Nishant Kumar
email: nishant.kumar@cnrs.pprime.fr

Franck Kerhervé

Laurent Cordier

Dynamic reconstruction of a numerical 2D cylinder wake flow using Data Assimilation

Keywords: data assimilation, dual ensemble Kalman filter, POD Reduced-Order Model, 2D-cylinder wake flow

Introduction

Despite decades of intensive research in shape optimization, aerodynamic mechanisms such as separation and mixing still represent an important source of energy expenditure in transport vehicles. The manufacturers have consequently developed, over the years, a range of strategies to improve the aerodynamic performance of their vehicles. One of these strategies consists in using active devices that can be actuated on and off based on the change of the inflow conditions. To drive such actuators, command laws and more globally valid controllers are required. To design these controllers, different strategies can be employed, one of which being the model-based approach. The later requires the knowledge of a dynamical model of the flow that one wants to control in real-time. In practice, the identification of such a model is challenging because it is generally obtained from limited spatial and/or time discrete informations.

This problem of state estimation is well-known in the field of meteorology where data is collected at multiple locations, with spatial and temporal scales varying in orders of magnitude, to estimate the evolution of the weather [START_REF] Talagrand | Variational Assimilation of Meteorological Observations with the Adjoint Vorticity Equation. I: Theory[END_REF]. Classical methodologies involve the use of data assimilation (DA) techniques where data is assimilated, when available, with a dynamical model to correct the state estimate obtained by the model, and eventually, to correct the model itself for further, more accurate predictions. In the specific context of aerodynamicrelated problems, the complexity resides essentially in the large number of degrees of freedom of the flow dynamics, in the broad range of scales to examine and in non-linearities driving the flow system. To design a physical model that can be managed in real-time applications, the Navier-Stokes equations are traditionally projected onto an appropriate reduced basis leading to a reduced-order model of the flow [START_REF] Rowley | Model Reduction for Flow Analysis and Control[END_REF]. Proper Orthogonal Decomposition (POD, [START_REF] Lumley | Atmospheric Turbulence and Wave Propagation. the structure of inhomogeneous turbulence[END_REF]) is typically used to design the orthonormal basis which captures the flow features which are believed essential, from an energetic point of view, in representing the flow dynamics. In general, this basis is then used in a Galerkin approach to derive a POD Reduced-Order Model (POD ROM) of the flow by projecting the Navier-Stokes equations onto the POD modes. Unfortunately, this dynamical system is generally not sufficiently accurate for application in flow control, and identification methods are then used to improve the prediction ability of the POD ROM [START_REF] Cordier | Calibration of POD Reduced-Order Models using Tikhonov Regularization[END_REF]. An alternative is to use the ROM in conjunction with DA techniques to design dynamic observers able to predict the flow state, provided limited information of the state itself is available [START_REF] Papadakis | Assimilation de données images : Application au suivi de courbes et de champs de vecteurs[END_REF].

In this paper, a particular stochastic DA method known as Dual Ensemble Kalman Filter (Dual EnKF) is applied to a numerical dataset of a 2D-cylinder wake flow at low Reynolds number (Re = 100). In the DA setup considered, time-resolved measurements of streamwise velocity component sampled at specific downstream locations serve as observations for the assimilation procedure. At the end of the assimilation window, the measurements are used to test the predictability of the DA method with respect to reproducing the flow dynamics beyond the window. In § 2, an overview of snapshot POD ( § 2.1), POD-ROM ( § 2.2), residual representation ( § 2.3), and Dual EnKF ( § 2.4) is given. POD Reduced-Order model is then applied to the cylinder wake flow in § 3. The results obtained from the application of DA to predict the dynamics of the numerical flow is discussed in § 4. Finally, the concluding remarks are given in § 5.

Model reduction and Ensemble Kalman Filter

In this section, we present the reduced-order modelling strategy and describe the Ensemble Kalman Filtering (EnKF) method employed for the reconstruction and prediction of the flow fields.

Snapshot POD

The proper orthogonal decomposition (POD) is a reduced-order model approach which was introduced to the fluid dynamics community as a mathematical tool to extract coherent structures from turbulent flow fields [START_REF] Lumley | Atmospheric Turbulence and Wave Propagation. the structure of inhomogeneous turbulence[END_REF]. It provides an objective algorithm to decompose a set of data into a minimal number of basis functions or modes to capture as much as possible the information contained in the data. It is a pure data-driven method and as such requires only the flow field data for modal decomposition. POD extracts the modes based on optimizing the mean square of the field variable under observation [START_REF] Cordier | Model Reduction, POD and Data Assimilation[END_REF].

Let χ be the spatial coordinates and t represents the scalar time, we assume that the unsteady component of the vector field u(χ, t) can be decomposed as

u(χ, t) -u(χ) = i a i (t)φ i (χ), ( 1
)
where u is the temporal mean of u. In (1), φ i and a i represent the spatial modes and the time-dependent projection coefficients, respectively. The POD method requires an ensemble of snapshots of any scalar (e.g. pressure, temperature) or vectorial (e.g. velocity, vorticity) fields defined at discrete spatial points χ i (i = 1, . . . , N χ ) and discrete time instants t k (k = 1, . . . , N t ). From this data set, we prepare snapshots of the flow field stacked in terms of a collection of column vectors u (χ i , t k ) where

u (χ i , t k ) = u(χ i , t k ) -u(χ i ) ∈ R Ns , i = 1, 2, . . . , N χ , k = 1, 2, . . . , N t (2)
where N s is the number of the spatial degrees of freedom of the data. For a fluid flow data, N s is equal to the number of grid points (N χ ) times the number of components (N c ) to be considered in the data. The snapshot vector can be stacked in matrix form as

X = [u (χ i , t 1 ) u (χ i , t 2 ) . . . u (χ i , t Nt )] ∈ R Ns×Nt . (3)
Next, the objective is to find the optimal basis vectors that can best represent the flow data in an average energetic sense. In other words, we seek the vectors φ POD i in (1) that can represent u (χ i , t k ) in an optimal manner and with the least number of modes. The solution to this problem can be determined by finding the eigenvectors ψ i and eigenvalues λ i from

X T Xψ i = λ i ψ i , ψ i ∈ R Nt (4)
where X T X ∈ R Nt×Nt is a temporal correlation matrix1 . The eigenvalues λ i are arranged in descending order such that λ 1 ≥ λ 2 ≥ . . . ≥ λ Nt ≥ 0. In most numerical cases, spatial size of the data is very large compared to the temporal size, N s N t . For this reason, the method of snapshots proposed by Sirovich [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. I. Coherent structures[END_REF] is preferred to the classical spatial POD method that leads to a spatial correlation matrix XX T of size N s [START_REF] Rowley | Model Reduction for Flow Analysis and Control[END_REF].

After obtaining the eigenvectors ψ i from the smaller eigenvalue problem (4), the (spatial) POD modes are recovered as

φ i = 1 √ λ i Xψ i ∈ R Ns , i = 1, 2, . . . , N t , (5)
which can be also written as

Φ = XΨ Λ -1/2 , (6)
where

Φ = [φ 1 φ 2 • • • φ Nt ] ∈ R Ns×Nt and Ψ = [ψ 1 ψ 2 • • • ψ Nt ] ∈ R Nt×Nt .
Note that these matrices are orthonormal, i.e. Φ T Φ = ΦΦ T = I and Ψ T Ψ = Ψ Ψ T = I (normality of X T X). The temporal coefficients are determined accordingly as

a i (t) = u (χ, t), φ i (χ) Ω , i = 1, 2, . . . , N t , (7)
where • Ω represents the spatial inner product defined as

v I (χ, t), v II (χ, t) Ω = Ω v I (χ, t) • v II (χ, t) dχ (8)
for two given vector fields v I and v II defined in a spatial domain Ω.

POD Reduced-Order Model (POD ROM)

A standard Galerkin projection is then used to derive the POD ROM. The projection of the incompressible Navier-Stokes equations onto the spatial POD modes φ i leads, after some algebraic manipulation [START_REF] Cordier | Calibration of POD Reduced-Order Models using Tikhonov Regularization[END_REF], to the following expression,

       ȧi (t) = C i + N Gal j=1 L ij a j (t) + N Gal j=1 N Gal k=j Q ijk a j (t)a k (t) + R i (t) a i (0) = u(χ, 0) -u(χ), φ i (χ) Ω i = 1, . . . , N Gal (9)
where N Gal is the number of POD modes kept in the Galerkin projection. The residual term R i reads

R i (t) = Nt j=N Gal +1 L ij a j (t) + Nt j=N Gal +1 Nt k=j Q ijk a j (t)a k (t). (10)
The coefficients C i , L ij and Q ijk are the constant, linear and quadratic coefficients of the model equation and can be expressed as functions of the spatial eigenfunctions φ i and the mean field u [START_REF] Cordier | Calibration of POD Reduced-Order Models using Tikhonov Regularization[END_REF]. When the residual term R i is neglected, the model reduces to a so-called reduced-order model (ROM). In practice, an energetic criterion can be used to determine the value of N Gal necessary to capture the essential flow dynamics. The relative information content (RIC) [START_REF] Bergmann | Optimal control of the cylinder wake in the laminar regime by trust-region methods and pod reduced-order models[END_REF] is defined as

RIC(N Gal ) = N Gal i=1 λ i Nt i=1 λ i . (11)

Representation of the residual

Due to the truncation in (9), the ROM is in general not sufficiently accurate to estimate correctly the long-term dynamics (see also the discussion in § 3.2). This truncation is equivalent to keep only the most energetic modes that are associated to the largest flow motions. As a consequence, the contribution of the viscous dissipation terms are neglected and the solution of ( 9) may diverge in time. To account for this truncation error, a subscale turbulence representation can be used to model the residual term R i . Rempfer and Fasel [START_REF] Rempfer | Evolution of three-dimensional coherent structures in a flat-plate boundary layer[END_REF] suggested to include this dissipative effect in a linear term as a "model eddy viscosity" i.e.

