Antoine Tavant
email: antoine.tavant@lpp.polytechnique.fr

Vivien Croes

Romain Lucken

Trevor Lafleur

Anne Bourdon

Pascal Chabert

The effects of secondary electron emission on plasma sheath characteristics and electron transport in an E Â B discharge via kinetic simulations

Keywords: Hall effect thruster, 2D particle-in-cell simulation, PIC simulation, cross-field electron transport, electron cyclotron drift instability, secondary electron emission, sheath model

Hall-effect thrusters, which are electrostatic devices based on an E B ´plasma discharge, have successfully been used as satellite propulsion systems for the last few decades. However, the presence of anomalous electron cross-field transport is still poorly understood, and involves complex and strongly coupled mechanisms such as azimuthal electron drift instabilities and intense secondary electron emission (SEE) from the thruster walls. The present work focuses on the relative importance of these two phenomena. We use a 2D particle-in-cell/Monte Carlo collision model configured to simulate the radial-azimuthal directions near the thruster exit plane. A constant radial magnetic field and axial electric field are imposed, and electron drift instabilities are observed in the azimuthal (E B

´) direction. A simplified SEE model is implemented and an extensive parametric study is performed to directly determine the effect on electron transport. It is found that, for the operating conditions used in our simulations, SEE enhances the near-wall electron mobility by a factor 2, while reducing the bulk plasma mobility by about 20% (due to electron cooling). However, the dominant contribution to anomalous electron transport is still observed to be caused by electron drift instabilities driven by the E B discharge configuration. SEE modifies the electron mobility profile, but the spatially-averaged value remains relatively constant. Three different operating regimes are identified depending on the SEE rate value: two that are stable, and a third which shows an oscillatory behaviour. In addition to electron transport, the kinetic simulations give new insight into the plasma sheath formation at the radial walls, and comparison with typical analytical sheath models demonstrate important differences.

Introduction

The development and understanding of Hall effect thrusters (HETs) is one of the driving forces in the research on E B ṕlasma discharges. HETs have been used for more than 40 years since the USSR Meteor satellite in 1974 [START_REF] Mazouffre | Electric propulsion for satellites and spacecrafts: established technologies and novel approaches Plasma Sources[END_REF] and numerous satellites have been successfully operated using HETs as primary or secondary propulsion system. Nevertheless, the full potential of HETs for space propulsion is not yet attained, as its physics is not clearly understood yet.

The typical HET consists of an annular ceramic channel [START_REF] Goebel | Fundamentals of Electric Propulsion: Ion and Hall Thrusters[END_REF], where an axial electric field E accelerates the ions and creates the thrust. A radial and intense magnetic field B, imposed by an external magnetic circuit, is applied in order to increase the residence time of the electrons, thus increasing the ionization of the propellant, usually xenon. Therefore, properly understanding the cross-field transport of electrons inside the channel would allow the improvement of the HETs. More complete descriptions of the HETs can be found in Morozov et al [START_REF] Morozov | Reviews of Plasma Physics ed B B Kadomtsev and V D Shafranov[END_REF], Adam et al [START_REF] Adam | Study of stationary plasma thrusters using two-dimensional fully kinetic simulations[END_REF], Goebel and Katz [START_REF] Goebel | Fundamentals of Electric Propulsion: Ion and Hall Thrusters[END_REF], Mazouffre [START_REF] Mazouffre | Electric propulsion for satellites and spacecrafts: established technologies and novel approaches Plasma Sources[END_REF], Boeuf [START_REF] Boeuf | Tutorial: physics and modeling of Hall thrusters[END_REF].

Many experimental studies have shown that the electron mobility along the channel axis is much higher than predicted by the classical mobility from drift-diffusion or Bohm diffusion [START_REF] Goebel | Fundamentals of Electric Propulsion: Ion and Hall Thrusters[END_REF][START_REF] Morozov | Reviews of Plasma Physics ed B B Kadomtsev and V D Shafranov[END_REF][START_REF] Meezan | Anomalous electron mobility in a coaxial Hall discharge plasma[END_REF][START_REF] Garrigues | Empirical electron crossfield mobility in a Hall effect thruster[END_REF][START_REF] Adam | simulation, and diagnostics of Hall effect thrusters Plasma[END_REF]. Based on both experimental and numerical studies [START_REF] Adam | Study of stationary plasma thrusters using two-dimensional fully kinetic simulations[END_REF][START_REF] Yoshikawa | Anomalous diffusion of a plasma across a magnetic field[END_REF][START_REF] Janes | Anomalous electron diffusion and ion acceleration in a low density plasma[END_REF][START_REF] Choueiri | Fundamental difference between the two Hall thruster variants[END_REF][START_REF] Smith | Time and space-correlated plasma potential measurements in the near field of a coaxial Hall plasma discharge[END_REF][START_REF] Scharfe | Comparison of hybrid Hall thruster model to experimental measurements[END_REF][START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF], an azimuthal instability raising from the E B ´drift of the electrons has been proposed as the main mechanism for the enhanced mobility. Lafleur et al [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: II. Kinetic model[END_REF] developed a theoretical model of this electron cyclotron drift instability (ECDI) and its effect on the electron mobility. The theory has been successfully confirmed by 1D and 2D PIC simulations [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF][START_REF] Lafleur | The role of instability-enhanced friction on 'anomalous' electron and ion transport in Halleffect thrusters Plasma Sources[END_REF][START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: I. Insights from particle-in-cell simulations[END_REF].

On the other hand, wall material is also known to change the behaviour of the HET discharge [START_REF] Gascon | Wall material effects in stationary plasma thrusters: I. Parametric studies of an SPT-100[END_REF], and experimental evidences of near-wall transport have been reported [START_REF] Goebel | Fundamentals of Electric Propulsion: Ion and Hall Thrusters[END_REF][START_REF] Zhurin | Physics of closed drift thrusters Plasma Sources[END_REF][START_REF] Kaganovich | Kinetic effects in a Hall thruster discharge[END_REF][START_REF] Sydorenko | Electron kinetic effects and beam related instabilities in Hall thrusters[END_REF]. The main physical phenomenon proposed to explain this effect from the walls is the intense secondary electron emission (SEE) arising from electron-wall collisions enhancing the electron mobility near the channel walls [START_REF] Morozov | Reviews of Plasma Physics ed B B Kadomtsev and V D Shafranov[END_REF][START_REF] Kaganovich | Kinetic effects in a Hall thruster discharge[END_REF][START_REF] Sydorenko | Electron kinetic effects and beam related instabilities in Hall thrusters[END_REF]. However, experiments and simulations [START_REF] Hirakawa | Electron transport mechanism in a Hall thruster[END_REF][START_REF] Boniface | Anomalous cross-field electron transport in a Hall thruster[END_REF][START_REF] Hagelaar | Role of anomalous electron transport in a stationary plasma thruster simulation[END_REF][START_REF] Meezan | Kinetic study of wall collisions in a Coaxial Hall discharge[END_REF][START_REF] Parra | A two-dimensional hybrid model of the hall thruster discharge[END_REF][START_REF] Smirnov | Electron crossfield transport in a miniaturized cylindrical Hall thruster[END_REF], as well as successfully operating wall-less and magnetic shielded HETs [START_REF] Mazouffre | Development and experimental characterization of a wall-less Hall thruster[END_REF][START_REF] Mikellides | Magnetic shielding of walls from the unmagnetized ion beam in a Hall thruster[END_REF], show that electron-wall collisions and SEE alone are not sufficient to explain the observed cross-field electron transport. Nevertheless, SEE is expected to significantly alter the plasma behaviour [START_REF] Hobbs | Heat flow through a Langmuir sheath in the presence of electron emission[END_REF], and consequently the study of the enhanced mobility due to ECDI along with the SEE is required to understand the relative importance of these two effects and their coupling.

Various numerical studies of the SEE effects have been performed using unidirectional simulations. A 1D kinetic simulation along the radial direction [START_REF] Sydorenko | Kinetic simulation of secondary electron emission effects in Hall thrusters[END_REF] has demonstrated the importance of SEE on the creation of electron beams between the walls. Fluid simulations [START_REF] Barral | Wall material effects in stationary plasma thrusters: II. Near-wall and in-wall conductivity[END_REF] highlighted the importance of SEE and electron backscattering for near-wall conductivity. However, those simulations did not model the azimuthal direction, and thus were unable to highlight the effects of the ECDI.

Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] used a 2D-3V PIC simulation in order to model the r-θ plane and study the coupling between SEE effects and ECDI effects on the HET operations. They have shown that SEE does not reduce the ECDI. However their model allows to simulate only the first microseconds of the discharge, and not to study a steady state. Finally, Taccogna et al [START_REF] Taccogna | Anomalous transport induced by sheath instability in Hall effect thrusters[END_REF] used a more complex 3D-3V PIC simulation to successfully study these effects. However, the authors used a scaling factor in order to decrease the size of the simulation domain, which affects the instability growth rate and amplitude, hence the electron transport [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: I. Insights from particle-in-cell simulations[END_REF]. Consequently, the study of the coupling between SEE effects and the self-consistently simulated azimuthal instability needs to be pursued. In particular, this study should be performed on long time scales and without any scaling.

As the work of Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] is very similar to the work presented here, a deeper discussion is warrented. Three main points of interest can be highlighted. First, the model used in [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] did not take into account axial plasma losses, and as a result, no steady state discharge could be reached. Instead the electron temperature, and hence the SEE rate, continuously increased with time [33, figures 17 and 18]. In this work we have proposed a different simulation model that approximately accounts for axial losses, and consequently, allows for the formation of a steady state plasma. Secondly, [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] used radial boundary conditions that mimic the floating nature of dielectric walls. In the present work we have chosen not to explicitly model a dielectric, but we still nonetheless model floating radial boundaries where the net current to the walls is zero; as occurs in real HETs. Thus no major differences are expected from this simplification. Lastly, we have neglected the curvature of the thruster channel. The authors in [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] have shown that the SEE process depends to some extent on the curvature as the radii of the inner and outer surfaces of the channel are different. Here we have again chosen to neglect this effect so as to focus just on the coupling between SEE and the ECDI, and to more clearly understand the fundamental physics involved.

Meanwhile, fluid simulations [START_REF] Garrigues | Empirical electron crossfield mobility in a Hall effect thruster[END_REF][START_REF] Barral | Wall material effects in stationary plasma thrusters: II. Near-wall and in-wall conductivity[END_REF][START_REF] Garrigues | Simulations of a miniaturized cylindrical hall thruster[END_REF] model the plasma-wall interaction using a sheath model with SEE given by fluid theory [START_REF] Hobbs | Heat flow through a Langmuir sheath in the presence of electron emission[END_REF][START_REF] Ahedo | Presheath/sheath model with secondary electron emission from two parallel walls[END_REF][START_REF] Ahedo | Partial trapping of secondaryelectron emission in a Hall thruster plasma[END_REF]. Experimental measurements of the sheath with SEE have been performed in simple conditions [START_REF] Langendorf | Effect of secondary electron emission on the plasma sheath[END_REF]. However, to the best of our knowledge, far too little attention has been paid on the validation of these sheath models in the condition of the HET.

The aim of the present investigation is to quantify by a parametric study the relative importance of the ECDI and the SEE on the electron mobility in HETs. This is achieved by using a 2D-3V PIC simulation code able to reach a steady state by setting a finite length in the third dimension, as described in our previous study [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF]. A simple SEE model [START_REF] Barral | Wall material effects in stationary plasma thrusters: II. Near-wall and in-wall conductivity[END_REF] is implemented in order to investigate the importance of ECDI compared to SEE effects. The electron cross-field transport measured in the simulation is compared to the kinetic model of Lafleur et al [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: II. Kinetic model[END_REF] over a wide range of values of the re-emission rate.

We will firstly describe the different numerical and physical models used in this work in section 2. Then sections 3 and 4 will describe the results obtained from the simulations and the comparisons with the predicted values concerning electron mobility, SEE rates and sheath characteristics.

Models

2D PIC simulation of a HET

The results presented have been obtained thanks to a 2D-3V Particle-in-cell/Monte-Carlo collisions (PIC/MCC) simulation code: LPPic. Details concerning the models and validations (benchmark, convergence and numerical heating) used for LPPic can be found in Croes et al [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF]. We simply recall that LPPic simulates the r -θ plane without any scaling method and does not consider the thruster curvature.

We note that the curvature in a real thruster can induce asymmetries in the plasma radial profile, as well as the relative magnitude of plasma-wall interactions [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF][START_REF] Domínguez-Vázquez | Particle modeling of radial electron dynamics in a controlled discharge of a Hall thruster Plasma Sources[END_REF]. Hence, the dispersion relation of the ECDI instability can be different in the regions near the inner and outer channel radii. While this may affect the results obtained, we have specifically chosen to neglect the curvature so as to better focus on the fundamental coupling of the SEE and the ECDI.

The bi-dimentionnal simulation domain uses a Cartesian coordinate system. The Ox direction corresponds to the 'radial' direction of the thruster (r), and the Oy direction corresponds to the 'azimuthal' direction (θ). As represented in figure 1, we impose a uniform magnetic field B 0 along the 'radial' direction, and a perpendicular electric field E 0 in the axial direction, normal to the simulation domain. Hence, the E B 0 0

´direction corresponds to the 'azimuthal' Oy direction. The electric field in the simulation domain x y E , ( ) is computed by solving Poisson's equation. The simulation domain is closed by walls in the Ox direction, while periodic boundaries are used along the Oy axis. The walls absorb all incident particles and have a fixed potential f=0 V. Although imposing such a potential may appear artificial, it is worth noting that the thruster anode and cathode are not modelled in the simulation, so the actual potential chosen on the radial boundaries is simply a convenient reference potential used in the solution of Poisson's equation. Due to the symmetric nature of the simulation domain, no net electron and ion current can flow to the radial boundaries, and thus they effectively represent floating walls. This is similar to the floating nature of more realistic dielectric walls, except that the net current must only be zero over the entire length of the wall, whereas with dielectric walls the net current is zero at each grid point along the wall. As the 'azimuthal' simulation distance is relatively small, this is not expected to play any significant role, and has been explicitly chosen so as to remove the added complexity of dielectrics. This allows a more transparent and focussed understanding of the coupling between the SEE effect, and the ECDI.

The Oz direction, normal to the (Ox, Oy) simulation plane is closed by fixing a physical length L z in order to reach a steady state. Particles crossing the lower and upper simulation domain boundaries are re-injected cold on the other side. Details and consequences of this finite length have already been described for a similar 1D case [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: I. Insights from particle-in-cell simulations[END_REF], and extensively discussed for the 2D case [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF][START_REF] Croes | Modelisation bidimensionnelle de la decharge plasma dans un propulseur de Hall[END_REF].

Unless otherwise stated, the operating conditions and numerical parameters used in order to run the simulations are listed in table 1. They have been chosen in order to respect the PIC conditions [START_REF] Birdsall | Plasma Physics via Computer Simulation[END_REF]. Under these conditions, it requires approximately 32 h with 360 CPUs to simulate 2.5×10 6 time-steps, which corresponds to approximately 10 μs of physical time.

In table 1, T i,0 and T e,0 are respectively the ion and electron temperatures at initialization, T see is the temperature of emission of secondary electrons, n 0 is the initial plasma density, and P n , T n and n g are respectively the neutral pressure, temperature, and density. The simulation parameters are Δt the time step, Δx the cell length, and N/NG the average number of particles/cell. Statistical convergence has been tested by increasing the number of particles/cell up to 340,

Xenon - L x ×L y ×L z 2.0×0.5×1.0 (cm 3) B 0 200 (G) E 0 2×10 4 (Vm -1 ) n 0 3×10 17 (m -3 ) Te ,0 5.0 (V) T i,0 0.1 (V) T see 1.0 (V) P n 1.0 (mTorr) T n 300 (K) n g 3.22×10 19 (m -3 ) Simulation parameter Value Unit Δt 4×10 -12 (s) Δx=Δy=Δz 2×10 -5 (m) N/NG 80 (part/cell) Simulation result Value Unit T e 40 (V) ω pe 3.1×10 10 (rad s -1 ) ω ce 3.5×10 9 (rad s -1 ) r Le 6×10 -4 (m)
although little effect on the plasma discharge is observed compared with 80 particles/cell. The average electron temperature T e , plasma frequency ω pe , electron gyrofrequency ω ce and electron Larmor radius r Le are given as representative values for the simulations. These values are also close to those used in the previous 1D PIC [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: I. Insights from particle-in-cell simulations[END_REF][START_REF] Boeuf | Rotating structures in low temperature magnetized plasmas-insight from particle simulations Front[END_REF] and 2D PIC simulations [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF], ensuring comparability. As observed in Lafleur et al [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: I. Insights from particle-in-cell simulations[END_REF], Croes et al [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF], using the operating conditions given by table 1, the wavelength of the ECDI is expected to be about 1.8 mm.

