Komsun Chaihan

Natthawat Semakul

Vinich Promarak

Thanh-Tuân Bui
email: tbui@cyu.fr

Nawee Kungwan

Fabrice Goubard
email: fabrice.goubard@cyu.fr

Tunable far-red fluorescence utilizing π-extension and substitution on the excited state intramolecular proton transfer (ESIPT) of naphthalene-based Schiff bases: A combined experimental and theoretical study

Keywords: Excited-state intramolecular proton transfer, Double hydrogen bond, Schiff base, Far-reded emission

    Tunable far-red fluorescence utilizing π-extension and substitution on the excited state intramolecular proton transfer (ESIPT) of naphthalene-based Schiff bases: A combined experimental and theoretical study Komsun Chaihan, Natthawat Semakul, Vinich Promarak, Thanh-Tuân Bui, Nawee Kungwan, Fabrice

Introduction

Organic fluorescent compounds with tunable and controllable emission have been gained much attention in recent years due to their versatilities that are relevance in various applications ranging from optoelectronics [START_REF] Park | Application of excited-state intramolecular proton transfer (ESIPT) principle to functional polymeric materials[END_REF][START_REF] Kim | White luminescence from polymer thin films containing excited-state intramolecular proton-transfer dyes[END_REF][START_REF] Park | A white-light-emitting molecule: frustrated energy transfer between constituent emitting centers[END_REF] to molecular sensors in chemistry and biology [START_REF] Choi | A dual channel fluorescence chemosensor for anions involving intermolecular excited state proton transfer[END_REF][START_REF] Klymchenko | Electrochromic modulation of excited-state intramolecular proton transfer: The new principle in design of fluorescence sensors[END_REF][START_REF] Sedgwick | Excited-state intramolecular proton-transfer (ESIPT) based fluorescence sensors and imaging agents[END_REF][START_REF] Zhao | Excited state intramolecular proton transfer (ESIPT): From principal photophysics to the development of new chromophores and applications in fluorescent molecular probes and luminescent materials[END_REF].

Particularly, organic compounds exhibiting emission in far-red to near infrared (NIR) region have emerged as functional materials for utilizations in selective detections of the harmful species in biological environment because of avoiding photodamage to the biological samples [START_REF] Ding | Recent advances in near-infrared II fluorophores for multifunctional biomedical imaging[END_REF][START_REF] Ding | Beyond 1000 nm emission wavelength: Recent advances in organic and inorganic emitters for deep-tissue molecular imaging[END_REF][START_REF] Li | Design of AIEgens for near-infrared IIb imaging through structural modulation at molecular and morphological levels[END_REF][START_REF] Mishra | Towards NIR-active hydroxybenzazole (HBX)based ESIPT motifs: A recent research trend[END_REF]. One of the functional materials with tunable emission is a bifunctional organic compound undergoing excited-state intramolecular proton transfer (ESIPT) through a strong intramolecular hydrogen bond (Intra-HB) between a proton donor (e.g. -OH, -NH 2 , and a newly discovered -SH [START_REF] Liu | Sulfur-based intramolecular hydrogen-bond: Excited-state hydrogen-bond on/off switch with dual room-temperature phosphorescence[END_REF][START_REF] Wang | Chapter open for the excited-state intramolecular thiol proton transfer in the room-temperature solution[END_REF]) in the vicinity of a proton acceptor (e.g. -N=, -C=O) [START_REF] Kwon | Advanced organic optoelectronic materials: Harnessing excited-state intramolecular proton transfer (ESIPT) process[END_REF]. An ESIPT involves with the structural change from the enol form (E) to the keto form (K) via a four-level photochemical process (E E* K* K E), leading to a red-shifted K* emission with a large Stokes shift that avoids an unwanted self-reabsorption and provides a distinct recognition even by the naked eyes [START_REF] Kwon | Advanced organic optoelectronic materials: Harnessing excited-state intramolecular proton transfer (ESIPT) process[END_REF][START_REF] Padalkar | Excited-state intramolecular proton-transfer (ESIPT)-inspired solid state emitters[END_REF]. Moreover, the ESIPT can exhibit either a single proton transfer (SPT) or a double proton transfer (DPT) depending on the number of Intra-HBs within ESIPT compounds [START_REF] Chaihan | Effect of number and different types of proton donors on excitedstate intramolecular single and double proton transfer in bipyridine derivatives: Theoretical insights[END_REF][START_REF] Chaihan | Tunable keto emission of 2-(2′hydroxyphenyl)benzothiazole derivatives with π-expansion, substitution and additional proton transfer site for excited-state proton transfer-based fluorescent probes: Theoretical insights[END_REF] as depicted in Scheme 1. The ESIPT compounds with single Intra-HB normally provide a single emission from the K* form or a dual emission from the tautomeric equilibrium between E* and K* forms. While those with double Intra-HBs can give various emission types originating from three different tautomeric forms; (i) di-enol (DE*) where no proton transfers, (ii) mono-keto (MK*) resulted by only one proton transfers, and (iii) di-keto (DK*) resulted by two protons completely transfer. In addition, ESIPT compounds with double Intra-HBs hypothetically exhibit the keto emission with larger Stokes shifts than those with single Intra-HB, which have triggered various interests in the white-light generators in a single compound [START_REF] Sun | Understanding solvent effects on luminescent properties of a triple fluorescent ESIPT compound and application for white light emission[END_REF], ratiometric sensors [START_REF] Song | A ratiometric fluorescent probe based on ESIPT and AIE Processes for alkaline phosphatase activity assay and visualization in living cells[END_REF], and particularly in the fluorescent molecular probes [START_REF] Song | A novel AIE plus ESIPT fluorescent probe with a large stokes shift for cysteine and homocysteine: Application in cell imaging and portable kit[END_REF]. Typically, the DPT process can proceed via two different mechanisms; (i) a concerted reaction channel where two protons in DE* form simultaneously transfer to yield the DK* form, and (ii) a stepwise reaction channel where one proton in DE* form transfers to give the MK* form first and then another proton transfers after completion of the first step to provide the DK* form (Scheme 1b). Most studied DPT processes would be a stepwise mechanism rather than a concerted one because the concerted proton transfer proceeds on a ridge of energy surfaces making this reaction channel highly infeasible [START_REF] Plasser | Excited-state diproton transfer in [2,2′bipyridyl]-3,3′-diol: The mechanism is sequential, not concerted[END_REF][START_REF] Dong | Determinant of ESIPT mechanism by the structure designed for symmetrical and unsymmetrical molecules[END_REF]. To date, ESIPT compounds exhibiting DPT have been reported both experimental and theoretical approaches in various systems, such as derivatives of bipyridines [START_REF] Chaihan | Tunable keto emission of 2-(2′hydroxyphenyl)benzothiazole derivatives with π-expansion, substitution and additional proton transfer site for excited-state proton transfer-based fluorescent probes: Theoretical insights[END_REF][START_REF] Zhao | Controlling excited state single versus double proton transfer for 2,2′-bipyridyl-3,3′-diol: Solvent effect[END_REF], benzazoles [START_REF] Yang | The effects of amino group meta-and parasubstitution on ESIPT mechanisms of amino 2-(2'-hydroxyphenyl) benzazole derivatives[END_REF][START_REF] Huang | Theoretical investigation of excited-state single and double proton transfer mechanisms for 2,5-bis(benzoxazol-2-yl)thiophene-3,4-diol[END_REF], flavones [START_REF] Meng | A two-channel mechanism for the asymmetric excitedstate intramolecular double proton transfer in a hydroxyflavone derivative[END_REF][START_REF] Serdiuk | Single and double intramolecular proton transfers in the electronically excited state of flavone derivatives[END_REF], quinolines [START_REF] Liu | Theoretical study of the excited state intramolecular double proton transfer and spectral behaviors of 7-hydroxyquinoline-8-carboxylic acid[END_REF], quinones [START_REF] Liu | Insight into the excited-state double proton transfer mechanisms of doxorubicin in acetonitrile solvent[END_REF][START_REF] Zhou | Mechanism for the excited-state multiple proton transfer process of dihydroxyanthraquinone chromophores[END_REF], benzaldehydes [START_REF] Chen | A study of the competitive multiple hydrogen bonding effect and its associated excited-state proton transfer tautomerism[END_REF], and salicylideneanilines [START_REF] Zhang | The mechanism of the excited-state proton transfer of salicylaldehyde azine and 2,2'-[1,4-Phenylenebis{(E)-nitrilomethylidyne}] bisphenol: Via single or double proton transfer[END_REF][START_REF] Shen | Excited state intramolecular single proton transfer mechanism of pigment yellow 101 in solid state: Experiment and DFT calculation[END_REF].

Among them, salicylideneaniline (SA) derivatives as a family of Schiff bases have fascinating abilities to form coordination complexes with metal ions and flexible skeletons as well as easily structural modifications [START_REF] Mayans | From mesocates to helicates: Structural, magnetic and chiro-optical studies on nickel(II) supramolecular assemblies derived from tetradentate Schiff bases[END_REF][START_REF] Da Silva | Schiff bases: A short review of their antimicrobial activities[END_REF][START_REF] Qin | Schiff bases: A short survey on an evergreen chemistry tool[END_REF]. These abilities lead SA derivatives to apply for ligands with multicolor emission for metal ion detections [START_REF] Abu-Dief | A review on versatile applications of transition metal complexes incorporating Schiff bases[END_REF][START_REF] Hu | AIE+ESIPT" bis Schiff-base ligands with multicolor emission and their corresponding Eu(III) complexes: Synthesis and properties research[END_REF][START_REF] Zheng | ESIPT-AIE active Schiff base based on 2-(2′-hydroxyphenyl)benzo-thiazole applied as multi-functional fluorescent chemosensors[END_REF], colorimetric/fluorescent dualchannel sensors [START_REF] Sun | Multifunctional AIE-ESIPT dual mechanism tetraphenylethene-based Schiff base for inkless rewritable paper and a colorimetric/fluorescent dual-channel Zn2+ sensor[END_REF] and proton-transfer molecular switches [START_REF] Jankowska | Electric field control of proton-transfer molecular switching: Molecular dynamics study on salicylidene aniline[END_REF]. Structurally, SA comprising a hydroxyphenyl moiety as a proton donor and an imine moiety as a proton acceptor (Scheme 2a, upper panel) absorbs energy at 340 nm and exhibits a keto emission peaked at 540 nm [START_REF] Mitra | Dynamics of photochromism in salicylideneaniline: A femtosecond spectroscopic study[END_REF][START_REF] Ziółek | An ultrafast excited state intramolecular proton transfer (ESPIT) and photochromism of salicylideneaniline (SA) and its "double" analogue salicylaldehyde azine (SAA). A controversial case[END_REF].