R i (t) = ν T i N Gal j=1 L ij a ROM j (t). ( 12
)
Finally, the POD ROM considered in this paper is thus given by,

       ȧROM i (t) C i + N Gal j=1 (1 + ν T i )L ij a ROM j (t) + N Gal j=1 N Gal k=j Q ijk a ROM j (t)a ROM k (t) a ROM i (0) = a i (0) i = 1, . . . , N Gal (13)
where the superscript ROM is used to separately identify the temporal coefficients obtained from the POD ROM as compared to the ones which are directly obtained from snapshot POD.

The model coefficient vector (C 1 . . . L 11 . . . Q 111 . . . )

T can be calibrated thanks to a least-mean-square procedure (L2 minimization) such as detailed in Perret et al. [START_REF] Perret | Polynomial identification of POD based low-order dynamical system[END_REF] or Cordier et al. [START_REF] Cordier | Calibration of POD Reduced-Order Models using Tikhonov Regularization[END_REF]. In the current paper, the model coefficient vector is calculated once and for all, and the estimation of the model eddy viscosity ν T i (i = 1, . . . , N Gal ) is done by DA.

Dual Ensemble Kalman Filter (Dual EnKF)

As the POD ROM is not sufficient in general to replicate the correct dynamics, we rely on data assimilation methods to correctly identify the model parameters. Data assimilation is a generic methodology which combines heterogeneous observations with the underlying dynamical principles governing the system under consideration to estimate at best the states and/or physical quantities parametrising the dynamics. Starting from a background solution and incoming imperfect information, an optimal estimation of the unknowns of the system is determined which takes into account the respective statistical confidences of the different observations. We consider a non-linear, time-discrete dynamical system given as

x t k+1 = M k,k+1 (x t k , θ k ) + η k , (14)
where the superscript t stands for true. Here, k = 0, . . . , N -1 is the time index with N being the total number of time steps. The propagation of the true state

x t k ∈ R Ns (where N s is the dimension of the state after spatial discretisation) from time t k to t k+1 is performed by a non-linear function M k,k+1 which also depends on the model parameters vector θ k at time t k . The model error η k ∼ N (0, Q k ) is considered to be Gaussian distributed with zero mean and covariance matrix Q k . The model is initialised with an initial background state x b 0 such that x t 0 = x b 0 with the corresponding initial covariance Q 0 . In conjunction with the model equation, an observation equation is used which allows the estimated state to be corrected thanks to the available measurements. The observation vector y o k+1 ∈ R No (where N o is the dimension of the observations) at time t k is related to the state by the relation

y o k+1 = H k+1 (x t k+1 ) + k+1 , (15)
where H k+1 denotes the observation function (which may be non-linear), and k+1 ∼ N (0, R k+1 ) the observation noise approximated as a Gaussian distribution with zero mean and covariance R k+1 .

If the model parameters are known to be true, a DA method is specifically used for the state estimation. On the other hand, in practice, we frequently encounter cases where the parameters are unknown or imprecise.

One of the possible approaches to solve this dual estimation problem is to use an iterative method where the filtering is alternatively applied to estimate the state and the parameters in the course of each iteration. More precisely, the model parameters are obtained from the corrected state at the previous time state and the new state is evaluated from the current analysed parameters. This method, formally known as Dual EnKF [START_REF] Moradkhani | Dual stateparameter estimation of hydrological models using ensemble Kalman filter[END_REF], consists of a sequential double prediction-correction scheme at the basis of the mechanism of evolution of the traditional Kalman filter, one for the update of the parameter vector, one for the update of the state variable.

Prediction-correction of the parameters vector

First, an ensemble of N e forecasted parameters is created using a forced random walk of the parameters set up according to a Gaussian law ξ k ∼ N (0, C k ) with zero mean and covariance matrix C k . During the prediction step, the ensemble of forecasted parameters is constructed as

θ f,(n) k+1 = θ a,(n) k + ξ (n) k n = 1, . . . , N e (16)
where the superscript (n) represents the n-th member of the ensemble of parameters, while the superscripts f and a denote the forecasted (predicted) and assimilated (corrected) values of the parameters, respectively. The predicted ensemble of parameters are then updated using the available observations. For this, the predicted state at the instant t k+1 is computed as,

x f,(n) k+1 = M k,k+1 (x a,(n) k , θ f,(n) k+1 ) n = 1, . . . , N e . ( 17
)
The predicted ensemble of observations is then obtained as,

y f,(n) k+1 = H k+1 (x f,(n) k+1 ) n = 1, . . . , N e . (18)
Next, the random observations are built to be centred around the actual observation y o k+1 following

y o,(n) k+1 = y o k+1 + o,(n) k+1 n = 1, . . . , N e (19)
where the covariance matrix of the observation error at the instant t k+1 is given as

R e k+1 = 1 N e -1 Ne n=1 ( o,(n) k+1 )( o,(n) k+1 ) T . ( 20
)
Given the random observations from (19) and the empirical estimator (20), the update of the predicted ensemble of parameters is obtained by the Kalman analysis step which reads as,

θ a,(n) k+1 = θ f,(n) k+1 + K θ,e k+1 (y o,(n) k+1 -y f,(n) k+1 ). ( 21
)
The Kalman gain allowing the correction of the parameter trajectories is expressed using the ensemble predictions as K θ,e k+1 = P f,e θy,k+1 (P f,e yy,k+1 + R e k+1 ) -1 .

In this expression, the terms P f,e θy,k+1 and P f,e yy,k+1 designate the unbiased empirical estimators of the cross-correlation matrix between the parameters and the observations, and the correlations between the observations, respectively.

Prediction-correction of the state variable

Once the predicted parameters have been updated, the second filter is implemented to estimate the state variable at the instant t k+1 by assuming the parameters as known. In other words, the parameters just corrected by the first filter are trusted. The following prediction-correction mechanism is identical to the classical ensemble Kalman filter. In the first step, the predicted ensemble of states at the instant t k+1 is built starting from the ensemble of assimilated states at the instant t k and the corrected ensemble of parameters at the instant

t k+1 i.e. x f,(n) k+1 = M k,k+1 (x a,(n) k , θ a,(n) k+1 ) + η (n) k . ( 23
)
For the second time, a forecast ensemble of observations is calculated as

y f,(n) k+1 = H k+1 (x f,(n) k+1 ). ( 24
)
At this stage, the observations are again considered as random variables and the covariance matrix of the observation error involved in calculating the Kalman gain is also empirically estimated. Next, the update of the predicted ensemble of the state is realised by Kalman analysis, given as

x a,(n) k+1 = x f,(n) k+1 + K e k+1 (y o,(n) k+1 -y f,(n) k+1 ). ( 25
)
The use of a Kalman gain permits the correction of the state trajectories as K e k+1 = P f,e xy,k+1 (P f,e yy,k+1 + R e k+1 ) -1 .

In this expression, P f,e xy,k+1 and P f,e yy,k+1 denote the unbiased empirical estimators of the cross-correlation matrix between the state and the observations, and the correlation matrix between the observations, respectively. This double prediction-correction scheme is sequentially repeated over time. Note that the step of prediction-correction of the system state can precede the operation on the parameters without consequence on the quality of the estimation. We can now return to the POD ROM and present the use of the Dual EnKF method to correctly identify the model parameters. In the DA problem, the temporal coefficients a ROM i (t k ) form the state vector x k . The POD ROM in [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. I. Coherent structures[END_REF] serves as the dynamical model M k,k+1 with the model eddy viscosity ν T i as parameters θ. The initial background condition x b 0 is same as the POD temporal coefficients. Finally, as will be discussed in § 4.1, the observations vector y o k is built using velocity measurements in the computational domain.

Following the so-called representers' formulation proposed by Evensen and Leeuwen [START_REF] Evensen | Assimilation of geosat altimeter data for the agulhas current using the ensemble kalman filter with a quasigeostrophic model[END_REF], the Kalman analysis equations ( 25) and ( 26) can be used to characterise the effect of the probe location on the estimated fields. The representers give an estimate of the expected value of the cross correlation between the state and measurements. The influence vectors correspond to the columns of P f,e xy,k+1 and the magnitudes are given by the product (P f,e yy,k+1 +R e k+1 ) -

). The influence vectors give the areas in the computational domain where the corrections from the measurements take place and the magnitudes give the level of correction [see 12]. This framework will be used in §4.1 to analyse the effect of the measurements' locations.

Dynamical modelling of a 2D-cylinder wake flow

In order to test the data assimilation framework in a simple numerical configuration, we choose to estimate the dynamics of a 2D-cylinder wake flow. The snapshots necessary to determine the POD modes (see § 2.1) are obtained from a numerical simulation. A POD reduced-order model is then built using the procedure described in § 2.2. This POD ROM serves as the dynamical model used for the application of the Dual EnKF method described in § 2.4.

Numerical simulation

The numerical simulations are performed using a finite-element based incompressible Navier-Stokes equations solver in FreeFem++ [START_REF] Hecht | New development in FreeFem++[END_REF]. The computational domain along with the dimensions and boundary conditions is shown in Fig. 1. All the dimensions have been parametrised with the diameter of the cylinder D, here set to one. For the purpose of mesh size control, the domain is divided into sub-domains and the automatic mesh generation offered by the buildmesh command in FreeFem++ is used. The mesh has 10263 vertices and 20360 triangles.

To solve the space discretised Navier-Stokes equations, an optimised Newton method is used. This approach is a variant of the classical Newton method for which the non-linear term is discretised semi-implicitly. The velocity and pressure variables are discretised using P 2 and P 1 finite element spaces, respectively. After convergence of the iterative procedure, the pressure field is such that the resulting velocity is divergence free.

The Reynolds number of the numerical simulation based on the cylinder diameter D is set to Re = 100. The snapshots are taken in a time range of [150, 250] s after all initial disturbances have been damped. The simulation generates N t = 1000 snapshots at a sampling frequency of f s = 10 Hz. The snapshots contain the information of the two components of the fluctuating velocity vector u = [u x , u y ] T . The number of degrees of freedom for the problem (size of the discretised state variables) is N s = 81772. The time evolution of the drag and lift coefficient are shown in Fig. 2. From the evolution, it is observed that after the initial transient phase, the periodic vortex shedding is observed from t = 100 s onwards. This implies that the snapshots will represent the dynamics of the post-transient stable vortex shedding over the whole assimilation window.

