Nazmiye Kılıç

Serkan Yeşilot

Selin Sariyer

Arpita Ghosh

Adem Kılıç

Ozlem Sel

Rezan Demir-Cakan

A Small Inorganic-Organic Material Based on Anthraquinone Decorated Cyclophosphazene as Cathode for Aqueous Electrolyte Zinc-Ion Batteries

Keywords: Aqueous electrolyte, Zn-ion battery, Anthraquinone, Cyclophosphazene, Small Inorganic-Organic Material

The small inorganic-organic material (hexakis(2-anthraquinonyloxy) cyclotriphosphazene -THAQ) is synthesized from the commercially available starting materials in a one-step reaction and its relevance as an aqueous zinc-ion battery cathode material is assessed. The THAQ structure is verified by using appropriate standard spectroscopic methods such as 31 P and 1 H Nuclear magnetic resonance spectroscopy and MALDI-TOF (Matrix-assisted laser desorption/ionizationtime-of-flight). The charge storage mechanism and the evolution of the interfacial properties of the THAQ are investigated through several ex-situ analysis (Fourier-transform infrared spectroscopy and X-ray photoelectron spectroscopy) and electrochemical quartz crystal microbalance, indicating both Zn 2+ and H + participation. The insoluble THAQ electrode demonstrates a remarkable electrochemical performance with over 150 mAh/g at 30 C, as well as an ultra-long term cycling (>30000 cycles) stability at ultra-high current rate (100 C). The outstanding electrochemical performance turns out to be governed by the multiple nucleophilic carbonyl active sites and increased π-π interaction of THAQ, and its lower band gap compared to the anthraquinone counterparts, verified by density functional theory calculations. Overall, this work is the first report revealing the nature of charge carries of inorganic-organic material system, specifically anthraquinone decorated cyclophosphazene, obtained by a facile and cost-effective methods that further demonstrates excellent electrochemical performance.

Introduction

The development of efficient energy storage systems, more specifically rechargeable batteries, is of great importance for the worldwide energy problem. Among the novel rechargeable batteries, lithium-ion batteries are widely investigated because of their high energy and power densities.

However, drawbacks such as safety problems (i.e., hazardous lithium dendrites and the use of organic-based electrolytes) and cost issues due to the limited lithium supply seriously restrict their large-scale applications. [START_REF] Fang | Recent Advances in Aqueous Zinc-Ion Batteries[END_REF] These limitations have directed the research fraternity to focus on higher safety and sustainable battery options in recent years. In this respect, aqueous ion batteries (AIBs) are thought to be a promising class of batteries for energy storage systems because of their lower price, higher operational safety, higher ionic conductivity, environmental benignity, and water stability. [START_REF] Wang | Recent advances in cathode materials of rechargeable aqueous zinc-ion batteries[END_REF] In particular, aqueous electrolyte Zn-ion batteries (ZIBs) have been the focus over the past years because of the low standard redox potential (0.76 V vs. standard hydrogen electrode (SHE)) and high theoretical capacity (820 mAh g -1 ) of the Zn anode that operates at mild acidic electrolytes.

For building high-performance ZIBs, manganese-based materials, vanadium-based materials, or Prussian blue analogs are the most widely investigated inorganic compounds that have been simply coupled with Zn metal anode. [START_REF] Zhang | Materials chemistry for rechargeable zinc-ion batteries[END_REF] Nevertheless, there are many roadblocks for practical applications, such as phase changes and insufficient stability of the inorganic cathodes upon cycling. [START_REF] Luo | Aqueous zinc ion batteries based on sodium vanadate electrode materials with long lifespan and high energy density[END_REF] Seeking for organic alternatives, quinones are carbonyl-based compounds and are ubiquitous in nature, which constitute a promising class of sustainable and environment-friendly electrode materials. [START_REF] Cui | Organic Cathode Materials for Rechargeable Zinc Batteries: Mechanisms, Challenges, and Perspectives[END_REF][START_REF] Li | Quinone Electrodes for Alkali-Acid Hybrid Batteries[END_REF][START_REF] Yu | A High-Rate Two-Dimensional Polyarylimide Covalent Organic Framework Anode for Aqueous Zn-Ion Energy Storage Devices[END_REF] Quinone compounds indeed have been widely presented as cathode active materials for organic batteries because of their high theoretical specific capacities and flexible molecule pattern.

Moreover, the working potential of the quinone-based electrodes can be tailored by altering the functional groups. [START_REF] Kundu | Organic Cathode for Aqueous Zn-Ion Batteries: Taming a Unique Phase Evolution toward Stable Electrochemical Cycling[END_REF][START_REF] Wang | An organic cathode with tailored working potential for aqueous Zn-ion batteries[END_REF][START_REF] Ba | Benzoquinone-Based Polyimide Derivatives as High-Capacity and Stable Organic Cathodes for Lithium-Ion Batteries[END_REF] However, as of many other organic based cathodes, quinone electrodes in organic electrolytes suffer from fast capacity fading and short cycle life due to their dissolution, while in aqueous medium, they have been proven to be stable with an excellent redox activity. [START_REF] Liang | Universal quinone electrodes for long cycle life aqueous rechargeable batteries[END_REF][START_REF] Lin | A high capacity small molecule quinone cathode for rechargeable aqueous zinc-organic batteries[END_REF][START_REF] Xie | Recent progress in aqueous monovalent-ion batteries with organic materials as promising electrodes[END_REF] Generally, ZIBs based on carbonyl compounds operate via reversible coordination of Zn 2+ ion with the carbonyl groups during the electrochemical process, encouraging electrochemical reversibility. [START_REF] Zhang | Charge Storage Mechanism of a Quinone Polymer Electrode for Zinc-ion Batteries[END_REF] In the light of acquired knowledge on organic-based electrode materials, numerous cathodes have been designed or modified for rechargeable ZIBs, presenting electrochemical behaviors which are suitable for coupling with zinc metal anodes. [START_REF] Cui | Organic Cathode Materials for Rechargeable Zinc Batteries: Mechanisms, Challenges, and Perspectives[END_REF] Small quinone compounds were also investigated as cathodes for aqueous ZIBs (i.e., 1,4-naphthoquinone (1,4-NQ), [START_REF] Kumankuma-Sarpong | Naphthoquinone-Based Composite Cathodes for Aqueous Rechargeable Zinc-Ion Batteries[END_REF] 9,10-phenanthrenequinone (9,10-PQ), [START_REF] Yang | Ultralong-Life Cathode for Aqueous Zinc-Organic Batteries via Pouring 9,10-Phenanthraquinone into Active Carbon[END_REF] calix [START_REF] Luo | Aqueous zinc ion batteries based on sodium vanadate electrode materials with long lifespan and high energy density[END_REF]quinone (C4Q), [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] 1,2naphthoquinone (1,2-NQ), [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] 9,10-anthraquinone (9,10-AQ), [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] or pyrene 4,5,9,10 tetraone (PTO), [START_REF] Cui | Research Progress of High-Performance Organic Material Pyrene-4,5,9,10-Tetraone in Secondary Batteries[END_REF]). For example, Zhao et al. fabricated a calix [START_REF] Luo | Aqueous zinc ion batteries based on sodium vanadate electrode materials with long lifespan and high energy density[END_REF]quinone (C4Q) molecule, where each benzoquinone unit is linked by a C-C single bond to form a bowl-like structure, exhibited a high specific capacity of 335 mAh g -1 . [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] Guo et al. reported PTO cathode with four active carbonyl groups with a high capacity of 336 mAh g -1 at 0.04 A g -1 [START_REF] Guo | An Environmentally Friendly and Flexible Aqueous Zinc Battery Using an Organic Cathode[END_REF]. A type of quinone compound, triangular phenanthrenequinone (PQ-Δ) structure, allowed the co-insertion of Zn 2+ and water delivering a 225 mAh g -1 at 30 mA g -1 . [START_REF] Nam | Redox-Active Phenanthrenequinone Triangles in Aqueous Rechargeable Zinc Batteries[END_REF] Beside these studies, many other types of quinonebased materials have been explored as cathodes. [START_REF] Han | The rise of aqueous rechargeable batteries with organic electrode materials[END_REF][START_REF] Dawut | High-performance rechargeable aqueous Zn-ion batteries with a poly(benzoquinonyl sulfide) cathode[END_REF][START_REF] Wang | Molecular Engineering of Covalent Organic Framework Cathodes for Enhanced Zinc-Ion Batteries[END_REF][START_REF] Nam | Conductive 2D metal-organic framework for high-performance cathodes in aqueous rechargeable zinc batteries[END_REF][START_REF] Gao | A high-performance aqueous rechargeable zinc battery based on organic cathode integrating quinone and pyrazine[END_REF] One of the most important reasons for the preference of small cathode molecule is easy synthesis and lower cost. To avoid the dissolution problem encountered in battery systems, small molecules were modified by relatively more complex and multiple synthesis steps. [START_REF] Lin | A high capacity small molecule quinone cathode for rechargeable aqueous zinc-organic batteries[END_REF] Therefore, there is still room for the development of new materials synthesized with a facile synthesis route providing both insolubility in electrolytes and high capacity in order to preserve the advantages of small cathode molecule to be used in battery systems. Thus, the aim of the study is to produce a small inorganic-organic material with the desired properties by an effortless and low cost synthesis method.

The phosphorous-nitrogen compounds known as phosphazenes attract attention due to their synthetic flexibility and organic-inorganic structures . [START_REF] Gleria | Phosphazenes: A worldwide insight[END_REF] The phosphazenes are used in a wide variety of applications due to their characteristic properties such as thermal stability, electrical conductivity, liquid crystallinity, and biomedical activity. [START_REF] Amin | Recent Research Progress in the Synthesis of Polyphosphazene and Their Applications[END_REF]28] Especially, the electrochemical/thermal stability of these compounds leads to their potential to be used for many energy storage systems as cathode [START_REF] Yeşilot | Phosphazene based star-branched polymeric cathode materials via inverse vulcanization of sulfur for lithium-sulfur batteries[END_REF][START_REF] Yeşilot | A Flame-Retardant and Insoluble Inorganic-Organic Hybrid Cathode Material Based on Polyphosphazene with Pyrene-Tetraone for Lithium Ion Batteries[END_REF], anode materials [START_REF] Dufek | Hybrid phosphazene anodes for energy storage applications[END_REF], or separators. [START_REF] Fei | Recent Progress with Ethyleneoxy Phosphazenes as Lithium Battery Electrolytes[END_REF] The hexachlorocyclotriphosphazene (HCCP) structure is one of the most well-known cyclic structures in the phosphazene chemistry formed by bonding of P and N atoms in a planar structure with D3h symmetry. HCCP derivatives can be prepared through substitution reactions of the chlorides with side groups. [START_REF] Chandrasekhar | Phosphazenes as scaffolds for the construction of multi-site coordination ligands[END_REF] The phosphorus atom can be easily organized to bear a wide range of different substituents and can redesign the material characteristics with respect to their applications. [START_REF] Gleria | Phosphazenes: A worldwide insight[END_REF]28] HCCP derivatives are also known to have high thermal stability and are extremely sensitive to nucleophilic reactions under basic conditions, so it was chosen, in the present work, as a core to provide the desired properties in efficient cathode materials.

