Tauqir Nasir

Alonso ||

Gamero-Quijano

Christelle Despas

Manuel Dossot

Grégoire Herzog
email: gregoire.herzog@univ-lorraine.fr

Alain Walcarius
email: alain.walcarius@univ-lorraine.fr.

Signal amplification by electro-oligomerisation for improved isoproturon detection

Keywords: Stripping Voltammetry, Modified Electrode, Mesoporous Silica Film, Raman Mapping, Aniline Oxidation

A novel concept is introduced for signal amplification in electrochemical sensing: the electrooligomerisation stripping voltammetry, which has been applied here to the improved detection of the isoproturon herbicide in spring waters as a proof-of-principle. It involves a potentiostatic accumulation step onto a glassy carbon electrode (at +1.5 V vs Ag/AgCl reference electrode for 300 s) leading to the formation of an oligomeric film, which is then detected by cathodic stripping square wave voltammetry (SWV). The presence and composition of the film are confirmed by confocal Raman spectroscopy. Its characterisation by cyclic voltammetry demonstrates the reversible nature of the electrodeposited material, confirming its interest for sensitive detection by SWV. Adding a mesoporous silica membrane with vertically oriented nanochannels further enhances the sensitivity of the sensor, exhibiting a linear response in the 10-100 µM concentration range. This effect was even more interesting for real media analysis thanks to the permselective properties of such nanoporous coating in rejecting interferences and/or surface fouling agents. The method should be applicable to other analytes that are usually not detectable by conventional accumulation/stripping voltammetry.

Introduction

The improvement of sensitivity and limits of detection of electrochemical sensors is part of the challenges of analytical chemists. Accumulation (or preconcentration) steps have been implemented to electrochemical techniques to push further the analytical performances of electrochemical sensors and match the concentration in real sample measurements. Stripping methods are among the most common strategies for preconcentration [START_REF] Wang | Stripping Analysis: Principles, Instrumentation and Applications[END_REF][START_REF] Brainina | Electroanalytical Stripping Methods[END_REF][START_REF] Cheraghi | A Review: Stripping Voltammetric Methods as a High Strategy for Trace Analysis of Ions, Pharmaceutical and Food Samples[END_REF][START_REF] Economou | Recent developments in on-line electrochemical stripping analysis-An overview of the last 12 years[END_REF]. They consist of the accumulation of the target analyte on the electrode surface by a faradaic process of anodic or cathodic nature, which is then followed by the reverse reaction under electrochemical monitoring. The quantitative analysis is provided by either stripping voltammetry or potentiometric stripping analysis, which signals are proportional to the concentration of the target analytes. Stripping methods are particularly suited for the detection of metal ions in aqueous samples [START_REF] Borrill | Addressing the practicalities of anodic stripping voltammetry for heavy metal detection: A tutorial review[END_REF][START_REF] Pujol | Electrochemical sensors and devices for heavy metals assay in water: The French groups' contribution[END_REF][START_REF] Niu | Review: Electrochemical Stripping Analysis of Trace Heavy Metals Using Screen-Printed Electrodes[END_REF][START_REF] Wang | Stripping-based electrochemical metal sensors for environmental monitoring[END_REF]. Unfortunately, many analytes (e.g. organic molecules) cannot be accumulated at the electrode surface by a reversible faradaic process and some approaches have been developed, such as adsorptive stripping voltammetry [START_REF] Vire | Adsorptive stripping voltammetry applied to drug analysis: A powerful tool[END_REF][START_REF] Van Den | Adsorptive cathodic stripping voltammetry of trace elements in sea water[END_REF][START_REF] Aleksic | Application of Adsorptive Stripping Voltammetry for the Determination of Selected Methoxyimino Cephalosporins in Urine Samples[END_REF][START_REF] Arrigan | Electrochemistry of proteins at the interface between two immiscible electrolyte solutions[END_REF], electrochemical generation of sparingly-soluble complexes prior to stripping analysis [START_REF] Fojta | Electrochemical Stripping Techniques in Analysis of Nucleic Acids and their Constituents[END_REF], or extractive stripping voltammetry [START_REF] Sys | Extractive Stripping Voltammetry at Carbon Paste Electrodes for Determination of Biologically Active Organic Compounds[END_REF]. Insertive stripping voltammetry has been also proposed for the detection of non-redox ions [START_REF] Suherman | Electrochemical Determination of Sodium Ions Using an Iron(III) Phosphate-Modified Electrode[END_REF]. Nevertheless, new strategies still need to be devised to achieve improved sensitivities, in particular for the detection of soluble contaminants in the environment.

For this study we selected isoproturon (N,N-dimethyl-N'-[4-(propan-2-yl)phenyl]urea) an electroactive herbicide [START_REF] Li | Molecularly Imprinted Electrochemical Luminescence Sensor Based on Enzymatic Amplification for Ultratrace Isoproturon Determination[END_REF][START_REF] Noyrod | The simultaneous determination of isoproturon and carbendazim pesticides by single drop analysis using a graphene-based electrochemical sensor[END_REF][START_REF] Manisankar | Utilization of sodium montmorillonite claymodified electrode for the determination of isoproturon and carbendazim in soil and water samples[END_REF][START_REF] Manisankar | Determination of pesticides using heteropolyacid montmorillonite clay-modified electrode with surfactant[END_REF][START_REF] Manisankar | Utilisation of polypyrrole modified electrode for the determination of pesticides[END_REF][START_REF] Manisankar | Electroanalysis of some common pesticides using conducting polymer/multiwalled carbon nanotubes modified glassy carbon electrode[END_REF], which can be oxidised to 4-isopropylaniline (4-(propan-2yl)aniline) [START_REF] Manisankar | Utilization of sodium montmorillonite claymodified electrode for the determination of isoproturon and carbendazim in soil and water samples[END_REF]. Isoproturon is mainly used in cereal crop production to prevent the growth of broad leaf weeds. It is water soluble with a half-life of up to one year in groundwater and up to 40 days in surface water [START_REF] Kulshrestha | The photochemical decomposition of the herbicide isoproturon[END_REF]. Isoproturon and its metabolites have been identified as carcinogenic and it has been banned in the European Union since 2016 due to its tendency to disrupt the endocrine system [START_REF] Erickson | EU bans endocrine disrupting herbicides[END_REF]. We propose here to use the electrochemical activity of 4isopropylaniline, product of the isoproturon oxidation, to improve the detection of isoproturon. Indeed, aniline derivatives are known to form electroactive polymers on the surface of electrodes [START_REF] Huang | Polyaniline , a Novel Conducting Polymer[END_REF]. Furthermore, protection of the electrode surface can be essential in terms of environmental sensing. In recent years, mesoporous silica film (MSF) electrodes have shown very promising results in the field of electrochemical sensors and separation [START_REF] Walcarius | Mesoporous Materials-Based Electrochemical Sensors[END_REF][START_REF] Yan | Vertically ordered silica mesochannel films: electrochemistry and analytical applications[END_REF][START_REF] Walcarius | Silica-based electrochemical sensors and biosensors: Recent trends[END_REF][START_REF] Zhou | Silica Nanochannel Membranes for Electrochemical Analysis and Molecular Sieving: A Comprehensive Review[END_REF][START_REF] Etienne | Molecular Transport into Mesostructured Silica Thin Films: Electrochemical Monitoring and Comparison between p 6 m , P6 3 / mmc , and Pm 3 n Structures[END_REF][START_REF] Nasir | Critical Effect of Film Thickness on Preconcentration Electroanalysis with Oriented Mesoporous Silica Modified Electrodes[END_REF][START_REF] Nasir | Mesoporous Silica Thin Films for Improved Electrochemical Detection of Paraquat[END_REF][START_REF] Vilà | pH-modulated ion transport and amplified redox response of Keggin-type polyoxometalates through verticallyoriented mesoporous silica nanochannels[END_REF]. Such membranes can be prepared by either electrochemically assisted self-assembly (EASA) [START_REF] Walcarius | Electrochemically assisted selfassembly of mesoporous silica thin films[END_REF][START_REF] Goux | Oriented Mesoporous Silica Films Obtained by Electro-Assisted Self-Assembly (EASA)[END_REF] or by Stöbersolution growth [START_REF] Teng | Highly Ordered Mesoporous Silica Films with Perpendicular Mesochannels by a Simple Stöber-Solution Growth Approach[END_REF], magnetically induced orientation [START_REF] Yamauchi | Magnetically induced orientation of mesochannels in mesoporous silica films at 30 Tesla[END_REF], epitaxial growth [START_REF] Richman | Vertically oriented hexagonal mesoporous films formed through nanometre-scale epitaxy[END_REF] or via an oil-induced co-assembly process [START_REF] Kao | A General Method for Growing Large Area Mesoporous Silica Thin Films on Flat Substrates with Perpendicular Nanochannels[END_REF]. The EASA method generally leads to the formation of defect-free film with a controlled thickness ranging from 50 to 150 nm [START_REF] Goux | Oriented Mesoporous Silica Films Obtained by Electro-Assisted Self-Assembly (EASA)[END_REF], and with pore size adjustable between 2 and 3 nm [START_REF] Vilà | Molecular Sieving with Vertically Aligned Mesoporous Silica Films and Electronic Wiring through Isolating Nanochannels[END_REF]. MSF have been used as molecular sieves, excluding molecular probes with dimensions greater than the pore diameter [START_REF] Vilà | Molecular Sieving with Vertically Aligned Mesoporous Silica Films and Electronic Wiring through Isolating Nanochannels[END_REF][START_REF] Lin | Ultrathin Silica Membranes with Highly Ordered and Perpendicular Nanochannels for Precise and Fast Molecular Separation[END_REF][START_REF] Serrano | Mesoporous silica thin films for molecular sieving and electrode surface protection against biofouling[END_REF]. Taking advantage of this size selectivity, the exclusion of small proteins have protected the electrode surface from fouling [START_REF] Serrano | Mesoporous silica thin films for molecular sieving and electrode surface protection against biofouling[END_REF][START_REF] Sun | Anti-Biofouling Isoporous Silica-Micelle Membrane Enabling Drug Detection in Human Whole Blood[END_REF].