Characterisation of POD ROM

The POD modes are calculated for the fluctuating velocity fields obtained from the snapshot database. The energy content of the modes is plotted in Fig. 3. We observe that the first ten modes together represent 99.98% of the energy. Therefore, for the further evaluation of the ROM, we reduce the number of modes to N Gal = 10, where N Gal N t . The most dominant temporal and spatial modes are plotted in Fig. 4 and Fig. 5, respectively. From now on, the coefficients and modes obtained by POD will be denoted with the superscript POD. The amplitude of the temporal coefficients a POD Finally, knowing the time evolution of the temporal POD modes, we can apply the identification method described in § 2.2. For the DA application, the initial parameters of the ROM are identified using the first 450 snapshots cor- As already discussed, the POD ROM is in general not sufficiently accurate to describe the original dynamics followed by the POD modes. This is illustrated in Fig. 6 where the ROM built using the parameter values obtained from the leastmean-square minimisation (see § 2.2) and without the model eddy viscosity, i.e. ν T i = 0 (for all i = 1, . . . , N Gal ) is employed to determine the time evolution of the POD coefficients hereafter denoted as a ROM-0 i . We observe that after certain time within the integration window, the true dynamics are not exactly captured by the identified ROM. This is the main motivation to use a DA method like the Dual EnKF to correct the values of the ROM parameters.

Results of Dual EnKF assimilation

The estimation of the state and model parameters is performed by tracking the dynamical system. The configuration of the observer and the estimation results are discussed in this section. Fig. 6: True a POD i (dashed) and estimated a ROM-0 i (solid) temporal POD modes (i = 1, 9) obtained with the identified POD ROM (N Gal = 10) for the 2Dcylinder wake flow at Re = 100. The parameters of the POD ROM are obtained with a least-mean-square minimization procedure.

Setup of observations

The observation vector y in (15) is obtained in the form of measurements of the streamwise velocity component u x at specific locations (probes) in the flowfield. The representers introduced in § 3.1 are shown in Fig. 7 for different probe configurations. Here, N o is equal to the number of probes. By comparing the influence fields of single probe (N o = 1) in Fig. 7a and Fig. 7b, it can be concluded that the measurements taken at points with relatively lower variations have lower contribution to the state correction. In this paper, the DA is performed with multiple probes (N o = 14) which gives a higher magnitude of influence than the single probes as shown in Fig. 7c.

The fluctuating component u x is obtained after subtracting the temporal mean of the measurements over the assimilation window. The measurements are sampled at the same frequency and instants as the flow snapshots. The spatial POD eigenfunction corresponding to each probe serves as the linear observation operator which maps the temporal POD modes obtained from the dynamical model to the observed fluctuation. The observation equation is reformulated as

u x (χ i , t k ) = N Gal j=1 φ POD j (χ i )a j (t k ) + k , (27)
where i = 1, . . . , N χ , k = 1, . . . , N t,DA , and u x ∈ R No .

State and parameter estimation

In order to separately identify the estimated values of the temporal POD coefficients in the assimilation window, these coefficients are denoted by a DA i (i = 1, . . . , N Gal = N s ). In addition, the size of the parameter space θ in terms of the model eddy viscosity ν T i is N p = N Gal . The initial values of the state and parameters and the level of covariances that are used to build the ensemble of the state, parameters and observation are assigned at the start of the assimilation window (see § 2.4). The state and parameters are initialised as a POD i (t * = 0) and ν T i = 10 -3 , respectively, for i = 1, . . . , N Gal . The covariance matrices are fixed for the whole assimilation window as Q k = 10 -8 I N Gal for the state and R k = 10 -6 I No for the observations. The parameter covariance matrix is fixed as C k = 10 -4 I Np . This choice of covariance levels reflects the relatively higher confidence in the dynamical and observation models as compared to confidence in the dynamical model parameters. The ensemble size of the state and parameter space is N e = 50.

The results of the Dual-EnKF assimilation for the POD-ROM is shown in window (t * ≤ 136.4) with the results obtained from the POD ROM as shown in Fig. 6, we observe that the ensemble Kalman filter enables the model to faithfully recover the trajectory corresponding to the POD modes a POD i . This reproduction of the true background trajectory is also important with respect to the parameter update. Indeed, by definition of the Dual EnKF method, the model parameters (model eddy viscosity), are also updated simultaneously under the influence of the state estimation. This parameter evolution within the assimilation window is shown in Fig. 9. All the parameters evolve to stable values by t * = 100.

In order to test the long-term stability of the time integration, the POD ROM with the updated model eddy viscosity obtained at the end of the assimilation (not to scale).

window is used to forecast the dynamics. The temporal modes obtained from this updated model are denoted as a ROM-DA i to highlight the corrections introduced by DA. The initial value for the forecast is taken at the end of the assimilation window as a POD i (t * = 136.4). The result is shown in Fig. 8 for t * > 136.45 in continuation to the assimilation. The evolution is compared with the true values a POD i and a ROM-0 i which is the temporal evolution of the POD ROM without the model eddy viscosity, i.e. ν T i = 0 for i = 1, . . . , N Gal . It is observed that the true values are faithfully reproduced and the characteristic instability of the POD ROM without the model eddy viscosity terms is no longer present. This effect is more prominent at the end of the forecast window and for the less energetically dominant mode (i = 9). The performance is quantified by calculating the rootmean-square error (RMSE) of the forecasted streamwise velocity obtained from both the models. By definition, we have

RMSE(t) = 1 N χ Nχ i=1 (u est x (χ i , t) -u x (χ i , t)) 2 , (28)
where u est x is the estimated velocity determined either with the coefficients a ROM-0 i or a ROM-DA i . The time evolution of RMSE is shown in Fig. 10. The gain in stability of the POD ROM with the assimilated values of the model eddy viscosity is evident from the bounded forecast error. Especially, for t * > 260, the RMSE is lower than the forecast error obtained from the ROM without the model eddy viscosity terms. This result highlights the robust long time performance of the assimilated ROM and also the ability of the assimilation method to provide the correct value of the parameters from sparse observations. Finally, once the temporal POD coefficients have been determined, the already known spatial POD eigenfunctions and mean field can be used in (1) to recover the instantaneous velocity fields as shown in Fig. 11.

Conclusion

The Dual EnKF data assimilation is a well-established method in meteorology. It combines different inhomogeneous sources of information (data, dynamical model) for the dual estimation of the true state and model parameters. This allows the prediction of the dynamical evolution with time. In this paper, this approach has been applied in a fluid mechanics context to a POD ROM of a 2D-cylinder wake flow at low Reynolds number (Re = 100). We observe that the initial POD ROM obtained from identification is not sufficiently stable to be used for the prediction of the long-term dynamics. Therefore, we apply the Dual EnKF algorithm to identify the model parameters. During the assimilation, the sparse observations of the streamwise velocity obtained from probes placed in the flow domain are used to simultaneously estimate the state and model parameters. These parameters correspond to the model eddy viscosity introduced in the Galerkin POD ROM to account for the mode truncation. At the end of the assimilation window, with the updated values of the parameters, the ability of the POD ROM to forecast the flow dynamics is characterised. The Dual-EnKF method provides a robust estimation of the state in the assimilation window by combining the results of the dynamical model (POD ROM) and the measurements. The parameters simultaneously evolve and attain stable values by the end of the assimilation procedure. The POD ROM with the updated parameters provides a stable and more accurate estimate of the long-term dynamics as compared to the ROM without the model eddy viscosity terms.

As an extension of current work, the influence of the initial covariance levels of the state, parameter and observations on the performance of the assimilation method must be studied. Also, the numerical test case studied here features a large fraction of energy captured by a few dominant modes. This limits the scope of improvement of the dynamical model by the application of DA. Therefore, in the future, a similar study will be performed on more chaotic experimental flow cases. Also, the influence field of the probes may be used to propose a methodology for optimal sensor placement.

Fig. 1 :

 1 Fig. 1: (a) Schematic of the computational domain and boundary conditions for the 2D-cylinder wake flow. (b) Finite-element discretisation of the domain obtained from FreeFem++.

2 )Fig. 2 :

 22 Fig. 2: Time evolution of drag coefficient (C d ), lift coefficient (C l ), and pressure drop across the cylinder's leading and trailing surfaces (dP ).

Fig. 3 : 1 and φ POD 2 )Fig. 4 :

 3124 Fig. 3: Eigenvalues spectrum of the POD modes.

4 Fig. 5 :

 45 Fig. 5: Spatial POD modes φ POD i corresponding to the four most energetic modes for the 2D-cylinder wake flow at Re = 100.

Fig. 7 :

 7 Fig. 7: Measurement influence fields for the streamwise velocity component for different probe configurations (a) N o = 1 located at (x * , y * ) = (10, 1), (b) N o = 1 located at (x * , y * ) = (5, 1), and (c) N o = 14 spanning the sub-domain x * ∈ {3.5, 5}, y * ∈ {-3, 3} with uniform separation of ∆x * = 1.5, ∆y * = 1.0. Note that all the figures have the same contour levels.

Fig. 8 .Fig. 8 :

 88 Fig. 8: Evolution of the temporal coefficients a i (i = 1, 9). The estimated a DA i

Fig. 9 :

 9 Fig. 9: Evolution of the model eddy viscosity ν T i (i = 1, . . . , 10) of the POD ROM in the assimilation window. The spread of the parameter ensemble throughout the assimilation stage is shown by the grey shaded area. The red shaded area in the bottom shows evolution of the variance of the parameter space θ = {ν T i } 10 i=1

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Time evolution of RMSE of the forecasted variables u x ROM-0 and u x ROM-DA .

For simplicity, we have considered the field data to be placed on a uniform grid. In general, the cell volume needs to be included in the formulation to represent the spatial inner product. The covariance matrix should therefore be written as X T W X where W holds the spatial weights.