Herein, a small inorganic-organic material based on anthraquinone decorated cyclophosphazene (THAQ) is synthesized and applied as a cathode material for aqueous ZIBs. THAQ offers potential cathode material properties with high thermal stability and insolubility in aqueous electrolytes, as well as multiple electroactive sites for ZIBs, which resulted in propitious electrochemical performance. Prior to the electrochemical studies, THAQ was characterized by using appropriate standard spectroscopic methods, including 31 P and 1 H NMR (Nuclear Magnetic Resonance) spectroscopy, MALDI-TOF, FT-IR (Fourier-transform infrared spectroscopy), TGA (Thermogravimetric analysis) and simple flame test. Thermal stability, flammability, and ignitability of THAQ and 2-hydoxyanthraquione (HAQ) were investigated by TGA and burning tests. The electrochemical performance as an aqueous ZIB cathode material was evaluated in 2 M ZnSO4 aqueous electrolyte. In-situ EQCM (electrochemical quartz crystal microbalance) with motional resistance monitoring (EQCM-R) has been used to probe the possible role of multispecies (Zn 2+ and H + ) in the charge compensation mechanism and the evolution of electrode/electrolyte interface during electrochemical cycling. [START_REF] Hillman | The EQCM: electrogravimetry with a light touch[END_REF][START_REF] Gavriel | Enhanced Performance of Ti3C2Tx (MXene) Electrodes in Concentrated ZnCl2 Solutions: A Combined Electrochemical and EQCM-D Study[END_REF][START_REF] Easley | A practical guide to quartz crystal microbalance with dissipation monitoring of thin polymer films[END_REF] The dissolution tests were carried out by UV-Vis spectroscopy in an aqueous solution for THAQ and AQ. Compositional and structural information are investigated by ex-situ X-ray diffraction (XRD), scanning electron microscopy (SEM), and energy-dispersive X-ray spectroscopy analyses (EDS). The Zn 2+ /H + ion coordination mechanism of THAQ was further examined by using ex-situ FT-IR, X-ray photoelectron spectroscopy (XPS) analyses and density functional theory (DFT) calculations.

Experimental Material

2-hydroxyanthraquinone (HAQ), anthraquinone (AQ), sodium hydride (60% suspension in mineral oil-NaH), hexane and tetrahydrofuran (THF) were purchased from Sigma-Aldrich. Hexachlorocyclotriphosphazene (HCCP) was also obtained from Sigma-Aldrich (98%) and further purified by vacuum sublimation before reaction. The deuterated solvent (DMSO(Dimethyl sulfoxide)-d6) for NMR spectroscopy was obtained from Merck. Analytical thin layer chromatography (TLC) was performed on Silica gel plates (Merck, Kieselgel 60, 0.25 mm thickness) with F254 indicator. All other chemicals and solvents were reagent grade quality and were obtained from commercial suppliers.

Synthesis of hexakis(2-anthraquinonyloxy)cyclotriphosphazene (THAQ)

HAQ (1.0 g, 4.46 mmol) and dry sodium hydride (NaH) (0.2 g, 4.46 mmol) were dissolved in dry THF (10 mL) in argon atmosphere at room temperature. The HCCP containing solution (0.24 g, 0.68 mmol in 4 mL THF) was taken into dropping funnel and added drop by drop to reaction mixture. The reaction mixture was stirred at the boiling point of solvent (~70-80 °C) for 96 h and in between controlled by thin layer chromatography (TLC). The reaction mixture filtered off and volatile materials were evaporated under vacuum. The resulting product was subjected to column chromatography on silica gel using hexane: ethyl acetate (1:1) as the mobile phase. The last eluate was designated as THAQ compound and was subjected to further analyses (0.214 g, yield 57%).

Material Characterization

31

P NMR and 1 H NMR spectra were conducted by NMR on a Varian INOVA 500-MHz spectrometer in DMSO-D6 (85% H3PO4 and TMS (Tetramethylsilane) as reference). Mass analysis was recorded by a Bruker MALDI-TOF (matrix-assisted laser desorption/ionization-time-offlight) spectrometer using 1,8,9-Anthracenetriol (Dithranol-DIT) as a matrix for THAQ. FT-IR spectra were recorded on a Bruker Alpha-P in ATR in the range of 650-4000 cm -1 . Functional groups of HAQ and THAQ were determined by FT-IR method. To qualitatively determine the presence of halogen, simple flame tests were carried out with a clean copper wire on a Bunsen burner. The thermal property of the THAQ and HAQ were investigated on Mettler Toledo TGA/SDTA 851 thermogravimetric analyser (TGA) at a heating rate of 10 C min -1 under argon flow (50 mL min -1 ) between 25 and 700 °C. The surface morphology and elemental analysis of the THAQ/KB was investigated by scanning electron microscopy (SEM) (FEI PHILIPS, XL30 SFEG SEM) coupled with energy-dispersive X-ray spectroscopy (EDS). XRD patterns of THAQ/KB electrode (pristine and full discharged), ZnSO4.7H2O, and (Zn4(OH)6SO4•0.5H2O were recorded by using a Bruker-D8 Advance diffractometer (Bruker AXS).

The burning tests of THAQ and HAQ were carried out on a Bunsen burner using ~2 mg of the sample with a spatula. The specific ignition time (SIT) and the self-extinguish time (SET) were determined by the burning test. Limiting oxygen index (LOI) value was calculated from TGA data using Van Krevelen and Hoftyzer's equation. [START_REF] Zhou | Polyphosphazenesbased flame retardants: A review[END_REF] LOI=17.5 + 0.4*Char yield(CR) Dissolution tests were also performed to evaluate the solubility of THAQ and AQ in the aqueous medium, for the pristine and discharged samples. All dissolution tests were recorded with a Shimadzu 2101 UV spectrophotometer with quartz cuvettes at room temperature. XPS spectra were recorded using a Specs Flex-Mod system with a monochromatic Al Kα X-ray source.

Elementary mapping was performed to reveal the chemical distribution of atoms on the surface of the THAQ electrode after the electrochemical cycling process.

Electrochemical Measurements

THAQ/KB electrode was prepared by mixing the Ketjen black (KB) and THAQ powder at a weight ratio of 70:30 (KB:THAQ) in a mortar for half an hour. The weight of active materials was set to 1.5-2 mg. Zinc metal anode, 2 M ZnSO4 aqueous electrolyte and THAQ/KB cathode were used in the Swagelok type cell configuration. All electrochemical performance tests were carried out employing Bio-Logic VMP-3 electrochemical workstation in a voltage range of 0.1-1.7 V (vs. Zn 2+ /Zn). Electrochemical impedance spectroscopy (EIS) was conducted at OCV in the frequency range from 1 MHz to 10 mHz with a potential perturbation amplitude of 5 mV, in two-electrode Swagelok cells where zinc metal was used as both working and reference electrodes. EQCM-R experiment were carried out using AT-cut gold-patterned 9 MHz quartz resonators (AWS, Valencia, Spain) as substrates. A homogeneous slurry containing the active material (THAQ) was prepared and drop-casted on the gold electrode (0.2 cm 2 ) of the quartz resonator. For the slurry preparation, the THAQ has been mixed thoroughly with Ketjen Black (KB) and PVdF(Polyvinylidene fluoride) in 24:56:20 (THAQ:KB:PVdF) ratio prior to be dispersing in NMP, in order to ensure the binding and the electrical conductivity of the resulting composite electrode. After drop-casting, the QCM resonators were dried in a hot-plate at 80 °C. The active mass loading (THAQ) has been kept as low as possible (~ 7.14 μg), to have a thin layer covering the gold electrode of the quartz resonator. As prepared electrodes were then subjected to electrochemical measurements performed with Bio-Logic electrochemical workstation (Bio-Logic SP200) coupled with a SEIKO QCM922A device, which permitted the frequency (f) and the motional resistance changes (Rm) to be simultaneously monitored. As prepared QCMs, platinum wire and Ag/AgCl electrode were used as the working, counter, and reference electrodes, respectively. All the EQCM tests have been performed in 2 M ZnSO4 aqueous electrolyte.

DFT Calculation

Theoretical calculations were performed to investigate the charge compensation mechanism of the THAQ. DFT calculations were performed by the Gaussian 16W software [START_REF] Frisch | Gaussian 16 Rev. C[END_REF] package to gain structural information of the molecules. On the basis of the optimized structure of THAQ molecule (B3LYP/6-31G(d,p) [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] and wB97XD/6-31G (d,p) [START_REF] Alidağı | Pyrene functionalized cyclotriphosphazene-based dyes: Synthesis, intramolecular excimer formation, and fluorescence receptor for the detection of nitro-aromatic compounds[END_REF]), the molecular electrostatic potential (MESP) method was applied to predict the electrostatic potential map of THAQ. The MESP analysis was also done to deduce the possible Zn/H uptake positions using the Gaussian 16W software package. The single point energy (SPE) and frequency calculations of involving Zn species were calculated by B3LYP basis set LANL2DZP level. [START_REF] Tie | Proton Insertion Chemistry of a Zinc-Organic Battery[END_REF]The final Gibbs free energy is the sum of electronic energy and thermal correction from the frequency analysis. Also, partial optimization (freeze atoms) was performed with a T* structure because of the chemical structure and interactions of THAQ. The binding energy (Eb) was obtained by calculating the Gibbs free energy difference between the total energy of the complex (𝐺 𝑇 * 𝐻𝑥𝑍𝑛𝑦 ) and the sum of individual energy of the ions (𝐺 H + , 𝐺 Zn 2+ ) and (GT*), respectively [START_REF] Gao | A high-performance aqueous rechargeable zinc battery based on organic cathode integrating quinone and pyrazine[END_REF], which was expressed as follows:

