Kevin Hurlbutt

Feliciano Giustino

Mauro Pasta
email: mauro.pasta@materials.ox.ac.uk

George Volonakis
email: yorgos.volonakis@univ-rennes1.fr

Electronic Structure and Electron-Transport Properties of Three Metal Hexacyanoferrates

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

Introduction

The metal hexacyanometallates, known more commonly as the Prussian blue analogs (PBAs), are promising active materials as electrodes for next-generation, sodium-ion batteries. [START_REF] Wang | Rhombohedral Prussian white as cathode for rechargeable sodium-ion batteries[END_REF][START_REF] Song | Removal of interstitial H 2 O in hexacyanometallates for a superior cathode of a sodium-ion battery[END_REF][START_REF] Wu | Highly Crystallized Na2CoFe(CN)6 with Suppressed Lattice Defects as Superior Cathode Material for Sodium-Ion Batteries[END_REF] PBAs are also important in potassium-ion [START_REF] Jiang | Building aqueous K-ion batteries for energy storage[END_REF] and all-solid-state 5 batteries, as well as in other electrochemical technologies including electrocatalysis, [START_REF] Yu | Unconventional CN vacancies suppress iron-leaching in Prussian blue analogue pre-catalyst for boosted oxygen evolution catalysis[END_REF] thermionics, [START_REF] Gao | Engineering the Electrochemical Temperature Coefficient for Efficient Low-Grade Heat Harvesting[END_REF] and smart windows. [START_REF] Shibata | Quick Response of All Solid Electrochromic Device[END_REF] They are a highly substitutable, and therefore tunable, material system. [START_REF] Hurlbutt | Prussian Blue Analogs as Battery Materials[END_REF] Their general formula is A x M J [M K (CN) 6 ], in which A is an alkali or alkali-earth metal, M and M are possibly distinct transition metals in oxidation states J and K, respectively. Appropriate compositions can be selected that undergo two electrochemical reactions within the stability window of conventional organic electrolytes, for specific capacities around 170 mA h g -1 at potentials of approximately +1.0 V versus the standard hydrogen electrode. [START_REF] Qian | Prussian Blue Cathode Materials for Sodium-Ion Batteries and Other Ion Batteries[END_REF] Their impressive rate capabilities and high ionic conductivities [START_REF] Wu | Diffusion-free Grotthuss topochemistry for high-rate and long-life proton batteries[END_REF][START_REF] Pasta | A high-rate and long cycle life aqueous electrolyte battery for grid-scale energy storage[END_REF] derive from the wide channels in their crystal structure, shown in Figure 1, for fast ion movement through the bulk.

This work addresses two conspicuously understudied areas of PBA science: electronic conductivity and first-principles modelling. First, past studies have carefully correlated chemical composition with materials properties including, for example, lattice parameter, [START_REF] Pasta | Manganese-cobalt hexacyanoferrate cathodes for sodium-ion batteries[END_REF] reduction potential, [START_REF] Scholz | The Formal Potentials of Solid Metal Hexacyanometalates[END_REF] and coefficient of thermal expansion. [START_REF] Wang | Large and tunable negative thermal expansion induced by a synergistic effect in M II 2 [M IV (CN) 8 ] Prussian blue analogues[END_REF] But there has never been a systematic study of the effect of composition on electronic conductivity. This is despite the fact that electronic conductivity is important in virtually every PBA-based technology. Early work on the composition NaFe[Fe(CN) 6 ] (also known as Prussian blue, the prototype compound) reported a value on the order of σ e = 10 -5 S m -1 , but it lacked structural characterization. [START_REF] Xidis | On the Electronic Conduction in Dry Thin Films of Prussian Blue, Prussian Yellow, and Everitt's Salt[END_REF] Studies on the effect of hydration state [START_REF] Tennakone | Experimental and theoretical study of electronic conduction in H 2 O-doped Prussian blue[END_REF] and nickel substitution at the nitrogen-coordinated site [START_REF] Shrivastava | Electron Conduction in Nanoparticle Agglomerates Limits Apparent N a + Diffusion in Prussian Blue Analogue Porous Electrodes[END_REF] gave similar values. The full range of values for the electronic conductivities of the PBAs remains unmeasured. Second, investigating their electronic properties using densityfunctional theory (DFT) is complicated because the PBAs are strongly correlated systems [START_REF] Wojde L | First principles calculations on the influence of water-filled cavities on the electronic structure of Prussian Blue[END_REF] and DFT notoriously over-delocalizes d orbital electrons in such systems. [START_REF] Himmetoglu | Hubbard-corrected DFT energy functionals: The LDA+U description of correlated systems[END_REF] Early PBA models addressed the challenge with an ad hoc strategy of using different pseudopotentials for the two iron ions in the formula unit, [START_REF] Wojde L | From cluster calculations to molecular materials: A mixed pseudopotential approach to modeling mixed-valence systems[END_REF] and later work suggested that the composition NaFe[Fe(CN) 6 ] is a half-metal. [START_REF] Wojde L | Prediction of half-metallic conductivity in Prussian Blue derivatives[END_REF][START_REF] Hegner | A Database of the Structural and Electronic Properties of Prussian Blue, Prussian White, and Berlin Green Compounds through Density Functional Theory[END_REF] Most recently, the voltage profiles for numerous compositions were explored by Ong and coworkers, [START_REF] Guo | Water Contributes to Higher Energy Density and Cycling Stability of Prussian Blue Analogue Cathodes for Aqueous Sodium-Ion Batteries[END_REF] and Henkelman and coworkers used the nudged-elastic band method to understand ion diffusion in manganese hexacyanoferrate. [START_REF] Xiao | Theoretical study of the structural evolution of a N a 2 F eM n(CN ) 6 cathode upon Na intercalation[END_REF] The majority of current research uses DFT+U. [START_REF] Ling | First-principles study of alkali and alkaline earth ion intercalation in iron hexacyanoferrate: The important role of ionic radius[END_REF] Despite the success and usefulness of DFT+U, it requires the careful selection of the U parameter to match a calculated property to a value taken from experiment or from a more advanced calculation. This complicates its application to new, hypothetical PBAs, so a rigorous, purely ab initio strategy for modelling PBAs is desirable.

Here the electronic structure and electron-transport properties are explored systematically in three of the most promising metal hexacyanoferrates for sodium-ion cathodes: manganese hexacyanoferrate, (Na x Mn[Fe(CN) 6 ]), iron hexacyanoferrate (Na x Fe[Fe(CN) 6 ]), and cobalt hexacyanoferrate (Na x Co[Fe(CN) 6 ]). Each material is studied in each of three oxidation states (x ∈ {0, 1, 2}) for a total of nine (9) compounds. The ground-state spin configuration for each compound is determined by comparing the total energy of the lowspin and high-spin phases. The projected density of states are calculated to examine the nature of the valence-and conduction-bands as well as to determine the compounds' band gaps. Finally, the charge-carrier effective masses are calculated. All the electronic properties are calculated within DFT employing either the generalized-gradient-approximation Perdew-Burke-Ernzerhof (PBE) functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] or the non-local, hybrid Heyd-Scuseria-Ernzerhof (HSE) functional. [START_REF] Heyd | Hybrid functionals based on a screened Coulomb potential[END_REF]

Results and discussion

Ground-state spin configurations

For PBAs, the carbon-coordinated metal ion assumes a low-spin (ls) electronic configuration while the nitrogen-coordinated metal is usually in a high-spin (hs) electronic configuration.

In order to treat the spin configuration of PBAs within spin-polarized DFT calculations we assume a ferromagnetic ordering between the two metal centers and fix the overall spin, per formula unit, of the material, and perform a total energy calculation. Figure 2 shows the difference in total energy between the (ls, hs) configuration and the (ls, ls) configuration. Seven compounds are found to be more stable in the (ls, hs) configuration, and only compounds Co[Fe(CN) 6 ] and NaCo[Fe(CN) 6 ] are more stable in the (ls, ls) configuration. The total energies calculated using the HSE functional correctly reproduce the experimentally observed spin configuration for all nine compounds. [START_REF] Wang | Rhombohedral Prussian white as cathode for rechargeable sodium-ion batteries[END_REF][START_REF] Song | Removal of interstitial H 2 O in hexacyanometallates for a superior cathode of a sodium-ion battery[END_REF][START_REF] Pasta | Manganese-cobalt hexacyanoferrate cathodes for sodium-ion batteries[END_REF] Contrast these results with those calculated by the PBE functional, which incorrectly energetically favors the (ls, ls) configuration for eight of the nine compounds. All calculations results reported hereafter are performed within the DFT-HSE. The compound NaMn[Fe(CN) 6 ] can attain two (ls, hs) electronic configurations: (Fe(t 5 2g ), Mn(t 3 2g e 2 g )) and (Fe(t 6 2g ), Mn(t 3 2g e 1 g )), with the latter lower in energy by less than 0.04 meV atom -1 .