Lakhdar Allagui

Bilel Chouchene

Thomas Gries

Ghouti Medjahdi

Emilien Girot

Xavier Framboisier

Abdesslem Ben

Haj Amara

Balan Lavinia

Lavinia Balan

Raphael Schneider
email: raphael.schneider@univ-lorraine.fr

Raphaël Schneider

Core/shell rGO/BiOBr particles with visible photocatalytic activity towards water pollutants

Keywords: BiOBr, Reduced graphene oxide, Photocatalyst, Charge separation

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

Introduction

In recent years, semiconductor photocatalysis has emerged as one of the most promising methods for energy conversion and environmental remediation [START_REF] Bhatkhande | Photocatalytic degradation for environmental applications -a review[END_REF][START_REF] Kumar | Review on modified TiO 2 photocatalysis under UV/Visible light: selected results and related mechanisms on interfacial charge carrier transfer dynamics[END_REF][START_REF] Ibhadon | Heterogeneous photocatalysis: Recent advances and applications[END_REF][START_REF] Lia | Semiconductor-based photocatalysts and photoelectrochemical cells for solar fuel generation: a review[END_REF][START_REF] Schneider | Understanding TiO 2 photocatalysis: mechanisms and materials[END_REF][START_REF] Ong | A review of ZnO nanoparticles as solar photocatalysts: Synthesis, mechanisms and applications[END_REF]. When considering air and wastewater treatment, photogenerated electrons and holes react with oxygen and water molecules adsorbed at the surface of the photocatalyst to generate reactive species that are able to degrade pollutants into water, carbon dioxide and inorganic salts. Thus, the efficient absorption of visible light and the separation of photogenerated charge carriers constitute key elements to develop photocatalysts exhibiting a high efficiency.

Bismuth oxyhalides BiOX (X = Cl, Br or I) are emerging photocatalysts due to their stability, low toxicity and high photocatalytic efficiency. These materials exhibit a layered structure composed of [Bi 2 O 2 ] 2+ slabs interleaved by double slabs of halogen ions [START_REF] Guan | Vacancy associates promoting solar-driven photocatalytic activity of ultrathin bismuth oxychloride nanosheets[END_REF][START_REF] Pan | Facet-dependent catalytic activity of nanosheet-assembled bismuth oxyiodide microspheres in degradation of bisphenol A[END_REF][START_REF] He | Review on nanoscale Bi-based photocatalysts[END_REF]. The internal static electric field generated by this layered structure improves both the separation and the transfer of photogenerated electron-hole pairs along the (001) direction, which enhance the photocatalytic activity [START_REF] Cheng | Engineering BiOX (X = Cl, Br, I) nanostructures for highly efficient photocatalytic applications[END_REF][START_REF] Ye | Recent advances in BiOX (X = Cl, Br and I) photocatalysts: synthesis, modification, facet effects and mechanisms[END_REF][START_REF] Du | Novel carbon dots/BiOBr nanocomposites with enhanced UV and visible light driven photocatalytic activity[END_REF]. Thus, the development of BiOX photocatalysts with exposed {001} facets is a currently highly studied research topic [START_REF] Zhang | High {001} facets dominated BiOBr lamellas: facile hydrolysis preparation and selective visible-light photocatalytic activity[END_REF][START_REF] Li | Hierarchical {001}-faceted BiOBr microspheres as a novel biomimetic catalyst: dark catalysis towards colorimetric biosensing and pollutant degradation[END_REF][START_REF] Li | Efficient visible light nitrogen fixation with BiOBr nanosheets of oxygen vacancies on the exposed {001} facets[END_REF][START_REF] Wu | Organic-free synthesis of {001} facet dominated BiOBr nanosheets for selective photoreduction of CO 2 to CO[END_REF]. Among bismuth oxyhalides, BiOBr is of high interest for photocatalytic applications. However, due to its relatively wide bandgap of 2.87 eV, BiOBr suffers from its modest visible-light photoresponse. Moreover, BiOBr exhibits a low surface adsorption ability [START_REF] Cao | An in situ Bi-decorated BiOBr photocatalyst for synchronously treating multiple antibiotics in water[END_REF].

Due to their good mechanical stability, excellent electron transport properties (up to 200000 cm 2 /V/s) that enhance photoinduced charge transfer and to their high specific surface area ca. 2600 m 2 /g), graphene-based materials are co-catalysts of high interest for various photocatalytic applications ranging from water remediation to fuel production [START_REF] Fan | Nanocomposites of TiO 2 and reduced graphene oxide as efficient photocatalysts for hydrogen evolution[END_REF][START_REF] Liu | UV-assisted photocatalytic synthesis of ZnO-reduced graphene oxide composites with enhanced photocatalytic activity in reduction of Cr(VI)[END_REF][START_REF] Yu | A noble metal-free reduced graphene oxide-CdS nanorod composite for the enhanced visible-light photocatalytic reduction of CO 2 to solar fuel[END_REF][START_REF] Han | Structural diversity of graphene materials and their multifarious roles in heterogeneous photocatalysis[END_REF][START_REF] Moussa | ZnO rods/reduced graphene oxide composites prepared via a solvothermal reaction for efficient sunlight-driven photocatalysis[END_REF][START_REF] Tatykayev | Synthesis of core/Shell ZnO/rGO nanoparticles by calcination of ZIF8/rGO composites and their photocatalytic activity[END_REF][START_REF] Reddy | Reduced graphene oxide wrapped ZnS-Ag 2 S ternary composites synthesized via hydrothermal method: Applications in photocatalyst degradation of organic pollutants[END_REF][START_REF] Xiang | Graphene-modified nanosized Ag 3 PO 4 photocatalysts for enhanced visible-light photocatalytic activity and stability[END_REF][START_REF] Xu | Noble metal-free RGO/TiO 2 composite nanofiber with enhanced photocatalytic H 2 -production performance[END_REF][START_REF] Li | Graphene-based heterojunction photocatalysts[END_REF][START_REF] Li | Graphene in Photocatalysis: A Review[END_REF].

Moreover, graphene materials allow to extend the visible light absorption of photocatalysts.

The development of new synthetic methodologies for the preparation of BiOBr-graphene composites has been the subject of numerous studies in recent years. Using graphene oxide (GO) or reduced graphene oxide (rGO), BiOBr spherical, plate-like or sheet-like particles associated or deposited at the surface of graphene sheets can be prepared via solvothermal, hydrothermal or precipitation methods [START_REF] Ai | Efficient visible light photocatalytic removal of NO with BiOBrgraphene nanocomposites[END_REF][START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF][START_REF] Song | Microwave-assisted synthesis of BiOBr/graphene nanocomposites and their enhanced photocatalytic activity[END_REF][START_REF] Zhang | Synthesis and photocatalytic activity of graphene/BiOBr composites under visible light[END_REF][START_REF] Liu | One-pot synthesis of graphene-BiOBr nanosheets composite for enhanced photocatalytic generation of reactive oxygen species[END_REF][START_REF] Wei | Advanced visible-light-driven photocatalyst BiOBr-TiO 2 -graphene composite with graphene as a nano-filler[END_REF][START_REF] Zhang | Synthesis of BiOBr-graphene and BiOBr-graphene oxide nanocomposites with enhanced visible light photocatalytic performance[END_REF][START_REF] Vadivel | Graphene oxide-BiOBr composite material as highly efficient photocatalyst for degradation of methylene blue and rhodamine-B dyes[END_REF][START_REF] Liu | Graphene sheets grafted three-dimensional BiOBr0.2I0.8 microspheres with excellent photocatalytic activity under visible light[END_REF][START_REF] Yang | Controllable in situ synthesis of BiOBr x I 1-x solid solution on reduced graphene oxide with enhanced visible light photocatalytic performance[END_REF][START_REF] Li | Synthesis of nitrogen-doped graphene-BiOBr nanocomposites with enhanced visible light photocatalytic activity[END_REF][START_REF] Jiang | Reduced graphene oxide as co-catalyst for enhanced visible light photocatalytic activity of BiOBr[END_REF][START_REF] Patil | Facile sonochemical synthesis of BiOBr-graphene oxide nanocomposite with enhanced photocatalytic activity for the degradation of Direct green[END_REF][START_REF] Rani | Photodegradation of methylene blue using magnetically reduced graphene oxide bismuth oxybromide composite[END_REF][START_REF] Yu | A three-dimensional BiOBr/RGO heterostructural aerogel with enhanced and selective photocatalytic properties under visible light[END_REF][START_REF] Alansi | Visible-light responsive BiOBr nanoparticles loaded on reduced graphene oxide for photocatalytic degradation of dye[END_REF][START_REF] Yu | A three-dimensional BiOBr/RGO heterostructural aerogel with enhanced and selective photocatalytic properties under visible light[END_REF][START_REF] Bao | Novel Z-scheme BiOBr/reduced graphene oxide/protonated g-C 3 N 4 photocatalyst: Synthesis, characterization, visible light photocatalytic activity and mechanism[END_REF]. The rGO/BiOBr composites were found to exhibit a high photocatalytic activity for the degradation of dyes (Methyl Orange, Methylene Blue, Rhodamine), of aromatic derivatives (phenol, nitrobenzene) and for the removal of nitrogen monoxide (NO) [START_REF] Ai | Efficient visible light photocatalytic removal of NO with BiOBrgraphene nanocomposites[END_REF][START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF][START_REF] Song | Microwave-assisted synthesis of BiOBr/graphene nanocomposites and their enhanced photocatalytic activity[END_REF][START_REF] Zhang | Synthesis and photocatalytic activity of graphene/BiOBr composites under visible light[END_REF][START_REF] Liu | One-pot synthesis of graphene-BiOBr nanosheets composite for enhanced photocatalytic generation of reactive oxygen species[END_REF][START_REF] Wei | Advanced visible-light-driven photocatalyst BiOBr-TiO 2 -graphene composite with graphene as a nano-filler[END_REF][START_REF] Zhang | Synthesis of BiOBr-graphene and BiOBr-graphene oxide nanocomposites with enhanced visible light photocatalytic performance[END_REF][START_REF] Vadivel | Graphene oxide-BiOBr composite material as highly efficient photocatalyst for degradation of methylene blue and rhodamine-B dyes[END_REF][START_REF] Liu | Graphene sheets grafted three-dimensional BiOBr0.2I0.8 microspheres with excellent photocatalytic activity under visible light[END_REF][START_REF] Yang | Controllable in situ synthesis of BiOBr x I 1-x solid solution on reduced graphene oxide with enhanced visible light photocatalytic performance[END_REF][START_REF] Li | Synthesis of nitrogen-doped graphene-BiOBr nanocomposites with enhanced visible light photocatalytic activity[END_REF][START_REF] Jiang | Reduced graphene oxide as co-catalyst for enhanced visible light photocatalytic activity of BiOBr[END_REF][START_REF] Patil | Facile sonochemical synthesis of BiOBr-graphene oxide nanocomposite with enhanced photocatalytic activity for the degradation of Direct green[END_REF][START_REF] Rani | Photodegradation of methylene blue using magnetically reduced graphene oxide bismuth oxybromide composite[END_REF][START_REF] Yu | A three-dimensional BiOBr/RGO heterostructural aerogel with enhanced and selective photocatalytic properties under visible light[END_REF][START_REF] Alansi | Visible-light responsive BiOBr nanoparticles loaded on reduced graphene oxide for photocatalytic degradation of dye[END_REF][START_REF] Yu | A three-dimensional BiOBr/RGO heterostructural aerogel with enhanced and selective photocatalytic properties under visible light[END_REF][START_REF] Bao | Novel Z-scheme BiOBr/reduced graphene oxide/protonated g-C 3 N 4 photocatalyst: Synthesis, characterization, visible light photocatalytic activity and mechanism[END_REF].