We investigated here the possibility to harness the electrochemical oxidation of 4isopropylaniline to form a redox active film, which can be used as an analytical signal for the detection of isoproturon. The mechanism of the formation of the redox active film was studied by cyclic voltammetry. With a sufficiently high oxidation potential, redox-active species were deposited on the electrode surface and were then analysed by confocal Raman spectroscopy.

Their identification was confirmed by theoretical Raman spectra calculated by Density Functional Theory (DFT). A calibration curve for isoproturon was then established using these redox active species electrodeposited through a mesoporous silica film, whose function was to protect the electrode surface from fouling.

Materials and methods

Chemicals and reagents

Tetraethoxysilane (TEOS, 98%, Alfa Aesar), [START_REF] Cheraghi | A Review: Stripping Voltammetric Methods as a High Strategy for Trace Analysis of Ions, Pharmaceutical and Food Samples[END_REF]

GCE modification with mesoporous silica films

Glassy carbon plates (SIGRADUR G plates, 20 mm × 10mm × 1 mm) were purchased from HTW (Germany) and used after polishing with alumina (3, 1, and 0.05 μm) on a polishing cloth followed by ultrasonication for 5 minutes in distilled water. Electrografting of APTES followed by mesoporous silica film formation was performed according to previously described methods [START_REF] Nasir | Mesoporous Silica Thin Films for Improved Electrochemical Detection of Paraquat[END_REF][START_REF] Nasir | Electrografting of 3-Aminopropyltriethoxysilane on a Glassy Carbon Electrode for the Improved Adhesion of Vertically Oriented Mesoporous Silica Thin Films[END_REF]. APTES electrografting is performed to increase the mechanical stability of mesoporous silica films at carbon electrodes. Sol for the silica film formation was prepared by mixing ethanol and aqueous NaNO3 (0.1 M) in a 1/1 v/v ratio. TEOS and CTAB (100 and 32 mM, respectively) were added to this solution and pH adjusted to 3 using hydrochloric acid.

This solution was stirred for 2.5 hours for proper hydrolysis and film was electrochemically deposited using galvanostatic condition (by applying current density j, of -0.74 mA cm -2 for 30s) [START_REF] Goux | Oriented Mesoporous Silica Films Obtained by Electro-Assisted Self-Assembly (EASA)[END_REF]. Hexaammine-ruthenium(III) was used as redox probe for cyclic voltammetric characterization of electrode modifications. Template was removed from silica films using 0.1 M HCl in ethanol and stirring moderately for 1 hour.

Electrochemical methods

All experiments were carried out using three electrode system. Experiments involving aqueous solutions were performed using glassy carbon electrode (GCE) as working, stainless steel rod as counter and Ag/AgCl/1 M KCl (purchased from Metrohm, Switzerland) as reference electrodes. Silver wire was used as pseudo reference for APTES electrografting and silica film deposition. Galvanostatic experiments were performed with a PGSTAT 100 apparatus from Ecochemie (Metrohm Autolab, Switzerland). Cyclic and square wave voltammetry measurements were carried out using an EMStat2 apparatus. Square wave voltammetric conditions used were: E deposition 1.5 V, T deposition 300 seconds, equilibrium time 5

seconds, E step 2 mV, Amplitude 50 mV and Frequency 10 Hz.

Raman spectroscopy

Raman confocal microscopy was performed using a Renishaw Qontor set-up equipped with a laser wavelength of 785 nm, a ×100 objective with 0.85 of numerical aperture, a Renishaw Centrus 0LL829 Peltier-cooled CCD detector stabilized at -70°C. The laser output power was 100 mW and sent through a neutral density filter of 0.0001% of transmission to ensure a maximum laser irradiance below 250 mW.cm -2 . For such experiments, the electrode material was changed to polycrystalline gold since glassy carbon electrodes gave strong sp 2 carbon bands at 1350 cm -1 (D-band) and 1590 cm -1 (G-band), which overlapped the Raman peaks of the deposit. The sample was found sensitive to laser heating effect and this illumination condition was optimum to obtain spectra with good signal to noise ratio and no detectable heating effect. The 1200 grooves per millimeter grating enabled a spectral resolution close to 1.5 cm -1 . For each point in the Raman maps, 5 spectra of 4 s of acquisition were averaged. A separation distance of 1 µm was used (both in X and Y directions) between each data points in the map. To spare time, only experimental data between 800 and 1800 cm -1 were collected, which corresponded to the spectral range available with the 1200 gr mm -1 grating at 785 nm without changing the grating position and increasing the acquisition time. Since most of the spectral changes were expected in this region (according to calculated DFT spectra), it was the best strategy to maintain reasonable acquisition time for Raman maps and benefit from spectral analysis.

Theoretical Raman spectra were calculated using Gaussian 09 and GaussView [44]. The quantum mechanical modelling method from geometry optimization and vibrational frequencies calculation was the density functional theory (DFT) using the B3LYP hybrid functional and the 6-311G++(d,p) basis set. A water solvation sphere was used (with a polarizable continuum model). The frequency of the calculated Raman modes were corrected using the equations given in reference [START_REF] Osaki | Quadratic scaling functions for obtaining Normal vibrational wavenumbers from the B3LYP calculation[END_REF]. These calculated spectra were used as input spectral components to the Renishaw Wire 5.2 software to perform component analysis maps based on non-negative least squares (NNLS) regression. To do so, the experimental Raman data were background corrected and cosmic peaks were removed (using the Renishaw Wire 5.2 software). Experimental Raman spectrum intensity was normalized from 0 to 1, as well as the intensity of calculated spectra from DFT computations. A NNLS method was used to calculate the weight of each DFT-calculated spectral component at all experimental data points in the Raman maps. We used the 5 possible spectral components (i.e., chemical compound)

isoproturon, 4-isopropylaniline, di(isopropyl)azobenzene ((E)-bis[4-(propan-2- yl)phenyl]diazene), di(isopropyl)-2-aminodiphenylamine ((1Z)-4-(propan-2-yl)-N 1 -[3- (propan-2-yl)phenyl]cyclohexa-3,5-diene-1,2-diimine), di(isopropyl)phenazine (2,7- di(propan-2-yl)phenazine).
One should notice that it was impossible to find experimental data points in the Raman maps corresponding to a unique component, at least two compounds were always present in each case. However, we found some locations richer in a given compound, which enabled us to extract some spectra and compare them to DFT calculated ones. These data are shown below, with the corresponding spectral weight maps, for each compound.

Chromatographic methods and real samples analysis

Both anionic and cationic compositions of real samples were determined by ion chromatography using a Metrohm 882 Compact IC plus instrument equipped with a high- directly in these samples spiked with isoproturon at selected concentrations.

Results and discussion

Oxidation of isoproturon

Figure 1 illustrates the electrochemical behaviour of isoproturon, as investigated by cyclic voltammetry at a bare GCE. On the first cyclic voltammogram (inset of Figure 1), an oxidation peak (noted I on Figure 1) at +0.93 V was observed and followed by an increase of the anodic current (noted II). The cathodic wave at + 0.35 V (noted III) was not apparent on the first scan.

