H Fakih

S Jacques

M.-P Berthet

F Bosselet

O Dezellus

J.-C Viala

The growth of Ti 3 SiC 2 coatings onto SiC by reactive chemical vapor deposition using H 2 and TiCl 4

Keywords: reactive CVD, Ti3SiC2, multilayer coating, solid-gas reaction, solid state diffusion

SiC/Ti3SiC2)n multilayered coatings composed of n layers alternately of continuous SiC and Ti3SiC2 sub-layers were prepared on carbon substrates. The method consists in depositing SiC by classical chemical vapor deposition (CVD) and then producing a new Ti-containing sublayer from a reaction between SiC and a H2/TiCl4 gaseous mixture by reactive CVD (RCVD), and repeating this sequence n times. In accordance with thermodynamics, this RCVD method produces only pure TiC sub-layers when a low dilution ratio of TiCl4 in H2 (R=[H2]/[TiCl4]) is used. With a high R value, the growth of pure Ti3SiC2 thin continuous sub-layers from solidgas reaction is observed for short deposition times. But for long deposition times, the simultaneous growth of additional Ti5Si3Cx and TiC sub-layers occurs from solid state diffusion when the initial Ti3SiC2 sub-layer thickness exceeds about 1 µm. In that case, the repeating of the sequence results in the further conversion of the additional phases into new Ti3SiC2 sublayers from solid state reactions involving TiSi2 as an intermediate phase and SiC.

Introduction

Titanium carbide-silicide Ti3SiC2 exhibits unique properties; it is soft and easily machinable, behaves plastically and maintains its strength at high temperature and is resistant to oxidation and thermal shock [START_REF] Barsoum | Synthesis and characterization of a remarkable ceramic: Ti 3 SiC 2[END_REF].

As a thin coating, Ti3SiC2 has interesting mechanical properties in prospect: it can deform through the formation of kink bands and delamination cracks [START_REF] Molina-Aldareguia | Kink formation around indents in laminated Ti 3 SiC 2 thin films studied in the nanoscale[END_REF][START_REF] Emmerlich | Growth of Ti 3 SiC 2 thin films by elemental target magnetron sputtering[END_REF]. It is also an excellent electrical and thermal conductor [START_REF] Barsoum | Synthesis and characterization of a remarkable ceramic: Ti 3 SiC 2[END_REF] and therefore can be considered for electrical contact layers and connectors at high temperature [START_REF] Wenzel | Diffusion barriers on titanium-based ohmic contact structures on SiC[END_REF]. In addition, this material exhibits a thermal expansion coefficient around 9 × 10 -6 °C-1 over the range 25 -1200°C [START_REF] Barsoum | Thermal properties of Ti 3 SiC 2[END_REF], which is halfway between SiC (around 4 × 10 -6 °C-1 ) and most metals (Ag: 19 × 10 -6 °C-1 , Cu: 17 × 10 -6 °C-1 , Fe: 12 × 10 -6 °C-1 …). This ternary compound that combines the advantage of both metals and ceramics could be used as an interlayer to join these two dissimilar materials for thermostructural applications.

However, few studies have reported the synthesis of Ti3SiC2 films. Thin coatings could be prepared recently by physical vapor deposition [START_REF] Palmquist | Magnetron sputtered epitaxial single-phase Ti 3 SiC 2 thin films[END_REF]. In earliest works, Ti3SiC2 coatings were deposited by chemical vapour deposition (CVD) from different gaseous mixtures such as TiCl4-SiCl4-CCl4-H2 [START_REF] Nickl | Gas-phase deposition in the titanium-silicon-carbon system[END_REF][START_REF] Goto | Chemically vapor deposited titanium silicon carbide (Ti 3 SiC 2 )[END_REF][START_REF] Pickering | CVD of Ti 3 SiC 2[END_REF] or TiCl4-SiCl4-CH4-H2 [START_REF] Racault | On the chemical vapor deposition of Ti 3 SiC 2 from TiCl 4 -SiCl 4 -CH 4 -H 2 gas mixtures. Part II. An experimental approach[END_REF]. CVD allows complex geometrical shape substrates to be coated but these studies highlighted that the deposition of pure Ti3SiC2 is difficult.

A previous thermodynamic analysis [START_REF] Jacques | Pulsed reactive Chemical Vapor Deposition in the C-Ti-Si system from H 2 /TiCl 4 /SiCl 4[END_REF] showed that it is theoretically possible to obtain Ti3SiC2 as a single solid phase by reactive CVD (RCVD) through the reaction of H2/TiCl4 gaseous mixture on solid SiC with a concentration ratio R ([H2]/[TiCl4]) equal to 4 (Fig. 1.a) if the correct reactant gas/solid ratio is achieved. But, the experimental procedure did not allow the growth of the pure ternary phase. This was attributed to the impossibility of exceeding several nanometers in thickness during the first growth of SiC by RCVD (the method used in the previous study) and so to control the gas to solid ratio through matter diffusion [START_REF] Jacques | Pulsed reactive Chemical Vapor Deposition in the C-Ti-Si system from H 2 /TiCl 4 /SiCl 4[END_REF].

In the present study, we propose to start with a micrometer-thick SiC film deposited by a classical CVD route (in which both silicon and carbon are provided by the gaseous phase) and then to use RCVD for the growth of Ti3SiC2 through SiC consumption. For that purpose, we have used a classical low pressure hot wall CVD reactor in which the gaseous pressure and flow are maintained constant during the growth of each layer. As long as solid SiC is available, the gas to solid ratio is therefore controlled by the thickness of the second growing layer, i.e. by the RCVD processing time.

Thermodynamic analysis

For the thermodynamic analysis, the computer program used was Gemini 2 and the thermodynamic data were from the Scientific Group Thermodata Europ (Saint Martin d'Hères, France). The temperature was of 1100°C, which is a maximum for our reactor chamber made of silica. The total pressure was set as low as 5 kPa in order to favor coating homogeneity. The calculation details are given in reference [START_REF] Jacques | Pulsed reactive Chemical Vapor Deposition in the C-Ti-Si system from H 2 /TiCl 4 /SiCl 4[END_REF].

With R = 4, Fig. 1a shows that, if the H2/TiCl4 gaseous reagent reacts in too large excess with SiC (i.e. more than 4000 moles of TiCl4 for one mole of SiC), the solid phase obtained at equilibrium is no longer Ti3SiC2 but TiC, silicon being vented in the by-product gaseous phase as silicon chlorides: SiClp (p = 2, 3 and 4). In the case of a very large excess of gaseous reagent, using an R ratio as high as R = 35 allows Ti3SiC2 to be obtained at equilibrium (Fig. 1b) according to (1):

But it can be also co-deposited with Ti5Si3Cx particularly if silicon can not be evacuated in the gaseous phase as SiCl2 (Fig. 1b) according to (2):

(6-x)SiC(s) + (14-3x)TiCl4(g) + 2(14-3x)H2(g) → (3-x)Ti3SiC2(s) + Ti5Si3Cx(s) + 4(14-3x)HCl(g) [START_REF] Molina-Aldareguia | Kink formation around indents in laminated Ti 3 SiC 2 thin films studied in the nanoscale[END_REF] where x is a constant value in the range 0≤ x≤ 1.1 [START_REF] Viala | Phase equilibria at 1000°C in the Al-C-S-Ti quaternary system: An experimental approach[END_REF][START_REF] Schuster | Design criteria and limitations for SiC-metal and Si 3 N 4 -metal joints derived from phase diagram studies of the systems Si-C-metal and Si-N-metal[END_REF].

Experimental procedure

Ti-Si-C layer growth

The coatings were deposited on polycrystalline graphite disks (12 mm in diameter, Ellor 30 from Carbone Lorraine France) placed inside the hot area of a CVD reactor chamber. The CVD apparatus included a horizontal silica tube (30 mm in inner diameter) heated by an electrical resistor furnace. The temperature was 1100°C within the hot area. It was regularly checked between two deposition runs with three Process Temperature Control Rings (type LTH from Ferro Electronic Materials B.V., Uden, The Netherlands) submitted to the same conditions in the reactor as the graphite disks. The total pressure was maintained in the apparatus at a constant value during the deposition of each kind of layer by a vacuum pump through a regulating valve and a pressure gauge. Liquid nitrogen traps permitted the condensation of corrosive by-products such as HCl. Mass flowmeters were used to control the partial gas flow rates of H2 and CH3SiCl3 (MTS).

In each case, a SiC layer was previously deposited by classical CVD from H2/MTS to be used as solid carbon and silicon sources for the following Ti-Si-C layer growth. For SiC deposition, the ratio [H2]/[MTS] was 4, the total gas flow rate was 250 sccm (standard cm 3 min -1 , i.e. at P = 101 kPa and T = 20°C) and the total pressure 3 kPa. Then, after evacuation, the H2/TiCl4 gaseous mixture was introduced in the reactor and the pressure was set at 5 kPa, which allowed the growth of Ti-Si-C layer(s) by SiC consumption. The titanium tetrachloride vapor was carried by hydrogen through a bubbler at room temperature. The reactant concentration ratio R ([H2]/[TiCl4]) was adjusted to 4 or 35 by adding pure H2 with an extra parallel hydrogen feed line and taking into account the value of TiCl4 partial pressure at room temperature. For instance, with a room temperature equal to 20°C, the TiCl4 partial pressure in the bubbler was 1.28 kPa. The H2 flow rate was set to 26 sccm through the bubbler and to 290 sccm in the second parallel hydrogen line. With a pressure equal to 5 kPa in the feed lines, the TiCl4 gas flow rate carried out from the bubbler was estimated to 9 sccm, which corresponded to R = 35.

To obtain R = 4, the H2 flow rate was set to 80 sccm through the bubbler (TiCl4 gas flow rate ≈ 19 sccm) and to 30 sccm in the second parallel hydrogen line. In each case, the deposition sequence (CVD treatment at 3 kPa with H2/MTS, evacuation, RCVD at 5 kPa with H2/TiCl4, and evacuation) was repeated several times in order to obtain a total coating thick enough for further characterization. A repetitive deposition sequence was designated as (Ss'/Tt')n were s was the number of minutes used for each previous SiC sub-layer deposition, t the time in minutes chosen for each RCVD reaction with H2/TiCl4 and n the total number of repeated sequences.

Characterization

The coated disk surfaces were characterized by X-ray diffraction (XRD: X'Pert Pro MPD from Panalytical, equipped with an X'Celerator detector monochromator, CuKα radiation, Almelo, The Netherlands). Then, the coated graphite disks were broken and the fracture surfaces of the coatings were observed with a scanning electron microscope (SEM: S800 from Hitachi, Tokyo, Japan). With sufficiently thick enough sub-layers (i.e. above 1 µm), cross section composition profiles were determined by electron probe micro-analysis (EPMA: CAMEBAX from CAMECA, Courbevoie, France). EPMA was operated under an accelerating voltage of 10 kV. After background subtraction, the counting rates obtained for Si and Ti were referred to those recorded under the same conditions on pure and freshly polished standards.

