown), and 15 mm behind the shock front for the case (VT-m) + (VT-m-D) + (VV-m-D) + (VT-a) + (VT-a-D) (Fig. 3, right).

Fig. 3

 3 Fig. 3 Boltzmann plots across the shock wave. Left figure : processes (VT-m) + (VT-m-D) + (VV-m-D) are included. Right figure : processes (VT-m) + (VT-m-D) + (VV-m-D) + (VT-a) + (VT-a-D) are included

Fig. 4 N

 4 Fig. 4 N and N 2 total mass fractions, and gas temperature along the stagnation line of the shock layer flow. Comparison between detailed (vibrational-specific, 2328 processes)) and global (Park, Dunn-Kang, Gupta, CORIA) chemical kinetics models. Right figure is a closeup view of the left figure


Marcello De Michele
email: m.demichele@brgm.fr

Daniel Raucoules

Julia De Sigoyer

Manuel Pubellier

Nicolas Chamot-Rooke

2 Sichuan earthquake (China) derived from Synthetic Aperture Radar: evidence for rupture on a blind thrust

Keywords: Satellite geodesy, Seismicity and tectonics, Intra-plate processes

published or not. The documents may come

Three-dimensional surface displacement of the 2008 May

I N T RO D U C T I O N

The Longmen Shan (LMS) thrust system bounds the eastern margin of the Tibetan plateau and is considered to be a Triassic transpressive zone that was reactivated during the course of the India-Asia collision [START_REF] Tapponnier | Active faulting and tectonics in China[END_REF][START_REF] Chen | Emplacement of the Longmen Shan Thrust-Nappe belt along the eastern margin of the Tibetan plateau[END_REF][START_REF] Arne | Differential exhumation in response to episodic thrusting along the eastern margin of the Tibetan Plateau[END_REF][START_REF] Densmore | Active tectonics of the Beichuan and Pengguan faults at the eastern margin of the Tibetan Plateau[END_REF][START_REF] Godard | Late Cenozoic evolution of the central Longmen Shan, eastern Tibet: Insight from (U-Th)/He thermochronometry[END_REF]. The LMS front is characterised by a steep topographic gradient displaying a relief of over 5 km within a distance of less than 50 km. Tectonically, the LMS front is a thrust system dominated by three major faults that strike SW-NE (for clarity, we will use nomenclature proposed by [START_REF] Chen | Emplacement of the Longmen Shan Thrust-Nappe belt along the eastern margin of the Tibetan plateau[END_REF]; from the inner to the outer range, respectively, these are the Wenchuan Fault (WF), the Beichuan Fault (BF) and the Guanxian-Anxian Fault (GF; Fig. 1). The range front displays three principal tectonic units (e.g. [START_REF] Burchfiel | A geological and geophysical context for the Wenchuan earthquake of 12 May 2008, Sichuan, People's Republic of China[END_REF][START_REF] Robert | Structural and thermal characters of the Longmen Shan (Sichuan, China)[END_REF]): a crystalline massif composed of Proterozoic basement and Neoproterozoic cover, a narrow foothills area composed of deformed Paleozoic and Mesozoic sediments and the weakly deformed Sichuan Basin sediments. Recent studies based on thermochronometry across the belt and balanced cross-sections indicate that crustal shortening along a thrust system is respon-sible for the high topography of the region [START_REF] Godard | Late Cenozoic evolution of the central Longmen Shan, eastern Tibet: Insight from (U-Th)/He thermochronometry[END_REF]Hubbard & Shaw 2009). This process has mainly affected the BF since 10-8 Ma, without substantial activation of the WF.