In this paper, we report the successful association of BiOBr sheets possessing exposed {001} facets with rGO via a hydrothermal process using Bi(NO 3 ) 3 and KBr as precursors of BiOBr.

Results show that rGO was found to deposit at the surface of BiOBr yielding rGO/BiOBr shell/core plate-like particles. The rGO (1%)/BiOBr composite exhibits a high photocatalytic activity under visible light irradiation in the photodegradation of the Orange II diazo dye but also for the removal of the acetaminophen drug from aqueous solutions.

Experimental section

Materials

Graphite powder (Merck), KMnO 4 (> 99%, Sigma), H 2 O 2 (30%, VWR Chemicals), sodium nitrate (> 99%, Sigma), H 2 SO 4 (reagent grade), L-ascorbic acid (reagent grade, Sigma), Bi(NO 3 ) 3 . 5H 2 O (>99.99%, Sigma), KBr (>99%, Sigma), NaOH (> 97%, Sigma), and ethanol were used as received without further purification. All solutions were prepared using Milli-Q water (18.2 MΩ.cm, Millipore) as solvent.

Synthesis of exfoliated graphene oxide

Graphite oxide was prepared by oxidation of graphite by a modified Hummer's method followed by an exfoliation [START_REF] Wang | Biocompatibility of graphene oxide[END_REF]. Briefly, graphite (2 g) and NaNO 3 (1 g) were mixed in a flask cooled at 4°C using an ice bath. Then, concentrated H 2 SO 4 (50 mL) was dropwise added to the flask and the mixture was stirred for 30 min. KMnO 4 (0.3 g) was then added and the mixture further stirred for 30 min. Next, KMnO 4 (7 g) was gradually added and the mixture further stirred for 1 h at a temperature below 20°C. The temperature was raised to 35°C and the solution was stirred for 2 h. Then, H 2 O (90 mL) was slowly added (the temperature increased to 70°C) and the mixture stirred for 15 min, followed by the addition of mixture of 7 mL H 2 O 2 and 55 mL of water to reduce the residual permanganate to soluble manganese ions. The resulting suspension was filtered, washed by centrifugation with 5% HCl (2500 rpm, 10 min, 4 times) followed by water (4000 rpm, 10 min, 5 times) and finally dried at 60°C for 24 h to obtain a brownish graphite oxide powder. The graphite oxide powder was dispersed in water (1 mg/mL) to obtain a suspension. Then, the suspension was ultra-sonicated for 1 h by probe sonication to obtain a graphene oxide suspension. Finally, the suspension was centrifuged at 5000 rpm for 15 min to collect graphene oxide (GO) pellets. The GO pellets were dried at 60°C for 24 h before use.

Synthesis of reduced graphene oxide

The reduction of GO into rGO was conducted according to the procedure described by Zhang et al. [START_REF] Zhang | Reduction of grapheme oxide via L-ascorbic acid[END_REF] with slight modifications. Briefly, GO (15 mg) was dispersed in 50 mL water at room temperature. L-Ascorbic acid (200 mg) was next added and the mixture stirred for 24 h under ambient conditions. rGO was recovered by centrifugation (4000 rpm for 15 min), washed with water, and dried overnight at 50°C.

Synthesis of BiOBr/rGO composites

Pure BiOBr was prepared according to reference [START_REF] Wu | Organic-free synthesis of {001} facet dominated BiOBr nanosheets for selective photoreduction of CO 2 to CO[END_REF]. rGO/BiOBr composites were prepared according to the following synthetic protocol. Bi(NO 3 ) 3 .5H 2 O (4 mmol) and rGO were dispersed under sonication in 36 mL of 1M HNO 3 . The loading in rGO was varied to prepare composites containing 0.5, 1, 2.5 and 5 wt% rGO relative to BiOBr. KBr (4 mmol) was dissolved into 12 mL of HNO 3 (1M). Then, the KBr solution was added dropwise to the Bi(NO 3 ) 3 -rGO mixture and the solution stirred for 30 min at room temperature. Next, the mixture was transferred into a 100 mL Teflon-lined autoclave and maintained at 160 °C for 12 h. After cooling, the solid was recovered by centrifugation (4000 rpm for 15 min), washed with absolute ethanol (4 x 20 mL) and dried at 70°C for 12 h.

Photocatalytic activity

The photocatalytic activities of BiOBr and rGO/BiOBr composites were evaluated in the degradation of the Orange II dye and of acetaminophen under visible light irradiation. All experiments were conducted at room temperature in a batch reactor equipped with a 400 nm cutoff filter. 30 mL of solution containing the catalyst and the pollutant (concentration of 10 mg/L for Orange II and 5 mg/L for acetaminophen) were magnetically stirred for 30 min in the dark to attain the adsorption-desorption equilibrium. For the degradation of Orange II, Sylvania LuxLine FHO T5 neon tubes were used as light source (light intensity at the surface of the Orange II solution = 8 mW/cm 2 ). At given intervals, 2 mL of the solution were extracted and centrifuged (15000 rpm for 2 min) to remove the photocatalyst. The concentration of Orange II was determined by measuring the maximum absorbance at 485 nm using a UV-visible spectrometer. For the degradation of acetaminophen, a Hg/Xe lamp was used as light source (light intensity at the surface of the acetaminophen solution = 20 mW/cm 2 ). The characteristic absorption peak of acetaminophen at 242 nm was used to determine the extent of photodegradation. Samples were also analyzed by HPLC using a Shimadzu model LC30 system (Shimadzu Corporation, Japan). 50 µL of sample were injected without column. The mobile phase consisted in water and acetonitrile in a 70/30 proportion (v/v) with 0.1 % formic acid (v/v). Samples were eluted at a flow rate of 0.2 mL/min. The UV signal was measured at 254 nm with a photometric diode array and using a cell with an optical path of 1 cm.

Reactive species produced during photocatalytic experiments were detected via in situ trapping experiments. p-Benzoquinone (BQ, O 2

•-radical scavenger), ammonium oxalate (AO, hole scavenger) and DMSO (electron scavenger) were added to the pollutant and photocatalyst solution before the adsorption-desorption phase. Tert-butanol (t-BuOH, • OH radical scavenger) was added to the solution before the irradiation phase. All scavengers were used at a 10 mM concentration except BQ that was used at a 0.1 mM concentration.

Characterization

Transmission electron microscopy (TEM) images were taken by placing a drop of the particles dispersed in water onto a carbon film-supported copper grid. Samples were studied using a Philips CM200 instrument operating at 200 kV. The X-ray powder diffraction (XRD) patterns were measured using Panalytical X'Pert Pro MPD diffractometer using Cu Kα radiation. Thermogravimetric analysis was conducted under O 2 atmosphere from room temperature to 1000°C at a heating rate of 10°C/min using a TGA/DSC1 STAR equipment (Mettler-Toledo)

A Micromeritics Tristar II Plus Surface Characterization Analyzer instrument was used to measure the Brunauer-Emmet-Teller (BET) surface areas of the samples using liquid nitrogen (-196 °C). Prior to the analyses, the samples were out-gassed overnight under primary vacuum at 40°C on the ports of the Micromeritics VacPrep 061 degasser followed by 4 h outgassing under primary vacuum on the analyse ports. The resulting isotherms were analysed using the BET method while the pore volume (Vpore) was determined using the total volume of nitrogen adsorbed at saturation.

All the optical measurements were performed at room temperature (20 ± 1°C) under ambient conditions. Absorption spectra of liquid samples were recorded on a Thermo Scientific Evolution 220 UV-visible spectrophotometer. The diffuse reflectance absorption spectra (DRS) were collected on a Shimadzu 2600-2700 UV-visible spectrophotometer.

BaSO 4 powder was used as a standard for baseline measurements and spectra were recorded in a range of 250-750 nm. Photoluminescence emission spectra were measured on a Horiba Fluoromax-4 Jobin Yvon spectrofluorimeter.

The photoelectrochemical performance of as-prepared samples was measured in a three To identify intermediates produced during the degradation of acetaminophen, samples analyzed by HPLC were introduced in a Shimadzu MS2020 model mass spectrometer.

Acquisition were done in ESI +/-mode, the scan was 50 to 500 uma in TIC conditions and three SIMS (152, 174 and 150 uma) measures were done simultaneously.