The chemical modifications including π-extensions and substitutions with electron-donating (EDG)/-withdrawing (EWG) groups on either or both hydroxyphenyl ring (R1) and phenyl ring (R2) of SA (Scheme 2a, upper panel) [START_REF] Fang | Excited-state charge coupled proton transfer reaction via the dipolar functionality of salicylideneaniline[END_REF][START_REF] Richey | Spectroscopy and mechanisms of the photo-and thermal reactions of photochromic anils[END_REF][START_REF] Ohshima | Photochromism, thermochromism, and solvatochromism of naphthalene-based analogues of salicylideneaniline in solution[END_REF] generally shift the keto emission peaks of SA derivatives to be longer wavelengths (550-570 nm) than that of SA (540 nm). However, keto emission peaks of SA derivatives with single Intra-HB are quite far from far-red region (~ 600-800 nm), which are limitations for fluorescent probes in the living cells [START_REF] Nolting | Near-infrared dyes: Probe development and applications in optical mole-cular imaging[END_REF][START_REF] Escobedo | NIR dyes for bioimaging applications[END_REF]. Hypothetically, there are various ESIPT compounds with double Intra-HBs exhibiting longer wavelengths of keto emission with larger Stokes shifts than those with single Intra-HB [START_REF] Meng | A two-channel mechanism for the asymmetric excitedstate intramolecular double proton transfer in a hydroxyflavone derivative[END_REF][START_REF] Chen | A study of the competitive multiple hydrogen bonding effect and its associated excited-state proton transfer tautomerism[END_REF][START_REF] Ziółek | An ultrafast excited state intramolecular proton transfer (ESPIT) and photochromism of salicylideneaniline (SA) and its "double" analogue salicylaldehyde azine (SAA). A controversial case[END_REF][START_REF] Van Benthem | Intramolecular hydrogen bonding. 4. Dual fluorescence and excited-state proton transfer in 1,5-dihydroxyanthraquinone[END_REF][START_REF] Kownacki | Single versus double proton transfer in the photochromic Schiff bases. Electronic spectroscopy and synthesis of model compounds[END_REF][START_REF] Ziółek | Excited state proton transfer and photochromism of an aromatic Schiff base. Pico-and femtosecond kinetics of the N, N'-bis (salicylidene)-p-phenylenediamine (BSP)[END_REF]. For Schiff base family, salicylaldehyde azine (SAA) containing two SA units without π-spacer (Scheme 2b, upper panel) was documented to show a red-shift of keto emission peak at 560 nm compared to that of SA (540 nm) [START_REF] Ziółek | An ultrafast excited state intramolecular proton transfer (ESPIT) and photochromism of salicylideneaniline (SA) and its "double" analogue salicylaldehyde azine (SAA). A controversial case[END_REF]. As well, bis(salicylidene)-p-phenylenediamine (BSP) and bis(salicylidene)-1,5-diaminonaphthalene (BSD) having phenyl and naphthyl rings as π-spacer, respectively, were reported to exhibit red-shifted keto emission peaks at 565 nm [START_REF] Ziółek | An ultrafast excited state intramolecular proton transfer (ESPIT) and photochromism of salicylideneaniline (SA) and its "double" analogue salicylaldehyde azine (SAA). A controversial case[END_REF][START_REF] Kownacki | Single versus double proton transfer in the photochromic Schiff bases. Electronic spectroscopy and synthesis of model compounds[END_REF][START_REF] Ziółek | Excited state proton transfer and photochromism of an aromatic Schiff base. Pico-and femtosecond kinetics of the N, N'-bis (salicylidene)-p-phenylenediamine (BSP)[END_REF], nevertheless, such keto emission peaks are still far from 600 nm. Whilst other ESIPT systems with double Intra-HBs providing the keto emission in far-red to NIR emission have been derivatives of flavones (~670 nm) [START_REF] Meng | A two-channel mechanism for the asymmetric excitedstate intramolecular double proton transfer in a hydroxyflavone derivative[END_REF], quinones (~620 nm) [START_REF] Van Benthem | Intramolecular hydrogen bonding. 4. Dual fluorescence and excited-state proton transfer in 1,5-dihydroxyanthraquinone[END_REF], and benzaldehydes (~660 nm) [START_REF] Chen | A study of the competitive multiple hydrogen bonding effect and its associated excited-state proton transfer tautomerism[END_REF]. However, ESIPT compounds containing double Intra-HBs with tunable fluorescence in farred to NIR emission (600-950 nm) [START_REF] Li | Progress in tuning emission of the excitedstate intramolecular proton transfer (ESIPT)-based fluorescent probes[END_REF] that is a genuine feature for utilization in fluorescent probes in biological systems [START_REF] Sedgwick | Excited-state intramolecular proton-transfer (ESIPT) based fluorescence sensors and imaging agents[END_REF] are still lacking to date. Therefore, the design of double Intra-HB systems with desirable emission is still essential to develop novel ESIPT-based fluorescent probes. Herein, we designed a series of BSD-based Schiff bases with different chemical modifications (Scheme 2c, lower panel) on the proton donor moieties (hydroxyphenyl ring); (i) a phenyl π-extension (NP-BSD), (ii) EDG substituents with diethylamino (DA-BSD) and julolidine (PQ-BSD), and (iii) an EWG substituent with fluorine (F-BSD) to moderate the position of the keto emission close to 600 nm. It is well known that the modification of πconjugation and introduction of EDG and EWG are straightforward methods to shift the absorption and emission spectra of chromophores at longer wavelengths [START_REF] Zhao | Excited state intramolecular proton transfer (ESIPT): From principal photophysics to the development of new chromophores and applications in fluorescent molecular probes and luminescent materials[END_REF][START_REF] Kwon | Advanced organic optoelectronic materials: Harnessing excited-state intramolecular proton transfer (ESIPT) process[END_REF][START_REF] Barbon | Effect of extended π conjugation on the spectroscopic and electrochemical properties of boron difluoride formazanate complexes[END_REF]. Particularly, a julolidine moiety at meta-position of BSD-based Schiff bases could show the improved optical properties, including red-shifted emission, high quantum yield and good photostability, owing to its restricted motion and electron-donating capability [START_REF] Xia | A julolidine-fused anthracene derivative: Synthesis, photophysical properties, and oxidative dimerization[END_REF][START_REF] Varejão | Synthesis and derivatization of julolidine: A powerful heterocyclic structure[END_REF]. In addition, a fluorine substituent at para-position could increase acidity of a proton donor driving tautomerization to yield the keto form, meaning that keto emission is located at the long wavelength, due to its electronwithdrawing inductive effect [START_REF] Siodła | Toward a physical interpretation of substituent effects: The case of fluorine and trifluoromethyl groups[END_REF]. In this design, the substitutions are introduced in different positions on hydroxyphenyl rings considered by the impact of substituent from Hammett parameter [START_REF] Hammett | The effect of structure upon the reactions of organic compounds. Benzene derivatives[END_REF][START_REF] Hansch | A survey of Hammett substituent constants and resonance and field parameters[END_REF] in which the EDG is placed on the meta-position and the EWG is placed on the para-position. The characteristics of BSD-based Schiff bases governed by the phenyl πextension and substitutions with EDG and EWG were experimentally characterized by spectroscopic techniques. Additionally, theoretical calculations were incorporated to give in-dept insights into different chemical modifications affecting the molecular structures, absorption and emission spectra, and proton transfer processes in all studied compounds.

Experiment and calculation

Chemical materials were obtained from Tokyo Chemical Industry (France) and solvents were purchased from Sigma-Aldrich and VWR International (France). All chemicals were used as received without further purification. 1 H and 13 C nuclear magnetic resonance (NMR) spectra of synthesized compounds were recorded on Bruker AVANCE III HD 600 (600 MHz for 1 H, 151 MHz for 13 C) and Bruker NEO TM 500 MHz (500 MHz for 1 H, 126 MHz for 13 C) spectrometers using deuterated dimethyl sulfoxide (DMSO-d 6 ) and chloroform (CDCl 3 ) as solvents. High-resolution (HR) matrix-assisted-laser-desorption/ionization (MALDI) mass spectra (MS) were obtained on Bruker Autoflex speed spectrometer using α-phenylcinnamic acid as a matrix. Melting points of all synthesized compounds were examined on Krüss KSP1N melting meter and the recorded data was uncorrected. Fourier-transform infrared (FT-IR) spectra were carried out for all synthesized compounds in the attenuated total reflectance (ATR) mode on Bruker-Tensor 27 FT-IR spectrometer. Single-crystal X-ray diffraction data of synthesized compounds were collected on Bruker D8 Venture single-crystal diffractometer. The crystal structures were refined and visualized using Olex2-1.5 and MERCURY program packages [START_REF] Dolomanov | OLEX2: A complete structure solution, refinement and analysis program[END_REF].

UV-Vis absorption spectra of synthesized compounds in solution were collected on PerkinElmer Lambda 1050 UV/Vis/NIR spectrometer using quartz cell. Steady state fluorescence spectra were measured using Edinburgh Instruments FLS 980 spectrometers.

Synthesis

BSD and its derivatives were synthesized according to a conventional condensation reaction [START_REF] Măluţan | A fluorescence emission, T-IR and UV-VIS absorption study of the some uranium (VI) Schiff bases complexes[END_REF]. Generally, 0.348 mmol of 1,5-diaminonaphthalene and 0.632 mmol of different aldehydes (as shown in Table 1) were suspended in 10 mL of absolute ethanol (EtOH). The suspension was added with a small amount ( 5 drops) of glacial acetic acid (CH 3 COOH) as catalyst and then refluxed at 85˚C for 4 h. Under heating and stirring, the color of the suspension gradually changed from yellow to orange or red. After completion, the reaction was cooled down and further added with water (30 mL). The precipitate was filtered off and washed with water followed by cold ethanol. The crude solid was purified by recrystallization from DMSO/CHCl 3 (v/v ≈ 4/1) at 4˚C to yield all compounds as yellowish-orange or reddish crystal.

Table 1. Results of the synthesis of BSD and its derivatives.