After correction for atomic number, absorption and fluorescence, the Si and Ti contents, in percent, were obtained within an error bar not higher than 1 at.%. The carbon content was obtained by subtracting the Ti and Si contents to 100%.

Transmission electron microscopy (TEM) was used (TOPCON 002B, Tokyo, Japan) for thin sub-layers utilizing bright field (BF) and selected area electron diffraction (SAED) techniques. The microscope was equipped with a Kevex energy dispersive spectroscopy (EDS) device for chemical analysis. The chosen spot size for EDS was 56 nm. Only titanium and silicon were considered in the coating analysis (not carbon). Quantitative composition measurements were obtained using the standard Cliff-Lorimer equation. The relative elemental concentration (CTi and CSi) are related to the measured X-ray peak surfaces above background (STi and SSi, respectively) by CTi/CSi = kSiTi SSi/STi, where kSiTi is the sensitivity or k-factor. The value of kTiSi was determined from Ti3SiC2 grains identified first by SAED in the samples. Ti3SiC2 is indeed known as a stoichiometric compound and can serve as a standard. To obtain cross sections for TEM, two pieces of broken coated graphite disks were stuck together with an epoxy resin (Micro-Measurements-Wishay adhesive, Reference M-Bond 610) and mechanically thinned. The thin sheets were then ion-milled (GATAN PIPS, USA) until the electron transparency.

Results

Experimental RCVD with R = 4

Figs. 2 and3 show that a 240 minutes long treatment of a SiC layer by a H2+TiCl4 gaseous mixture with a molar ratio H2/TiCl4 = 4 (R = 4) results in the growth of a TiC coating of about 3 µm in thickness. Fig. 4 shows the SEM observation of a multilayered coating obtained with shorter treatments. SAED patterns confirm that these RCVD treatments of SiC layers lead also to the formation of pure titanium carbide (Fig. 5). Furthermore, the presence of pores can be noticed in the first one-third part of the TiC layers near the previously deposited SiC layer (Figs. 4 and5).

RCVD with R = 35 4.2.1 "Thick layers"

As shown by the thermodynamic analysis, a way to favor the formation of Ti3SiC2 by RCVD is to increase the R ratio by dilution in hydrogen. A first multilayered coating was deposited with R = 35 by following three sequences: (S60'/T90')3. The XRD pattern presented in Fig. 7 reveals the presence of five phases in the multilayered film: SiC, TiC, Ti3SiC2, TiSi2 and Ti5Si3Cx. EPMA coupled with SEM observation allows the different phases to be localized across the coating (Fig. 8). As a general trend, the coating consists of a succession of continuous single-phased sub-layers. But surprisingly, even though the three CVD sequences were identical, the three resulting sub-layer sequences found in the whole coating are different from each other. From the carbon disk substrate surface, the coating (about 30 µm thick)) can be described as follow: The first sequence is made of one SiC sub-layer covered with a Ti3SiC2 then with a TiSi2 thin sub-layer (about 1 µm) and then with a last sub-layer of global composition Tix-Siy-Cz that is likely to contain a mixture of TiC and Ti3SiC2. The second diffraction pattern, which indicates that the basal crystallographic planes are oriented perpendicular to the substrate [START_REF] Tang | Analysis on preferred orientation and purity estimation of Ti 3 SiC 2[END_REF]. This result is similar to the previous CVD studies [START_REF] Nickl | Gas-phase deposition in the titanium-silicon-carbon system[END_REF][START_REF] Goto | Chemically vapor deposited titanium silicon carbide (Ti 3 SiC 2 )[END_REF][START_REF] Pickering | CVD of Ti 3 SiC 2[END_REF][START_REF] Racault | On the chemical vapor deposition of Ti 3 SiC 2 from TiCl 4 -SiCl 4 -CH 4 -H 2 gas mixtures. Part II. An experimental approach[END_REF]

Discussion

The use of different durations for the RCVD treatments allows the relationship between spatial and time sequence to be drawn. With R = 35, the rapid growth of pure Ti3SiC2 is first observed at the very beginning of the treatment. As long as the treatment duration t does not exceed a critical time denoted as tcrit (t < tcrit, with 10 min < tcrit' < 30 min), the pure Ti3SiC2 layer still grows but at a reduced rate: 1.1 µm thick in 3 minutes to only 1.3 µm thick in 10 minutes. Then, the very last unit of each multi-layered coating (the one not covered by a new SiC film) must be considered to describe the spatiotemporal growth, i.e. the sixth Ti-Si-C layer in (S30'/T30')6 and the third one in (S60'/T90')3. For t > tcrit, a second layer made of Ti5Si3Cx grows at a fast rate: 1.9 µm thick in 30 minutes and 6.0 µm thick in 90 minutes. At the same time, the first Ti3SiC2 sub-layer located near the previously deposited SiC still grows very slowly (1.7 µm thick in 30 minutes and 2.8 µm thick in 90 minutes). For a reaction time longer than 30 minutes, a third and thin TiC sub-layer also develops between SiC and Ti3SiC2.

These results tend to show that a one step growth of a pure Ti3SiC2 layer thicker than 1.5 µm is impossible by RCVD. A way to circumvent this limit and obtain a thick pure Ti3SiC2 film should be to repeat short SiC/Ti3SiC2 coating sequences by reducing each SiC sub-layer deposition duration till it is totally consumed, i.e. by realizing (Ss'/Tt')n with t ≤ tcrit and s<t.

Experiments are planned to check up on that assumption.. For "thick" Ti-Si-C sub-layers prepared with t > tcrit, if a new SiC film is deposited on the three-phased TiC/Ti3SiC2/Ti5Si3Cx coating, two new solid state reactions can develop consistently with the Ti-Si-C phases diagram [START_REF] Jacques | Pulsed reactive Chemical Vapor Deposition in the C-Ti-Si system from H 2 /TiCl 4 /SiCl 4[END_REF][START_REF] Wakelkamp | Phase relations in the Ti-Si-C system[END_REF][START_REF] Racault | On the chemical vapour deposition of Ti/sub 3/SiC/sub 2/ from TiCl/sub 4/-SiCl/sub 4/-CH/sub 4/-H/sub 2/ gas mixtures: Part I. A thermodynamic approach[END_REF][START_REF] Du | Experimental investigation and thermodynamic calculation of the titanium-silicon-carbon system[END_REF]. On the one hand, SiC reacts with Ti5Si3Cx to form Ti3SiC2 and TiSi2. On the other hand, the TiC thin layer formed onto SiC is slowly converted into Ti3SiC2 by reaction with Ti5Si3Cx and TiSi2. Development of these solidstate reactions explains why the brittle Ti5Si3Cx phase is no more present in the sub-layers covered by SiC. The texture difference between a Ti3SiC2 sub-layer located near the previously deposited SiC film and a Ti3SiC2 sub-layer located near the next SiC film (Fig. 8 and9) evidences the difference between both mechanisms: mainly solid-gas (SiC-H2/TiCl4) reaction for the first one and solid-solid (Ti5Si3Cx-SiC) reaction for the second one.

At first sight, in the case of solid-gas reactions, this work seems to raise discrepancies between thermodynamic calculations and experimental results. The reasons for this discrepancies will be analyzed in the following by considering different cases: low R value (i) or high R value (ii) and for a high R value: thin coatings (iia) and thick coatings (iib). When R = 4 (i), Fig. 1a shows that we should obtain pure Ti3SiC2 rather than the TiC obtained experimentally (Fig. 5). The previously calculated deposition diagram [START_REF] Jacques | Pulsed reactive Chemical Vapor Deposition in the C-Ti-Si system from H 2 /TiCl 4 /SiCl 4[END_REF] showed that to obtain TiC at equilibrium, R should be less than 2.3 for a co-deposition with Ti3SiC2 and equal to about 0.5 for pure TiC growth. This first seeming discrepancy could originate from a mere shift between the R ratio used for calculation and the experimental ratio and more particularly by a shift between the ratio corresponding to the gaseous system H2/TiCl4 introduced in the CVD reactor and the actual ratio for H2 and TiCl4 molecules really reaching the SiC substrate surface where the reaction takes place. As H2 is nearly one hundred times less dense than TiCl4 gas, after convection through the reactor, the gas may be poorer in hydrogen at the bottom part of the hot area where the carbon disk substrates are set. Also, the diffusion rates of hydrogen across the boundary layer at the substrate surface could be different from the one of titanium chloride, which might be responsible for a change of the R ratio.

When R = 35 and when only thin layers are deposited by RCVD during short deposition times (iia), the precursor gas phase is in direct contact with uncovered SiC solid substrate. As a gaseous phase is less dense than a solid phase, this case should correspond to a small quantity of TiCl4 reacting with SiC, i.e. n(TiCl4)<n(SiC). In that case, as long as SiC is still present at equilibrium, Ti3SiC2 should be co-deposited with TiSi2 (Fig. 1b). Experimentally, if thin Ti3SiC2 layers could be actually obtained in these conditions, TiS2 was never detected in thin coatings realized in less than 10 minutes (Fig. 14). Once again, if a decrease of the R ratio is admitted, the thermodynamic calculations that predict the growth of pure Ti3SiC2 by partial consumption of SiC when R is less or equal to 20 (and superior to 2.3) are in accordance with the experimental result.

Finally, for thick coatings prepared with a high R value (iib), after the formation of pure Ti3SiC2, the growth of Ti3SiC2 and Ti5Si3Cx is observed as predicted by the thermodynamic calculations for n(TiCl4)>n(SiC) (Fig. 1b) and R > 6.7 [START_REF] Jacques | Pulsed reactive Chemical Vapor Deposition in the C-Ti-Si system from H 2 /TiCl 4 /SiCl 4[END_REF]. This point can be achieved when the Ti3SiC2 sub-layer is thick enough (~1.5 µm) to isolate the SiC substrate from the gaseous molecules, which results in the vanishing of SiC from the system at equilibrium according to thermodynamics. However, the simultaneous formation of both ternary compounds does not result from the expected co-deposition since the phases are not mixed in one layer but separated in two distinct continuous sub-layers. It is worth noting that, once a continuous reaction layer has formed at the substrate surface by RCVD, this layer acts as a diffusion barrier. Further reaction between the substrate and the gas phase can then only proceed by solid state volume diffusion through the barrier layer. Even if local equilibria still exist at the substrate/layer and layer/gas interfaces, chemical potential gradients take place in the barrier layer and the underlying substrate is no more in equilibrium with the gas phase [START_REF] Kirkaldy | Diffusion behavior in ternary multiphase systems[END_REF][START_REF] Van Loo | Multiphase diffusion in binary and ternary solid state systems[END_REF]. Thus, the thermodynamic analysis presented in this paper and in reference [START_REF] Jacques | Pulsed reactive Chemical Vapor Deposition in the C-Ti-Si system from H 2 /TiCl 4 /SiCl 4[END_REF] is actually no more valid when t >> tcrit, and particularly it can no longer be used to account for the experimental results when (TiCl4(g))>>n(SiC(s)).