Figure 1 presents the azimuthal electric field with an azimuthal length of L y =10 mm with no SEE. The instability is clearly seen and the dominant wave length obtained by Fourier transform is λ=1.44 mm. Reducing the azimuthal length to L y =5 mm did not alter the ECDI characteristics i.e. its frequency, wavelength, and amplitude. Hence, and in order to reduce the simulation time of the study, the length L y =5 mm is used in the rest of the study.

Figure 2 illustrates the radial density profiles of the electrons and ions. We can see that the centre of the simulation is quasi-neutral, and that the sheath thickness is of the order of the millimetre.

SEE model

The system detailed in section 2.1 has already been tested and studied in Croes et al [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF] without SEE. In the present work, a simple linear model [START_REF] Barral | Wall material effects in stationary plasma thrusters: II. Near-wall and in-wall conductivity[END_REF][START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF][START_REF] Smirnov | Electron crossfield transport in a low power cylindrical Hall thruster[END_REF][START_REF] Sydorenko | Breakdown of a space charge limited regime of a sheath in a weakly collisional plasma bounded by walls with secondary electron emission[END_REF] is implemented for the SEE process. In this model, the incident electron kinetic energy is used to estimate the re-emission probability as illustrated in figure 3 and given by:  between 4 and 200 V. This allows us to compare the plasma discharge behaviour with various re-emission strengths from the walls. For example, 50 V *  = is a good approximation for BnSiO 2 walls [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF]. Al 2 O 3 , a more emissive material, would be better modelled with 17 V *  = , while Graphite, less emissive, is better modelled with 280 *  = [START_REF] Gascon | Wall material effects in stationary plasma thrusters: I. Parametric studies of an SPT-100[END_REF]. As observed in figure 3, the probability of re-emission can be higher than 1 which means that a single incident electron can extract more than one secondary electron from the wall. In figure 3, a Maxwellian electron energy distribution function (EEDF) of temperature T 45 V e = is also plotted to compare the relative position of SEE probability and of the EEDF for a typical energy of electrons in HETs:

min , 1 , 1 max 0 0 *    s s s s = + - ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ( ) ( ) ( )
where m v e e 1
f 2 T exp T . 2 e e e
,Maxw 1 2

   p = - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ( ) ( ) ( )
We can observe that the saturation of the linear model appears in the high energy tail of the EEDF, therefore involving only a small part of the electrons. The secondary electrons are reinjected with a temperature T see following a Maxwellian distribution in the two directions parallel to the wall, and a Maxwellian flux in the direction normal to the wall. The SEE rate is defined by integrating the emission probability over the half distribution function, giving:

f v f v v f v v v v d d 3
*  s s s = + - ¯( ) ( ) ( )
In the simulations, the measured SEE rate, PIC s ¯, is assessed using the discretized form of equation [START_REF] Morozov | Reviews of Plasma Physics ed B B Kadomtsev and V D Shafranov[END_REF], that is to say by counting the number of electrons re-emitted by the walls, and dividing it by the number of electrons hitting the wall. It is important to note that even if an individual electron can emit more than one secondary electron, s ¯has to be smaller than one to sustain the discharge.

Sheath model

The sheath model featuring SEE processes has been historically studied by Hobbs and Wesson [START_REF] Hobbs | Heat flow through a Langmuir sheath in the presence of electron emission[END_REF], but is still an active research topic nowadays [START_REF] Ahedo | Partial trapping of secondaryelectron emission in a Hall thruster plasma[END_REF]. The sheath is often considered to be collision-less and isothermal, while the plasma is composed of hot Maxwellian electrons and cold ions. A third population of electron-induced secondary electrons is also present in the sheath, and the re-emission rate s ¯is assumed to be constant. The SEE process modifies the potential drop in the sheath as [START_REF] Hobbs | Heat flow through a Langmuir sheath in the presence of electron emission[END_REF]:

m m T ln 1 2 5 e i e sheath f s p D = -  ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ( ¯) ( )
with m i the ion mass, and T e the electron temperature in the direction parallel to the magnetic field, thus normal to the walls. Adding a pre-sheath drop of T 2 e [START_REF] Ahedo | Presheath/sheath model with secondary electron emission from two parallel walls[END_REF], the total potential drop to the wall is:

m m T 2 T 1 2 ln 1 2 .
6

e e i e sheath f f s p D = D + = + -   ⎡ ⎣ ⎢ ⎢ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎤ ⎦ ⎥ ⎥ ( ¯) ( )
At this point, it is important to note that the temperature is defined in our work from the kinetic definition:

T m q n f v v U v 3

Mobility

The electron mobility in the transverse direction of the simulation domain is defined, and calculated, as:

U E 10 z 0 m = ( ) with U v z ez ,
= á ñ the average electron velocity in the axial direction. In the PIC simulation, equation ( 10) is used to calculate PIC m . The classical drift-diffusion theory of the mobility transverse to a magnetic field predicts a mobility classical m expressed as [START_REF] Meezan | Anomalous electron mobility in a coaxial Hall discharge plasma[END_REF]:

q m , 1 1 e m m c e classical 2 2 m n n w = + | | ( ) with q B m ce e 0 w = | |
the electron cyclotron frequency, and ν m the electron-neutral momentum transfer collision frequency.

Lafleur et al [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: II. Kinetic model[END_REF][START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: I. Insights from particle-in-cell simulations[END_REF] have shown that anomalous transport due to an enhanced electron-ion friction force induced by the ECDI may be modelled by the following effective mobility:

n E n E 1 1 2 ce m e y eff classical 0 0 m m w n d d = - á ñ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ( )
with n 0 the mean electron density, δn e the fluctuation of the electron density and δE y the fluctuation of the azimuthal electric field. In our simulation set-up and assuming that the saturation of the instability is due to ion trapping, the mobility can be expressed as:

R q n E 1 , 1 3 ce m ei eff sat classical 0 0 m m w n = + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ | | ( )
where [START_REF] Meezan | Anomalous electron mobility in a coaxial Hall discharge plasma[END_REF] 14

R q n E q n v c q n v c L T 4 6 T 4
d d = -á ñ =  » | | | • ( )| | | ( )
is the saturated electron-ion friction force. In equation [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF], v i is the local ion drift velocity, c q m T s e i 1 2

= (| |

) is the Bohm speed, and the spatial derivative has been approximated across the axial simulation direction and it was assumed that there are no derivatives in the radial and azimuthal directions.

The ion outlet velocity along (Oz) in the simulation is:

v q V m 2 15 iz z i 1 2 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ | | ( )
with V z =E 0 L z the total potential difference in the axial direction. Neglecting the collision frequency against the cyclotron frequency, equation (13) can be simplified to:

T V B 4 3 . 1 6 e z eff sat 1 2 0 m = ( ) ( )
Equation [START_REF] Lafleur | The role of instability-enhanced friction on 'anomalous' electron and ion transport in Halleffect thrusters Plasma Sources[END_REF] shows that for the simple 2D PIC geometry used here, the enhanced mobility is only a function of the electron temperature. However, this is not true in general. We can note that eff m can be estimated at each grid-point of the simulation domain, similarly to PIC m . On the contrary eff sat m can only be globally calculated over the whole simulation domain.

In section 4, comparisons between PIC m the measured mobility in the simulations and the predicted values of eff m and eff sat m will be presented to quantify the relative contributions of the ECDI and SEE to the electron cross-field transport.

