11-15 Avril 2022, Marseille

In this work, a new method is presented in order to locate high order exceptional points (EP) arising in eigenvalue problems. EP correspond to a particular tuning of some complex-valued parameters which render the problem degenerate. These non-Hermitian degeneracies have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems (PT-symmetry, thermo-acoustic or fluid-structure instability, etc.) and their numerical solution. For applications dealing with dissipative acoustic waveguides, strong modal attenuation can be achieved close to EP and a maximum of attenuation occurs at EP of high order corresponding to the coalescence of more than two modes. The method is based on the automatic computation of the successive derivatives of some selected eigenpairs with respect to the parameters so that, after recombination, regular functions can be constructed. This algebraic manipulations permit to build a reduced order model allowing i) to quickly solve the eigenvalue problem for other parameters values, ii) to follow modal branches, iii) to locate higher order EPs. The method is applied to the particular case of a circular duct with a locally reacting liner at its wall which admittance varies with azimuthal position. This particular application allows to gradually test the proposed method with EP of increasing order.

Introduction

Acoustic treatments are often used to attenuate sound propagation in guiding structures generally encountered in HVAC, exhaust devices, and aircraft engines. In many applications, the walls of the waveguide are acoustically treated with absorbing materials, and it is usual for the analysis to distinguish between two classes of liners depending whether the latter is locally like perforated plate bounded with honeycomb or non-locally reacting like porous materials. In both cases, the propagation of sound waves is best described in terms of duct acoustic modes and the existence of exceptional point (EP) is known to be connected with very strong modal attenuation [START_REF] Tester | The optimization of modal sound attenuation in duct, in the absence of mean flow[END_REF][START_REF] Bi | New insights into mode behaviours in waveguides with impedance boundary conditions[END_REF][START_REF] Xiong | Sound attenuation optimization using metaporous materials tuned on exceptional points[END_REF][START_REF] Qiu | Optimality analysis of bulk-reacting liners based on mode-merging design method[END_REF][START_REF] Perrey-Debain | Mode coalescence and the green's function in a two-dimensional waveguide with arbitrary admittance boundary conditions[END_REF]. At the EP, eigenvalues and associated eigenvectors coalesce at a branch point singularity in the parametric space. Moreover, in the vicinity of the EP, perturbation analysis show that eigenvalues take the form of a Puiseux series [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] i.e. a series in fractional power of the parameter. From practical point of view, this strong attenuation can be explained by a transmission dominated by surface waves [START_REF] Bi | New insights into mode behaviours in waveguides with impedance boundary conditions[END_REF][START_REF] Xiong | Sound attenuation optimization using metaporous materials tuned on exceptional points[END_REF].

Eigenvalue problems are ubiquous in all physics branch, parametric eigenvalue problems have thus been the subject of intensive research. In acoustic waveguides, the usual parameters can be the frequency or the wall impedance. Most approaches are based on first or second perturbation order [START_REF] Lin | A state-ofthe-art review on theory and engineering applications of eigenvalue and eigenvector derivatives[END_REF]. Recently, high order perturbation methods have been proposed to optimize the numerical efficiency when the parameter appears explicitly in the numerical model. In [START_REF] Nennig | A high order continuation method to locate exceptional points and to compute puiseux series with applications to acoustic waveguides[END_REF][START_REF] Ghienne | Beyond the limitations of perturbation methods for real random eigenvalue problems using exceptional points and analytic continuation[END_REF] the bordered matrix [START_REF] Andrew | Derivatives of eigenvalues and eigenvectors of matrix functions[END_REF] is used for obtaining the derivatives and the adjoint vector is used in [START_REF] Orchini | Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems : Semisimple eigenvalues[END_REF][START_REF] Mensah | Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems : simple eigenvalues[END_REF]. It is noteworthy that the bordered matrix approach is limited (at least in this form) to simple eigenvalue whereas [START_REF] Orchini | Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems : Semisimple eigenvalues[END_REF] extend the adjoint method for semisimple eigenvalue, ie a repeated eigenvalue with different eigenvectors. However, high order Taylor expansion of the eigenvalue is still limited by the presence of branch point singularity. In [START_REF] Nennig | A high order continuation method to locate exceptional points and to compute puiseux series with applications to acoustic waveguides[END_REF], it has been proposed to recombine the Taylor expansion of the eigenvalue to build analytic functions which are not affected by the singularity at the EP. This algebraic manipulation allows to extend the radius of convergence of the eigenvalue approximation [START_REF] Ghienne | Beyond the limitations of perturbation methods for real random eigenvalue problems using exceptional points and analytic continuation[END_REF], the localization of EP using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order.

EP of order N (EPN) require more parameters and the their localization become more challenging than EP2. In addition, the eigenvalue sensitivity increases also with N and the eigenvalue may become strongly dependent on the rounding error and split [START_REF] Ryu | Classification of multiple arbitrary-order non-hermitian singularities[END_REF]. Despite this, practical realization have been done [START_REF] Ding | Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization[END_REF], although, to the best authors knowledge, no generic algorithm exists for finding them. The route to obtained high order EP is generally to work on simple (or reduce) system and combine them [START_REF] Zhong | Hierarchical construction of higher-order exceptional points[END_REF].

The aim of this work is to propose i) a new perturbation method for eigenvalue problem and ii) a generic algorithm to locate EPN. For this purpose, a new framework, based on analytic functions combining more than two eigenvalues to improve the radius of convergence and to take into account several parameters, is proposed. This can be done through the new concept of partial characteristic polynomial (PCP), which consists on reconstruct a part of the characteristic polynomial 1 from the Taylor expansion of some selected eigenpairs obtained at an initial value of the parameters. To compute the successive derivatives, the bordered matrix [START_REF] Andrew | Derivatives of eigenvalues and eigenvectors of matrix functions[END_REF] is used here as in [START_REF] Nennig | A high order continuation method to locate exceptional points and to compute puiseux series with applications to acoustic waveguides[END_REF] (but other approaches can be used). For new parameters values, the eigenvalues can be obtained quickly as the roots of a low order polynomial. The EP of order N can then be found by imposing that the PCP and its N -1 successive derivatives with respect to the eigenvalue vanish. The main ingredient of the method are given in sec. 2, an application to a 2D waveguide illustrating the accuracy and the interest of the proposed method is presented in sec. 3.

Partial characteristic polynomial

Let us consider a pair of eigenvalue, denoted by λ -(ν) and λ + (ν), dependant on the scalar parameter ν. In [START_REF] Ghienne | Beyond the limitations of perturbation methods for real random eigenvalue problems using exceptional points and analytic continuation[END_REF], it has been shown that the auxiliary functions

g(ν) = λ + + λ -, (1)
h(ν) = (λ + -λ -) 2 , (2)
are analytic even if the two eigenvalues present a branch point singularity and that their Taylor series (denoted by T • ) is not convergent. In the same way, the product of the eigenvalues is also an analytic function of ν. It follows that analytic auxiliary functions (AAF) scheme,

λ ± (ν) = g(ν) ± √ h(ν) 2 ≈ T g (ν) ± √ T h (ν) 2 , (3)
behaves generally better than Puiseux or Taylor series to reconstruct eigenvalue loci as a function of ν. In the vicinity of its EP, the Puiseux series provides the good branch structure of the eigenvalue Riemann surfaces. Nevertheless, the implicit function theorem shows that Puiseux series radius of convergence is limited by the presence of the next EP even if this EP involved the same pair of eigenvalue, while in this specific case AAF scheme remains analytic and is able to accurately reconstruct the eigenvalues (but not directly the branches).

Getting the eigenvalue from the analytic auxiliary function is equivalent to find the roots of the polynomial

P(λ) = (λ -λ + )(λ -λ -) = λ 2 -(λ + + λ -)λ + λ + λ -, (4)
whom coefficients are analytic functions of the parameter ν (the explicit dependency of ν has been dropped for clarity).

In particular, in (3), the link between h and the polynomial discriminant is obvious. This approach can be generalized with the concept of partial characteristic polynomial and to vectorial parameters ν. For an analytic matrix function L(λ(ν), ν), its characteristic polynomial P(λ, ν) is also an analytic function. Assuming that P(λ, ν) can be split into

P(λ(ν)) = λ-λ 1 (ν) . . . λ-λ (ν) . . . λ-λ L (ν) • f (λ(ν)) (5)
where λ stands for all the eigenvalues (counting their multiplicities) present in a disk of the λ-complex plane and f is an analytic function vanishing outside the disk. We call this set Λ and it contains L = |λ| elements. We denotes by

P(λ, ν) = λ -λ 1 (ν) . . . λ -λ (ν) . . . λ -λ L (ν) (6)
the partial characteristic polynomial of L(λ(ν), ν). Using the Vieta's formulas, the coefficient of this polynomial can be expressed as analytic function of the eigenvalue.

P(λ, ν) = a L (ν)λ L + a L-1 (ν)λ L-1 + • • • + a 0 (ν), (7)
where

a k = c∈C k i∈c λ i (ν), (8)
and the set C k = {all ways to chose (k -|Λ|) different eigenvalues in Λ}. For instance, for a quintic polynomial build using a set of 5 eigenvalues, we get

a 5 = 1, (9a)
a 4 = -λ 0 -λ 1 -λ 2 -λ 3 -λ 4 (9b) a 3 = λ 0 λ 1 + λ 0 λ 2 + λ 0 λ 3 + λ 0 λ 4 + λ 1 λ 2 + λ 1 λ 3 (9c) + λ 1 λ 4 + λ 2 λ 3 + λ 2 λ 4 + λ 3 λ 4 , a 2 = -λ 0 λ 1 λ 2 -λ 0 λ 1 λ 3 -λ 0 λ 1 λ 4 -λ 0 λ 2 λ 3 (9d) -λ 0 λ 2 λ 4 -λ 0 λ 3 λ 4 -λ 1 λ 2 λ 3 -λ 1 λ 2 λ 4 -λ 1 λ 3 λ 4 -λ 2 λ 3 λ 4 , a 1 = λ 0 λ 1 λ 2 λ 3 + λ 0 λ 1 λ 2 λ 4 + λ 0 λ 1 λ 3 λ 4 + λ 0 λ 2 λ 3 λ 4 (9e) + λ 1 λ 2 λ 3 λ 4 , a 0 = -λ 0 λ 1 λ 2 λ 3 λ 4 . (9f)
Because each polynomial coefficient a k (k = 0, . . . , L -1) is analytic, their Taylor series be obtained thanks to multivariate Leibniz' rule applied to (8) from the Taylor series of each eigenvalue.