𝐸 𝑏 = 𝛥𝐺 = (𝐺 𝑇 * 𝐻𝑥𝑍𝑛𝑦 ) -(𝐺 𝑇 * ) -𝑥𝐺 𝐻+ -𝑦𝐺 𝑍𝑛2+

Results and discussion

THAQ was synthesized by one-step reaction of hexachlorocyclotriphosphazene and 2hydroxyanthraquinone, in the presence of NaH in THF (Figure 1a). The 31 P NMR spectrum of THAQ exhibited a single peak at 7.15 ppm, indication of substitution of all chlorine atoms of HCCP (Figure 1b). The complete substitution was further supported by the disappearance of the HCCP peak, which is supposed to be at approximately 19 ppm, as well as the absence of specific chlorine color in the simple flame test (inset Figure 1a). The simple flame test is used to qualitatively examine the presence of certain elements [START_REF] Lamb | The Copper Flame Test For Halogen In Air.1[END_REF][START_REF] Sanger | Flame Tests: Which Ion Causes the Color?[END_REF], especially metals and halogens in chemical compounds. Specific green color of chlorine in HCCP was no longer observed for that of THAQ. In the 1 H NMR spectrum, the peaks located at 8.08 and 6.60 ppm suggested the five types of aromatic H atoms (also integration ratios 1:1:1:2:2) in THAQ (Figure 1b). In addition, the exact m/z value of 1475.05 was found by MALDI TOF, which is in-agreement with the calculated theoretical value of 1474.18, ascertaining the successful synthesis of the THAQ (Figure 1c). The FT-IR spectra of THAQ and HAQ are shown in Figure 1d. The characteristic vibrations of C=O and C=C peaks are located at 1677 cm -1 and 1588 cm -1 , respectively, which correspond to anthraquinone side groups. [START_REF] Li | Polysulfone grafted with anthraquinone-hydroanthraquinone redox as a flexible membrane electrode for aqueous batteries[END_REF] The characteristic vibrations peaks of HCCP at 1176 and 1152 cm - 1 clearly belongs to P=N and P-N vibrations. [START_REF] Yeşilot | A novel polyphosphazene with nitroxide radical side groups as cathode-active material in Li-ion batteries[END_REF] Compared with the FT-IR spectrum of HAQ, the 11 vibrations of the hydroxyl group (-OH) (3339 cm -1 ) disappeared in the FT-IR spectrum of THAQ, instead new peaks appear that are assigned to the P=N (1176 cm -1 ), P-N (1132 cm -1 ) and P-O-C (923 cm -1 ). The thermal stability of THAQ and HAQ were compared with TGA performed between 25-700 ℃ (Figure 2a). The first obvious weight loss of THAQ is observed at approximately 120 °C and 59.2 wt.% remained up to 700 °C, versus only 2.2 % for HAQ. The specific ignition time (SIT), specific self-extinguish time (SET), and the limiting oxygen index (LOI) values, which determine the flammability and ignitability of materials, were determined by burning test (Figure 2b) and TGA. [START_REF] Zhou | Polyphosphazenesbased flame retardants: A review[END_REF] The calculated LOI of the THAQ and HAQ are ~41 and ~18, respectively, determined by the residue char yield from the TGA data. THAQ can be classified as self-extinguishing materials (LOI: 28-100). [START_REF] Amarnath | Eco-Friendly Halogen-Free Flame Retardant Cardanol Polyphosphazene Polybenzoxazine Networks[END_REF] The SIT and SET values of THAQ and HAQ were measured to be 5.23 -5.11 and 3.99 -9.17 s, as shown in Figure 2d. In addition, vibrations of P=N and P-O-C bonds are observed in the FT-IR spectrum of THAQ after the burning test (Figure 2c). [START_REF] Yeşilot | Halogen-Free Polyphosphazene-Based Flame Retardant Cathode Materials for Li-S Batteries[END_REF] meaning that THAQ molecule was preserved its chemical structure to a certain extent upon burning. As a result, the thermal stability of THAQ is acceptable and reliable for battery applications. Redox behavior of the THAQ/KB (weight ratio of 30:70) cathode was evaluated by using cyclic voltammetry (CV) between a voltage range of 0.1 and 1.7 V vs. Zn 2+ /Zn (Figure 3a). A pair of redox peaks at scan rate of 1 mV s -1 were observed at 0.42 and 0.65 V (vs Zn 2+ /Zn) for the reduction and oxidation, respectively, which was ascribed to the reversible (de)coordination reaction between the charge carrier species and carbonyl group (C=O) of quinone based active materials. [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] After the initial CV cycle, overlapping consecutive CV curves indicate the high stability of THAQ/KB electrode. Moreover, in order to compare the redox behavior of THAQ/KB, commercially available anthraquinone (AQ) was used with the same carbon amount (quoted as AQ/KB) electrode. Similarly, AQ/KB electrode demonstrated one reduction and one oxidation peak at 0.38 and 0.73 V vs. Zn/Zn 2+ . [START_REF] Wang | An organic cathode with tailored working potential for aqueous Zn-ion batteries[END_REF] The redox peak separation of AQ is more significant in comparison with the THAQ composite (Figure S1) that is ascribed to the lower electronic conductivity of AQ resulting from its wider band gap in agreement with the DFT calculations to be discussed below. Electrochemical performance of THAQ/KB electrode was further investigated by galvanostatic discharge/charge measurements. THAQ/KB cell exhibited stable discharge voltage profiles at different current densities (Figure 3b). The plateau-like voltage behavior of the composite cathode is favorable for attaining a stable voltage output. [START_REF] Kumankuma-Sarpong | Naphthoquinone-Based Composite Cathodes for Aqueous Rechargeable Zinc-Ion Batteries[END_REF][START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] C-rate performance of the THAQ/KB composite cathode delivered discharge capacity of ca. 157, 150, 158, 160, 163 and 167 mAh g -1 at a current density of 120, 60, 30, 20, 10 and 5C, respectively, where, 1 C corresponds to ~36 mA g -1 . During the rate capability performance test, this electrode showed high discharge capability as well as high Coulombic Efficiency (CE) (~99%). However, the lower CE obtained at relatively lower current density (~95% at a current density of 5C) could be originated from involvement of the proton during the charge compensations. [START_REF] Emanuelsson | An All-Organic Proton Battery[END_REF] Furthermore, cycling stability performance results of THAQ/KB and AQ/KB electrodes at a current density of 30 C (1.1 A g -1 ) were comparatively presented in Figure 3d. Although AQ/KB composite electrode provided higher first discharge capacity of 277 mAh g -1 than THAQ/KB (initial discharge capacity of 174 mAh g -1 ), the cathode exhibited poorer cycling performance. Remarkably, THAQ/KB retained its discharge capacity of 144 mAh g -1 with a capacity retention of 82.5% after 1000 cycles.

Compared with THAQ/KB electrode, the deteriorating cycle performance of AQ/KB composite could be due to the dissolution of the host material in the aqueous medium. [START_REF] Wang | An organic cathode with tailored working potential for aqueous Zn-ion batteries[END_REF][START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] Furthermore, ultra-fast current density applied to the THAQ/KB can be seen at Figure 3e, in which over 100 mAh g -1 capacity was obtained at 100 C during long-term cycling (>30000). Especially, when organic electrode materials utilized in aqueous electrolyte ZIBs were compared, THAQ/KB cathode stands out with its ultra-long-term cycling (30000 cycles) stability and capacity retention (63 %) at high current rates (3600 mA g -1 ), exhibiting superior electrochemical performance (Table S1). Electrochemical impedance spectroscopy (EIS) profile was investigated to reveal the reason for the better rate capability of the synthesized THAQ electrode compared with the commercial AQ.

In Figure S2c, R1, R2, R3 components of the proposed equivalent circuit refer to electrolyte resistance (Re), solid-electrolyte interface resistance (RSEI) and charge-transfer resistance (Rct), respectively. [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] The charge transfer resistances obtained by fitting the EIS data of THAQ (Figure S2a) and AQ (Figure S2b) were found to be 38.4 Ω and 903.3 Ω. THAQ/KB composite cathode, which resulted in reasonably lower charge transfer resistance, allowing faster ions transportation during discharge/charge processes in accordance with the electrochemical performance.

Dissolution of organic electrode materials especially discharged products is a major drawback, which causes a rapid capacity decay during cycling. Therefore, the solubility of the THAQ/KB and AQ/KB electrodes were tested in an aqueous solution. The pristine and discharge states of all solutions were colorless in the water, indicating that THAQ/KB and AQ/KB are not dissolved in the aqueous system (inset Figure S3). Further UV-vis spectroscopy tests of THAQ/KB and AQ/KB were performed to compare their solubility and it was observed that while THAQ/KB did not show any absorption peak, AQ/KB demonstrated an absorption peak at around 270 nm revealing its slight solubility in aqueous medium (Figure S3). Consequently, the insoluble nature of THAQ ensured advanced cycling stability in the aqueous battery systems compared to the AQ counterparts.

Numerous studies have been conducted to clarify the charge storage mechanism of organic active materials in aqueous electrolyte zinc-ion batteries. These studies suggest that charge compensation in organic cathodes can be contributed by several mechanisms, including (i) only Zn 2+ coordination, [START_REF] Guo | An Environmentally Friendly and Flexible Aqueous Zinc Battery Using an Organic Cathode[END_REF] (ii) individual H + intercalation, [START_REF] Tie | Proton Insertion Chemistry of a Zinc-Organic Battery[END_REF] or (iii) H + /Zn 2+ co-insertion. [START_REF] Lin | A high capacity small molecule quinone cathode for rechargeable aqueous zinc-organic batteries[END_REF][START_REF] Wang | Molecular Engineering of Covalent Organic Framework Cathodes for Enhanced Zinc-Ion Batteries[END_REF][START_REF] Gao | A high-performance aqueous rechargeable zinc battery based on organic cathode integrating quinone and pyrazine[END_REF][START_REF] Wang | Binding Zinc Ions by Carboxyl Groups from Adjacent Molecules toward Long-Life Aqueous Zinc-Organic Batteries[END_REF][START_REF] Na | Effects of Zn2+ and H+ Association with Naphthalene Diimide Electrodes for Aqueous Zn-Ion Batteries[END_REF]Therefore, herein, the electrochemical reaction mechanism of the THAQ electrode was investigated by various ex-situ characterization tools at the pristine, discharged, and charged states (Figure 4a). Moreover, theoretical calculations of were performed to investigate the electrochemical properties of the THAQ and AQ. Firstly, ex-situ FT-IR and XPS analyses were performed in which C=O bonds are regarded as active sites to the host charge carrier ions (Figures 4b and4c). The peak intensity of C=O was monitored by ex-situ FT-IR at the pristine, discharged and charged states of THAQ. The peak at 1677 cm -1 assigned to the stretching vibration of the C=O bonds decreased in intensity and the vibration of C-O bonds at 1379 cm -1 appeared upon discharge. [START_REF] Vizintin | Probing electrochemical reactions in organic cathode materials via in operando infrared spectroscopy[END_REF] During the charging step the reverse change occurred for the C-O. XPS analyses were used to further investigate the mechanism of the charge storage during the cycling process. The survey XPS spectra of THAQ/KB cathode material confirmed the presence of C, O, P, N for the pristine sample (Figure 4c). Zn-2p XPS spectra of the THAQ/KB indicate the appearance of the divalent state Zn element after the first discharge down to 0.1 V, while it becomes much weaker when the electrode is recharged to 1.7 V (Figure 4d). This observation may indicate a slight Zn 2+ trapping which is in line with the lower initial CE value in Figure 3c. Besides, the C1s XPS spectra of the pristine sample in Figure 4e demonstrates two main peaks attributed to C=O (288.6 eV) ,C=C (284.7 eV) bonds , belonging to the anthraquinone side groups. [START_REF] Li | Rechargeable Aqueous Polymer-Air Batteries Based on Polyanthraquinone Anode[END_REF] In addition, the peak, which slight π-π interaction of the THAQ, is located at 291.2 eV. [START_REF] Zeng | π -π interaction between carbon fibre and epoxy resin for interface improvement in composites[END_REF] After the first discharge to 0.1 V, the C=O peak has diminished, while that corresponding to C-O, located at 287.2 eV, becomes more prominent, indicating an interaction with the charge carries ions. On the other side, during recharge remaining C=O bond is still observed that is consistent with the FT-IR spectra of the THAQ/KB, corresponding to a potential trapping of the charge carrier ions.