At this point, the diffusion path concept and solid state reaction-diffusion models [START_REF] Kirkaldy | Diffusion behavior in ternary multiphase systems[END_REF][START_REF] Van Loo | Multiphase diffusion in binary and ternary solid state systems[END_REF][START_REF] Dibkov | Growth kinetics of chemical compound layers[END_REF] appear as the most appropriate tools to describe further development of the RCVD process. The parabolic curve presented in Fig. 6 for a low R value is thus typical of a diffusion mechanism.

(The plateau observed for very long time deposition can be attributed to the development of pores in TiC by Kirkendall effect that end up by cuting off the diffusion path.) The introduction of inert markers should be useful in further work for better understanding of the growth of thick Ti-Si-C layers by RCVD.

Conclusion

The RCVD method allows a sub-micrometer SiC surface layer to be converted into Ti3SiC2 in a few minutes if a sufficient dilution of the titanium tetrachloride vapor in hydrogen is used to avoid the direct growth of TiC. These results accord with thermodynamics if a shift between the R ratio used for calculation and the experimental ratio is admitted. Hence, with periodic changes of the gases introduced in the CVD reactor, it is possible to deposit (SiC/Ti3SiC2)n multilayered coatings as long as short enough treatment times are used for the growth of each Ti3SiC2 thin sub-layer. For long deposition times, the RCVD treatment of SiC results in the formation of TiC and Ti5Si3Cx additional sub-layers through solid state diffusion.

The alternation of these thick coatings with SiC layers results in the further conversion of the additional brittle phases into new Ti3SiC2 sub-layers from solid state reactions that involve TiSi2 as an intermediate phase.

Legends

Fig. 6

 6 Fig. 6 presents the change of the TiC layer thickness versus the duration of the treatment which can be interpolated by a parabolic curve (square root of time) except for the last point at 240 minutes.

Fig. 1 .

 1 Fig. 1. Calculated quantities of solid species produced at equilibrium from {H2/TiCl4} gaseous mixture reaction on 1 mole of SiC for R = 4 (a) and R = 35 (b) versus the quantity of TiCl4 (R = H2/TiCl4, 1100°C, 5 kPa).

Fig. 2 .Fig. 3 .

 23 Fig. 2. SEM observation of (S150'/T240')1 coating deposited with R = 4. Fig. 3. EPMA of (S150'/T240')1 coating deposited with R = 4.

Fig. 4 .

 4 Fig. 4. SEM observation of (S20'/Tt')4 coating deposited with t = 15, 25, 35 and 45 minutes and with R = 4.

Fig. 5 .Fig. 6 .

 56 Fig. 5. BF TEM observation and SAED pattern (insert) of the fourth bi-layer (S20'/T45') of (S20'/Tt')4 coating {t' = 15', 25', 35' and 45'} deposited with R = 4. Fig. 6. Thicknesses of TiC layers deposited with R = 4 versus treatment durations.

Fig. 7 .

 7 Fig. 7. XRD pattern of(S60'/T90')3 deposited with R = 35.

Fig. 8 .

 8 Fig. 8. SEM observation and EPMA of (S60'/T90')3 deposited with R = 35.

Fig. 9 .

 9 Fig. 9. SEM observation of (S30'/T30')6 coating deposited with R = 35.

Fig. 10 .

 10 Fig. 10. XRD pattern of (S30'/T30')6 deposited with R = 35.

Fig. 11 .

 11 Fig. 11. BF TEM observation and SAED patterns (inserts) of sub-layer n°5 of (S30'/T30')6 deposited with R = 35.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. TEM EDS analysis of Ti-Si-C sub-layer n° 2 (a), sub-layer n° 5 (b) and sub-layer n° 6 (c) of (S30'/T30')6 coating deposited with R = 35. (carbon concentration not represented) Fig. 13. SEM observation of (S10'/T10')n (a) and (S3'/T3')n (b) deposited with R = 35.

Fig. 14 .FiguresFig. 1

 141 Fig. 14. XRD pattern of (S3'/T3')n deposited with R = 35.

  Fig. 1. Calculated quantities of solid species produced at equilibrium from {H2/TiCl4} gaseous mixture reaction on 1 mole of SiC for R = 4 (a) and R = 35 (b) versus the quantity of TiCl4 (R = H2/TiCl4, 1100°C, 5 kPa).

Fig. 2 .

 2 Fig. 2. SEM observation of (S150'/T240')1 coating deposited with R = 4.

Fig. 3 .

 3 Fig. 3. EPMA of (S150'/T240')1 coating deposited with R = 4.

Fig. 4 .

 4 Fig. 4. SEM observation of (S20'/Tt')4 coating deposited with t = 15, 25, 35 and 45 minutes and with R = 4.

Fig. 5 .

 5 Fig. 5. BF TEM observation and SAED pattern (insert) of the fourth bi-layer (S20'/T45') of (S20'/Tt')4 coating {t' = 15', 25', 35' and 45'} deposited with R = 4.

Fig. 6 .

 6 Fig. 6. Thicknesses of TiC layers deposited with R = 4 versus treatment durations.

Fig. 7 .

 7 Fig. 7. XRD pattern of(S60'/T90')3 deposited with R = 35.

Fig. 8 .Fig. 9 .

 89 Fig. 8. SEM observation and EPMA of (S60'/T90')3 deposited with R = 35.

Fig. 10 .

 10 Fig. 10. XRD pattern of (S30'/T30')6 deposited with R = 35.

Fig. 11 .

 11 Fig. 11. BF TEM observation and SAED patterns (inserts) of sub-layer n°5 of (S30'/T30')6 deposited with R = 35.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. TEM EDS analysis of Ti-Si-C sub-layer n° 2 (a), sub-layer n° 5 (b) and sub-layer n° 6 (c) of (S30'/T30')6 coating deposited with R = 35. (carbon concentration not represented)

Fig. 14 .

 14 Fig. 14. XRD pattern of (S3'/T3')n deposited with R = 35.

SiC(s) +

TiCl4(g) + 5 H2(g) → Ti3SiC2(s) + SiCl2(g) + 10 HCl(g)[START_REF] Barsoum | Synthesis and characterization of a remarkable ceramic: Ti 3 SiC 2[END_REF]

sequence is similar to the first one but here the last sub-layer clearly corresponds to Ti3SiC2.

The third sequence is quite different from the others: after the SiC sub-layer, a thin TiC sublayer (about 1 µm in thickness) is first observed before the Ti3SiC2 sub-layer. Then the sequence ends with a thick (about 4 µm) Ti5Si3Cx sub-layer. Each Ti3SiC2 sub-layer detected in the coating is about 2-3 µm thick.

A second multilayered coating was deposited with R = 35 but, in this case, the duration chosen for the SiC(s) + H2/TiCl4(g) RCVD reaction was divided by three: (S30'/T30')6. SEM observation confirms the presence of the six (SiC/Ti-Si-C) bi-layers in the coating (Fig. 9). The corresponding XRD pattern is similar to the previous one except for the TiC phase that was not detected (Fig. 10). A TEM observation performed in the area of sub-layer n°5 reveals the presence of Ti3SiC2 crystals (Fig. 11). The total coating thickness is 35 µm but here the thickness of each sub-layer is barely enough for EPMA analysis. So, Si and Ti EDS composition measurements were performed in the TEM on the Ti-Si-C sub-layers n°2, n°5 and n°6 of the thin foil (Fig. 12). This multilayer is quite similar to (S60'/T90')3. Roughly, Ti-Si-C sub-layer n°2 is only made of Ti3SiC2, Ti-Si-C sub-layer n°5 is made of three sub-layers: Ti3SiC2/TiSi2/Ti3SiC2, and the last Ti-Si-C sub-layer is made of two sub-layers: Ti3SiC2/ Ti5Si3Cx.

"Thin layers"

By reducing more the duration of each deposition sequence, it is possible to obtain submicrometer thin sub-layers as shown in Fig. 13 for (S10'/T10')n and (S3'/T3')n. In both cases, the XRD patterns only reveal the presence of SiC and especially Ti3SiC2 (Fig. 14). In that case, the formation of TiC, Ti5Si3Cx and TiSi2 is avoided and the resulting deposited film is a (SiC/Ti3SiC2)n multilayered coating. It is important to note that from SEM observations, the Ti3SiC2 coatings are characterized by plate-like microstructure with grains oriented perpendicular to the substrate surface. This orientation is confirmed by XRD: the {008} peak is missing whereas the {110} reflection appears stronger than for a calculated powder


Guilherme Costa

Ayres Tolentino

Jean Vianei Leite

Mathieu ; Rossi

Olivier ; Ninet

Guillaume ; Parent

Jonathan Blaszkowski

Guilherme C ; Tolentino

Jean V ; Leite

Modelling of magnetic anisotropy in electrical steel sheet by means of cumulative distribution functions of Gaussians

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.

This approach has been introduced in the 90s for the study of isothermal remanent magnetization acquired by rock samples containing magnetic minerals [1,2,3].

The main drawback of this approach lies in the fact that a given magnetization curve is derived from one Gaussian function only, which is itself defined by three fixed parameters only [1], leading to non-negligible differences when compared to experimental data.

Moreover, this approach has never been adapted to account for the anisotropic behavior of electrical steel.

In this paper, we propose to enhance this method by using a function which results from the sum of four Gaussian functions, as illustrated in Fig. 1. To account for the anisotropy, all the parameters used to define the Gaussian functions depend on the angle between the magnetic field H and the rolling direction of the steel, denoted θ. The work presented in this paper shows multiple originalities and improvements. Firstly, it shows that using more than one Gaussian function allows for a better approximation of first magnetization curves (Fig. 2).

Secondly, it shows that considering θ-dependent parameters to define the Gaussian function allows the method to account for magnetic anisotropy (Fig. 2). Thirdly, it highlights that those parameters follow a polynomial trendline.

References: [1] D. J. Robertson and D. E. France, Physics of the Earth and Planetary Interiors, vol. 82, no. 3-4, pp. 223-234 (1994) [2] H. Stockhausen, Geophysical Research Letters, vol. 25, no. 12, pp. 2217-2220(1998) [3] D. Heslop, G. McIntosh, and M. J. Dekkers, Geophysical Journal International, vol. 157, no. 1, pp. 55-63 (2004) Fig. 1 First magnetization curve derived from a Gaussian function defined as the sum of four subfunctions.

2022 Joint MMM-Intermag

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Fig. 2

 2 Fig.2shows the comparison between first magnetization curves for three different magnetization directions (θ) obtained from experimental data (M530-50A electrical steel), from the original model based on the use of one Gaussian function only and from the proposed enhanced model.

Fig. 2 Manuscript

 2 Fig. 2 Comparison between curves obtained from experimental data (markers), from the original model (dashed) and from the enhanced proposed model (solid).