These correspond to different oxidation states: (Fe(III), Mn(II)) versus (Fe(II), Mn(III)), respectively. Experiments have shown that below 225 K, the (Fe(t 6 2g ), Mn(t 3 2g e 1 g )) phase is favored, in agreement with our DFT calculations; at higher temperatures, (Fe(t 5 2g ), Mn(t 3 2g e 2 g )) is observed. [START_REF] Tokoro | A large thermal hysteresis loop produced by a charge-transfer phase transition in a rubidium manganese hexacyanoferrate[END_REF] Furthermore, operando X-ray diffraction (XRD) [START_REF] Bie | A novel K-ion battery: hexacyanoferrate(II)/graphite cell[END_REF] and ex situ X-ray absorption near-edge structure (XANES) 4 studies of room-temperature batteries clearly show that the manganese is first reduced from manganese(III) to manganese(II) and then the iron is reduced form iron(III) to iron(II). )), which is also consistent with experimental findings. [START_REF] Wang | Rhombohedral Prussian white as cathode for rechargeable sodium-ion batteries[END_REF] The compounds' electronic configurations as calculated within HSE are shown in Figure 3. are more stable in the (ls, ls) configuration; both contain cobalt(III, ls). On further reduction to Na 2 Co[Fe(CN) 6 ], the cobalt becomes cobalt(II) and switches to high spin. This is in agreement with electrochemical observations of a spin transition over the second reduction plateau during battery discharge. [START_REF] Kurihara | Electronic structure of hole-doped transition metal cyanides[END_REF] NaCo[Fe(CN) 6 ] has the smallest energy difference between the two spin configurations at 4 meV atom -1 . This small energetic difference is consistent with studies that observe facile photo-induced spin transition in this material. [START_REF] Cammarata | Charge transfer driven by ultrafast spin transition in a CoFe Prussian blue analogue[END_REF] These experiments show that the spin transition is accompanied by a charge transfer from the iron to the cobalt. The calculated atom-projected magnetic moment vanishes on both the iron atom and the cobalt atom in the ground state, which is consistent with a non-spin-polarized electronic configuration of t 6 2g for iron and t 6 2g for cobalt. In the higher energy (ls,hs) state, the atom-projected magnetic moments are µ Fe = 1.051 µ B and µ Co = 2.753 µ B (where µ B is the Bohr magneton). This is consistent with an electronic configuration of t 5 2g for iron and t 5 2g e 2 g for cobalt, shown in Figure 3, and supports the experimental model of charge-transfer-accompanied spin transition. [START_REF] Cammarata | Charge transfer driven by ultrafast spin transition in a CoFe Prussian blue analogue[END_REF] All other energy differences are greater (in magnitude) than the thermal excitation of 26 meV atom -1 at 300 K.

In terms of crystal structure, the fully oxidized compound Mn[Fe(CN) 6 ] has a tetragonal geometry. This is the result of a Jahn-Teller distortion arising from the electronic configuration on the manganese, Mn(t 3 2g e 1 g ). For the cobalt materials, Co[Fe(CN) 6 ] is cubic and the reduced compounds are rhombohedrally distorted. For all compounds, the main difference between the (ls, ls) and the (ls, hs) configurations is a higher unit-cell volume for the (ls, hs) configuration and longer M N -N bond length (where M N is Mn, Fe, or Co). This is due to the increased density of spin-aligned electrons. Finally, comparing structural optimization between DFT-PBE with the hybrid HSE functional, we found that both result in basically the same geometry for all studied compounds.

These calculated geometries are consistent with two important phase changes that have been observed experimentally. First, fully desodiated Mn[Fe(CN) 6 ] has been shown to be tetragonal. which, as discussed, is tetragonal). The calculated geometries are already rhombohedral in the x = 1 phase. We attribute the early onset of the rhombohedral distortion to the fact that the simulated compounds are completely free of vacancies and crystalline water, both of which are known to stabilize the cubic structure over the rhombohedral structure. [START_REF] Moritomo | Structural phase diagram of Mn-Fe cyanide against cation concentration[END_REF] We show the simulated XRD patterns and crystal structures for the iron compounds in Figure

SI.1.

Atomic-projections of valence-and conduction-band states

The projected density of states (PDOS) for the ground-state spin configuration of the nine materials as calculated with the HSE functional are shown in Figure 4. For Mn III [Fe III (CN) 6 ],

the valence-band states arise from spin-down iron t 2g orbitals, which are nearly degenerate with the spin-up manganese e g orbitals and cyanide orbitals. The conduction-band states are stem from spin-down, iron t 2g orbitals. The valence-band states for NaMn II [Fe III (CN) 6 ] originate from the spin-up, manganese e g orbitals, in accordance with Figure 3 and the conduction-band states still arise from spin-down, iron t 2g orbitals. These become occupied on further reduction to Na 2 Mn II [Fe II (CN) 6 ], in which the valence-band states stem from degenerate spin-up iron t 2g orbitals, spin-up manganese e g orbitals, spin-down iron t 2g orbitals, and cyanide orbitals. The conduction-band states originate from spin-down manganese t 2g orbitals and cyanide orbitals.

In Fe III [Fe III (CN) 6 ], the valence-band states arise from spin-down, carbon-coordinatediron t 2g orbitals. The conduction band bottom stems from t 2g orbitals of both iron atoms, which is consistent with the t 5 2g and t 3 2g spin configuration shown in Figure 3. Upon reduction,

NaFe III [Fe II (CN) 6 ] also has its valence-band states originating from the carbon-coordinatediron, but the conduction-band states now solely originate from the nitrogen-coordinatediron t 2g orbitals. This can be explained due to the change of the carbon-coordinated-iron In eight of the nine compounds either the valence-band or conduction-band states (or both) arise from t 2g orbitals of the carbon-coordinated iron. For all three fully oxidized materials (x = 0), both the valence and conduction bands stem from the carbon-coordinated iron. This suggests that the carbon-coordinated iron participates significantly in the conduction of charge-carrying electrons and holes through the metal hexacyanoferrate lattice, especially for battery electrodes in the fully charged state. Sluggish kinetics caused by low electronic or ionic conductivity have been suggested as the cause of low coulombic efficiency, especially in manganese hexacyanoferrate. [START_REF] Fiore | Paving the Way Toward Highly Efficient High-Energy Potassium-Ion Batteries with Ionic-Liquid Electrolytes[END_REF] Research efforts to increase conductivity have focused on doping the nitrogen-coordinated element. [START_REF] Jiang | Building aqueous K-ion batteries for energy storage[END_REF] However, if the low coulombic efficiency is caused by lower conductivity in the fully oxidized lattice, then these results suggest that experiment work must refocus on doping the carbon-coordinated iron to improve cycle lives of PBA batteries.

All nine materials have band gaps with values, as calculated with the HSE functional,

shown in Table

Charge-carrier effective masses and charge distributions

The effective mass of an electron is defined as the second derivative of the electron energy with respect to the wavevector k about the minimum according to

m * ij = h2 (∂ 2 /∂k i ∂k j ) -1 ,
in which i and j are the indices for the three crystallographic directions, is the energy eigenvalue, h is the reduced Planck's constant, and m * ij is the (i, j) component of the effective mass tensor. [START_REF] Filip | Band Structures and Carrier Effective Masses of CH 3 N H 3 P bI 3 and Hypothetical Perovskites of the Type AP bI 3 : A = N H 4 , P H 4 , AsH 4 , and SbH 4[END_REF] The partial derivatives are calculated numerically using a first-order, finitedifference approximation, and diagonalization of the effective-mass tensors gives the effective masses along the three principal axes. (The extremum k points of the valence-band tops and conduction-band bottoms are in Table SI The effective masses are very heavy due to the fact that these compounds are ionic crystals, as shown in the band-decomposed charge densities for the valence and conduction bands of Figure 5. Eight of the nine compounds have significant valence-band charge density on the carbon-coordinated iron, and six of nine compounds have significant conductionband charge density on the carbon-coordinated iron. This is in agreement with the PDOS data, and underscores the need for investigation of the effect of doping at this site for tuning PBA electronic conductivity. Also of note is that the valence band in Prussian blue, NaFe[Fe(CN) 6 ], is localized on the carbon-coordinated iron while the conduction band is localized on the nitrogen-coordinated iron. This is consistent with experimental observation of charge transfer between the two iron atoms on excitation of a valence electron to the conduction band. [START_REF] Robin | The Color and Electronic Configurations of Prussian Blue[END_REF] In general, the charge density is highly localized on the transition-metal ions. Together, the heavy effective masses and the ionic character of the electronic structure imply that small-polaron hopping is likely an important mechanism for charge conduction in these compounds. First-principles studies have shown the importance of small-polar hopping for electronic conductivity in battery materials including lithium iron phosphate [START_REF] Fisher | Insights into defect association, transport mechanisms, and doping behavior[END_REF] and the (manganese, cobalt) oxide family. [START_REF] Wang | Polaron states and migration in F-doped Li 2 M nO 3[END_REF] Tennakone and coworkers used small-polar hopping to interpret experimental data on the complicated relationship between crystalline water, temperature, and electronic conductivity in Prussian blue (NaFe[Fe(CN) 6 ]). 17 These theoretical results rationalize Tennakone and coworkers' application of the small-polaron-hopping mechanism to understand their data; these results also call for future work to model and understand polarons in PBAs in greater detail.