The Sichuan earthquake ruptured two parallel thrust faults along the LMS front, generating one of the largest seismic ruptures ever documented in a continental thrust event [START_REF] Dong | Surface rupture and co-seismic displacement produced by the Ms 8.0 Wenchuan earthquake of May 12th, 2008, Sichuan, China: eastward growth of the Qinghai-Tibet Plateau[END_REF]. Field campaigns integrated with seismic profiles indicate that at least two imbricate structures (BF, GF) ruptured during the earthquake (e.g. [START_REF] Liu-Zeng | Co-seismic ruptures of the 12 May 2008, Ms 8.0 Wenchuan earthquake, Sichuan: east-west crustal shortening on oblique, C 2010 The Authors[END_REF][START_REF] Xu | Coseismic reverse-and oblique-slip surface faulting generated by the 2008 Mw 7.9 Wenchuan earthquake[END_REF]. As corroborated by further data interpretation based on synthetic aperture radar (SAR) amplitude images [START_REF] De Michele | The Mw 7.9, 12 May 2008 Sichuan earthquake rupture measured by sub-pixel correlation of ALOS PALSAR amplitude images[END_REF]Hashimoto et al. 2009;[START_REF] Kobayashi | Locations and types of ruptures involved in the 2008 Sichuan earthquake inferred from SAR image matching[END_REF], BF ruptured over an ∼230 km long segment while GF ruptured over a ∼70 km long segment. Field observations indicate that the slip mechanism was a variable combination of thrust and right lateral displacement. However, the 3-D rupture geometry and the associated surface motion are difficult to measure in the field due to both the inaccessibility of most of the damaged areas and the rapid erosion of the fault scarps. A number of teams worldwide, for instance [START_REF] Hashimoto | Coseismic Deformation from the 2008 Wenchuan, China, Earthquake Derived from ALOS/PALSAR Images[END_REF], [START_REF] Shen | Slip maxima at fault junctions and rupturing of barriers during the 2008 Wenchuan earthquake[END_REF], [START_REF] Hao | Coseismic surface-ruptures and crustal deformations of the 2008 Wenchuan earthquake Mw7.9, China[END_REF], [START_REF] Kobayashi | Locations and types of ruptures involved in the 2008 Sichuan earthquake inferred from SAR image matching[END_REF]Feng et al. (2010), have worked with SAR data to partially overcome the difficulties associated with field data collection. As proposed by [START_REF] Shen | Slip maxima at fault junctions and rupturing of barriers during the 2008 Wenchuan earthquake[END_REF]), Feng et al. (2010) and [START_REF] Hao | Coseismic surface-ruptures and crustal deformations of the 2008 Wenchuan earthquake Mw7.9, China[END_REF], conventional L-band Differential SAR Interferometry (DInSAR) partly overcomes these difficulties by being able to retrieve the surface displacement field from space, in the line-of-sight (LOS) direction of the satellite. [START_REF] Shen | Slip maxima at fault junctions and rupturing of barriers during the 2008 Wenchuan earthquake[END_REF] used DInSAR and GPS data to retrieve the longer wavelength surface displacement field and used it to constrain a model of coseismic slip distribution on the fault plane at depth. The interferometric phase signals decorrelate quickly close to the seismic rupture, mainly due to the high displacement gradient exceeding the theoretical limit of 1/4 of the wavelength per pixel and the GPS network is too sparse to capture short wavelength surface displacement. Therefore, coseismic surface displacement in the near field is not well resolved. Moreover, ionospheric-related phase delay on the L-band interferograms hampers a unique interpretation of the surface displacement field (Raucoules & de Michele 2009). Feng et al. (2010) overcame this problem by carefully choosing SAR couples that are less affected by ionospheric-related phase delay (acquired between 68 and 499 days after the quake). This approach does improve the data quality, but the problem of near-field interferometric phase signal decorrelation still holds.