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Yulong Zhong

Thomas A Sobiech

Brice Kauffmann

Bo Song

Xiaopeng Li

Yann Ferrand

Ivan Huc

Bing Gong

Hollow helices as high-affinity hosts: Single and double helical pseudofoldaxanes

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

A major aim in molecular recognition is the development of structurally and/or functionally tunable hosts capable of tailoring ionic and molecular guests. 1 Since the discovery of crown ethers, a bewildering array of disc-like macrocyclic hosts containing two-dimensional (2D) binding cavities have been created. 2 Hosts such as cryptands, [START_REF] Dietrich | Comprehensive Supramolecular Chemistry[END_REF] cavitands, 4 and various cages 5 containing deepened cavities are also known to have drastically enhanced binding affinity and selectivity for various guest species.

Tube-like macrocycles such as cyclodextrins, 6 calixarenes, 7 cucurbit[n]urils, 8 pillar[n]arenes, 9 and other systems 10 provide a class of hosts offering three-dimensional (3D) cavities with legs or walls defining their inner surfaces. [START_REF] Ariga | Supramolecular Chemistry-Fundamentals and Applications[END_REF] Many such hosts, especially those with covalently locked, overall rigid conformations and non-deformable cavities, show spectacular recognition capability. For example, cucurbit[n]urils exhibit remarkably tight binding affinities (up to 10 17 M -1 in water) for rigid cationic guests. [START_REF] Rekharsky | Proc. Natl. Acad. Sci[END_REF] Tubular structures with cylindrical cavities of adjustable lengths (or depths) may serve as hosts with unique binding and transport capability, leading to new understanding of hostguest interaction. For example, hosts with deep cavities may be accommodate long, rodlike guests and provide fundamental understanding on the role played by multiple non-covalent forces in host-guest binding. Elongated cavities spanning the lipid bilayers can serve as transmembrane channels that facilitate mass transport with selectivity and large flux, [START_REF] Schoch | [END_REF]14 and allow the identification of guest species without relying on specific guest binding. Several self-assembling organic nanotubes are known. [15][16][17][START_REF] Jonkheijm | [END_REF][19][20][21] The majority of such nanotubes, while showing many interesting binding and transport properties, have undefined length, deformable shapes, and low stability that limit their study and applications. Molecular tubes with non-deformable inner cavities reminiscent of those of carbon nanotubes may be constructed by extending the covalent frameworks of rigid tubular macrocycles such as cucurbit [n]urils. Such a possibility, while fascinating, remains to be realized until daunting synthetic challenges are addressed.

A conceptually feasible strategy for constructing molecular tubes is the folding of synthetically accessible oligomers into helices. 22 With helical cavities, such foldamers can serve as hosts for recognizing ionic and molecular guests. Examples of foldamer-based hosts for neutral small molecules were reported by Lehn,23 Moore, 24 Inouye,25 Li, 26 Huc, 27 and Jeong; 28 those for recognizing cations were described by Lehn,29 Chen, 30 Fox, 31 Gong, 32 and Zeng 33 ; and hosts for anions were created by Jeong, 34 Craig, 35 Flood, 36 Guichard, 37 Berryman, 38 and Gong. 39 Except for a few systems, 26,27,32,33 the majority of foldamer hosts known thus far are those undergoing guest-induced folding driven by binding enthalpy. Association constants (K) between such foldamers and guests including ions and small molecules are typically from 10 2 to 10 4 M -1 (up to 10 7 M -1 ) in organic solvents such as chloroform and acetonitrile; and from 10 3 to 10 4 M -1 (up to 10 6 M -1 ) in water-containing solvents with the binding driven by hydrophobic effects. As noncyclic hosts, helical foldamers exhibit unique binding behavior. For example, they can wind around rod-like dumbbell-shaped guests to give host-guest complexes, dubbed by Huc and Ferrand as foldaxanes, that possess properties both similar to and different from traditional rotaxanes. Cartoons on the right illustrate the helical, cyclic, and rodlike shapes of H8, H16, 1, and the two cationic guests. (b) Illustrations of the observed 2:1 complex between 1 and OV 2+ , the anticiapted 2:1 complex between H8 and OV 2+ , and the anticipated 1:1 complex between H16 and OV 2+ or TB 2+ . While a few short foldamers adopting stably folded conformations are known, 26,27,32.33 multi-turn helical foldamers with stable, preorganized cavities capable of binding common organic guests, i.e., those with size larger than the diameter (~4 Å) of linear alkyl chains, 41 are rare. 42,44,45 One class of foldamers with highly stable helical conformations and sufficiently large cavity are represented by general structure Hn (Figure 1a, top), which we first proposed and subsequently established. 22 With their backbones being constrained by highly favorable three-center hydrogen bonds, 43 oligoamides Hn of different lengths, such as the 8-residue H8 and the 16-residue H16 (Figure 1a), all fold into cavity-containing, "hollow" helices. 44 Oligoamides folding into helices of up to 3 turns were synthesized recently, with the crystal structure of H16 revealing a compact helix having ~6.6 residues per turn and a non-deformable inner pore of ~9 Å across. 45 Independent of their lengths, helices Hn were found to be stable in solvents of different polarity and at elevated temperatures. 44d Thus, by synthetically adjusting the length of oligoamides Hn, stable hollow helices with tunable but defined lengths and inner cavities of a fixed diameter are obtained. With inward-pointing amide oxygens decorating their inner walls, the cylindrically-shaped cavities of helices Hn are highly electronegative and strongly H-bonding, which are distinctly different from the hydrophobic cavities of known high-affinity hosts such as cucurbit[n]urils. As molecular tubes with electronegative cavities having multiple preorganized amide oxygens, our hollow helices are expected to strongly bind cationic guests of suitable sizes.

Sharing the same rigid backbone with oligoamides Hn, sixresidue macrocycles 1 (Figure 1a, middle) were found to strongly bind octyl viologen (OV 2+ ) (Figure 1a, bottom) in their preorganized electronegative cavity in solvents of high polarity. 46 Since the cavity of one molecule of 1 is too "thin" to match the length of the bipyridinium segment of guest OV 2+ , two molecules of 1 were observed to stack to provide a cavity with a sufficient depth for binding the cationic rod (Figure 1b, left). The cationic guest threads through the cavities of the two stacked macrocycles, leading to 2:1 complex 12•OV 2+ with overall association constants (Ktotal) of ~10 11 M -2 in DMSO/CHCl3 (1/1, v/v) and ~10 8 M -2 in DMF.

The high affinity of macrocycles 1 for guest OV 2+ suggests that helical oligoamides Hn, with their electronegative cavities, should also strongly bind this and other rodlike cationic guests. To match the bipyridinium rod of guest OV 2+ , a short (~1 turn) helix needs to stack into a self-assembling host consisting of two or more helical molecules. In contrast, a long helix providing a sufficiently deep cavity that matches the cationic segment will serve as a unimolecular host.

To probe the possibility of tailoring the size (length) of guests with helical oligoamides Hn, oligoamides H8 and H16 were studied for their binding with guest OV 2+ . Based on the 2:1 binding of macrocycles 1 with OV 2+ , oligoamide H8, as a helix of ~1.2 turns, was expected to bind guest OV 2+ to give a 2:1 complex (Figure 1b, middle). The ~2.5-turn helix H16 provides a cavity with a depth (~9 Å) that largely matches the length of the bipyridinium segment (~9.8 Å, between the two N + CH2carbons), and was anticipated to bind guest OV 2+ in a 1:1 ratio (Figure 1b, right). As a unimolecular host, H16 was expected to bind OV 2+ with lowered entropic cost and thus higher binding affinity than that between helix H8 and the same guest. To further demonstrate the role of positive charges and the bulkiness of the guest played in host-guest binding, cationic guest 1, 4-bis(trimethylammonium)benzene (TB 2+ ) (Figure 1a, bottom), with a length (~9.0 Å, between the hydrogens of the two N + (CH3)3 groups) that is the same as the depth of the cavity of helix H16, is designed and examined for its interaction with H16.

Hn O O O N H R O Me O Me O N H R O O O N H R O Me O Me O N H R O O O N H R O Me O Me

Results and discussion

Binding process and stoichiometry

The host-guest interactions between oligoamides H8 or H16 and OV 2+ was first probed with 1 H NMR titration experiments.

In DMSO-d6/CDCl3 (3/7, v/v), titrating H8 with 0 to 2 equiv of OV 2+ resulted downfield shift of the resonances of aromatic protons b1 (Figure S1), b2 through b7 of H8, and protons α and β of OV 2+ (Figure 2a), along with the upfield shift of the signal of proton b8 (Figure 2a). Among the signals of aromatic protons b2 through b8, those of protons b3, b5, b7, and b8 remained well dispersed with increasing proportion of OV 2+ (Figure 2a). With more than one equivalent of OV 2+ , the resonances of aromatic protons b2 through b8 (Figure 2a) show insignificant shifts, while the signals of protons α and β of OV 2+ continue to move upfield, approaching those of free OV 2+ ion with increasing proportions of the guest. Plotting the chemical shifts of protons b3, b5, b7, and b8 of H8 against the ratio of OV 2+ reveal a linear dependence that changes abruptly at one equiv of OV 2+ (Figure 2b), indicating that H8 and OV 2+ bind in a 1:1 ratio that is corroborated by Job's plot (Figure 2c). The observation of only one set of 1 H NMR signals with varying proportions of OV 2+ suggests that the free and bound host and guest undergo rapid exchange on the 1 H NMR time scale. Since the 1 H resonances of H8 exhibit insignificant shift with ≥ 1 equiv of OV 2+ , the equilibrium must have shifted toward the presumable 1:1 complex as the dominant species.

Titrating H16 with OV 2+ led to significant changes in the aromatic and amide region from 5.9 to 10.4 ppm (Figure 3). With < 1 equiv of OV 2+ , the region conatining the resonances of internal aromatic protons b1 through b16 is found to contain many (>>16) signals that indicate the presence of free H16 and the 1:1 complex of H16 and OV 2+ . With 1 equiv of OV 2+ , only the 16 new signals attributed to the 1:1 complex, i.e., those appeared upon titrating H8 with OV 2+ , remain. With more than one but less than two equiv of OV 2+ , the 16 peaks attributed to the 1:1 complex and multiple new peaks are found in this region; with > 2 equiv of OV 2+ , the second set of 16 new peaks remain and the peaks of the 1:1 complex completely disappear, suggesting the presence of only the 1:2 complex of H16 and OV 2+ . In addition, the signals of protons α and β belonging to free OV 2+ at 9.4 and 8.7 ppm are observed with > 2 equiv of OV 2+ . These observations indicate that the binding of H16 with OV 2+ happens stepwise. The formation of the 1:1 complex occurs first as up to 1 equiv of OV 2+ is added, followed by the appearance of the 1:2 complex with ≥ 1 equiv of OV 2+ . The presence of ≥ 2 equiv of OV 2+ completely drives the equilibrium toward the side of the 1:2 complex. In the presence of > 2 equiv of OV 2+ , the unchanged signals of H16 and the simultaneous presence of both the free and bound OV 2+ suggest that H16 strongly binds OV 2+ , with the free and bound OV 2+ undergoing no or slow exchange on the NMR time scale.

