Ana Gabriela Porras Gutiérrez

Joceline Zeitouny

Antoine Gomila

Bénédicte Douziech

Nathalie Cosquer

Françoise Conan

Olivia Reinaud

Philippe Hapiot

Yves Le Mest

Corinne Lagrost

Nicolas Le Poul

Insights into water coordination associated with the Cu(II

Introduction

Reorganizational processes coupled to electron transfer are of particular importance in many chemical and biological systems associated to the Cu II /Cu I reaction.

1 Such processes are due to crystal field forces and Jahn-Teller effects when passing from Cu I (d 10) to Cu II (d 9 ). They involve most of the time the coordination/decoordination of one or several donor ligands together with geometric changes. The Cu I (d 10) ion is quite "plastic" toward its environment and can readily adopt 2-, 3-or 4-coordinated geometry.

Contrariwise, the Cu II (d 9 ) ion has a strong preference for tetragonal environments, leading to 5-and 6-coordinated cores, and Cu II can exert a strong electronic/geometric effect on its surroundings in the absence of sterical constraints. Many studies have been devoted to understand/control this chemical upheaval with the variation of the electronic/geometric/steric environment of the copper centre, using mainly synthetic Cu complexes as model systems.1 a One of the most popular system is the mononuclear complex [Cu II (TMPA)(L)] 2+ (L=solvent), based on the tripodal tetradentate ligand TMPA (TMPA = tris(2-pyridylmethyl)amine), which is a ligand of choice for mimicking metallo enzymes of relevance to oxygen activation, for instance. This complex and its derivatives have been studied in depth over the past years since they react at Cu I redox state with dioxygen to form transient Cu n O 2 reactive species.

2,3,4

Recent works have shown that these derivatives were able to activate dioxygen, leading to the catalytic oxidation of organic substrates as homogeneous catalysts 5,6,7 , or to reduce O 2 into H 2 O by heterogeneous catalysis. 8 Also, in acidic aqueous media, catalytic properties towards reduction of nitrite ions into nitric oxide were emphasized by voltammetric studies for [Cu I (TMPA)(H 2 O)] + . 9 In another connection, Le Mestet al. have used the nitrilo Cu complex [Cu II (TMPA)(CH 3 CN)] 2+ for evidencing secondary-sphere effects at the redox level by comparison with a more sophisticated complex. The chemical modifications associated to the electronic transfer are usually detected by complementing spectroscopic studies (NMR, EPR, UV-Vis...) with electrochemical techniques (often cyclic voltammetry). In almost all cases, the electrochemical studies of [Cu II (TMPA)(L)] 2+ were performed in coordinating solvents (DMF, CH 3 CN, H 2 O) and/or in presence of coordinating anion (Cl -, N 3 -). The purpose of these studies was essentially to compare the donating effect of each ligand by correlation with the thermodynamic standard potential value E 0 in the same solvents, the redox potential being highly influenced by electronic, steric and geometric factors.

So far, none of the voltammetric studies in solution have ever intended to investigate the chemical mechanisms associated to the Cu II/I electron exchange. Particularly the water ligand(s) decoordinationprocess at Cu I redox state has not been explored by electrochemistry, despite its high interest. Indeed, water plays an important role in many enzymatic processes. 11 In particular, it was shown that water could inhibit oxygen reaction in cytochrome c oxidase models.

12 Also, it can affect the redox potential of proteins through secondary sphere effects.

13

Recently, water molecules in the active site of Cu-nitrite reductase, were shown to specifically protect from nitrite coordination at Cu(II) and favor catalytic activation at Cu(I) by displacement of the water ligand.

14 Water dissociation is thus of fundamental importance in the activation process. Here we propose to explore such a phenomenon with water as an exogenous ligand with a TMPA Cu complex.

Herein, with the help of electrochemistry combined to EPR/UV-Vis spectroscopies, we report a detailed study of the behaviour of the [Cu II (TMPA)(H 2 O)](CF 3 SO 3 ) 2 complex upon electron transfer in different non-coordinating media including dichloromethane (CH 2 Cl 2 ), tetrahydrofuran (THF), propylene carbonate (PC) and imidazolium-based room temperature ionic liquids (RTILs) (Scheme 1). This study is aimed at investigating the chemical processes associated to the Cu II/I electron transfer reaction, particularly the possible release of the water molecule at the Cu I redox state in terms of kinetics and thermodynamics. Ionic liquids are liquid at room temperature (or close to) and are composed entirely of ions. Generally consisting in a combination of bulky organic cations and weakly coordinating anions, they possess several typical properties such as ionic conductivity, wide electrochemical window, high viscosity, high polarity, high thermal stability, negligible vapour pressure and the ability to dissolve a large range of organic or inorganic compounds. 15 A major advantage of the ionic liquids is the ease of change of the anion/cation combination, allowing the preparation of highly "tunable" media especially designed for a specific application. As such, they nowadays cover a wide range of applications including synthetic chemistry, catalysis, (bio) analytical sensors, and material science.

15,16

They have been widely used as solvents in fundamental electrochemical reactions. 17 However ionic liquids have been scarcely considered as solvents for analysing the electrochemical reactivity of coordination metal complexes. 18 While they can represent advantageous alternative to conventional solvents due to their intrinsic properties, their specific nature may also influence the coordination properties of the complexes. Thus, some studies have explored the coordinating properties of the anion from the ionic liquids towards metallic cations (Cu 2+ , Li + ). 19,20 Nevertheless, the influence of the RTILs as secondary sphere on the coordination properties of metallic complexes associated to their redox properties has not been investigated so far. ) is coordinated to the Cu centre as previously reported for the bis-perchlorato analogous complex. 9 Cu-N distances in the trigonal plane are almost equal (2.06 Å), and N-Cu-N angles are close to the ideal 120° value. In the axial part, Cu-N( 4) and Cu-O(7) distances are shorter (2.01 Å and 1.95 Å respectively) depicting strong interaction of the amine and the water ligands with the metal ion. These structural features are similar to those found for the perchlorato derivative [Cu(TMPA)(H 2 O)](ClO 4 ) 2 ( = 0.96), 9 emphasizing the minor role of the counter anions on the structural properties at the solid state.

Results and Discussion

Medium

[H2O] /ppm g (A /10 -4 cm -1 ) g (A/10 -4 cm -1 ) max /nm ( /M -1 cm -1 ) ‡ max /nm ( /M -1 cm -1 ) # CH2Cl2 dry 200 [a] 1.99 (66) 2.21 ( 103) 695 ( 54) 940 (249)

CH2Cl2 wet 1900 [b] 2.00 (66) 2.21 ( 105) 694 ( 53) 898 (317) THF dry 50 [c] 2.00 (64) 2.20 ( 106) 677 ( 67) 892 ( 185) THF wet 1900 [d] 2.00 (64) 2.20 ( 106) 667 ( 120) 846 ( 215) PC dry 150 [e] 2.00 (63) 2.20 ( 106) 689 ( 69) 895 ( 183) PC wet 6800 [f] 2.00 (68) 2.20 (105) 682 ( 80) 870 (210)

[BMIm][NTf2] dry 50 [g] 1.99 (62) 2.20 (103) 676 ( 52) 892 (120)

[BMIm][NTf2] wet 3600 [h] [j]

[j] 675 (64) 880 (175)

[BMIm][PF6] dry 77.5 [i] [k]

[k] [j] [j] [Et3BuN][NTf2] [j]
2.00 2.20 (105)

[j]

[j]

Solid state

[j] [j] [j]
647 826

Corresponding concentration in mol L -1 : [a] 1.5 10 -2 , [b] 1.4 10 -1 , [c] 2.5 10 -3 , [d] 9.5 10 -2 , [e] 9.9 10 -3 , [f] 4.5 10 -1 , [g] 4 10 -3 , [h] 2.9 10 -1 , [i] 5.8 10 -3 . [j] not determined. [k] pseudo-axial spectrum gx = 2.21 (117), gy =2.17 (117); gz = 1.99 (72). ‡ dxz,yz dz2transition; # dx2-y2,xydz2transition.

Spectroscopic studies in solution.