Additionally, elemental mappings of the THAQ/KB cathode surface display the first dischargecharge process from a macroscopic view (Figure 4f). The spatial distribution of C, O and P elements is similar in both discharge and charge states. However, the intensiveness of Zn mapping decreased from the discharge state to charge, demonstrating participation of Zn 2+ into charge compensation process. In addition, a typical flake-like morphology of zinc sulfate hydroxide Zn4(OH)6SO4•xH2O (x varying between 0.5 and 1) was observed in SEM image of the fully discharged THAQ/KB electrode in 2 M ZnSO4 electrolyte (Figure S4). More significantly, EDS (Energy dispersive X-ray spectroscopy) (Figure S4) analysis revealed the elemental composition of the first discharged THAQ/KB which possesses the mole ratio of Zn:S about 4, implying the formation of Zn4(OH)6SO4•xH2O (ZHS) structure. Furthermore, the new intense peaks at 12.4 ᵒ , 32.9 ᵒ and 35.3 ᵒ are observed in the XRD pattern of the discharged electrode (Figure S5). These peaks formed after the discharge are indexed as zinc sulfate hydroxide (Zn4(OH)6SO4•0.5H2O, JCPDS number 00-044-0674), [START_REF] Wang | Molecular Engineering of Covalent Organic Framework Cathodes for Enhanced Zinc-Ion Batteries[END_REF] arising from protons' participation to charge compensation process. As it is well-known, the consumption of protons induces local pH increase, then OH - reacts with ZnSO4 and H2O to form the so-called ZHS as a by-product. [START_REF] Pan | Reversible aqueous zinc/manganese oxide energy storage from conversion reactions[END_REF] The results of these ex-situ FT-IR, XPS and XRD analyses strengthen the idea that the coordination of both Zn 2+ cations and as well as H + occurs during discharge of the THAQ/KB electrode. In order to further investigate the interfacial charge storage properties of the THAQ electrodes, EQCM has been employed as a piezoelectric sensor coupled to electrochemical measurements.

We have simultaneously monitored the frequency variation (Δf) during the redox process of the THAQ composite thin film. The current and Δf responses of the EQCM measurement carried out at 25 mV s -1 are shown in Figure 5a. A pair of pronounced redox peaks can be seen in the CV profiles (Figure 5a) in agreement with the Figure 3a. A significant frequency decrease and increase were observed, during a cathodic and an anodic scan, respectively (Figure 5a). This would imply a major mass increase/decrease during the reduction and oxidation process in Figure 5a. It is noted that these Δf patterns can be correlated to mass variations (Δm), only if the gravimetric conditions are satisfied, then the Sauerbrey equation: Δf = -ks × Δm, where ks is the experimental sensitivity coefficient, can be applied. [START_REF] Hillman | The EQCM: electrogravimetry with a light touch[END_REF][START_REF] Lemaire | Making Advanced Electrogravimetry as an Affordable Analytical Tool for Battery Interface Characterization[END_REF][START_REF] Sauerbrey | Verwendung von Schwingquarzen zur Wägung dünner Schichten und zur Mikrowägung[END_REF] During the commence of reduction the Δf changes are small, whereas the prominent decrease can be observed when the potential is approaching -0.48 V. During the reverse scan, again a noticeable frequency increase can be seen in the vicinity of the oxidation peak around -0.42 V. The overall Δf profiles may indicate a charge compensation process by a major cationic response, however the Δf magnitudes are rather high. This observation has prompted us to study the plausible contributions, leading to such high Δf values. Figure 5b depicts the simultaneously measured Δf and motional resistance change (ΔRm) as a function time, during the EQCM measurements. No significant changes in the profiles (values going back to initial during the reverse scan) ensure the stability of the active material upon cycling. Figure 5c shows an example Δf vs. ΔQ profile (at 25 mV s -1 ), corresponding to the reduction scan of the EQCM response in Figure 5a. This profile can be used to estimate the average mass per electron (MPE) using the following equation: 𝑀. 𝑃. 𝐸. = 𝑛 × 𝐹 × ∆𝑚 ∆𝑄 ⁄ where, n is the number of the electrons transferred, F is the Faraday's constant, assuming that the microbalance works as a gravimetric probe. From the MPE analyses, an estimation of the molar mass of the species which participate in the charge compensation during the redox process can be drawn, if there is no other contribution to the frequency changes. The MPE analysis of the middle region shown in Figure 5c lead to a significantly high value, i.e., 859 g mol -1 . This high MPE value indicates that another electro-assisted process contributes to the frequency variations, in agreement with our previous work on organic electrodes and also in literature reports. [START_REF] Sariyer | Aqueous Multivalent Charge Storage Mechanism in Aromatic Diamine-Based Organic Electrodes[END_REF][START_REF] Efremova | EQCM study of intercalation processes into electrodeposited MnO2 electrode in aqueous zinc-ion battery electrolyte[END_REF] Remarkably high ΔRm values in Fig. 5b attests this result that the Δf values cannot be attributed to solely a cation coordination process of THAQ active material.

Overall, the frequency profiles provided by the EQCM demonstrates that the charge compensation process is intertwined with the ZHS formation during the reduction, related to the local pH changes (as a result of coordination of H + with the THAQ, in agreement with the predictions of the DFT calculations discussed below). In general, such phenomena leads to a viscosity increase at the vicinity of the electrode/electrolyte interface, characterized by the motional resistance (Rm) increase during the EQCM measurements, indicating that the acoustic wave has difficulty to pass through the newly formed interface. [START_REF] Lemaire | Making Advanced Electrogravimetry as an Affordable Analytical Tool for Battery Interface Characterization[END_REF][START_REF] Sariyer | Aqueous Multivalent Charge Storage Mechanism in Aromatic Diamine-Based Organic Electrodes[END_REF] The high ΔRm/Δf ratio in Figure 5b With the intention of better understanding of the structural arrangement and predicting Zn 2+ /H + ions coordination mechanism in the THAQ molecule, theoretical calculations were carried out using the density functional theory (DFT) with the B3LYP/6-31G(d,p) [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] and B3LYP

/LANL2DZP [START_REF] Tie | Proton Insertion Chemistry of a Zinc-Organic Battery[END_REF] basis set in the gas phase. The optimized geometry of the THAQ demonstrated that the distance between anthraquinone units in the THAQ structure is less than 4Å (Figure S6a).

Hence, the structure was re-optimized using wB97XD/6-31G (d,p) levels which is a more suitable method to determine the non-covalent π-π interactions (Figure S6b-S6c). [START_REF] Alidağı | Pyrene functionalized cyclotriphosphazene-based dyes: Synthesis, intramolecular excimer formation, and fluorescence receptor for the detection of nitro-aromatic compounds[END_REF] In addition, the peak at around 25.65° at the XRD spectrum of THAQ could be attributed to π-π interaction resulting from the polar carbonyl groups in anthraquinone structure (Figure S6d) [START_REF] Tang | Tailoring π-Conjugated Systems: From π-π Stacking to High-Rate-Performance Organic Cathodes[END_REF][START_REF] Xu | Pseudocapacitive anthraquinone modified with reduced graphene oxide for flexible symmetric all-solid-state supercapacitors[END_REF]. The molecular orbitals of AQ and THAQ are shown in Figure 6a, including the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for the comparison of their structural stability. The LUMO energy level of THAQ (-2.958 eV) is much lower than that of AQ (-2.792 eV), indicating a higher electron affinity and, thus higher reduction potentials of THAQ. [START_REF] Yeşilot | A Flame-Retardant and Insoluble Inorganic-Organic Hybrid Cathode Material Based on Polyphosphazene with Pyrene-Tetraone for Lithium Ion Batteries[END_REF][START_REF] Ye | High-rate aqueous zinc-organic battery achieved by lowering HOMO/LUMO of organic cathode[END_REF] Furthermore, due to the increased π-π interaction with the THAQ structure, it provides a lower band gap (3.96 eV) compared to the AQ structure (4.20 eV), thus leads to higher electrical conductivity, and as a consequence faster charge transfer, which is beneficial for the improvement of electrochemical properties. [START_REF] Xu | Electrochemical oxidation of π-π coupling organic cathode for enhanced zinc ion storage[END_REF] Moreover, faster zinc ion transfer coefficient was promoted by also Warburg diffusion process calculation (Figure S7), indicating 3.19×10 -16 cm 2 s -1 for THAQ/KB and 4.46×10 -17 cm 2 s -1 for AQ/KB electrode. In addition, the MESP mapping method was used to determine the electroactive regions of the THAQ structure with an optimized geometry obtained by considering π-π interactions (Figure 6b. It is well known that this method is performed to investigate the active site for the cation uptake in which more positive sites (toward blue color) are leaning to react with a nucleophilic reagent, whereas the more negative sites (reddish color) prefer to react with an electrophilic reagent. [START_REF] Zhang | Extended π-Conjugated System in Organic Cathode with Active C═N Bonds for Driving Aqueous Zinc-Ion Batteries[END_REF] Zn 2+ /H + ions can be assumed as electrophile during the discharge process, that is why the sites with a higher negative region are preferable to other sites for the cation uptake in the discharge reaction. [START_REF] Chen | Two-Dimensional Organic Supramolecule via Hydrogen Bonding and π-π Stacking for Ultrahigh Capacity and Long-Life Aqueous Zinc-Organic Batteries[END_REF] Therefore, the red region indicates strong chemical affinity for cation coordination. [START_REF] Zhao | High-capacity aqueous zinc batteries using sustainable quinone electrodes[END_REF] Based on the results from MESP mapping, it was demonstrated that electronegative regions of the THAQ molecule are mainly located around C=O groups, which are the main active sides for cation coordination (Figure 6b, S8). The structure of THAQ has possible sites for multistep cation coordination. The coordination steps of the THAQ structure of zinc ions were constructed based on the MESP method with the single point energy calculations and the side group arrangements (Figure S9).