The final step is to solve the partial polynomial P(λ, ν) for a given range of ν to recover the λ i (ν) (i = 0, . . . , L -1). Depending of the number of eigenvalue in Λ, the roots can be obtained numerically or analytically if the polynomial is lower than 4. In practice, the numerical burden due to the numerical roots is neglectable when compared to eigenvalue computation of matrices obtained after standard discretization methods like finite element method. This representation also allow to locate EP.

Higher order EP location

EP corresponds generically to multiple roots of the PCP for complex symmetric problems [START_REF] Seyranian | Coupling of eigenvalues of complex matrices at 16 ème Congrès Français d'Acoustique 11-15 Avril 2022, Marseille diabolic and exceptional points[END_REF]. If the first derivative ∂ λ P(λ; µ, ν) vanishes, it corresponds to an EP2. If the second derivatives also vanishes it corresponds to an EP3 and so one. It is noteworthy that PCP derivative with respect to the eigenvalue also vanishes for semisimple eigenvalue. Thus to be sure to have an EP further checks may be required for instance by inspecting Puiseux series coefficients [START_REF] Welters | On explicit recursive formulas in the spectral perturbation analysis of a Jordan block[END_REF].

To locate the finite set of EP3, we need to solve the multivariate polynomial system

             P(λ; ν) = 0, ∂ λ P(λ; ν) = 0, ∂ 2 λ P(λ; ν) = 0, (10)
with ν ∈ C 2 . The Bezout theorem (see [START_REF] Wise | Algorithm 801 : Polsys_plp : A partitioned linear product homotopy code for solving polynomial systems of equations[END_REF] for instance) ensures that this system has only discrete solutions.

The system can be solved using

• Newton solver. Since such approach need an initial guess, a mesh parametric space is required and the solver need to be called for all starting points. This approach is generally fast but may miss some roots. • Homotopy solver [START_REF] Wise | Algorithm 801 : Polsys_plp : A partitioned linear product homotopy code for solving polynomial systems of equations[END_REF][START_REF] Nennig | A python wrapper to the fortran package polsys_plp that solve polynomial systems with homotopy method[END_REF]. This family of method are able to find all the solution of (10) numerically. The basic idea is to exploit analyticity to move slowly from the solution of a trivial similar problem to the solution of the real problem. Both approaches are efficient, but due to the truncated Taylor series, the number of spurious solutions may be huge. An estimation can be obtained from the Bezout Number [START_REF] Wise | Algorithm 801 : Polsys_plp : A partitioned linear product homotopy code for solving polynomial systems of equations[END_REF].

To filter the spurious roots, comparing solution for two truncation orders in the a k Taylor series is generally sufficient. It could also be combined with an estimation of radius of convergence PCP terms.

2D duct application 3.1 Problem statement

We consider a two-dimensional acoustic waveguide of infinite length and unit width lined with two admittances µ and ν, as described in Fig. 1. This reference problem have been already studied in [START_REF] Perrey-Debain | Mode coalescence and the green's function in a two-dimensional waveguide with arbitrary admittance boundary conditions[END_REF] and can lead to a finite set of EP3 or a continuum of EP2. Their values are useful to validate the proposed approach. In the duct, the acoustic pressure satisfies the Helmholtz equation (e -iωt convention is adopted here) ∆p + k 2 a p = 0, [START_REF] Nennig | A library to locate exceptional points and to reconstruct eigenvalues loci[END_REF] where k a = ω/c a is the wavenumber, c a is the sound speed and ω is the angular frequency. On both walls, the liner is assumed to be locally reacting which implies that the pressure must satisfy the Robin boundary condition ∂ y p = -µp, at y = 0 and, ∂ y p = νp, at y = 1 [START_REF] Nennig | A high order continuation method to locate exceptional points and to compute puiseux series with applications to acoustic waveguides[END_REF] Using invariance along the waveguide x-axis, the modal analysis is performed by assuming that the pressure field can be written in the separable form p = φ(y)e iβx . Here, function φ(y) is the mode shape and β the axial wavenumber. The weak formulation associated with the Helmholtz equation gives

- 1 0 ∂ y ψ • ∂ y φ dy + (k 2 a -β 2 ) 1 0 ψφ dy + µψ(0)φ(0) + νψ(1)φ(1) = 0, ( 13
)
where ψ stands for the test function. Once the variational formulation is discretized with linear Lagrangian finite element, we obtain a generalized eigenvalue problem of the form To be consistent with our notation, we put λ = β 2 and ν = (µ, ν). Here, the vector x contains the finite element nodal values of the acoustic pressure, the matrix Γ i (i = 1, 2) stems from the admittance boundary condition and K and M are the standard stiffness and mass matrices respectively. We can easily calculate formally the partial derivative of ( 14) with respect to ν components and λ required to get the eigenvalue derivative as described in [START_REF] Nennig | A high order continuation method to locate exceptional points and to compute puiseux series with applications to acoustic waveguides[END_REF].

L λ(ν), ν x(ν) = -K + (k 2 a -λ)M + ν 1 Γ 1 + ν 2 Γ 2 x(ν) = 0. ( 14
)

Results

This approach is implemented in EasterEig [START_REF] Nennig | A library to locate exceptional points and to reconstruct eigenvalues loci[END_REF] open source framework dedicated to perturbation of eigenvalue problem. We use 200 linear Lagrange element in the cross section of the duct.

To illustrate the numerical efficiency of eigenvalue reconstruction based on PCP, the absolute error E between direct and PCP computation is given in Fig. 2 for increasing values of the perturbation parameter δ such ν = ν 0 + e 0.3i δ. The comparison is performed by combining the Taylor series of 2, 4, 6 or 8 eigenvalues (sort by modulus) computed at ν 0 = (7.01265 -4.76715i, 2.89872 -2.47i). It can be observed that increasing the number of eigenvalues in the PCP strongly enhance the reconstruction quality. This can be explained by a mutual regularization of the branch point singularity in eigenvalue Taylor series. For this 1D problem, the prediction is valid up to 2 digits for perturbation parameter equal to 10, spanning most of interesting admittance values with a single direct computation.

The second aspect of the proposed approach concerns the EP location. Again, we start from ν 0 = (7.01265 -4.76715i, 2.89872 -2.47i) and combine the 6 first eigenvalues. The Homotopy method is used to solve [START_REF] Nennig | A python wrapper to the fortran package polsys_plp that solve polynomial systems with homotopy method[END_REF]. Solution is a set containing the triplets (λ, µ, ν). The found ν values that yields to EP3 are presented in Fig. 3. Note that because of the symmetry of the problem, the role

Figure 1 -

 1 Figure 1 -Lined duct with two admittances.

Figure 2 -

 2 Figure 2 -Evolution of error on eigenvalue reconstruction according to the number of mode in the PCP and the modulus of the perturbation parameter δ.

The characteristic polynomial cannot be computed for big matrices obtained after standard discretization methods like FEM.

ème Congrès Français d'Acoustique 11-15 Avril 2022, Marseille

of ν and µ is interchangeable thus the solution are found by pairs. It can be shown that several EP3 are accurately identified among the spurious roots when two truncation order are compared.

Conclusion

A new eigenvalue perturbation method has been proposed. The method is based on the partial characteristic polynomial concept. This approach allows to improve the radius of convergence of eigenvalue reconstruction for parametric problems depending on several parameters. This representation also permits to locate higher order EP.

This generic approach can be applied on circular ducts [START_REF] Koch | Attenuation of sound in multi-element acoustically lined rectangular ducts in the absence of mean flow[END_REF][START_REF] Zorumski | Multiple eigenvalues of sound-absorbing circular and annular ducts[END_REF]. For instance EP3 have been found for wall presenting two lined parts separated by rigid wall portions. Work is on going to better understand the merging mechanism and the link with strong attenuation as observed in 2D ducts [START_REF] Perrey-Debain | Mode coalescence and the green's function in a two-dimensional waveguide with arbitrary admittance boundary conditions[END_REF].

The method can also be applied to other parametric eigenvalue problems like those arising in thermoacoustic [START_REF] Mensah | Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems : simple eigenvalues[END_REF][START_REF] Orchini | Perturbation theory of nonlinear, non-self-adjoint eigenvalue problems : Semisimple eigenvalues[END_REF], in topological acoustic waveguides [START_REF] Coutant | Subwavelength su-schrieffer-heeger topological modes in acoustic waveguides[END_REF] or in structural dynamics.


ABOUT SOME REMARKABLE PROPERTIES OF GENERALIZED CANONICAL ANALYSIS Gilbert SAPORTA Institut Universitaire de Tech~ologie 143, Avenue de Versailles -75016 PARIS France

The method and the criterion propose? by J.D. Carroll for generalizing canonical analysis to more than two groups of variables, provide a very general model which unify various techniques of multidimensional data analysis. Let X. 1 = 1 1,2, •. , p be p sets of m. variables on n 1 individuals ; each X. 1 will be considered ~sa nxm. matrix. Various 1 attempts [START_REF] Horst | the prediction of personal adjustment" Social Science Rescarch Council. On the quantification of qualitative data from the mathematico -statistical point of view[END_REF] ; [START_REF] Kettenring | Canonical analysis of several sets of variables[END_REF] , [START_REF] Masson | Liaison entre plusieurs ensembles de variables et codages de données qualitatives[END_REF])has been made to generalize canonical analysis when p ~ 3 by finding directly p-uples of canonical variables s .. (s. = X. a.) satisfying to some -1 -1 1 -1 criteria of optimality. Most of them lead to tricky algorithms and the s. , generally, are no longer solutions of an eigenequation.

-1

Carroll's method consists in finding an auxiliary variable, the most correlated with the p groups, in the following sense:

(1) p I i= 1 R 2 (!., X. ) 1s maximal 1 where R 2 is the square multiple correlation coefficient. This leads to the eigenequation:

(2) f: .

p I i=l ' -1 ' X. (X.X:) X. z = 1 1 1 1 À z À max where I -1 I_ X. (X. X.) X.:-A. 1 1 1 1 1 1s the orthogonal
-1

• It may be noted [START_REF] Ten | Optimizing factorial invariance[END_REF])that this solution (equivalent Then, applying Carroll's generalized canonical analysis top nominal variables (each X. is the array of the indicator variables th 1 of the categories of the i variable ;f.) leads exactly to the 1 principal components of scales analysis of Guttman (1941), to the optimal quantification of Hayashi ( 1950), to the Benzecri-Lebart (1972) multiple correspondence analysis (See also [START_REF] Bouroche -G | Generalized canonical analysis of qualitative data[END_REF] and to the optimal scaling method Homals of J. De [START_REF] De | A unified treatment of the weighting problem[END_REF] which are all equivalent.

Since the multiple correlation coefficient with the indicator variables is nothing but a correlation ratio, all these methods may be presented in a fairly simple way: they consist in finding a numerical variable z maximizing

t n 2 (~ ; ~i) i=l
We have thus a general criterion for analyzing sets of nominal or numerical variables, the l;. where l;.. = X. a.