In the subsequent cycles, a pair of peaks (III and III') rose continuously at +0.35 V (III) and +0.40 V (III') from the second scan up to 100 scans. Finally, a second pair of peaks rose at +0.21 V (IV) and +0.26 (IV'). Peak I was attributed to the oxidation of isoproturon [START_REF] Manisankar | Utilization of sodium montmorillonite claymodified electrode for the determination of isoproturon and carbendazim in soil and water samples[END_REF][START_REF] Manisankar | Determination of pesticides using heteropolyacid montmorillonite clay-modified electrode with surfactant[END_REF][START_REF] Manisankar | Utilisation of polypyrrole modified electrode for the determination of pesticides[END_REF][START_REF] Manisankar | Electroanalysis of some common pesticides using conducting polymer/multiwalled carbon nanotubes modified glassy carbon electrode[END_REF],

which led to the formation of 4-isopropylaniline. This product can be subsequently oxidised (peak II) to a radical cation leading to further oxidation [START_REF] Huang | Polyaniline , a Novel Conducting Polymer[END_REF]. The shape of peak I changed with the number of scans. At first, peak I seemed to be the result of an oxidative process controlled by diffusion of species in solution, before changing to an adsorption peak A magnification of this potential region is shown in Figure SI1. This evolution of the peak shape was probably due to a change in the electron transfer mechanism. For the first cycles, isoproturon molecules were oxidised directly at the electrode surface, whereas, once an electroactive film was formed on the electrode surface, isoproturon was oxidised through the film. Peak I potential and the onset of the electropolymerisation potential were shifted to lower values upon repetitive cycling, indicating that direct oxidation of isoproturon and subsequent oxidation required lower energy (Figure SI2). peaks III and III' were not observed. Figure 2A shows the 5 th scans of two series of cyclic voltammograms. The potential was scan up to +1.5 V, allowing the electro-oligomerisation to occur. On this curve, peaks III and III' were not as well defined as after 100 cycles but nevertheless present. When the change of potential scan direction was set at +1.2 V, these two peaks were absent (curve b on Figure 2A). As a control experiment, the electrochemical behaviour of 100 µM 4-isopropylaniline was investigated under the same conditions than isoproturon, (Figure 2B). 4-isopropylaniline was oxidised at + 1.3 V and the continuous cycling of the potential window ranging from -0.15 V to + 1.5 V led to the rise of two poorly resolved electrochemical processes at E1/2 = +0.376 V and +0.288 V (Figure 2B, curve b), which were in the similar potential region than the one observed for isoproturon. This control experiment confirmed that 4-isopropylaniline could be an intermediate product of the oxidation shown as peak I (Figure 1), which is further oxidised.

Modified electrode characterisation

The deposit formed on the electrode surface was characterised by electrochemistry and by confocal Raman microscopy. For the electrochemical characterisation, a GCE was modified by cycling 100 times between 0 and + 1.5 V in an electrolyte solution of 0.1 M NaClO4 (pH 6)

containing 100 µM isoproturon. The modified electrode was then rinsed with DI water and transferred to an electrochemical cell filled with 0.1 M NaClO4 (pH 6). Cyclic voltammogram in the absence of isoproturon (i.e., blank electrolyte solution) still showed the two pairs of peaks III/III' and IV/IV', confirming that the electrode surface has been modified and coated with an electrochemically active component through repetitive cycling (Figure 3). The increase in capacitance (measured at +0.6 V) from 5.9 µF before repetitive cyclic voltammetry, to 47.6 µF also confirmed the electrode surface modification. The peak charge for III', with a value of 24.4 µC, is much larger than the charge for IV' with a value of 8.5 µC. ), scan rate = 100 mV s -1 , of a GCE subjected to 100 CV cycles in isoproturon medium and then transferred into a blank electrolyte solution (curve a, the electrode was modified under the same conditions as in Figure 1), and of an electrode cycled in the absence of isoproturon as a control experiment (curve b, all conditions, other than isoproturon concentration were the same as curve a).

The origin of these two pairs of peaks is unlikely to be due to the presence of poly(4isopropylaniline) at the electrode surface. The peak-to-peak separation of both III/III' and IV/IV' is close to 60 mV while, in the case of polyaniline, the peak to peak separation is closer to 120 mV [START_REF] Lacroix | Electrolyte Effects on the Switching Reaction of Polyaniline[END_REF][START_REF] Gawli | 3D Polyaniline Architecture by Concurrent Inorganic and Organic Acid Doping for Superior and Robust High Rate Supercapacitor Performance[END_REF]. These peaks are more likely to be due to the formation of an electroactive film resulting from the oxidation of 4-isopropylaniline at the electrode surface than to the formation of a truly conducting polymer. Scheme 1 presents three possible redox active dimers:

1 di(isopropyl)azobenzene, 2 di(isopropyl)-2-aminodiphenylamine, 3 di(isopropyl)phenazine
that could be formed at the electrode surface. The electrochemical activity of similar aniline dimers was reported for azobenzene [START_REF] Komorsky-Lovrić | Measurements of redox kinetics of adsorbed azobenzene by "a quasireversible maximum" in square-wave voltammetry[END_REF], 2aminodiphenylamine [START_REF] Cotarelo | Synthesis and Characterization of Electroactive Films Deposited from Aniline Dimers[END_REF], and phenazine [START_REF] Bailey | Electrochemistry and Photopotentials of Phenazine in Methanol Solutions[END_REF]. The nature of the deposit formed on an electrode surface was investigated ex-situ by confocal Raman microscopy (Figure 4). The spatial distribution of the three compounds 1, 2 and 3 of Scheme 1 was shown using the weight of the NNLS analysis performed on the Raman hyperspectral data. Calculated DFT corrected spectra of compounds 1, 2 and 3 were used as spectral source (i.e. components) for the analysis.

The Raman shift of the calculated DFT spectra was corrected using experimental and theoretical spectra of isoproturon (Figure SI3). The three dimers differed from the way monomers reacted with one another. It could be through the formation of either an azo group (dimer 1), or of an amine bridge (dimer 2) or of a phenazine structure (dimer 3). A distinct Raman shift was predicted for each of these three possible bindings (Figure SI4). For dimer 1, a peak at 1142 cm -1 was attributed to stretching of the C-N bond coupled to the in-plane bending of aromatic C-H, while the C=N-C asymmetric stretching and the deformations of the phenyl rings were responsible for the peak at 1540 cm -1 in the predicted spectrum for dimer 2. The phenazine structure of dimer 3 gave a peak at 1406 cm -1 due to symmetrical C-N=C bending coupled to the stretching of the aromatic C=C. Dimers 1-3 could be differentiated from the starting monomer, 4-isopropylaniline, which showed a peak at 1620 cm -1 , caused by the NH2 symmetrical stretching and the aromatic C=C stretching. Raman peaks predicted by DFT computations for compounds 1, 2 and 3 were found in the experimental data with very good agreement. Higher weight indicated a larger quantity of the compound was found at the data point analysed in the map. It was not possible to find a location in the Raman map which corresponded to a unique chemical compound, so additional peaks coming from the other possible species were always present in addition to those assigned to the specific compound analysed in the map. However, Raman spectra proved that the intermediate species 1, 2 and 3 were present on the electrode surface with patchy spatial distributions. It also seemed that the patches for each compound were not located in the same area of the electrode. Supporting information reported the same kind of analysis for isoproturon and 4-isopropylaniline, and it showed (Figures SI4-5) that isoproturon was still present on the electrode surface, and 4isopropylaniline could also be found as an intermediate species. Therefore, Raman data agreed well with electrochemical data and confirmed well the chemical intermediates proposed to explain the mechanism of the isoproturon oxidation.