M E T H O D O L O G Y

A SAR system sends radar pulses to the ground and measures both the amplitude and the phase of the backscattered signal. Here we use the radar amplitude data to construct ascending and descending correlograms following the subpixel correlation methodology (hereafter named SPC for short) described in [START_REF] Michel | Measuring ground displacements from SAR amplitude images: application to the Landers earthquake[END_REF], which is today a widely used technique for retrieving coseismic surface displacements (e.g. [START_REF] Peltzer | The Mw7.1, Hector Mine, California earthquake: surface rupture, surface displacement field, and fault slip solution from ERS SAR data[END_REF][START_REF] Fialko | Three-dimensional deformation caused by the Bam, Iran, earthquake and the origin of shallow slip deficit[END_REF]. SPC has already been applied to L-band SAR data over the Sichuan earthquake [START_REF] De Michele | The Mw 7.9, 12 May 2008 Sichuan earthquake rupture measured by sub-pixel correlation of ALOS PALSAR amplitude images[END_REF]Hashimoto et al. 2009;[START_REF] Kobayashi | Locations and types of ruptures involved in the 2008 Sichuan earthquake inferred from SAR image matching[END_REF]), but only with ascending data, therefore yielding a non-unique interpretation of the sense of displacement if one does not have a priori information on dip angles and slip mechanisms. Here, we use ascending and descending data. To retrieve the 3-D surface displacement in the near field of the earthquake rupture, we use both L-band SAR data from the Japanese Space Agency Phase Array L-band SAR sensor (PALSAR) onboard the ALOS satellite and C-band SAR data from the European Space Agency Advanced-SAR (ASAR) sensor onboard the ENVISAT satellite (Tables 1 and2). We have used the following approach; we run the subpixel correlation analysis on the Single Look Complex radar images (SLC) using the correlator implemented in the GAMMA processor (Wegmueller & Werner 1997). First, we compute the correlograms by using a 64 × 128 correlation window size. This provides us with a measurement of the surface displacement that is homogeneous over a cell of ∼300 × 400 m for PALSAR and ∼500 × 500 m for ASAR. We arrange the correlator sampling frequency to provide an independent measurement every ∼64 pixels in the range direction and 128 pixels in the azimuth direction. Secondly, we use the satellite ancillary information to put the correlograms in the original SLC geometry, as specified by the GAMMA correlation procedure (called 'offset tracking'). Thirdly, the correlograms are projected in the DEM geometry (orthorectification).

SPC is less precise than conventional DInSAR interferometry but presents several advantages. C-and L-band correlograms are negligibly sensitive to tropospheric phase delay and are not limited by high surface displacement gradients. Further, we carefully estimated and reduced the ionospheric contribution to the subpixel offset in L-band correlograms using the method described in Raucoules & de Michele (2010).

Each ascending and descending correlogram provides two components of the coseismic offset field, one in the azimuth direction and one in the LOS direction of the satellite, with a precision of up 1/20th of the radar image pixel size (Fig. S1). Thus, we have four non-colinear independent radar observations that we invert to retrieve the east, north and up components of the near-field coseismic surface displacement.

We proceed as follows; let be the unitary vectors associated with the LOS (u r(k) ) and (u az(k) ) azimuth directions for each of the two acquisitions (ASAR descending and PALSAR ascending). We use the following rule: the LOS values are positive when we move away from the sensor, azimuth value are positive when acquisition time increases. u r and u az can be easily defined using the incidence angles and the angles between the orbits and the local meridian. We therefore obtain a four-components data vector:

d = ⎛ ⎜ ⎜ ⎜ ⎝ az (1) az (2) los (1) los (2) ⎞ ⎟ ⎟ ⎟ ⎠
az represents the two azimuth offsets and los represents the LOS offsets. Subscripts (1) and (2) refer to the two acquisitions.

From the unitary vectors, we can define a matrix A:

A = ⎛ ⎜ ⎜ ⎜ ⎝ u t az (1) u t az (2) u t r (1) u t r (2) ⎞ ⎟ ⎟ ⎟ ⎠ .
The relation between the obtained four-components data and the three-components displacement vector is provided by:

d = Av (1) v = ⎛ ⎝ dx dy dz ⎞ ⎠
is the displacement vector in north, east, up coordinates. We therefore use a weighted least-squares formulation (e.g. Björk 1996) to derive the displacement vector from the data.

V = (A t A) A t d, (2)
is the weight matrix, the weights are the signal-to-noise ratio (SNR) values that we estimate during the correlation procedure. For simplicity, we use a diagonal matrix which elements are the SNR correlation values associated to the offset measurement, which allows taking into account the inhomogeneous quality of the measurement.

The results are shown in Fig. 1. We validated our results against field measurements from Xu et al. ( 2009), finding differences with mean ∼0.1 m and standard deviation 0.4 m (Fig. 2), thus showing that SAR provides a reliable estimate of the displacement field from space. The 3-D surface displacement measurements provide us with a unique data set for this earthquake. In particular, we have overcome the need for a direct reliable measurement of the thrust displacement component at the surface-a parameter that is generally difficult to measure in the field.