Conclusions

Calculations from first principles using DFT to both the PBE-and HSE-level of theory identify important features of the electronic structure of three metal hexacyanoferrates in each of three oxidation states that are technologically relevant for next-generation battery cathodes. Comparison of the total energies of the (low spin, low spin) versus the (low spin, high spin) configurations as calculated by the HSE functional reproduces experimentally observed spin configurations in all nine compounds. This strategy provides a purely ab initio way of modelling PBA electronics. The PDOS and band decomposed charge density reveal the importance of the carbon-coordinated iron onto which conduction-and valence-bands states in many cases, confirming this as a promising site for substitutional doping to design more or less electronically conductive PBAs. The materials are all gapped with band gaps ranging from 1.90 eV to 4.94 eV. The charge-carrier effective masses are calculated, with median values of 2.60 m e for holes and 1.94 m e for electrons. These very heavy effective masses, along with the highly ionic character of the compounds, suggest that small-polaron hopping may be an important mechanism for electron transport. Future work will center around modeling polarons in PBAs in greater detail. These results are important contributions to the study of PBA electronic conductivity and first-principles modeling, and this work lays the theoretical groundwork to incorporate electronic conductivity into the repertoire of tunable PBA properties.

Methods

All calculations were carried out using DFT as implemented in the Vienna ab inito Simulation Package (VASP). [START_REF] Kresse | Theory of the crystal structures of selenium and tellurium: The effect of generalized-gradient corrections to the local-density approximation[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF] A Γ-centered 4x4x4 k-point grid and a plane-wave kinetic-energy cutoff of 520 eV were used. Electronic and ionic convergence criteria were 10 -5 eV and 0.05 eV Å-1 , respectively. Full geometric relaxation was performed on all materials with both the PBE and HSE functionals separately. For the HSE functional, we used a screening parameter of 0.2 Å-1 and a mixing of 25% Fock exchange with 75% PBE exchange. The effective masses were calculated using a first-order finite-difference approximation of the second partial derivative of the energies at 18 zero-weight k points about the band extremum with a discretization of the local reciprocal space of 0.01 • 2π/a. Band-decomposed charge densities were calculated using the band and k point that were used for the effective mass calculations.

Figure 1 :

 1 Figure 1: The PBA crystal structure. Two transition metals are octahedrally coordinated to cyanide ligands forming a cubic geometry (space group Fm3m) with wide channels for fast ionic conduction. Shown here is the intermediate oxidation state with one sodium ion (in the interstitial position) per formula unit. Application of an appropriate electrical potential results in oxidation (sodium removal) or reduction (sodium addition).

Figure 2 :

 2 Figure2: Energy differences between spin configurations. Negative differences correspond to energetic stability of the (ls, hs) state. Black, blue, and green bars correspond to Mn, Fe, and Co, respectively. Within each color, the three bars represent 0, 1, or 2 Na-ions per formula unit from left to right. The inset shows the results using the PBE functional.

Figure 3 :

 3 Figure 3: Electronic configurations. The three t 2g orbitals and two e g orbitals are formed from the five d orbitals of an octahedrally coordinated transition metal. The red orbitals correspond to carbon-coordinated iron, while black, blue, and green states refer to manganese, nitrogen-coordinated iron, and cobalt orbitals, respectively.

  On reduction to NaMn[Fe(CN) 6 ], the sodium ion occupies the center of the face separating two subcubes (i.e. the cubic interstice formed by eight cyanide ligands and four of each lattice metal ion), as shown in Figure SI-1. This relaxes the tetragonal distortion, but introduces a slight rhombohedral distortion. The lattice vectors are changed by about 2 • away from the value of 60 • for the face-centered cubic geometry. The fully reduced compound Na 2 Mn[Fe(CN) 6 ] has its second sodium ion in a different face of the cube and is also rhombohedrally distorted. For the iron materials, the fully oxidized compound Fe[Fe(CN) 6 ] is cubic. The reduced compounds NaFe[Fe(CN) 6 ] and Na 2 Fe[Fe(CN) 6 ] have their sodium ions in the cube faces and are rhombohedrally distorted.

Figure 4 :

 4 Figure 4: Projected density of states (PDOS) of the ground-state spin configuration to HSE. The top, middle, and bottom rows correspond to Mn, Fe, and Co, respectively. The left, middle, and right columns correspond to 0, 1, and 2 Na-ions per formula unit, respectively. The energy axis is referred to the Fermi level (gray, dotted line) and occupied states are shaded light gray.

Figure 5 :

 5 Figure 5: Band-decomposed charge densities. For each compound, the band-decomposed charge-density isosurface is plotted for the valence band (purple, left) and the conductionband (orange, right). The manganese, iron, and cobalt compounds are in the top, middle, and bottom rows, respectively. The number of sodium ions (gold) per formula unit are 0 (fully oxidized), 1, and 2 (fully reduced) from left to right.

  critical sodium concentration between 1 and 2 sodium ions per formula unit, 1 < x critical < 2.[START_REF] You | High-quality Prussian blue crystals as superior cathode materials for room-temperature sodium-ion batteries[END_REF][START_REF] Pramudita | Sodium uptake in cell construction and subsequent in operando electrode behaviour of Prussian blue analogues, F e[F e(CN ) 6 ] 1-x • yH 2 O and F eCo(CN ) 6[END_REF] Below x critical the compounds are cubic (except for Mn[Fe(CN) 6 ],

[START_REF] Song | Removal of interstitial H 2 O in hexacyanometallates for a superior cathode of a sodium-ion battery[END_REF]

Second, Na x Mn[Fe(CN) 6 ], Na x Fe[Fe(CN) 6 ], and Na x Co[Fe(CN) 6 ] all become rhombohedral above a

  spin configuration to the fully filled t 6 2g . Further reduction to Na 2 Fe II [Fe II (CN) 6 ] breaks the symmetry of the spin-down nitrogen-coordinated-iron t 2g orbitals, one of which becomes occupied. Both the valence-band and the conduction-band states thus stem from spin-down orbitals. Similarly in Co III [Fe III (CN) 6 ], both the conduction-and valence-band states arise from spin-down, iron t 2g orbitals. For the reduced material, NaCo III [Fe II (CN) 6 ], both Co-and Fe-sites are in the same fully filled t 6 2g spin configuration; hence, it is not spin-polarized. Valence-band states originate from the iron t 2g orbitals and the conduction-band states arise from cobalt e g orbitals (and some cyanide states). The spin transition of the cobalt on reduction to Na 2 Co II [Fe II (CN) 6 ] changes the electronic structure substantially. The valenceband states stem from degenerate spin-up and spin-down iron t 2g orbitals. The valence band originate from spin-down cobalt t 2g orbitals.

  . The gaps range from 1.90 eV to 4.94 eV. Even the smallest bang gap is too large for appreciable populations of thermally generated carriers at room temperature. The calculated band gap for Prussian blue, NaFe III [Fe II (CN) 6 ], is 1.90 eV, versus an experimental value of 1.75 eV. 37 The material NaCo[Fe(CN) 6 ] has a direct band gap; the rest have indirect band gaps. PDOS calculated with the PBE functional are shown in Figure SI.2.

	Table 1: Band gaps calculated using the HSE functional
		M[Fe(CN) 6 ] NaM[Fe(CN) 6 ] Na 2 M[Fe(CN) 6 ]
	Mn	2.46	2.88	4.94
	Fe	1.96	1.90	3.33
	Co	3.00	3.33 *	3.80

* direct band gap

  .1 and 2, and the shapes of the energy isosurfaces near the extreme k points are shown in Figures SI.3 through 5). The isotropic average ) for the compounds are given in Table SI.3. The hole effective masses range from 2.06 m e in NaCo[Fe(CN) 6 ] to 3.62 m e in Na 2 Mn[Fe(CN) 6 ] with a median value of 2.60 m e for all nine compounds. The electron effective masses range from 0.73 m e in Fe[Fe(CN) 6 ] to 4.95 m e in Co[Fe(CN) 6 ] with a median value of 1.94 m e for all nine compounds.

	effective mass (m * = ( 1 3 Σ i m * -1 i	) -1

Acknowledgement

We gratefully acknowledge the use of the following computer resources: ARCHER UK National Supercomputing Service (http://www.archer.ac.uk); the University of Oxford Advanced Research Computing (ARC) facility (http://dx.doi.org/10.5281/zenodo.22558); and PRACE for awarding time on Beskow based in Sweden at KTH.

Supporting Information Available

Supplementary data are available online.