N+ N+ CH2-n-C7H15 n-C7H15-CH2 α β µ OV 2+
of H8 and OV 2+ contains two peaks given by the 2:1 complex (H8)2•OV 2+ , along with a third peak corresponding to the 1:1 complex H8•OV 2+ (Figure S2a). The spectrum of the 1:1 mixture of H8 and OV 2+ (Figure S2b) reveals a major peak for H8•OV 2+ , the 1:1 complex, and another peak of (H8)2•OV 2+ , the 2:1 complex. Surprisingly, a peak corresponding to (H8)2•(OV 2+ )2, the 2:2 complex, which cannot be distinguished from the 1:1 complex by NMR, is also observed. In the spectrum of the 1:2 mixture of H8 and OV 2+ (Figure S2c), the ions of the 1:1 complex H8•OV 2+ (dominant), 2:1 complex (H8)2•OV 2+ (much weaker), and 2:2 complex (H8)2•(OV 2+ )2 are detected. In contrast, the 1:2 complex H8•(OV 2+ )2 could not be clearly detected in the spectra of the mixtures. These observations suggest that H8 can bind with OV 2+ in both 1:1 and 2:2 ratios. Complex (H8)2•OV 2+ , which was observed in the mass spectra of all three mixtures, seems to be the intermediate between the 1:1 and 2:2 complexes.

The ESI-QTOF spectrum of the 1:1 mixture of H16 and OV 2+ reveals the presence of only 1:1 complex H16•OV 2+ (Figure S3a). With the proportion of OV 2+ being doubled, the 1:2 mixture of H16 and OV 2+ gives a mass spectrum containing peaks of both the 1:1 complex H16• OV 2+ and 1:2 complex H16•(OV 2+ )2 (Figure S3b), which suggests that increasing the ratio of OV 2+ drives the complexation of the second guest OV 2+ into the cavity of H16. Results from studies based on 1 H NMR titrations and mass spectrometry indicate that the originally expected complexation stoichiometry and processes of H8 and H16 with OV 2+ shown in Figure 1 need to be revised. As shown in Figure 4, the complexation of H8 for OV 2+ involves the initial formation of the 1:1 complex, followed by the binding of the second molecule of H8 to give the 2:1 complex which, by binding the second OV 2+ , yields the 2:2 complex (Figure 4a). The 1:1, 2:1, and 2:2 complexes of H8 and OV 2+ undergo rapid exchange as shown by the observation of only one set of signals throughout the 1 H NMR titration (Figure 2a). The binding of H16 with OV 2+ follows a clear stepwise path, with the 1:1 and 1:2 complexes, and the unbound H16 and OV 2+ undergoing slow exchange (Figure 4b). The 1:1 complex exists as the only host-guest species with up to 1 equiv of OV 2+ , and the 1:2 complex as the only complex with > 2 equiv of OV 2+ .

Binding strength and thermodynamic parameters

The binding of OV 2+ with H8 and H16 was then probed with isothermal titration calorimetry (ITC) which, in addition to determining the association constants (K), also provides the corresponding thermodynamic parameters including changes of enthalpy (ΔH), entropy (ΔS), and free energy (ΔG) as shown in Table 1. In MeOH/CHCl3 (3/7, v/v) at 35 o C, the ITC thermogram (Figure S4a(i)) for the binding of H8 with OV 2+ is consistent with a 1:1 binding ratio, with an association constant over 10 6 M -1 that reflects both the 1:1 and 2:2 binding modes. In the same solvent, the complexation of H16 for OV 2+ gives an association constant (K1) over 10 8 M -1 for the first binding event, followed by that (K2) of the second binding event that is three orders of magnitude smaller than K1 (Figure S4b). The high affinity of the first binding event approaches the upper limit of ITC measurements, leading to K1 with a significant error. In the more polar DMSO/CHCl3 (1/1, v/v) at 45 o C, the binding of H8 with OV 2+ gives an apparent association constant of ~10 5 M -1 (Figure S4(ii)), while the affinities of H16 and OV 2+ are also reduced (Figure S5), with K1 over 10 7 M -1 and K2 over 10 4 M -1 that are in the range allowing accurate ITC measurements.

For the complexation of H16 and OV 2+ , the much stronger first binding step than the second step precluded the determination of the binding parameters with one ITC titration. Instead, in DMSO/CHCl3 (1/1), K1, along with ΔS1 and ΔH1, was obtained by first titrating H16 (50 μM) with OV 2+ •PF6 -(0.5 mM) (Figure S5a); the much smaller K2, along with ΔS2 and ΔH2, was obtained by titrating the 1:1 mixture of H16 (3 mM) and OV 2+ •PF6 -(3 mM) with OV 2+ •PF6 -(30 mM) (Figure S5b). The binding of H16 for OV 2+ , with its second binding event being three orders of magnitude weaker than the first one, gives interaction factors α 47 of 0.005 in MeOH/CHCl3 (3/7) and 0.002 in DMSO/CHCl3 (1/1). Such remarkable negative cooperativity 47 reflects the unfavorable stuffing of the second OV 2+ guest into the cavity of H16. These observations demonstrate that the inner cavity of H16 offers a highly electronegative environment that not only overcomes the repulsion between the two cationic guests but also provides additional driving force for the formation of the 1:2 complex.

In MeOH/CHCl3 (3/7), the first binding event of H16 and OV 2+ is entropically driven, which reflects the desolvation of, i.e., the release of methanol molecules from, the cavity of H16 upon binding the first OV 2+ . The second binding event is enthalpically driven, due to the electrostatic attraction that drives the binding of the second OV 2+ into the desolvated cavity. In contrast, in DMSO/CHCl3 (1/1), both first and second binding events of H16 and OV 2+ are enthalpically driven. As an aprotic solvent, DMSO is not able to effectively solvate the electronegative cavity of H16. The poorly solvated cavity of H16, with multiple amide carbonyl groups as preorganized binding sites, is amenable to accommodating the cationic guest via attractive electrostatic interaction.

The dominant role played by electrostatic interaction is verified by the binding of rodlike guest TB 2+ which, like guest OV 2+ , is rigid and carries two positive charges. ITC shows that (Figure S6), in DMSO/CHCl3 (1/1), the 1:1 binding of H16 and TB 2+ , with an association constant (K) over 10 7 M -1 that is the same as that of binding the first OV 2+ with H16, is driven predominantly by a favorable (negative) enthalpy change that results from the strong electrostatic interaction between the two positive charges of guest TB 2+ and the negative cavity of H16 (Table 1). Unlike the pairwise binding of OV 2+ , the binding of a second TB 2+ ion with H16 was not observed. The bulkiness of the two trimethylammonium groups and the small aromatic surface of TB 2+ are the most likely reasons that hinder the cramming of two guests TB 2+ inside the cavity of the H16. Thus, by performing structural tuning on the guest, the binding stoichiometry involving host H16 can be adjusted and controlled.

Crystal structures of complexes (H8)2•(OV 2+ )2 and H16•(OV 2+ )2

Single crystals of the complexes of OV 2+ with H8 and H16 were obtained via liquid-liquid diffusion of methanol into a dichloromethane solution in an NMR tube, which revealed the existence of 2:2 complex (H8)2•(OV 2+ )2 and 1:2 complex H16•(OV 2+ )2 in the solid state. Similar to that of OV 2+ with H8, the binding of OV 2+ into the cavity of H16 is driven by C-H•••O interactions involving 14 of the 16 aromatic CH groups of the OV 2+ ions and the amide carbonyls of H16, with an average H•••O distance of 2.39 Å that is shorter than that (2.45 Å) in complex (H8)2•(OV 2+ )2 (Figure 5b, top). N + •••O distances of 2.97, 3.10, 3.12, and 4.30 Å, which are overall shorter than those in complex (H8)2•(OV 2+ )2, are found between each of the pyridinium N + atoms and its nearest amide carbonyl oxygen of H16, indicative of charge-dipole interactions that are stronger than those in complex (H8)2•(OV 2+ )2. Unlike the offset alignment of the bipyridinium segments of the two OV 2+ ions in the cavity of double helix (H8)2, the bipyridinium segments in the cavity of H16 slide much less along their long axes and show a larger extent of overlap. The two aligned cationic rods have their four aromatic rings being partially stacking with an average stacking distance of 4.0 Å, indicating weak but noticeable stacking interaction (Figure 5b, bottom). To avoid the N + atoms directly facing one another, the bipyridinium segments cross over one another, with an angle of ~42 o between their long axes. The N + •••N + distances of 4. 31 and 4.43 Å between the ends of the two bipyridinium units, are much shorter than those than in complex (H8)2•(OV 2+ )2.

The crystal structures of complexes (H8)2•(OV 2+ )2 and H16•(OV 2+ )2 reveal that the two guests OV 2+ in each complex adjust their alignment to optimize their interaction with each host. Compared to that in complexes (H8)2•(OV 2+ )2, the cationic segments of OV 2+ ions in complex H16•(OV 2+ )2, being confined in the more compact cavity of H16, have their ends being placed more closely as shown by shorter N + •••N + distances, leading to a quadruply charged, shorter dimer that engages in stronger interactions with the host as shown by the shorter C-H•••O and N + •••O distances. Additional insights into this unique hostguest system are gained by comparing the crystal structure of H16 alone 45 with that in complex H16•(OV 2+ )2, which indicates that the helical host fine-tunes its folded structure upon binding OV 2+ . Compared to the ~9-Å diameter of cavity of helix H16 alone, 45 the cavity diameter of helix H16 in complex H16•(OV 2+ )2 increases to over 9.5 Å, presumably to better match the size of the bound OV 2+ dimer. Such host-guest mutual adaption is made possible by the self-assembling nature of the dimeric guest and the limited conformational flexibility of the host, leading to enhanced host-guest interaction shown by the high binding affinities of H16 for OV 2+ and TB 2+ .

Thus, by overcoming the otherwise strong electrostatic repulsion between guests OV 2+ , the electronegative cavities of double helix (H8)2 and helix H16 are able to bind a pair of guest OV 2+ in high affinity. The pairwise binding of the dicationic guests shown by H8 and H16 is precedented by few known hostguest systems. In fact, few hosts are known to allow two or more viologen-based guests, i.e., four or more positive charges, to be placed in close proximity in the same cavity. Cucurbit [8]uril (CB [8]), although having a cavity large enough to accommodate two molecules or ions with sizes comparable to that of the bipyridinium segment of viologens, was reported to form a 1:1 complex with methyl viologen (MV 2+ ). The complexation of one CB [8] for two MV 2+ ions and similar guests has not been realized, presumably because of the strong electrostatic repulsion between the two dicationic guests in the cavity. 48 To the best of our knowledge, the only known 1:2 complexes having a dimer of a viologen-based guest residing in the cavity of a host were reported by Chen et al. 49 With the binding affinities being presented as "average association constants" (Kav) of ~10 3 M -1 in CD3CN/CDCl3 (1/1, v/v), these 1:2 complexes involve a host based on a triptycene-based macrotricycle that provides a partially rigid binding cavity deeper than those of common macrocycles. The guests are viologens carrying β-hydroxyethyl or γ-hydroxypropyl end chains. In this system, the terminal hydroxyl groups of these guests seemed to play a critical role in driving the formation of the 1:2 complex since other guests based on dialkyl viologens could only form 1:1 complexes with this host.