Amplified signal for isoproturon detection

In previous reports [START_REF] Noyrod | The simultaneous determination of isoproturon and carbendazim pesticides by single drop analysis using a graphene-based electrochemical sensor[END_REF][START_REF] Manisankar | Utilization of sodium montmorillonite claymodified electrode for the determination of isoproturon and carbendazim in soil and water samples[END_REF][START_REF] Manisankar | Determination of pesticides using heteropolyacid montmorillonite clay-modified electrode with surfactant[END_REF][START_REF] Manisankar | Utilisation of polypyrrole modified electrode for the determination of pesticides[END_REF][START_REF] Manisankar | Electroanalysis of some common pesticides using conducting polymer/multiwalled carbon nanotubes modified glassy carbon electrode[END_REF], electrochemical detection was based on the direct oxidation of isoproturon in acidic media (0 < pH < 1). In addition, isoproturon oxidation was an irreversible electrochemical process and was not improved by pulsed voltammetry techniques. The electrochemical behaviour of isoproturon shown in section 3.2 suggested that its electrochemical detection could be amplified through the electro-oligomerisation at the electrode surface. In addition, the reversible nature of the couple III/III' favoured the use of square wave voltammetry as the detection method. For further analytical investigation, isoproturon signal amplification was achieved through electro-oligomerisation at + 1.5 V for an optimized time of 300 s (Figure SI6) before detection by square wave voltammetry from 0.8 to 0.0 V. Square wave voltammetry were recorded for increasing isoproturon concentrations ranging from 10 to 100 µM (Figure 5A). The peak current was linear with the concentration from 10 to 80 µM before levelling off. The sensitivity achieved through signal amplification is much higher than the one with isoproturon direct oxidation (black triangles in Figure 4A). These results showed that the isoproturon signal was amplified by the electrooligomerisation, acting as a preconcentration step at the electrode surface. The signal amplification was possible at pH close to neutral, which constituted an advantage over direct oxidation methods, conducted in strongly acidic conditions [START_REF] Noyrod | The simultaneous determination of isoproturon and carbendazim pesticides by single drop analysis using a graphene-based electrochemical sensor[END_REF][START_REF] Manisankar | Utilization of sodium montmorillonite claymodified electrode for the determination of isoproturon and carbendazim in soil and water samples[END_REF][START_REF] Manisankar | Determination of pesticides using heteropolyacid montmorillonite clay-modified electrode with surfactant[END_REF][START_REF] Manisankar | Utilisation of polypyrrole modified electrode for the determination of pesticides[END_REF][START_REF] Manisankar | Electroanalysis of some common pesticides using conducting polymer/multiwalled carbon nanotubes modified glassy carbon electrode[END_REF]. The nature of the counterion (ClO4 -, NO3 -, SO4 2-and Cl -) of the supporting electrolyte was also varied and its impact on the current associated to the III/III' peaks (Figure SI7). In all cases, oxidation II occurred although peaks III and III' were ill-defined in the presence of chloride ions. The order of the oxidation currents for II with various electrolytes agrees with the literature data available for poly(aniline): SO4 2-> NO3 -> ClO4 -> Cl - [START_REF] Mandic | The influence of counter-ions on nucleation and growth of electrochemically synthesized polyaniline film[END_REF][START_REF] Ariza | Nitrate and chloride transport through a smart membrane[END_REF]. In the presence of chloride ions, oxidative current at the positive end of the potential window was much higher than with the other ions (Inset of Figure SI7). This could be due to the formation of chlorine at such potentials. Sodium nitrate was selected as a background electrolyte as nitrate ions are more likely to be found in the environment than perchlorate ions.

Furthermore, the analysis of real samples by electrochemical sensors remained challenging because of the presence of interfering species. These are responsible for non-specific adsorption and hence of the hampering of the electrochemical signal. Recent works have reported the use of electrodes modified with mesoporous silica thin films to protect the electrode surface from adsorption of interfering species [START_REF] Zhou | Silica Nanochannel Membranes for Electrochemical Analysis and Molecular Sieving: A Comprehensive Review[END_REF][START_REF] Nasir | Mesoporous Silica Thin Films for Improved Electrochemical Detection of Paraquat[END_REF][START_REF] Serrano | Mesoporous silica thin films for molecular sieving and electrode surface protection against biofouling[END_REF][START_REF] Sun | Anti-Biofouling Isoporous Silica-Micelle Membrane Enabling Drug Detection in Human Whole Blood[END_REF][START_REF] Yan | Direct electrochemical analysis in complex samples using ITO electrodes modified with permselective membranes consisting of vertically ordered silica mesochannels and micelles[END_REF][START_REF] Cheng | Simultaneous label-free and pretreatment-free detection of heavy metal ions in complex samples using electrodes decorated with vertically ordered silica nanochannels[END_REF][START_REF] Zhou | Electrochemical detection of Alzheimer's disease related substances in biofluids by silica nanochannel membrane modified glassy carbon electrodes[END_REF][START_REF] Zhou | In Vivo Monitoring of Oxygen in Rat Brain by Carbon Fiber Microelectrode Modified with Antifouling Nanoporous Membrane[END_REF]. We have investigated here the signal amplification approach described above through a mesoporous silica film (MSF) generated onto the electrode surface. Previous studies suggested that this could be possible as some conductive polymers were successfully electrodeposited through mesoporous silica thin films [START_REF] Montilla | Hybrid solgelconducting polymer synthesised by electrochemical insertion : tailoring the capacitance of polyaniline[END_REF][START_REF] Salinas-Torres | All electrochemical synthesis of polyaniline/silica sol-gel materials[END_REF][START_REF] Gamero-Quijano | Vertically Aligned and Ordered One-Dimensional Mesoscale Polyaniline[END_REF]. At the lowest isoproturon concentrations (10 <

[isoproturon] < 40 µM), there was no significant difference between the peak current achieved at a GCE (red squares on Figure 4B) and the one achieved at a MSF electrode (blue circles on Figure 4B). However, when [isoproturon] ≥ 40 µM, the peak currents at MSF electrodes were significantly greater than at the GCE. This could be explained by the higher concentration of oxidized species in the vicinity of the electrode and resulting in the electrodeposition of a larger number of oligomers at the MSF surface and hence improving the signal amplification.

Analysis of real samples

Spring water samples were analysed by ion chromatography to estimate their ionic strength.

The anion concentrations were measured at 24, 8, and 47 µM with retention times of 6.3, 9.1, and 17.4 minutes for Cl -, NO3 -, and SO4 signal was observed at the bare GCE (Figure 6A). As the isoproturon concentration increased, a peak is observed at the bare GCE but it remained significantly lower than the peak current measured at the MSF electrodes (Figure 6B). For a 70 µM isoproturon concentration, the peak current at the MSF electrode is -1.50 µA while it was -0.39 µA at the bare GCE. The greater peak observed at the MSF electrode than at the bare GCE can be attributed to a double effect.

Firstly, the MSF can act as a molecular sieve on the electrode surface, limiting electrode surface fouling. In the case of this water sample, the protection offered by the MSF is not complete as peak currents are 6-9 times lower than with clean laboratory solutions. In addition to this role of surface protection, local concentration of oxidized species within MSF might be higher than at a bare electrode and hence favour the oligomerization and the preconcentration as already described for Figure 5B. This improved signal amplification is different from the one already reported for paraquat by our group, as isoproturon is a neutral species and is not accumulated in the anionic films unlike cationic probes such as paraquat [START_REF] Nasir | Mesoporous Silica Thin Films for Improved Electrochemical Detection of Paraquat[END_REF]. The method is also conceptually different from other signal amplification approaches such as those developed in electrochemical biosensors or immunosensors [START_REF] Song | Disposable electrochemical aptasensor array by using in situ DNA hybridization inducing silver nanoparticles aggregate for signal amplification[END_REF][START_REF] Huo | Strategic Applications of Nanomaterials as Sensing Platforms and Signal Amplification Markers at Electrochemical Immunosensors[END_REF][START_REF] Cho | Current technologies of electrochemical immunosensors: Perspective on signal amplification[END_REF][START_REF] Zhang | Recent Advances on Electrochemical Sensors for the Detection of Organic Disinfection Byproducts in Water[END_REF][START_REF] Kumar | Engineered Nanomaterial Assisted Signal-amplification Strategies for Enhancing Analytical Performance of Electrochemical Biosensors[END_REF], and it offers the advantage of being simpler and does not require the use of expensive biological materials to modify the electrode surface.

Conclusions

Isoproturon is oxidized at the surface of a GCE to form 4-isopropylaniline, which is further oxidized to form electroactive active oligomers, presenting reversible electrochemical behaviour. This electrochemical signal increased with the number of cycles between -0.15 and +1.5 V, modifying the electrode surface. The presence of these products on the electrode surface was established by electrochemical studies and confirmed by ex-situ Raman spectroscopy mapping. The variety of the Raman spectra supported the products heterogeneity of the successive oxidation of isoproturon and 4-isopropylaniline at the electrode surface. Three dimers of 4-isopropylaniline were suggested as possible products of the oxidation reactions, which were confirmed by comparison of their DFT calculated and experimental spectra.