Computational calculation

Geometric structures of all synthesized compounds in di-enol (DE), mono-keto (MK), and di-keto (DK) forms were fully optimized in the ground (S 0 ) state and the first singlet-excited (S 1 ) state using density functional theory (DFT) and time-dependent density functional theory (TD-DFT), respectively, with the use of B3LYP/TZVP level. This B3LYP/TZVP level has been validated to be a suitable method for describing electronic properties of synthesized compounds compared to other methods as given in Section S2.1 of the Electronic Supplementary Information (ESI). Chloroform (CHCl 3 ) was incorporated through polarizable continuum model (PCM) using the integral equation formalism variant (IEF-PCM) [START_REF] Cammi | Remarks on the use of the apparent surface charges (ASC) methods in solvation problems: Iterative versus matrix-inversion procedures and the renormalization of the apparent charges[END_REF][START_REF] Cancès | A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics[END_REF] in consistent with the experiment. Vertical excitation energies were calculated from all optimized structures with ten low-lying absorbing and emitting electronic transitions to simulate the absorption and emission spectra, respectively. In addition, frontier molecular orbitals (FMOs) are analyzed to visualize the electronically excited-state behaviors of all synthesized compounds. Occurrences of SPT and DPT processes of all compounds are predicted using the PT barriers and reaction energies. All calculations were carried out using the Gaussian 09 program suit [63].

Results and discussion

Synthesis and structural characterization

BSD and its derivatives are synthesized by a conventional condensation of aromatic diamines (as primary amines) with aromatic aldehydes in an equivalent ratio of 1:2 yielding symmetric Schiff bases with double Intra-HBs (as outlined in Table 1). The synthesized compounds are obtained as yellowish-orange or reddish crystals after recrystallization. The yields of products are dependent on the used types of aldehydes. In the cases of aldehydes as salicylaldehyde, 2-hydroxy-1-naphthaldehyde and 5-fluorosalicylaldehyde, high yields over 90% are observed for BSD, NP-BSD, and F-BSD, respectively. Whereas those as 4-(diethylamino)

salicylaldehyde and 8-hydroxy-julolidine-9-carboxaldehyde provide rather low yields (54% and 30% for DA-BSD and PQ-BSD, respectively) due to the aldehyde's stability in the reaction mediums. These low products might be from the protonation of aromatic tertiary amine by acetic acid yielding the oxidative product (azane oxide) [START_REF] Roberts | Basic principles of organic chemistry[END_REF], as observed in the previous synthesis of polyanils [START_REF] Frath | Polyanils and polyboranils: Synthesis, optical properties, and aggregation-induced emission[END_REF]. The molecular structures of synthesized compounds are confirmed by 1

Crystal structures

To get insights into the molecular organizations of all studied compounds, their suitable single crystal structures are obtained from co-solvent recrystallization with DMSO/CHCl 3 (for BSD and PQ-BSD) and CHCl 3 /hexane (for NP-BSD, DA-BSD, and F-BSD), and then characterized using a single-crystal X-ray diffractometer. Most of studied compounds forms crystal structures in the monoclinic crystal system with space groups of P2 1 /c for BSD, PQ-BSD and F-BSD, P2 1 /n for DA-BSD, and the orthorhombic crystal system with a space group of Pca2 1 for NP-BSD (Section S1.4 of ESI). As depicted in Figure 1, the main skeleton of single molecular structures of all compounds is not planar with dihedral angles between the planes of the central naphthalene-diimine moiety and the hydroxyaromatic moiety in the range of 37-43˚ and such angles change upon modification of π-extension and substitutions with EDG and EWG.

Typically, the changes of dihedral angles are not solely rationalized by an X-ray single crystal analysis, thereby the optimized S 0 -structures are analyzed for all compounds as given in Section S2.3 of ESI. In all cases, the DE structure is the stable form in S 0 state with C1C2N1C3 and C6C7N2C8 dihedral angles in the range of 41.6-43.3˚ (Table S2.

Absorption spectra

The steady-state electronic absorption spectra of all studied compounds are characterized in CHCl 3 solution with concentration of ~1х10 -5 M at room temperature as shown in Figure 2.

Their maximum wavelengths of absorption spectra with molar absorption coefficient ( ) are listed in Table 2. UV-Vis absorption spectra of all compounds are observed with major absorption peaks in the range of 370-420 nm. Note that the measured absorption peaks of BSD (370 nm) and DA-BSD (404 nm) agree well with the previous reports by Kownacki et al. at 377 nm [START_REF] Kownacki | Single versus double proton transfer in the photochromic Schiff bases. Electronic spectroscopy and synthesis of model compounds[END_REF] and Frath et al. at 404 nm [START_REF] Frath | Polyanils and polyboranils: Synthesis, optical properties, and aggregation-induced emission[END_REF], respectively. From Table 2, it is worth to notice that the molar absorption coefficient ( ) of BSD at 13.38 x 10 4 M -1 •cm -1 is greater than that of BSP at 3.50 x 10 4 M -1 •cm -1 [START_REF] Ziółek | An ultrafast excited state intramolecular proton transfer (ESPIT) and photochromism of salicylideneaniline (SA) and its "double" analogue salicylaldehyde azine (SAA). A controversial case[END_REF] which might be from the π-extension inducing the effect changing from the phenyl to the naphthalene. The value of DA-BSD at 8.08 x 10 4 M -1 •cm -1 is comparable to the reported value at 5.78 x 10 4 M -1 •cm -1 by Frath et al [START_REF] Frath | Polyanils and polyboranils: Synthesis, optical properties, and aggregation-induced emission[END_REF]. However, when comparing among the BSD derivatives, the value is decreased upon chemical modifications (compared to BSD) particularly the NP-BSD and F-BSD. This finding might be from the introduction of the phenyl π-extension and substitutions (both EDG and EWG) at the end sites of ESIPT units somehow reduces the value, whereas the π-extension in the π-spacer between the two ESIPT units normally increases this value [START_REF] Ziółek | An ultrafast excited state intramolecular proton transfer (ESPIT) and photochromism of salicylideneaniline (SA) and its "double" analogue salicylaldehyde azine (SAA). A controversial case[END_REF]. As seen in Figure 2, all modifications make absorption peaks of BSD derivatives to be red-shifted (378-420 nm) compared to that of BSD (370 nm), in which phenyl π-extension and EDGs (diethylamino and julolidine) induce absorption peaks of NP-BSD (402 nm), DA-BSD (404 nm) and PQ-BSD (420 nm) to be longer wavelengths than that of F-BSD (378 nm). This is to say that the π-extension and EDG substitution significantly cause the red-shift in absorption, whereas the EWG substitution makes absorption to be slightly redshifted. However, the red-shifts induced in the absorption maxima of BSD derivatives not only are relevant to their molecular structures governed by such modifications, but also related to their electronic properties corresponding to frontier molecular orbitals and vertical excitation energies. To get in-depth insights into the effect of modifications on electronic properties, vertical excitation energies are analyzed by using DE form for all compounds (Table S2.2 of ESI). Based on our calculations, excitation energies of all compounds are associated with their absorption peaks (Table 2). The main contribution of excitation energy is corresponded to the electronic transition from the highest occupied molecular orbital (HOMO) to the lowest unoccupied molecular orbital (LUMO) with oscillator strengths in the range of 0.854-1.693 (Table 2). The red-shift in absorption peaks of BSD derivatives can be rationalized by the shift of HOMO-LUMO energy levels affected by the certain modifications. For example, the phenyl π-extension (NP-BSD) moderately destabilizes HOMO energy to be higher but slightly stabilizes LUMO energy to be lower. And EDGs (DA-BSD and PQ-BSD) notably raise HOMO energies and moderately give rise LUMO energies to be higher. As well, EWG (F-BSD) slightly stabilizes

Prediction of proton transfer occurrence

Based on the results of molecular structures, Intra-HBs are formed along the two O-H•••N coordinates suggesting the possibility of proton transfers in all studied compounds.

However, the molecular structures obtained from single crystal analysis are in S 0 state which is not sufficient information to predict the proton transfer occurrences. Thus, optimized S According to Figure 4 and Table S2.7, energies of PT processes in S 0 state of all studied compounds increase as functions of lengthening O-H bonds, resulting in large PT barriers in the range of 2.99-5.61 kcal/mol with highly endothermic whereas PT barriers in S 1 state are in the range of 2.03-4.29 kcal/mol with exothermic or slightly endothermic. These findings suggest that the PT process could be difficult to take place in S 0 state but possible to occur in S 1 state which are correlated to the Intra-HB strength. To verify that the Intra-HB becomes strengthened upon photoexcitation, the changes of structural parameters (Section S2. S2.3). As well, the O-H stretching frequencies are shifted to lower energies in S 1 state than that in S 0 state with deviations in the range of 82-129 cm -1 (Table S2.5), promoting the proton transfer in all compounds.

Despite rather high PT barrier in S 1 state, energies of Frank-Condon (FC) excitations are enough to surpass the 1 st PT barrier (Figure 4), implying that the SPT process is possible for all compounds regardless of modifications. Judging from the degrees of PT barriers and reaction energies, substitutions with EDGs and EWG do not significantly affect the 1 st PT barriers of DA-BSD (3.36 kcal/mol), PQ-BSD (3.33 kcal/mol), and F-BSD (3.23 kcal/mol) but destabilize their keto energies to be slightly endothermic. Conversely, the phenyl π-extension provides rather low the 1 st PT barrier for NP-BSD (2.03 kcal/mol) and highly exothermic reaction (-1.38 kcal/mol) compared to BSD. Thus, the SPT process may be more effective in NP-BSD but less effective in DA-BSD, PQ-BSD and F-BSD compared to that of BSD. On the assumption that SPT process is easily completed in all compounds, the DPT process might be rather difficult to occur due to high the 2 nd PT barriers (3.74-4.29 kcal/mol) and highly endothermic reaction (1.19-1.86 kcal/mol). This can be rationalized by slightly shortening Intra-HBs (Table S2.4) and small redshifts of O-H stretching (Table S2.6) in MK form compared to DE form (associated with the DPT process or the 2 nd PT). Interestingly, the 2 nd PT barrier of NP-BSD is low (2.18 kcal/mol) after completion of SPT process with highly exothermic (-1.73 kcal/mol), suggesting that the DPT process could be possibly observed in NP-BSD. Overall, the phenyl π-extension stabilizes the keto energy of NP-BSD making both SPT and DPT processes to be exothermic, whereas the substitutions with EDGs (diethylamino and julolidine) and EWG (fluorine) destabilize the keto energies particularly in DK* form leading to only SPT process possible in DA-BSD, PQ-BSD, and F-BSD compared to that of BSD [START_REF] Dong | Determinant of ESIPT mechanism by the structure designed for symmetrical and unsymmetrical molecules[END_REF][START_REF] Wang | Unraveling photoexcitation dynamical behavior for bis (salicylidene)-1,5-diaminonaphthalene (BSD) system[END_REF]. Moreover, the Intra-HB is strengthened for the nitro and cyano groups [START_REF] Shapenova | Resonance assisted hydrogen bonding phenomenon unveiled through both experiments and theory: A new family of ethyl nsalicylideneglycinate dyes[END_REF][START_REF] Barboza | Photophysical transformations induced by chemical substitution to salicylaldimines[END_REF] however, the EWG with fluorine substitution does not significantly increase the strength of Intra-HB because of its very weak electron-withdrawing ability. Furthermore, it is worth to mention that the π-electron delocalization especially phenyl πextension is important for fine-tuning the strength of Intra-HBs [START_REF] Barboza | Photophysical transformations induced by chemical substitution to salicylaldimines[END_REF][START_REF] Sobczyk | Interrelation between H-bond and pi-electron delocalization[END_REF] that can be further supported by the stronger Intra-HBs of NP-BSD at 0.033 a.u. compared to that of BSD at 0.024 a.u. (see the hydrogen bonding energy from topology analysis as given in Table S2.8).