Results

A parametric study is conducted using the SEE model detailed in section 2.2 with 0.5 0 s = and 2.9 max s = while the value of *  is varied from 4 to 200 V. For every simulation presented hereafter, we observe azimuthal oscillations as shown in figure 1. The simulations reach a steady state, characterized by stable oscillations, in less than 2 μs in every case. The ECDI characteristics are consistent with previous studies [START_REF] Croes | 2D particle-in-cell simulations of the electron drift instability and associated anomalous electron transport in hall-effect thrusters Plasma Sources[END_REF], validating the kinetic theory of Lafleur et al [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: II. Kinetic model[END_REF]. The physical values presented in this section are averaged in space over the azimuthal direction and in time between t=5 μs and t=10 μs.

Figure 4 compares the value of the measured SEE rate PIC s ¯obtained from the PIC simulations with Maxw s ¯from equation (4) using T e measured in the whole simulation domain using equation [START_REF] Garrigues | Empirical electron crossfield mobility in a Hall effect thruster[END_REF]. We can see that in the PIC simulations, PIC s ¯decreases monotonically from 1 for low *  to a value close to 0 s , as expected, for high *  . This conv- ergence to 1 has already been observed and commented by Sydorenko et al [START_REF] Sydorenko | Kinetic simulation of secondary electron emission effects in Hall thrusters[END_REF]. For 40 V *   we observe that PIC cr s s > ¯¯. It corresponds to a regime of non-monotonic potential profile. Indeed, when the SEE is too intense the charge density close to the wall becomes negative to the point where the sign of the electric field changes. Hence, the plasma potential profile forms a well close to the wall, as already observed in Sydorenko et al [START_REF] Sydorenko | Breakdown of a space charge limited regime of a sheath in a weakly collisional plasma bounded by walls with secondary electron emission[END_REF]. In this low *  limit, the theory of a standard monotonic sheath is not valid.

However, equation ( 4) is expected to predict well s ¯for 40 V *  > . Yet, figure 4 shows that the predicted values of Maxw s ¯using equation (4) and the electron temperature measured in the simulations are much higher than the PIC simulation results.

In order to check the sheath model, figure 5(a) presents as a function of the SEE rate measured in the PIC simulations PIC s ¯, the evolution of the total potential drop f D measured between the wall and the centre of the domain, normalized to the electron temperature T e given in figure 5(b). It is compared to equation [START_REF] Meezan | Anomalous electron mobility in a coaxial Hall discharge plasma[END_REF], normalized to T e as well. In PIC simulations, the electron temperature shown in figure 5(b) decreases monotonically with the SEE rate s ¯, from 45 V for low emission to around 30 V for high emission. Figure 5(a) shows that the value of T e f D derived from PIC simulations follows the same trend as the theoretical value from equation [START_REF] Meezan | Anomalous electron mobility in a coaxial Hall discharge plasma[END_REF], that is a monotonic decrease with s ¯. The PIC results seem to show a transition between two different regimes around 0.8 PIC s = ¯. This is discussed further in section 3.3.

However, on figure 5(a), the values differ quite significantly for both the high and low limits. Indeed, for high s the expected potential drops to negative value, while we observe that T e f D tends towards 1 in the simulations. This difference can be explained by the fact that the model described in section 2.3 is only valid for a monotonic sheath, while for high SEE rate ( 40 V *  <

) we observe a potential well near the wall as previously discussed. Hobbs and Wesson [START_REF] Hobbs | Heat flow through a Langmuir sheath in the presence of electron emission[END_REF] showed that under these conditions of high emission, the total potential drop to the wall goes down to T e and not to zero or below, as we observe in figure 5(a).

For low values of s ¯, the measured sheath drop is lower than the value predicted by almost one third. This observation is more surprising, as the sheath is monotonic for these parameters, and should hence be well predicted.

Figures 5 and4 show that the sheath model proposed in section 2 is not satisfactory when confronted to the PIC simulation results. An assumption used in section 2 is that the sheath is monotonic and at a steady state. For high SEE rate ( i.e.

V *  <

) this hypothesis is not respected, hence differences were expected. However, for low SEE rate this hypothesis is respected. The plasma-wall interaction model of section 2 uses two other main hypotheses: a Maxwellian distribution of the electrons, and an isothermal plasma. Both hypotheses will be confronted to the kinetic simulation data in the next sections.

Sheath electron energy distribution

Using the kinetic information of the PIC simulations, we present in figure 6 the mean electron energy probability functions (EEPF) in the case 200 V *  =

. Figure 6(a) shows the projections of the EEPF in the bulk of the simulations along the three directions. These projections are compared to the Maxwellian probability function of the same kinetic temperature calculated using equation [START_REF] Garrigues | Empirical electron crossfield mobility in a Hall effect thruster[END_REF]. Figure 6(b) shows the total EEPF for both the bulk and the sheath populations. The sheath length is defined as the location where the ions reach the Bohm speed, and is about 0.4 mm.

We see in figure 6(a) that the electron distribution function is not Maxwellian. In particular the high energy tails are depleted. In order to evaluate the effect of the non-Maxwellian EEPF, we numerically integrate the EEPF from the PIC data using equation (3). The results (not shown) do not differ significantly from the Maxwellian values of equation (4). Hence, we can conclude that even if the Maxwellian hypothesis is not respected in the PIC simulations, it is not enough to explain the differences observed in figure 4. ) in red dotted line, and (b) for the bulk and the sheath populations.

Figure 6(b) presents the EEPF for the bulk population as well as for the sheath population. We can see that the sheath population is colder than the population at the centre, which could explain the difference of figure 4. This effect is assessed in the next section. Surprisingly, we do not observe in figure 6 secondary electron beams in the radial direction, unlike previous observations of Sydorenko et al [START_REF] Sydorenko | Kinetic simulation of secondary electron emission effects in Hall thrusters[END_REF] with a 1D model or Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] with a 2D model. On the other hand, we do observe SEE beams when we artificially remove the ECDI by forcing E y =0 V m -1 [START_REF] Croes | Modelisation bidimensionnelle de la decharge plasma dans un propulseur de Hall[END_REF]. Hence, it seems that the ECDI, when simulated in 2D, quickly thermalizes the secondary electrons emitted from the walls. In addition, we can see in figure 6(a) that the radial EEPF is close to the EEPF in the other directions. For instance, for

200 V *  = we measure T 41.5 V e =  while T 49 V e = ^.
The cause of the difference with Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] is not yet clear, but there the results were only obtained at early times where saturation of the instability has not necessarily been reached, and no electron loss in the axial direction was accounted for. It is possible that at steady state, the electrons in our simulations have had sufficient time to become more isotropic. The energy transfer from the axial and azimuthal directions to the radial direction is not clearly understood yet [START_REF] Janhunen | Evolution of the electron cyclotron drift instability in two-dimensions[END_REF], but we believe that it is due to the instability.

Corrected SEE rate

We can calculate the sheath electron temperature using equation [START_REF] Garrigues | Empirical electron crossfield mobility in a Hall effect thruster[END_REF] from the EEPF obtained in the sheath as shown in figure 6(b). Figure 7 shows the values of the SEE rate measured in the PIC simulation compared to the values obtained by equation (4) using this time the sheath electron temperature. As the calculation of the EEPF is very computationally costly, we have calculated its value for only 4 simulations corresponding to 10, 25, 50, 200 V *  = { } . We observe from figure 7 that using the sheath electron temperature in equation (4), the SEE rate is well predicted (error less than 5%), except for 10 V *  = where a large error is observed.

As previously discussed, the case 10 V *  = presents a very high SEE rate which leads to a potential well close to the wall. Consequently, some secondary electrons emitted at low energy would be reflected back to the wall. Hence, since equation (4) does not take into account this local effect, it is not surprising that the SEE rate calculated using the mean electron temperature in the sheath is too high.

To summarise, figure 7 shows that when the sheath potential profile is monotonic, the SEE rate can be well predicted by equation (4) using T e in the sheath, which is lower than in the centre of the domain. The energy flux inducing the SEE is not well described if we use the electron temperature of the bulk as in the isothermal sheath model. This explains the overestimations of s ¯and f D presented in figures 4 and 5. Hence, the isothermal hypothesis used in the sheath model of section 2.3 is denied by the PIC simulations.