Results and discussion

Characterization of rGO/BiOBr photocatalysts

rGO/BiOBr composites were prepared by varying the loading in rGO (0.5, 1, 2.5 and 5 wt%)

during the hydrothermal synthesis of BiOBr conducted at 160°C for 12 h. Thermogravimetric analysis (TGA) was used to get information about the thermal stability and to determine the rGO content in rGO/BiOBr composites (Fig. S1). The weak weight loss (< 0.5%) observed by heating until 150°C originates from the removal of water molecules adsorbed at the surface of the photocatalysts. Two apparent mass loss can be observed at higher temperatures for all materials. The first one occurring between 200 and 550°C originates from the decomposition of rGO, while the second one between 600 and 700°C can be attributed to the decomposition of BiOBr into Bi 2 O 3 . These results are in agreement with previous reports [START_REF] Song | Microwave-assisted synthesis of BiOBr/graphene nanocomposites and their enhanced photocatalytic activity[END_REF]. The first weight loss was used to determine the rGO content in rGO/BiOBr composites (Fig. S1b). For theoretical loadings in rGO of 0.5, 1, 2.5 and 5 wt%, the actual loadings were determined to be 0.29, 0.76, 1.32 and 1.78%, respectively.

The diffraction pattern of BiOBr shows intense and sharp signals which are indicative of its good crystallinity (Fig. 1a). These peaks can be indexed to a tetragonal phase with a space group of P4/nmn (129). The FT-IR spectrum of BiOBr shows a signal located at 500 cm -1 corresponding to the Bi-O vibration and this peak is present in all rGO/BiOBr composites (Fig. 1b) [START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF]. Weak signals centered at 1721 and 1605 cm -1 corresponding to the C=O vibration and to the skeletal vibration of rGO, respectively, can also be observed for the composites loaded with 2.5 and 5% rGO, indicating that GO was not fully reduced after the treatment with ascorbic acid followed by the hydrothermal process. The TEM images confirm the sheet-shaped structure of BiOBr and rGO/BiOBr composites (Fig. 4a-b). As shown in Fig. 4b, rGO deposits at the BiOBr sheets, indicating that a composite associating rGO and BiOBr with close contact was successfully prepared. The association of rGO and its relative uniform dispersion at the surface of BiOBr sheets was demonstrated by the energy dispersive X-ray spectroscopy (EDS) mapping (Fig. 5), in which a thin layer of C can be observed at the surface of BiOBr (Fig. 5e). This result is in accordance with our previous report showing that rGO deposits at the surface of nanocrystals during their solvothermal or hydrothermal synthesis [START_REF] Moussa | ZnO rods/reduced graphene oxide composites prepared via a solvothermal reaction for efficient sunlight-driven photocatalysis[END_REF][START_REF] Tatykayev | Synthesis of core/Shell ZnO/rGO nanoparticles by calcination of ZIF8/rGO composites and their photocatalytic activity[END_REF]. An angle of 45° between the (200) and

(110) planes of BiOBr was also determined from the associated selected-area electron diffraction pattern (SAED) (inset of Fig. 4b), confirming that the favoured growth direction is along the [001] orientation. The HR-TEM image of the rGO (1%)/BiOBr sample shows continuous lattice fringes with an interplanar spacing of 0.27 nm corresponding to the (110) atomic plane of tetragonal BiOBr (d-spacing of 0.277 nm for the (110) reflection) (Fig. 4c).

The fringes with spacing of 0.32 nm correspond to the (002) plane of rGO, which further confirms that rGO was successfully associated to BiOBr. shows the SAED pattern. The surface element composition and the chemical state of elements present in the rGO (1%)/BiOBr composite were investigated by XPS (Fig. 6). The survey XPS spectrum shows that the sample is only composed of Bi, O, Br and C elements (Fig. S2). Two signals located at 160.17 and 165.48 eV corresponding to Bi 4f 7/2 and Bi 4f 5/2 , respectively, can be observed in the high resolution spectrum of Bi [START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF]. The splitting between these signals is of 5.31 eV, which confirms that Bi is present in the +3 oxidation state in the composite (Fig. 6a). The O 1s peak can be deconvoluted into two signals located at 530.78 and 533.18 eV, which can be attributed to O in BiOBr [START_REF] Liu | Graphene sheets grafted three-dimensional BiOBr0.2I0.8 microspheres with excellent photocatalytic activity under visible light[END_REF] and to O of C-O bonds in rGO [START_REF] Moussa | ZnO rods/reduced graphene oxide composites prepared via a solvothermal reaction for efficient sunlight-driven photocatalysis[END_REF], respectively (Fig. 6b). The Br 3d 5/2 and Br 3d 3/2 peaks appear at 69.10 and 70.12 eV, respectively (Fig. 6c). Finally, the C 1s signal can be deconvoluted into two peaks at 284.92 and 286.70 eV, corresponding to graphitic carbons and C-O bonds (alcohol, acid and epoxide functions) (Fig. 6d) [START_REF] Moussa | ZnO rods/reduced graphene oxide composites prepared via a solvothermal reaction for efficient sunlight-driven photocatalysis[END_REF][START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF].

These results further confirm that rGO/BiOBr composites were successfully synthesized via the hydrothermal method developed in the work. The UV-visible absorption spectra of BiOBr and of the rGO/BiOBr composites are shown in Fig. 7. rGO exhibits a dark grey color and a continuous absorption over the whole visible region. As can be seen, the association of BiOBr with rGO increases the visible light absorption likely due to the strong chemical bonds between these materials. The bandgaps of rGO/BiOBr composites were determined by plotting (αhν) 1/2 vs hν (where α is the absorption coefficient and hν is the photon energy) and by extrapolating the linear part of the curves to the x axis (Fig. 7b). The bandgap of BiOBr is 2.70 eV, which is in accordance with previous reports [START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF][START_REF] Zhang | Synthesis and photocatalytic activity of graphene/BiOBr composites under visible light[END_REF]. Upon increasing the rGO content in the composites, a continuous decrease of the bandgap is observed (2.66, 2.60, 2.48 and 2.33 eV for composites loaded with 0.5, 1, 2.5 and 5% rGO, respectively), indicating a higher sensitivity of the composites to visible light which should be beneficial to the photocatalytic activity. The decrease of bandgap observed for rGO/BiOBr composites likely originates from the formation of Bi-C bonds during the hydrothermal process via the bonding of some unpaired electrons on the surface of BiOBr, which is in accordance with previous reports [START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF].

Photocatalytic degradation of Orange II

The photocatalytic activity of BiOBr and rGO/BiOBr composites was first evaluated in the degradation of the Orange II dye under visible light irradiation (intensity of 8 mW/cm 2 ). A series of control experiments show that no bleaching of Orange II occurred in the absence of the photocatalyst or without light irradiation. After 30 min stirring in the dark to reach the adsorption-desorption equilibrium (ca. 10% adsorption of the dye at the surface of the photocatalyst), light was turned on. The photocatalytic was found to depend on the rGO content in the composite and the highest activity towards the degradation of Orange II was observed for the rGO (1%)/BiOBr composite (Fig. 8a). The increase of the rGO loading (2.5 or 5%) was found to be detrimental on the photocatalytic activity likely due to the decrease of The photodegradation of Orange II follows the pseudo-first-order model ln (C 0 /C) = kt, where C 0 and C are the concentrations of the dye at t = 0 and t, respectively, and k is the rate constant (Fig. S3). The rate constants determined for rGO/BiOBr with loadings in rGO of 0.5, 1, 2.5 and 5% are 0.029, 0.040, 0.024 and 0.028 min -1 , respectively. These values confirm that the rGO (1%)/BiOBr material is the most effective for the bleaching of Orange II.

Although dyes, amounts of catalysts used and light sources are not similar to those used in this work, the rGO (1%)/BiOBr favorably compares with graphene oxide-modified BiOBr photocatalysts described in the literature which allow between 75 and 100% dye bleaching after ca. 75-120 min visible light irradiation [START_REF] Ai | Efficient visible light photocatalytic removal of NO with BiOBrgraphene nanocomposites[END_REF][START_REF] Tu | One-pot synthesis, characterization, and enhanced photocatalytic activity of a BiOBr-graphene composite[END_REF][START_REF] Song | Microwave-assisted synthesis of BiOBr/graphene nanocomposites and their enhanced photocatalytic activity[END_REF][START_REF] Zhang | Synthesis and photocatalytic activity of graphene/BiOBr composites under visible light[END_REF][START_REF] Liu | One-pot synthesis of graphene-BiOBr nanosheets composite for enhanced photocatalytic generation of reactive oxygen species[END_REF][START_REF] Wei | Advanced visible-light-driven photocatalyst BiOBr-TiO 2 -graphene composite with graphene as a nano-filler[END_REF][START_REF] Zhang | Synthesis of BiOBr-graphene and BiOBr-graphene oxide nanocomposites with enhanced visible light photocatalytic performance[END_REF][START_REF] Vadivel | Graphene oxide-BiOBr composite material as highly efficient photocatalyst for degradation of methylene blue and rhodamine-B dyes[END_REF][START_REF] Liu | Graphene sheets grafted three-dimensional BiOBr0.2I0.8 microspheres with excellent photocatalytic activity under visible light[END_REF][START_REF] Yang | Controllable in situ synthesis of BiOBr x I 1-x solid solution on reduced graphene oxide with enhanced visible light photocatalytic performance[END_REF][START_REF] Li | Synthesis of nitrogen-doped graphene-BiOBr nanocomposites with enhanced visible light photocatalytic activity[END_REF][START_REF] Jiang | Reduced graphene oxide as co-catalyst for enhanced visible light photocatalytic activity of BiOBr[END_REF][START_REF] Patil | Facile sonochemical synthesis of BiOBr-graphene oxide nanocomposite with enhanced photocatalytic activity for the degradation of Direct green[END_REF][START_REF] Rani | Photodegradation of methylene blue using magnetically reduced graphene oxide bismuth oxybromide composite[END_REF][START_REF] Yu | A three-dimensional BiOBr/RGO heterostructural aerogel with enhanced and selective photocatalytic properties under visible light[END_REF][START_REF] Alansi | Visible-light responsive BiOBr nanoparticles loaded on reduced graphene oxide for photocatalytic degradation of dye[END_REF][START_REF] Yu | A three-dimensional BiOBr/RGO heterostructural aerogel with enhanced and selective photocatalytic properties under visible light[END_REF][START_REF] Bao | Novel Z-scheme BiOBr/reduced graphene oxide/protonated g-C 3 N 4 photocatalyst: Synthesis, characterization, visible light photocatalytic activity and mechanism[END_REF].