The ).Moreover, the addition of water at room temperature did not induce any modification of the frozen solution spectra (see SI for each solvent). Hence, taking into account that the EPR data are not affected at all by the presence of water whatever the solvent and that the donor ability of THF is much weaker compared to water, it can reasonably be concluded that H 2 O is the fifth ligand in axial position in THF as well as in CH 2 Cl 2 and PC, and that the solid state aqua TBP structure determined by XRD is kept in solution. The same conclusion could be drawn for the ionic liquids, as the EPR responses of the complex were almost identical to those obtained in CH 2 Cl 2 , THF and PC (Fig. 2A). Moreover, for the three different ionic liquids media, the recorded spectra were very similar, evidencing that neither the nature of the cation (imidazolium vs ammonium) nor that of the anion (NTf 2 -vs PF 6 -

) do modify the coordination properties of the Cu II TMPA complex in the ionic liquids. In the case of the anions, this result deserves to be emphasized because previous studies showed that the NTf 2 -anion, despite its non-coordinating nature, is able to bind ligand-deficient metal complexes through various nitrogen and/or oxygen binding modes. 22 As for non-coordinating solvents, addition of acetonitrile to the ionic liquids yielded a change of the EPR response, indicating the substitution of H 2 O by CH 3 CN in the Cu II coordination sphere. UV-Vis absorption analysis is in good concordance with EPR results. In all media, d-d transition bands in the 850-950 nm wavelength range, with a shoulder at ca. 680 nm were observed. This is indicative of a large contribution of trigonal bipyramidal TBP geometry in solution, as confirmed by the solid state spectrum of the complex (Table 2). Interestingly, the addition of water induces a significant shift of the  max value, between 10 and 50 nm in all media (see SI and Table 2).

For instance, in THF, a shift of 46 nm is observed by addition of 1 % of water (Fig. 2B).This clearly shows, in agreement with EPR data, that it is not correlated to the coordination of the solvent. Possibly, this effect is due to the increased polarity of the solution and solvation of the counterion and/or water. To summarize, the spectroscopic solution studies show that H 2 O remains coordinated to the cupric center in all the non-coordinating media starting from the solid-state characterized [Cu II (TMPA)(H 2 O)] 2+ complex. The geometric features of the Cu complex in solution are not significantly perturbed by the change of medium (TBP conformation) and water content. This point indicates that all voltammetric studies will be initiated with the same (geometry, coordination

) complex [Cu II (TMPA)(H 2 O)] 2+
and is prerequisite for a comparison of the redox behavior and associated mechanisms in the different media. as typical examples of ionic liquids. In addition to their non-coordinating nature, the solvents exhibit different properties of viscosity, polarity (Table 3) for allowing a comparative study of the influence of the medium on the redox behaviour of [Cu II (TMPA)(H 2 O)] 2+ . The water content in the solvents was varied to examine the possible effect of the presence of water on the chemical processes associated to the electron transfer reaction for this aqua complex. For sake of clarity, solvents were denominated as "dry" and "wet" according to the water concentration contained in the medium as described in Table 2. . When decreasing the scan rates, the redox system appears less reversible (i pa /i pc < 1) whereas an additional irreversible oxidation peak becomes visible at more positive potential (ca. 0.4 V vs Fc) on the reverse scan (Fig. 3). For ionic liquids media, an additional redox system of less intensity is also visible at -0.61 V and is due to the presence of residual chloride ions (from ionic liquids precursor), leading to the formation of a [Cu II TMPA(Cl)] + complex. ‡ The amount of water in the media has a strong influence on the electrochemical behaviour of the copper complex. As the concentration of water is increasing, the anodic peak disappears while the redox system at E 0 1 tends to be more reversible, even at the lowest scan rates (Fig. 4). The electrochemical behaviour of [Cu

II (TMPA)H 2 O]

2+ in CH 2 Cl 2 is similar to those described above but it is complicated by a reaction between the cuprous complex, formed during the electrochemical reduction of the initial Cu II complex, and dichloromethane to produce [Cu II (TMPA)(Cl)] + as previously described by Karlin and co-workers. 26 A second reversible redox process could be then observed at -0.75 V, as in ionic liquids (E° = -0.61 V), except that it is not due, this time, to the presence of residual chloride ions (Fig. 5).

Under mild conditions, the Cu I complex reacts in a stoechiometric manner with alkyl or benzyls halides to produce near quantitative yields of coupled products. 26 The copper center acts as the halogen atom abstractor and the halide Cu II complex [Cu II (TMPA)Cl] + is formed. This chloro-Cu II complex is then reversibly reduced at -0.75 V explaining the occurrence of the second reversible redox process in CH 2 Cl 2 . The formation of the chloro complex is fully corroborated by the addition of chloride to the solution, which leads to the formation of a reversible system at the same potential. For each medium, under dry conditions (Table 4), the diffusion coefficient (D) value for [Cu II (TMPA)(H 2 O)] 2+ was estimated from current peak values by using the Randles-Sevcik equation. 27 The diffusion coefficient decreases as the viscosity of the medium is increasing, in agreement with the Stokes-Einstein relationship (1) :

(1

)
where D is the diffusion coefficient,  is the dynamic viscosity of the medium, k B is the Boltzmann constant, T is the absolute temperature, a 0 is the hydrodynamic radius of the species, and  is a numerical value being equal to 6 or 4 depending if the "stick" or the "slip" conditions apply. It can be noted that the Stokes-Einstein equation applies in ionic liquids in the case where the diffusing species are similar in size with the ions of the IL but was shown not to apply for small molecules such as dihydrogen or sulphur dioxide. 28 A non Stokesian behaviour has been even evidenced for concentrated iodide in ionic liquids. 29 In more recent works, Licence and co-workers have demonstrated that the Stokes-Einstein equation describes the behaviour of electroactive species such as ferrocenemethanol and ferrocenemethylimidazole in ionic liquids but the numerical value lies somewhere between 4 and 6.

30 As can also be observed, the [Cu II (TMPA)(H 2 O)] 2+ 2+ vs the media.

As shown in Table4, the value of the redox potential E 0 1 is highly dependent on the medium and on the amount of water present in that medium. Addition of water results in a large negative shift of the potential values except in CH 2 Cl 2 . In this latter medium, the "dry" medium still contains a large amount of water regarding the water saturation value (1.5 10

-1 mol L -1

) in CH 2 Cl 2 , then the further addition of water does not lead to a considerable shift of the potential (20 mV). For the other media, a 120-140 mV shift of the apparent potential is observed. More surprising is the relative order of the apparent redox potential in the different media (Table 4). It varies according

E 0 1 (THF) <E 0 1 (PC) <E 0 1 (CH 2 Cl 2 ) <E 0 1 ([BMIm][NTf 2 + ≤ E 0 1 ([BMIm][PF 6 ]
). As the polarity is increased (for instance THF to PC), the stabilization should be higher for the dication (Cu(II)) than for the monocation (Cu I ). As a consequence, a negative shift of the redox potential is expected. 33 As shown in Table 4, the E 0 1 values varies with the media but not in the trend expected on the only basis of the dielectric constants values (PC > [BMIm][NTf 2 + ≥ *BMIm+*PF 6 ] > CH 2 Cl 2 > THF). Moreover, if we compare the "dry" THF medium with the "dry" ionic liquids media -because those media contain a similar amount of water-it can be observed that the apparent redox potential is positively shifted in the ionic liquids media (E 0 = 60 mV). This effect is even stronger (140 mV) when the fifth ligand bound to Cu is a chloride ion (-0.75 V vs -0.61 V). Such a situation suggests that the Cu I complex is better stabilized than the Cu II cation. This observation can be related to specific solvation effect in ionic liquids media. 34 Ionic liquids are organised media and the Cu I complex is likely to induce much less disorder than its corresponding Cu II analogue, leading to an apparent stabilization of the cupric complex (entropic factor).

Medium effects on the kinetics of the reduction process of [Cu

II (TMPA)(H 2 O)] 2+ .