This is an important result because the sheath model described in section 2.3 is used in several fluid simulations [START_REF] Barral | Wall material effects in stationary plasma thrusters: II. Near-wall and in-wall conductivity[END_REF][START_REF] Ahedo | Presheath/sheath model with secondary electron emission from two parallel walls[END_REF][START_REF] Ahedo | Partial trapping of secondaryelectron emission in a Hall thruster plasma[END_REF]. Hence, further work is needed to develop an improved, non-isothermal, sheath model in order to better account for the SEE in fluid simulations and theoretical models.

Sheath oscillations

The evolution of the SEE rate averaged over time and the whole computational domain as a function of *  (figure 4) is smooth, decreasing monotonically from 1 s ~to 0 s when *  increases. To go further in the analysis, we have also studied the time evolution of s ¯for various values of *  . Figure 8 presents the temporal evolution of the space average PIC s ¯for three different values of *  , corresponding to three different regimes we have identified. In regimes I and III, PIC s ¯reaches a steady state after a few microseconds. Regime I, with low *  , is characterized by a saturation of s ¯at a value between cr s ¯and 1, which leads to a non-monotonic potential profile. Regime III, for high *  , is characterized by a steady state with a SEE rate lower than cr s ¯. This is the standard regime described in section 2. The transition between these two stable regimes (monotonic and non-monotonic sheath) passes by regime II, an oscillating mode between the two stable regimes. As shown in figure 9 9). Consequently, we observe a few oscillations before the steady-state is reached, as these cases are close to the bifurcation to the relaxation oscillation regime. Similar oscillations have been observed in the 1D simulation of Sydorenko et al [START_REF] Sydorenko | Breakdown of a space charge limited regime of a sheath in a weakly collisional plasma bounded by walls with secondary electron emission[END_REF]. The oscillations occur because the mean electron energy evolves slowly during the two stable states: when cr s s > ¯¯the electron heat flux to the wall is higher, hence decreasing the electron energy. When the mean electron energy is lower than some threshold, the sheath changes quickly to a stable state where the electron heat flux to the wall is lower, hence increasing the mean electron energy. When the electron energy passes another threshold, the sheath shifts to the first state. However, a detailed theory of the relaxation oscillations needs to be done. 12) and ( 16) respectively. The average is the same as the one used in figures 4 and 5, i.e. an average both in time and space.

Electron enhanced mobility

We observe in figure 9 m still provides a reasonable approximation of the electron enhanced mobility even with very high SEE rates.

Figure 10 presents the radial profiles of the mobility measured in the PIC simulations PIC m on the left, and on the right the mobility predicted due to the ECDI eff m for three values of *  , obtained by averaging the mobility in the azi- muthal direction during 5 s m of the simulation. The three chosen values correspond to the three regimes observed in figures 9 and 8: high SEE (regime I), low SEE (regime III) and the transition regime (regime II). The results obtained without SEE are also shown.

We observe in figure 10 that the mobility measured in the centre decreases by 20% as *  decreases, i.e. the SEE rate increases. This observation is consistent with the evolution of eff m as well as with the evolution of eff sat m in figure 9. This is due to the decrease of the electron temperature due to the SEE as seen in figure 5(b). We note in passing that the above results are in apparent disagreement with the observations obtained in Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF], where anomalous transport was found to be higher at lower values of *  (see for example figure 24 in Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF]. However, since no axial loss mechanism was included in those simulations, a steady state solution was not reached, and as shown in figure 23 in Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF], the anomalous transport rate is still increasing with time for large values of ; *  even at the end of the simulation. Since the SEE rate decreases with an increased *  , this results in less electron cooling which causes the bulk electron temperature to increase (this would be further exacerbated by a lack of any axial loss mechanism). By considering the linear growth rate of the most dominant mode of the instability [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: II. Kinetic model[END_REF], it is found that a higher electron temperature reduces the growth rate. Thus, a longer time is needed to reach a given instability amplitude. Since the instability amplitude largely determines the level of anomalous transport, the difference of our results discussed above with those in Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] may only be apparent, and related to the specific time chosen in Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] to make the observations. If the simulations in Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] could have been run for longer times, it is possible that the anomalous transport rate would have saturated and that similar conclusions to our results above would have been obtained.

On the other hand, the mobility close to the wall, also referred to as near-wall mobility in the literature, increases by almost a factor of 2 with the SEE rate. This near-wall mobility is not observed on the effective mobility shown in figure 10, confirming that it is due to SEE.

These observations of figures 9 and 10 show that when the SEE rate increases, the electron axial mobility in the centre decreases slightly, while the near-wall mobility increases. As a result, even if the SEE rate do vary significantly, the overall mobility measured in the simulations, which takes into account both bulk and near-wall mobilities, is almost constant.

Conclusion

In this article a study of the effects of SEE from the walls of an E B ´discharge were presented. It was observed that the anomalous electron cross-field transport in the bulk plasma decreases by 20% as the effective SEE coefficient increases. This appears to occur as a direct consequence of electron cooling, and demonstrates a coupling between SEE and the electron drift instability. The observation of reduced electron transport in the plasma bulk due to a lower electron temperature is in good agreement with the kinetic theory predictions of Lafleur et al [START_REF] Lafleur | Theory for the anomalous electron transport in Hall effect thrusters: II. Kinetic model[END_REF]. However, when the SEE coefficient increases from 0 to almost 1, the near-wall mobility increases almost by a factor 2. As a result, the total mobility in the simulations is found to remain almost constant. Consequently, in our simulations the main effect of SEE is to decrease the electron temperature by both decreasing the potential drop at the wall, and through the injection of cold electrons into the discharge. Overall though the dominant contribution to anomalous electron transport is caused by the electron drift instability.

The plasma characteristics in the simulations have been compared to a typical plasma sheath model that accounts for SEE. The effective SEE coefficient observed in the simulations appears much lower than the predicted value from this model, even though no electron beams have been observed, contrary to Sydorenko et al [START_REF] Sydorenko | Kinetic simulation of secondary electron emission effects in Hall thrusters[END_REF]. Hèron and Adam [START_REF] Héron | Anomalous conductivity in Hall thrusters: effects of the non-linear coupling of the electroncyclotron drift instability with secondary electron emission of the walls[END_REF] also observed such beams, but do not compare the measured SEE rate to a theoretical value. However, we observed that the electron temperature is lower in the sheath than in the bulk plasma, which when accounted for, correctly explains the discrepancies observed in the SEE coefficient and potential drop at the wall. Further work is needed to develop an improved, non-isothermal, sheath model in order to better account for plasma-surface interactions in fluid simulations and theoretical models.

Three regimes have been identified when varying the effective SEE coefficient. For low SEE rates, the plasma sheath at the walls has a typical monotonic spatial profile. This region is stable. For high SEE rates, a potential well is observed to form near the wall due to space charge effects associated with intense electron emission. This saturated regime is also found to be stable. However, for intermediate SEE rates, an oscillatory behaviour is observed where the potential profile changes between those for a standard monotonic sheath, and the saturated sheath. The main trigger for this oscillatory behaviour is not yet understood, but this regime appears similar to the relaxation sheath oscillations observed before by Sydorenko et al [START_REF] Sydorenko | Breakdown of a space charge limited regime of a sheath in a weakly collisional plasma bounded by walls with secondary electron emission[END_REF].

In order to improve the realism of the simulation, two limitations of the current study should be addressed: (1) the radial walls are conducting instead of insulating (although they still float electrically like an insulator), (2) the simulation domain neglects the curvature of the thruster channel. Both of these points are expected to alter the equilibrium values [START_REF] Domínguez-Vázquez | Particle modeling of radial electron dynamics in a controlled discharge of a Hall thruster Plasma Sources[END_REF], but not necessarily the overall conclusion on the electron transport and the SEE effects.

Figure 1 .

 1 Figure 1. Schematic representation of the 2D simulation domain. Overlaid is a snapshot of the azimuthal electric field E y taken at t=10μs. Lengths used are: L y =10 mm, and L x =20 mm.