-f. 1 -1 and V. . = x' }{:

1] i J

  onto the Xi gives then the canonical variates s . and z is both the mean and the first principal component of the -1

  to solutions proposed by HORST, Mc DONALD and generalization of canonical analysis because, to successive eigenvalues ~ are not orthogonal when p > 2. of equation (2) others) is an improper if the z (k) corresponding -(k) are orthogonal the s. -1 . .. / ... Nevertheless, the main interest of Carroll's_'1Ilethod stay in its criterion and 1n finding ~ variates (more than the~.metric which takes into account the grouping of variables.

  D. Carroll nor others gave a simple equation for we prove here that the canonical variables can be obtained as the eigenvector of an np x np matrix whose blacks are the products of projectors A. A.1J; this 1s a generalization of a seemingly new property of ordinary canonical analysis. From this result, it can be aasily shown that the i;. satisfy a mean orthogonality property:

  theoretical interest since it is of dimension np , the solution is in fact perforned in dimension Emi by solving the equation (


Jean-Christophe M Monbaliu
email: jc.monbaliu@uliege.be|www.citos.uliege.

Julien Legros
email: julien.legros@univ-rouen.fr

Will the next generation of chemical plants be in miniaturized flow reactors ?

For decades, a production paradgim based on centralized, stepwise, large scale processes has dominated the chemical industry horizon. While effective to meet an ever increasing demand for high value-added chemicals, the so-called macroscopic batch reactors are also associated with inherent weaknesses and threats; some of the most obvious ones were tragically illustrated over the past decades with major industrials disasters and impactfull disruptions of advanced chemical supplies. The COVID pandemic has further emphasized that a change in paradigm was necessary to sustain chemical production with an increased safety, reliable supply chains and adaptable productivities. More than a decade of research and technology development has led to alternative and effective chemical processes relying on miniaturised flow reactors (a.k.a. micro and mesofluidic reactors). Such miniaturised reactors bear the potential to solve safety concerns and to improve the reliability of chemical supply chains. Will they initiate a new paradigm for a more localized, safe and reliable chemical production?

Introduction

Recent global events are leading the world to rethink some models which, until recently, seemed irreplaceable. The COVID pandemic has pulverised economic patterns of some manufactured goods, among which the severe limit in fine chemical supply, and therefore of active pharmaceutical ingredients (APIs), has been highlighted. Significant shortages of strategic medicines were thus cruelly felt, jeopardising the capacity of national health systems to treat their fellow citizens. Most of the so-called "industrialized nations" have shown the limitations of their industrial apparatus in this emergency situation. Moreover, the Russian-Ukrainian conflict has further compounded this situation with an urgent need for rapid and radical change in reorganizing manufacturing.

Over the last twenty years, the production of fine chemicals, such as APIs and their intermediates (most of which have fallen into the public domain), has been almost entirely relocated to remote countries for both reasons of cost and lower environmental/safety constraints, which is intrinsically linked to the production method for these compounds: macroscopic batch reactors. Indeed, the use of batch reactors means that, in order to produce more, their capacity must be increased, leading to high investments and significant workforces, as well as significant safety concerns. This logic has led to the dismantling of chemical production facilities in numerous countries.

Unlike other sciences that have revolutionised their concepts over the last few decades, organic synthesis (the centrepiece of drug manufacturing) has used roughly the same tools since the 1950s and is still based on knowledge that is often empirical. However, there is now an alternative technology to this synthesis in large reactors: miniaturised continuous flow reactors. Where conventional production requires very large installations, flow synthesis uses a production tool with the size of household appliances. [1][2][3][4][5][6][7][8][9] This "miniature factory" is both modular and mobile; it could be used to prepare different molecules of interest at different sites according to local needs.

Chemical synthesis: thinking small to produce more… and better

As evoked above, the tools for producing fine chemicals have not been subjected to major changes and the transfer of a chemical reaction from lab-scale to production (scale-up) goes through the increase of reactor size from milliliters to several thousands of liters. The quantity of chemical produced is thus related to the capacity of the reactor (hence the name of "batch synthesis"). In contrast, miniaturized continuous-flow reactors consist of reacting chemicals (either liquid, gaseous or solid) in motion in tubes (or channels) of micrometric or millimetric diameter, without ever interrupting the reaction (at least theoretically, maintenance may impose periodic interruptions). The intrinsic features of this technology make it possible to solve many problems encountered in chemical synthesis. Fast mixing and efficient heat transfers are now well-established assets of flow reactors and often deployed for reactions sensitive to local stoichiometry, 10 biphasic systems, 11,12 strongly exothermic and high-temperature processes with sensitive materials. 13 Flow chemistry also brings novel challenges to the Chemist and Chemical Engineer, such as for handling solids and high viscosity materials. Viscosity is commonly dealt through higher process temperatures or with appropriate additives. For solids, major advances were recently reported with oscillatory reactors for the handling of slurries/heterogeneous catalysts. 14 Alternatively, the use of packed-bed reactors with immobilized heterogeneous regents or catalysts is very common. Marginal, yet effective solutions for handling solids may also rely on process temperatures above their melting points. 15 Although counter-intuitive, miniaturised flow-through devices greatly accelerates and simplify the scale-up of a process since a flow-through reactor allows for a quantity of material proportional to its operating time, avoiding reactor resizing and inevitable redesign of reaction conditions. 1,16 Note that the quantity of material involved at a given time remains the same regardless of scale, which is very valuable for reactions that are difficult to control on a large scale, or for the manipulation of toxic species (vide infra). In order to produce substantial amounts of compounds, some companies even commercialize flow devices with an increasing number of reactor plates and high flow rates (Scheme 1). Consequently, numbering-up or scaling-out strategies can be easily deployed for accessing larger scales in flow. 7 Typical outputs for miniaturized reactors can drastically vary, hence also conditioning their end-use: microreactors with a typical internal diameters of ~100 µm and an internal volume of a few 100 µL are usually associate with throughputs of a ~100 µL/min; mesofluidic reactors with an internal diameter of up to 800 µm are usually associated with total flow rates of about 10 mL/min, and are well suited in lab environments. Larger with internal dimensions of about 1 mm can accommodate flow rates of several hundred mL/min (pilot scale) and up to several L/min (commercial scale). Accessing the scalability and selecting an appropriate strategy to reach larger scales in flow is often a critical point. The selection of a strategy, namely numbering-up of scaling-out (Scheme 1) ultimately depends on the end product, the target market and internal organization protocols. For instance, radiopharmaceuticals, the shelf-life of which is very limited, are often produced on the spot with microreactors of limited volumes being more than sufficient to produce the required dosages. Scaling up a process to access pharmaceuticals will already be very successful with a few kilograms per day of active ingredient. Other specialty or bulk chemicals often require much larger scales. One of the major breakthrough in flow-microreactors is the development of the so-called "flash chemistry" by Yoshida. [START_REF] Yoshida | Flash chemistry: fast organic synthesis in microsystems[END_REF][START_REF] Yoshida | [END_REF] The uniqueness of this approach relies on the control of very fast reactions, in virtue of the very short residence time t R permitted in the microreactor, allowing to perform "impossible chemistries" (ie in batch reactors). [START_REF] Yoshida | [END_REF] Indeed in a flow reactor, the reaction time is a direct function of the reactor size according to t R = (volume of the reactor)/(flow rate), and miniaturized flow reactors allow to attain very short reaction times: a few minutes to some milliseconds [19][20][21] and even below. 22 This breakthrough is however often connected to a hard-to-eradicate misconception, namely, that flow conditions only applies to reactions with inherent fast kinetics. This misconception relates to the small internal volume of flow reactors, which is often associated with short residence/reaction times. Flow microreactors offer several options for process intensification, among which the temperature and pressure is often leveraged to significantly accelerate slow reactions. It is quite common to find in the actual state of the art striking examples with reaction processes taking hours in batch while reaching completion within minutes or less under intensified flow conditions. 13,23,24 Moreover, a key distinction between batch and flow conditions is the composition of the reaction medium. In a macroscopic batch reactor, the chemical composition of the reaction medium evolves with time as the conversion of reactants/substrates to products increases, hence labelling these processes as "time-resolved". Under flow conditions, the conversion varies along the reaction path, thus increasing (and ideally reaching completion at the outlet of the flow reactor) and thus defining "space-resolved" conditions. An easy fix to this apparent dilemma, which might be one of the most confusing aspects of flow chemistry for neophytes, relies on the concept of residence time. The latter correlates space and time and offers a very handy metric to connect the volume of the reactor, the overall velocity of fluids going through it and the time required to leave the reactor. Regardless of the regime in a flow reactor (laminar, transitional or turbulent), any point in the reactor thus corresponds to a specific state of progress of the reaction (Scheme 2). This feature avoids undesired byproducts due to consecutive competitive reactions between reagents and products, with high benefits for chemical selectivity toward a desired single reaction product. 25,26 The emergence of photochemistry in flow over the past decade has also tremendously benefited from this feature, with a much finer control on irradiation time even for large scale applications, 27 and there a much reasons to believe that electrochemistry in flow is improved likewise. [28][29][30] Scheme 2. Evolution of the chemical composition of a reactor: batch (up) vs flow (down). This approach is especially fruitful for the ultra-fast halogenlithium exchange where unstable chemical entities can be formed, previously untamed under classical conditions. 21,31,32 The generation and trapping of such entities in an integrated system allow thus to extend the chemical space to new transformations and novel molecular architectures (Scheme 3). Noteworthy, such very fast reactions in very small reactors naturally induce high productivity (quantity of product/time) and high space-time yields (productivity/reactor volume), which are key points for cost-effective delivery of chemicals.