Effect of gyrus folding angle on peak electric field in cerebral cortex

Keywords: Dosimetry (computational), All Frequencies, Completed

Reported transcranial current stimulation (tCS) effects on cerebral activity leave numerous questions open regarding the involved mechanisms of action. A number of studies have identified clear physiological effects, however it remains to be understood how to take into account interindividual variability to tCS: For example, to what extent does individual brain geometry, such as cerebral tissues morphology, impact this prediction? Here, we aimed at evaluating how the geometry of cortical tissue surfaces impacts the charge distribution and therefore the electric fields induced by tCS.

Introduction

Non-invasive brain stimulation (NIBS) is gaining increasing interest in neuroscience. Indeed, countless studies have suggested therapeutic potential in the field of drug-refractory depression [START_REF] Boggio | A randomized, double-blind clinical trial on the efficacy of cortical direct current stimulation for the treatment of major depression[END_REF]], Parkinson's disease [START_REF] Lee | Does transcranial direct current stimulation improve functional locomotion in people with Parkinson's disease? A systematic review and meta-analysis[END_REF], and stroke [START_REF] Fregni | Transcranial direct current stimulation of the unaffected hemisphere in stroke patients[END_REF], among others. NIBS can be induced using at least two electrodes that cause a transcranial current flow, which can be either alternative (tACS) or direct (tDCS). Effects of tCS on cerebral activity depend directly on the electric field generated in brain tissues [START_REF] Bikson | Effects of uniform extracellular DC electric fields on excitability in rat hippocampal slices in vitro: Modulation of neuronal function by electric fields[END_REF][START_REF] Radman | Role of cortical cell type and morphology in subthreshold and suprathreshold uniform electric field stimulation in vitro[END_REF][START_REF] Rahman | Cellular effects of acute direct current stimulation: somatic and synaptic terminal effects[END_REF].

Since the brain shape is non-deterministic, personalized therapy is laborious and expensive, requiring individual magnetic resonance imaging (MRI) to tailor and predict the therapeutic effect. The distribution of E-field in heterogeneous brain tissues varies indeed significantly as a function of individual shape and size [START_REF] Truong | Computational modeling of transcranial direct current stimulation (tDCS) in obesity: Impact of head fat and dose guidelines[END_REF][START_REF] Laakso | Inter-subject variability in electric fields of motor cortical tDCS[END_REF]. Moreover, the electric field in any disordered heterogeneous media will exhibit local maxima and minima in vicinity of convex and concave features and, in particular, close to wedge-shaped gyri as a direct consequence of Gauss's law. The goal of this study was therefore to quantify the influence of cortical gyri shape on the distribution of the local electric field.

Methods

The MRI data used in this study was from the ICBM model: an averaged geometry of a large number of healthy human brains [START_REF] Fonov | Unbiased average age-appropriate atlases for pediatric studies[END_REF]. In order to build a numerical model based on this geometry, our first step was the segmentation of tissues using the ROAST pipeline [START_REF] Huang | ROAST: An open-source, fully-automated, realistic volumetric-approachbased simulator for TES[END_REF]. Then, an axial slice was chosen to accurately represent the cross-sectional anatomy of the brain. The resulting slice is presented in Figure 1, and consists of the raw MRI data (Figure 1a) and the segmented representation (Figure 1b). The segmented slice was then outlined using the Inkscape vector editing software. As shown in Figure 1c, Bezier curves were defined for the boundary of each tissue to obtain a complete vector image suitable for finite element modeling in COMSOL Multiphysics.

The dielectric parameters were attributed according to the Cole-Cole model [Gabriel et al., 1996]. At 1 kHz, the following tissue properties were defined as: skin [σ = 2.0 × 10 -4 (S/m); ε r = 1.1 × 10 3 ]; scalp [σ = 2.0 × 10 -2 (S/m); ε r = 2.7 × 10 3 ]; cerebrospinal fluid [CSF; σ = 2.0 (S/m); ε r = 1.1 × 10 2 ]; grey matter [GM; σ = 9.0 × 10 -2 (S/m); ε r = 1.6 × 10 5 ]; white matter [WM; σ = 6.0 × 10 -2 (S/m); ε r = 7.0 × 10 4 ]; where σ is the conductivity and ε r is the relative permittivity. The conductive gel for electrodes was modelled by two rectangles intersecting with the surface of the skin (Figure 1d). The insulator boundary condition (J • n = 0) was assigned to the skin surface. The outer boundaries of gel modelled the electrodes with V = 0 for the ground and I 0 = 2 mA for the terminal (Figure 1d). The next step was to isolate a gyrus close to electrodes where a wedge angle can be varied with a minimal E-field disturbance from surrounding tissues. For the variation of the angle, three control points (A, B, C) were defined (Figure 2b). (B, C) remained fixed while A was moved to adjust the angle (Figure 2b). The degree of this angle θ can be evaluated from the coordinates of (A, B, C) by calculating their Euclidian distance. For this, we used Al-Kashi's theorem where d(X, Y) denotes the Euclidian distance between the two points X and Y. Ten angles θ n ∈ [68.1, 143.7] were established, as shown in Figure 2b. Normalized electric field was evaluated on the wedge point A.

Results and Discussion

Figure 2a highlights a non-homogeneous distribution of the electric field along the interfaces between tissues. As hypothesized, these variations were more pronounced at the interfaces with locally strong convex or concave features. Figure 2c presents the local E-field distribution on a gyrus as a function of the wedge angle. Clearly, the E-field amplitude on the CSF-GM decays exponentially as a function of the wedge angle (Figure 3). This dependence can be accurately interpolated as E = 0.5933 e -0.017θ (coefficient of determination R 2 = 0.9986), where E is the E field and θ is the wedge angle. The effect of the wedge angle was significant: locally, the electric field doubled for an angle from 143° to 68°. Importantly, this variation in amplitude can affect the focality of neurostimulation in cortical regions.

Conclusion

This study did show the significant influence of gyri shape, especially its wedge angle, on the local distribution of E-field in the cerebrospinal fluid. More than an observation, it is actually an exponential dependence that was identified. Those results emphasize the importance of taking into account subject-specific brain morphology, which could explain at least in part the lack of consistency between numerous tCS studies. Having improved estimates of electric field distributions in target regions could also assist in calibrating stimulation intensities to facilitate the comparison of outcomes between subjects.

Figures

Figure 1 .

 1 Figure 1. Model formulation using a 2D cerebral axial slice (a) Source image from ICBM model (105 out of 189 images); (b) ROAST classification; (c) Phantom obtained with vectorization of tissues; (d) Final model including gel and electrodes.

Figure 2 .

 2 Figure 2. (a) Electric field distribution on the global surface with an input current of 2mA at 1 kHz; (b) Variation of the wedge angle θ from 68° to 143°; and (c) Associated electric field distribution.

Figure 3 .

 3 Figure 3. Effect of the wedge angle on the maximum electric field.

Acknowledgment

This study was supported in part by the LabEx CominLabs "PKSTIM" and in part by the French region of Brittany through the SAD "EM-NEURO."


Wassim El Housseini

François Lapicque

Alain Walcarius

Dr Mathieu Etienne
email: mathieu.etienne@univ-lorraine.fr

Wassim El Housseini

A hybrid electrochemical flow reactor to couple H 2 oxidation to NADH regeneration for biochemical reactions

Keywords: β-nicotinamide adenine dinucleotide, [Cp*Rh(bpy)Cl] +, gas diffusion electrode, flow reactor, electrochemical regeneration

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

NAD + /NADH cofactor and its phosphorylated derivative (NADP + /NADPH) are important biological cofactors that play a critical role as electron shuttles in biological systems. [START_REF] Sorci | Compr. Nat. Prod. II Chem. Biol[END_REF] They can also be applied as an electron carrier in enzymatic systems catalyzing stereo-and regioselective reactions having a great biotechnological potential, e.g. dehydrogenases or cytochrome P450. [START_REF] Chenault | [END_REF] However, NADH and its derivatives in enzymatic systems are expensive, a so the regeneration of the enzymatically-active reduced form of the cofactor is strongly required.

Four methods for regeneration of the NAD(P)H cofactor can be employed: (i) chemical and photochemical regenerations, (ii) biological regeneration, (iii) enzymatic regeneration, (iv) electrochemical regeneration or electroenzymatic regeneration. [START_REF] Zhang | Funct. Electrodes Enzym. Microb[END_REF] To quantify and compare the efficiency of each method, two criteria are usually defined: the turnover number (TN in h -1 ) and the total turnover number (TTN). TN, which is homogeneous to a frequency, expresses the velocity at which the cofactor is regenerated. TTN provides information on the number of moles of product formed per mole of cofactor during the entire synthesis reaction of the targeted product. [START_REF] Chenault | [END_REF] The average TTN of the enzymatic cofactor in the different regeneration methods has been reported to be in the order of 105 using chemical regeneration, [START_REF] Devaux-Basseguy | [END_REF] 21 with photochemical regeneration, [5] between 100 000 and 150 000 using enzymatic regeneration, [6][7][START_REF] Zhang | Funct. Electrodes Enzym. Microb[END_REF] and between 101 and 3760 by electrochemical regeneration. [START_REF] Jones | [END_REF][10][11] The difference between the TTN values obtained in enzymatic and electrochemical regenerations appears to be due to the nature of the a NAD + 88.2 €/g, NADH 206 €/g, NADP + 355 €/g, NADPH 1330 €/g (A chemical provider catalog consulted the 12/12/2020) catalytic reaction that is used, i.e. homogeneous versus heterogeneous, and of the mediator used in enzymatic and indirect electrochemical regenerations respectively.