2 2 

 2 = | | is the electron kinetic energy with m e the electron mass and v e the electron velocity vector, *  and 0 s two parameters, and max s the maximum re-emission probability.In this work we chose 2

Figure 2 .

 2 Figure 2. Radial profiles of the electron density n e and the ion density n i averaged in the azimuthal direction for the same parameters as figure 1.

Figure 3 .

 3 Figure 3. Representation of the SEE model described in section 2.2. Overlaid is a Maxwellian electron energy distribution function of temperature T 45V e = . The dashed line corresponds to 35.04 V *   = = .

  v x the radial components of v e , Ω is the half space v n a unit vector towards the wall, and f v e e ( ) is the electron velocity distribution function. Hence n e x º + for the upper wall, and n e x ºfor the lower. Using equation (1) and assuming a Maxwellian electron distribution, we can calculate T e Maxw s ¯( ) as detailed in the appendix. Neglecting the saturation at max s of equations (1) and (3) reduces to:

  U the mean velocity vector, q the electron charge, and n e the electron density. This definition allows us to extend the definition of T e for non-Maxwellian populations and to use it directly in the PIC simulation. Similarly, the parallel electron temperature used in equation (5) can be expressed by integrating f v e e ( ) along the radial direction: the radial components of U. We can easily observe that equation[START_REF] Boeuf | Tutorial: physics and modeling of Hall thrusters[END_REF]

Figure 4 .

 4 Figure 4. SEE rates averaged over time and space over the whole computational domain as a function of the re-emission parameter ò * . Comparison of PIC s ¯derived from the PIC simulations and Maxw s ¯, the Maxwellian theory of equation (4) with T e measured in the PIC simulations.

Figure 5 .

 5 Figure 5. (a) Plasma potential drop to the wall normalized by the electron temperature T e f D as a function of the SEE rate PIC s ¯for the PIC simulations and the theory of equation (6); (b) the electron temperature measured in the PIC simulation as a function of the SEE rate.

Figure 6 .

 6 Figure 6. Electron energy probability functions for 200 V *  = . (a) in the bulk of the plasma, compared with Maxwellian probability of the same temperature (T 49 V e =) in red dotted line, and (b) for the bulk and the sheath populations.

Figure 7 .

 7 Figure 7. SEE rates as a function of the re-emission parameter ò * . Comparison of PIC s ¯derived from the PIC simulations and Maxw s ¯, the Maxwellian theory of equation (4) with the sheath temperature.

Figure 9

 9 Figure 9 presents the evolution, as a function of *  , of the average value of the measured mobility in the simulation PIC m as well as the effective mobility eff m , the saturated estimate eff sat m , and the classical mobility classical m , from equations (11), (12) and (16) respectively. The average is the same as the one used in figures 4 and 5, i.e. an average both in time and space.We observe in figure9that ,

8 .

 8 Evolution as a function of time of the averaged SEE rate s īn the 3 regimes observed, 2 stable and one with oscillations. The light green zones correspond to the periods when cr s s > ¯¯.

Figure 9 .

 9 Figure 9. Electron mobility as a function of the parameter *  . PIC m in blue is directly measured in the PIC simulations, while classical m , eff m and eff sat m respectively in purple, red and green, are calculated respectively with equations (12), (16) and[START_REF] Choueiri | Fundamental difference between the two Hall thruster variants[END_REF]. Three regimes are identified by I, II and III.

Figure 10 .

 10 Figure 10. Radial profiles of: (left) the measured mobility in the simulation ; PIC m (right) the effective mobility in the simulation eff m from equation (12), bothaveraged in the azimuthal direction and in time on 5 s m .

Table 1 .

 1 Standard operating and numerical parameters used in the 2D PIC simulations of an HET. The simulation results are given as representative values.

	Physical parameter	Value	Unit
	Gas

Acknowledgments

The authors would like to thank Franois Pechereau, Abdoul Wahid C Mainassara, Thomas Charoy, Roberto Martorelli and Stephan Zurbach for many useful discussions, as well as for their help and support in the development of the LPPic simulation code. VC and AT acknowledge financial and technical support from a Safran Aircraft Engines doctoral research award. This work has been partially funded byANR (n°ANR-16-CHIN-003-01) and Safran Aircraft Engines with the project POSEIDON. The authors were granted access to the HPC resources of CINES (under the allocation A0020510092 and A0040510092 made by GENCI) and of CERFACS.

Appendix

The exact expression of the SEE rate using equations (1)-( 3) can be found to be:


Clément Larquet

Yannick Klein

David Hrabovsky

Andrea Gauzzi

Clément Sanchez

Sophie Carenco
email: sophie.carenco@sorbonne-universite.fr

Tunable Magnetic Properties of (Gd,Ce) 2 O 2 S Oxysulfide Nanoparticles

Keywords: gadolinium oxysulfide, nanoparticles, lanthanides, tunable magnetic moment

Nanoparticles with strong paramagnetic responses are of prime interest for advanced MRI imaging. To replace Gd expensive and toxic complexes, Gd-based nanoparticles have emerged as a viable solution for efficient and harmless MRI contrast agents. Gadolinium oxysulfide nanoparticles could represent suitable candidates for MRI imaging and bimodal imaging thanks to their excellent properties as host matrix and chemical stability, but their magnetic properties at the nanoscale have been hitherto poorly investigated especially in the case of ultrafine nanoparticles (< 10 nm), where surface effects and ligands can significantly affect the magnetic behavior. In the present work, we synthesized and characterized bimetallic (Gd,Ce) 2 O 2 S nanoparticles and demonstrated that they are paramagnetic over a wide temperature range including the body one. The mixture of Gd and Ce magnetic centers enables a fine control of the magnetic properties up to high Ce concentrations (80 %) and over a large range of magnetic moments, while photoemission properties are guaranteed up to 20 % of Ce owing to a regular dispersion of the Ce centers. The present study on bimetallic oxysulfide nanoparticles with high concentrations of two lanthanides shows that (Gd,Ce) 2 O 2 S nanoparticles are viable candidates as tunable nanoscaled agents for bimodal imaging.

Introduction

Lanthanides can exhibit high magnetic susceptibility, which is of major interest for chemicals to be injected in a living organism. For instance, Gd III complexes are used as positive contrast agents in magnetic resonance imaging (MRI) due to the 4f 7 electronic configuration of the ion (µ eff = 7.94 µ B ). Gadolinium ions in molecular complexes remain toxic because of polarizing effects and competition with calcium. Special hydrosoluble complexes were then developed to mitigate the toxicity of Gd III . [START_REF] Tóth | Relaxivity of MRI Contrast Agents[END_REF] An alternative to lanthanide complexes is offered by lanthanide nanoparticles which display better relaxations because, in these particles, several thousand atoms are concentrated in a little volume. Iron oxide nanoparticles have also been used as negative contrast agents, however they often produce artefacts in the MRI images. [START_REF] Bulte | [END_REF] Gd 2 O 3 nanoparticles were found to have similar or better relaxation properties than Gd complexes, without the drawbacks of iron oxides. These particles were then chosen for the precise visualization of locally injected cells. 3,4 Doped gadolinium oxide nanoparticles were then employed for bimodal imaging (MRI and luminescence). 5 Because of their very good luminescence properties, similar results are expected for oxysulfide Gd 2 O 2 S nanoparticles. Bimodal agents are especially useful to probe the chemical environment of the nanoparticles by analyzing the changes of luminescence properties (wavelength, lifetime, etc.). These properties can be measured during short times and their analysis can be coupled with data obtained for long times and precise localization by means of magnetic resonance imaging. 6 In future applications, improving the size and morphology control of Gd 2 O 2 S nanoparticles will be a key to make them compatible with biomedical applications, for reproducibility and regulation purposes. Prior to this, a better understanding of their intrinsic properties has to be reached. Indeed, although the literature on Ln 2 O 2 S nanoparticles synthesis is now well documented, [7][8][9][10][11][12][13][14] only a few reports on the properties of lanthanide oxysulfide nanoparticles are available. In 2012, He et al. synthesized and characterized the Eu 2+x O 2 S nanorods obtained using a single source precursor prepared in situ by hot injection of diethylammonium diethyldithiocarbamate (dea-ddtc) at 320 °C in a mixture of europium oleate, oleylamine and 1,10-phenanthroline and dodecanethiol. 15 Non-stoichiometry is attributed to the presence of Eu II in the solid and was already observed for the bulk phase in the 1960's by Ballestracci and Quezel who estimated that 1 % of the Eu atoms are divalent from neutron diffraction and magnetic measurements. 16,17 confirmed the potential use of Gd 2 O 2 S:Yb,Er magnetic and upconverting phosphors for bimodal imaging. Both MRI imaging and infrared imaging could be performed with the same set of nanoparticles, and the nanomaterials were shown to be non toxic.