The development of processes under flow conditions in miniaturized reactors is often presented as a solution for reducing their environmental footprint and to increase their sustainability. While the arguments associated with the reduced needs for energy, improved homogeneity and purity of reactor effluents and increased inherent safety definitively speak for themselves, the simple transposition of batch conditions under flow does not guarantee de facto sustainability. [START_REF]Sustainable Flow Chemistry[END_REF][START_REF] Hessel | [END_REF][35][START_REF] Ferlin | 2 Flow Chemistry -Applications[END_REF] The development of flow processes for replacing older batch processes must also be seen as an opportunity to redefine the chemistry in light of the assets of flow technology. [START_REF] Rogers | [END_REF]

Reactors downsized, safety increased

Performing chemical reactions at large scale in macroscopic batch reactors comes with safety concerns, which arise at different stages of the process. Safety issues are associated with stockpiling and shipment of large volumes of starting materials and intermediates with a high risk of accidental chemical spilling. Many industrial incidents are associated with thermal runaway in large vessels, with potentially dramatic impacts on the surrounding operators and environment. One obvious solution to significantly reduce chemical hazard upon processing chemicals relies on downsizing the internal size of the reactor -such strategy is not compatible with macroscopic batch reactors within the context of a centralized, global production scheme, since a reduction in size would be deleterious to the production scale. However, this does not apply to flow reactors that are continuously operated. The reduced internal volume guarantees a safer inherent safety: [START_REF] Chen | 1 Flow Chemistry -Fundamentals[END_REF] in case of rupture or a runaway at a given time, only a minimal amount of chemicals would be released to the environment, hence also improving the direct operational safety. In addition to the direct positive impact on safety of downsizing reactors, the high heat transfer efficiency of flow reactors also enables either the handling of highly exothermic processes or to suppress the formation of problematic byproducts. Along with an improved heat transfer deeply rooted into the much larger surface/volume ratio of flow reactors, structural robustness of the reactor wall is another important factor that unlocks safe use of high pressure for routine operations.

Besides the internal and structural features of flow reactors, the concepts of reaction telescoping (or concatenation) [START_REF] Britton | [END_REF]40 and of chemical generator 41 have been widely developed under flow conditions to enable the safe handling of toxic or highly unstable materials, even at large scale (Scheme 4). A chemical generator in flow feeds upon feed solutions of widely available, stable and non-toxic materials, which upon mixing in a first flow module will react and produce discrete amounts of a highly reactive or unstable chemical. The concatenation of the first flow module to a second, downstream flow module, enables to consume right on the spot the reactive/toxic species with another substrate, hence drastically reducing the inherent risk of processing such chemicals. Such concepts were widely documented by Kappe and coworkers and has become an invaluable tool to implement conditions and chemicals that would otherwise not be allowed in conventional processing units. [41][42][43][44][45][46] This concept is used likewise for the generation of explosive intermediates, such as peroxides, diazo species and azides, hence minimizing the chemical risk with only discrete amounts of explosive materials being generated and consumed right away. [47][48][49][50][51][52][53][54][55] On another note, the absence of head-space in flow reactors also allows the implementation of dangerous conditions, such as the use of oxygen with flammable solvents 56,57 or to stabilize reactive gaseous species in solution. 51

Scheme 4. Reaction concatenation/telescoping in flow: illustration of a chemical generator

Devices for the disposal of toxic chemicals

As stated above, the wide diversity of chemical processes can be labelled as constructive processes and aim at increasing the molecular diversity and the added value of compounds, starting from widely available and affordable building blocks. For instance, the manufacture of API clearly fits in such definition, where complex reaction sequences or costly catalysts and reagents are acceptable, as long as the added value of the final target leaves room for comfortable margins. [58][START_REF] Taylor | Pharmaceuticals in the Environment[END_REF][START_REF] Brennan | Process Industry Economics[END_REF] Destructive processes are different in essence. Regardless of the end application, the latter aims at the physical or chemical destruction of molecular entities to annihilate safety concerns through a chemical modification. [START_REF] Picard | [END_REF] In the context of the chemical disposal of chemical warfare agents (CWAs), addedvalue (in the sense of their production costs and not for their end application) molecular entities with an acute toxicity are neutralized through chemical modification of their backbone toward lower value, lower toxicity entities that can be next safely handled for disposal. [START_REF] Picard | [END_REF] It comes without saying that such processes inherently come with a reverse economical scheme. An ideal destructive process would therefore rely not only on affordable reagents and catalysts, but also on robust and straightforward process conditions that are amenable to chemical threats of high societal impact. 5,[START_REF] Picard | [END_REF] The design of affordable and robust conditions is not an easy task. [START_REF] Brennan | Process Industry Economics[END_REF] The affordability of a process comes mostly from the chemicals (ideally off-the-convenience-store) and the operating conditions (ideally room temperature and low pressure), as well as from the process technology that is relied on. The robustness of a process results from the combination of a thorough understanding of the mechanisms at stakes with process technologies that enable steady process conditions. Besides, within the specific context of the neutralization of CWAs, additional important criteria include a potential mobility and high operational safety to make it deployable anywhere with minimal resources. [START_REF] Picard | [END_REF]62 Miniaturized flow reactors meets all these criteria and they have been shown to allow safe generation/handling of hazardous compounds such as oxidants. 41,[63][64][65] Therefore, flow devices are safe systems for the controlled oxidative neutralization of CWA, since the selectivity of the process is a key aspect to afford harmless compounds. This has been successfully performed with commercial oxidizing reagents 66,67 or by generating more reactive/unstable oxidants upstream in the flow system. 68,69 The fine control of the conditions in the miniaturized reactor avoids the formation of undesired and harmful overoxidized products (Scheme 5).

Strong bases can also be generated/safely used in dedicated flow systems for the same purpose. 70,71 Noteworthy, the treatment of methyl paraoxon in a concatenated sequence (basic hydrolysis/reduction/acylation) allows the cleavage of the side chain of this toxic pesticide and its transformation into the API paracetamol. 71 Scheme 5. Safe flow neutralization of CWA and pesticides.

Pharmaceutical sovereignty and factory 5.0

Following up the conclusions of the 2007 Green Chemistry Institute roundtables that identified the next priorities for investment and development in the pharmaceutical production, a thrust toward the implementation of miniaturized flow chemistry reactors was witnessed at the R&D level. A few years later, when the regulatory authorities such as the US Food and Drug Administration (US FDA) started to advocate continuous manufacturing at all stages of an API development, hence leading to a progressive adoption of flow reactors at the production scale. With more than 15 years of technology development, mature flow processes are now thriving. This momentum was further increased in the current context with a succession of global crises causes regular supply disruption of fine chemicals along with prices volatility, with dramatic consequences in all economic sectors. The most visible impacted one is undoubtedly the health system which has undergone shortages in essential drugs, such as anesthetics for intensive care units. To circumvent this issue the relocation of APIs on domestic soils using the "same old recipes" seems unrealistic, and even non-desirable, for several reasons. Obviously, it will not be economically viable to reinvest in large macroreactors to produce old essential -and no longer costeffective-API that are in the public domain. Even if public health care is a national priority, it might not be bearable on a medium/long-term. More, such factories can only be dedicated to well-established processes for API synthesis with zero impact on research and drug discovery, whereas the emergence of new diseases will require new drugs. Major shortages have triggered a thrust toward re-shoring the production of drug substances to Western countries. However, extremely restrictive environmental and safety policies (e.g., REACH in the European Union) are incompatible with outdated large-scale, stepwise macroscopic batch settings. Besides, in many countries, citizens are now much reluctant to live next door to big chemical plants, which are associated with environmental and health issues. Advanced flow technologies have emerged over the last decade as a robust and viable solution to address both environmental and safety concerns. In this context, a new generation of chemical plants based on miniaturized flow reactors is certainly most promising. Indeed, flow devices allow faster transfer to laboratory to production than batch technology, which is highly desirable for drug discovery. Moreover, not only the reactors are miniaturized in flow systems, but also the usual analytical tools have been significantly downsized such as benchtop-sized NMR as most impressive achievement. 72 Thus, the reacting flow passing in a continuous fashion through the analytical tool (inline analysis), real-time information is provided on the reaction course, which offers the possibility to adapt the continuous reaction parameters, mostly through modification of the flow rates (change in stoichiometry, residence time), temperature and pressure.

In a 2016 visionary article (3 years before COVID crisis and subsequent drug shortage), Jamison, Jensen and Myerson already proposed a reconfigurable "fluidic household appliance" for the on-demand synthesis of essential drugs according to U.S. Pharmacopeia standards, namely diphenhydramine, lidocaine, fluoxetine (as hydrochloride salts) and diazepam (Scheme 6). 73 The reconfigurable MIT setup was then further improved with an extended scope, more versatile fluidic modules and with advanced automation, robotics and artificial intelligence. 74,75 Scheme 6. Reconfigurable flow platform for the on-demand synthesis of some APIs from Jamison, Jensen and Myerson. 73 Copyright 2016 The American Association for the Advancement of Science.

More recently, Khan and Wu demonstrated that, while counter-intuitive at first glance, multi-step solid-phase synthesis could be implemented in an automated flow system to perform a six-step synthesis of prexasertib in 65% isolated yield after 32 h of continuous execution (Scheme 7). 76 A library of 23 analogues of this kinase inhibitor was easily synthesized by feeding the platform inlets with different reagents. Scheme 7. Automated synthesis of prexasertib and derivatives enabled by continuousflow solid-phase synthesis (SPS). 76 Miniaturized flow reactors offer thus the possibility to perform numerous chemical experiments with few interventions, and provide numerous data to be collected and stored as digital files, also with alternative trends to the more conventional linear approach including radial and cyclic approaches. 77 The merging of flow reactors with advanced automation and process analytic technology (PAT) significantly strengthens the robustness and versatility of flow processes, either at the R&D or production scales, and provide invaluable tools for fast optimization. 78,79 Combined to intelligent algorithms, it would ultimately lead to autonomous flow platforms able to self-plan and -execute its own synthetic pathway to designated target molecules. [79][80][81] Whereas this road might take some times, efficient black-box algorithms have proven their effectiveness. These algorithms do not require information on the reaction studied and focus on identifying an optimum (yield) rather than generating data to develop reaction models. 82,83 For example, Felpin reported a 4-step continuous flow synthesis of a potent anxiolytic compound (FGIN-1-27) from inexpensive and commercially available starting materials (Scheme 8). The use of inline analyses and modified Nelder-Mead optimization algorithm, to assist the decision-making process significantly minimized the number of experiments required in optimization campaigns. 84 Further digitalization of chemical processes is expected to become the emerging trend for the next few years. 85

Conclusions

The last decade has seen the emergence of miniaturized flow reactors as a possible alternative to classical batch processes for chemical synthesis. Whereas this technology initially appeared as an exotism, or simply a transient technological fashion like the chemical world already experienced in the past (with microwaves for example), several recent events transformed it into a new paradigm. As often during major crises, a full redesign of several unchanging patterns was required to overcome important issues, especially drug shortages. The re-shoring of fine chemicals can only be restored with a long-term viable approach breaking with the former recipes. Moreover, this reshoring needs to adapt to the new life model of citizens who do not want to live with a massive chemical plant on the next door. Therefore, continuous flow miniaturized reactors penetrated into industrial research centers with significant momentum over the past five years, as witnessed with frequent job ads emphasizing the need for talents in flow chemistry, through media and press releases, patents or with primary scientific literature in less frequent occurrences. Industrial research centers include flow chemistry both at the R&D and production levels, in an ultimate effort to accelerate the lab-to-market transitions. Some examples of recent public disclosures include the "step-in" announcements of Angelini Pharma (CDMO, Italy) and Medichem (Spain) with substantial investments in lab, pilot and production flow reactors for development and manufacturing of custom APIs.