Photocatalytic degradation of acetaminophen, catalyst stability and influence of pH

Pharmaceuticals are emerging pollutants that can be found as trace amounts among others in drinking water, groundwater, soils and sediments. Acetaminophen (ACT), also called paracetamol, is one of the most commonly used drug in the world due to its analgesic and antipyretic properties [START_REF] Gotootos | Kinetic study of acetaminophen degradation by visible light photocatalysis[END_REF][START_REF] Chang | Photocatalytic degradation of acetaminophen in aqueous solutions by TiO 2 /ZSM-5 zeolite with low energy irradiation[END_REF][START_REF] Fan | Degradation of acetaminophen in aqueous solution under visible light irradiation by Bi-modified titanate nanomaterials: morphology effect, kinetics and mechanism[END_REF]. Between 58 and 68% of ACT is excreted from the body after ingestion [START_REF] Muir | Comparative bioavailability of aspirin and paracetamol following single dose administration of soluble and plain tablets[END_REF]. Due to its modest biodegradability, ACT can be found at concentrations up to 65 µg/L in effluents. Moreover, ACT has been demonstrated to be toxic towards bacteria and plants [START_REF] Gotootos | Kinetic study of acetaminophen degradation by visible light photocatalysis[END_REF][START_REF] Chang | Photocatalytic degradation of acetaminophen in aqueous solutions by TiO 2 /ZSM-5 zeolite with low energy irradiation[END_REF][START_REF] Fan | Degradation of acetaminophen in aqueous solution under visible light irradiation by Bi-modified titanate nanomaterials: morphology effect, kinetics and mechanism[END_REF].

The rGO (1%)/BiOBr was also found to be effective for the degradation of ACT as shown in Fig. 9a. After 105 min visible light irradiation, the removal efficiency of ACT is of 93% while it is only of 55% using pure BiOBr (Fig. 9b). The k values are 0.006 and 0.003 min -1 for rGO (1%)/BiOBr and BiOBr, respectively (Fig. S4). After the photodegradation of ACT by the rGO (1%)/BiOBr catalyst, the total organic carbon (TOC) decreased by ca. 53% after 105 min of irradiation, further indicating that a great part of ACT was decomposed into CO 2 and H 2 O.

Recently, numerous photocatalysts like TiO 2 , TiO 2 /graphene, C-modified TiO 2 or Ti-Zr MOFs have been reported for the degradation of ACT under UV or simulated solar light irradiation [START_REF] Lin | Hollow mesoporous TiO 2 microspheres for enhanced photocatalytic degradation of acetaminophen in water[END_REF][START_REF] Tao | Enhanced photoactivity of graphene/titanium dioxide nanotubesfor removal of Acetaminophen[END_REF][START_REF] Xiong | Decomposition of acetaminophen (Ace) using TiO 2 /UVA/LED system[END_REF][START_REF] Gomez-Aviles | C-modified TiO 2 using lignin as carbon precursor for the solar photocatalytic degradation of acetaminophen[END_REF][START_REF] Gomez-Aviles | Mixed Ti-Zr metal-organic-frameworks for the photodegradation of acetaminophen under solar irradiation[END_REF][START_REF] Nasr | Photocatalytic degradation of acetaminophen over Ag, Au and Pt loaded TiO 2 using solar light[END_REF][START_REF] Yan | Fabrication of In 2 S 3 /Zn 2 GeO 4 composite photocatalyst for degradation of acetaminophen under visible light[END_REF]. The degradation efficiencies vary from 40 to 95% for irradiation times between 1 and 4 h. The photodegradation of ACT under visible light has been more rarely investigated.

Catalysts like In 2 S 3 /Zn 2 GeO 4 or Ag-, Pt-or Au-loaded TiO 2 that allow 95 and 100% degradation of ACT after 6 and 3 h irradiation, respectively, have recently been described [START_REF] Nasr | Photocatalytic degradation of acetaminophen over Ag, Au and Pt loaded TiO 2 using solar light[END_REF][START_REF] Yan | Fabrication of In 2 S 3 /Zn 2 GeO 4 composite photocatalyst for degradation of acetaminophen under visible light[END_REF]. Compared with these catalysts, the rGO (1%)/BiOBr photocatalyst exhibits a significantly higher activity (93% degradation in 105 min under visible light irradiation), indicating that the association of rGO with BiOBr promotes the generation of reactive species able to decompose ACT.

The stability of the rGO/BiOBr material for the degradation of ACT was evaluated by recycling the photocatalyst. After each run, the catalyst was recovered by centrifugation and reused without any treatment (Fig. 9c). No significant decrease of the photocatalytic activity was observed after the fifth reuse indicating that the rGO/BiOBr did not photocorrode during the repeated degradations of ACT, which further proves its high potential for water decontamination.

The pH of the aqueous solution is one key parameter influencing the photocatalytic activity since the charge of the catalyst and of the pollutant and thus their electrostatic interactions but also the positions of valence and of the conduction bands of the photocatalyst depend on pH [START_REF] Chiou | Influence of operating parameters on photocatalytic degradation of phenol in UV/TiO 2 process[END_REF][START_REF] Moussa | Growth of ZnO nanorods on graphitic carbon nitride gCN sheets for the preparation of photocatalysts with high visible-light activity[END_REF]. The point of zero charge (pzc) of the rGO (1%)/BiOBr catalyst is of 5.11, value slightly higher to that pure BiOBr (4.81) (Fig. S5a). The pH of the ACT solution was adjusted using 0.1 M NaOH or HCl before conducting photocatalytic experiments and the results are depicted in Fig. S5b. At pH values below the pzc, a decrease of the photocatalytic activity efficiency of the rGO (1%)/BiOBr composite is observed (78 and 89% degradation at pH 2.5

Mechanism

The specific surface area and the porosity of BiOBr and of rGO/BiOBr composites were characterized via N 2 adsorption-desorption isotherms and the results are given in Table 1.

The isotherms are of type III according to the IUPAC classification, indicating the presence of mesopores (Fig. 10a). The first hysteresis loop observed in the lower relative pressure range (0.4 < P/P 0 < 0.8) is related to the pores originating from the association of BiOBr sheets into BiOBr nanoflowers while the second one observed at higher relative pressure (0.9 < P/P 0 <1) is associated to the pores present between the secondary particles, which is in accordance with previous reports [START_REF] Jia | Simple solvothermal routes to synthesize 3D BiOBr x I 1-x microspheres and their visible-light-induced photocatalytic properties[END_REF][START_REF] Yu | Microstructures and photoactivity of mesoporous anatase hollow microspheres fabricated by fluoride-mediated self-transformation[END_REF]. These results are consistent with the SEM images of BiOBr and rGO/BiOBr composites shown in Fig. 3.

Pure BiOBr exhibits a BET surface area of 7.38 m 2 /g. A significant increase of the BET surface area is observed for the rGO (0.5%)/BiOBr material (13.36 m 2 /g). Further increasing the loading in rGO from 1 to 5% induces a gradual decrease of the BET surface area (from 9.88 to 3.61 m 2 /g), likely originating from the deposition of rGO at the periphery of BiOBr sheets as

shown by TEM analyses. The rGO (1%)/BiOBr material exhibits the highest photocatalytic activity but does not possess the highest BET surface area, which suggests that the BET surface area does not play a key role on the catalytic efficiency of the rGO/BiOBr samples.

The Barrett-Joyner-Hallenda (BJH) method was used to determine the pore volumes (Fig. 10b). Results obtained show that the pore volume was not significantly affected by the loading with rGO but a continuous increase of the pore size was observed. The small pores with sizes varying from 7.52 to 11.11 nm are likely associated to the stacked nanosheets in BiOBr nanoassemblies. The photocatalytic efficiency also depends on the recombination rate of photogenerated electron/hole pairs. If the recombination is fast, less time is allowed to electrons and holes to react with oxygen and water molecules adsorbed at the surface of the catalyst and thus the production of the reactive species responsible for the photodegradation is decreased.

The charge carrier separation and their lifetime were first investigated by photoluminescence (PL) spectroscopy (Fig. 11a). After excitation at 312 nm, pure BiOBr exhibits a broad PL ranging from 375 to 550 nm. After association with rGO, the shape of the PL signal is not altered but its intensity markedly decreased indicating that the electron/hole pair recombination in BiOBr is inhibited by the presence of rGO.

This was further evidenced by the transient photoelectrochemical (PEC) responses as these measurements correlate with the recombination efficiency of photogenerated charge carriers [START_REF] Moussa | Growth of ZnO nanorods on graphitic carbon nitride gCN sheets for the preparation of photocatalysts with high visible-light activity[END_REF]. The photocurrent densities of rGO/BiOBr composites are in good accordance with their photocatalytic activities, the separation of electron-hole pairs being the most efficient in the rGO (1%)/BiOBr composite (Fig. 11b). Under visible light irradiation, the photocurrent measured for BiOBr deposited on FTO is of ca. 0.00085 mA. After association with 1 wt% rGO, the photocurrent increased to ca. 0.001 mA, indicating that the rGO (1%)/BiOBr composite more efficiently transfers the photogenerated charges to the working electrodes and thus that the separation of charge carriers is enhanced. Finally, the photocurrent quickly increases and remains almost stable when the light is turned on. A good reproducibility was also observed for all materials during the switching on and off cycles, indicating that the photoresponses are stable and reversible.

Electrochemical impedance spectroscopy was also used to get deep insight into the charge transport behavior in the absence of light excitation. All Nyquist plots exhibit semi-circle like curve and the diameters can be used to assess the impedance (Fig. 11c). The smallest semicircle is observed for the rGO (1%)/BiOBr sample, indicating that the association of BiOBr with 1% rGO can effectively reduce the charge transfer resistance at the material surface/interface. This result indicates a strongly improved electronic conductivity in the non-photoexcited state, which is beneficial for the charge separation during photocatalysis. To determine the role played by the various reactive species generated during the photocatalytic process, different scavengers of these species were added during the degradation of ACT and the results are shown in Fig. 12a •-radicals and holes play a key role in the photocatalytic process.