Despite the heterogeneity of the electrode surface, the amount of redox-active oxidation products, which were electrodeposited on the electrode surface, increased with time. The electrodeposition step was thus used as a preconcentration step and the electrochemical signal increased linearly with the isoproturon concentration, offering greater analytical possibilities than with direct oxidation of isoproturon. Finally, this approach was successfully tested with an electrode surface protected with a mesoporous silica film for the detection of isoproturon in spiked river water samples. This signal amplification approach could be extended to a number of methylurea-based herbicides that present a similar chemical structure (e.g. diuron, chlortoluron, fenuron, linuron, monolinuron, metobromuron, metoxuron, monuron, siduron, neburon).

  pressure pump, sequential (Metrohm CO2 suppressor MCS) and chemical (Metrohm suppressor MSM II for chemical) suppression modules and a conductivity detector. The anion separation was performed on a MetrosepA Supp 4-250/4.0 column packed with polyvinyl alcohol particles functionalized with quaternary ammonium group (9 µm particles diameter) and preceded by a guard column (Metrosep A supp 4/5 guard) and an RP 2Guard column to remove traces of organic compounds. The mobile phase consisted of a solution of Na2CO3 (1.8 mmol L -1 ) and NaHCO3 (1.7 mmol L -1 ) in ultrapure water (18.2 MΩ cm at 293 K). The flow rate was 1 mL min -1 and the sample loop volume was 20 µL. The cation separation was performed on a Metrosep C6-150/4.0 column packed with silica gel particles functionalized with carboxyl groups (5 µm particles diameter) and preceded by a guard column (Metrosep C6/4.0 guard) and an RP 2Guard column to remove traces of organic compounds. The mobile phase consisted of a solution of HNO3 (1.7 mmol L -1 ) and picolinic acid (1.7 mmol L -1 ) in ultrapure water (18.2 MΩ cm at 293 K). The flow rate was 0.9 mL min -1 and the sample loop volume was 5 µL. The MagIC Net™ 2.3 Professional chromatography software controlled the 882 Compact IC plus instrument and its peripherals. Real samples were spring waters collected in the watershed of the Strengbach River (Vosges, France) and were stored at 4°C and used for electrochemical experiments after addition of 0.1 M NaNO3 as supporting electrolyte. pH of real samples was found to be 6.4. The electrochemical stripping analyses were performed

Figure 1 :

 1 Figure 1: Repetitive cyclic voltammetry of 100 µM isoproturon in 0.1 M NaClO4 (pH 6) at a bare glassy carbon electrode, scan rate = 100 mV s -1 . Only the 1 st , 10 th , 20 th , 30 th , 40 th , 50 th , 60 th , 70 th , 80 th , 90 th , and 100 th cycles are shown for clarity reasons.

Figure 2 :

 2 Figure 2: (A) 5 th cyclic voltammograms obtained for 100 µM isoproturon in 0.1 M NaClO4 (pH 6), scan rate = 100 mV s -1 . Vertex potential was + 1.5 V (curve a in red) or + 1.2 V (curve b in orange). (B) 100 th scan after cyclic voltammetry in the presence of 100 µM isoproturon (curve a in red), 100 µM 4-isopropylaniline (curve b in blue) in 0.1 M NaClO4 (pH 6), scan rate = 100 mV s -1 . Control experiment in 0.1 M NaClO4 (in the absence of redox probe) is shown as curve c.

Figure 3 :

 3 Figure 3: Cyclic voltammogram in 0.1 M NaClO4 (pH 6), scan rate = 100 mV s -1 , of a GCE subjected to 100 CV cycles in isoproturon medium and then transferred into a blank electrolyte solution (curve a, the electrode was modified under the same conditions as in Figure1), and of an electrode cycled in the absence of isoproturon as a control experiment (curve b, all conditions, other than isoproturon concentration were the same as curve a).

Scheme 1 :

 1 Scheme 1: Possible redox active dimers formed by the 4-isopropylaniline oxidation; 1: di(isopropyl)azobenzene, 2: di(isopropyl)-2-aminodiphenylamine; 3: di(isopropyl)phenazine.

Figure 4 : 1 , 2 , and 3 of scheme 1 .

 4121 Figure 4: Confocal Raman spectroscopy obtained after modification by repetitive cyclic voltammetry in the presence of 100 µM isoproturon. Raman mapping (left) spectra for compounds 1, 2, and 3 of scheme 1. Coloured scales indicated the magnitude of the weight for each compound in the map. Spectra on the right show in black (curves b) the calculated DFT corrected spectrum of compounds 1, 2 and 3 (from top to bottom) and in red (curves a) an example of experimental spectrum in the map indicated by the white crosshair in the corresponding map on the left.

Figure 5 :

 5 Figure 5: (A) Square wave voltammetry of 10-100 µM isoproturon in 0.1 M NaNO3 at a bare GCE using direct oxidation peak (blank) and the signal amplification by electrooligomerisation. (B) Calibration curves for isoproturon (10 -100 µM) using direct oxidation

Figure 6 :

 6 Figure 6: (A) Square wave voltammetry of 20 (left) and 70 (right) µM isoproturon in a spring sample (added electrolyte: 0.1 M NaNO3) obtained at a GCE (black curve) and at MSF electrode. (B) Calibration curves for isoproturon (10 -100 µM) using signal amplification by electro-oligomerisation at GCE () and MSF electrodes (•).

  2-, respectively. Cation concentration were determined at 143 µM for Na + , 0.3 µM for K + , 50 µM for Ca 2+ and 22 µM for Mg 2+ with retention times of 7.5, 13.4, 16.2, and 23.4 min, respectively. In order to conduct electrochemical measurements, 0.1 M NaNO3 was added to the samples. Spring water samples were then spiked with various isoproturon concentrations to be analysed by the signal amplification method described in section 3.2. Both bare GC and MSF electrode were used for these analyses. For a 20 µM isoproturon concentration, a peak of -0.47 µA is measured at a MSF electrode while no

Acknowledgements

The authors thank the French PIA project "Lorraine Université d'Excellence" (Reference No. ANR-15-IDEX-04-LUE). T.N. is grateful to the Higher Education Commission of Pakistan for funding his Ph.D. We also thank Ms Claire Genois for the ion chromatography determination of ion concentrations and Dr Neus Vila for her help on the 4-isopropylaniline distillation.

Content of the supporting information section:

Characterisation of isoproturon peak oxidation by cyclic voltammetry, effect of electrolyte nature; Raman control experiments and DFT calculations; Time dependence of square wave voltammetry peak currents.


Jean-Pierre Boeuf

Masayuki Takahashi

Rotating spokes, ionization instability, and electron vortices in partially magnetized 𝑬 × 𝑩 plasmas

Regions of enhanced light emission rotating in the azimuthal direction are present in various 𝑬 × 𝑩 plasma devices. A kinetic model reveals that these "rotating spokes" are due to electron heating and enhanced ionization localized along a double layer at the interface between a region of large electric field and a quasi-equipotential region close to the anode potential. Electrons drifting along this interface are heated due to 𝛁𝐵 drift in the large electric field region. The formation of electron vortices due to the velocity shear in the double layer also contributes to electron heating. The possibility for spoke motion in the retrograde 𝑬 × 𝑩 direction as well as in the 𝑬 × 𝑩 direction, observed experimentally, can be reproduced and explained.

In plasma devices such as magnetrons 1 and Hall thrusters 2 an external magnetic field 𝑩 is placed perpendicular to the applied electric field 𝑬 to confine the electrons and allow ionization and plasma formation at low gas pressure. These devices are cylindrically symmetric, with 𝑬 × 𝑩 in the azimuthal direction to allow efficient electron confinement. In these "partially magnetized plasmas" electrons are strongly magnetized (electron Larmor radius 𝜌 𝑒 small with respect to the plasma size 𝐿 , 𝜌 𝑒 ≪ 𝐿 ) while ions are not (𝜌 𝑖 ≳ 𝐿). Turbulence and instabilities resulting in anomalous transport and in the formation of coherent structures are present in partially magnetized 𝑬 × 𝑩 plasmas as in fusion or space plasmas but the physics of these instabilities is specific due the difference in the magnetization of electrons and ions 3 and to the existence of localized ionization regions. An example of instability that has been thoroughly studied in the recent years is the Electron Cyclotron Drift Instability (ECDI). Particle-In-Cell (PIC) simulations predict that the ECDI is present and controls electron transport in the exhaust region of Hall thrusters where a large axial electric field in the quasineutral plasma extracts and accelerates positive ions [4][5][6][7] . This azimuthal kinetic instability results from electron Bernstein waves Doppler-shifted towards lower frequencies by the large electron drift velocity and was first identified in the context of collisionless shocks in space plasmas 8,9 .