A technique for the regeneration of NADHor NADPHcofactor must allow TTN larger than 1000 in order to be economically viable. [12,13] However, the technique that enables to reach the highest TN or TTN is not always the most economical due to the loss of the cofactor, or its thermal, chemical, or photochemical degradation. [13] In fact, none of the regeneration methods cited above appear to offer a fully satisfactory solution to the problem of NADH regeneration that fulfils economic and environmental criteria for its application. [18] It is often reported that the cost of the technology or the products used for the regeneration of this cofactor (particularly enzymes used in enzymatic regeneration) are in the same order of magnitude or even higher than the cost of the cofactor and present problems related to their possible inhibition effects. [START_REF] Kane | Design and Production of Electrochemical Microreactors: Application to the Electroenzymatic Regeneration of NADH and Potentialities in Synthesis[END_REF] Therefore, TTN is not the only criterion for the economic evaluation of the process. The optimization of the reactor design, the limitation of the formation of by-products such as the inactive form of NADH, and the prevention of the NADH degradation during the process, all play an important role in the regeneration technique of NADH. [START_REF] Cantet | Process for the Regeneration of the NADH Cofactor by the Simultaneous Use of an Electrode and a Hydrogenase[END_REF] The regeneration of NADH through electrochemical techniques is one of the most promising and controllable approaches that are used for the reduction of NAD + . [START_REF] Chenault | [END_REF][START_REF] Fukuzumi | [END_REF]23] It offers an easy separation of products of the reaction, if catalysts are immobilized, and it allows as well the control of the potential and the progress of the reaction. [24] The direct regeneration of NADH at bare electrodes only occurs at very negative potentials leading to the inactive dimerized form of NADH formed through one electron transfer instead of two electrons exchanged in the indirect regeneration of the NADH cofactor. [25] The usage of a mediator leads to the active form NADH with a potential lower than the potential needed in the direct regeneration. [Cp*Rh(bpy)Cl] + (chloro(2,2′-bipyridyl) (pentamethylcyclopentadienyl)-rhodium (III)) acting as a redox mediator has been shown to be the best non-enzymatic catalyst for the selective regeneration of NADH. [15,26,27] The mechanism of NADH regeneration consists first in the reduction of the [Cp*Rh(bpy)Cl] + (M ox ) complex to [Cp*Rh(bpy)] (M red1 ) through two electron transfers, [28] which can be globally written as:

(1)

The reduced complex is protonated to rhodium hydride complex ([Cp*Rh(bpy)H] + , M red2 ): [28] (2) NAD + can then reduce to 1,4-NADH with the hydride form of the complex (M red2 ) acting as a mediator: [29] (3) Development of continuous flow reactors has occupied a wide range of interest by chemical engineers for over one decade due to their high efficiency when being compared with batch reactors. [30] In continuous flow reactors, reactants are continuously fed into the reactor and emerge as a continuous stream of products. [31] These continuous flow reactors are more and more used by chemists for the synthesis of fine chemicals such as NADH. [START_REF] Löwe | Proceeding Third Int. Conf. Microreact. Technol. (IMRET 3)[END_REF] Electrochemical developments in continuous flow reactors have contributed to the reduction in energy consumption, to the improvement of material transport, and to increased conversion and selectivity [START_REF] Britton | [END_REF] through more refined cell design and technology. It is anticipated that the synthesis in flow will be coupled in the future with new analytical techniques for online monitoring of the process and speeding up the isolation and the purification. [34] As shown in equations ( 2) and (3), the reactions are associated with proton transfer. The oxidation of hydrogen on platinum appears attractive for the generation of protons required at the biocathode for electrochemical synthesis offering the possibility to be applied as a reducing agent in a hybrid reactor combining a fuel cell and a redox flow device. This reaction is highly selective, producing no toxic or hazardous species. [35] The electrochemical regeneration of fine chemicals, e.g. NADH, by indirect electrochemical reaction coupled with the oxidation of hydrogen is limited by the solubility of hydrogen in the electrolyte. [36] Gas diffusion electrodes (GDEs) have proven to boost current density by overcoming the poor solubility of hydrogen in aqueous electrolytes [37] and reducing gas-liquid mass transfer constraints. [37] Recently, GDEs have been applied to bioelectrochemical synthesis in continuous flow reactors [38,39] and evaluated for anodic NAD + regeneration coupled with energy production. [40] While being compared to chemical catalysts, enzymes are known as efficient biocatalysts offering high competitive processes and becoming well recognized in organic synthesis and biotechnology. [41,42] The electrochemical regeneration of the NADH cofactor through a rhodium complex mediator can be coupled to a bioconversion for the synthesis of fine chemicals and to assess whether the regenerated NADH cofactor is in its active form. [24] Besides, for the sake of long term stability and reusability, enzymatic catalysts have to be immobilized [43] as for example by adsorption on carbon materials. [44] In the present work, we combined the above approaches, namely the electrochemical regeneration of NADH mediated by a rhodium complex, the application of continuous flow reaction, and hydrogen oxidation using a GDE, in a single reactor with recirculation of the electrolytic solution. This approach relies upon the electrochemical reactor coupling efficiently the reduction of NAD + to the anode oxidation of hydrogen. The proposed process has been optimized by varying operating factors, e.g. the concentrations of rhodium complex and NAD + , the pH, the activation procedure for the carbon-based electrode, as well as the cell operating conditions. Finally, the conversion of pyruvate to lactate by L-lactic dehydrogenase (LDH) has been carried out in the process after insertion of an enzymatic module in series with the electrochemical reactor, to check the activity of the in-situ regenerated NADH cofactor. The purpose of this study is to evaluate the contribution of this redox flow technology on the performance of NADH regeneration with a well described rhodium electrocatalyst. [15]

Materials and methods

Chemical and reagents

β-nicotinamide adenine dinucleotide (NAD + , >98%) and β-nicotinamide adenine dinucleotide reduced dipotassium salt (NADH, >97%), K 2 HPO 4 (99%), KH 2 PO 4 (99%), HCl (37 %), Tris HCl (99 %), L-lactic dehydrogenase (LDH) from bovine heart (≥ 250 units/mg protein, 144,000 Da.), pyruvate (98 %) and lactate (98 %) were from Sigma-Aldrich. Phosphate buffer (0.1 M) was used to investigate the electrocatalytic properties of the rhodium complex towards NAD + reduction and was reported to be the most appropriate for such experiments [45] . Multiwall carbon nanotubes (MWCNT, NC7000™ series) were from Nanocyl (Belgium) and carbon papers (SpectraCarb 2050L-0550 Carbon Paper) were acquired from Fuel Cell Store (USA). MWCNT were dispersed in a solution of ethanol (96 %). All solutions were prepared with high purity water (18 MΩ cm) from a Purelab Option water purification system.

Rhodium complex synthesis

[Cp*Rh(bpy)Cl] + complex was prepared in a CHCl 3 solution after stoichiometric reaction between the 2,2-bipyridine and the pentamethylcyclopentadienyl rhodium(III) chloride dimer (RhCp*Cl 2 ) 2 . After addition of the reagents, the solution has been stirred for three hours under air and at room temperature then allowed to settle overnight at 8 °C. The solid complex was recovered by filtration by Buchner filtration. [24] 2.3. Electrode pretreatment A piece of graphite felt (GFD4.6EA, SGL, Germany) being 4 cm x 4 cm x 0.4 cm in dimensions was used as cathode in the reactor (flow cell in Scheme 1). The as-received material is hydrophobic which can limit the good distribution of solution in the reactor. The graphite felt (GF) piece was treated to provide a suitable wettability using washing in successive solutions: [START_REF] Sivasankar | Microbial Fuel Cell Technology for Bioelectricity[END_REF] the GF piece was first immersed in ethanol and submitted to ultrasounds for 15 minutes at 25 °C, then, the process was repeated with 80:20, 60:40, 40:60 and 20:80 ethanol-water solutions and ending with pure water.

Enzymatic carbon paper with MWCNT (LDH-CP-MWCNT) preparation

A piece of SpectraCarb carbon paper being 1.5 cm x 1.5 cm x 0.05 cm was used as the enzyme support for the conversion of pyruvate to lactate using L-lactate dehydrogenase. The MWCNTcontaining CP was prepared following the same process for a bucky paper electrode preparation reported in literature. [START_REF] Hussein | [END_REF] 10 mg MWCNT were dispersed in 50 mL ethanol by ultrasonication for 5 h. Afterwards, the suspension was decanted and vacuum filtered using the CP as filter. Then, the material was dried at 60 °C overnight. This process has been repeated three times to obtain the desired number of MWCNT layers. Finally, an LDH enzyme was adsorbed on the MWCNT surface by depositing 50 µL suspension of LDH (500 units, i.e. 2 mg) and keeping for 2 hours at a temperature of 3°C.