Depending on the Zn coordination, the energy values decreased thermodynamically and the stability of the structures advanced, as expected. [START_REF] Nam | Redox-Active Phenanthrenequinone Triangles in Aqueous Rechargeable Zinc Batteries[END_REF][START_REF] Sun | Aqueous zinc batteries using N-containing organic cathodes with Zn2+ and H+ Co-uptake[END_REF] Depending on the arrangement of the anthraquinone side groups, there are two different arrangements between the units (Figure 6c). The As it was pointed above in Figure 6c the sides, green highlighted circles are more plausible for Zn coordination. Therefore, one further point of interest is the coordination mechanism of H + that was performed with DFT calculations. [START_REF] Tie | Proton Insertion Chemistry of a Zinc-Organic Battery[END_REF][START_REF] Wang | Binding Zinc Ions by Carboxyl Groups from Adjacent Molecules toward Long-Life Aqueous Zinc-Organic Batteries[END_REF] The coordination mechanism is shown by using the simplified structure of T* (Figure 7a). In Figure 7b, three main coordination types of ions have been illustrated (H + , Zn 2+ or H + -Zn 2+ ). According to the binding energies, H + ion is quite dominant in the coordination steps, especially for the T*-H1 (-1.43 eV) to T*-H4 (-12.088 eV) (Table S2).

When compared with H + , the possibility of Zn 2+ ion is coordination is lower. On the other side, the probability of the formation in steps containing both Zn 2+ and H + ions is also much lower. DFT calculations indicate that the binding of H4 with T* was shown to be more likely than the other Zn 2+ containing configurations. However, under experimental conditions Zn ions involvement is more likely to occur where the concentration of Zn 2+ is much higher than H + in 2 M ZnSO4 electrolyte which leads to a competition between Zn 2+ and H + . [START_REF] Wang | Binding Zinc Ions by Carboxyl Groups from Adjacent Molecules toward Long-Life Aqueous Zinc-Organic Batteries[END_REF] Additionally, the Zn-O (~570 cm -1 ) and O-H (3450 cm -1 ) vibrations [START_REF] Gao | A high-performance aqueous rechargeable zinc battery based on organic cathode integrating quinone and pyrazine[END_REF][START_REF] Chen | Two-Dimensional Organic Supramolecule via Hydrogen Bonding and π-π Stacking for Ultrahigh Capacity and Long-Life Aqueous Zinc-Organic Batteries[END_REF] in the ex-situ FT-IR analysis (Figure S10) and the formation of Zn4(OH)6SO4•0.5H2O in the ex-situ XRD analysis are observed (Figure S5) demonstrating that experimentally both ions coordination is plausible.

Conclusion

In summary, a facile and straightforward one-step synthesis of a small inorganic-organic material (THAQ) from the commercially available starting materials, resulting in a HCCP-bound six anthraquinone structure has been demonstrated. The charge storage mechanism and interfacial processes occurring at the electrode-electrolyte interface were systematically investigated through several ex-situ and in-situ experimental methods and corroborated with the theoretical calculations (DFT). Multiple redox-active sites and insoluble nature in aqueous electrolyte render THAQ electrode a superior cathode material for aqueous ZIBs with ultra-long cycling (>30000 cycles) stability at high current density (3.6 A g -1 ) in comparison to the commercial anthraquinone counterparts (AQ). This superior performance was associated with the increased π-π

Figure 1 .

 1 Figure 1. a) Synthetic route of THAQ (inset simple flame test results of HCCP and THAQ). The characterization of THAQ in b) 31 P and 1 H NMR spectra of THAQ (in d6-DMSO), c) MALDI-

Figure 2 .

 2 Figure 2. a) TGA of HAQ and THAQ, b) burning test of HAQ and THAQ, c) FT-IR spectrum of

Figure 3 .

 3 Figure 3. Electrochemical performance of THAQ/KB electrode a) CV profiles at a scan rate of 1 mV.s -1 , b) Discharge/charge profiles at different current densities, c) Discharge/charge capacity at

Figure 4 .

 4 Figure 4. a) Time-dependent evolution of first discharge and charge cycle for THAQ/KB electrode

Figure 5 .

 5 Figure 5. EQCM analyses of THAQ composite film deposited on the Au electrode of a 9MHz

  indicates the formation of ZHS at the electrode/electrolyte interface and is fully in line with the ex-situ SEM and XRD analyses on the discharged THAQ/KB samples (Figure S4 and S5).

Figure 6 .

 6 Figure 6. a) HOMO and LUMO plots and band gaps of AQ and THAQ b) MESP of THAQ

distance between the anthraquinones is approximately 3 - 4 ÅFigure 7

 347 Figure 7. a) The simplified structure of THAQ (T*) and b) The calculated ∆G energy for T* ion

Acknowledgments

The authors appreciate the financial support from TUBITAK (119N054). TUBITAK ULAKBIM, High Performance and Grid Computing Center (TR-Grid e-Infrastructure) is thanked for the calculations data in the computational chemistry details of this paper.

interaction in THAQ structure. Theoretical analyses based on DFT reveals the lower band gap of THAQ (3.96 eV) compared to the AQ structure (4.20 eV). The reversible ion coordination of THAQ was verified by monitoring the carbonyl groups using the time-dependent FT-IR and ex-situ XPS. Involvement of proton towards the charge compensation was further identified by the ex-situ XRD measurement, which demonstrates the formation of well-known ZHS resulting from the local pH increase. EQCM analyses are well-consistent with XRD analyses, where the reversible ZHS formation/dissolution can be identified indicating both Zn 2+ and H + participation.

Overall, this work is the first report revealing the nature of charge carries of anthraquinone decorated cyclophosphazene obtained by a facile and cost-effective method and demonstrates the development of small cathode molecule based on inorganic-organic material systems to be used in aqueous ZIBs with excellent electrochemical performance.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Zengyu Zhang

Shiqing Huang

Chen-Yuan Li

Liang-Liang Zhao

Wei Liu

Mohand Melaimi

Guy Bertrand
email: gbertrand@ucsd.edu

Xiaoyu Yan

The triplet state of deprotonated mesoionic carbene Breslow intermediates is thermally accessible: Distal difunctionalization of aldehydes

INTRODUCTION

N-heterocyclic carbenes (NHCs), especially thiazol-2-ylidenes A 1 and 1,2,4-triazol-5-ylidenes B, [START_REF] Enders | Preparation, Structure, and Reactivity of 1,3,4-Triphenyl-4,5-dihydro-1H-1,2,4-triazol-5-ylidene, a New Stable Carbene[END_REF] have long been known as organocatalysts for aldehyde transformations, such as the benzoin condensation, Stetter reaction, cycloadditions, Claisen rearrangement, etc. 3 , 4 , 5 , 6 , 7 (Fig 1a). These reactions occur via electron-pair-transfer processes through the so-called Breslow Intermediates (BIs). [START_REF] Breslow | On the Mechanism of Thiamine Action. IV. Evidence from Studies on Model Systems[END_REF][START_REF] Berkessel | Aldehyde Umpolung by N-Heterocyclic Carbenes: NMR Characterization of the Breslow Intermediate in its Keto Form, and a Spiro-Dioxolane as the Resting State of the Catalytic System[END_REF][START_REF] Berkessel | Characterization of the Key Intermediates of Carbene-Catalyzed Umpolung by NMR Spectroscopy and X-Ray Diffraction: Breslow Intermediates, Homoenolates, and Azolium Enolates[END_REF][START_REF] Paul | Breslow Intermediates from a Thiazolin-2-ylidene and Fluorinated Aldehydes: XRD and Solution-Phase NMR Spectroscopic Characterization[END_REF][START_REF] Wessels | Formation of Breslow Intermediates from N-Heterocyclic Carbenes and Aldehydes Involves Autocatalysis by the Breslow Intermediate, and a Hemiacetal[END_REF] The role of the BI in single electron transfer (SET) pathways was also established a long time ago in biosynthesis, 13 and expanded in the last decade to NHC-organocatalysis. 14 , 15 , 16 , 17 Among the important differences between the electron-pair-transfer and the SET pathways is that the first relies on the BI, whereas the second involves the deprotonated BI (BI -). [START_REF] Regnier | What Are the Radical Intermediates in Oxidative N-Heterocyclic Carbene Organocatalysis?[END_REF][START_REF] Delfau | Critical Assessment of the Reducing Ability of Breslow-type Derivatives and Implications for Carbene-Catalyzed Radical Reactions[END_REF][START_REF] Ishii | N-Heterocyclic Carbene-Catalyzed Decarboxylative Alkylation of Aldehydes[END_REF][START_REF] Ishii | N-Heterocyclic Carbene-Catalyzed Radical Relay Enabling Vicinal Alkylacylation of Alkenes[END_REF][START_REF] Li | Radical Acylfluoroalkylation of Olefins through N-Heterocyclic Carbene Organocatalysis[END_REF][START_REF] Kim | N-Heterocyclic carbene-catalyzed deaminative cross-coupling of aldehydes with Katritzky pyridinium salts[END_REF]24 Also noteworthy is the recent development of NHC/photoredox dual catalysis [START_REF] Liu | Light opens a new window for N-heterocyclic carbene catalysis[END_REF] involving acyl azolium salts BI + , [START_REF] Bay | Combined Photoredox and Carbene Catalysis for the Synthesis of Ketones from Carboxylic Acids[END_REF][START_REF] Meng | Cooperative NHC and Photoredox Catalysis for the Synthesis of β-Trifluoromethylated Alkyl Aryl Angew[END_REF][START_REF] Meng | Benzylic C-H Acylation by Cooperative NHC and Photoredox Catalysis[END_REF][START_REF] Liu | Direct α-Acylation of Alkenes via N-Heterocyclic Carbene, Sulfinate and Photoredox Cooperative Triple Catalysis[END_REF][START_REF] Zuo | Cooperative NHC/Photoredox Catalyzed Ring-Opening of Aryl Cyclopropanes to 1-Aroyloxylated-3-Acylated Alkanes[END_REF][START_REF] Yu | Aroyl Fluorides as Bifunctional Reagents for Dearomatizing Fluoroaroylation of Benzofurans[END_REF][START_REF] Bay | Light-Driven Carbene Catalysis for the Synthesis of Aliphatic and α-Amino Ketones[END_REF][START_REF] Bayly | High-throughput photocapture approach for reaction discovery[END_REF]34,[START_REF] Bay | Synthesis of Cyclohexanones by a Tandem Photocatalyzed Annulation[END_REF][START_REF] Sato | Light-Driven N-Heterocyclic Carbene Catalysis Using Alkylborates[END_REF] as well as few reports [START_REF] Ren | Carbene-Catalyzed Alkylation of Carboxylic Esters via Direct Photoexcitation of Acyl Azolium Intermediates[END_REF][START_REF] Huang | Suzuki-type cross-coupling of alkyl trifluoroborates with acid fluoride enabled by NHC/photoredox dual catalysis[END_REF][START_REF] Mavroskoufis | N-Heterocyclic Carbene Catalyzed Photoenolization/Diels-Alder Reaction of Acid Fluorides[END_REF] dealing with their excited state (BI +* ), both of which being the doubly oxidized deprotonated BI (Fig 1b). Obviously, the latter processes are based on the oxidizing power of the BI + and the BI +* , whereas NHC-organocatalysis via SET reaction depends on the reducing ability of BI -in the ground state. Herein we report that the excited triplet state of BI -(BI -* ) derived from 1,2,3-triazol-5-ylidenes (mesoionic carbenes, MIC) D 40 , 41 is thermally accessible, allowing for the catalytic distal difunctionalization of aryl aldehydes. At the formyl side, an oxidative esterification reaction takes place, concomitant with reduction of the aryl side. Here, three types of chemical transformation are demonstrated, namely hydro(deutero)defunctionalization, hydrogenation and reductive alkylation. (Fig. 1c).