A longer wavelength instability, much easier to observe experimentally, is characterized by the rotation of a region of enhanced light emission ("rotating spoke") at relatively low velocities (less than 10 km/s). Such moving macroscopic plasma non-uniformities have been reported in most 𝑬 × 𝑩 plasma devices. The first studies date back to the 1960's when rotating-plasma machines were investigated as possible devices for fusion applications 10 . The goal was to generate a large radial electric field E in the quasi-neutral plasma, for example in a coaxial electrode system with a strong axial magnetic field B. The plasma was expected to rotate at the velocity E/B. It was found experimentally that there was a maximum "burning voltage" of the discharge, i.e. a maximum plasma rotation velocity 11 . This limitation of the rotation velocity was attributed to an abnormal ionization mechanism 12 that was related to the concept of critical ionization velocity (CIV) introduced by Alfven [START_REF] Alfven | On the Origin of the Solar System[END_REF] . In a model proposed Piel, Möbius and Himmel [START_REF] Piel | [END_REF] , the anomalous ionization was associated with the formation and rotation of an azimuthal double layer where electrons were heated by a two-stream instability. More recently, a PIC simulation 15,16 of a cylindrical magnetron discharge showed similarities with this model but the reason for the presence of the double layer was not elucidated and the question of the nature of electron heating in the double layer was not clarified.

Rotating spokes have also been observed in Hall thrusters [17][18][19][20] and in pulsed planar magnetrons used in plasma processing (HiPIMS, for High Power Impulse Magnetron Sputtering) 1,21 . In some papers on Hall thrusters 17 or magnetrons 22 the CIV concept has been invoked to explain the presence of rotating spokes but there is no consensus on the physics of the rotating spokes in spite of important recent progress in the diagnostic and qualitative interpretation of these phenomena 23,24 . An intriguing feature of experiments on 𝑬 × 𝑩 devices is that spokes do not always rotate in the 𝑬 × 𝑩 direction. Retrograde motion (in the -𝑬 × 𝑩 direction) has also been observed in the low current regime of HiPIMS or in direct current (DC) magnetrons 23,24 .

In this paper we present simulations of a recently published experiment by Ito, Young, and Cappelli 25 on a simple DC magnetron discharge, where self-organized structures rotating in the -𝑬 × 𝑩 direction were observed and analyzed with a high framing rate camera. On the basis of simulation results in conditions around this experiment and of published experimental results in other conditions, we propose a new insight into the physics of rotating spokes associated with an ionization instability in 𝑬 × 𝑩 plasmas. This new model of the physics of rotating spokes shares some qualitative aspects with but is clearly distinct from the CIV model of Ref. [START_REF] Piel | [END_REF] .

The experiment of Ito, Young and Cappelli consists of a small planar DC magnetron discharge in argon (gap 2 mm, diameter 5 mm, pressure 20 Pa) where the light emission is observed with a fast camera through a transparent planar anode. The magnetic field is generated by a ring magnet 25 . Its component parallel to the electrodes varies from around 1 T at the cathode surface to about 0.1 T at the anode surface. given time in the conditions described in the text, corresponding to the experiment of Ref. 25 . The maximum values are respectively 4×10 17 m -3 , 260 V, 15 eV, 2×10 25 m -3 s -1 . The spoke (ionization region), plasma non-uniformity and potential structure move together up in the -𝑬 × 𝑩 direction at about 10 km/s while the electron vortices (local maxima in the electron temperature and ionization rate, and associated "holes" in the electron density) move down in the 𝑬 × 𝑩 direction at a velocity about ten times larger. Movies available in Supplemental Material at (URL).

The relatively large magnetic field and pressure are related to the small dimensions of the device and according to classical discharge scaling laws similar features should be observed for dimensions 10 times larger and magnetic field and pressure 10 times smaller.

The two-dimensional (2D) PIC-MCC model described in previous papers 15,16,26 has been used to simulate this experiment. The simulation domain is defined by the axial and azimuthal directions 𝑥 ̂ and 𝑦 ̂. The axial length of the domain (cathode-anode distance) is 𝑑 = 2 mm , and the azimuthal length is 𝑤 = 4 mm. Periodic boundary conditions are assumed in the azimuthal direction. The magnetic field (in the -𝑧̂= -𝑥 ̂× 𝑦 ̂) direction is perpendicular to the simulation domain, and its magnitude 𝐵 varies axially according to 𝐵(𝑥) = 𝑎exp(-𝑥 2 In these conditions the simulation predicts the formation of an ionization instability (region of enhanced ionization, or spoke) moving in the -𝑬 × 𝑩 direction. When the simulation is performed with an azimuthal dimension of 8 mm instead of 4 mm, two similar spokes are present in the simulation domain and move in the -𝑬 × 𝑩 direction. The spoke velocity is around 10 km/s. These features (retrograde motion, spoke velocity, and periodicity) are in good agreement with the experiments of Ref. 25 .

Figure 1 shows the space distribution of the main plasma properties at a given time in the simulation at steady-state. The plasma presents a strong azimuthal non-uniformity that creates a distortion of the equipotential lines. This distortion separates the plasma in two regions: one region of low electric field (lighter color region in Figure 1b) whose potential is close to the anode potential and a region of large axial electric field between the cathode and the quasi-equipotential region. A double layer is formed at the boundary between the two regions. We call this boundary (the ABCDE line in Figure 1b) "the interface" in the following. Note that the electric field is perpendicular to the equipotential lines so that the 𝑬 × 𝑩 electron drift follows the equipotential contours. Since the interface intercepts the anode, electrons are lost along this line (segments AB and DE of the interface on Figure 1b). The formation of the double layer at the interface is the consequence of the electron deficit due to losses at the anode not being entirely balanced by ionization along this line. We see on Figure 1d that ionization is enhanced along the interface and on the cathode side of the equipotential region (left side of the BC segment). It is interesting to note that in these conditions, most of the overall ionization takes place in this region, i.e. the plasma is sustained by the ionization instability. Ionization in the cathode region (generally important in glow discharges) is small in these conditions because of the large magnetic field and small collisional electron velocity parallel to the electric field in the cathode region leading to insignificant electron heating. The breaking of quasi-neutrality in the double layer can be clearly seen in Figure 2, which displays the axial profiles of the electron density, ion density, axial electric field, and electron temperature at the three azimuthal locations indicated by the white horizontal dashed lines in Figure 1a.

One reason for the enhanced electron heating and ionization in the BC region is cross-field electron transport associated with the axial variations of magnetic field (𝛁𝐵 drift [START_REF] Chen | Introduction to Plasma Physics and Controlled Fusion[END_REF] ) in the large electric field region on the left of the BC line. This is illustrated in Figure 3. This figure shows an example of electron trajectory (Figure 3a) in the vicinity of the interface, on the left side of the BCD segment, and the variations of the electron energy over the same trajectory (Figure 3b). The mean value of the electron energy increases by about 5 eV when the electron moves parallel to BC in the spoke front and decreases when the electron moves parallel to CD. The consequence of this electron heating is the intense ionization that takes place on the left side of the BC region in Figure 1d. The 𝛁𝐵 electron drift velocity, given by 27 𝒗 𝛁𝐵 = 1 2 ⁄ 𝜌 𝑒 𝑣 ⊥ 𝑩 × 𝛁𝐵/𝐵 2 (𝑣 ⊥ is the electron velocity in the axial-azimuthal plane) is directed in the azimuthal direction, downward along the BC segment and upward along CD. The electron heating along BC is given by 𝜕 𝑡 𝜀 ⊥ = -𝒗 𝛁𝐵 . 𝑬 = 𝜀 ⊥ 𝐸 𝑦 (𝐵𝐿) ⁄ where 𝜀 ⊥ is the electron kinetic energy perpendicular to the magnetic field and 𝐿 = |𝐵 𝜕 𝑥 𝐵 ⁄ |. The electron heating frequency 𝛼 = 𝐸 𝑦 (𝐵𝐿) ⁄ can be as large as 10 9 s -1 in the vicinity and on the left side of the BC line (𝐸 𝑦 𝐵 ≈ 5 × 10 5 m/s ⁄ and 𝐿 ≈ 0.5 mm) leading to significant heating (the electron transit time along BC is in the ns range). This simple estimation is confirmed numerically on Figure 3b, where the red line corresponds to the electron mean energy calculated by integrating the expression above along the electron trajectory: 𝜀 ⊥ = 𝜀 ⊥0 -∫ 𝒗 𝛁𝐵 . 𝑬𝑑𝑡. This clearly shows that the 𝛁𝐵 drift plays an important role in electron heating in the conditions of Figure 1.