Solution structures of the complexes probed with twodimensional NMR spectroscopy

As revealed by 1D 1 H NMR spectroscopy, ITC measurements, and X-ray crystallography, the binding of guest OV 2+ with oligoamides H8 and H16 leads to 2:2 complex(H8)2•(OV 2+ )2 and 1:2 complex H16•(OV 2+ )2, respectively. The solution structures of these complexes were probed with two-dimensional (NOESY) 1 H NMR experiments.

Figure 6 shows the partial NOESY spectra of H8 and OV 2+ (1:1) recorded in DMSO-d6/CDCl3 (7/3, v/v) at 45 o C. Strong NOEs between aromatic protons b2 through b8 of H8 and aromatic protons α and β (Figure 6a), and methylene protons μ (Figure 6b) of OV 2+ , are revealed. NOEs between protons b1 and α, β, and μ are also detected (Figure S7). In contrast, no NOE is observed between protons α, β, or μ of OV 2+ and aromatic protons c1-c8, i.e., the "exterior" protons of H8. These observations demonstrate that the OV 2+ ions reside in the cavity of the double helix. An NOE between protons d of of H8 and proton b2 confirms that H8 remain folded when binding guests OV 2+ (Figure 6c). Another NOE involving protons d and β is also noticed (Figure 6c), while no NOE can be found between protons d and α, which suggests that, the two molecules of H8 have their C-termini placed in the interior and their N-termini located at the two ends of double helix (H8)2. This observation is consistent with what is revealed by the crystal structure of complex (H8)2•(OV 2+ )2, i.e., in solution, the two molecules of H8 constitute a symmetrical double helix in which the C-terminal residues are placed near each other (Figure 6d).

The backbone of H16 follows an N-to-C direction, leading to an unsymmetrical cavity for helix H16. Upon binding with H16, the two groups of otherwise equivalent aromatic protons of OV 2+ give four discrete 1 H NMR peaks, suggesting that the two OV 2+ ions in the cavity of H16 are desymmetrized. Likewise, protons μ, i.e., those of the methylene groups directly attached to the N + atoms of OV 2+ , also give multiple signals. Assisted by COSY and NOESY spectra, the aromatic protons of OV 2+ corresponding to the four 1 H NMR signals are labeled as α, β, γ, and δ. Four 1 H resonances can be attributed to methylene protons μ of OV 2+ , which is due to the desymmetrization of the bound OV 2+ ions and the methylene protons being diastereotopic in a helical cavity. The methylene protons corresponding to these four resonances are labeled as μ1, μ1', μ2, and μ2'. The two halves of the deysmmetrized OV 2+ ion are defined by protons α, β, and μ1/μ1', and protons γ, δ and μ2/μ2', respectively.

The NOESY spectrum of H16 and OV 2+ (1:2) recorded in DMSO-d6/CDCl3 (4/6, v/v) at 45 o C reveals different NOEs between aromatic protons b1 (Figure S8), b2 through b16 of H16 and protons α, δ (Figure 7a), β, γ (Figure 7b), and μ1/μ1', μ2/μ2' (Figure 7c) of OV 2+ . Strong NOEs are found between protons α and protons b2, b3, b4, b5, b6, and b7 (Figure 7a) and b1 (Figure S8) that belong to the N-terminal half of H16. The intensities of NOEs involving protons α and protons b8 to b14 follow a descending order, with no NOEs being detected between protons α and protons b15 or b16. Compared to protons α, protons δ exhibit the opposite trend in the strength of their NOEs with aromatic protons b1-b16 of H16 (Figure 7a). Strong NOEs are found between protons δ and protons b9 through b16 that belong to the C-terminal half of H16, with NOEs involving protons δ decreasing consecutively from protons b8 to b3, and disappearing with protons b2 (Figure 7a) and b1 (Figure S8). The NOEs observed between protons α and δ of OV 2+ , and protons b1 through b16, along with the absence of NOE between the aromatic protons of OV 2+ and protons c1-c16 of H16, suggesting that, similar to what is revealed by the crystal structure of complex H16•(OV 2+ )2, in solution, the two OV 2+ ions are aligned along the long axis of the cylindrical cavity of helix H16.

In addition to NOEs between the aromatic protons of guests OV 2+ and those of H16, methylene protons μ also show NOE contacts that corroborate the solution structure of complex H16•(OV 2+ )2. Protons μ1/μ1' have obvious NOE contacts with protons b1 (Figure S8) and b2-b7 (Figure 7c), while NOEs between protons μ2/μ2' and protons b10-b16 are noticeable (Figure 7c), which indicate that protons μ1/μ1' and μ2/μ2' are placed near the N-and C-termini of H16, respectively.

Therefore, the two ends of the OV 2+ ion, as represented by protons α and μ1/μ1', and δ and μ2/μ2', are placed near the Nand C-termini, respectively, of H16. In such a complex, protons β and γ of OV 2+ must be placed near the middle of helix H16. Indeed, protons β and γ have NOE contacts of similar strength with most internal aromatic protons b of H16 (Figure 7b). The absence of NOE between protons β and protons b15 and b16, and between protons γ and protons b2 and b16 (Figure 7b) provides additional evidence supporting the alignment of the OV 2+ ions in the cavity of helix H16 in solution (Figure 7d).

NOESY spectrum clearly demonstrates the binding of H16 and guest TB 2+ (Figure S9). Strong NOEs between protons b1 through b16 of H16, and aromatic protons α and methylene protons μ of TB 2+ , are detected, confirming that the cationic guest resides in the cavity of H16.

Conclusions

Aromatic oligoamides H8 and H16 fold into hollow helices with non-deformable electronegative cavities that strongly bind rodlike dicationic guests OV 2+ and TB 2+ . The 8-residue H8, a helix of ~1.2 turns, assembles into double helix (H8)2 with a cavity capable of accommodating two OV 2+ ions that align in an offset fashion. The resultant 2:2 complex, (H8)2•(OV 2+ )2 undergoes rapid exchange with the unbound host and guest. The 16residue H16, which can be regarded as a covalent dimer of H8, serve as a unimolecular host with a 3D cavity that strongly binds guest OV 2+ . The high stabilities of the resultant 1:1 complex H16•OV 2+ and 1:2 complex H16•(OV 2+ )2 are reflected by the lack of exchange between the complexes and with the unbound host and guest. The strong binding of H16 for OV 2+ is confirmed by ITC. The extraordinary affinity of H16 for the first OV 2+ ion and much weaker yet significant binding for the second OV 2+ ion point to highly negative cooperativity. In contrast, helix H16 binds only one bulky guest TB 2+ with the same affinity as that of H16 with the first OV 2+ ion, indicating that binding stoichiometry can be adjusted by tuning the structure (bulkiness) of the guest.

X-ray crystallography provides atomic details for complexes (H8)2•(OV 2+ )2 and H16•(OV 2+ )2, which reveals the mutual adaption of the helical hosts and the dimeric guest. Double helix (H8)2 provides a cavity that fully accommodates the two OV 2+ ions. The crystal structure of complex H16•(OV 2+ )2 reveals that the two bound OV 2+ ions undergo more compact alignment than those in (H8)2•(OV 2+ )2, perhaps to better match the cavity of helix H16. In comparison to that of helix H16 alone, the cavity of H16 in complex H16•(OV 2+ )2 is slightly enlarged to better accommodate the two OV 2+ ions.

In solution, the binding of OV 2+ ions in the cavity of duplex (H8)2 or helix H16, the head-on alignment of H8 in duplex (H8)2, and the desymmetrization of the otherwise symmetrical guest OV 2+ in the cavity of H16 are clearly demonstrated by two-dimensional (NOESY) 1 H NMR spectra. The binding of H16 and guest TB 2+ in solution is also been confirmed by NOESY spectrum.

Complexes (H8)2•(OV 2+ )2, H16•(OV 2+ )2 and H16•TB 2+ are single and double helical pseudofoldaxanes featuring two and one axle components, respectively, with few known precedents. The presence of discrete complex H16•(OV 2+ )2 demonstrates that hollow helices, as high-affinity hosts with 3D binding pockets, lead to host-guest complexes with significantly enhanced stability. The resultant pseudofoldaxanes, with foldaxane-like stability, are formed by simple mixing of the molecular components. By being able to tailor the size (length) and/or shape of guests, hollow helices, with their ready synthetic tunability, are new hosts that are uniquely different from hosts based on most macrocycles, resulting in host-guest complexes with adjustable sizes, stability, and binding stoichiometry.

  University at Buffalo, The State University of New York Buffalo, New York 14260, USA. E-mail: bgong@buffalo.edu b.

40

 40

Figure 1 .

 1 Figure 1. (a) Structures of aromatic oligoamide Hn, macrocycle 1, guests OV 2+ and TB 2+ .Cartoons on the right illustrate the helical, cyclic, and rodlike shapes of H8, H16, 1, and the two cationic guests. (b) Illustrations of the observed 2:1 complex between 1 and OV 2+ , the anticiapted 2:1 complex between H8 and OV 2+ , and the anticipated 1:1 complex between H16 and OV 2+ or TB 2+ .

  (CH 2 ) 2 O(CH 2 ) 2 CH(CH 3 ) 2 1a: R = -(CH 2 ) 2 O(CH 2 ) 2 CH(CH 3 ) 2 1b: R = -(CH 2 CH 2 O) 3

Figure 2 .

 2 Figure 2. Oligoamide H8 titrated with 0 to 2 equiv of guest OV 2+ •(PF6 -)2 in DMSO-d6/CDCl3 (3/7, v/v) at 45 o C. (a) 1 H NMR spectra (400 MHz), (b) Changes in chemical shifts (δH) of aromatic protons b3, b5, b7, and b8 of H8 vs the equiv of OV 2+ •(PF6 - )2 and, (c) Job's plot based on the chemical shifts of proton b8 of H8 in the presence of different ratio of OV 2+ •(PF6 -)2.

Figure 3 .

 3 Figure 3. 1 H NMR spectra of H16 titrated with 0 to 3 equiv of OV 2+ •(PF6 -)2 in DMSO-d6/CDCl3 (2/3, v/v) at 45 o C.

Figure 4 .