Finally, flow devices have an intrinsic connection with automatization since the evolution of a reaction in a flow reactor is controlled by continuous parameters (flow rate) , which are easily adjustable by algorithms/computers with the help of and process analytic technology. Whereas artificial intelligence will allow to self-plan a chemical synthesis from the reaction scheme to its execution, it is even possible to design its miniaturised flow system by CAD/additive manufacturing, and therefore to fully conceive a flow synthesis from the device and chemical route to advanced chemical scaffolds. 70,[86][87][88] Regarding the flow syntheses of API molecules that have been sorely missing during the COVID crisis, it can be noted the recent reports on the synthesis of anesthetics propofol [89][90][91] and ketamine. 92 The synthesis of remdesivir, a broad-spectrum antiviral agent approved against SARS-CoV has also been described, [93][94][95] showing thus the increasing role of flow reactors for the implementation of the miniaturised chemical plants of the future.

Scheme 1 .

 1 Scheme 1. Chemical synthesis, from lab to production: batch vs flow strategy.

Scheme 3 .

 3 Scheme 3. Flash chemistry in flow microreactors.

Scheme 7 .

 7 Scheme 7. Continuous flow synthesis of FGIN-1-27 enabled by in-line19 F NMR analyses and optimization algorithms. Copyright 2021 Royal Society of Chemistry.84

Acknowledgements JL acknowledges the European France(Manche)England crossborder cooperation program INTERREG V A "SmartT", cofinanced by ERDF, for financial support, as well as the University of Rouen Normandy, INSA Rouen Normandy, the Centre National de la Recherche Scientifique (CNRS), Labex SynOrg (ANR-11-LABX-0029), Carnot Institute I2C, the graduate school for research XL-Chem (ANR-18-EURE-0020 XL CHEM) and the Région Normandie. JCMM acknowledges the University of Liège and the F.R.S.-FNRS (Incentive grant for scientific research MIS F453020F).

Notes and references

Conflicts of interest

There are no conflicts to declare.


Maria Ivanova

Thomas Poisson

Philippe Jubault

Julien Legros
email: julien.legros@univ-

Flow platform for the synthesis of benzodiazepines

Keywords: diazepam, acylation, methylation, oxidation, drugs

Benzodiazepines possess a wide spectrum of valuable pharmacological effects and are among the leading drugs, commonly prescribed ones for various pathologies. Herein we describe the continuous flow synthesis of six benzodiazepines from aminobenzophenones: diazepam, fludiazepam, nordazepam, nitrazepam, clonazepam, oxazepam.

The 1,4-benzodiazepine-2-ones backbone is an essential pharmacophore in the pharmaceutical industry, 1 and the first benzodiazepine was developed in 1960 by Hoffmann-La Roche and marketed as Librium (Scheme 1). [START_REF] Sternbach | The Benzodiazepine Story[END_REF] This class of compounds exhibits a highly significant clinical activity and finds various therapeutic applications as anxiolytics, 3-7 hypnotic, [START_REF] Hallahan | Benzodiazepine and Hypnotic Prescribing in an Acute Adult Psychiatric In-Patient Unit[END_REF][START_REF] Scharf | Comparative Amnestic Effects of Benzodiazepine Hypnotic Agents[END_REF] antiarrhythmics, [START_REF] Selnick | Class III Antiarrhythmic Activity in Vivo by Selective Blockade of the Slowly Activating Cardiac Delayed Rectifier Potassium Current IKs by (R)-2-(2,4-Trifluoromethyl)-N-[2-Oxo-5-Phenyl-1-(2,2,2-Trifluoroethyl)-2,3-Dihydro-1H-Benzo[e][1,4]Diazepin-3-Yl]Acetamide[END_REF] vasopressin antagonists, 11 HIV reverse transcriptase inhibitors, [START_REF] Breslin | Synthesis and Anti-HIV-1 Activity of 4[END_REF] and cholecystokinin antagonists, [START_REF] Castro | 5-(Piperidin-2-Yl)-and 5-(Homopiperidin-2-Yl)-1,4-z z : -Affinity, Basic Ligands for the Cholecystokinin-B Receptor[END_REF] among others. Some studies even demonstrated the antiproliferative effect of benzodiazepines against cellular tumors. [START_REF] Dourlat | Novel 1,4-Benzodiazepine Derivatives with Antiproliferative Properties on Tumor Cell Lines[END_REF] This huge, continuously expanding class, might be as well a key synthon in the synthetic pathways of other fused ring compounds, such as triazolo-, oxadiazolo-, oxazino-, or furanobenzodiazepine, for instance. [START_REF] Insuasty | Tetrahydro-5H-Naphtho[END_REF] In spite of the widespread adoption of various dibenzodiazepine scaffolds as molecules possessing psychoactive activity to treat anxiety disorders, the pioneering work of Schmutz in the early 1960s remains to be a significant contribution in the area of the benzodiazepine pharmacophore synthesis. [START_REF] Hunziker | Diazepine, -Oxazepine Und -Thiazepine. 3. Mitteilung Über Siebengliedrige Heterocyclen[END_REF][START_REF] Hunziker | Amino-5H-Dibenzo[END_REF][START_REF] Schmutz | Über in 11-Stellung Amino-Substituierte Dibenzo[b,f]-1, 4-Thiazepine Und -Oxazepine. 9. Mitteilung Über Siebengliedrige Heterocyclen[END_REF] Scheme 1. Some relevant examples of benzodiazepines On another hand, continuous flow chemistry is an attractive tool that can allow chemists to access efficiently to highly demanded active pharmaceutical ingredients. [START_REF] Domokos | Integrated Continuous Pharmaceutical Technologies-A Review[END_REF][START_REF] Hughes | Applications of Flow Chemistry in the Pharmaceutical Industry-Highlights of the Recent Patent Literature[END_REF][START_REF] Gioiello | The Medicinal Chemistry in the Era of Machines and Automation: Recent Advances in Continuous Flow Technology[END_REF][START_REF] Ferlin | Sustainable Flow Approaches to Active Pharmaceutical Ingredients[END_REF][START_REF] Bloemendal | Continuous One-Flow Multi-Step Synthesis of Active Pharmaceutical Ingredients[END_REF][START_REF] Gérardy | Continuous Flow Organic Chemistry: Successes and Pitfalls at the Interface with Current Societal Challenges: Continuous Flow Organic Chemistry: Successes and Pitfalls at the Interface with Current Societal Challenges[END_REF][START_REF] De Angelis | Development of a Continuous Flow Synthesis of Propranolol: Tackling a Competitive Side Reaction[END_REF] In this context, the flow synthesis of benzodiazepines has been reported by several groups. Thus, Baxendale described a flow synthesis of benzodiazepines as a combination of a S N Ar reaction to produce a nitrodiarylamine intermediate, which upon reduction of the nitro group subsequently underwent a cyclocondensation (Scheme 2a). [START_REF] Baumann | Synthesis of Riboflavines, Quinoxalinones and Benzodiazepines through Chemoselective Flow Based Hydrogenations[END_REF] Thompson and colleagues demonstrated the ability of MS analysis and droplet reaction to guide microfluidic synthesis, performing continuous synthesis of diazepam in two steps on a microscale (Scheme 2b). [START_REF] Ewan | Multistep Flow Synthesis of Diazepam Guided by Droplet-Accelerated Reaction Screening with Mechanistic Insights from Rapid Mass Spectrometry Analysis[END_REF] In 2016, the group of Jamison described the preparation of diazepam in flow, starting from the acylation of 5-chloro-2-(methylamino) benzophenone with neat bromoacetyl chloride in NMP, followed by an intramolecular cyclization reaction upon addition of a stream of NH 3 in a MeOH/H 2 O mixture. This method implied an elevated pressure (250 psi) and temperatures (90 °C and 130 °C; Scheme 2c). [START_REF] Adamo | On-Demand Continuous-Flow Production of Pharmaceuticals in a Compact, Reconfigurable System[END_REF] One year later, the same group depicted a 2-steps process, including the amidation reaction between a solution of benzophenone and neat chloroacetyl chloride, followed by a reaction with aqueous ammonia solution; in-system dilution to avoid clogging and sonicating bath and a static mixer (Scheme 2d). [START_REF] Bédard | Minimizing E-Factor in the Continuous-Flow Synthesis of Diazepam and Atropine[END_REF] Recently, the report on an automated multistep chemical synthesizer AutoSyn was disclosed, showing the syntheses of ten FDA approved drugs, including diazepam (Scheme 2e). [START_REF] Collins | Fully Automated Chemical Synthesis: Toward the Universal Synthesizer[END_REF] Herein, within the framework of our research program dedicated to the API continuous flow manufacturing, 31,[START_REF] Ivanova | Continuous Flow Synthesis of Celecoxib from 2-Bromo-3,3,3-Trifluoropropene[END_REF]33 we describe a general synthesis of 1,4-benzodiazepine-2-one scaffolds. Our flow synthesis of the benzodiazepines is shown in Scheme 2. The installation does not require an additional special equipment or high temperature and allows the access to 6 different benzodiapines, the largest number of such APIs accessible by a single platform approach reported so far. To demonstrate the synthetic utility of our process, we performed a reaction on a larger 24 mmol scale with 2-amino-5-chlorobenzophenone 1b, using Syrris Asia pump, and the corresponding product 3b was isolated in 94% yield.