We note on Figure 3a that the electric potential presents local maxima of several volts above the anode potential in the quasi-equipotential region. These maxima are associated with the formation of electron vortices due to the electron velocity shear in the double layer. These vortices (apparent as electron density "holes" and local maxima of the electron mean energy and ionization rate in Figure 1) move along the double layer at the 𝐸 𝐵 ⁄ velocity where 𝐸 is an average value of the electric field in the double layer. The physics is similar to that of the diocotron instability observed in pure electron plasmas [START_REF] Kabantsev | [END_REF] except that the electron vortices form here in a local potential maximum rather than minimum of the potential, due to the positive space charge of the double layer. This is the reason why the vortices form "holes" instead of "clumps" [START_REF] Kabantsev | [END_REF] in the electron density, as can be seen in Figure 1a. Electron hole vortices were already observed in simulations of cylindrical magnetrons 15,16 (see Fig. 11 of Ref. 16 ). The distortion of the potential in the electron vortex can also contribute to electron heating as can be seen in Figure 3. Note however that the electron trajectory of Figure 3 is calculated for an electric potential distribution assumed to be constant in time. The local potential maxima move along the double layer and the electron energy balance is actually more complex and is the result of an electron-wave interaction (see Supplemental Material at URL). Nevertheless we can see on Figure 1c and Figure 1d that the formation and motion of the electron vortices contribute to electron heating since local maxima of both electron mean energy and ionization rate appear at the vortex locations. Electron heating due to 𝛁𝐵 drift is important in the conditions considered and seems to be responsible for the stability of the spoke. In the absence of magnetic field gradient, electron heating associated with electron vortices and electron-wave interaction in the double layer may be sufficient to generate spokes as can be inferred from the simulation results of Refs.[ 15,16 ].

The reason for the spoke motion in the -𝑬 × 𝑩 direction in the conditions of Figure 1 can be understood simply by the fact that the ionization region is located on the side of the plasma and potential maxima that is opposite to the 𝑬 × 𝑩 direction. The plasma and ionization region (spoke) apparently "move" in the -𝑬 × 𝑩 direction, i.e. in the region where new electrons and ions are generated by ionization. This is illustrated in Figure 4a. The question now is, why spoke rotation is also observed in the 𝑬 × 𝑩 direction in other experiments (HiPIMS 23,24 ). We performed simulations in the same conditions as Figure 1 but with 𝐵(0) = 0.5 T instead of 1T (the electron current density emitted by the cathode was imposed and equal to 5 A/m 2 in this simulation). In these conditions the simulations predict spoke rotation in the +𝑬 × 𝑩 direction as illustrated in Figure 4b. The main differences between the two cases are 1) the position of the ionization region with respect to the plasma density, and 2) the direction of the ion flow in the spoke front. In Figure 4a new electrons and ions are generated in and above the location of maximum plasma density and ions are not expelled from the front by the electric field. In Figure 4b ions are expelled azimuthally from the front faster than they are generated by ionization. This can be simply formulated by writing the ion continuity equation 𝜕 𝑡 𝑛 𝑖 = 𝑆 -𝛁.𝑛 𝑖 𝒗 𝒊 at the starting point of the arrows representing the ion flux in the spoke front in Figure 4 (𝑆 is the ionization rate, 𝒗 𝒊 the ion mean velocity). In Figure 4a, 𝑆 > 𝛁.𝑛 𝑖 𝒗 𝒊 so 𝜕 𝑡 𝑛 𝑖 > 0 and the front moves in the upward (-𝑬 × 𝑩) direction. The opposite is true for Figure 4b.

The conditions of Figure 4b are likely to be met when the azimuthal component of the electric field in the spoke front is larger than the axial component and/or for larger plasma densities. In any case the simulation results indicate that the same physics (electron heating along a double layer) is responsible of spoke rotation in the retrograde 𝑬 × 𝑩 direction as well as in the 𝑬 × 𝑩 direction. The main results can be summarized as follows. The small cross-field electron mobility is responsible for an instability (presumably of the Simon-Hoh type 26 in its initial phase) that leads to an azimuthal distortion of the plasma potential and density. This distortion allows some electrons to flow to the anode along an equipotential line connecting the cathode and anode regions. A double layer forms along this line, which lies at the interface between an equipotential region close to the anode potential and a lower potential region. Electron flow along this interface due to 𝑬 × 𝑩 drift. The double layer is generated by the drop in electron density due to electron losses to the anode along this line. Electron gain energy due to 𝛁𝐵 induced crossfield transport while drifting along the double layer toward regions of increasing electric field. The formation of electron vortices due to the velocity shear in the double layer also contributes to electron heating. Ionization is therefore enhanced in that region. The spoke can rotate in the -𝑬 × 𝑩 as well as in the +𝑬 × 𝑩 depending on the position of the plasma non-uniformity with respect to the ionization region and on the relative values of the axial and azimuthal components of the electric field in the spoke.

The main similarities of the mechanism described above with the CIV model of Ref. [START_REF] Piel | [END_REF] is the presence of a double layer at the spoke front and the enhanced electron heating and ionization in that region but the reason for the formation of a double layer, the mechanism of electron heating, and the spoke velocity are distinct from those of the CIV model. Finally, we note that the conclusions of recent experiments 23,24 and of the present model are converging on a number of points: existence of a double layer at the spoke front, energization of electrons around the double layer, possible rotation in the ±𝑬 × 𝑩 direction.

Figure 1 :

 1 Figure 1: Contours of constant (a) electron density, (b) plasma potential, (c) electron mean energy, and (d) ionization rate at agiven time in the conditions described in the text, corresponding to the experiment of Ref.25 . The maximum values are respectively 4×10 17 m -3 , 260 V, 15 eV, 2×10 25 m -3 s -1 . The spoke (ionization region), plasma non-uniformity and potential structure move together up in the -𝑬 × 𝑩 direction at about 10 km/s while the electron vortices (local maxima in the electron temperature and ionization rate, and associated "holes" in the electron density) move down in the 𝑬 × 𝑩 direction at a velocity about ten times larger. Movies available in Supplemental Material at (URL).

  , with 𝜎 = 0.35𝑑. 𝑎 and 𝑏 are chosen so that 𝐵(0) = 1 T and 𝐵(𝑑) = 0.1 T. The discharge is sustained by ionization (calculated self-consistently in the Monte Carlo module) and secondary electron emission at the cathode due to ion impact. The argon electron-neutral and ion-neutral scattering cross-sections are the same as in Refs.15,16 . An effective secondary emission coefficient 𝛾 is used in the simulation. The net flux Γ 𝑒 of electrons leaving the cathode is related to the ion flux Γ 𝑖 by Γ 𝑒 = -𝛾Γ 𝑖 , i.e. an electron coming back to the cathode is re-emitted. 𝛾 = 0.005 is used in the simulations. Electrons are emitted according to a semi-Maxwellian distribution at a temperature 𝑇 𝑒 =2 eV. The applied voltage in the results presented here was 𝑈 𝑑 = 260 𝑉. The simulations were started with a uniform plasma density 𝑛 𝑒 = 𝑛 𝑖 = 2 × 10 16 𝑚 -3 . Most simulations were performed on a 128x256 grid with about 300 particles per cell at steady-state (reached typically in less than 20 s).

Figure 2 :

 2 Figure 2: Axial profiles of the electron density, ion density, axial electric field and electron mean energy at different azimuthal positions indicated by the dashed lines in Figure 1.

Figure 3 :

 3 Figure 3: (a) Equipotential contours, and example of electron trajectory at a given time (same conditions as Figure 1), close to the double layer, on the left side of the BCD line of Figure 1. (b) Variations of the electron energy along the same trajectory. The initial electron velocity, at x=1.6 mm, y=3.4 mm, is 𝑣 𝑥 = 10 6 m/s, 𝑣 𝑦 = 0. The red line is the averaged energy calculated assuming electron heating due 𝛁𝐵 drift.

Figure 4 :

 4 Figure 4: Equipotential contours (grey), and lines of constant plasma density and ionization rate corresponding to 70% of their maximum values. The green arrows indicate the ion flux direction in the spoke front. (a) spoke rotation in the -𝑬 × 𝑩 direction, case of Figure 1, (b) rotation in the +𝑬 × 𝑩 direction, same conditions as Figure 1 with 𝐵(0) = 0.5 T instead of 1T. Movies available in Supplemental Material at (URL).