A further analyse of the electrochemical behaviour of the [Cu II (TMPA)(H 2 O)] 2+ complex allows to explore in more details the dynamics of the process namely regarding its kinetics. First, we will consider the apparent heterogeneous electron-transfer kinetics of the process in "dry" and "wet" media. One problem for measuring the associated rate constant with cyclic voltammetry is the large resistance resulting from low conductive media as those used in this work and which is somewhat difficult to mitigate with the positive feedback compensation feature included in most of the commercially available potentiostat. A way to overcome this difficulty is to use electrochemical impedance spectroscopy (EIS) that permits to determine the apparent heterogeneous electron-transfer kinetic rate constant from a charge transfer resistance (R ct ) measurement separated from the uncompensated solution resistance (R unc ). EIS was performed in the different media for dry and wet conditions. # As depicted in Fig. 6 for [BMIm][NTf 2 ], the addition of water systematically leads to a variation of the Nyquist Plots (Z" vs Z' where Z" and Z' correspond to the imaginary and real part of the impedance respectively). Reasonable fits of the experimental curves were obtained on the basis of a classical equivalent circuit modelling the electrochemical cell including a constant phase element (CPE) (instead of a pure double layer capacitance) and a Warburg impedance (W) (related to the diffusion limiting process) (Scheme 2). The Nyquist plots fitting provides a determination of R ct values then of the apparent electron transfer rate constant k app in the different media, under "dry" and "wet" conditions. Results are gathered in Table 5 for comparison purpose. For any of the media used in this work, the overall electrochemical process appears as a very slow process since the reported k app values are small. The addition of water results in an enhancement of the apparent electron-transfer kinetics, indicating that the overall kinetics depends on the water concentration in the medium. This suggests that the reduction process of the [Cu II (TMPA)(H 2 O)] 2+ complex involves chemical reactions (H 2 O coordination/decoordination) that are faster than the electrochemical reaction as previously described by Laviron in a square scheme mechanism.

35,36

Scheme 2. Equivalent circuit used for experimental impedance simulation. Runc: uncompensated solution resistance; Rct: charge transfer resistance; CPE: constant-phase element; W: Warburg impedance.

Table 5 Apparent heterogeneous electron-transfer kinetic rate constant (kapp/cm s -1 )values in different media (dry/wet) from Nyquist plots fitting.

"dry" medium "wet" medium

[H2O]/ mol L -1 kapp/ cm s -1 [H2O]/ mol L -1 kapp/ cm s -1
CH2Cl2 1.5 10 -2 1. 4 10 -4 1.4 10 -1 7.1 10 -4 THF 2.5 10 -3 2.3 10 -4 9.4 10 -2 4.1 10 -4 PC 9.9 10 -3 2.3 10 -4 4.5 10 -1 6.7 10 -4

[BMIm][NTf2] 4 10 -3 5 10 -5 2.9 10 -1 9 10 -5

[BMIm][PF6] 5.8 10 -3 1 10 -5 2.0 10 -1 8 10 -5

The comparison of k app values between organic solvents and the ionic liquids clearly shows (for both dry and wet conditions) a slower electron transfer process in the IL media. This effect was already reported for different electroactive species.

31,37

From a kinetic point of view, IL could affect the activation energy, through solvent-solute interactions, dynamic solvent effects, as well as the entropic term because of the strong structuration of the media. In the framework of the Marcus-Hush theory, outer-sphere electron transfer kinetics could be expressed under its Arrhenius form (equation (2)):

(2

)
where  is the probability of electron tunnelling (1 in case of adiabicity), K p is the equilibrium constant for a precursor model,  L is the longitudinal relaxation time, G* is the activation energy.

A recent theoretical work on the use of Marcus theory to redox processes in RTILs has predicted that lower rate constants obtained in ionic liquid vs usual organic solvents could be due to solvent dynamics, i.e the  L parameter from the pre-exponential factor in equation (2). 38 Indeed, screening times are supposed to be higher in ionic liquids than in organic solvents. It has been suggested that this could result from slow translational motion of the ions in the ionic liquid media whereas fast longitudinal relaxation occur in organic solvents.

Voltammetric simulations

The cyclic voltammetry experiments as described in Fig. 3 are in agreement with an electron transfer associated with a slow (at the cyclic voltammetry timescale) decoordination of the H 2 O axial ligand when Cu II is reduced to Cu I . The anodic peak located at ca. 0.4 V vs

Fc is H 2 O-related and arises upon the reduction of Cu II complex to Cu I complex. This irreversible oxidation peak could be reasonably ascribed to the cuprous complex with no water molecule bounded to Cu I . This is supported by the fact that the oxidation potential corresponding to the formed [Cu I (TMPA)] + is hardly affected by the media since a value E p ~ 0.4 V vs Fc is found in all the four media.

Notice that this system remains totally irreversible for all the scan rates, up to 20 V s -1

, and its oxidation occurs at a potential around 0.7 V more positive than the aqua complex. We could derive that, contrarily to [Cu I (TMPA)] + , the binding of water to [Cu II (TMPA)] 2+ is fast and totally irreversible.

1/2 * * 1 app exp 4 p L G G k K RT RT                   
Scheme 3. Formal square scheme mechanism for the exchange of water associated with electron transfer for the [Cu II (TMPA)(H2O)] 2+ complex.

The electrochemical process can be then described as in scheme3 . Similarly, the reversibility increases with an addition of water in the media. These two observations imply that the reaction of [Cu I (TMPA)] + with H 2 O is relatively slow and equilibrated. These conclusions are illustrated by the derived rate constants of Table 6. On the whole, there is little change with the solvent. We just notice that the H 2 O binding is more favoured and faster in the ionic liquid than in the two other classical solvents. On one hand, the sluggishness of the decoordination process in all media may be associated to the fact that pyridyl moities can be very labile at Cu I redox state when an exogenous ligand is bounded, as previously reported. 39, 42 Indeed, the pyridyl decoordination may induce a reinforcement of the metal-water bond, thus disfavouring water departure. On the other hand, the higher kinetics for the water binding reaction at Cu I detected for the imidazolium-based ionic liquids vs THF and PC may be ascribed to the existence of specific domains in these media which significantly favour the diffusion of water molecules despite high viscosity. 43 This behaviour is reminiscent of channel enzymes 44 and could be exploited in catalytic reactions involving fast water coordination/decoordination.

Conclusions

The coordination behaviour of a mononuclear complex [Cu II (TMPA)(H 2 O)] 2+ , simple mimic of metallo enzymes was investigated in different non-coordinating solvents, including IL, by combining electrochemistry and spectroscopic techniques. Only a few studies report on the fundamental behaviour of inorganic metal complexes in IL. The coordination geometry of the Cu II complex remains trigonal bipyramidal in all the media with the water ligand being axially coordinated to Cu II . In particular, the anions of the ionic liquids were not found to coordinate the copper centre. The dissociation mechanism of the water axial ligand upon electron transfer in terms of kinetics and thermodynamics was studied in different media. The overall process is highly dependent on the water concentration, with a slow electron transfer reaction, slower in ionic liquids than in the conventional solvents, as already reported for different electroactive species. The decoordination of the water molecule was detected by cyclic voltammetry from the monoelectronic reduction of the Cu II complex. The increase of the water content and/or decrease of the timescale of the experiment resulted in the disappearance of the oxidation peak related to the water-free Cu I complex. Numerical simulations of the experimental CVs suggest that the binding/unbinding of water at the Cu(I) redox state is relatively slow and equilibrated in all media. Moreover, the binding of water at Cu I is somewhat faster in the ionic liquids than in the non-coordinating conventional solvents, while the unbinding process is weakly sensitive to the nature of the solvents.

In summary, the ionic liquids exhibit promising properties in comparison to classical solvents, to investigate redox systems which are Cu-enzymes models: (i) no coordination of the anion to the metal ion at any redox state, (ii) enhanced water flow/exchange, and (iii) tunable molecular structure (shape and bulkyness). This last property can be exploited for investigating host-guest reactions with biomimetic systems incorporating cavities such as calix [6]arene,1 b-c,10 since adequate design of the RTILs can preclude competitive solvent hosting and thus promote better access of the substrate to the cavity.