The major role of O 2

•-radicals was further demonstrated by removing O 2 from the ACT solution by Ar bubbling before conducting the photodegradation of ACT (Fig. S5). In these experimental conditions, the degradation efficiency was reduced to 53%.

The electrochemical potentials of the valence band (VB) and of the conduction band (CB) of BiOBr are of high importance to understand the photodegradation mechanism over the rGO/BiOBr composite. E VB was estimated according to the following equation :

E VB =  -E e + 0.5 Eg
where  is the absolute electronegativity of BiOBr (6.176), E e is the energy of free electrons on the hydrogen scale (ca. 4.5 eV) and Eg is the bandgap (2.60 eV for the rGO (1%)/BiOBr composite) [START_REF] Xu | The absolute energy positions of conduction and valence bands of selected semiconducting minerals[END_REF]. The VB and CB edge potentials of BiOBr were estimated to be 2.976 and 0.376 eV vs NHE, respectively (Fig. 12b).

On the basis of the previously described experimental data, the photocatalytic mechanism can be explained as follows (Fig. 12c). Under visible light irradiation, the electrons in the VB of BiOBr are excited to the CB, leaving holes in the VB. Due to its p-type semiconductor nature, the Fermy energy level of BiOBr is located near the VB edge. The photogenerated electrons in the CB of BiOBr easily transfer to rGO due to the high working function of this material (0.08 eV vs NHE) and due to the internal electron field at the rGO/BiOBr interface [START_REF] Zhu | Enhanced photocatalytic hydrogen evolution performance based on Ru-trisdicarboxybipyridine-reduced graphene oxide hybrid[END_REF]. The Schottky barrier at the interface of BiOBr and rGO inhibits the electron/hole recombination in BiOBr [START_REF] Chen | Preparation of a novel TiO 2based p-n junction nanotube photocatalyst[END_REF][START_REF] Ai | Efficient visible light photocatalytic removal of NO with BiOBrgraphene nanocomposites[END_REF], as previously demonstrated by PL and PEC measurements.

Due to the ability of rGO in storing and shuttling electrons, the electrons collected by rGO can react with oxygen molecules adsorbed at the surface of the photocatalyst to produced

Photodegradation intermediates of ACT

The intermediates formed during the photodegradation of ACT were identified by flow injection analysis-mass spectrometry (FIA-MS) in ESI mode. Among the products identified is hydroquinone (m/z = 110) whichis likely formed via an ipso-substitution of the amide function by O 2

•-radicals generated during photocatalysis followed by the rupture of the hydroperoxyde bond (Fig. 13). Hydroxyquinol (m/z = 126) could also be detected as an intermediate, indicating a further hydroxylation of the aromatic ring (Fig. S7). Next, hydroquinone and hydroxyquinol are oxidized into mono-and dicarboxylic acids by cleavage of the aromatic ring. These results are in agreement with the photocatalytic pathways described in the literature for ACT [START_REF] Tao | Enhanced photoactivity of graphene/titanium dioxide nanotubesfor removal of Acetaminophen[END_REF][START_REF] Moctezuma | Photocatalytic degradation of paracetamol: Intermedites and total reaction mechanism[END_REF].

Conclusion

In this paper, a hydrothermal method allowing the deposition of rGO at the surface of BiOBr sheets with exposed {001} facets was developed. The rGO (1%)/BiOBr composite exhibits high photocatalytic activity under visible light irradiation for the degradation of Orange II and acetaminophen. On the basis of the characterization results, the enhanced photocatalytic activity of the rGO (1%)/BiOBr composite originates from the increased visible light absorption and from the effective charge separation and transportation due to the strong association of rGO with BiOBr sheets. This work highlights the potential of graphenebased materials for the design of heterostructured photocatalysts with high activity for environmental applications.

  electrode cell with Pt wire as the counter electrode, a saturated Ag/AgCl electrode as reference electrode and FTO glass coated with BiOBr or rGO/BiOBr materials with a surface area of 1 cm 2 as the working electrode. 0.1 M Na 2 SO 4 aqueous solution was used as electrolyte. The photocurrent densities were evaluated by measuring the current with and without 100 mW/cm 2 light illumination provided by a 300 W Xe arc lamp fitted with an AM 1.5G filter. The photocurrent response was recorded at a constant potential of + 0.30 V vs. the reference electrode by using a SP150 BioLogic potentiostat. Electrochemical impedance spectra (EIS) measurements were performed in the same workstation by applying a sinusoidal AC perturbation of 10 mV with a frequency range from 500 kHz to 0.01 Hz under open circuit potential conditions.

Fig. 1 .

 1 Fig. 1. (a) XRD patterns and (b) FT-IR spectra of BiOBr and rGO/BiOBr composites.

Fig. 2 .Fig. 3 .

 23 Fig. 2. (a) XRD Raman spectra of BiOBr and rGO/BiOBr composites and (b) magnification of the 1250-1750 cm -1 region for rGO (2.5%) and rGO (5%)/BiOBr composites.

Fig. 4 .

 4 Fig. 4. (a-b) TEM and (c) HR-TEM images of the rGO (1%)/BiOBr composite. The inset of (b)

Fig. 5 .

 5 Fig. 5. (a) TEM image of the rGO (1%)/BiOBr composite, (b-e) EDX elemental mapping images of Bi, Br, O and C elements, respectively.

Fig. 6 .

 6 Fig. 6. High-resolution XPS spectra of Bi 4f, O 1s, Br 3d and C 1s for the rGO (1%)/BiOBr composite.

Fig. 7 .

 7 Fig. 7. (a) UV-vis absorption spectra of BiOBr and of rGO/BiOBr composites. (b) Plots of (αhν) 1/2 vs hν for the determination of the bandgaps of the photocatalysts.

  photocatalytically active BiOBr due to the deposition of rGO at its surface as previously shown by TEM. The time-dependent UV-vis absorption spectra of Orange II during its photodegradation with the rGO (1%)/BiOBr composite show a gradual decrease of the maximum absorption peak located at ca. 485 nm and the degradation efficiency reached 97% after 90 min irradiation. Meanwhile, the color of the dye solution changed from orange to transparent and no new absorption signals were observed, further indicating the effective degradation of Orange II.

Fig. 8 .

 8 Fig. 8. (a) Photocatalytic performances of BiOBr and rGO/BiOBr composites for the degradation of Orange II under visible light irradiation (intensity of 8 mW/cm 2 ). (b) UV-vis absorption spectral changes during the photodegradation of Orange II with the rGO (1%)/BiOBr composite

and 4 . 5 ,

 45 respectively, after 105 min irradiation). The optimal degradation of ACT was observed at pH values ranging from 5.5 to 9.5, due to the association via hydrogen bonds between ACT and the negatively charged surface of the catalyst. The modest photocatalytic activity (60% degradation avec 105 min irradiation) observed at pH 11.5 likely originates from the repellency of the negatively charged ACT (pK phenol = 9.38) with the negatively charged photocatalyst.

Fig. 9 .

 9 Fig. 9. (a) Temporal UV-vis absorption changes during the degradation of ACT under visible light irradiation (intensity of 20 mW/cm 2 ). (b) Photocatalytic degradation efficiencies of ACT

Fig. 10 .

 10 Fig. 10. (a) Nitrogen adsorption-desorption isotherms of BiOBr and rGO/BiOBr composites and (b) the corresponding pore size distributions.

Fig. 11 .

 11 Fig. 11. (a) PL emission spectra, (b) transient photocurrent responses and (c) EIS Nyquist plots of BiOBr and rGO/BiOBr composites.

O 2 •

 2 -. In the meantime, holes in the VB of BiOBr directly oxidize ACT into ACT + and, to a much lower extent, react with H 2 O molecules to give • OH radicals.

Fig. 12 .

 12 Fig. 12. (a) Influence of scavengers on the photodegradation of ACT using the rGO (1%)/BiOBr catalyst under visible light irradiation, (b) Schematic band structure of the rGO

Fig. 13 .

 13 Fig. 13. Proposed degradation route of ACT mediated by the rGO (1%)/BiOBr photocatalyst.

Table 1 .

 1 BET, pore volume, and pore size for BiOBr and rGO/BiOBr composites.

	Sample	BET	Pore volume	Pore size
		(m²/g)	(cm 3 /g)	(nm)
	BiOBr	7.38 ± 1.04	0.029	7.91
	rGO (0.5%)/BiOBr	13.36 ± 1.93	0.030	7.52
	rGO (1%)/BiOBr	9.88 ± 0.90	0.025	8.22
	rGO (2.5%)/BiOBr	4.40 ± 0.67	0.022	9.93
	rGO (5%)/BiOBr	3.61 ± 0.56	0.027	11.11

  [START_REF] Moussa | ZnO rods/reduced graphene oxide composites prepared via a solvothermal reaction for efficient sunlight-driven photocatalysis[END_REF][START_REF] Tatykayev | Synthesis of core/Shell ZnO/rGO nanoparticles by calcination of ZIF8/rGO composites and their photocatalytic activity[END_REF][START_REF] Moussa | Growth of ZnO nanorods on graphitic carbon nitride gCN sheets for the preparation of photocatalysts with high visible-light activity[END_REF]. Tert-butanol (t-BuOH) used as hydroxyl • OH radical scavenger and DMSO used as electron scavenger only weakly

	affect the photodegradation of ACT (87 and 93% degradation after 90 min irradiation,
	respectively), indicating that • OH radicals and electrons only play a minor role in ACT
	removal. p-Benzoquinone (BQ) used as O 2	•-radical scavenger and especially ammonium
	oxalate (AO) used as hole scavenger were found to be much more deleterious to the
	degradation of ACT (31 and 12% degradation after 90 min irradiation, respectively). These
	results demonstrate that O 2

Acknowledgements

This work was supported partly by the french PIA project « Lorraine Université d'Excellence », reference ANR-15-IDEX-04-LUE. The authors thank Dr. Khalid Ferji and Steve Pontvianne for zeta potential measurements and TOC analyses, respectively. Appendix A. Supplementary data