Emission spectra

The emission spectra of all compounds measured in CHCl 3 solution with concentration of ~1х10 -5 M at room temperature are shown in Figure 5. Their maximum wavelengths of emission spectra with fluorescence quantum yield ( ) are given in Table 3. Upon excitation corresponding to their absorption maxima, emission peaks of BSD and its derivatives are observed in the range of 498-582 nm and the shapes of their emission spectra are different. For BSD, a yellow-green emission peak is observed at 565 nm with a large Stokes shift of 9328 cm -1 compared to its absorption peak. This large Stokes shift with no spectral overlap between The absolute fluorescence quantum yield ( ) in dilute CHCl 3 solution of all synthesized compounds recorded on an integrating sphere are in the range of 0.83-3.51% (Table 3). Low values are found for BSD (0.89%), NP-BSD (0.83%), and F-BSD (0.93%) whereas high values are found for DA-BSD (2.06%) and PQ-BSD (3.51%). The degree of values may be relevant to the electron-donating ability of modifications [START_REF] Xia | A julolidine-fused anthracene derivative: Synthesis, photophysical properties, and oxidative dimerization[END_REF]. However, all synthesized Schiff base compounds have rather low values compared to other ESIPT systems about 5-10% [START_REF] Benelhadj | White emitters by tuning the excited-state intramolecular proton-transfer fluorescence emission in 2-(2′-hydroxybenzofuran)benzoxazole dyes[END_REF][START_REF] Padalkar | Highly emissive excited-state intramolecular proton transfer (ESIPT) inspired 2-(2′-hydroxy)benzothiazolefluorene motifs: spectroscopic and photophysical properties investigation[END_REF], which might be caused by the skeleton rotation after proton transfer. Based on potential energies curves of scanning NCCC torsion angles (Figure S2.11), small different energies between S 1 and S 0 (less than 2.0 eV) are estimated for all compounds at the critical angle (90˚), suggesting the conical intersection [START_REF] Chaihan | Effect of number and different types of proton donors on excitedstate intramolecular single and double proton transfer in bipyridine derivatives: Theoretical insights[END_REF][START_REF] Barboza | Photophysical transformations induced by chemical substitution to salicylaldimines[END_REF][START_REF] Daengngern | Theoretical investigation of 2-(iminomethyl)phenol in the gas phase as a prototype of ultrafast excited-state intramolecular proton transfer[END_REF][START_REF] Jankowska | Tailoring the Schiff base photoswitching -A non-adiabatic molecular dynamics study of substituent effect on excited state proton transfer[END_REF][START_REF] Muriel | A theoretical study of the photodynamics of salicylidene-2-anthrylamine in acetonitrile solution[END_REF] might be easily accessible after PT process.

This internal conversion phenomenon from S 1 to S 0 may lead to nonradiative transition resulting in the observed low . Accordingly, the observed keto emission peaks of all compounds are mainly from the mono-keto form. Among synthesized compounds, F-BSD exhibiting unique properties including far-red keto emission peaked at 582 nm and a large Stokes shift of 9273 cm -1 which are potential for fluorescent probes because of avoiding photodamage to the living cells and providing a distinct recognition even by the naked eyes.

Scheme 1 .

 1 Scheme 1. Molecular structures and possible tautomeric species of ESIPT compounds with (a)

Scheme 2 .

 2 Scheme 2. Upper panel: molecular designs of (a) Schiff bases with chemical modifications on

H & 13 C

 13 NMR, HR MALDI-TOF MS, and FT-IR spectroscopic techniques as given in Section S1.1-S1.3 of the ESI. In the 1 H NMR spectra, large downfield shifts ( > 10 ppm) of the -OH protons are observed, suggesting the existence of Intra-HB in all synthesized compounds[START_REF] Massue | Synthesis of luminescent BPh2-coordinated 2-(2′-hydroxyphenyl)benzoxazole (HBO)[END_REF]. Note that the signals of -OH protons in 1 H NMR spectra of BSD and F-BSD appear as broad singlet because the proton is exchangeable through -S=O•••H-O in DMSO-d6. The 13 C NMR spectra are not obtained in NP-BSD and DA-BSD because of their low solubility in CDCl 3 or DMSO-d6 even at high temperature[START_REF] Frath | Polyanils and polyboranils: Synthesis, optical properties, and aggregation-induced emission[END_REF]. The lack of solubility also makes 1 H NMR spectra of NP-BSD and DA-BSD to present the signals of interferences. The FT-IR spectra of all synthesized compounds display a broad signal of O-H stretching mode in the range of 2500-3200 cm -1 , supporting the Intra-HB (O-H•••N) formation in these compounds. Moreover, the structures of BSD and its derivatives are further accessible by the single-crystal X-ray diffraction analysis.

  photoexcitation which will be discussed in Section 3.4.

Figure 1 .

 1 Figure 1. X-ray single crystal structures of BSD and its derivatives with the thermal ellipsoids

Figure 2 .

 2 Figure 2. UV-Vis absorption spectra of BSD and its derivatives in CHCl 3 .

  in the following order; PQ-BSD (2.84 eV) < NP-BSD (2.86 eV) < DA-BSD (2.93 eV) < F-BSD (3.00 eV) < BSD (3.07 eV). As depicted in Figure3, electron densities on HOMO and LUMO are mainly attributed to the π and π* characters, respectively, on the central naphthalene moiety in DE form as observed in derivatives of BSD[START_REF] Dong | Determinant of ESIPT mechanism by the structure designed for symmetrical and unsymmetrical molecules[END_REF][START_REF] Wang | Unraveling photoexcitation dynamical behavior for bis (salicylidene)-1,5-diaminonaphthalene (BSD) system[END_REF][START_REF] Miengmern | Aggregation-induced emission enhancement (AIEE) of N,N′-bis(salicylidene)-pphenylenediamine Schiff base: Synthesis, photophysical properties and its DFT studies[END_REF][START_REF] Hao | Insight into the new excited-state intramolecular proton transfer (ESIPT) mechanism of N,N′-bis(salicylidene)-p-phenylenediamine (p-BSP)[END_REF]. Note that electron distributions from HOMO to LUMO in BSD and its derivatives are quite different. The EDGs with diethylamino (DA-BSD) and julolidine (PQ-BSD) induce electron densities localized on electron-donating moieties in HOMO moving to central naphthalene moiety in LUMO which is assigned to the partial intramolecular charge transfer character, whereas phenyl π-extension (NP-BSD) and EWG with fluorine (F-BSD) show dominant π→π* character like in the electron distributions of BSD. Such characteristics of electron-donating ability may lead to the large red-shift of absorption particularly in PQ-BSD compared to other compounds.

Figure 3 .

 3 Figure 3. Frontier molecular orbitals (HOMO and LUMO) with energy levels and electron

1 -

 1 structures are analyzed to further use for construction of potential energy curves (PECs) along PT coordinates as shown in Figure 4. In all cases, the studied compounds can exhibit either or both SPT (generating MK* form) and DPT (generating DK* form) corresponding to the 1 st PT (DE→MK) and 2 nd PT (MK→DK), respectively. To predict the SPT and DPT processes, there are two important factors to be considered; (i) the PT barriers are obtained from energy differences between the highest point relative to the initial point on PECs which are constructed from the optimized DE form (for the 1 st PT) and MK form (for the 2 nd PT) in S 0 and S 1 states by scanning O-H bond lengths with a step of 0.05 Å, and (ii) the reaction energies ( E) are calculated from different energies between optimized keto and enol structures; E DE→MK = E MK -E DE and E MK→DK = E DK -E MK . The relative PT barriers and E values of all compounds are listed in TableS2.7.

Figure 4 .

 4 Figure 4. Calculated potential energy curves along PT coordinates in S 0 and S 1 states for (a)

  photoexcitation, the changes of structural parameters (Section S2.3 of ESI) and the spectral shifts

  emission and absorption curves (Figure S1.24 of ESI) suggests the occurrence of ESIPT process in BSD. The yellow-green emission peak at 565 nm agrees well with calculated MK* emission at 524 nm (Figure S2.6 of ESI) which is in line with calculations by Dong et al. (552 nm in acetonitrile) [22]. The spectral shifts (blue-shift and red-shift) and shapes of emission spectra are significantly controlled by different modifications compared to BSD. The phenyl π-extension (NP-BSD) and EDG with diethylamino (DA-BSD) show a dual emission with main emission peaks at 516 and 498 nm and their shoulders at around 540 and 530 nm, respectively. These two emission bands of NP-BSD and DA-BSD are from different ESIPT species as shown in Figure S2.7 (for NP-BSD) and S2.8 (for DA-BSD). For NP-BSD, emission bands at 516 and 540 nm are from MK* and DK* forms. And for DA-BSD, emission bands at 498 and 530 nm are from MK* and DE* forms, respectively. These results are accordance with their ESIPT occurrences as given in Section 3.4. The emission peaks of NP-BSD and DA-BSD at 516 nm and 498 nm areblue-shifted with small Stokes shifts of 5496 and 4672 cm -1 , respectively. The EDG with julolidine (PQ-BSD) provides a broad emission spectrum with maximum wavelength at 565 nm with a Stokes shift of 6110 cm -1 , which is not different from that of BSD. This large Stokes shift implies that PQ-BSD is one of the ESIPT characteristic and the observed emission peak is from MK* species (FigureS2.9) supported by its PEC result (Figure4) in which only SPT process could be possible. Note that the broad emission spectrum is due to characteristics of julolidine moiety as found in the previous work by Nano et al[START_REF] Nano | Panchromatic luminescence from julolidine dyes exhibiting excited state intramolecular proton transfer[END_REF]. The EWG with fluorine substituent (F-BSD) gives a far-reded emission band peaked at 582 nm with a large Stokes shift of 9273 cm -1 that is the evidence of the ESIPT occurrence in F-BSD. This far-red emission peak is in well agreement with calculated MK* emission (FigureS2.10) which is in line with its PEC result (Figure4). As illustrated in Figure6, electron densities of all compounds are different, and their spectral shifts (blue-shift and red-shift) can be rationalized by keto energies related mostly to LUMO energy levels. The phenyl π-extension and EDG (diethylamino and julolidine) destabilize LUMO energy level of MK* to be higher than that of BSD leading to inefficiency of spectral shift in emission spectra (blue-shift) in NP-BSD, DA-BSD, and PQ-BSD. Whereas the EWG (fluorine) stabilizes LUMO energy level in MK* form inducing a red-shifted MK* emission in F-BSD.