R E S U LT S

We observed that coseismic displacement along the GF occurred mainly as a thrust movement, whereas the BF underwent more complex motion involving thrusting and dextral displacement, in agreement with previous field observations [START_REF] Dong | Surface rupture and co-seismic displacement produced by the Ms 8.0 Wenchuan earthquake of May 12th, 2008, Sichuan, China: eastward growth of the Qinghai-Tibet Plateau[END_REF][START_REF] Liu-Zeng | Co-seismic ruptures of the 12 May 2008, Ms 8.0 Wenchuan earthquake, Sichuan: east-west crustal shortening on oblique, C 2010 The Authors[END_REF][START_REF] Xu | Coseismic reverse-and oblique-slip surface faulting generated by the 2008 Mw 7.9 Wenchuan earthquake[END_REF]. Three major trends were clearly identified in the displacement field, highlighting a complex strain partitioning. First, in the epicentral area (S-1), surface displacement on BF was accommodated mostly by an oblique thrust mechanism. Average motion on this segment reached ∼3.5 m vertical, ∼3.5 m horizontal-parallel to the fault strike with the dextral slip mechanism and ∼4.8 m horizontal-perpendicular to the fault strike (Fig. 2). No rupture was observed in the field along the SW segment of GF. Nevertheless, transect profiles aa (Fig. 3) reveal a small but significant thrust movement close to the southwestern prolongation of GF. Secondly, in the area 30 km northeast of the epicentre (S-2), strain is fully partitioned between BF and GF. Here, surface displacement on BF shows an almost purely dextral strike-slip motion reaching up to ∼3 m. On GF, surface displacement occurs with an almost pure thrust motion, reaching ∼2 m of average vertical displacement and ∼2.3 m of average horizontal displacement. This drastic lateral change in the style of motion reflects differential shortening of the thrust sheets. The differential shortening is associated with the reactivation of the Xiaoyudong tear fault (XF, Fig. 1), a ∼6 km long left-lateral strike-slip boundary between the BF and GF segments. This observation is corroborated by field measurements (Liu-Zeng [START_REF] Wang | Structural control of rupturing of the Mw7.9 2008 Wenchuan Earthquake, China[END_REF]. The surface rupture occurring on the Gaochuan segment of the BF, between S-2 and S-3, changes strike to ∼N80E for about 11 km (Fig. 1). In our data, we observe that this segment ruptured with oblique dextral motion that is partitioned into strike-slip (∼3 m), vertical (∼1 m) and perpendicular to the fault (∼0.5 m) (Fig. 2).

I N T E R P R E TAT I O N A N D M O D E L L I N G

First, we ask why strain in the LMS system is partitioned between BF and GF in its south-western section and not in its north-eastern section.