RESULTS AND DISCUSSION

To gain further insight into the electronic properties of BIs and BI -s derived from A, B, C [START_REF] Arduengo | A stable crystalline carbene[END_REF] and MICs D, we performed density functional theory (DFT) calculations at the PWPB95-D3/def2QZVPP//B3LYP-D3(BJ)/6-31G** + SMD (THF) level of theory.

The adiabatic singlet-triplet gaps for BIs derived from classical NHCs A-C are greater than 37.0 kcal/mol, whereas for the mesoionic carbene (D)BI, 43 it is only 23.7 kcal/mol (Fig. 2). The same trend is observed for the singlet-triplet gap of the corresponding deprotonated BIs (BI -s). They vary from 26.1 to 28.7 kcal/mol for NHCs A-C, but the gap is strikingly as low as 4.0 kcal/mol for the mesoionic analogue (D)BI -. The small S/T gap for (D)BI and (D)BI -is due to the polarized exocyclic CC bond, 43 which in the singlet state leads to a repulsion between the negatively charged carbon and adjacent oxygen. The SOMO of (D)BI -* has contributions from the π orbital of the triazole ring, the carbonyl and, interestingly, the phenyl ring. The SOMO-1 is mainly located on the π* orbital of the triazole ring and on the carbonyl group. From these computational results, we were confident that the very low singlet-triplet gap predicted for (D)BI - would allow access to the triplet form (D)BI -* without photoexcitation. Moreover, since the SOMO is at least partially located on the phenyl group of the aldehyde moiety, it implied that the ring has a radical anion character. Thus, we envisioned that if a leaving group is present on the aryl ring, it could be eliminated, affording a biradical (D)BR*. DFT calculations predict that such compounds would have a singlet ground state 14.5 kcal/mol lower in energy than the triplet and should therefore be regarded as the zwitterionic compound (D)ZW, which should allow for distal difunctionalization of aryl aldehydes. We capitalized on the potent leaving group ability of sulfonyl groups through single-electron reduction [START_REF] Chu | Desulfonylation via Radical Process: Recent Developments in Organic Synthesis[END_REF] to check our hypothesis. We first reacted 4-sulfonylbenzaldehyde 1a with methanol in the presence of 10 mol% of a 1,2,3-triazolium salt as precatalyst, using a base to generate in situ the corresponding mesoionic carbene D and to deprotonate the ensuing Breslow Intermediate. With 4,5-unsubstituted triazolium Cata and Catb, methyl benzoate 2aa was formed in 42 and 38% yield, respectively (Figure 3, entries 1-2). The leaving group was quantitatively recovered as benzenesulfinic acid (PhSO2H). The yield in 2aa significantly increased to 65 and 74% by using ester substituted triazolium salts Catc and Catd, (entries 3-4), which have proven to be the best catalysts for the previously reported MIC-catalyzed arylacylation of alkynes via SET. [START_REF] Liu | Mesoionic Carbene-Breslow Intermediates as Super Electron Donors: Application to the Metal-Free Arylacylation of Alkenes[END_REF] Classical-NHCs were less efficient; only very low yields were obtained (entries 5-7). Base and solvent screening showed t-BuOK and t-BuOMe (MTBE) to be the best choices (entries 8-12). Reducing the amounts of MeOH or decreasing the reaction temperature gave lower yields (entries 13-15). A control experiment in the dark gave the same yield, which revealed that this reaction did not proceed via photoexcitation (entries 16). Note that the transformation of the 4-sulfonylbenzaldehyde 1a into methyl benzoate 2aa, involved a reductive desulfonylation at the aryl group and an oxidative esterification at formyl carbon, and is therefore overall redox-neutral. The scope of the MIC-catalyzed hydrodefunctionalization reaction was investigated under the optimized conditions. Treatment of 1a with CH3OD and CD3OD gave deuterated products 2aa-D1 and 2aa-D4, respectively. Note that the only partial deuteration at the phenyl ring is due to the presence of the formyl-H and MIC + -H, which are exchangeable. Different nucleophiles were next employed. The yields decreased as the chain length and steric hindrance of the alcohols increase. Primary alcohols gave benzoate 2aa-2ae in 30-71% yields. For secondary alcohols, only cyclopropanol gave the corresponding benzoate 2af in 37% yield, while other alcohols gave trace products. When ethanethiol was employed with ethanol as proton source, S-ethyl benzothioate 2ag could be isolated in 47% yield. We extended our investigations to sulfonyl-substituted aryl aldehydes 1b-1e. When bromo, trifluoromethyl and methyl substituents were embedded on 4-sulfonylbenzaldehyde, using CH3OD as deuterium source, the corresponding deuterodesulfonylation products 2b-D1 -2d-D1 were isolated in 43-80% yields with deuteration levels of 50-53% at the 4-position. 5-Sulfonyl-2-thiophenecarbaldehyde 1e gave deuterated product 2e-D1 in 73% isolated yield. Other sulfonyl groups, such as methylsulfonyl (1f) and aminosulfonyl (1g) were converted less efficiently, giving benzoate 2aa in 30 and 42% yield, respectively. Interestingly, the sulfonyl group can be placed in a more distal position, as shown by the desulfonylation of 1h, 1i and 1j which gave 2h and 2i in 27, 38 and 25 % yield, respectively.

The scope of the MIC-catalyzed hydrodefunctionalization and esterification of aryl aldehydes is not restricted to the sulfonyl functionality. Esters can also be used as the leaving group, as shown by the formation of the toluoate 2i in 21% yield. Fluoride is also an attractive leaving group, since the reductive defluorination of trifluoromethylarenes is a versatile strategy for the construction of difluoromethylarenes. [START_REF] Chen | Selective Single C(sp3)-F Bond Cleavage in Trifluoromethylarenes: Merging Visible-Light Catalysis with Lewis Acid Activation[END_REF] When 4-trifluoromethyl substituted aryl aldehydes 1l were employed, the corresponding 4-difluoromethylbenzoates 2l was formed in 42% isolated yields. The 4-deuterodifluoromethylbenzoate 2l-D1 was obtained in 40% yield and 60% deuteration level, using CH3OD as deuterium source. When the trifluoromethyl group was placed in the ortho-position, the corresponding We were curious to test whether the MIC-catalyzed distal difunctionalization of aryl aldehydes could be extended to other types of reduction processes. We first chose the hydrogenation reaction of azo compounds (Fig. 5). Under the optimized conditions, in the presence of methanol as the NuH agent, 4-azobenzaldehyde 3a was converted into the corresponding hydrazine 4a in 90% yield. By using CD3OD as deuterium source, the deuterated product 4a-D3 was obtained in 87% yield with a high deuteration level of 95%. Ethanol can also act as the NuH, affording 4b, which was isolated in 67% yield.

Azo derivatives with different substituents 3c-3f also gave hydrazine derivatives 4c-f in 47-76% yields. The azo group can also be placed in meta-and even a more distal position, giving the target products 4g-4h in 50-61% yields. Note that these results allow to rule out a two-electron pathway via p-or o-quinodimethane. [START_REF] Dai | NHC-Catalyzed ε-Umpolung via p-Quinodimethanes and Its Nucleophilic Addition to Ketones[END_REF] These hydrogenation reactions can also be extended to other unsaturated substrates. When 4-benzoylbenzaldehyde 5 was used, the benzoyl group was reduced to the α-hydroxylbenzyl group while the formyl group was oxidized to an ester, giving 6 which was isolated in 40% yield. A dearomatization reaction was also observed for 9-anthraldehyde 7, giving 9,10-dihydroanthracene-9-carboxylate 8 in 32% yield. Lastly, we wanted to demonstrate that the process could allow for C-C bond formation, therefore adding complexity to aryl aldehydes (Fig. 6). To this end, we reacted 4-iodobenzaldehyde 9 under the optimized catalytic conditions with alkenes 10 in the presence of methanol. Gratifyingly, 2-vinylpyridine derivatives (10a-c), 2-vinylbenzothiazole (10d), and even methyl methacrylate (10e) reacted smoothly with 4-iodobenzaldehyde 9, generating the desired products 11a-e in 37-55% yields. Note that in these cases, the second inversion of state could occur either just after the cleavage of the carbon-iodine bond, or after the addition to the alkene.

CONCLUSIONS

Photochemical reactions involving direct excitation of a substrate to its triplet state are well explored. [START_REF] Hoffmann | Photochemical Reactions as Key Steps in Organic Synthesis[END_REF] Recently, this area experienced a renaissance by exploiting a photosensitizer as a catalyst for energy transfer. 51 , 52 , 53 This work shows that the combination of aryl aldehydes with carbenes leads to low S/T adducts (Breslow Intermediates and their deprotonated forms) due to the radical-stabilization effect of carbenes. 54 The S/T gap is especially small with mesoionic carbenes, which results in their low activation energy and viability of thermal excitation. The resulting triplet state of deprotonated Breslow Intermediates enables tandem distal functionalization of aryl aldehydes. The reactivity of these biradicals is significantly different from that of the aforementioned photo-excited triplet substrates, which have a high activation energy, high reactivity and limited reaction patterns (Fig. 7). Synthetic strategies, based on the thermal accessibility of triplet states of other carbene-derived intermediates, such as azolium enolates, dienolates and homoenolates, are under current investigation.

Figure 1

 1 Figure 1 A) Carbenes considered in this study; B) Breslow Intermediate (BI), deprotonated Breslow Intermediate (BI -), and acyl azolium intermediate in the ground state (BI + ) and the excited state (BI +* ); C) mesoionic carbene (D)-catalyzed distal difunctionalization of aryl aldehydes via the excited state of deprotonated Breslow Intermediate (D)BI -* .