Electrochemical reactor

Scheme 1 shows the electrochemical reactor used in this study. A Pt/C gas diffusion electrode (Paxitech France) with a Pt content at 0.2 mg.cm -2 , was used as anode and installed on a graphite bipolar plate, with grooved 1 x 1 mm 2 channels forming three channel clusters for continuous hydrogen introduction and distribution in the anode chamber. The gas diffusion electrode allows uniform and constant production rates of protons and electrons for a given flow rate of hydrogen. 40 mL of reaction solution, containing 0.1 M phosphate buffer and various amounts of NAD + and [Cp*Rh(bpy)Cl] + , were introduced to the glass storage tank. Temperature was at ambient level and unless specified otherwise, the initial pH was adjusted to 7.2. The solution was circulating continuously using a peristaltic pump through a graphite felt (4 cm x 4 cm x 0.4 cm), known as a good electrical conductor and acting as a three-dimensional cathode, with homogeneous distribution of the solution: this configuration is expected to be more efficient than a carbon foil as used in former works. [15] A Nafion membrane (N-212) was placed between the two electrodes for separation of the two compartmentsin particular to avoid anode flooding from the solution treated at the cathodeand to permit a uniform proton transport rate from the anode to the cathode. Two copper ending plates allowed mechanical cohesion of the cell and electrical connection to the potentiostat (Bio-Logic VSP3). Continuous nitrogen bubbling was allowed in the storage tank using a sintered glass sparging system for saturating the gas with water at ambient temperature to avoid undesired evaporation of water during the NADH Cyclic voltammograms and batch electrolysis were performed in a two-electrode configuration, the hydrogen anode acting also as reference electrode. The usage of a two-electrode configuration where the anode acts as reference can be considered here because of the very low polarization of gas diffusion anode with current densities below 10 mA cm -2 . All potentials are thus given versus H + /H 2 couple, noted ref. (H 2 ). Potential scan rate was fixed 5 mV s -1 . In electrolysis tests, the solution was sampled at regular intervals (each 5 or 10 minutes) and the conversion of NAD + to NADH was followed by UV absorption at 340 nm after suitable dilution (10 times). For the production of NADH, two protons are produced by hydrogen oxidation while one proton is consumed by NAD + reduction, according to the stoichiometry of both reactions. In consequence, the pH slightly decreased during electrosynthesis, but owing to the buffer concentration, the pH change was never larger than 0.4 unit. The biological activity of the cofactor was then confirmed by reaction with a lactate dehydrogenase immobilized in the enzymatic cell.

Enzymatic cell

Scheme 2 shows the enzymatic cell used for the conversion of pyruvate to lactate connected in series with the electrochemical reactor (Scheme 1). The solution containing the rhodium complex, NADH, and the pyruvate was fed to the enzymatic cell through a graphite plate with grooved 1 x 1 mm 2 channels forming three channel clusters for its homogeneous distribution.

The solution then percolated through a carbon felt that maintains the stability of the enzymes immobilized on the MWCNT-CP, which was put in contact with the felt. A gas diffusion layer, known as hydrophobic, has been placed in contact with the rear side of the LDH-MWCNT-CP preventing the adsorption of the solution, thus ensuring the stability of the enzymes even at high solution flow rates. The whole assembly was stacked between two polymeric (PMMA) plates for mechanical cohesion and to prevent solution leakage. The solution leaving the enzymatic cell with a higher lactate content, was pumped to the glass storage provided by N 2 degassing before returning to the electrochemical reactor. Samples taken at regular intervals were analyzed by HPLC using a Laminex separation column with 0.05 M H 2 SO 4 as eluant.

Results and discussion

Evidence of electrochemical NADH Regeneration

The catalytic response of the GF electrode for the NADH regeneration was evaluated by cyclic voltammetry (Figure 1A). In the presence of 250 µM [Cp*Rh(bpy)Cl] + (curve b), a redox signal is observed, showing a cathodic peak at -0.39 V and an anodic peak at -0.27 V. The intensity of the cathodic peak is approximately double that of the anodic peak. The latter corresponds to the oxidation of the reduced active form of the rhodium complex I ([Cp*Rh(bpy)H] + : M red2 ).

However, the first reduced form remains inactive ([Cp*Rh(bpy)]: M red1 ) and unsuitable for redox reactions. Addition of 2.5 mM NAD + to the above solution (curve c) led to a large increase in the cathodic peak caused by the electrocatalytic reduction of NAD + by [Cp*Rh(bpy)H] + and not by the direct reduction of NAD + which is observed at much cathodic potential values (and usually gives non-active forms of the cofactor, i.e. dimers). [25,[START_REF] Gorton | Encycl. Electrochem[END_REF] The cathodic peak was observed at a potential slightly more negative (-0.45 V) than that observed with the reduction of the complex only. The cause of this potential shift can be a limitation associated to the electron transfer kinetics on the graphite felt electrode [24] or to other ohmic losses in the configuration of the system. The effect of the rhodium complex concentration on the intensity of the cathodic peak has been investigated at a fixed concentration of NAD + (2.5 mM); it shows that this intensity increases steeply with the mediator concentration before levelling off at higher concentrations (Figure 1B).

The corresponding CV responses are presented in Figure S1 in Supporting Information.

The rhodium complex concentration and the cathode potential in electrolysis tests are important operating parameters that should be adjusted for optimizing the process of the regeneration of NADH. The performance of an electrochemical process is based on two criteria: conversion X measured by UV-visible absorption for NADH regeneration (Figure 1C), and the faradaic efficiency expressing the fraction of electrons involved in the key electrochemical reaction (Figure 1D). ; = 2; F=96500 C/mol; V = 40 mL At -0.4 V vs ref.

(H 2 ) and with an initial concentration of NAD + of 2.5 mM, the NADH was totally regenerated within 30 min for a rhodium complex concentration of 125 µM (curve c) and 250 µM (curve b). However, while decreasing the concentration of the rhodium complex to 62.5 µM, the conversion was limited to 0.62 after this lapse of time (curve d). At this potential, the faradaic efficiency is higher, attaining 92% by using the lowest concentration of the rhodium complex (62.5 µM) and 80 % with 250 µM of rhodium complex (see data b-d in Figure 1D). In order to increase the conversion with a rhodium complex concentration of 62.5 µM with 2.5 mM NAD + , tests were made at -0.5 V vs ref.

(H 2 ). Total conversion was observed within 20 min (see curve a in Figure 1C), with a faradaic efficiency of 96 % (data a in Figure 1D) which is actually far larger than the values usually reported in the relevant literature in the range 50-70 %, [15] indicating the high selectivity and production rate of the process. Working at -0.5 V seems thus to be very promising based on the total conversion within an optimal time and the high faradaic efficiency. One explanation of these better performances can be found in Figure S1 of the Supporting Information (blue curve). The maximum cathodic current was observed around -0.5

V. Applying -0.5 V vs ref.

(H 2 ), led thus to the highest conversion rate. On the contrary, applying -0.4 V vs ref.

(H 2 ) led to a much lower current and consequently to a lower conversion rate. Note that the ratio between rhodium complex and NAD + was also found critical to reach complete conversion, as doubling the cofactor concentration to 5 mM with 62.5 µM [Cp*Rh(bpy)Cl] + led to incomplete conversion (Figure S2 in Supporting Information). We suppose that a high concentration of NADH is hindering the catalytic activity of the rhodium catalyst for NAD + reduction. Actually, we do not know if this effect is reversible or irreversible. In any case, high concentration of cofactor is not meaningful for the electrosynthetis application that is targeted here.

Finally, to validate that the total regeneration of 2.5 mM of NADH was achieved in the presence of 62.5 µM of the rhodium complex and at a potential of -0.5 V vs ref, the biochemical conversion of 2.5 mM pyruvate to lactate was carried out in the presence of lactate dehydrogenase in the solution. Results are reported in Figure S3 of the Supporting Information, they show that indeed 2.5 mM lactate were produced within 200 min.

Optimization of hydrogen and solution flows

Flow of hydrogen in the anodic compartment and solution flow through the cathode chamber represent also important operating conditions in the electrochemical regeneration of NADH.