While no surface rupture was directly observed in the displacement field along the frontal GF at locations 103 • 7 and 104 • 6 [START_REF] Dong | Surface rupture and co-seismic displacement produced by the Ms 8.0 Wenchuan earthquake of May 12th, 2008, Sichuan, China: eastward growth of the Qinghai-Tibet Plateau[END_REF][START_REF] Liu-Zeng | Co-seismic ruptures of the 12 May 2008, Ms 8.0 Wenchuan earthquake, Sichuan: east-west crustal shortening on oblique, C 2010 The Authors[END_REF][START_REF] Xu | Coseismic reverse-and oblique-slip surface faulting generated by the 2008 Mw 7.9 Wenchuan earthquake[END_REF], our transects in Fig. 3 suggest that a small but significant amount of slip at these locations might have occurred at depth on a structure that was activated by the earthquake. As post-seismic displacement can be substantially ruled-out over the period of observation [START_REF] Shen | Slip maxima at fault junctions and rupturing of barriers during the 2008 Wenchuan earthquake[END_REF], the pattern of displacement in Fig. 3 would be expected for a coseismic rupture on a blind thrust, possibly at the northern continuation of the GF (Fig. 1) and locally on the SE marginal fault [START_REF] Chen | Emplacement of the Longmen Shan Thrust-Nappe belt along the eastern margin of the Tibetan plateau[END_REF]. We verify this hypothesis by comparing the observed data with an elastic dislocation model [START_REF] Okada | Surface deformation due to shear and tensile faults in a half space[END_REF] at the locations of transects aa , bb , cc . As our SAR results represent near-field surface displacement measurements (±25 km from the faults), we can resolve the upper crust fault slips and geometry (down to ∼12 km depth). The aim of the modelling is to answer the question of how the newly acquired data modify our previous knowledge. In other words, we wish to show that slip at depth on the BF and GF only cannot fully explain the set of observations. We therefore start the modelling procedure with a forward approach. We used the physical theory of [START_REF] Okada | Surface deformation due to shear and tensile faults in a half space[END_REF], implemented in the code described by [START_REF] Briole | Inversion of geodetic data and seismicity associated with the Friuli earthquake sequence (1976-1977) Italy[END_REF], for predicting the results of observations. We started with a set of a priori information on the geometry and slip of the BF and GF. BF and GF geometry was inferred from the structural models of [START_REF] Jia | Structural model of 2008 Mw7.9 Wenchuan earthquake in the rejuvenated Longmen Shan thrust belt, China[END_REF] and Hubbard & Shaw (2009). These models are based on seismic reflection profiles. We also used the mean slip at depth on the BF and GF, as derived in previous works by [START_REF] Hashimoto | Coseismic Deformation from the 2008 Wenchuan, China, Earthquake Derived from ALOS/PALSAR Images[END_REF], [START_REF] Shen | Slip maxima at fault junctions and rupturing of barriers during the 2008 Wenchuan earthquake[END_REF], [START_REF] Hao | Coseismic surface-ruptures and crustal deformations of the 2008 Wenchuan earthquake Mw7.9, China[END_REF]Feng et al. (2010) who inverted InSAR, GPS and field measurements. In the second step, we used a trial and error approach to explore the model parameters space and slightly adjust the model parameters to ameliorate the fit to the measurements (Table 3). At the end of this step, we noted that the fit to the data in terms of the basic shape of the profile was unsatisfactory. Therefore, we progressed by falsification (e.g. [START_REF] Tarantola | Popper, Bayes and the inverse problem[END_REF]). We considered the BF and GF geometry to be fixed and introduced a new fault in the model, representing a rupture on a blind thrust. We tested a number of blind fault geometries. For each model, we compared the predictions to actual observations and rejected models with the highest root-mean-square error (RMSE). Generally, we looked for a model configuration whose prediction jointly best-fit both vertical and horizontal surface displacement observations at transect locations (Table 3, Fig. 3). The blind thrust model parameter space was sampled following [START_REF] Ellis | Coseismic fault slip of the 2008 Mw 7.9 Wenchuan earthquake estimated from InSAR and GPS measurements[END_REF] who used elastic dislocation theory to study the evolution of first-order topography in the presence of blind thrusts and their relation to different parameters such as fault dip, fault depth and amount of slip. Once we were satisfied with the forward model configuration, we considered it fixed and inverted for slip only. The inversion is based on the Tarantola-Valette method [START_REF] Tarantola | Inverse problems = quest for information[END_REF] implemented in the code inverse 5 described by [START_REF] Briole | Inversion of geodetic data and seismicity associated with the Friuli earthquake sequence (1976-1977) Italy[END_REF]. The inversion was run against two end-member models, with and without blind thrust. We estimated the discrepancy between the model results and the observations by defining an error percentage calculated using the formula described by [START_REF] Battaglia | Location of tremor sources and estimation of lava output using tremor source amplitude on the Piton de la Fournaise volcano: 1. location of tremor sources[END_REF] and also by calculating the RMSE. By adding a buried fault in the dislocation model, the percentage error decreased from 53.8 to 41.7, corresponding to an improvement in the RMSE from 0.43 m to 0.29 m.