Figure 5 .

 5 Figure 5. Normalized emission spectra of BSD and its derivatives in CHCl 3 .

Figure 6 .

 6 Figure 6. Frontier molecular orbitals (HOMO and LUMO) with energy levels and electron

  HRMS (MALDI-TOF): calcd for C 24 H 19 N 2 O 2 [M+H] + : 367.1447, found: 367.2269. .6, 130.0, 129.2, 126.1, 121.0, 113.9, 113.1, 109.0, 106.5, 50.2, 49.8, 27.3, 22.0, 21.1, 20.4. HRMS (MALDI-TOF): calcd for C 36 H 37 N 4 O 2 [M+H] + : 557.2917, found: 557.4339.

	Bis(5-fluorosalicylidene)-1,5-diaminonaphthalene (F-BSD). Yellowish-orange crystal.
	Melting point: 287 °C. IR (ATR): ν = 1145, 1244, 1317, 1483, 1580, 1618, 2775, 3057 cm -1 .
	1 H NMR (600 MHz, DMSO-d 6 ): δ = 9.02 (s, 2H), 8.14 (d, J = 8.4 Hz, 2H), 7.67-7.69 (m, 4H),
	7.48 (d, J = 7.3 Hz, 2H), 7.33-7.37 (m, 2H), 7.06-7.08 (m, 2H). 13 C NMR (151 MHz, DMSO-
	d 6 ): δ = 162.7, 156.9, 155.6 (d, J = 234.8 Hz), 146.5, 128.9, 127.4, 122.0, 121.0 (d, J = 23.6 Hz),

). Yellowish-orange crystal. Melting point: 221 °C. IR (ATR): ν = 1205, 1275, 1400, 1576, 1611, 2750, 3047 cm -1 . 1 H NMR (600 MHz, DMSO-d 6 ): δ = 12.96 (brs, 2H), 9.04 (s, 2H), 8.12 (d, J = 8.4 Hz, 2H), 7.79 (d, J = 7.3 Hz, 2H), 7.68 (t, J = 7.8 Hz, 2H), 7.47-7.51 (m, 4H), 7.02-7.06 (m, 4H). 13 C NMR (151 MHz, DMSO-d 6 ): δ = 164.7, 160.7, 146.4, 134.1, 132.9, 128.9, 127.4, 121.7, 120.3, 119.8, 117.2, 115.7.

Bis(2-hydroxy-1-naphthylmethylene)-1,5-diaminonaphthalene (NP-BSD). Reddish crystal. Melting point: >350 °C. IR (ATR): ν = 1207, 1287, 1340, 1582, 1605, 2652, 3051 cm -1 . 1 H NMR (500 MHz, CDCl 3 ): δ = 15.67 (s, 2H), 9.58 (s, 2H), 8.32 (d, J = 8.5 Hz, 2H), 8.23 (d, J = 8.5 Hz, 2H), 7.91 (d, J = 9.1 Hz, 2H), 7.80 (d, J = 8.2 Hz, 2H), 7.66 (t, J = 7.8 Hz, 2H), 7.57 (t, J = 7.7 Hz, 2H), 7.45 (d, J = 7.2 Hz, 2H), 7.40 (t, J = 7.4 Hz, 2H), 7.24 (d, J = 9.4 Hz, 2H). HRMS (MALDI-TOF): calcd for C 32 H 23 N 2 O 2 [M+H] + : 467.1760, found: 467.2749. Bis(4-diethylamino-2-hydroxyl-1-phenylmethylene)-1,5-diaminonaphthalene (DA-BSD). Dark-red crystal. Melting point: 302 °C. IR (ATR): ν = 1130, 1236, 1346, 1579, 1618, 2690, 2963, 3069 cm -1 . 1 H NMR (500 MHz, CDCl 3 ): δ = 13.92 (s, 2H), 8.51 (s, 2H), 8.17 (d, J = 8.6 Hz, 2H), 7.50 (t, J = 7.8 Hz, 2H), 7.23 (d, J = 8.8 Hz, 2H), 7.19 (d, J = 7.2 Hz, 2H), 6.26-6.30 (m, 4H), 3.43 (q, J = 14.2, 7.1 Hz, 8H), 1.24 (t, J = 7.1, 12H). HRMS (MALDI-TOF): calcd for C 32 H 37 N 4 O 2 [M+H] + : 509.2917, found: 509.4296. Bis(8-hydroxyjulolidine-9-methylene)-1,5-diaminonaphthalene (PQ-BSD). Dark-red crystal. Melting point: 320 °C. IR (ATR): ν = 1177, 1305, 1348, 1576, 1634, 2700, 2934, 3040 cm -1 . 1 H NMR (600 MHz, CDCl 3 ): δ = 14.03 (s, 2H), 8.44 (s, 2H), 8.18 (d, J = 8.5 Hz, 2H), 7.48 (t, J = 7.9 Hz, 2H), 7.16 (d, J = 7.3 Hz, 2H), 6.83 (s, 2H), 3.26-3.28 (m, 8H), 2.82 (t, J = 6.4 Hz, 4H), 2.72 (t, J = 6.2 Hz, 4H), 1.96-2.01 (m, 8H). 13 C NMR (126 MHz, CDCl 3 ): δ = 161.5, 159.1, 147.0, 146120.8 (d, J = 7.6 Hz), 118.6 (d, J = 7.5 Hz), 117.0 (d, J = 23.2 Hz), 115.6. HRMS (MALDI-TOF): calcd for C 24 H 17 F 2 N 2 O 2 [M+H] + : 403.1258, found: 403.2558.

Table 2 .

 2 Maximum wavelengths of absorption spectra ( abs ) with molar absorption coefficient ( ) and vertical excitation energies with oscillator strengths (f) of DE form computed at TD-B3LYP/TZVP level in CHCl 3 for BSD and its derivatives.

		Experiment		TD-DFT calculation
	Compound	(nm)	(10 4 M -1 •cm -1 )	(nm)	f	MOs contribution
	BSD	370	13.38	404	0.787 HOMO→LUMO (98%)
	NP-BSD	402	2.47	433	0.925 HOMO→LUMO (97%)
		320		354	0.300 HOMO-1→LUMO+1 (80%)
	DA-BSD	404	8.08	424	1.499 HOMO→LUMO (97%)
		330		343	0.873 HOMO-1→LUMO+1 (93%)
	PQ-BSD	420	7.35	437	1.469 HOMO→LUMO (97%)
		370		358	0.806 HOMO-1→LUMO+1 (93%)
	F-BSD	378	2.66	413	0.766 HOMO→LUMO (98%)

Table 3 .

 3 Maximum wavelengths of emission spectra ( emis , nm) with fluorescence quantum yield 1 ( , %), Stokes shifts (cm -1 ) and emission energies with oscillator strengths (f) computed at TD-2 B3LYP/TZVP level in CHCl 3 for BSD and its derivatives.

	3
	Experimental
	Compound

* H→L represents HOMO→LUMO.

[START_REF] Zhou | Mechanism for the excited-state multiple proton transfer process of dihydroxyanthraquinone chromophores[END_REF]

5. Acknowledgements K. Chaihan gratefully acknowledges the Royal Golden Jubilee Ph.D. Program Scholarship (PHD/0005/2560) and the Eiffel Excellence Scholarship Program (919741J) for the fellowship. The authors would like to thank Laboratoire de Physicochimie des Polymères et des Interfaces (LPPI), CY Cergy Paris Université, Cergy, France for supporting chemicals and working space to do the synthesis and Vidyasirimedhi Institute of Science and Technology (VISTEC), Rayong, Thailand for providing instrument facilities. This research work was partially supported by Chiang Mai University. Department of Chemistry, Faculty of Science and Graduate School of Chiang Mai University, Chiang Mai, Thailand are also acknowledged.

CRediT authorship contribution statement


A Cannici
email: aron.cannici@onera.fr

D Passiatore
email: donatella.passiatore@poliba.it

L Sciacovelli
email: luca.sciacovelli@ensam.eu

P Cinnella
email: paola.cinnella@sorbonne-universite.fr

A PRIORI TESTS OF TURBULENCE MODELS FOR COMPRESSIBLE FLOWS

The present work reports the results of a priori tests of Reynolds-Averaged Navier-Stokes (RANS) models based on Direct Numerical Simulations (DNS) data of zero-pressure-gradient flat-plate turbulent boundary layers. The DNS database covers a wide range of thermodynamic operating flow conditions, from supersonic (M ∞ = 2.25) up to the high-enthalpy hypersonic regime (M ∞ = 12.48). The most common RANS closures and compressibility corrections in literature are assessed against the exact terms from the DNS. Particular attention has been paid to closure models for the turbulent heat fluxes and the dilatational dissipation, as well as to the analysis of turbulence/chemistry and turbulence/vibrational relaxation interactions for the high-enthalpy simulations.