The used nitrogen was humidified as expressed above. In addition, it is noted that in operation of membrane fuel cells, the reacting gases are often humidified to improve the membrane conductivity and protons transport in the vicinity of the Pt clusters. [START_REF] Goswami | [END_REF] In consequence, the hydrogen used in first tests has been humidified as well. Figure 2 shows the effect of hydrogen humidification on the conversion of 2.5 mM NAD + with 250 µM rhodium complex, at -0.4 V vs ref. (H 2 ). With humidified hydrogen, the conversion at room temperature, was observed to be limited to 0.55 and 0.50 with 10 (Figure 2, curve b) and 20 mL.min -1 H 2 respectively (Figure 2, curve a) after 30 min conversion. Dismantling the cell and examination of its components showed that the GDE catalyst was soaked, thus not suitable anymore for H 2 oxidation. With fully humidified hydrogen, the water activity in the ionomer becomes very large which may favor water flux from the (liquid phase) cathode medium, resulting in severe flooding of the anode. The effect of the non-humidified hydrogen flow rate in the anodic compartment was then examined in a series of experiments using increased flow rates of pure hydrogen from 1 mL.min - 1 to 20 mL.min -1 at -0.5 V vs ref.

(H 2 ). As shown by Figure 3A, higher conversion was actually allowed upon increased hydrogen flow rate. The conversion was very low with 1 mL.min -1 H 2 (curve a, 0.12 after 15 min), but was total within the same time period upon 20 mL.min -1 hydrogen (curve d). As a matter of fact, the amount of hydrogen consumed at the anode is far lower than 1 cm 3 .min -1 after Faraday's law. Therefore, the lower performance of the cell at flow rates below 20 mL.min -1 is presumably due to small leakage in the hydrogen circuit and in the 96% for all tested hydrogen flow rates, partly owing to 100 % current efficiency for hydrogen oxidation on a GDE and because this faradaic efficiency is certainly controlled by the conditions applied in the cathodic compartment, i.e. flow, potential and concentrations. The main limitation comes from the anodic side being the amount of hydrogen that can be oxidized at the GDE to provide all electrons needed for the regeneration of NADH.

To confirm this hypothesis, the effect of the solution flow rate on the conversion was also followed with a positive effect on the conversion rate as shown in Figure 3B. Larger solution flow rates favor its uniform distribution within the porous electrode and mass transport between species, thus improving the overall conversion rate. The conversion was 0.78, with an average reaction rate of 7.08 mmol.L -1 .s -1 , after 15 min of electrolysis with the solution flowing at 4 mL.min -1 (curve a); however, the conversion at the same time with a solution flow rate of 10 mL.min -1 attained 0.93 with an average reaction rate of 8.16 mmol.L -1 .s -1 (curve b). In contrast with what was observed with the effect of hydrogen flow rate, the faradaic efficiency of the reaction increased with the solution flow rate, passing from 83 % at 4 mL.min -1 to 96 % at 20 mL.min -1 (curve c). These data clearly show that the faradaic efficiency is also controlled by flow conditions in the cathodic compartment.

Effect of pH on NADH regeneration

The effect of pH on the NADH regeneration was studied in 0.1 M of phosphate buffer with different ratio of K 2 HPO 4 (99%) and KH 2 PO 4 (99%) for pH between 4 and 7.2 and upon addition of 0.1 M of tris-HCl for the pH of 9. The current variations (Figure 4A) have very similar profiles for pH 6.5, 7.2, and 9 featuring an asymptote after 15 min operation corresponding to nearly complete conversion. However, at pH 4 the current decreases far less in absolute value than in neutral or slightly alkaline media. The conversion (Figure 4B) and the faradaic efficiency (Figure 4C), in relation with the current variations (Figure 4A), are increasing with the solution pH. In the acidic medium, the conversion attained 60% after 20 min, with a faradaic efficiency near 72 %. By contrast, full conversion was obtained within 20 min for all other pH values, whereas the faradaic efficiency increased from 87 % at pH 6.5 to 98 % at pH 9 showing the increased selectivity of the electrocatalytic reaction when proton concentration is lowered. It is also reported in the literature that NADH is unstable in acidic environment while NAD + is unstable in alkaline media. [51] Thus, as a compromise between stability of the molecules and acceptable faradaic efficiency, the optimal pH for NADH regeneration was taken in the range 6.5 Figure 5, very similar conversion rates were observed for Rh complex concentrations ranging from 12.5 to 62.5 µM, with differences in average reaction rates below 13%. For example, the average rate of the reaction using 2.5 mM NAD + and 62.5 µM [Cp*Rh(bpy)Cl] + is 8.16*10 -4 mol.L -1 .s -1 fairly similar to the average reaction rate with 0.5 mM NAD + and 12.5 µM [Cp*Rh(bpy)Cl] + (7.8*10 -4 mol.L -1 .s -1 ). Measured in the first 5 min of the reaction, TN is in this case 370 h -1 , far larger than that reported in the literature, at room temperature (25 °C), for rhodium complex in solution from 0.68 to 6.3 h -1 . [15,52] While applying the chemical regeneration of the NADH cofactor with this rhodium complex, reduced by action of phosphite species, [53] the TN obtained was equal 21 h -1 using the same operational conditions of temperature and pH. In such conditions, the conversion is total in all studied cases and the ratio between [NAD + ] and [Cp*Rh(bpy)Cl] + corresponds to the TTN. This TTN can only be increased here if the regenerated cofactor is consumed continuously by an enzymatic reaction, a development briefly examined in this study and described in the next section with lactate dehydrogenase as example.

Turnover number

Enzymatic systems applied continuously with the electrochemical regeneration of the NADH depend on the amount of enzymes used, in relation to the quantity of substrate to be converted, which rules the time required for given bioconversion. For that purpose, it was important to evaluate the stability of the regenerated NADH with time. Results showed that 98 % of the regenerated NADH after 72 hours were still stable and active as shown in Figure S4 of the Supporting Information.

Application of regenerated NADH in an enzymatic reaction

After having determined the optimal parameters for the efficient regeneration of NADH in the flow electrochemical reactor, integrating an enzymatic reaction in the process was attempted to validate the production of the active form of the NADH cofactor regenerated in the above mediated redox process. This is indeed important for the practical application of such bioelectrocatalytic system [24] and for the evaluation of its possible use in the production of fine chemicals as expected from biocatalysis in biotechnology. [41] For this purpose, Lactate dehydrogenase was used here since it is a widely available NAD-dependent enzyme, whose substrate and products can be easily analyzed. The first step in this work was to demonstrate the stability and the selectivity of LDH enzymes immobilized on the chosen support in the presence of dissolved rhodium complex and NAD + /NADH species (reported in Figure S5 in Supporting Information). The second step was to prove that NADH was regenerated with a high purity and exhibiting a high activity towards the bioconversion of pyruvate to lactate in comparison to the commercial one (reported in Figures S6 to S9 in Supporting Information). The third step was the application of the regenerated NADH for the conversion of pyruvate to lactate in the presence of the rhodium complex. It was assumed that the bioconversion occurs at the surface of the LDH-MWCNT-CP. For this reason, it was important to evaluate the optimal amount of LDH that has to be added on MWCNT. 500 units of LDH were found sufficient (> 200 U/cm² CP-MWCNT)

for the efficient bioconversion of pyruvate to lactate, as reported in Fig. S10 of the Supporting Information). In a previous report from the literature, 1400 U of LDH were added to a 50 mL solution containing 2 mM NAD + and 20 mM pyruvate. [15] Scheme 3. Schematic representation of the electrochemical reactor in series with the enzymatic cell including Rhmediated electrochemical and enzymatic reactions with the LDH-MWCNT-CP system. hours, with a current regularly decreasing from 21 mA to 16 mA after 5 hours electrolysis (Fig. 6B).

The concentration of the enzymatic cofactor decreased rapidly in the 10 first minutes of the run (Fig. 6C) to less than 0.1 mM, due to its quantitative LDH-catalyzed consumption. In the meanwhile, the lactate concentration raised to 0.56 mM, corresponding to approx.1.12 times of the initial NADH concentration of the NADH cofactor, i.e. with little significant NADH regeneration. Afterward, the NADH concentration slowly increased, attaining 0.17 M after 4 hours, whereas pyruvate was efficiently converted to lactate (Fig. 6D), with equilibrated constant rates between electrochemical and enzymatic processes. In the last hour, the lower pyruvate concentration rendered the enzymatic reaction slower, resulting in increased NADH concentration (0.41 mM after 5 hours). Conversion was nearly total after 5 hours, with a 24.7 mM lactate final concentration, corresponding approximately to 50-times of the initial NADH concentration. The faradaic efficiency was higher than 90% during the first hours of reaction, but decreased abruptly during the last hour to 61 %, i.e. below than that obtained without enzymes (96 %). Apparently, the decrease in NAD + concentration in the flow reactor when NADH was not consumed anymore by LDH (because of the full conversion of pyruvate to lactate), provided more opportunities for side electrochemical reactions, e.g. reduction of oxygen traces or hydrogen evolution. A total turnover number of more than 2000 was obtained for the rhodium complex after 5 h, which is the highest value ever obtained with rhodium complex in solution, compared with those in former electroenzymatic synthesis tests for the bioconversion at room temperature, reported between 55 and 214. [54], [55] Without the application of an enzymatic reaction, the highest TTN for the rhodium complex applied in the electrochemical regeneration of the NAD(P)H cofactor was 400. [56] In addition, the TN in the period from 1 to 4 hour with enzymes was near 300 h -1 , in the same order of magnitude but slightly lower than the value of 370 h -1 obtained without enzymes in the presence of a higher concentration of NAD + (2.5 mM reported in Figure 5 versus 0.5 mM in Figure 6); TTN obtained for the enzymatic reaction was equal to 6.