D I S C U S S I O N A N D C O N C L U S I O N S

The most striking result is that without adding an additional rupture on a blind thrust on the range front, the observed surface displacement cannot be satisfactorily reproduced (Fig. 3). This activated blind structure in the range front in the north-eastern sector possibly coincides with previously mapped Quaternary structures [START_REF] Chen | Emplacement of the Longmen Shan Thrust-Nappe belt along the eastern margin of the Tibetan plateau[END_REF][START_REF] Jia | Structural model of 2008 Mw7.9 Wenchuan earthquake in the rejuvenated Longmen Shan thrust belt, China[END_REF][START_REF] Robert | Structural and thermal characters of the Longmen Shan (Sichuan, China)[END_REF] and may be a north-eastern continuation of the GF. Modelled slip on the blind thrust reaches 2.7 m, which is close to the amount of slip found on the GF. Hence, we suggest that where surface displacement evolves from oblique thrust motion to almost pure strike-slip on the BF, crustal shortening is accommodated both on the GF in the south-western sector and on the blind structure at depth in the north-eastern sector. This structural configuration might give rise to the two push-up like structures found in both the epicentral area as well as in the area northeast of Beichuan Town, where vertical displacement reaches a maximum (the two red patches in Fig. 1). In these two areas, strain is not fully partitioned between the BF and the GF. Therefore, transpressional shortening might be accompanied by vertical thickening. Partitioning of transpressional strain can occur when stress is applied oblique to pre-existing structures (e.g. [START_REF] Jones | Strain partitioning in transpression zones[END_REF]. On a larger picture, we observe that the displacement field recorded during the Sichuan earthquake is compatible with oblique stress applied to inherited ∼N50 • oriented thrust faults within an overall ENE-WSW compression, as also suggested by the maximum horizontal stress derived from nearby drilling observations in the Sichuan basin [START_REF] Heidbach | The World Stress Map -Release[END_REF] and the work of [START_REF] Liu-Zeng | Co-seismic ruptures of the 12 May 2008, Ms 8.0 Wenchuan earthquake, Sichuan: east-west crustal shortening on oblique, C 2010 The Authors[END_REF]. Although there are certainly blind structures in the Sichuan basin, very little is known about them (e.g. Jia et al. 2009). A good statistical fit does not guarantee a good, or even a reasonable, description of reality. Our model rather represents one solution that cannot be disregarded. This solution strengthens the hypothesis of a blind rupture at depth on the range front and provides a new argument in favour of an out-of-sequence thrusting event [START_REF] Morley | Out of Sequence Thrusts[END_REF], as first proposed by [START_REF] Liu-Zeng | Co-seismic ruptures of the 12 May 2008, Ms 8.0 Wenchuan earthquake, Sichuan: east-west crustal shortening on oblique, C 2010 The Authors[END_REF], for the LMS. This result calls for further investigation of the seismic activity on hinterland to marginal thrusts, which have a twofold importance: for understanding the long-term evolution of the mountain belt and for reassessing seismic hazard in the area.

Figure 1 .

 1 Figure 1. Surface displacement distribution of the Sichuan Earthquake, obtained by SPC of radar amplitude images. Vectors: horizontal displacement components. Colours: vertical displacement components. Surface rupture trace (in plain red) after de Michele et al. (2009) and Xu et al. (2010). BF, Beichuan Fault; GF, Guanxian-Anxian Fault; WF, Wenchuan Fault; XF, Xiaoyudong Tear Fault; GS, Gaochuan Segment. The background topography (semi-transparent) is the Digital Elevation Model from the Shuttle Radar Topography Mission. Transect locations are shown; aa , bb , cc represent the stacked profiles envelope.

Figure 2 .