Figure 2 A

 2 Figure 2 A) Singlet/triplet gap of BIs and BI -s derived from carbenes A-D; B) SOMO and SOMO-1 of (D)BI -*, isovalue is 0.05; C) Postulated reaction sequence leading to the formation of the zwitterionic compound (D)ZW, allowing for difunctionalization of aryl aldehydes.

Figure 3

 3 Figure 3 Optimization of the reductive desulfonylation and oxidative esterification. a Reaction carried out in 0.3 mmol scale. Yields were determined by 1 H spectroscopy with CH2Br2 as internal standard. Isolated yield is given in parentheses. b 1.0 equiv

2 -

 2 difluoromethylbenzoate 2m was obtained in 22% yield. Interestingly, when two trifluoromethyl groups were placed on the ortho-and para-positions, -4-trifluoromethylbenzoate 2n in 38% yield. Using CH3OD as deuterium source, the deuterated analogue 2n-D1 was isolated in 35% yield with 70% deuteration level.

Figure 4

 4 Figure 4 Scope of the MIC-catalyzed hydrodefunctionalization and esterification of aryl aldehydes. Reactions were carried out in 0.3 mmol scale. b 3 equiv. EtOH, 2.5 equiv t BuOK, and 1.0 equiv EtSH were used. Isolated yields are given. c Temperature was changed from -15 o C to 0 o C. d Catc (20 mol%) was used as catalyst, 90 o C for 5 h. e Catc (20 mol%) was used as catalyst, 50 o C for 5 h.

Figure 5

 5 Figure 5 MIC-catalyzed hydrogenation reactions. Reactions carried out in 0.3 mmol scale. Isolated yields are given. a CD3OD was used. b EtOH was used.

Figure 6 C

 6 Figure 6 C-C bond formation, adding complexity to aryl aldehydes.

Figure 7

 7 Figure 7 Carbenes, especially MICs, lead to adducts featuring low-lying triplet states.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 22171284 and 21602249), the Public Computing Cloud Platform, Renmin University of China, and the NSF (CHE-2153475)


F R Éd

Éric Goualard
email: frederic.goualard@irin.univ-nantes.fr

TOWARDS GOOD C ++ INTERVAL LIBRARIES: TRICKS AND TRAITS

Despite its usefulness for overcoming floating-point arithmetic defects, manipulating imprecise data, and performing non-linear global optimization, interval arithmetic is not yet a first-class type in any computer language. Some attempts are underway for Fortran and Java, not for C++. However, some add-on C++ interval libraries already exist though they do not respect the C++ standard library spirit since they are not generic (one cannot choose the type for the interval bounds). The design and implementation of JAIL, a C++ templated library based on the "traits" C++ programming technique, are presented in this paper; parameterization is shown to enhance the reliability, the versatily, and the portability of the library across multiple platforms. The impact on efficiency of parameterization through templated classes and of some implementation tricks is also investigated.

Introduction

Following Moore's Law, computers performances nearly double every 18 months. But, due to the recourse to floating-point arithmetic for numerical programs, it only means they compute plain wrong results faster and faster. In the sixties, another Moore 1 conceived of interval arithmetic as a means to compute with floats while controlling the accuracy of the results. Roughly speaking, interval arithmetic consists in replacing any real constant by an interval containing it, using extensions of real operations to intervals, and by preserving the inclusion of any intermediate result through outward rounding of the floating-point bounds. Interval arithmetic (shortened hereafter to IA) enjoys some very important properties:

• IA ensures completeness, such that no solution is ever lost. Consequently, any computation that leads to empty intervals for the domain of the variables involved is a proof for the non-existence of solutions; • many classical numerical algorithms, such as Gauss-Seidel's or Newton's, have been adapted to intervals, thereby benefiting from new properties. For example, the interval Newton algorithm has been proved to be convergent whenever the starting domains contain a solution. What is more, condition criteria-based on Miranda's and Brouwer's theorems-may be goualard-ascm2000: submitted to World Scientific on November 9, 2009 1 used to prove the existence and uniqueness of a solution in a computed interval; • IA permits reasoning on sets of values, thus allowing the devising of efficient procedures for nonlinear global optimization.

Proper implementation of interval arithmetic demands floating-point numbers properties and rounding facilities that are available from all IEEE 754 2 compliant computers-the great majority of them-since 1984. However, despite the above-mentioned qualities of interval arithmetic, there still does not exist any language providing intrinsic support for it-a mandatory condition for performance and correctness issues. Are the proposals for integrating interval support into Fortran 2000 a and Java 3 a harbinger of change? At least not for a widely used language such as C ++ , whose adopted standard does not state anything about IA, while defining a generic class for complex arithmetic (i.e. a template of class that can be instantiated with different float formats, depending on the required precision). Some C ++ interval libraries already exist 4,5 though they do not respect the spirit of standard C ++ libraries since they do not provide generic classes but a "hard-wired" class for intervals whose bounds are usually doubles.

In this paper, we present the design and implementation of JAIL b , a templated C ++ interval library, and we exhibit the benefits obtained from parameterization, namely: a runtime speed-up over non-templated libraries; a seamless integration into the C ++ Standard Template Library; an enhanced reliability and portability over different platforms; a better versatility, by allowing the use of different floating-point formats for the interval bounds, depending on the required precision. The impact of some implementation tricks is also investigated.

The rest of the paper is organized as follows: basics of interval arithmetic are briefly presented in Section 2; C ++ techniques to be used in designing JAIL are described in Section 3; Section 4 then introduces JAIL and presents some of the techniques used to improve both its speed and correctness; some experimental results are given and analysed in Section 5; last, directions for future researches are discussed in Section 6.

Interval arithmetic

An interval is a connected set of reals. While it may be closed or open, we will only consider closed intervals in the following, that is sets of the form:

a See http://interval.usl.edu/F90/f96-pro.asc b JAIL stands for "Just Another Interval Library".

goualard-ascm2000: submitted to World Scientific on November 9, 2009 2

I = [a .. b] = {c ∈ R | a c
b}, where a and b must be floating-point numbers for intervals to be stored and manipulated on computers. Floatingpoint arithmetic features that are relevant to interval arithmetic are presented in Section 2.1 while interval arithmetic is surveyed in Section 2.2. Due to lack of space, both topics are but sketched in the following sections. The reader is referred to Goldberg's survey 6 and Moore's book 1 for thorough presentations of floating-point arithmetic and interval arithmetic.

Floating-point arithmetic

The finite nature of computers precludes them from representing the whole set R of real numbers. Modern computers handle floating-point numbers (or simply, floats) of various precisions, whose formats are precisely defined by the IEEE 754 standard 2 . A set F ⊂ R of floats in one of these given precisions must be closed under all supported operations in order not to break the computation stream. Practical consequences are:

• the compactification of F with two infinities (-∞ and +∞) with the rules: ∀a ∈ F, a > 0 : a/0 = +∞ and -a/0 = -∞. By symmetry, the IEEE 754 standard defines two zeroes (-0 and +0, with -0 = +0), such that: ∀a ∈ F, a > 0 : a/+∞ = +0 and a/-∞ = -0. In the sequel, let F ∞ be the set F ∪ {-∞, +∞}; • the introduction of special "numbers", NaNs c , that are the result of any otherwise undefined operation, such as extracting the square root of a negative number, or divisions like 0/0 and +∞/+∞; • the specification of rounding operations: given x ∈ R \ F a real number, the IEEE 754 defines four policies for rounding x to a float: rounding towards +∞, -∞, 0, and the nearest-even. The only ones of interest to us are rounding towards +∞ and towards -∞ that can be formally defined by:

Rounding towards -∞: ⌊ ⌋ : R -→ F ∞ , x -→ max{y ∈ F ∞ | y x} Rounding towards +∞: ⌈ ⌉ : R -→ F ∞ , x -→ min{y ∈ F ∞ | x y}
The IEEE 754 standard also stipulates that the real result of any of the five operations {+, -, ×, ÷, √ } must be correctly rounded, that is rounded by excess or by default to the nearest floating-point number. Due to the Table maker's dilemma 6 , no requirement of the sort is imposed on transcendental functions, whose precision is thus implementationdependent.

c NaN stands for "Not A Number ".

goualard-ascm2000: submitted to World Scientific on November 9, 2009 3

Real and floating-point intervals

In this section, we will consider both the real interval set I R (i.e. intervals whose bounds are real numbers) and the floating-point interval set I F (i.e. intervals whose bounds are floats).

Real intervals. Real interval arithmetic is defined as follows: given by

I 1 = [a .. b] and I 2 = [c .. d] two intervals in I R , • ∈ {+, -, ×,
I 3 = {x • y | x ∈ I 1 , y ∈ I 2 } and I 4 = { ϕ (x) | x ∈ I 1 }.
From monotony considerations, one can easily deduce the following formulae from the above definitions:

[a .. b] + [c .. d] = [a + c .. b + d], [a .. b] -[c .. d] = [a -d .. b -c] [a .. b] × [c .. d] = [min(ac, ad, bc, bd) .. max(ac, ad, bc, bd)] [a .. b] ÷ [c .. d] = [min(a/c, a/d, b/c, b/d) .. max(a/c, a/d, b/c, b/d)], 0 / ∈ [c .. d] exp([a .. b]) = [exp(a) .. exp(b)]
Note that division may also be defined such that dividing by an interval spanning 0 is permitted. See Kahan's extended interval arithmetic 7 .