Experimental Materials and methods

Anhydrous "extra-dry" dichloromethane (H 2 O < 30 ppm, Acros), tetrahydrofuran (99.85%, Acros), and propylene carbonate (99.7%, Sigma-Aldrich) were used as received and kept under N 2 in the glovebox. CH 2 Cl 2 , THF and PC were dried on activated molecular sieves in the glovebox before use. All chemicals were of reagent grade and were used without purification. The supporting electrolyte NBu 4 PF 6 was synthesized from NBu 4 OH (Fluka) and HPF 6 (Aldrich). It was then purified, dried under vacuum for 48 hours at 100° C, then kept under N 2 in the glovebox. EPR spectra were run at T=150 K on a Bruker Elexsys spectrometer (X-band). EPR simulation was carried on the XSophe-Sophe-Xpreview software and afforded the quantitative determination of the g  and g // parameters as well as the related hyperfine coupling constants A  and A // . 45 UV-Vis-NIR spectroscopy was performed with a JASCO V-670 spectrophotometer and an integration-sphere accessory for solid-state measurements. The electrochemical studies were performed in a glovebox (Jacomex) (O 2 < 1 ppm, H 2 O < 1 ppm) with a home-designed 3-electrodes cell (Working electrode: glassy carbon disk, reference electrode: Pt in Fc + /Fc solution, counter electrode: Pt wire). The glassy carbon electrode was carefully polished before each voltammetry experiment with a 3 µm alumina aqueous suspension and ultrasonically rinsed in water then acetone. The concentration in Cu complex was 10 -3 mol L -1 except in the ionic liquid where the concentration was set to 3 10 -3 mol•L -1

, in order to get a better signal-to-noise ratio. All potential values are given against the ferrocene/ferrocenium couple. Cyclic voltammetry and impedance spectroscopy measurements were performed with an Autolab electrochemical analyzer (PGSTAT 302 potentiostat/galvanostat from Eco Chemie B.V., equipped with the GPES/FRA software). For cyclic voltammetry, scan rates were varied between 0.02 and 5 V s -1

. For impedance spectroscopy, a 10 mV rms sinusoidal signal was applied along with the dc potential, which was set at the E 0 value previously determined by cyclic voltammetry. The frequency range was varied from 0.1 Hz to 10 kHz. The charge transfer resistance value (R ct ) was extracted from the experimental Nyquist plots on the basis of the equivalent circuit. The apparent rate constant k app was evaluated from the simplification of the Butler-Volmer equation at small overpotential (equation ( 3)), assuming that  = 0.5 and C O = C R at the equilibrium potential.

[R is the perfect gas constant, T is the absolute temperature, n the number of electrons exchanges, F the Faraday constant, A is the electrode area, C O and C R are the bulk concentrations of the oxidized and reduced species,  is the symmetry barrier coefficient and was taken as 0.5].

(3)

Synthesis of the ligand and Cu complex

The TMPA ligand (TMPA = tris(2-pyridylmethyl)amine) was prepared according to a previously published procedure.

46

The synthesis of the bis-triflate Cu complex [Cu II (TMPA)(H 2 O)](CF 3 SO 3 ) 2 1 was operated by addition of equimolar amounts of Cu II (CF 3 SO 3 ) 2 to a solution of the TMPA ligand (toluene), leading to the formation of a blue powder. After evaporation of the solvent and washing with diethyl ether, the compound was dissolved in THF. Slow evaporation of the solvent afforded blue crystals identified by X-ray diffraction analysis as the mono-aqua copper (II) complex, [Cu II (TMPA)(H 2 O)](CF 3 SO 3 ) 2 .

Synthesis of the ionic liquids

The room temperature ionic liquids

10

 10

Fig. 1 .

 1 Fig. 1.A) ORTEP representation of [Cu II (TMPA)(H2O)](CF3SO3)2 (50 % probability ellipsoids shown). Hydrogen and both triflate anions are omitted for clarity. B) Schematic representation of the [Cu II (TMPA)(H2O)] 2+ complex.

23Fig. 2 .

 2 Fig. 2. A) EPR spectra at T = 150 K of [Cu II (TMPA)(H2O)] 2+ (10 -3 mol L -1 ) in a) "dry" CH2Cl2 ([H2O] = 1.5 10 -2 M (200 ppm)) and b) "dry" *BMIm+*NTf2]([H2O] = 4 10 -3 M (50 ppm)). B) UV-Vis spectra of [Cu II (TMPA)(H2O)] 2+ (1.5 10 -3 mol L -1 ) in dry (green) and wet (blue) THF.

Fig. 3 .

 3 Fig. 3. Cyclic voltammetry of [Cu II (TMPA)(H2O)] 2+ (~ 10 -3 mol L -1 ) at a glassy carbon disk electrode at 0.02 V s -1 (left) and 0.5 V s -1 (right) in (A, B) "dry" THF ([H2O] = 2.5 10 -3 mol L -1 ), (C,D) "dry" PC (*H2O] = 9.9 10 -3 mol L -1 ) and (E,F) "dry" *BMIm+*NTf2] ([H2O] = 4 10 -3 mol L -1 ).

Fig. 4 .

 4 Fig. 4. Cyclic voltammetry of [Cu II (TMPA)(H2O)] 2+ (~ 3-5 10 -3 mol L -1 ) at a glassy carbon disk electrode at v = 0.02 V s -1 in (A) "wet" THF (*H2O] = 9.5 10 -2 mol L -1 ), (B) "wet" PC (*H2O] = 4.5 10 -1 mol L -1 ) and (C) "wet" *BMIm+*PF6] ([H2O] = 1.9 10 -1 mol L -1 ).In THF, PC, [BMIm][NTf 2 ] and [BMIm][PF 6 ], cyclic voltammetries display very similar patterns while the electrochemical behaviour in CH 2 Cl 2 exhibits some slight differences (vide infra). For the four first media, a chemically reversible redox system located between E 0 1 =

Fig. 5 .

 5 Fig. 5.Cyclic voltammetry of [Cu II (TMPA)(H2O)] 2+ (~ 10 -3 mol L -1 ) at a glassy carbon disk electrode at v = 0.1 V s -1 in "dry" CH2Cl2.

Fig. 6 .

 6 Fig. 6. Nyquist plots (Z" vs Z') from EIS measurements at a glassy carbon electrode of *Cu II (TMPA)(H2O)] 2+ in [BMim][NTf2] upon addition of water (Eac = 0.01 V, 0.1 Hz << 10 4 Hz). A) [H2O] = 4 10 -3 mol L -1 (50 ppm) (), [H2O] = 6 10 -2 mol L -1 (700 ppm) (), and [H2O] = 2.9 10 -1 mpl L -1 (3600 ppm) (•); B)[H2O] = 2.9 10 -1 mol L -1 (3600 ppm) exp. data and simulated curve on the basis of the equivalent circuit in Scheme 1; C) [H2O] = 6 10 -2 mol L -1 (700 ppm) exp. data and simulated curve; D) [H2O] = 4 10 -3 mol L -1 (50 ppm) exp. data and simulated curve.

Table 1 )

 1 . Clearly, none of both triflate anions (CF 3 SO 3 -

Table 1

 1 Selected bond distances and angles for [Cu II (TMPA)(H2O)](CF3SO3)2. See Supporting Information part for details

		Distance /Å		Angle /°
	Cu-N(1)	2.060(3)	O(7)-Cu-N(1)	98.5(1)
	Cu-N(2)	2.058(3)	O(7)-Cu-N(2)	94.8(1)
	Cu-N(3)	2.069(3)	O(7)-Cu-N(3)	99.9(1)
	Cu-N(4)	2.011(3)	O(7)-Cu-N(4)	176.9(1)
	Cu-O(7)	1.945(8)	N(1)-Cu-N(2)	119.9(1)
			N(1)-Cu-N(3)	117.3(1)
			N(1)-Cu-N(4)	82.4(1)
			N(2)-Cu-N(3)	117.4(1)
			N(2)-Cu-N(4)	82.2(1)
			N(3)-Cu-N(4)	82.2(1)

Table 2

 2

	EPR data (150 K) and UV-Vis data for [Cu II (TMPA)(H2O)](SO3CF3) in the different media.