Conclusion

In summary, we have demonstrated the full potential of a bioelectrochemical reactor for the regeneration of NADH that couples efficiently the oxidation of hydrogen to the reduction of NAD + mediated by a rhodium complex: high activity with a TN of 370 h -1 for TTN value near 40 was shown.

The efficient regeneration of NADH featured with a complete conversion and with a high faradaic yield can be reached by fixing different parameters for the used reactor at their optimal values: hydrogen and solution flow rates (20 mL.min -1 ), NAD + to [Cp*Rh(bpy)Cl] + concentration ratio (40), pH (close to neutrality, to avoid NADH and NAD + degradation).

Based on these optimized parameters for the regeneration of the NADH cofactor, adding an enzymatic reaction for the bioconversion of pyruvate to lactate by the LDH-CP-MWCNT showed its high efficiency and stability with a high TTN for the cofactor electrochemical regeneration and the enzymatic reaction of 2000 and 6.3*10 6 , respectively. In addition, no deactivation of both the rhodium complex and the LDH was observed here (a possible deactivation is reported in the literature [55] ).

The electrochemical reactor and the enzymatic cell, presented in Schemes 1 and 2 respectively, are in principle scalable. While the performances regarding TN and TTN of the rhodium catalyst in solution are larger than those reported in the literature, one must be aware that TN and TTN for the catalyst are still lower than enzymatic based electrogeneration associated to surface immobilization. [11,57] Moreover, TTN for the cofactor in the electroenzymatic synthesis remain largely lower than the most recent achievements with closely confined enzymatic catalyst also immobilized on the surface of porous electrodes. [58] As a perspective, the immobilization of the rhodium complex allowing much higher TTN (up to 12 000) [26,59] has to be applied for this flow reaction and associated more closely with the enzymatic reaction in order to promote the best cycling of both the catalyst and the enzymatic cofactor and potentially easier purification of the synthesized molecules. In this way, one could promote the application of redox flow technologies in quite relevant industrial topic: dehydrogenases allow for the development of blocks of chiral synthesis comprising various functions e.g. alcohols, hydroxy acids, aldehydes and ketones having a large interest in cosmetics, food, and pharmaceutical industries. [5] Besides, lactate production from lactate dehydrogenase, the illustrative example that is considered in this study, is widely used in various applications e.g. food, cosmetics, pharmaceutical, and chemical industries [6], [7]

Scheme 1 .

 1 Scheme 1. Schematic view of the flow electrochemical reactor. Solution flow is on the left side and hydrogen flow is on the right side. A proton-conducting Nafion membrane is separating the gas diffusion electrode, for hydrogen oxidation, from the graphite felt electrode, for NADH production by electrocatalytic reduction of NAD + with [Cp*Rh(bpy)Cl] + complex.

Scheme 2 .

 2 Scheme 2. Schematic view of the enzymatic cell. Solution flows from the right side.

Figure 1 .

 1 Figure 1. (A) Cyclic voltammograms recorded at 5 mV.s -1 successively with the graphite felt electrode in (a) buffer solution (b) after addition of 250 µM [Cp*Rh(bpy)Cl] + and (c) after addition of 2.5 mM NAD + . (B) Influence of the rhodium complex concentration on the cathodic current peak measured by CV. (C and D) Influence of the catalyst concentration and the electrolysis potential on (C) the conversion rate of 2.5 mM NAD + to NADH and (D) the faradaic efficiency; (a) 62.5 µM [Cp*Rh(bpy)Cl] + at -0.5 V vs ref. (H 2 ); (b) 250 µM [Cp*Rh(bpy)Cl] + at -0.4 V vs ref. (H 2 ); (c) 125 µM [Cp*Rh(bpy)Cl] + at -0.4 V vs ref. (H 2 ); (d) 62.5 µM [Cp*Rh(bpy)Cl] + at -0.4 V vs ref. (H 2 ).All experiments were carried out in 0.1 M phosphate buffer solution at pH 7.2 under nitrogen atmosphere with a solution and hydrogen flow rates of 20 mL.min -1 each.

Figure 2 .

 2 Figure 2. Influence of humidified hydrogen on the production of NADH at -0.4 V vs ref. (H 2 ) All experiments were done in 0.1 M phosphate buffer solution at pH 7.2 in the presence of 2.5 mM NAD + and 250 µM [Cp*Rh(bpy)Cl] + under nitrogen atmosphere with a solution flow rate of 20 mL.min -1 and (a) humidified hydrogen with a flow rate of 20 mL.min -1 , (b) humidified hydrogen with a flow rate of 10 mL.min -1 , (c) non-humidified hydrogen with a flow rate of 20 mL.min -1 .

Figure 3 .

 3 Figure 3. Influence of (A) hydrogen flow rate ((a) 1 (b) 5, (c) 10 and (d) 20 mL.min -1 ) and (B) solution flow rate ((a) 4, (b) 10 and (c) 20 mL.min -1 ) on the conversion of NAD + . Experiments were carried out in 0.1 M phosphate buffer solution at pH 7.2 in the presence of 1 mM NAD + and 25 µM [Cp*Rh(bpy)Cl] + , under nitrogen atmosphere, at an operating potential of -0.5 V vs Ref. (H 2 ) with either (A) a constant solution flow rate or (B) a constant hydrogen from rate of 20 mL.min -1 each.

B

  flow cell, with possible maldistribution of the fuel gas in the bipolar plate channel at too low flow rates. Besides, the faradaic efficiency of NADH regeneration remains unchanged at approx.

Figure 4 .

 4 Figure 4. Influence of the solution pH on (A) the current, (B) the conversion and (C) the faradaic yield for the mediated transformation of NAD + to NADH. All experiments were performed in 0.1 M phosphate buffer solution in the presence of 1 mM NAD + and 25 µM [Cp*Rh(bpy)Cl] + under nitrogen atmosphere with solution and hydrogen flow rates of 20 mL.min -1 each.

9 A

 9 Regeneration of NADH using the same NAD + over Rh concentration ratio was carried out at various concentration levels, at -0.5V vs ref. (H 2 ). For a [NAD + ]/[Rh] ratio of 40, as shown in

Figure 5 . 1 M

 51 Figure 5. Studying the reduction of NAD + using the same ratio of NAD + and [Cp*Rh(bpy)Cl] + (40) and the electrolysis potential (-0.5 V vs H 2 oxidation) using different concentrations of NAD + and [Cp*Rh(bpy)Cl] + : (a) 2.5 mM NAD+ and 62.5 µM [Cp*Rh(bpy)Cl] + , (b) 1.25 mM NAD + and 31.25 µM [Cp*Rh(bpy)Cl] + , (c) 1 mM NAD + and 20 µM [Cp*Rh(bpy)Cl] + , (d) 0.5 mM NAD + and 12.5 µM [Cp*Rh(bpy)Cl] + . All experiments were performed in

Scheme 3

 3 Scheme 3 represents the electrochemical reactor in series with the enzymatic cell in the

Figure 6 .

 6 Figure 6. (A) Current recorded after the addition of 25 mM pyruvate recorded using 12.5 µM [Cp*Rh(bpy)Cl] + in solution (a) without the enzymatic cell and with 0.5 mM of NAD + and (b) with the enzymatic cell. (B) Current, and (C) NADH evolution recorded with time in test (b). (D) Time variations of the concentrations of (a) pyruvate, (b) lactate, and (c) the sum of pyruvate and lactate, as determined by HPLC. Enzymatic reaction (Figures 6A (b), 6B, 6C, and 6D) was carried out in 0.1 M PBS solution at pH 7.2, with 0.5 mM NADH and 12.5 µM [Cp*Rh(bpy)Cl] +

  3*10 6 showing the good stability of the flow device combining the electrochemical reactor and the enzymatic cell. No conversion was observed in the absence of NADH (Figure S11 of the Supporting Information). The lactate produced by the redox flow bioreactor displayed a very good purity, over 99% as determined by 1H NMR (Figures S12-S13 of the Supporting Information).

Acknowledgement. We gratefully acknowledge CNRS for funding of the project NADP/H 2 . We also thank Neus Vilà (LCPME) for help in the rhodium complex synthesis, Steve Pontvianne and Xavier Framboisier (LRGP) for development of the HPLC analytical method and Olivier Fabre at the LCPM laboratory for the NMR measurements. Supporting Information. Cyclic voltammetry of electrochemical reduction of NAD + in the presence of various concentrations of [Cp*Rh(bpy)Cl] + ; an example of non-complete conversion; validation of the flow enzymatic reactor with short discussion of the reported data, illustration of optimal LDH quantity for the bioconversion of pyruvate, control experiments regarding NADH regeneration and stability, 1H NMR analysis of NADH and lactate.