A weakness of interval arithmetic is that it is only sub-distributive:

∀I 1 , I 2 , I 3 ∈ I R : I 1 × (I 2 ± I 3 ) ⊆ I 1 × I 2 ± I 1 × I 3 .
On the other hand, its strength comes from the following theorem. Theorem 1 (IA fundamental theorem 1 ) Given F (X 1 , . . . , X n ) a rational expression on intervals X 1 ,. . . , X n , the following does hold for all

X ′ 1 , . . . , X ′ n ∈ I R : X ′ 1 ⊆ X 1 , . . . , X ′ n ⊆ X n ⇒ F (X ′ 1 , . . . , X ′ n ) ⊆ F (X 1 , . . . , X n )
Consequently, it is possible to compute the range of a rational expression over a given domain by merely evaluating its extension over intervals. Example 1 (Computing the range of a function) Given the function f (x) = (x + 1/2) 2 -1/4 and the interval I = [-1 .. 1], computing the range of f over I is as easy as evaluating the interval expression

I + [1/2 .. 1/2]) 2 - [1/4 .. 1/4], which leads to f (I) = [-1/4 .. 2].
Due to sub-distributivity, the form considered for evaluating f over intervals is important for tightness of the result. For example, evaluating f as x(x+1) would lead to

f (I) = [-2..2].
Floating-point intervals. In order to effectively compute with intervals, one has to make the shift from real intervals to floating-point intervals. This is achieved by outward rounding of real bounds to float bounds in order to ensure completeness. For example, the preceding definitions become:

[a .. b] + [c .. d] = [⌊a + c⌋ .. ⌈b + d⌉] [a .. b] -[c .. d] = [⌊a -d⌋ .. ⌈b -c⌉] . . . exp([a .. b]) = [⌊exp(a)⌋ .. ⌈exp(b)⌉]
goualard-ascm2000: submitted to World Scientific on November 9, 2009 4

Due to IEEE 754 correct rounding requirement for +, -, ×, ÷, √ , the computed intervals for these operations are the smallest representable. This is not so for the other operators such as exp, whose precision is not specified. As a consequence, ⌊exp(a)⌋ may be a value greater than the true result, depending on the accuracy of the exp function on a given platform. In order to overcome this problem, implementors of interval libraries usually rely on epsilon-inflation: a small quantity is added or subtracted from the value of exp(a), depending on the intended rounding direction. However, the preceding formulae cannot be directly used for implementing floating-point interval arithmetic. For example, consider intervals

I 1 = [-∞ .. 3] and I 2 = [-1 .. 0].
Applying blindly the formula for the product of I 1 and I 2 leads to:

I1 × I2 = [ min(⌊-∞ × -1⌋, ⌊-∞ × 0⌋, ⌊3 × -1⌋, ⌊3 × 0)⌋.. max(⌈-∞ × -1⌉, ⌈-∞ × 0⌉, ⌈3 × -1⌉, ⌈3 × 0)⌉)]
IEEE 754 specifies that -∞ × 0 must be a NaN. Since NaNs are unordered, the result of I 1 ×I 2 depends on the implementation of min and max functions.

It is then crucial to avoid the arising of NaNs by testing bound values prior to float multiplication.

C ++ techniques

The C ++ techniques that are used in the design of JAIL are now presented in this section. Generic programming through templated classes is first introduced; the "traits 8 " method for defining interfaces much like signatures is then described.

Templated classes

A C ++ templated class is a pattern for an actual class that is parameterized by a type. It is not a valid class until instantiated. For example, one can instantiate a templated class list to define lists of integers list<int> or lists of characters list<char>. The whole C ++ standard library is composed of templated classes (containers such as lists, hash tables, . . . ) and templated functions (sorting procedures,. . . ). Parameterization through templated class allows factorizing code for many types. It may however lead to weird situations. For example, consider Prog. 1 for a C ++ class MyNumber whose intended use is to provide a means to count the number of calls to arithmetic operators. The class is parameterized by the type T of the actual values to be manipulated. Execution of Prog. 1 will print values 0.70867 (i.e. cos(5.5)) and 0 on the screen, where the 0 comes from the fact that the call to cos for b induces goualard-ascm2000: submitted to World Scientific on November 9, 2009 5 the promotion of the integer value 6 to a double for calling the "double cos(double)" function of the standard library at line 18, and then a cast of the result back to an int for returning a MyNumber<int> object.

Prog. 1. The MyNumber class without "traits" One would indeed like to have the ability to return a MyNumber<double> object from a MyNumber<int> for some operators, and specify which function cos() to use depending on type T. The "traits" technique devised by Myers 8 is an elegant solution to these requests.

The "traits" technique

A trait is a templated class collecting types and method names acting as an interface. Prog. 2 presents the MyNumber class implemented with traits. The traits class defines a type Real and a cos() function whose input and output have the same type by default. Lines 6 through 11 present a specialization of the trait for short ints. The MyNumber class then uses the traits class to know what should be the cos() function to use and what is the type of its result, depending on its parameter T. As a consequence, Prog. 2 prints the value 0.96017 (i.e. cos( 6)), since the cos() function for MyNumber<short> knows that it must invoke function "float cos(short)" and return a MyNumber<float> as a result.

The traits technique then permits optimizing the code by specializing the class for some types (for example, we use here a "float cos(short)" that should hopefully be faster than the standard "double cos(double)" function. Reliability is also enhanced since the user must explicitly specify which functions to call in the MyNumber methods, thereby avoiding ambiguities.

Design and implementation of JAIL

The heart of the JAIL library is the templated class Interval, that contains most of the interval operators and relations defined in the Basic Interval goualard-ascm2000: submitted to World Scientific on November 9, 2009 6 Arithmetic Specification 9 .

Prog. 2. The MyNumber class with "traits" The Interval class may be instantiated with any floating-point format by specializing the JailRealTraits trait (see Prog. 3) that specifies all constants and functions used in implementing interval functions. In particular, the trait requires specifying downward and upward rounded versions for each function whose accuracy is not specified by the IEEE 754 standard (e.g. cosDn() and cosUp()). Besides enhancing the versatility of the library by permitting generic programming with intervals, the use of traits also increases its reliability when porting it on different platforms by making explicit the need for correctly rounded functions. Depending on the possibilities at hand on a given platform, one can define a function such as cosDn() by using epsilon-inflation or by directly calling a cos() function in a mathematical library in which it is known to be correctly rounded. What is more, the amount of epsilon-inflation may thus be easily parameterized according to the precision of the available functions without modifying the Interval class itself.

Prog. 3. JAIL's trait specialized for the double type 1 struct JailRealTraits<double> { 2 typedef double JailReal; 3 static inline double NextFloat(double x); 4 static const double MAXREAL; 5 static inline double cosDn(double x); 6 static inline double cosUp(double x); 7 . . .

};

Modern FPUs are usually pipe-lined: at one time, the FPU contains several instructions at different decoding stages. Modifying the rounding direction for computing interval bounds requires flushing all these instructions, then modifying a flag before admitting new instructions in the pipe-line. Such an interruption of the stream greatly lessens performances. In order to overgoualard-ascm2000: submitted to World Scientific on November 9, 2009 7 come this problem, JAIL uses only rounding towards +∞ by relying on the following IEEE 754 guaranteed property: ⌊x • y⌋ = -⌈(-x) • y⌉, for • ∈ {+, -, ×, ÷} For the square root, we use the following property (the proof is easily done by using the definition of ⌈ ⌉ twice):

∀x ∈ F, x < 0 : - x √ -x √ -x
In order to avoid using the negation twice, we have chosen to directly represent the left bound of an interval by its opposite (e.g. the interval [-6..4] is internally represented by the pair 6, 4 . The code for computing the sum of two intervals is then as follows:

Addition(in I 1 , I 2 ; out I 3 = I 1 + I 2 ) begin RoundUp()

I 3 ← -(I 1 + I 2 ) I 3 ← I 1 + I 2 end
One thus halves the number of rounding direction switching. But one can do better still, provided that each function that modifies the rounding direction (basically, output routines) reset it to "towards +∞". It is then possible to switch the rounding towards +∞ only once at the beginning of the computation. This is what we call the trust on mode since the library trusts functions for preserving the rounding direction. By contrast, the trust off mode requires switching the rounding towards +∞ for each interval operation.

NaNs propagate along all the computation, as the empty interval do. We have then decided to use [N aN .. N aN ] for representing the empty interval (actually, suffices for one of the bounds to be a NaN). Since NaNs are not comparable, it is then necessary to negate tests such as test for emptiness: The test (-I 1 I 1 ) is false whenever one of the bounds is a NaN or the right bound is smaller than the left bound.

Since NaNs are only characterized by the the value of the exponent (suffices for the mantissa to be different from 0), one can use the bits of their mantissa for coding information recording when and why an empty interval appeared.

goualard-ascm2000: submitted to World Scientific on November 9, 2009 8

In order to validate our implementation choices, we have developed five versions of our library: a templated version using the trust on mode and another using the trust off mode, and three non-templated versions (one in trust on mode, the second in trust off mode, and the last using downward rounding for computing left bounds and upward rounding for right bounds). We have then used a set of intervals exercising the whole code of the operators tested. Table 1 presents the computation times for this benchmark. One may easily see that the templated version using the trust on mode is the fastest of all. However, the compiling time suffers greatly from the use of templates (this is a well known drawback that should be made less accurate in the future with new generations of C ++ compilers). We have also compared JAIL with other available (non-templated) interval libraries, using the same benchmark as above. The results are summarized in Table 2. They show that the higher versatility of JAIL (genericity of the Interval class) does not prevent it from achieving good performances since it is the fastest of all the libraries tested on the elementary operations. There is still room for improvement as for the trigonometric functions, though. Note also that Profil is the fastest for computing powers, but it returns wider intervals (use of an adaptation of the Russian peasant method to exponentiation).

Conclusion

We have shown that a generic interval library for C ++ can be obtained at no extra cost as for the computation time, thereby permitting to seamlessly goualard-ascm2000: submitted to World Scientific on November 9, 2009 9 integrate it into the standard C ++ library. What is more, the use of templates and traits enhances the reliability of the resulting library.

Future works on JAIL include implementing the whole Basic Interval Arithmetic Specification 9 , and integrating directed interval arithmetic 10 , a variant of interval arithmetic that permits computing inner-approximations of the range of functions.

  ÷}, and ϕ ∈ { √ , exp, ln, cos, . . . }, real intervals I 3 = I 1 • I 2 and I 4 = ϕ(I 1 ) are defined

  isEmpty(in: I 1 ; out: I 1 == ∅) begin return (¬(-I 1 I 1 )) end

Table 1 .

 1 Impact of the implementation choices for JAIL

		Parameterized version	Non-parameterized version
		trust on trust off t. on t. off Dn/Up
	x + y	19204	24175	34445	41281	49280
	x -y x × y x ÷ y √ x	17526 17220 33105 12829	24812 22559 33563 14451	34543 29709 39561 21412	41311 35383 40634 22942	42840 38121 41792 26693
	x 2	7309	9378	13827	15778	18071
	x -10 . . . x 10	13141	13913	15997	16572	11632
	e x	23915	23182	26262	24810	24545
	log x	18404	17934	21069	20769	20860
	cos x	19218	19577	19951	20421	20766
	sin x	21289	21628	22220	22581	32663
	tan x	14807	14970	15660	15608	16013
	compilation	11200	13300	4200	4100	4000
		Times in milliseconds on a SUN UltraSparc 1/167 MHz with egcs 2.91.66

Table 2 .

 2 Comparing interval arithmetic libraries on an UltraSparc 1/167 MHz Results in milliseconds with egcs 2.91.66 † Parameterized version with trust on

		Profil	RVInterval	fi lib	JAIL †
	x + y	26362	53400	48479	19204
	x -y x × y x ÷ y √ x	26317 25951 35521 321568	53102 40274 38245 12382	47637 35758 39538 23853	17526 17220 33105 12829
	x 2	16426	26819	14133	7309
	x -10 . . . x 10	6698	25747	-	13141
	e x	201797	23599	11605	23915
	log x	119980	20195	10940	18404
	cos x	109844	7963	6295	19218
	sin x	91883	16982	5953	21289
	tan x	41657	-	4117	14807