Then, to reach the final targets, the acylated products should be subjected to cyclisation, resulting in the formation of 1,4-benzodiazepine-2-ones, 34 and wich might then undergo a Nmethylation or a C-acylation, 35 as precursors of various benzodiazepins. The flow system for these transformations was set up using a R-series Vapourtec® flow reactor (R2+ pumps, R4 heater/cooler module) with two pumps. Thus, the next step involved nucleophilic substitution of the bromine by a convenient methanolic solution of NH 3 , followed by the dehydrative cyclisation toward the formation of the benzodiazepine 5. There was a slight formation of 2-amino-benzophenone 1, ie the starting material of the preceding step, resulting from the transamidification side reaction rather than the bromine substitution (8-24%). We have performed numerous trials to optimize this transformation and avoid the undesired reaction For instance, we diluted in various ways a commercially available 7 M solution of NH 3 in MeOH, but it did not influence the outcome of the reaction. Pleasingly, the ratio product/byproduct increased according to the temperature from 30 to 110 °C, whereas exceeding the latter resulted in lower yield. It was also preferable to use a larger reactor (10 mL) with a fast flow rate, rather than a small reactor (2-5 mL) along with a low flow rate. The total flow rate of 0.5 mL/min (0.25 mL/min + 0.25 mL/min) was the most beneficial (t R = 20 min). DMF was superior to toluene and 1,4-dioxane, and the maximum concentration of the substrate was 0.2 M in DMF. The solutions of ammonia in different solvents were also tried, among 7 M methanol, 0.4 M THF, 2 M ethanol, the former gave the best results. The reaction was performed on 4 mmol scale and the product 5a was obtained in a very good 94% yield (0.814 g/h, 0.081 kg/h/L) (Scheme 4). PTFE reactor (10 mL) at 120 °C, (t R =27 min), the Nitrazepam 5c was obtained in a modest 36% yield (0.190 g/h, 0.019 kg/h/L) (Scheme 4). Note that the overall yield of the nitrazepam was 31% over 2 steps. Finally, this protocol was applied to the synthesis of clonazepam 5d, which was isolated in a good 64% yield and a 37% yield over 2 steps. From the class of 3-hydroxy-1,4-benzodiazepines, oxazepam 15 and lorazepam are widely used for the treatment of anxiety. [START_REF] Biyani | Development of a Continuous Flow Synthesis of Lorazepam[END_REF] To synthesize 3-hydroxy derivatives of 1,4-benzodiazepines, first of all it is necessary to address the direct acetoxylation of the 3 position of the benzodiazepine backbone. As an oxidant we aimed at finding one, which could easily fill in a column reactor and not create a clogging. Among K 2 S 2 O 8 , NaNO 2 , MnO 2 and BaMnO 4 , the latter was the easiest to deal with.

The solution of benzodiazepine 5b, iodine 12, KOAc 11 in acetic acid (0.2 M) was flowed through a column reactor heated at 90 °C at 0.05 mL/min, the column was filled with mixture of BaMnO 4 (2 equiv) and silica (1 g total). The residence time of the transformation was 30 min, after that the solution was diluted with EtOAc, washed with the solutions of sodium thiosulfate and bicarbonate, extracted with EtOAc, passed through a shot plug of silica and evaporated, the product 13 was obtained in an excellent yield 91%, based on 2 mmol scale (0.18 g/h) (Scheme 7).

Scheme 7. Acetoxylation reaction of the 3-position of the 1,4-benzodiazepine ring (2 mmol scale). [a]

Yield of isolated product.

The obtained product 13 was converted by basic ester hydrolysis at 0 °C, followed by acidification with AcOH, into oxazepam 15, in a yield of 69%, based on 0.5 mmol scale (0.89 g/h, 0.089 kg/h/L; Scheme 8).

Conclusion

To conclude, robust flow chemical routes towards the valuable bioactive entities 1,4-benzodiazepine-2-ones has described. The protocol needs minimum requirements and could be applied for a great number of starting materials, providing a reliable route to this vital class of compounds. The use of flow chemistry translated to increased safety and shorter processing times. On a global standpoint, we performed the complete syntheses of 6 important 1,4-benzodiazepine-2-ones, with overall yields ranging from 31 to 75%. In the current post-pandemic situation, these syntheses offer a straightforward platform for the continuous flow synthesis of strategic entities 1,4-benzodiazepine-2ones.

Scheme 2 .

 2 Scheme 2. State of the art and present work.

Scheme 4 .

 4 Scheme 4. Formation of benzodiazepines using NH 3 in MeOH (4 mmol scale). [a] Yield of isolated products. [b] Overall yield.

Scheme 5 . 4 -,Scheme 6 .

 546 Scheme 5. Synthesis of nitrazepam and clonazepam using HMTM/NH 4 OAc (2 mmol scale). [a] Yield of isolated products. [b] Overall yield.

Scheme 8 .

 8 Scheme 8. Hydrolysis of 13 into oxazepam (0.5 mmol scale). [a] Yield of isolated product. [b] Overall yield.

Acknowledgements

This work is part of the API program supported by the European Union through the operational program FEDER/FSE 2014-2020 (no. 20E04976) and FNADT-DRACCARE program 2020-2023 ((N)15179). This work has been partially supported by University of Rouen Normandy, INSA Rouen Normandy, the Centre National de la Recherche Scientifique (CNRS), Eu -------. T.P. thanks the Institut Universitaire de France (IUF) for support.


Qiao Liu

Mariela Alicia Brites Helú

Alain Walcarius

Liang Liu
email: liang.liu@cnrs.fr

Visualization of working electrode reactivity from an electrochromic counter electrode

Keywords: electro-optical imaging, electrochromic counter electrode, WO3, local electrochemical reactivity, ensembles/arrays of ultramicroelectrodes

Understanding local electrochemical reactivity of surfaces is crucial for electrode design and applications at device level, in the fields of energy storage, biosensing, or electrocatalysis. In this work, we attempt to optically map the active distribution and local electrochemical reactivity of surfaces not by "looking" at the working electrode (WE), but by measuring optical change of the counter electrode (CE). The concept is based on an electrochromic CE that changes color with electron transfer accompanying ion intercalation. Because the CE is placed in parallel with the WE and maintains a micrometer distance, the optical transmittance of CE reduces locally at positions corresponding to the active area of WE by using a suitable anodic potential. The change can be captured dynamically in the form of video, which offers information on the transient behavior of the system. The spatial resolution of CE imaging increases as the distance between WE and CE decreases, likely due to the de-homogenization of current distribution on CE. The reported method could avoid the modification of neither the WE nor the electrolyte, which may be generic for measuring and imaging "invisible" electrochemical reactions.

Introduction

Measuring local electrochemical reactivity of electrodes is highly important for fundamental understanding of the materials as well as engineering of electrochemical devices [START_REF] Ma | In situ mapping of activity distribution and oxygen evolution reaction in vanadium flow batteries[END_REF][START_REF] Feng | Probe the localized electrochemical environment effects and electrode reaction dynamics for metal batteries using in situ 3D microscopy[END_REF][START_REF] He | Local charge transfer within a covalent organic framework and Pt nanoparticles promoting interfacial catalysis[END_REF][START_REF] Almalla | Development of layer-by-layer assembled thin coatings on aluminium alloy AA2024-T3 for high resolution studies of local corrosion processes[END_REF][START_REF] Yang | Imaging the electrochemical impedance of single cells via conductive polymer thin film[END_REF]. This is usually achieved by scanning electrochemical probe techniques, such as scanning electrochemical microscopy (SECM) and scanning electrochemical cell microscopy (SECCM) [START_REF] Polcari | Scanning electrochemical microscopy: A comprehensive review of experimental parameters from 1989 to[END_REF][START_REF] Ebejer | Localized high resolution electrochemistry and multifunctional imaging: Scanning electrochemical cell microscopy[END_REF]. With the recent development of nanoelectrodes and nano-capillaries, the spatial resolution of SECM and SECCM could reach tens of nanometers. However, to generate a map of electrochemical signals, the probe has to scan over the sample surface, which is intrinsically slow. It may take from minutes to hours, making it impossible to map transient electrochemical processes at high temporal resolution.

On the other hand, optical imaging is much faster with modern cameras. Even without special camera, it typically takes tens of milliseconds to acquire a static image, which is already several orders of magnitude less as compared with scanning probe techniques. Moreover, one may easily achieve good spatial resolution up to the diffraction limit of the light (<1 µm) without special optics. Thus, optical imaging is ideal for high throughput measurements and is widely used in industry [START_REF] Guo | Microtube electrodes for imaging the electrochemiluminescence layer and deciphering the reaction mechanism[END_REF][START_REF] Guerrette | Fluorescence coupling for direct imaging of electrocatalytic heterogeneity[END_REF]. Nevertheless, most of the electrochemical reactions cannot be directly "seen". It is essential to quantitatively convert the electrochemical reactions to measurable optical signals. For example, Tao and co-workers have developed electrochemical surface plasmon resonance microscopy (SPRM), where they derived quantitative links between SPR intensity and local electrochemical signals, i.e. current density, double-layer capacitance and charge transfer resistance [START_REF] Shan | Imaging local electrochemical current via surface plasmon resonance[END_REF][START_REF] Lu | Plasmonic-based electrochemical impedance spectroscopy: Application to molecular binding[END_REF]. The technique has been used to monitor the electrocatalytical activity of single nanoparticles in real-time [START_REF] Fang | Plasmonic imaging of electrochemical oxidation of single nanoparticles[END_REF][START_REF] Liu | Plasmonic measurement of electron transfer between a single metal nanoparticle and an electrode through a molecular layer[END_REF][START_REF] Wang | Fast electrochemical and plasmonic detection reveals multitime scale conformational gating of electron transfer in Cytochrome c[END_REF][START_REF] Jiang | Optical imaging of phase transition and Li-Ion diffusion kinetics of single LiCoO 2 nanoparticles during electrochemical cycling[END_REF]. Note that SPRM is usually performed on thin gold film electrodes for the sake of getting significant SPR signals. This would limit practical applications due to the potential interference from the oxidation of gold under anodic bias and/or the instability of gold films in acidic media. Another strategy is to "look" at reactions that involve photoresponsive molecules, such as electrochemical fluorescent microscopy or electrochemiluminescence microscopy [START_REF] Guerrette | Fluorescence coupling for direct imaging of electrocatalytic heterogeneity[END_REF][START_REF] Cao | Immuno-electrochemiluminescent imaging of a single cell based on functional nanoprobes of heterogeneous Ru(bpy)3 2+ @SiO2 /Au Nanoparticles[END_REF]. The former is based on the electrochemical production of molecules that can either excite or quench fluorescence, and the latter is based on the excitation of luminophore that emits light. In both cases, the optical signal intensity is linked to the quantity of electrochemically generated photo-responsive or photo-active species, which reflect the electrochemical reaction kinetics on the working electrode (WE). Yet, they are limited to a few special reactions.