 4 Figure 4. Schematic illustrations of the complexation processes of (a) oligoamide H8 and (b) oligoamide H16 with guest OV 2+ .

a

  Interaction factor α = 4K2/K1. (α > 1: positive cooperativity; α < 1: negative cooperativity; α = 1 no cooperativity). b Ktotal = K1  K2.

  The structure of complex (H8)2•(OV 2+ )2 shows that the two molecules of H8 form a double helix in which the two helical strands pair in an antiparallel orientation, i.e., the N end of one strand aligns with the C end of the other strand, with the C ends of the two H8 molecules being placed in the middle of the double helix. The bipyridinium segments of the two OV 2+ ions bind into the cavity of the double helix (Figure 5a, top) and engage in C-H•••O interactions involving 14 of the 16 aromatic C-H groups of the two OV 2+ ions and the amide carbonyls of H8, with an average H•••O distance of 2.45 Å. N + •••O distances of 3.02, 3.20, 3.65, and 3.98 Å are found between each of the pyridinium N + atoms and its nearest amide carbonyl oxygen of H8, indicative of strong charge-dipole interactions. To reduce the repulsion between the two OV 2+ ions, the two bipyridinium segments in the cavity of the double helix align in an offset way, with their long axes crossing over one another with an angle of ~27 o and the N + •••N + distances between the ends of the two bipyridinium units being 5.52 Å and 5.98 Å (Figure 5a, bottom). Two of the four pyridinium rings of the OV 2+ ions face each other with an average distance of ~4.3 Å, indicative of very weak, if any, stacking interaction.

Figure 5 .

 5 Figure 5. Crystal structures of (a) complex (H8)2•(OV 2+ )2 (top), along with the two OV 2+ ions in the cavity of double helix of H8 (bottom) and, (b) complex H16•(OV 2+ )2 (top), along with the two OV 2+ ions in the cavity of helix H16 (bottom). The N + •••N + distances between the termini of the two OV 2+ ions in each complex are highlighted with red dashed lines.

Figure 6 .

 6 Figure 6. Partial NOESY spectra (500 MHz, mix time: 500 ms) of H8 (5 mM) and OV 2+ •(PF6 - )2 (5 mM) in DMSO-d6/CDCl3 (7/3, v/v) recorded at 45 o C reveal NOEs between (a) protons α and β of OV 2+ and protons b2 through b8 of H8, (b) protons μ of OV 2+ and protons b2 through b8 of H8, and (c) protons d and b2, and protons d and β. (d) Illustration of the 2:2 complex of H8 and OV 2+ , with the major NOEs between the internal aromatic protons b1 through b8 of H8 and protons α (shown in yellow) and β (shown in green) of OV 2+ being indicated with double-headed arrows.

Figure 7 .

 7 Figure 7. Partial NOESY spectra (500 MHz, mixing time: 500 ms) of H16 (3 mM) and OV 2+ •(PF6 -)2 (6 mM) recorded in DMSO-d6:CDCl3 (4/6, v/v) at 45 o C reveal NOEs between protons b2 through b16 of H16 and protons (a) α and δ, (b) β or γ, and (c) μ1/μ1' and μ2/μ2' of OV 2+ . (d) Illustration of the 1:2 complex of H16 and OV 2+ , with the major NOEs between the internal aromatic protons b1 through b16 of H16 and protons α and δ of OV 2+ being indicated with double-headed arrows.

Table 1 .

 1 Association constants, thermodynamic parameters and interaction factors (α), for the complexation of hosts H8 and H16 toward guests OV 2+ and TB 2+ obtained from ITC experiments.

	Host	Guest	Solvent	Temp ( o C)	-ΔH (kcal/mol)	TΔS (kcal/mol)	K (or K1, K2) (M -1 )	Ktotal b (M -2 )	α a
	H8	OV 2+	MeOH/CHCl3 (3/7, v/v)	35	8.2 ± 0.1	1.3 ± 0.1	(5.5 ± 1.0) x 10 6	-	-
	H16	OV 2+	MeOH/CHCl3 (3/7, v/v)	35	2.7 ± 0.1 (ΔH1) 6.1 ± 0.1 (ΔH2)	9.4 ± 0.4 (ΔS1) 1.9 ± 0.1 (ΔS2)	(3.6 ± 2.0) x 10 8 (K1) (4.6 ± 0.3) x 10 5 (K2)	(1.7 ± 1.1) x 10 14	0.005

Acknowledgements

The acknowledgements come at the end of an article after the conclusions and before the notes and references.

Author Contributions

Y. L. Z. and T. A. S. performed the major part of the experimental work including NMR data collection, ITC measurements, along with data analysis. Y. L. Z., B. S. and X. P. L. performed mass spectral studies. B. K. and Y. F. performed crystal growth, X-ray data collection, crystal structure elucidation and refinement. B. G. conceptualised and supervised the study. I. H. contributed to X-ray study and helped revise the manuscript. B. G. and Y. L. Z. wrote the manuscript. All authors commented the data and reviewed the manuscript.

Conflicts of interest

There are no conflicts to declare.


Jonathan Jung
email: jonathan.jung@univ-pau.fr

Ibtissem Lannabi
email: ibtissem.lannabi@univ-pau.fr

Vincent Perrier
email: vincent.perrier@inria.fr

This paper deals with the numerical resolution of a linear wave system using the Godunov scheme with a centered discretization of the pressure gradient. The interest in such schemes is motivated by the low Mach number accuracy problem. We have shown that for both steady and unsteady flows, an oscillatory mode appears in the numerical solution. This can be explained by the loss of the Total Variation Diminishing property on the characteristic variables. Moreover, we have illustrated numerically that the long time numerical solution does not converge to the expected steady state. In addition, the existence of the oscillatory mode in the numerical solution jeopardizes the mesh convergence rate of the scheme.

On the convergence of Godunov scheme with a centered discretization of the pressure gradient

Introduction

It is well-known that explicit Godunov-type schemes for solving the Euler equations fail, in general, to be accurate at low Mach number [START_REF] Guillard | On the behaviour of upwind schemes in the low Mach number limit[END_REF], in particular, on Cartesian grids. The most common argument to explain this failure is based on an asymptotic analysis of both the continuous equations and the numerical scheme. Initially, the low Mach number accuracy problem was explained via a single-scale asymptotic analysis that takes into account only the convective waves [START_REF] Guillard | On the behaviour of upwind schemes in the low Mach number limit[END_REF]. Afterwards, both acoustic and convective waves were taken into account by performing a two-scale asymptotic analysis [START_REF] Müller | Low-Mach-number asymptotics of the Navier-Stokes equations. Floating, Flowing, Flying: Pieter J. Zandbergen's Life as Innovator[END_REF][START_REF] Jung | Steady low Mach number flows: identification of the spurious mode and filtering method[END_REF][START_REF] Bruel | A low Mach correction able to deal with low Mach acoustics[END_REF], which led to a coupling of the zeroth-order momentum and the first-order pressure through a wave system. A natural question that arises when considering a discretization of a Cauchy-problem for a wave system which has a long-time limit, is whether a discretization of this system also has a long-time limit and whether this long-time limit is consistent with that of the continuous system [START_REF] Jung | Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems[END_REF]. In [START_REF] Jung | Steady low Mach number flows: identification of the spurious mode and filtering method[END_REF], a link was elaborated between the low Mach number accuracy problem and the long-time consistency problem of a discrete wave system: a Godunov-type scheme is low Mach number accurate if and only if the discrete linear wave system derived from the two-scale asymptotic analysis is long-time limit consistent.

To ensure accuracy at low Mach number for the numerical resolution of the Euler equations on Cartesian grids, several fixes have been proposed. In this paper, we are interested in fixes of type [START_REF] Dellacherie | Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number[END_REF][START_REF] Rieper | A low-Mach number fix for Roe's approximate Riemann solver[END_REF]. These schemes are also known to be two-scale asymptotic consistent with a discretization of the wave system, in which a centered discretization of the pressure gradient is used [START_REF] Bruel | A low Mach correction able to deal with low Mach acoustics[END_REF]. These schemes will be called pressure-centered type schemes. Since these fixes may introduce other issues, for instance, the appearance of an oscillatory mode (or checkerboard mode) in the numerical solution, it is interesting to address the study of the long-time limit of the wave system. Consider the linear wave system

# B τ p p `1 p ρ ∇ x ¨p u " 0, B τ p u `p κ∇ x p p " 0, (1)
solved on a domain Ω, associated with the initial condition p ppτ " 0, xq " p p 0 pxq and p upτ " 0, xq T " p u 0 pxq T , where p p denotes the pressure, p u denotes the velocity field, p κ, p ρ ą 0 and p c " a p κ{p ρ is the sound velocity. Let T h " Ť i Ω i denotes a mesh of the domain Ω, on which a cell-centered finite volume discretization is applied. We denote by |Ω i | the volume of the cell Ω i , by |Γ ij | the size of the side Γ ij linking the cell Ω i to the cell Ω j , by n ij the unit normal outgoing from the cell Ω i and going to the cell Ω j , by V int piq the set of the cells sharing a face with the cell Ω i , by V b piq the set of the boundary sides of Ω i . We consider the numerical scheme

|Ω i | ˜p U n`1 i ´p U n i δτ ¸`ÿ jPVintpiq |Γ ij |F θ ´p U n i , p U n j , n ij ÿ jPV b piq |Γ ij |F b ´p U n i , p U b , n ij ¯" 0, (2)
where δτ denotes the time-step, p U n i " pp p n i , p u n i q T is the vector of the discretized variables, p U b " pp p b , p u b q T is a state weakly imposed on the boundary of the domain Ω, F θ ´p U n i , p U n j , n ij ¯is the numerical flux inside the domain and is given by

F θ ´p U n i , p U n j , n ij ¯" ¨1 p ρ p u n i `p u n j 2 ¨nij `p c 2 `p p n i ´p p n j p κ p p n i `p p n j 2 n ij `θ p c 2 ``p u n i ´p u n j ˘¨n ij ˘nij ‹ ', (3)
and F b ´p U n i , p U b , n ij ¯is the numerical flux on the boundary of Ω. The parameter θ P t0, 1u is introduced in order to consider both the Godunov (θ " 1) and the pressure-centered (θ " 0) schemes. Note that (2) with (3) is L 2 stable for θ " 1 with CFL 1 and for θ " 0 with CFL 0.5, see [START_REF] Dellacherie | Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number[END_REF]. In section 1, periodic boundary conditions will be considered, so V b piq " ∅ for all i. In section 2, wall and Steger-Warming boundary conditions will be considered. We denote by V wall piq{V SW piq the set of the boundary sides of Ω i on which wall/Steger-Warming boundary conditions are imposed:

V b piq " V wall piq Y V SW piq. The flux F b ´p U i , p U b , n ij ¯is then given by F b ´p U i , p U b , n ij ¯" $ ' & ' % ˆ0 p κp p i n ij `p c pp u i ¨nij q n ij ˙, if j P V wall piq, F θ"1 ´p U i , p U b , n ij ¯, if j P V SW piq. (4)
In section 1, we show that the Total Variation Diminishing (TVD) property is lost with the pressure-centered scheme.