INTRODUCTION

The study of high-speed flows is of critical importance in the field of aeronautics and astronautics. Vehicles travelling at supersonic and hypersonic speeds, such as reentry capsules and ballistic missiles, experience extreme mechanical and thermal loads due to aerodynamic effects; fundamental understanding of the flow conditions through high-fidelity numerical simulations or experimental measurements allows for accurate prediction of those loads. As the flow transitions to turbulence, a significant increase in skin friction and wall heat fluxes occurs, making it even more crucial to accurately describe the flow behavior. Generally, numerical simulations of turbulent flows for industrial applications are mainly conducted using Reynolds-Averaged Navier-Stokes (RANS) models, most of which have been designed and calibrated for incompressible flows. When dealing with compress-ible configurations, the most common approach is to account for variable density and transport properties by Favre-averaging the Navier-Stokes equations. This gives rise to several unclosed terms not present in the RANS equations for incompressible, constant-property flows, which are very often neglected. While this approach provides acceptable results at low supersonic Mach numbers, it becomes unsuitable when compressibility and/or non-equilibrium effects become relevant. Attempts of developing RANS closures well-suited to the compressible regime exist in the literature, but they are derived from relatively-simple canonical configurations, such as homogeneous isotropic turbulence or mixing layers. Relevant examples in literature are the works of Sarkar et al. [START_REF] Sarkar | The analysis and modelling of dilatational terms in compressible turbulence[END_REF] and Zeman [START_REF] Zeman | Dilatation dissipation: the concept and application in modeling compressible mixing layers[END_REF] which introduced the concept of dilatational dissipation for compressible turbulence and incorporated it in a closure model for mixing layers. Later on, Wilcox [START_REF] Wilcox | Dilatation-dissipation corrections for advanced turbulence models[END_REF] extended this concept to flat-plate boundary layers and proposed a modification of the k-ω model for compressible flows. Catris and Aupoix [START_REF] Catris | Density corrections for turbulence models[END_REF] introduced a density scaling in the diffusion terms of turbulent transport equations to account for variable flow properties across high-speed boundary layers due to friction heating. More recently, Pecnick and Patel [START_REF] Pecnick | Scaling and modelling of turbulence in variable property channel flows[END_REF] investigated variable-property scalings for non-adiabatic turbulent channel flows, and obtained a new scaling of the diffusion terms [START_REF] Otero | Turbulence modelling for flows with strong variations in thermo-physical properties[END_REF] rather similar to that of [START_REF] Catris | Density corrections for turbulence models[END_REF]. An extensive assessment of turbulence models for compressible turbulent flows was conducted in the review paper of Roy [START_REF] Roy | Review and assessment of turbulence models for hypersonic[END_REF], showing that while some of the turbulence models do provide reasonable predictions for the surface pressure, the predictions for surface heat flux are generally poor, and often in error by a factor of four or more. Overall, closure models for the compressible RANS equations remain an open subject of research. The closure problem is even more complex for high-enthalpy regimes, where thermochemical non-equilibrium effects such as dissociation/recombination and vibrational relaxation processes may play an important role in the overall flow dynamics and have to be accounted for. The closure of the additional unclosed terms for turbulent transport of chemical species or vibrational energy introduces additional uncertainties in RANS approaches, which are not wellquantified yet, due to the lack of reference data. Performing physical experiments in these extreme flow conditions is generally a costly or even infeasible task, whence the interest of leveraging high-fidelity approaches and specifically Direct Dumerical Simulations (DNS). DNS is computationally expensive and usually limited to simple geometries and low Reynolds numbers, and this is even more true for high-enthalpy hypersonic flows, for which additional transport equations for the chemical species in the reacting mixture and for thermal non-equilibrium must be solved alongside the standard equations for mass, momentum, and total energy. Despite these limitations, DNS represents an invaluable resource for fundamental understanding of turbulence dynamics and for the assessment of RANS models. Recently, our team has generated some of the very first DNS for high-enthalpy hypersonic flows out of chemical or thermochemical equilibrium [START_REF] Passiatore | Finite-rate chemistry effects in turbulent hypersonic boundary layers: A direct numerical simulation study[END_REF][START_REF] Passiatore | Thermochemical non-equilibrium effects in turbulent hypersonic boundary layers[END_REF]. The results showed the presence of a marked interaction between turbulence and non-equilibrium conditions; specifically, vibrational relaxation and chemical dissociation processes are promoted and sustained by the onset of turbulent motions. As a consequence, the classical approach of neglecting turbulence/thermochemistry interactions underestimates the source terms for the chemical production rates and the translational-vibrational energy exchanges. Additionally, the lack of properly-designed closures for the turbulent fluxes of the species and vibrational energy equations lead us to investigate the behaviour of the simplified existing models for these configurations, motivating the present study. Precisely, the objective of this work is to quantify the deficiencies of RANS models by means of a priori tests using a database of several DNS of zero-pressure-gradient flat-plate turbulent boundary layers (TBL) spanning from supersonic to high-enthalpy hypersonic conditions [START_REF] Sciacovelli | Assessment of a high-order shockcapturing central-difference scheme for hypersonic turbulent flow simulations[END_REF][START_REF] Passiatore | Finite-rate chemistry effects in turbulent hypersonic boundary layers: A direct numerical simulation study[END_REF][START_REF] Passiatore | Thermochemical non-equilibrium effects in turbulent hypersonic boundary layers[END_REF]. The structure of the paper is as follows: the numerical databases are presented in section 2; section 3 presents the averaged equations and the main closure models investigated. Results are then shown in section 4 and conclusive remarks and perspectives are listed in section 5.

NUMERICAL DATABASES

Five different DNS databases are considered in this work, spanning a wide range of thermodynamic conditions from the supersonic (M ∞ = 2.25) up to the hypersonic, high-enthalpy regime (M ∞ = 12.48). Details about the free-stream and boundary conditions, the number of grid points and the extent of the computational domains are given in table 1. In the M2 and M6 runs, air is modeled as a single-species thermally-and calorically-perfect gas (see [START_REF] Sciacovelli | Numerical investigation of high-speed turbulent boundary layers of dense gases[END_REF]), whereas simulations M10F, M10C and M12 make use of a thermally-perfect five-species mixture of N 2 , O 2 , NO, N and O. Non-catalytic wall conditions are considered for all the multi-species cases. Run M10C uses finite-rate chemistry models [START_REF] Sciacovelli | Assessment of a high-order shockcapturing central-difference scheme for hypersonic turbulent flow simulations[END_REF], whereas M10F is performed with a frozen-chemistry assumption (unpublished data set; a similar calculation using the same gas model and a somewhat different numerical setting can be found in [START_REF] Passiatore | Finite-rate chemistry effects in turbulent hypersonic boundary layers: A direct numerical simulation study[END_REF]). Lastly, case M12 considers the contextual presence of vibrational and chemical nonequilibrium. All the test cases are spatially-evolving boundary layers initiated at a specific inlet Reynolds number; transition to turbulence is triggered by means of suction and blowing strategy. The interested reader might refer to [START_REF] Sciacovelli | Numerical investigation of high-speed turbulent boundary layers of dense gases[END_REF][START_REF] Sciacovelli | Assessment of a high-order shockcapturing central-difference scheme for hypersonic turbulent flow simulations[END_REF][START_REF] Passiatore | Thermochemical non-equilibrium effects in turbulent hypersonic boundary layers[END_REF] for details about the governing equations and the numerical schemes used in the simulations. In order to carry out comparisons, profiles were extracted at streamwise stations corresponding to well-developed turbulent flow. Due to the very different flow conditions, the friction Reynolds numbers Re τ = ρ w u τ δ 99 /µ w of the considered simulations do not match. Nevertheless, profiles of the turbulent statistics were found to exhibit reasonable selfsimilarity at the chosen stations. Some properties of the stations selected for the analysis are given in table 2. Figure 1 displays wall-normal profiles of the turbulent Mach number M t = u ′2 i /c (u i being the velocity fluctuations and c the speed of sound), the normalized temperature fluctuations and the density fluctuations in the top, middle and bottom panel, respectively, against the semi-locally scaled wall distance y ⋆ . In all cases, M t is shown to peak in the buffer layer and its values get larger with the increasing free-stream Mach number. The much larger M t registered for the M12 case is due to the presence of a significant wall cooling imposed by the isothermal wall condition, which is also responsible of the peak of temperature fluctuations in the inner region. Furthermore, the strong near-wall temperature gradient leads in turn to increased density fluctuations towards the wall. The global maximum of the thermodynamic fluctuations is achieved close to the boundary layer edge, owing to the strong intermittency caused by the alternation of turbulent bulges and laminar-like flow regions. M10F and M10C runs display larger variations with respect to M12 because of the much smaller free-stream temperature. Since chemical dissociation reactions are of endothermic nature, the mean temperature values T for M10C tend to be smaller than those for M10F, leading to higher fluctuation levels for the former.

Table 1: Parameters of the DNS considered for the tests. M ∞ , T ∞ and P ∞ denote the free-stream Mach number, temperature and pressure, respectively. T w lists the isothermal/adiabatic wall boundary condition for each case; N x , N y and N z represent the number of grid points in the streamwise, wall-normal and spanwise direction; L x , L y and L z are the extents of the computational domains, δ 99 and δ * in being the thickness and the displacement thickness of the boundary layer at the inflow section, respectively.

Case

M ∞ T ∞ [K] P ∞ [Pa] T w [K] N x N y N z L x L y L z M2 [

EQUATIONS AND MODELS

Favre-averaged Navier-Stokes

The basis of compressible turbulence models are the Favre-averaged (i.e., density-weighted) compressible Navier-Stokes (NS) equations:

∂ ρ ∂t + ∂ (ρ u i ) ∂ x i = 0 (1) ∂ ρ u i ∂t + ∂ ρ u i u j ∂ x j = - ∂ p ∂ x i + ∂ ∂ x j τ i j -ρu ′′ i u ′′ j (2)
∂ ∂t ρ E + ρu ′′ i u ′′ i 2 + ∂ ∂ x j ρ u j H + ρu ′′ i u ′′ i 2 = - ∂ ∂ x j q j + ρu ′′ j h ′′ + ∂ ∂ x j τ i j u ′′ i -ρu ′′ j 1 2 u ′′ i u ′′ i + ∂ ∂ x j τ i j -ρu ′′ i u ′′ j u j (3)
where t and x i denotes the time and space coordinates, u i the velocity components, p the pressure and τ i j the viscous stress tensor; E = e + 1 2 u i u i the specific total energy (e being the specific internal energy), H the total enthalpy and q j the heat flux. In the momentum and total energy equations ( 2)-( 3), the Favre-averaged Reynolds stress tensor τ R i j = ρu ′′ i u ′′ j appears; other unclosed terms are the turbulent kinetic energy ρk = 1 2 ρu ′′ i u ′′ i , the turbulent heat flux q t j = ρu ′′ j h ′′ , the molecular diffusion τ i j u ′′ i and the turbulent transport of turbulent kinetic energy

ρu ′′ j 1 2 u ′′ i u ′′ i .
When the operating conditions are such that the flow enters in a chemical non-equilibrium regime, air can no longer be considered as an homogeneous thermally and calorically-perfect gas. A mixture of five species is considered, namely, N 2 , O 2 , NO, N and O; the NS equations are therefore complemented with species conservation equations, here given in instantaneous form:

∂ ρ n ∂t + ∂ (ρ n u j ) ∂ x j = - ∂ ρ n u D n j ∂ x j + ωn (4)
where ρ n = ρY n is the partial density of the n-th species (Y n being mass fraction), u D n j the n-th species diffusion velocity and ωn the chemical production rate. In thermal non-equilibrium conditions, vibrational relaxation processes should also be considered. Following the two-temperature (2T) model of Park [START_REF] Park | Two-temperature interpretation of dissociation rate data for N2 and O2[END_REF], these processes are taken into account by adding a conservation equation for the vibrational energy e V :

∂ ρe V ∂t + ∂ ρe V u j ∂ x j = ∂ ∂ x j -q V j - NM ∑ m=1 ρ m u D m j e V m + NM ∑ m=1 (Q TVm + ωm e V m ) (5)
Here, NM is the number of molecular species (NM = 3 in the current analysis), q V j the vibrational contribution of the heat flux and e V m the vibrational energy per unit of volume of the m-th species. The latter is given by

e V m = θ m R m exp (θ m /T V ) -1 , (6)
θ m being the characteristic vibrational temperature of each molecule, R m the gas constant and T V the vibrational temperature. Lastly, Q TV and ωm e V m are source terms denoting translational-vibrational energy exchanges and the vibrational energy variations due to chemical production/depletion, respectively. Additional information about the 2T model can be found in [START_REF] Passiatore | Thermochemical non-equilibrium effects in turbulent hypersonic boundary layers[END_REF]. Favre-averaging of equations ( 4) and ( 5) results in ad-ditional unclosed terms, not given here for the sake of brevity. The most relevant ones, i.e. turbulent transport of species and vibrational energy and the Favre-averaged source terms, are discussed in Sections 4.4 and 4.5.