  Cl 2 , THF and PC) and in three different ionic liquids where both anions and cations were varied ([BMIm][NTf 2 ], [BMIm][PF 6 ] and [Et 3 BuN][NTf 2 ]) (Scheme 1). Addition of coordinating species such as H 2 O and CH 3 CN was performed at room temperature for a full characterization of the coordinating environment around Cu II , with a particular focus on the nature of the fifth exogenous ligand. Whatever the non-coordinating media, [Cu II (TMPA)(H 2 O)]2+ displays an EPR reversed axial spectral pattern with g  >g // ≈ 2.0 (Fig.2Aand SI). The hyperfine coupling constants A  have small values in the range 100-120 G (Table2). These features are typical of a Cu II complex in a TBP geometry with a dz 2 ground state, as previously observed for the nitrilo

	coordination properties of [Cu(TMPA)(H 2 O)](CF 3 SO 3 ) 2 were examined by Electron Paramagnetic Resonance (EPR) and UV-Visible
	spectroscopies in three non-coordinating solvents (CH 2 analogue [Cu II (TMPA)(CH 3 CN)]

2+ in frozen acetonitrile. The g and A values are slightly different than those obtained in acetonitrile (g  = 2.19, g // = 2.02, A  = 113 10 -4 cm -1

Table 3

 3 Density (d), dynamic viscosity (), static (s) and optical (opt) dielectric constant values for different media at T = 20°C Calculated from the square of the refractive index.

	Medium	d /g cm -3  /mPa s	s	opt b
	CH2Cl2	1.33	0.4	9.1	2.03
	THF	0.89	0.5	7.6	1.98
	PC	1.19	2.5	64.9	2.02
	[BMIm][NTf2]	1.43	52	11.5 a	2.04
			40 (130 ppm H2O)
			33.8 (4180 ppm H2O)
	[BMIm][PF6]	1.35	308	11.4
	[a] from ref. 24 [b]

25 Electrochemical studies of the [Cu II (TMPA)(H 2 O)] 2+ complex. Cyclic voltammetric experiments were performed in the different solvents containing NBu 4 PF 6 as supporting salt, as well as in [BMIm][NTf 2 ] and in [BMIm][PF 6 ]

Table 4

 4 Selected electrochemical parameters for [Cu II (TMPA)(H2O)] 2+ in the different media. E /V vs Fc

		"Dry" medium		"Wet" medium
		10 3 [H2O]/	10 6 D/	E 0 1 /V	10 3 [H2O]/	E 0 1 /V
		mol L -1	cm 2 s -1		mol L -1
	CH2Cl2	15	6	-0.31	140	-0.33
	THF	2.5	2.9	-0.38	94	-0.52
	PC	9.9	0.58	-0.36	450	-0.48
	[BMIm][NTf2]	4	0.06	-0.25	290	-0.38
	[BMIm][PF6]	5.8	0.004	-0.22	200	-0.33
	complex diffuses very slowly in the two ionic liquids, D = 6 10	-12 m	2 s -1 in [BMIm][NTf 2 ] and D = 0.4 10	-12 m	2 s -1 in [BMIm][PF 6 ]. For
	comparison, the diffusion coefficient of inorganic metal complexes such as ferrocene or ferrocenemethanol have been reported as D =
	6.3 -46 10 -12 m 1.3 10 -12 m 2 s -1 in [BMIm][PF 6 ]) or for a large polyoxometalate ion 2 s -1 in [EMIm][NTf 2 ] ( = 34 mPa s) 31,32 or D = 25 10 -12 m 18c [Mo 6 O 19 ] 2 s -1 in [BMIm][NTf 2 ] and D = 5 10 -12 m 2 s -12 m -1 in [BMIm][NTf 2 ] and D = 2 s -1 in [BMIm][PF 6 ]. 2-(D = 1.4 10 -12 m 2 s

30b

Lower diffusion coefficient values have been found for a rhenium tetrazolato complex 18b (D = 5.8 10 -1 in [BMIm][PF 6 ]).

Table 6

 6 Kinetics and thermodynamic parameters of the dissociation reaction of [Cu I (TMPA)(H2O)] +[a]

	Medium	kf / s -1	kb /L mol -1 s -1	K /L mol -1
	THF	0.15	4	26
	PC	0.02	4	200
	[BMIm][NTf2]	0.08	20	250
	[a] Estimations from simulations of experimental voltammograms (See Fig. S1 and Supplementary Information part).

  [BMIm][NTf 2 ], [BMIm][PF 6 ] and [Et 3 BuN][NTf 2 ] were prepared according to standard procedures 47 from mixing aqueous LiNTf 2 (Sigma-Aldrich or Solvionic) or NaPF 6 (Sigma-Aldrich), and aqueous BMImCl (Sigma-Aldrich or Solvionic) or Et 3 BuNCl. The samples were purified by repeated washing with H 2 O, filtered over neutral alumina and silica. Prior to each experiment, vacuum pumping carefully dried RTIL overnight and then dried over molecular sieves at least for 48 h. The amount of water was measured by Karl Fischer titration (Karl Fischer 652 Metrohm).

	c t R		(1 O n F Ak C 2 2 app RT		)   R C

Acknowledgments

This research was supported by CNRS, Ministère de la Recherche et de l'Enseignement Supérieur (A. G.), Conseil Régional de Bretagne (J. Z.) and Agence Nationale pour la Recherche (ANR-2010-BLAN-0714, Cavity-zyme (Cu)). Dr F. Michaud is thanked for XRD analyses and Mohammad Kassem for the synthesis of the Cu II complex from tmpa ligand.

Notes and references

a Laboratoire de Chimie, Electrochimie Moléculaires et Chimie Analytique, CNRS, UMR 6521, Université de Brest, 6 av. Le Gorgeu, 29328 Brest cedex, France; Fax: +33298017001; Tel: +33298016168; E-mail: nicolas.lepoul@univ-brest 946244). See DOI: 10.1039/b000000x/. ‡Addition of chloride ions leads to the increase of the intensity of the redox system at -0.61 V vs Fc/Fc + . #The Edc potential was systematically determined from the cyclic voltammetric curve before ac impedance measurement by taking the mid-potential value for different water concentrations.


Virginie Daru

Diana Baltean-Carlès

Catherine Weisman

Philippe Debesse

Gurunath Gandikota

Two-dimensional numerical simulations of nonlinear acoustic streaming in standing waves

Keywords: acoustic streaming, standing wave, numerical simulation, nonlinear streaming regime

Numerical simulations of compressible Navier-Stokes equations in closed twodimensional channels are performed. A plane standing wave is excited inside the channel and the associated acoustic streaming is investigated for high intensity waves, in the nonlinear streaming regime. Significant distortion of streaming cells is observed, with the centers of streaming cells pushed towards the end-walls. The mean temperature evolution associated to the streaming motion is also investigated.

Introduction

Acoustic streaming is generated inside a two-dimensional channel as a consequence of the interaction between a plane standing wave and the solid boundaries. It consists of a mean second order flow produced mainly by shear Email address: virginie.daru@limsi.fr (Virginie Daru) forces within the viscous boundary layer along the solid walls. This motion was initially studied by Rayleigh [START_REF] Rayleigh | On the circulation of air observed in Kundts tubes, and on some allied acoustical problems[END_REF] in the case of wide channels, in which the boundary layer thickness is negligible in comparison with the channel width. This streaming flow is characterized by four steady counter-rotating vortices outside the boundary layer, nowadays referred to as Rayleigh streaming. The vortices develop along the half wavelength of the standing wave. Along the central axis of the channel, the streaming motion is oriented from acoustic velocity nodes to antinodes. Inside the boundary layer four additional vortices are created simultaneously, with the streaming motion oriented from acoustic velocity antinodes to nodes along the inner walls of the tube [START_REF] Nyborg | Acoustic streaming[END_REF][START_REF] Schlichting | Berechnung ebener periodischer Grenzschicht-strommungen [calculation of plane periodic boundary layer streaming[END_REF].