 2 Figure 2. Along-fault-strike evolution of the offset on the BF and GF ruptures as measured by SPC, assuming a ∼N50E fault strike except for GS (the grey segment on top of the figure) whose strike is ∼N80E. Data are sampled every ∼1 km. (a) vertical offset; (b) offset parallel to the fault strike; (c) offset perpendicular to the fault strike. The latter is often extremely difficult to measure in the field. Orange dots: ground measurements from Xu et al. (2009) that we averaged over ∼1 km to perform a validation of our radar results. et al. 2009; Xu et al. 2009). Thirdly, in the NE segments on the BF (S-3, S-3a), surface displacement gradually evolves from oblique thrust motion along S-3, with an average displacement of ∼4 m in the vertical and ∼3 m in the horizontal, to almost pure dextral strike-slip motion displaying up to ∼5 m displacement along S-3a, suggesting an evolution from oblique thrust to almost purely dextral strike-slip, consistent with seismological data[START_REF] Wang | Structural control of rupturing of the Mw7.9 2008 Wenchuan Earthquake, China[END_REF]. The surface rupture occurring on the Gaochuan segment of the BF, between S-2 and S-3, changes strike to ∼N80E for about 11 km (Fig.1). In our data, we observe that this segment ruptured with oblique dextral motion that is partitioned into strike-slip (∼3 m), vertical (∼1 m) and perpendicular to the fault (∼0.5 m) (Fig.2).

Figure 3 .

 3 Figure 3. Vertical and perpendicular components across-the-strike of the coseismic surface displacement measured at transect locations aa , bb , cc , versus model results from elastic dislocation modelling. Black dots: radar measurements (with uncertainties); red profile: predicted surface displacement without rupture on a blind thrust; blue profile: predicted surface displacement with rupture on a blind thrust. For each transect, we show the model geometry (refer to the text for the model configuration).

  provided by JAXA and ESA through the French CIEST agreement (Cellule d'Intervention et d'Expertise Scientifique et Technique) and the International Charter on Space and Major Disasters. We are thankful to Cecile Lasserre, Yann Klinger and Jerome van der Woerd for constructive comments, and to Pierre Briole for his advice on the elastic dislocation model.

Table 1 .

 1 PALSAR data acquisition characteristics.

	PALSAR		Mode FBS	Pixel size Range/
	Track	Date	(LOS 34 • )	Azimuth (m)
	471	2008/02/292008/05/31	Ascending	4.68/3.15
	472	2008/01/312008/06/17	Ascending	4.68/3.15
	473	2008/02/172008/05/19	Ascending	4.68/3.15
	474	2008/04/052008/06/05	Ascending	4.68/3.15
	475	2007/12/212008/06/22	Ascending	4.68/3.15
	476	2008/04/082008/05/24	Ascending	4.68/3.15

Table 2 .

 2 ASAR data acquisition characteristics.

	ASAR		Mode IS2	Pixel size Range/
	Track	Date	(LOS 23 • )	Azimuth (m)
	18	2006/11/152008/05/28	Descending	7.9/3.9
	247	2007/11/162008/08/22	Descending	7.9/3.9
	290	2008/03/032008/06/16	Descending	7.9/3.9

Table 3 .

 3 Elastic dislocation model parameters.

			Length; width	Depth of the fault
		Dip	(km)	top edge (km) a	E-N (km) b	Slip (mm)
	Segments (aa )
	BF (1)	32 •	50; 10	0.1	369.437; 3441.742	6120 r
	BF (2)	20 •	50; 3	5.29	361.437; 3449.742	2300 r
	Bt	70 •	40; 4	2.8	373.837; 3437.342	2430 r
	Segments (bb )
	BF (1)	60 •	64; 5	0.2	403.275; 3480.488	820 r; 3120 ds
	BF (2)	40 •	64; 5	4.33	401.507; 3482.255	800 r; 3000 ds
	GF (1)	40 •	40; 5	0.1	411.689; 3470.074	2900 r
	GF (2)	20 •	40; 5	3.21	410.981; 3472.781	2500 r
	Bt	70 •	30; 4	2	414.475; 3469.288	2700 r
	Segments (cc )
	BF (1)	45 •	70; 19	0.1	461.566; 3543.533	2100 r; 2100 ds
	Bt	30 •	50; 6	5	464.204; 3536.441	2300 r
	Note: Bt, Blind thrust; r, reverse motion; ds, dextral strike-slip motion.

a Vertical depth of the segment top edge. b East and North coordinates are in UTM zone 48 and refer to the middle point of the top edge of the fault segment.

C 2010 The Authors, GJI, 183, 1097-1103 Geophysical Journal International C 2010 RAS

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