Turbulence closure models

Throughout this work we solely focus on so-called eddyviscosity models, largely employed in engineering applications. These models rely on the so-called Boussinesq approximation to express the constitutive law for the Reynolds stresses:

ρτ R i j = 2µ t S i j - 1 3 S kk δ i j -ρ 2 3 kδ i j (7)
where µ t is the eddy viscosity, S i j the mean strain-rate tensor and δ i j the Kronecker symbol. Such an approximation reduces the modeling problem to the calculation of a single scalar, µ t . Hereafter we consider so-called two-equation models, which express µ t as a function of two turbulent variables allowing to determine a turbulent length and velocity scale. For each of them a transport equation is written in order to calculate the eddy viscosity. The first variable is generally the turbulent kinetic energy k, for which an exact transport equation can be derived from the momentum equation (see section 4.2).

The turbulent heat fluxes are generally modelled through a "turbulent Fourier law":

q t j = - µ t c p Pr t ∂ T ∂ x j (8)
where Pr t is the turbulent Prandtl number, most often assumed to be constant equal to 0.9 for turbulent air flows. In practice, this parameter is case-dependent and varies throughout the flow. In [START_REF] Passiatore | Thermochemical non-equilibrium effects in turbulent hypersonic boundary layers[END_REF], a "vibrational Fourier law" is proposed to model the corresponding term in the vibrational energy equation. Similarly, turbulent mass fluxes arising in the Favreaveraged species transport are modelled through a turbulent Fick law by introducing again a, supposedly constant, turbulent Schmidt number Sc t . For passive scalar transport, the turbulent Schmidt and Prandtl number coincide, so that a common approximation consists in setting Sc t,n ≈ 0.9 for all species.

A PRIORI TESTS

The validity of the above-mentioned assumptions, as well as supplementary assumptions used to close the turbulent kinetic energy transport equation are assessed against the exact unclosed terms extracted from the DNS statistics.

Eddy viscosity and Reynolds stresses

First, we discuss the validity of linear constitutive laws adopted for the Reynolds stresses, turbulent heat fluxes, and mass fluxes. The constitutive relation adopted for the Reynolds stresses is a linear eddy viscosity model (LEVM), of the form µ t = ρC µ ℓ t v t , where ℓ t and v t are the characteristic length and velocity scales of turbulent structures, respectively, and C µ a model constant to be defined. The eddy viscosity coefficient is computed according to the SST k-ω model [START_REF] Menter | Improved two-equation k-omega turbulence models for aerodynamic flows[END_REF]. The model transport equations in conservative form are

∂ ρk ∂t + ∂ ρk u j ∂ x j =τ R i j ∂ u i ∂ x j -β * ρωk (9)
+ ∂ ∂ x j (µ + σ k µ t ) ∂ k ∂ x j ∂ ρω ∂t + ∂ ρω u j ∂ x j = ργ µ t τ R i j ∂ u i ∂ x j -ρβ ω 2 (10)
+ ∂ ∂ x j (µ + σ ω µ t ) ∂ ω ∂ x j + 2 (1 -F 1 ) ρσ ω2 ω ∂ k ∂ x j ∂ ω ∂ x j
In figure 2 we report the "exact" eddy viscosity coefficient, computed as:

µ t ≈ -ρ u ′′ v ′′ ∂ u ∂ y -1 (11)
and the modelled µ t computed from the DNS data:

µ t = ρa 1 k max (a 1 ω, SF 2 ) (12)
where we use the k and ω resulting from the DNS statistics. For the simple sheared flow at stake, the shear stress dominates the wall normal components, and the LEVM eddy viscosity (top panel) follows reasonably well the exact one for cases M2, M6 and also for the frozenchemistry case M10F, indicating the effects of increasing friction heating are well-captured by the model overall. More significant deviations are observed for the finiterate chemistry, adiabatic wall case M10C, for which the SST-model overpredicts the eddy viscosity significantly. For such a case, the endothermic chemical reactions drain part of the energy supplied by the mean field, leading to reduced turbulent kinetic energy compared to the corresponding frozen case [START_REF] Passiatore | Finite-rate chemistry effects in turbulent hypersonic boundary layers: A direct numerical simulation study[END_REF] and altering to some extent the near-wall characteristic time scales. The largest discrepancies are obtained for the M12 case, for which thermal non-equilibrium effects combine with severe wall cooling. For this case, not only the near-wall profiles differ, but even larger deviations are observed at the boundary layer edge. In this region, turbulent velocity fluctuations vanish, but still fluctuating stress persist as an effect of the strong acoustic radiation outside the boundary layer. Note that since k and ω are directly obtained from DNS data, the above-mentioned discrepancies can be attributed to the expression used for µ t , and specifically to the empirical function F 2 . In contrast, as expected for a LEVM,

Turbulent Kinetic Energy equation

Next, we assess the validity of models used for the closure of the turbulent kinetic energy equation. In the compressible case, this takes the form:

ρ ∂ k ∂t + ρ u j ∂ k ∂ x j = ρτ R i j ∂ u i ∂ x j -τ i j ∂ u ′′ i ∂ x j (13)
+ ∂ τ ji u ′′ i -1 2 ρu ′′ j u ′′ i u ′′ i -p ′ u ′′ j ∂ x j -u ′′ i ∂ p ∂ x i + p ′ ∂ u ′′ i ∂ x i
In the above, several unclosed terms appear, some of which are counterparts of those already present in the in- compressible k formulation. Some other, related to compressibility effects, also need to be modelled. The term

∂ ∂ x j ( 1 2 ρu ′′ j u ′′ i u ′′ i )
, referred-to as turbulent transport (TT) of k, is traditionally modeled by means of a gradient approximation:

∂ ∂ x j 1 2 ρu ′′ j u ′′ i u ′′ i ≈ ∂ ∂ x j µ t σ k ∂ k ∂ x j ( 14
)
where σ k is a closure coefficient. For high-M flows, density gradients influence the logarithmic region of the velocity profile and must be accounted for in order to recover the correct near-wall behaviour. A compressibility correction for density variations was initially proposed by Catris & Aupoix [START_REF] Catris | Density corrections for turbulence models[END_REF]:

∂ ∂ x j 1 2 ρu ′′ j u ′′ i u ′′ i ≈ ∂ ∂ x j 1 ρ µ τ σ k ∂ ρk ∂ x j (15)
and later refined by Otero et al. [START_REF] Otero Rodriguez | Computational fluid dynamics for non-conventional power cycles: Turbulence modelling of supercritical fluids and simulations of high-expansion turbines[END_REF]:

∂ ∂ x j 1 2 ρu ′′ j u ′′ i u ′′ i ≈ 1 ρ ∂ ∂ x j 1 ρ µ τ σ k ∂ ρk ∂ x j (16)
In figure 3, we compare the two models against the exact term. For M2, the models exhibit little differences, since flow property variations are relatively small. The discrepancies tend to increase at higher Mach numbers, with the uncorrected model underpredicting the peak in the buffer layer. Of note, the Catris-Aupoix and Otero models are essentially superposed for all cases. For the M12 case, all the models predict the position of the first peak in the buffer layer but underestimate its magnitude. The second peak position and magnitude are not predicted accurately by any of the models. Overall, based on a priori tests, the corrections do not seem to improve the agreement with the DNS significantly. Next, we examine the k dissipation. When dealing with compressible flows, the dissipation ε can be expressed decomposed in two parts:

ρε = ρε s + ρε d = µ ρ ρω ′′ i ω ′′ i + 3 µ ρ ρϑ ′′ ϑ ′′ (17)
ω i and ϑ being the vorticity components and the velocity divergence, respectively. The solenoidal dissipation ε s coincides with the dissipation in the incompressible limit, whereas the dilatational dissipation ε d appears only in compressible flows. For a two-equations model, the transport equations have to be adjusted to account for ε d and this is done by considering a closure model for the term. A few models for ε d have been proposed in the literature, mostly based on DNS databases for homogeneous isotropic turbulence or free-shear flows [START_REF] Wilcox | Dilatation-dissipation corrections for advanced turbulence models[END_REF][START_REF] Zeman | Dilatation dissipation: the concept and application in modeling compressible mixing layers[END_REF][START_REF] Sarkar | The analysis and modelling of dilatational terms in compressible turbulence[END_REF]. In both cases, the dilatation dissipation is assumed to take the general form [START_REF] Gatski | Modeling compressibility effects on turbulence[END_REF]:

ε d = β F(M t )ε s (18)
with M t the turbulent Mach number and β a modelling constant. Figure 4 reports profiles in semi-local scaling of the exact ε d /ε s ratio from the DNS and its modelled counterpart according to the above-mentioned models. The analysis of such ratio from DNS allows to better quantify the magnitude of compressibility effects. ε d assumes maximum values of the order of ≈ 4-5%ε s at the wall, and becomes larger at the edge of the boundary layer where ε s decays faster. Nevertheless, it is generally much smaller in the rest of the boundary layer; neglecting ε d can therefore be considered a reasonable assumption. This does not seem to remain valid for M12, as the wall cooling increases the significance of compressibility effects with ε d =≈ 7-8%ε s . The results for the three models considered show their poor performance for all cases. This is a consequence of the simple functional relationship [START_REF] Xiang | Turbulence-chemistry interaction models with finite-rate chemistry and compressibility correction for simulation of supersonic turbulent combustion[END_REF] assuming proportionality to ε s through a function of M t only, leading the models to peak where M t peaks. Overall, the values tend to be largely overpredicted and the discrepancies increase with M ∞ . As reported by [START_REF] Wilcox | Dilatation-dissipation corrections for advanced turbulence models[END_REF], the role of models for ε d mostly consists in increasing the overall dissipation of the RANS model, but all of them fail to describe correctly all its physical dependencies on flow parameters.