To target more general "invisible" reactions, one needs to add photo-responsive molecules (or their precursors) or luminophore and design special reaction routes [START_REF] Long | Electrochemical methodsimportant means for fabrication of fluorescent nanoparticles[END_REF][START_REF] Richter | Electrochemiluminescence (ECL)[END_REF]. This would inevitably change the target system of interest. From the optical measurement aspect, all the approaches above focus on the WE sample, that is, capturing the image by directly "looking" at the optical change of WE as a result of electrochemical reactions. In an electroanalytical system, it is natural for electrochemists to study the interface between WE and the electrolyte, whereas another interface of counter electrode (CE) is often neglected. While CE is usually designed not to interfere with WE, in some cases it may also purposely regulate the electric field distribution by controlling its size and position relative to the WE [START_REF] Okumura | Effect of counter electrode in electroformation of giant vesicles[END_REF][START_REF] Pech-Canul | Influence of counter electrode positioning on solution resistance in impedance measurements of reinforced concrete[END_REF]. A typical example is direct mode SECM, where a microelectrode serving as CE is positioned close to macro-sized WE sample. This spatially localizes the electrochemical reactions in the vicinity under the micro-CE by regulating the current flux. It is widely used for patterning surfaces by local electrochemical deposition or etching, and the shape of pattern almost replicates that of the micro-CE when the distance between the sample (WE) and the microelectrode (CE) is comparable to the radius of the micro-CE [START_REF] Wuu | High Resolution deposition of polyaniline on Pt with the scanning electrochemical microscope[END_REF]. Similarly, by placing a planar CE in parallel with a planar WE at short distance, the current distribution on WE which reflects the local electrochemical reactivity, would also cause inhomogeneous current distribution on CE. Owen group [START_REF] Brace | A parallel optical screen for the rapid combinatorial electrochromic analysis of electrochemical materials[END_REF] reported that the color of WO3 electrochromic CE locally changes following the exposed area of WE, and the absorption corresponds to the catalytic reactivity of the WE. They used WE array, in which each electrode had 1 mm diameter and the gap was 3 mm between adjacent electrodes.

In this work, we attempt to extend the concept of visualizing electrochemical reactions on WE indirectly through the optical change of the CE. The experimental configuration consists of a planar WE (interdigitated electrode, microelectrode array, and micro-Au disk sputtered on ITO), a WO3-coated ITO as CE, and pure water as electrolyte. The two electrodes are separated by a PTFE spacer to control the distance. This would get rid of potential interference from the inhomogeneity of filter paper used in literature [START_REF] Brace | A parallel optical screen for the rapid combinatorial electrochromic analysis of electrochemical materials[END_REF]. The two-electrode electrochemical cell is fixed on an optical microscope, where a complementary metal oxide semiconductor (CMOS) camera is synchronized with potentiostat for capturing the dynamic color change of the CE. The latter is observed to be correlated with the electrochemical reactivity distribution of WE.

Experimental section

Materials and electrode preparation

Hydrogen peroxide (H2O2, 30%), ammonium hydroxide (NH4OH, 25%), tungsten powder (W, 99.99%), and platinum on carbon powder (Pt/C, 10 wt.% loading) were purchased from Sigma-Aldrich. All reagents were of analytical grade and used without further purification. Deionized water obtained from a Millipore water purification system (≥18 MΩ cm, 25 °C, Milli-Q, Millipore Co.) was used in all experiments.

The WO3 electrochromic CE was prepared by electrodeposition on ITO (Thickness: 0.5 mm, 10 Ω/sq, area: 15 × 15 mm 2 , SPI Supplies, France) from peroxo-tungstic acid (PTA) solution. The latter was prepared by dissolution of 1.8 g of W metallic powder in 20 ml of H2O2 and 20 mL of deionized water [START_REF] Deepa | Electrodeposited tungsten oxide films: annealing effects on structure and electrochromic performance[END_REF]. The mixture was constantly stirred for 24 h in a cold bath (between 0 and 10 ℃). Then, 0.05 g of Pt/C powder was added into the colorless solution to catalyze the decomposition of excess H2O2 under string until obtaining a yellow-colored solution. Next, the obtained PTA solution was further centrifuged at 7000 rpm for 8 min in order to remove the precipitate, while the supernatant was recovered and kept in the refrigerator for further use. The electrodeposition was carried out in a conventional three-electrode system, with ITO as WE, an Ag/AgCl commercial RE, and a Pt wire as CE. Before electrodeposition, the ITO surface was cleaned by sonication in acetone, ethanol, H2O2/NH3•H2O (2:1 vol. ratio) and water. The WO3 film was electrodeposited by scanning cyclic voltammetry between 0 and -0.6 V (vs. Ag/AgCl) at a scan rate of 2 mV/s for 10 cycles. After electrodeposition, the film was rinsed with water and dried in air. The as-prepared WO3 film showed dark blue color due to the electrochemical reduction, but it was rapidly oxidized and bleached after a few seconds of exposure in air [START_REF] Mansouri | Ab initio investigation on the effect of transition metals doping and vacancies in WO3[END_REF]. The WO3-coated ITO was stored under ambient condition for use.

The WEs studied include interdigitated Au electrodes (200 µm band with 200 µm gap, Metrohm), Au-disk array on ITO and Au microelectrode array. The Au-disk array was deposited by sputtering Au on masked ITO for 120 s at 35 mA current to achieve thickness of ca. 60 nm (Quorum sputter coater, Q150 TS). The mask was printed on SLA 3D printer (Asiga) with 200 μm diameter holes, and was removed after sputtering.

The Au microelectrode array was prepared by sealing Au wires (25 µm diameter, 99.9%, ChemPur, Germany) in UV-cured resin. First, the structural body of the array with holes of ca. 150 µm diameter was 3D printed from UV DLP resin. The Au wires were then inserted in the holes under stereomicroscope, and the holes were closed by filling the same resin for printing. Afterwards, the structure was cured in UV for at least 1 hour.

The electrical connection of the Au wires was made by attaching to a Cu sheet with silver epoxy (M.G. Chemicals, UK). The final step was to polish the Au microelectrode array on 0.1 μm abrasive disc (Buehler, USA) to ensure a smooth surface with Au wires well exposed.

Electrochemical cell assembly for optical imaging

The electrochemical cell was assembled in a sandwich configuration by pressing the WE, the spacer and the WO3/ITO CE together. The electrodes were fixed in a 3D printed frame by two magnets (Fig. 1A). Polytetrafluoroethylene (PTFE) spacers of thickness 0.005 mm -0.1 mm (Goodfellow, UK) were used to separate WE from CE and control the distance between them. A square hole of 2 mm × 2 mm was made for filling the deionized water electrolyte. After assembly, the electrochemical cell was placed under optical microscope for imaging. The optical imaging system is illustrated in Fig. 1B. A red LED of 623 nm wavelength was used as light source (Thorlabs). The objectives were 4× (NA = 0.10) and 10× (NA = 0.25) from Olympus, Japan. The imaging was carried out by a CMOS camera (Thorlabs) and a cubic elevator platform to adjust the spatial position of the sample device.

Results and discussion

The reactions occurring at a counter electrode cannot be neglected when the CE is placed close to WE, and this will affect the electrochemical response of the cell. Thus, for analyzing such system, one needs to consider the electrochemical reactions on both WE and CE. Moreover, when the WE and CE are different, either in size or in reactivity, the current distribution shall also be taken into consideration. For example, in direct mode SECM, the microelectrode determines the current distribution on the large-scale substrate and localizes the reactions [START_REF] Wuu | High Resolution deposition of polyaniline on Pt with the scanning electrochemical microscope[END_REF]. Another example is Pt electrocatalyst on carbon substrate. The Pt may locally catalyze the electrochemical oxidation of methanol, yielding higher current density which triggers deeper color change of an electrochromic CE [START_REF] Brace | A parallel optical screen for the rapid combinatorial electrochromic analysis of electrochemical materials[END_REF].

The current distribution can be analyzed by Nernst-Planck Equation, which considers the ohmic drop of electrolyte, the reaction kinetics on both WE and CE, and the mass transport of the reactive species [START_REF] Popov | A mathematical model of the current density distribution in electrochemical cells[END_REF][START_REF] Liu | 2-D mathematical modeling for a large electrochromic window-Part I[END_REF]. Depending on the system, it can be simplified by neglecting some of the factors. The simplest case is primary current distribution, which assumes the electrochemical reactions on WE and CE interfaces very fast (Nernst boundary condition) and thus considers only the ohmic drop of electrolyte. The secondary current distribution takes into account the reaction kinetics of WE and CE. In most cases, the kinetics depend on the concentration of reactive species near the electrodes, so the mass transport shall also be coupled. The tertiary current distribution further adds the migration of species, which could be non-negligible when using weakly supported electrolyte.

Here, we demonstrate an example where an interdigitated Au WE is placed in parallel with a WO3 electrochromic CE. Deionized water is used as electrolyte. By applying an anodic bias on WE, H2O is oxidized on WE and WO3 is reduced on CE. The scheme is shown in Fig. 2A. The reactions are depicted as follows:

WE: 2H2O → O2 + 4H + + 4e - CE: WO3 + 𝑥H + + 𝑥e-→ H𝑥WO3
The reduced form HxWO3 is blue-colored and can be visualized by naked eyes or under optical microscope. Fig. 2B illustrates the color change of the system during a linear sweep voltammetry scan. At 1.3 V, the current is very low (ca. 3.85 µA) and the WO3 CE stays transparent. This suggests that the reactions on both WE and CE are negligible.

As the applied voltage increases beyond 1.5 V, the current sharply increases and the WO3 CE starts to turn blue. Moreover, the blue strips on CE fall almost exactly above the Au strips on WE. This indicates that the current distribution on CE follows that on WE. In another word, the shape of WE can be "projected" to CE. Linear sweep voltammetry of the electrochromic device from 0 to 2.0 V at a scan rate of 0.05 V/s, the insets: electrochromic photos of CE at 1.3 and 1.6 V.

The color change of CE was quantitatively measured under optical microscope with a CMOS camera. The electrochemical cell was fixed under optical microscope with the optical focus on the WE. When scanning the voltage from 0 to 2 V (WE vs. CE), images were taken at various moments, which correspond to different applied voltage thanks to the synchronization. According to the scheme in Fig. 1B, the light irradiates the WE surface and reflects back to the camera. In the optical pathway, it goes through CE and electrolyte for two times (incite and reflected light). Assuming that the optical properties of the WE and the electrolyte do not change during the experiment, the light intensity captured by the camera reflects only the electrochromic effect of the CE. The transmittance of CE can thus be derived by dividing the gray value of the colored state by that of the bleached state. In this work, the image taken at 0 V was considered as bleached state for the normalization of gray values. It is seen that the normalized image at 1.3 V shows no strip (Fig. 3A), which agrees well with no visible color change in Fig.

2B.

As the applied voltage increases to 1.6 V, the strips start to appear (Fig. 3B). This suggests local coloration of the CE in the areas above the Au strips, which differs from those between them. As the applied voltage further increases to 1.9 V, the contrast between colored strips and the gaps becomes higher (Fig. 3C). Fig. 3D The color change of CE was also measured as a function of time by applying a constant voltage between WE and CE. Fig. 4 shows the images captured at 1, 5 and 15 s when applying 1.9 V on WE vs. CE. It is seen that the normalized gray value of the colored strip, which corresponds to the Au strip on WE, gradually decreases with time.