In the case of wide channels, Menguy and Gilbert [START_REF] Menguy | Non-linear Acoustic Streaming Accompanying a Plane Stationary Wave in a Guide[END_REF] showed that streaming itself can be linear (case of slow streaming) or nonlinear (case of fast streaming), and both regimes are characterized by a reference nonlinear Reynolds number Re N L = (M × y 0 /δ ν ) 2 reflecting the influence of inertial effects on the streaming flow (M is the acoustic Mach number, M = U max /c 0 , with U max the maximum acoustic velocity inside the channel and c 0 the initial speed of sound, y 0 is the half width of the channel and δ ν the viscous boundary layer thickness). Most analytical streaming models have been established in the case of slow streaming, characterized by Re N L ≪ 1. They are based on successive approximations of the nonlinear hydrodynamic equations and have been derived for arbitrary values of the ratio y 0 /δ ν , taking into account the variations of heat conduction and viscosity with temperature [START_REF] Hamilton | Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width[END_REF], and the existence of a longitudinal temperature gradient [START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF]. In the case Re N L = O(1), Menguy and Gilbert [START_REF] Menguy | Non-linear Acoustic Streaming Accompanying a Plane Stationary Wave in a Guide[END_REF] derived an asymptotic model for streaming flow inside wide cylindrical resonators, with no mean temperature gradient, and showed a distortion of streaming patterns due to inertia effects. However, this model does not cover the strongly nonlinear streaming regime (Re N L ≫ 1), and does not explain the nonlinear effects on acoustic streaming recently observed in several experimental works [START_REF] Moreau | Measurements of inner and outer streaming vortices in a standing waveguide using laser doppler velocimetry[END_REF][START_REF] Nabavi | Analysis of regular and irregular acoustic streaming patterns in a rectangular enclosure[END_REF][START_REF] Thompson | Influences of a temperature gradient and fluid inertia on acoustic streaming in a standing wave[END_REF], where the temperature gradient along the resonator wall has a significant influence.

Numerical simulations in the linear regime, yielding results for non idealized geometries, were performed in the specific cases of thermoacoustic refrigerators [START_REF] Marx | Computation of the mean velocity field above a stack plate in a thermoacoustic refrigerator[END_REF] or in annular resonators [START_REF] Boufermel | Velocity of Mass Transport to Model Acoustic Streaming: Numerical Application to Annular Resonators[END_REF] and solved the dynamics of the flow without taking heat transfer into account.

Simulations in the nonlinear regime were first performed by Yano [START_REF] Yano | Turbulent acoustic streaming excited by resonant gas oscillation with periodic shock waves in a closed tube[END_REF], who studied the acoustic streaming associated with resonant oscillations with periodic shock waves in tubes with aspect ratio (width over length) very large (0.1). He solved the full 2D Navier-Stokes equations with an upwind finite-difference TVD scheme and showed the existence of irregular vortex structures and even turbulent streaming for high streaming Reynolds numbers (based on a characteristic streaming velocity, the tube length, and the kinematic viscosity, R s = U s L/ν). This is a different configuration than our configuration, since it considers low frequency acoustic waves in wide tubes with respect to their length and focuses on turbulent streaming.

Simulations of acoustic streaming in the linear and nonlinear regime, taking heat transfer into account, in a two-dimensional rectangular enclosure, were performed by Aktas and Farouk [START_REF] Aktas | Numerical simulation of acoustic streaming generated by finite-amplitude resonant oscillations in an enclosure[END_REF]. In their study, the wave is created by vibrating the left wall of the enclosure and the full compressible Navier-Stokes equations are solved, with an explicit time-marching algorithm (a fourth order flux-corrected transport algorithm) to track the acoustic waves.

Their numerical results are in agreement with theoretical results in the linear regime and show irregular streaming motion in the nonlinear regime, but they show the existence of irregular streaming at small values of Re N L , in contradiction with experiments cited above. Moreover, these simulations do not analyze the deformation of the streaming cells until they split onto several cells.

We propose in this work to conduct numerical 2D compressible simulations for studying the origin of the distortion of streaming cells (of Rayleigh type) that were experimentally observed. Calculations are performed for channels with aspect ratios ranging from 0.01 to 0.07, and the coupling between streaming effects and thermal effects in the channel (existence of a mean temperature gradient) is also investigated.

Problem description and numerical model

We consider a rectangular channel of length L and half width y 0 , initially filled with the working gas. In order to initiate an acoustic standing wave in the channel, it is shaken in the longitudinal direction (x), so that an harmonic velocity law is imposed, V(t) = (V (t), 0) T , with V (t) = x p ω cos(ωt), ω being the angular frequency and x p the amplitude of the channel displacement.

The channel being undeformable, the flow can be modeled by the compressible Navier-Stokes equations expressed in the moving frame attached to the channel, so that a forcing source term is added. The model reads:

             ∂ρ ∂t + ∇ • (ρv) = 0 ∂ρv ∂t + ∇ • (ρv ⊗ v) + ∇p = ∇ • ( τ ) -ρ dV dt ∂ρE ∂t + ∇ • (ρEv + pv) = ∇ • (k∇T ) + ∇ • ( τ v) -ρv • dV dt (1)
where v = (u, v) T is the flow velocity, E = e + 1 2 v • v is the total energy, with e = p (γ-1)ρ the internal energy, γ the specific heat ratio, τ = -2 3 µ(∇ • v) Ī +2µ D the viscous stress tensor of a Newtonian fluid, D the strain tensor, µ the dynamic viscosity, k the thermal conductivity. The thermo-physical properties µ and k are supposed to be constant. The gas is considered as a perfect gas obeying the state law p = rρT , where T is the temperature and r is the perfect gas constant corresponding to the working gas. The physical boundary conditions employed in the moving frame are: no slip and isothermal walls.

The model is numerically solved by using high order finite difference schemes, developed in Daru and Tenaud [START_REF] Daru | High Order One-step Monotonicity-Preserving Schemes for Unsteady Compressible flow Calculations[END_REF]. An upwind scheme, third order accurate in time and space, is used for convective terms, and a centered scheme, second order, is used for diffusion terms. More detail about the scheme and computations showing its good qualities can be found in Daru and Gloerfelt [START_REF] Daru | Aeroacoustic computations using a high order shock-capturing scheme[END_REF], Daru and Tenaud [16]. This scheme can be derived up to an arbitrary order of accuracy for convective terms in the case of a scalar equation. Here the third order scheme is selected, after having done several comparisons using higher order schemes (up to the 11th order), that have shown that third order gives sufficient accuracy for a reasonable CPU cost. In cases where shock waves are present, the scheme can be equipped with a flux limiter (MP), preserving monotonicity, intended for suppressing the parasitic numerical oscillations generated in the shock region, while preserving the accuracy of the scheme in smooth regions. However, the flows considered here are always low Mach number flows. Although traveling shock waves are a main feature of the flow for high acoustics levels, as noticed by several authors [START_REF] Merkli | Thermoacoustic effects in a resonance tube[END_REF], they are of weak intensity and the numerical oscillations are very small and do not spoil the solution. Thus the MP limiter, which is expensive in terms of CPU cost, was not activated in these calculations.

For solving the 2D Navier-Stokes equations, the scheme is implemented using

Strang splitting. This reduces the formal accuracy of the scheme to second order. However, numerical experiments have shown that a very low level of error is still achieved.