Recently, we succeeded in synthesizing crystalline Gd 2(1-y) Ce 2y O 2 S bimetallic oxysulfide nanoparticles (Figure S1). Although Ce 2 O 2 S nanoparticles are sensitive to air and water and degrade completely within a few days, the bimetallic nanoparticles remain stable in air during several months even if they contain high amounts of cerium (up to y = 80 %). The synthesis in organic medium at relatively low temperature (310 °C) enabled the formation of a Gd 2 O 2 S -Ce 2 O 2 S solid solution, whereas classical solid-state syntheses would lead to a mixture of the two compounds. To the best of our knowledge, bimetallic oxysulfides containing high proportion of the two ions have not been previously studied, which motivates the present systematic study of their magnetic properties. Specifically, we report on the paramagnetic behavior of Gd 2(1-y) Ce 2y O 2 S bimetallic oxysulfide nanoparticles down to 5 K. We are interested in the tunable magnetic moment, which is due to the different number of lone electrons of gadolinium (4f 7 ) and cerium (4f 1 ). The present study also raises the question of the size effect on the magnetic properties of the Gd 2 O 2 S nanoparticles.

Results and discussion

Bulk lanthanide oxysulfides Ln 2 O 2 S (Ln = lanthanide) are paramagnetic compounds at ambient temperature. At low temperatures, the compounds with Ln = Gd, Tb, Dy, Ho and Yb order antiferromagnetically at Néel temperatures between 2.5 K (for Ho) and 7.7 K (for Tb) and exhibit Weiss constants, θ, between -8 K (for Ho) and -18 K (for Gd). The other compounds, including Ce 2 O 2 S, remain paramagnetic down to 2 K. 17 Gd 2 O 2 S is also paramagnetic in a wide range of temperature but orders antiferromagnetically at very low temperature. 16 Ballestracci et al. studied the Ln 2 O 2 S bulk compounds by neutron diffraction and estimated the Néel temperature (T N ) of Gd 2 O 2 S to be 5.7 K. In the paramagnetic phase, its Curie-Weiss temperature (T CW ) is -18 K. The magnetic moment of Gd atoms (7.92 μ B ) was very close to these of the isolated paramagnetic ion (7.94 μ B ). In 2014, Biondo et al. revisited the properties of Ln 2 O 2 S oxysulfides (Ln = Sm, Eu, Gd). 18 In the case of Gd 2 O 2 S, the reported properties are close to those of Ballestracci et al., i.e. T N = 5.8 K, T C = -21.3 K. This indicates the existence of a geometric magnetic frustration due to the triangular lattice. 19 In the Gd 2(1-y) Ce 2y O 2 S nanoparticles, Ce III luminescence could be detected and measured between 420 and 650 nm for the samples with y = 2 % and 20 % (Figure S4). This shows the relevance of these objects for further luminescence application in the visible range.

In view of the above results, we expect similar paramagnetic properties in Gd 2(1-y) Ce 2y O 2 S nanoparticles, which display a tunable concentration of cerium (Figure S2) and keep the Ln 2 O 2 S structure up to y = 80 % (Figure S3). Specifically, the paramagnetic response of the powders should decrease with y. Nevertheless, the coupling between gadolinium and cerium ions is possible because both ions have partially occupied f orbitals. Zero-field cooling (ZFC) curves were measured by vibrating sample magnetometry (VSM) on the powders and plotted in Figure 1. A conventional paramagnetic response was observed in the whole 5 -350 K range. The field-cooling (FC) curves were identical to the ZFC ones. We were able to fit the χ mol (T) curves using a generalized Curie's law, as demonstrated in Figure 2. The constant contribution χ 0 is orders of magnitude smaller than the total susceptibility at low temperature (for y = 0 -80 %: at 5 K, χ mol > 150 χ 0 ; at 100 K, χ mol > 10 χ 0 ). In a paramagnetic system, this term typically describes core diamagnetism or Pauli paramagnetism. In the present case, χ 0 takes into account the contribution of the sample holder and of the organic ligands on the surface of the nanoparticles. For all compositions, the value of θ falls in the 0 -2 K range, thus indicating negligible magnetic fluctuations. The values of the susceptibility and of the magnetic moments were corrected to take into account the oleate organic ligands (30 % of the sample in mass). Within a simple model of paramagnetic impurities, the above analysis explains well the linear dependence of the square    of the magnetic moment on composition (Figure 3). The linear dependence of (μ eff ) 2 vs. y is therefore consistent with a picture of solid solution of gadolinium and cerium atoms in the nanoparticles. This linear dependence is observed up to y = 60 %. For concentrations above 60 %, a downward deviation from the linear dependence was observed. We attribute this to the partial oxidation of Ce III into Ce IV , as discussed by some of us in a previous work. 14 This deviation is even more parked for y > 80 %, in the range of composition where a degradation of the nanoparticles occurs. Because the y = 100 % sample (Ce 2 O 2 S nanoparticles) is highly sensitive in air, no magnetic measurement was carried out.

Ce III Ce IV

Conclusion

In the present work, we have carried out a systematic study of the magnetic properties of Gd 2(1-y) Ce 2y O 2 S in a wide range of composition, y = 0 -90 %. We have found that the nanoparticles are paramagnetic in the whole 5 -350 K temperature range studied and the temperature dependence of the susceptibility is well described by a generalized Curie law.

Our data analysis indicates that the magnetic moment linearly decreases with Ce concentration, as expected from a simple ionic model suitable to describe the electronic configuration of the 4f shell. The above results indicate that the magnetic properties of the Gd 2(1-y) Ce 2y O 2 S nanoparticles are easily tuned by chemical composition in a wide range, y = 0 -80 %. For y > 60 %, Ce oxidation is significant and leads to a decrease of the expected magnetic moment. Remarkably, the Gd 2 O 2 S nanocrystals do not exhibit antiferromagnetic order down to 5 K, contrary to the bulk, which displays an order at 5.7 K. The reason for this difference, which we tentatively attribute to size effects, will be the object of a future work.

Altogether, the combination of tunable paramagnetic and photoluminescence properties is promising for the development of Gd 2(1-y) Ce 2y O 2 S nanoparticles for bimodal imaging applications. Indeed, Ce III cations emit light even for relatively high doping concentrations.

Future directions of work include the study of size-effect on the magnetic properties, which should be performed on nanoplates with a narrower width distribution. They also encompass the evaluation of in vitro and in vivo toxicity of the nanoparticles, which will be reported in due time. (0.25 mmol), S 8 (0.032 mmol), Na(oleate) (0.50 mmol), OM (17 mmol), OA (2.5 mmol) and ODE (32.5 mmol) were placed in a 100 mL three-neck flask at room temperature. The brown solution was heated to 120 °C under vacuum for 20 min to remove water and other impurities with low boiling points. The mixture was then heated to 310 °C and stirred at this temperature for 30 min under purified N 2 . The transparent solution gradually became turbid starting from 280 °C. Then the mixture was left to cool to room temperature under purified N 2 . The nanoparticles were isolated using ethanol and washed at least three times using a THF/ethanol (1/5) mixture to remove the remaining reagents and organic matter. A 40 to 90 mg amount of dried Gd 2(1-y) Ce 2y O 2 S particles was thus obtained depending on the initial cerium loading.

Experimental section

Synthesis of Gd

It should be noted that a significant decrease in the reaction yield (measured after several washings of the nanoparticles in air) is observed along with the amount of cerium introduced in the nanoparticles: starting from 0.5 mmol of lanthanide precursor, a yield of around 100 % vs. Gd (95 mg of powder) was obtained for y = 0 % (Gd 2 O 2 S) while a yield of 48 % (45 mg) was obtained for y = 70 % (Gd 0.6 Ce 1.4 O 2 S).