Turbulent heat flux

In this section we assess the behavior of constitutive relations for the turbulent heat flux term q t j = ρu ′′ j h ′′ that appears in the averaged total energy equation. For the boundary layers at stake, we estimate an "exact" turbulent Prandtl number from the DNS data, as follows:

Pr t ≈ ρu ′′ v ′′ ρv ′′ T ′′ ∂ T ∂ y ∂ u ∂ y -1 (19)
As mentioned, Pr t is most often assumed to be constant and equal to 0.9 for TBL. More accurate results can be obtained by modelling Pr t and letting it vary; one example is the empirical relation of Kays and Crawford [START_REF] Kays | Convective heat and mass transfer[END_REF]:

Pr t = 1 2Pr t∞ + CPe t √ Pr t∞ (20) -(CPe t ) 2 1 -exp - 1 CPe t √ Pr t∞ -1
where C = 0.3, Pr t∞ = 0.9 for air and Pe t = Pr µ t µ is a turbulent Peclet number. Another example is the model presented by Cebeci [START_REF] Cebeci | A model for eddy conductivity and turbulent Prandtl number[END_REF] 5 the comparison between the exact Pr t , the models and Pr t = 0.9 is shown. It appears that both the models and the constant assumption fail to accurately reproduce the trend of Pr t for all the cases considered. The Kay-Crawford model provides a better representation of the near wall behavior, but neither model captures the "bump" located at y/δ 99 < 0.2, which is particularly accentuated for the M12 case. In the outer region, the exact values are below 0.9.

Pr t = κ κ θ 1 -exp (-y + /A) 1 -exp (-y + /B) (21)

Mass transport and turbulencechemistry interaction

In this section we assess models of unclosed terms in the species transport equations. We first focus on the turbulent mass transport term ρu ′′ j Y ′′ n (n being the number of species) for a turbulent boundary layer, and we plot the "exact" Schmidt numbers, computed from DNS data as: The results are reported in figure 6 for the species considered in the M10C and M12 cases. Similarly to Pr t , Sc t,n exhibit a bump in the logarithmic region, more pronounced for M12, and are below the standard value in the outer region, the 0.9 value can be considered as an average across the boundary layer. The profiles do not differ much but the values are clearly affected by M ∞ .

Sc t,n = ρu ′′ v ′′ ∂ Y n /∂ y ρv ′′ Y ′′ n ∂ u/∂ y (
The influence of species mass fraction and temperature fluctuations on the species production rates, i.e. the intensity of turbulence-chemistry interactions (TCI), is quan- tified through the species interaction Dahmköler number: where ω n (T, ρ n ) is a source term which represents the production or depletion of the n-th species in the mixture due to chemical reactions. This parameter represents the chemical production due to turbulent fluctuations caused by the non zero difference between ω n (T, ρ n ) and ω n (T , ρ n ), since the expression for ω n is nonlinear. A simplification often adopted in the RANS framework is to consider ω n (T, ρ n ) ≈ ω n (T , ρ n ), which is an acceptable approximation only if the turbulence-chemistry interactions are limited. Due to the lack of models for the source terms in hypersonic TBL, an approach used for combustion applications is tested. We consider the model of [START_REF] Xiang | Turbulence-chemistry interaction models with finite-rate chemistry and compressibility correction for simulation of supersonic turbulent combustion[END_REF], based on the Eddy Dissipation Concept (EDC); the chemical source term is then expressed as:

Da I n = ω n (T, ρ n ) -ω n T , ρ n ω n (T, ρ n ) w , (23)
ω n ≈ γ * ω n (T , ρ n ). (24)
Here, γ * ≈ 9.7( νε k 2 )

3 4 represents the fine-scale structure volume fraction, i.e. the fraction of the volume in which chemical reactions take place which is assumed to be in the region where turbulent kinetic energy is quasi-steady. Figure 7 shows the profile of Da I n for the different species of the mixture; N is not shown being present in negligible amounts. The indicator is overall small but assumes large values close to the wall for the adiabatic case M10C, and in the buffer region for the wall-cooled case M12, where turbulent fluctuations are significant. To gain a better overview of TCI, the profiles of ω n (T, ρ n ), ω n T , ρ n and the models are shown in figure 8. It can be observed that the assumption ω n (T, ρ n ) ≈ ω n (T , ρ n ), is reasonably accurate while the EDC does not prove to be satisfactory.

Vibrational Turbulent Heat Flux and Turbulence-Vibrational Relaxation Interaction

We complete the analysis by investigating the behavior of unclosed terms in the averaged vibrational energy equation. As the heat flux was found to have an important part in the vibrational energy budget, modeling this term appears critical in a RANS approach. In [START_REF] Passiatore | Thermochemical non-equilibrium effects in turbulent hypersonic boundary layers[END_REF], a gradient transport model is proposed, based on a vibrational turbulent Prandtl number Pr V t defined in [START_REF] Baranwal | Vibrational turbulent prandtl number in flows with thermal non-equilibrium[END_REF]. Its "exact" value is extracted from DNS as:

Pr V t = ρu ′′ v ′′ ∂ T V /∂ y ρv ′′ T ′′ V ∂ u/∂ y , (25)
such that the model reads:

ρu ′′ j e ′′ V = µ t Pr V t ∂ e V ∂ x j . (26)
The plot of figure 9 shows a behavior similar to Pr t with a bump in the logarithmic region. In this case, the value 0.9 is reasonably well recovered in the outer region. Finally we assess the intensity of the interaction between turbulence and thermal relaxation, through the indicator introduced in [START_REF] Passiatore | Thermochemical non-equilibrium effects in turbulent hypersonic boundary layers[END_REF]:

Q I TV = Q TV (T, T V , ρ, p,Y n ) -Q TV T , T V , ρ, p, Y n Q max TV , (27)
where Q TV is a source term accounting for vibrational energy production/depletion due to translational-vibrational energy transfers and Q max TV is the maximum wall-normal value of at the selected station. Figure 10 shows differences up to ≈ 15% in the inner layer, indicating a strong interaction between turbulence and thermal nonequilibrium. While for ω a few models do exist, allowing to quantify the intensity of TCI, no models for Q TV are available, which is potentially critical for RANS modeling of hypersonic flows out of thermal equilibrium.

CONCLUSIONS

We reported a priori tests of RANS closures and compressibility corrections available in literature for various TBL configurations. Reference data were extracted from DNS databases ranging from the supersonic to the hypersonic regime, including conditions of thermochemical non equilibrium. Direct extensions of incompressible closures for unclosed terms in the turbulent kinetic energy equation are show to be inadequate. Unfortunately, the few compressibility correction available in the literature are not found to capture DNS data much better, especially at high Mach numbers and in presence of wall cooling. The gradient approximation, commonly used to model turbulent heat fluxes, was found unable to predict the stream-wise component of various terms, pointing out limitations in the Reynolds analogy, on which the model is based. Similar results were found for the vibrational turbulent heat flux, while models adapted from the ones for turbulent heat fluxes were shown to provide acceptable results. Other closure approaches specific to this term may be developed. High-temperature effects were found not to alter the validity of classical closure models. Nevertheless, the new unclosed terms in the species and vibrational energy equations need to be modeled accordingly. Further work on many aspects of compressible turbulence modelling is required, in particular in the development of closures that properly account for compressibility, for instance by means of machine learning techniques.

Figure 2 :

 2 Figure 2: Exact µ + t and Reynolds stresses from DNS versus their modelled counterparts.

Figure 3 :

 3 Figure 3: Turbulent transport of k, comparison of the exact term with the models. From top to bottom: M2, M6, M10C and M12.

Figure 4 :

 4 Figure 4: ε d /ε s ratio of the exact term with the models. From top to bottom: M2, M6, M10C and M12.

  log 10 Pr) i-1 with κ = 0.41, κ θ = 0.46, A = 26, C 1 = 34.96, C 2 = 28.79, C 3 = 33.95, C 4 = 6.3, and C 5 = -1.186. In Figure

Figure 6 :

 6 Figure 6: Exact Sc t from DNS.

Figure 7 :

 7 Figure 7: Species interaction Dahmköler number profile for the M10C and M12 simulations.

  ρn) (DNS) ωn(T , ρn) Model

Figure 8 :

 8 Figure 8: Species interaction Dahmköler number profile for the M10C and M12 simulations.

Figure 9 :

 9 Figure 9: Exact Pr V t from DNS.

Figure 10 :

 10 Figure 10: Translational-Vibrational source term interaction indicator for the M12 simulation.

Table 2 :

 2 Data from the DNS at the streamwise stations selected for the analysis. Streamwise Reynolds number Re x , friction Reynolds number Re τ , momentum-thickness-based Reynolds number Re θ , friction Mach number Ma τ , and grid resolutions in the streamwise (∆x + ), wall-normal (∆y + ) and spanwise (∆z + ) directions, respectively.

	16]	2.25	65	2422	T w = 120.2 8000 300 512 1600 δ ⋆ in	100 δ ⋆ in	62.83 π δ ⋆ in
	M6 [15] M10F M10C [16] M12 [11]	6 10 10 12.48 594.3 78 350 350	2421.3 T w = 422.5 7700 300 400 75.2 δ 99 2.13 δ 99 3596.0 Adiabatic 5520 256 240 8000 δ ⋆ in 320 δ ⋆ in 3596.0 Adiabatic 5520 256 240 8000 δ ⋆ in 320 δ ⋆ in 4656.0 T w = 1800 9660 480 512 3000 δ ⋆ in 120 δ ⋆ in	2.17 δ 99 100 π δ ⋆ in 100 π δ ⋆ in 30 π δ ⋆ in
			Case	Re x • 10 -6	Re τ	Re θ	Ma τ	∆x + ∆y + ∆z +
			M2 [16]	2.56		589.6 3531 0.075 4.4 0.59 2.7
			M6 [15]	11.8		401.8 5960 0.122 3.8 0.26 2.1
			M10F	48.1		213.5 10875 0.154 3.7 0.42 3.4
			M10C [16]	48.1		244.1 10411 0.157 4.5 0.50 4.1
			M12 [11]	15.2		984.3 5491 0.213 7.6 0.67 4.5