Let us describe our numerical procedure. The system (1) can be written in vector form :

∂w ∂t + ∂ ∂x (f -f v ) + ∂ ∂y (g -g v ) = h ( 2
)
where w is the vector of conservative variables (ρ, ρu, ρv, ρE) T , f and g are the inviscid fluxes f = (ρu, ρu 2 + p, ρuv, ρEu + pu) T and g = (ρv, ρuv, ρv 2 + p, ρEv + pv) T , f v and g v being the viscous fluxes

f v = (0, τ xx , τ xy , k ∂T ∂x + uτ xx + vτ xy ) T , g v = (0, τ xy , τ yy , k ∂T ∂y + uτ xy + vτ yy ) T .
The source term reads h = (0, -ρ dV dt , 0, -ρu dV dt ) T . Denoting w n i,j the numerical solution at time t = nδt and grid point (x, y) = (iδx, jδy), we use the following Strang splitting procedure to obtain second order of accuracy every two time steps :

w n+2 i,j = L δx L δy L δy L δx w n ij ( 3
)
where

L δx (resp. L δy ) is a discrete approximation of L x (w) = w + δt(-f x + f v x + h) (resp. L y (w) = w + δt(-g y + g v y )
). The 1D operators being similar in the two directions, we only describe the x operator. The scheme is implemented as a correction to the second order MacCormack scheme. It consists of three steps, as follows :

w * i,j = w n i,j -δt δx (f i+1,j -f i,j -f v i+1/2,j + f v i-1/2,j ) n + δt h n i,j w * * i,j = w * i,j -δt δx (f i,j -f i-1,j -f v i+1/2,j + f v i-1/2,j ) * w n+1 i,j = 1 2 (w n i,j + w * * i,j ) + C x i+1/2,j -C x i-1/2,j (4)
The viscous fluxes are discretized at each interface using centered second order finite differences formulae. The corrective term C x i+1/2,j -C x i-1/2,j provides the third order accuracy and the upwinding for the inviscid terms. Let us de-

fine ψ i+1/2,j = 1 6 ∑ 4 l=1 { |ν l i+1/2,j |(1 -ν l i+1/2,j )(1 + ν l i+1/2,j )δα l i+1/2,j • d l i+1/2,j } ,
where ν l = δt δx λ l , λ l and d l are the eigenvalues and eigenvectors of the Roeaveraged jacobian matrix A = df dw [START_REF] Roe | Approximate Riemann solvers, parameter vectors and difference schemes[END_REF], and δα l is the contribution of the l-wave to the variation (w n i+1/2,j -w n i-1/2,j ). Using the function ψ, the corrective term reads :

C x i+1/2,j =    -ψ n i+1/2,j + ψ n i-1/2,j if ν i+1/2,j ≥ 0 ψ n i+3/2,j -ψ n i+1/2,j if ν i+1/2,j < 0 (5)
This completes the description of the numerical method.

We are interested in the acoustic streaming generated by the interaction of the imposed plane standing wave and the channel wall. Resonant conditions are imposed, for which L = λ/2, λ = c 0 /f being the wave length, c 0 the speed of sound for initial state and f the vibration frequency of the channel. It is known [START_REF] Hamilton | Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width[END_REF] that boundary layers develop along the walls, with thickness δ ν = √ 2ν/ω, ν being the kinematic viscosity ν = µ/ρ 0 , and ρ 0 the density at initial state. Depending on the value of the ratio y 0 /δ ν , several patterns of streaming can appear: Rayleigh-type streaming in the central region, and boundary layer type streaming near the longitudinal walls. The boundary layer is of small thickness and must be correctly resolved by the discretization mesh.

After several trials, we have determined that a value of 5 points per boundary layer thickness is sufficient for reasonable accuracy of the simulations. The results obtained using 10 points per boundary layer thickness show very small differences with the former, the maximum value of the differences being less than 3%. All results presented below are thus obtained using a cartesian mesh of rectangular cells of constant size δx and δy, composed of 500 points in the axial direction x, and of 5 × y 0 /δ ν points in the y direction normal to the axis. In the considered geometry, this leads to cells such that δy ≪ δx.

The flow being symmetrical with respect to the x axis (at least in the range of parameters treated), only the upper half of the channel was considered.

Also, the scheme being fully explicit, the time step δt is fixed such as to satisfy the stability condition of the scheme which can be written as:

δt ≤ 1 2 min(δy 2 /ν, δy 2 /(k/ρ 0 c 0 ), δy/c 0 ) (6)
As shown in Equation ( 6), the first two limiting values δy 2 /ν and δy 2 /(k/ρ 0 c 0 ) are related to the viscous and thermal conduction terms, and the third one δy/c 0 is related to the acoustic propagation. In all cases considered here, the time step limitation is acoustic, ie δt ≤ 1 2 δy/c 0 . Taking δt = 1 2 δy/c 0 and δy = δ ν /5, this results in a number of time steps N T per period of oscillation

proportional to √ L, N T = 1/(f δt) = 10 √ 2πc 0 ν √ L.
Since transients of several hundreds of periods may be needed in order to reach stabilized steady streaming flow, simulations are very costly, and one must rely on numerical schemes that are sufficiently accurate in both space and time.

Finally, the mean flow is obtained from calculating a simple mean value for each physical quantity (velocity, pressure, temperature) over an acoustic period. The mean velocity obtained is the so-called Eulerian streaming velocity. The Lagrangian streaming velocity, associated to the streaming mass transport, could also be computed. The difference between them is significant only in the boundary layer, and in the case of wide channels the two velocities are almost the same. In order to observe the mechanism of cell distortion, either one of these velocities can be monitored.

Numerical results

We consider a channel initially filled with air at standard thermodynamic conditions, p 0 = 101325Pa, ρ 0 = 1.2kgm -3 , T 0 = 294.15K. The thermo-physical properties of air are µ = 1.795 10 -5 kgm -1 s -1 and k = 0.025Wm -1 K -1 . Also for air, γ = 1.4 and r = 287.06Jkg -1 K -1 . The Prandtl number P r is equal to 0.726. This results in an initial speed of sound c 0 = 343.82ms -1 .

For a 1m long channel, the limiting time step would correspond to N T ≈ 25000 √ L, that is 25000 iterations per period. Since transients of several hundreds of periods may be needed in order to reach stabilized streaming flow, several millions of iterations are necessary for each simulation. Considering these numerical constraints, a shorter channel is considered, with L = 8.59mm. This corresponds to a high-frequency wave, with f = 20000Hz.

The resulting boundary layer thickness is δ ν = 1.54 10 -5 m. The time step δt = 8 10 -9 s is chosen in order to satisfy the numerical stability condition, corresponding to 6250 time iterations per period for a mesh involving 5 grid

x p (µm) points across the boundary layer thickness. The acoustic velocity produced in the channel depends on the amplitude of the channel displacement and on the ratio y 0 /δ ν . It varies approximately linearly with the amplitude of the channel displacement, for a given ratio y 0 /δ ν . Table 1 summarizes the different parameter values corresponding to the simulations that are presented thereafter.

As mentioned earlier, the parameter identified as relevant in describing the regularity of streaming flow is the nonlinear Reynolds number Re N L introduced by Menguy and Gilbert [START_REF] Menguy | Non-linear Acoustic Streaming Accompanying a Plane Stationary Wave in a Guide[END_REF]. In this paper we used a slightly different definition for Re N L , because the definition of the viscous boundary layer thickness is different. Our Reynolds number corresponds to half of that of Menguy and Gilbert [START_REF] Menguy | Non-linear Acoustic Streaming Accompanying a Plane Stationary Wave in a Guide[END_REF].

We first present results concerning the main acoustic field in the channel, for a small value of Re N L corresponding to slow streaming. In Figure 1( This can be seen in Figure 3(right) showing the acoustic velocity signal at channel's center as a function of time counted by the number of periods elapsed. In Figure 3(a)(right) the signal contains only one frequency, but for all other cases, there are shock waves and the acoustic velocity signal is distorted in a "U" shape, because of the presence of odd harmonics (3, 5, etc). c,d) also confirm the direction of the displacement of the streaming cells' centers. This distortion of streaming cells was already observed in experiments in rectangular or cylindrical geometries in wide channels [START_REF] Moreau | Measurements of inner and outer streaming vortices in a standing waveguide using laser doppler velocimetry[END_REF][START_REF] Nabavi | Analysis of regular and irregular acoustic streaming patterns in a rectangular enclosure[END_REF][START_REF] Thompson | Influences of a temperature gradient and fluid inertia on acoustic streaming in a standing wave[END_REF]. Nabavi, Siddiqui and Dargahi [START_REF] Nabavi | Analysis of regular and irregular acoustic streaming patterns in a rectangular enclosure[END_REF] described it as irregular streaming and detected a critical nonlinear Reynolds number Re N L = 25 that separates regular and irregular streaming, which is in agreement with our simulations. In the literature there is to our knowledge no other theoretical or numerical study confirming measurements in these streaming regimes. With the weakly nonlinear model of Menguy and Gilbert [START_REF] Menguy | Non-linear Acoustic Streaming Accompanying a Plane Stationary Wave in a Guide[END_REF] the streaming can be calculated for a maximum value of Re N L = 2 (in our definition), while the numerical simulations of Aktas and Farouk [START_REF] Aktas | Numerical simulation of acoustic streaming generated by finite-amplitude resonant oscillations in an enclosure[END_REF] show the existence of multiple streaming cells for a low value of Re N L = 1.4, which is in contradiction with our results and with experiments. Moreover, these numerical simulations [START_REF] Aktas | Numerical simulation of acoustic streaming generated by finite-amplitude resonant oscillations in an enclosure[END_REF] do not analyse in detail the transition from two exterior streaming cells to more streaming cells.