In section 2, a problem in which the continuous wave system has a long-time limit is addressed. We show that the long-time limit of the numerical solution contains an oscillatory mode, which we are able to isolate. Lastly, we show that the long-time limit consistency is lost.

Loss of the TVD property

In this section, we are interested in the one-dimensional first-order wave system (1), solved on an infinite domain Ω, using a finite volume scheme of the form ( 2)-( 3). Let us recall that the TV norm of v " pv i q iPN is given by

TVpvq " ř j |v j`
TV ´p C n`1 ¯¯ď TV ´p C n ¯¯, @ p C n ¯.
• The explicit pressure-centered scheme (2)-(3) with θ " 0 is not TVD, i.e, for all δτ P s0, δx{p2p cqr , there exists an initial condition `p p 0 , p u 0 ˘for which

TV ´p C 1 ¯¯ą TV ´p C 0 ¯¯.
Proof. Regarding the conservation of the TVD property for the explicit Godunov scheme, it follows immediately since the Godunov scheme written in terms of the characteristic variables is an upwind scheme. We refer to [START_REF] Dellacherie | Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system[END_REF], for a detailed proof.

Regarding the loss of the TVD property for the explicit pressure-centered scheme, it is obtained by considering the following initial condition: p p 0 {2 " p u 0 {p2p ρp cq where p u 0 will be taken such that it has a discontinuity at some point. For example, we take the discrete initial condition defined by p u 0 j " ´1 for j ą k and p u 0 j " 1 for j ď k. We thus obtain

TV ´p C 1 ´¯" p cδt 2δx ÿ j ˇˇˇˇp u 0 j`2 ´3p u 0 j`1 `3p u 0 j ´p u 0 j´1 2p ρp c ˇˇˇˇ" δt p ρδx ˇˇp u 0 k ´p u 0 k`1 ˇ" 2δt p ρδx ą 0 " TV ´p C 0 ´¯,
which ends the proof.

In order to numerically illustrate the loss of the TVD property we consider the one dimensional Riemann test case for unsteady flows, associated with the first-order wave system (1). The initial condition considered for the selected test case is the following

p p 0 L " p u 0 L " 1, p p 0 R " p u 0 R " ´1, (5)
with final time equal to 0.1, p c " p ρ " 1 and a CFL number equal to 0.45. With the chosen initial condition, we remark that the characteristic variable p C ´is initialized as zero. In Figure 1, we observe that with the Godunov scheme, the uniformity of p C ´is preserved over time since the Godunov scheme is TVD on the characteristic variables. However, with the pressure-centered scheme, the uniformity of this characteristic variable is not preserved. Moreover, oscillations appear on the characteristic variables, as well as on the pressure and velocity. Note that the same phenomenon is observed when considering an initial uniform characteristic variable p C `. In Figure 2, the plot of the TV norms is presented. It highlights that the TV norms of the two characteristic variables are preserved for the Godunov scheme but increase for the pressure-centered scheme. Similar results were obtained by considering an implicit time integration for both the Godunov and the pressure-centered schemes, but these results are not presented in this paper. 2 Loss of mesh convergence of the numerical solution in the long time limit

In this section, we are interested in the linear wave system (1), solved on a bounded domain Ω, with wall and Steger-Warming boundary conditions. In this framework, the long-time limit of the solution of both the continuous wave system (1) and the numerical schemes ( 2)-( 3)-( 4) with θ P t0, 1u exists (see [START_REF] Jung | Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems[END_REF]). This limit can be identified thanks to a Hodge-Helmholtz Decomposition (HHD). Indeed, at the continuous level, by considering a vector field p u P L 2 pΩq defined on a bounded domain Ω with a smooth boundary BΩ, p u can be uniquely decomposed into a sum of an irrotational component (which is a gradient of a scalar potential function), and a divergence free one (which is a curl of a vector potential function). The proof for this decomposition in the case where p u satisfies wall/Steger-Warming boundary conditions can be found in [START_REF] Jung | Steady low Mach number flows: identification of the spurious mode and filtering method[END_REF] and is based on solving a Poisson equation with Neumann boundary conditions. At the discrete level, a similar exact decomposition is obtained on triangular grids in [START_REF] Jung | Steady low Mach number flows: identification of the spurious mode and filtering method[END_REF], see also [START_REF] Arnold | A uniformly accurate finite element method for the Reissner-Mindlin plate[END_REF][START_REF] Dellacherie | The influence of cell geometry on the Godunov scheme applied to the linear wave equation[END_REF][START_REF] Dellacherie | Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system[END_REF][START_REF] Dellacherie | Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity[END_REF] for other boundary conditions. By considering a piecewise constant vector field p u h , its discrete HHD is given by p u h " P ψ rp u h s `Pϕ rp u h s, where P ψ rp u h s is the curl of a continuous P 1 potential vector, P ϕ rp u h s is the gradient of a Crouzeix-Raviart scalar potential and P ψ rp u h s ¨n " p u b ¨n on the boundary (see [START_REF] Jung | Steady low Mach number flows: identification of the spurious mode and filtering method[END_REF]Proposition 5], for details about the proof). Back to the identification of the long-time limit, at the continuous level, it is identified as the uniform pressure p p b and the divergence free component of the HHD of the initial velocity p u 0 . On triangular meshes, with the Godunov scheme (( 2)-( 3)-( 4) with θ " 1), the long-time limit of p p h and p u h is consistent with the one obtained at the continuous level: it corresponds to p p b and P ψ rp u 0 h s. Since the long-time limit of the solution of the pressure-centered scheme ((2) with θ " 0) has not been identified yet, our objective in this section is to characterize it numerically. Unfortunately, the Cartesian grid case cannot be addressed due to the non existence of a discrete HHD on such a grid (at least adapted to such boundary conditions).

The test case selected for this study is the scattering of a first-order wave system by a cylinder. The computational domain is an annulus rr 0 , r 1 s ˆr0, 2πr, where r 0 " 0.5 and r 1 " 5.5. The boundary conditions considered for this test are: a wall boundary condition on the internal circle of radius r 0 and a Steger-Warming boundary condition on the external circle of radius r 1 with p p b " 0 and p u b " p1, 0q T . The initial condition is set to p p 0 " 0 and p u 0 " p0, 0q T . It is worth emphasizing that the exact steady solution is known [START_REF] Jung | Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems[END_REF] and is given by p p exact pr, θq " 0 and

p u exact pr, θq " r 2 1 r 2 1 ´r2 0 ˆ1 ´r2 0 r 2 cosp2θq, ´r2 0 r 2 sinp2θq
˙T . The mesh type considered is an unstructured triangular mesh generated using gmsh, with a characteristic length of 2πr 0 {4N on the internal circle and 2πr 1 {4N on the external circle. The mesh selected for the next figures is produced with N " 40 and contains 1332 triangular cells. The test is run with a CFL number equal to 0.45 and

p ρ " p κ " 1.
In what follows, numerical results presented correspond to a solution for which the pressure and velocity residuals have converged with a numerical precision of 10 ´14 .

In Figure 3, the iso-contours of the L 2 norm of the long-time limit of the exact and numerical velocity fields are plotted. We observe that the numerical velocity field computed using the Godunov scheme is close to the exact velocity field, while the one obtained with the pressure-centered scheme is also close but polluted by spurious oscillations. Using the long-time consistency result proved in [START_REF] Jung | Steady low Mach number flows: identification of the spurious mode and filtering method[END_REF][START_REF] Jung | Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems[END_REF] and recalled above, the long-time limit of the numerical velocity field p u 8 h computed using the Godunov scheme is P ψ rp u 0 h s, the divergence free component of the discrete HHD of the initial velocity 1 p u 0 h . If p u 8 h is the long-time limit of the numerical velocity field, we define

p u 8,Spurious h " p u 8 h ´Pψ rp u 0 h s. (6)
The difference ( 6) is plotted in Figure 4 for the Godunov and the pressure-centered schemes. With the Godunov scheme, the difference is zero, as expected, while with the pressure-centered scheme, ( 6) is a spurious oscillatory mode and its scale is of order 10 ´1, which is not negligible compared to the magnitude of the velocity field on the boundary which is of magnitude 1. In order to study the long-time consistency of the scheme, a grid convergence study is performed. To do so, unstructured triangular meshes obtained with N P t12, 20, 40, 80, 160, 320u containing 96, 332, 1332, 5154, 20546, 83810 cells respectively, are considered. In Figure 5, the L 2 norm of the error between the exact solution and the long-time numerical solution computed using the Godunov and the pressure-centered schemes is plotted. We observe that with both schemes, the long-time limit of the pressure matches a uniform pressure, which is consistent with the long-time limit of the continuous pressure, as explained in [START_REF] Jung | Long time behavior of finite volume discretization of symmetrizable linear hyperbolic systems[END_REF]. However, regarding the long-time velocity field, the order of convergence is 1 with the Godunov scheme, while the pressure-centered scheme does not converge. This provides numerical evidence that the presence of the spurious mode jeopardizes the long-time limit consistency for the velocity. h is equal to 0 for this test case (note that P ψ rp u 0 h s is not equal to 0 due to non homogeneous boundary conditions)

Conclusion

In this paper, the study of the numerical resolution of a first-order wave system using the pressure-centered scheme and explicit time integration is addressed. We have shown that, with this type of scheme,

• the Total Variation Diminishing property is lost.

• Oscillations appear in the unsteady numerical solution.

• The long-time numerical solution contains a spurious mode in the velocity field, which is an oscillatory mode.

• The mesh convergence is compromised due to the existence of this spurious mode.

Further study needs to be conducted for the pressure-centered scheme with implicit time integration. Moreover, the study was done only for triangular cell geometries, thus it needs to be extended to the quadrangular case.

p 2 ´p u 2p ρp c and p p 2

 2 1 ´vj |. In what follows, the characteristic variables p `p u 2p ρp c are denoted by p C ´and p C `, respectively. The following proposition is presented. Proposition 1. (Total Variation Diminishing property) • The explicit Godunov scheme (2)-(3) with θ " 1 is TVD on the characteristic variables p C ´and p C `under the CFL condition 0 ď p cδτ {δx ď 1, i.e

Figure 1 :

 1 Figure 1: One-dimensional Riemann problem: characteristic variables at t " 0.1.

Figure 2 :

 2 Figure 2: One-dimensional Riemann problem: TV norm of the characteristic variables as a function of time iterations.

Figure 3 :

 3 Figure 3: Numerical velocity field computed using the Godunov and the pressure-centered schemes: Exact (left), Godunov (middle), Pressure-centered (right).

Figure 4 :

 4 Figure 4: Difference (6) computed using the Godunov and the pressure-centered schemes: Godunov (left), pressurecentered (right).

Figure 5 :

 5 Figure 5: L 2 of the error between the exact solution and the long-time numerical solution computed using the Godunov and the pressure-centered schemes as a function of the mesh step.