Ansar Calloo
email: ansar.calloo@cea.fr

APPLICATION OF ANDERSON ACCELERATION TO THE NEUTRON TRANSPORT EQUATION

This work investigates the solution of the multigroup neutron transport equation with a discrete ordinates method. More specifically, it focuses on the k-eigenvalue problem of the equation. In this case, the variables of interest are the largest eigenvalue (k eff ) and the corresponding eigenmode is called the fundamental mode. Mathematically, this problem is usually solved using the power iteration method [START_REF] Lewis | Computational Methods for Neutron Transport[END_REF]. However, the convergence of this algorithm can be very slow, especially if the dominance ratio is high as is the case in some reactor physics applications. Thus, the power iteration method has to be accelerated in some ways to improve its convergence.

One such acceleration is the Chebyshev acceleration method [START_REF] Ferguson | Optimized Iteration Strategies and Data Management Considerations for Fast Reactor Finite Difference Diffusion Theory Codes[END_REF] which has been applied to legacy codes. In recent years, nonlinear methods have been applied to solve the k-eigenvalue problem. Nevertheless, these are often compared to the unaccelerated power iteration method. Hence, the goal of this paper is to apply the Anderson acceleration method to the power iteration method, and compare its performance to the Chebyshev acceleration method.

I. MATHEMATICAL BACKGROUND

Neutron transport equation

The multigroup discrete-ordinate transport equation is written in operator form as:

Hψ = Fψ (1)
where

• H = Ω n .∇ψ g n (r) + Σ g t ψ g n (r) - 1 2π G g =1 N n =1 ω n Σ g →g s (r, Ω n • Ω n ) ψ g n (r) • F = χ(E) 4π G g =1 N n =1 ω n νΣ f ψ g n (r)
The solution to Equation ( 1) is physical, i.e., positive, if the characteristics of the problem is such that λ = 1

is the smallest eigenvalue of the following:

Hψ = λFψ (2)
Therefore, the critical reactor core is obtained by establishing an equilibrium between fission sources and the removal term, without external sources, and is equivalent to solving for the largest eigenvalue k eff such that (k eff , ψ) is the solution to:

H -1 Fψ = k eff ψ (3)

Power Iteration

The Power Iteration (PI) method consists in iterating on the fission sources to solve the eigenvalue problem and is as follows:

Algorithm 1: Power Iteration 1 while k (n+1) eff -k (n) eff k (n) eff < k eff and Fψ (n+1) -Fψ (n) Fψ (n) < ψ do 2
Apply source iteration for new flux

Hψ (n+1) = 1 k (n) eff Fψ (n) 3
Compute the new eigenvalue k

(n+1) eff = k (n) eff Fψ (n+1) , Fψ (n+1) Fψ (n) , Fψ (n) 4 end
The eigenvector ψ (n) can be decomposed on the basis of eigenvectors of matrix

H -1 F as ψ (n) = λ n α 1 ψ 1 + O( λ 2 λ 1 n
). For thermal reactors, the dominance ratio is usually above 0.95, and the PI method converges slowly. Thus, over the years, several acceleration methods have been conceived among which are rebalance techniques, Chebyshev acceleration and more recent nonlinear methods [START_REF] Fichtl | Nonlinear acceleration of S n transport calculations[END_REF].

Chebyshev acceleration

This consists of modifying the initial problem into an equivalent one with a smaller dominance ratio to improve the convergence of the PI algorithm. The Chebyshev acceleration consists in applying PI method on a polynomial of H -1 F such that the dominance ratio κ = λ 2 /λ 1 is as small as possible without changing the eigenvectors of the latter. Thus, the starting point is to expand ψ (n) on the basis of eigenvectors of

A = H -1 F k (n) eff as a polynomial P l Given that P l (A)ψ = m i=1 α i P l (A)ψ i and that A l ψ = l i=1 α i λ i ψ i : P l (A)ψ = α 1 P l (λ 1 )ψ 1 + P l (λ 1 )(α 2 P l (λ 2 ) P l (λ 1 ) ψ 2 + • • • + α m P l (λ m ) P l (λ 1 ) ψ m ) (4)
and by minimising the fractional part of the former expression, the polynomial P l obtained may be applied to accelerate the PI [START_REF] Ferguson | Optimized Iteration Strategies and Data Management Considerations for Fast Reactor Finite Difference Diffusion Theory Codes[END_REF]. One such polynomial is obtained from the Chebyshev polynomial defined 2/5

as

P n (x) = C n ( 2x κ -1) C n ( 2 κ -1)
where

C n (x) = cos(n arccos(x)) if |x| < 1 or C n (x) = cosh(narccosh(x))
otherwise. Following the recurrence property of the Chebyshev polynomials, the acceleration step is hence expressed as: (1) , and ρ (n+1) = 2 2 -σ 2 ρ (n) . The main drawback of this method is that it requires a priori knowledge of λ 1 and the dominance ratio κ; its efficiency depends on an appropriate estimate for these. Algorithm 2 describes the Chebyshev acceleration as applied.

Fψ (n+1) = ρ (n+1) γ λ 1 Fψ (n+1) -ρ (n+1) (1 -γ)Fψ (n) + (1 -ρ (n+1) )Fψ (n-1) (5)
with γ = 2 2 -κ , σ = κ 2 -κ , ρ (1) = 1, ρ (2) = 2 2 -σ 2 ρ
Algorithm 2: Chebyshev acceleration

1 while k (n+1) eff -k (n) eff k (n) eff < k eff and Fψ (n+1) -Fψ (n) Fψ (n) < ψ do 2
Apply Power Iteration as in Algorithm 1

3

Compute an estimation of κ as τ (n+1) = (n+1)

(n) , (n) = relative error at iteration n 4

Accelerate source vector

5 if τ (n+1) < 1 & τ (n+1) -τ (n) < 0.01 & accelerate = False then 6 accelerate = T rue (flag to control acceleration phase) 7 α = 2 τ (n+1) , ρ (n+1) = 1 α -1 8 Fψ (n+1) = αρ (n+1) Fψ (n+1) -ρ (n+1) Fψ (n) 9 end 10 if accelerate = T rue then 11 ρ (n+1) = 4 4(α -1) -ρ (n) 12 Fψ (n+1) = αρ (n+1) Fψ (n+1) -ρ (n+1) Fψ (n) + (1 -αρ (n+1) + ρ (n+1) )Fψ (n-1)
13 end 14 end

Anderson acceleration

In this research effort, the Anderson acceleration [START_REF] Anderson | Iterative procedures for nonlinear integral equations[END_REF] is applied to the PI algorithm. This method has been successfully applied recently for accelerating the eigenvalue problem by [START_REF] Fichtl | Nonlinear acceleration of S n transport calculations[END_REF][START_REF] Calef | Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem[END_REF] and is described by Algorithm 3.

It ressembles a GMRES method, except that the subspace projection is orthogonalised using only the number of vectors prescribed by the history length of the method. Thus, the subspace spanned does not guarantee a theoretical convergence of the method but past works [START_REF] Fichtl | Nonlinear acceleration of S n transport calculations[END_REF][START_REF] Calef | Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem[END_REF] have shown that it converges in practical situations where the PI converges.

The application to the PI method is implemented in our case by substituting the function f in Algorithm 3 by the Algorithm 1. Successive iterations of the Anderson algorithm evaluates the PI only once. Unlike

past works, this investigation differs in its application of the Anderson acceleration to the source vector, and not the flux, similar to the Chebyshev acceleration described previously. Furthermore, the history length is arbitrarily set to 2 to compare to the amount of data required by the Chebyshev acceleration method.

Algorithm 3: Anderson acceleration

1 Given u (0) and a history length of m ≥ 1 2 u (1) = f (u (0) )

mn ) which solves the following minimisation problem

Compute new estimate for unknown vector

11 end

II. SLAB PROBLEM

The case considered for the numerical test is monogroup 1D slab problem, with a fuel zone of 200 cm enclosed between two reflector regions of 5 cm each. The problem is solved using the unaccelerated PI, and PI accelerated by the previously described acceleration methods. The stopping criteria for the iterative schemes are set to 1.e-6.

III. CONCLUDING REMARKS

The Anderson acceleration method has been successfully applied to the 1D problem considered in this work. During this work, multigroup cases have also been tested, and the results remain still very similar to the observations made with the monogroup case. The aim is to extend it to further multidimensional problems and compare its behaviour with respect to the Chebyshev acceleration.