According to Menguy and Gilbert [START_REF] Menguy | Non-linear Acoustic Streaming Accompanying a Plane Stationary Wave in a Guide[END_REF], the fluid inertia causes distortion of streaming cells for large values of Re N L . This was also verified through our simulations. For Re N L = O(1), the Mach number is still small (the wave is almost a mono-frequency wave) and the mean temperature difference inside the channel is smaller than 0.1K (the mean temperature gradient is negligible). The approximations of the model by Menguy and Gilbert [START_REF] Menguy | Non-linear Acoustic Streaming Accompanying a Plane Stationary Wave in a Guide[END_REF] still apply here, so we can say that the distortion is caused only by inertial effects. When

Re N L increases, periodic shocks appear and the mean temperature gradient becomes important in our simulations. In their experimental study, Thompson, Atchley and Maccarone [START_REF] Thompson | Influences of a temperature gradient and fluid inertia on acoustic streaming in a standing wave[END_REF] show the existence of some distortion of the streaming field that are not predicted by existing models of the literature in the nonlinear regime. They do not relate this distortion to fluid inertia but rather to the influence of the mean temperature field, and more specifically of the axial temperature gradient induced through a thermoacoustic effect along the horizontal walls of the resonating channel. In an experimental case with no shock waves, Merkli and Thomann [START_REF] Merkli | Thermoacoustic effects in a resonance tube[END_REF] showed that a mean temperature gradient is established inside the tube so that heat is removed close to the velocity antinodes, i.e. at the location of largest viscous dissipation, and heat is produced close to velocity nodes, along the lateral walls. Similar a)

x

Conclusions

The numerical simulations performed demonstrate the transition from regular acoustic streaming flow towards irregular streaming, in agreement with existing experimental data. These are the first simulations, to our knowledge, in good alinement with experiments of nonlinear streaming regimes.

Results show a sizable influence of vertical boundary layers for the chosen configuration. There is also intricate coupling between the mean temperature field and the streaming flow. This coupling effect will be the object of future work. Also, extension of current results in configurations with larger channels is currently in progress.

Figure 1 :

 1 Figure 1: a) Acoustic velocity at the channel's center, as a function of time counted by the number of periods elapsed. b) Mean horizontal velocity and mean temperature variation, on the channel's axis at x = λ/8. Case Re N L = 0.041 (y 0 /δ ν = 10, M = 0.02).

Figure 2 :

 2 Figure 2: a) U st as a function of y/y 0 at x = λ/8, numerical (present study) and analytical [20] results. b) Dimensionless mean fluctuating pressure, p -p 0 along the channel's axis. Case Re N L = 0.041 (y 0 /δ ν = 10, M = 0.02).

Figure 3 (

 3 Figure 3(left) shows the streamlines of the streaming velocity field over the whole length and only over the top half width of the channel. As expected, in the case of small Re N L number values (Figure3(a)), four symmetric streaming cells develop over the length and the half width of the channel: two cells in the boundary layer, and two cells in the core of the channel, identified in the literature as Rayleigh streaming. These results are in agreement with the predictions of analytical models of streaming flows[START_REF] Hamilton | Acoustic streaming generated by standing waves in two-dimensional channels of arbitrary width[END_REF][START_REF] Bailliet | Acoustic streaming in closed thermoacoustic devices[END_REF], and with experimental measurements[START_REF] Moreau | Measurements of inner and outer streaming vortices in a standing waveguide using laser doppler velocimetry[END_REF]. The only noticeable difference is the slight asymmetry of cells with respect to the vertical lines x = λ/8 and x = 3λ/8, due to the presence of vertical boundary layers. Indeed these boundary layers are accounted for in the present simulations but are neglected in the analytical models, and are very far from the measurement area in the experiments.Several simulations have shown that this asymmetry is independent of Re N L as long as the value of the latter remains small with respect to 1.For Re N L > 1, the steady streaming flow is established after the same characteristic time as in the linear case. The recirculation cells become very asymmetric as Re N L increases, and streaming flow becomes irregular (Figure3(b,c,d,e)(left)). This was also observed experimentally (with PIV measurements) by Nabavi, Siddiqui and Dargahi[START_REF] Nabavi | Analysis of regular and irregular acoustic streaming patterns in a rectangular enclosure[END_REF] in a rectangular enclosure.The centers of all streaming cells (boundary layer cells as well as central cells) are displaced towards the ends of the resonant channel, close to the boundary layers next to the vertical walls. PIV measurements byNabavi,

Figure 4 (

 4 a) shows the x variation along the channel's central axis y = 0, of the axial dimensionless streaming velocity component, using as reference velocity the Rayleigh streaming reference velocity [2, 5], u Rayleigh = 3 16 U 2 max /c 0 . There is a clear modification of the velocity profiles as Re N L increases: the sine function associated to slow streaming becomes steeper next to the channel's ends. The slope to the curve at the channel's center (acoustic velocity node) becomes smaller as Re N L increases, then becomes close to zero (curve parallel to the longitudinal axis) for a critical value between 13 and 27, and then changes sign, which indicates the emergence of new streaming cells (Figure 4(a)). Another consequence of the distortion of streaming cells can be observed on the acoustic streaming axial velocity profiles along the width of the channel, shown in Figure 4(b,c,d). The parabolic behavior in the center of the channel at x = λ/8 disappears as Re N L increases (see Figure 4(b)), as a consequence of displacement of the center of each streaming cell toward the velocity node. Figures 4(

Figure 3 :

 3 Figure 3: Streamlines of mean flow on the top half of the channel (left) and acoustic velocity signal at channel's center as a function of time counted by the number of periods elapsed (right) a) Re N L = 0.041 (y 0 /δ ν = 10, M = 0.02). b) Re N L = 4.257 (y 0 /δ ν = 10, M = 0.206). c) Re N L = 9.469 (y 0 /δ ν = 40, M = 0.077). d) Re N L = 12.636 (y 0 /δ ν = 20, M = 0.178). e) Re N L = 27.384 (y 0 /δ ν = 20, M = 0.262). Lengths are normalized with L.

Figure 4 :

 4 Figure 4: Horizontal mean velocity component U st , normalized with 3 16 U 2 max /c 0 for the 5 cases of Figure 3. a) U st along the channel's central axis. b),c) and d) U st as a function of y/y 0 for several sections x.

Figure 5 :

 5 Figure 5: Mean temperature field on the top half of the channel, a) Re N L = 0.041. b) Re N L = 4.257. c) Re N L = 12.636. d) Re N L = 27.383. The difference between minimum and maximum values of temperature is (respectively) : a) ∆T = 0.039K, b) ∆T = 6.39K, c) ∆T = 19.4K, d) ∆T = 44K.

Table 1 :

 1 Values of the parameters of the simulations.

		5	10	50	80	100
	U max (m/s)	7.0	26.45	61.11	70.94	89.96
	y 0 /δ ν	10	40	20	10	20
	y 0 /L	0.0180 0.0718 0.0359 0.018 0.0359
	Re N L	0.041	9.469	12.636 4.257 27.384

Preprint submitted to Wave MotionDecember 11, 2012

Acknowledgements

The authors wish to acknowledge many fruitful discussions with Joël Gilbert and Hélène Bailliet.