Moreover, the areas above the gap between Au strips also start to be colored after ca. 5 s. Considering that the distance between WE and CE is 25 µm, even 1 s would be sufficient for the H + generated on WE diffusing to the CE. Thus, the enhancement in coloration with time is likely related to the kinetics of WO3 coloration on CE, which is known to be relatively slow in the order of a few seconds [START_REF] Bohnke | Proton insertion into thin films of amorphous WO3: kinetics study[END_REF]. However, as the coloration of WO3 is driven by intercalation of H + in the film, the lateral propagation of H + may result in the expansion of colored strips at long time. This may explain the decrease in the normalized gray values, or the increase in coloration, of the gap with time (Fig. 4C). As the local color change on CE is determined by the current distribution in the electrochemical cell, the distance between WE and CE is likely to play an important role. Fig. 5 shows the images taken at 1.5 V for 5 s when using a spacer of different thickness. It can be seen that with a distance of 0.2 mm from WE and CE, there is no visible color strip on CE (Fig. 5A), while reducing the thickness of the PTFE spacer, strip starts to appear (Fig. 5B). Moreover, the current response decreases as the WE-CE distance increases (Fig. S1), which could be explained by the resistivity of the electrolyte as well as the longer distance for mass transport of H + . The normalized gray values of both the strip and the gap are homogeneous as the distance increases up to 0.2 mm (Fig. 5C), meaning that the color change of CE becomes less significant. When the distance is less than 0.05 mm, the normalized gray value of the strip becomes significantly lower than that of gap. This suggests that WE and CE being close is more favorable for projecting the current distribution on WE to CE. The imaging of CE is further challenged with higher spatial resolution using Au microelectrode array as WE. Fig. 6A shows the optical image of the microelectrode array, where each Au microelectrode has a diameter of 25 μm and the gap between adjacent electrodes varies from 20 to 250 μm. By scanning the voltage between WE and CE from 0 to 2 V with 10 µm WE-CE distance, the current shows a sharp increase at ca. 1.5 V, which corresponds to the onset of electrochemical oxidation of water (Fig. S2).

Meanwhile, the color above the areas of Au microelectrodes also starts to change, and the coloration becomes more significant as the applied voltage further increases (Fig. 6B). Note that electrodes 7 and 8, which are located very close to each other (20 µm), become less distinguishable at high voltage (Fig. 6C). This lateral propagation of color is consistent with the observations on Au interdigitated electrodes. It suggests that the distance between adjacent active spots on WE would affect the electrochromic imaging from CE. Experimentally, by controlling proper voltage, it is still possible to visualize The examples above, i.e. Au interdigitated WE and Au microelectrode array, are partially conductive. They could be considered as extreme cases where the WE has high contrast between reactive (Au electrodes) and non-reactive (insulator) areas, yet a question arises whether the difference originates from electrical conductivity rather than electrochemical reactivity. To address this, we demonstrate that the visualization from CE could also be extended to samples with different conductive materials. By scanning voltage of an Au-sputtered ITO electrode (photo shown in Fig. 7A) against a WO3 CE with 25 μm distance and H2O as electrolyte, images are acquired as shown in Fig. 7B.

After normalizing images with that taken at open circuit voltage (ca. 0.005 V), the micro-disks are almost invisible on the CE when the applied voltage is lower than ca.

1.0 V. When the voltage rises to ca. 1.1-1.3 V, the micro-disks become distinguishable.

It is interesting to see that the edges of the disks have lower normalized gray values than the centers and the edges (Fig. 7C). This will be discussed later. As the voltage increases to higher than 1.4 V, the color change of CE over the disks is more significant with a decrease in normalized gray value, and it also becomes more uniform. This suggests that higher current is passing through Au than through ITO, which could be related to their different electrocatalytic reactivity for the oxidation of water. However, when the voltage is further increased to 1.8 V, the gaps between Au disks also start to change color. This could be attributed to two possible effects: one is the lateral charge propagation due to higher current and lateral mass transport of H + , which is similar as observed from partially conductive samples (Fig. 3 and4); the other is the reduced contrast in electrocatalytic reactivity between Au and ITO, as ITO could also catalyze water oxidation at high potentials. Throughout the work, the light intensity (gray value) captured by the camera is systematically normalized with that measured from WO3 CE at bleaching state. The normalized gray value can be linked to the transmittance of the WO3 CE only under preassumptions that the transmittance of water and the reflectance of WE remain unchanged during the measurement. The former is acceptable when the generation of gas bubbles is insignificant. This is why we did not test higher voltage. However, the latter could be questionable as the optical properties of Au surface were reported to vary under applied potential [START_REF] Shi | Electrochemical impedance imaging on conductive surfaces[END_REF]. Fig. 8 shows a control experiment for Fig. 7 where the WO3 CE was replaced by bare ITO, which is considered non-electrochromic before its irreversible reduction. It is seen that the current is much lower at the same applied voltage, and the redox peaks disappear. This could be explained by the removal of redox reactions of WO3. The optical images with normalized gray values look very similar for different applied voltage. Yet, the edges of Au disks could still be distinguished especially when the voltage is high, although it has much less contrast to the Au disk and the gap as compared with Fig. 7. One should be very careful in interpreting this edge effect. On one hand, the optical pathway at the junction between Au and ITO is very complex due to their different reflectivity. This might cause some artifacts in imaging. On the other hand, it could also be electrochemically much more active due to Schottky effect [START_REF] Wu | Proton transport enabled by a field-induced metallic state in a semiconductor heterostructure[END_REF]. This would result in an improvement in catalyzing water oxidation, and could explain the appearance of normalized gray value change at edges prior to Au disks in Fig. 7. The electrochromic reaction of WO3 CE requires the intercalation of H + (Fig. 2A). A question then arises: Can pure water provide enough H + for the color change? With 10 -7 mol/L concentration in pure water, it is clearly not enough for inducing the clear color change observed in this work. However, one should consider the oxidation of water on WE that generates H + . The H + generated on WE can easily reach CE with mass transport since the two electrodes are positioned close to each other. It is similar as generation-collection mode of SECM [START_REF] Sánchez-Sánchez | Scanning electrochemical microscopy. 60. quantitative calibration of the SECM substrate generation/tip collection mode and its use for the study of the oxygen reduction mechanism[END_REF]. Note that the electrolyte is weakly supported, migration could also play an important role. Considering that H + is positively charged, migration facilitates its transfer from WE to CE when applying positive bias on WE. This is beneficial for localizing the current distribution on CE.

The direct projection of WE reactivity to CE color change is only an ideal case. The lateral charge propagation on CE is unavoidable, and even a more general situation. It could occur when applying high voltage (Fig. 3), or applying voltage for long time (Fig. 4), or having two reactive spots close to each other on WE (Fig. 6). While the former two could still be optimized with experimental conditions, the latter depends on the property of the sample thus may limit the application of the method. A better way would be the quantitative analysis of the reactivity of WE from the optical signals on CE by modeling. This is beyond the scope of the present work, but let's discuss briefly some prerequisites. The model should consider the electrochemical reaction kinetics on both WE and CE, the mass transport of species in the electrolyte including diffusion and migration, and the lateral boundary conditions. Mathematically, two types of lateral boundary conditions are convenient to work with. One is symmetry or no-flux, where the lateral flux of species at boundary is zero. This is applicable for the cell boundary regulated by PTFE spacer, and also for highly symmetrical systems like Au interdigitated electrodes. Another type is semi-infinite diffusion, where the concentration of species always equals to the initial condition (bulk) at the boundary. This is applicable for studying reactive sites that are well apart, with an inert background, such as points 1-6 in Fig. 6.

From the experimental aspect, several improvements can also be foreseen. The optical signal acquisition could be carried out at multiple wavelengths to improve the sensitivity in capturing the electrochromic change of CE. The electrochromic material for CE could also be optimized for higher color switching speed. This could eliminate the kinetic effect of CE by replacing with Nernst boundary condition, which would greatly ease the analysis of transient behavior of WE.

Conclusion

Fig. 1 .

 1 Fig. 1. (A) The photos of cell. (B) Schematic diagram of the imaging system.

Fig. 2 .

 2 Fig. 2. (A) Schematic diagram of the reaction mechanism of a gold interdigitated WE

  shows the variation of normalized gray values across the strips (in horizontal direction in the images). The normalized gray values of the areas above Au strips decrease as the applied voltage increases, indicating that the absorption of WO3 CE increases. Although the normalized gray values of the areas above gaps between Au strips also decrease obviously after 1.6 V, it could be neglected compared to the normalized gray values of the strips. Moreover, the width of the colored strips is almost identical to that of the Au interdigitated electrodes. This indicate that one may identify conductive areas of WE from the local color change of electrochromic CE.

Fig. 3 .

 3 Fig. 3. (A-C) the images for optical intensity changes on the CE at a voltage of 1.30 V,

Fig. 4 .

 4 Fig. 4. The effect of time on the color change on the CE under a voltage of 1.9 V at

Fig. 5 .

 5 Fig. 5. The effect of distance between WE and CE on the optical change of CE when

Fig. 6 .

 6 Fig. 6. (A) Optical micrograph of Au microelectrode array. The scale bar (white): 50

Fig. 7 .

 7 Fig. 7. (A) Optical micrograph of Au disks sputtered on ITO. Scale bar (white): 100 μm. (B) The images for optical change of CE with increasing voltage at the WE made of sputtered Au array on ITO (Scan rate: 0.05 V/s). Scale bar (white): 200 μm. (C) The variation of normalized gray value of the two micro-disks on CE (the yellow dotted frame in Fig. 7B) with increasing the voltage. The edges (1-4) correspond to the edges of the two micro-Au disk on the WE in Fig. 7A, respectively.

Fig. 8 .

 8 Fig. 8. The contrast experiment: the optical change of the cell device (bare ITO as CE)

  With a WO3 functionalized CE in parallel and close to the surface of a WE, the reactivity distribution on WE can be successfully projected to the electrochromic CE in the form of local color change, which is then captured by optical imaging. Model systems involving Au interdigitated electrodes, Au microelectrode array, or Au sputtered on ITO as WE, and WO3 film as CE, are demonstrated. Pure water is used as electrolyte to have high resistance, and the generation of H + by oxidation on WE. The latter is essential for driving the coloration of WO3 CE. The WE is placed in parallel to CE with PTFE spacer controlling the distance. It is seen that the distance needs to be comparable as the size of the target of observation. The effect of voltage and time are studied for inhibiting the lateral color dispersion on CE. This preliminary work opens the door for mapping the reactivity of WE by the optical change of CE. As the signals are captured from the CE, the method does not require specific choice or modification of neither the WE nor the electrolyte. This could be advantageous over optical methods based on direct observation of WE. Possible improvement in both analytical and experimental aspects are discussed.

Acknowledgement

Qiao Liu acknowledges Chinese Scholarship Council for supporting the Ph.D. study.

The financial support from LCPME is highly appreciated.