Figure 1 :

 1 Figure 1: Molar susceptibility χ mol of Gd 2(1-y) Ce 2y O 2 S powders measured by vibrating sample

Figure 2 :

 2 Figure 2: (A) Generalized Curie law. Fit of the ZFC molar susceptibility χ mol (T) of the

Equation 1 :Equation 2 :

 12 Expression of the Curie constant. Equation 2 describes the relation between the effective magnetic moment, the Landé factor and the angular momentum quantum number. Expression of the effective magnetic moment. From Equation 1 and Equation 2, the magnetic moment is proportional to the square root of the Curie constant's, as indicated in Equation 3. Equation 3: Approximated expression of the square effective magnetic moment, obtained with N = 6.02 x 10 23 mol -1 , k B = 1.38 x 10 -16 erg/K and μ B = 9.27 x 10 -21 erg/G.

Figure 3 :

 3 Figure 3: Experimental dependence of the square magnetic moments of Gd 2(1-y) Ce 2y O 2 S

  He et al. also described magnetic properties of europium oxysulfide nanoparticles and confirmed the Eu II :Eu III ratio, even if the estimated average composition Eu 2.11 O 2 S would correspond to about 15 % of Eu II . In 2013, Ajithkumar et al.

2(1-y) Ce 2y O 2 S nanoparticles

  ) and cerium acetylacetonate hydrate (Ce(acac) 3 •xH 2 O; 99.9 %) were purchased from Strem Chemicals. All products were used as received without further purification.In a typical synthesis of GdCeO 2 S, Gd(acac) 3 •xH 2 O (0.25 mmol), Ce(acac) 3 •xH 2 O

	Bimetallic oxysulfides were prepared via a solvothermal reaction including a mixture of
	organic solvents and metallic complexes. Oleylamine (OM; technical grade, 70 %), oleic acid
	(OA; technical grade, 90 %), sulfur (S 8 ; ≥ 99.5 %) and sodium oleate (Na(oleate); ≥ 99 %)
	were purchased from Sigma-Aldrich. 1-Octadecene (ODE; technical grade, 90 %) was
	purchased from Acros Organics. Gadolinium acetylacetonate hydrate (Gd(acac) 3 •xH 2 O;
	99.9 %

Y.-Z. Huang, L. Chen and L.-M. Wu, Cryst. Growth Des., 2008, 8, 739-743.

Acknowledgements

This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE led by Sorbonne Université. Sorbonne Université, CNRS, and the Collège de France are acknowledged for financial support.

Vibrating Sample Magnetometry (VSM)

The magnetization of the nanoparticles was measured between 5 K and 350 K by vibrating

Scanning electron microscopy (SEM) and energy dispersive X-ray spectroscopy (EDS)

A small amount of powder was deposited on a carbon adhesive tape on a scanning electron microscope sample holder. A layer of carbon was deposited on the samples for electronic conductivity. EDS analyses were performed on a SEM HITACHI S-3400N at 10 kV. Titanium was chosen as reference and analyses were performed on three to five different zones on each sample.

X-ray Powder Diffraction (XRD)

The different X-ray diffraction patterns of dry powders were measured on a Bruker D8 diffractometer using Cu Kα radiation at 1.5406 Å. Typical diffractograms were collected with steps of 0.05 ° and a scanning speed of 5 s/point. The backgrounds of the patterns were subtracted using the EVA software. When low noise Si monocristals sample holders were used, the angular position 2θ was corrected by adjusting the sample height (correction around 0.5 to 1 mm).

Table of Content Image

Luminescent (Gd,Ce) 2 O 2 S oxysulfide nanoparticles show tunable magnetic properties as a function of their cerium content.


Clément Larquet

Djamila Hourlier

Anh-Minh Nguyen

Almudena Torres-Pardo

Andrea Gauzzi

Clément Sanchez

Sophie Carenco
email: sophie.carenco@sorbonne-universite.fr

Thermal Stability of Oleate-Stabilized Gd2O2S Nanoplates in Inert and Oxidizing Atmospheres

Keywords: Gadolinium oxysulfide nanoparticles, organic ligands, structural stability, thermogravimetric analysis, mass spectrometry

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Introduction

Owing to their tunable optical emission and absorption properties, lanthanide oxysulfides Ln 2 O 2 S (Ln = lanthanide and yttrium) have found applications in various devices such as phosphors for cathode ray tubes, [1][2][3][4] scintillators, [5][6][7] and lasers (emission [8][9][10] and absorption 11,12 ). Following the first studies carried out in the 60's on the synthesis, morphology and doping of bulk samples, these compounds have been recently obtained also in the form of nanoparticles by means of various synthesis routes. This progress has renewed the hype for lanthanide oxysulfides, because these nanoparticles are promising for bimodal imaging (magnetic and optical), a key technology in nanomedicine. 13 Ln 2 O 2 S nanoparticles are synthesized either in an aqueous or organic medium, which requires two different sulfidation methods. In water, sulfidation is performed in a subsequent step by heating the precursors at high temperature (T> 600 °C) using either sulfur vapor, [14][15][16] CS 2 formed in situ (S 8 in carbon) 17,18 or H 2 S. 19 In organic medium, sulfidation is achieved at milder temperatures and in a single step. For instance, in a mixture of oleylamine, oleic acid and 1-octadecene, Ln(acac) 3 reacts with elemental sulfur at 310 °C in inert atmosphere.

Oleylamine helps dissolving the elemental sulfur and forming reactive sulfide species. 20 With the help of an alkaline source, Ln 2 O 2 S nanoplates surrounded by organic ligands are obtained. [21][22][23][24] One open question concerns the thermal stability of nanoscaled Ln 2 O 2 S, a factor that may limit the practical use of these nanomaterials. For instance, the absorption of light (X-rays in scintillators, visible light for laser absorption) is often accompanied by strong local heating.

Owing to the high-temperature sulfidation process, the nanoparticles obtained in aqueous medium are typically larger than 50 nm and their crystal structure is similar to that of the bulk phase. [25][26][27] This accounts for a high thermal stability up to 1200 °C observed for most lanthanides (cerium excluded). 28 On the other hand, the small Ln 2 O 2 S nanocrystals (below 30 nm) obtained in organic media differ markedly from bulk crystals. Typical feature are the non-stoichiometry, 29 the presence of alkyl-chains at the surface 21 and the capability of self-assembling into wires 30 . For example, gadolinium oxysulfide easily form small non-stoichiometric Gd 2 O 2 S x nanoplates (8 nm x 1.5 nm) with x = 0.4 -0.6, which self-assemble in solution into nanowires. 21,22 In this report, we studied the thermal stability of gadolinium oxysulfide nanoparticles covered with organic ligands in inert and oxidizing atmospheres. In inert atmosphere, we discovered a multi-step behavior with temperature that leads to the phase segregation of bulk oxysulfide Gd 2 O 2 S and Gd 2 O 3 . This is explained by the non-stoichiometry (x ≠ 1) of the Gd 2 O 2 S x nanocrystals. Thanks to Fourier transform infrared spectroscopy (FTIR) and thermogravimetric analysis coupled with mass spectrometry (TGA-MS), we were able to determine the nature of the ligands, their amount on the surfaces of the nanocrystals and their coordination modes on the surface gadolinium atoms.

Under oxidizing atmosphere, a few steps of transformation lead to the formation of both bulk gadolinium oxide and gadolinium oxysulfate. Moreover, we could isolate ligand-free nanoparticles by controlling the thermal decomposition of the ligands in oxidizing atmosphere. This process removes the highly hydrophobic character of the nanocrystals, and makes them suitable for practical use, eg. in water or biological media.

Results and discussion

Gd 2 O 2 S nanoparticles synthesis and structure Gd 2 O 2 S nanoparticles were synthesized following an optimized protocol described in the experimental section and previously reported by our group. 22 Briefly, Gd(acac) 3 •xH 2 O (0.5 mmol) was mixed with elemental sulfur (0.5 equiv. S vs. Gd), Na(oleate) (1 equiv. vs.

Gd) in a mixture of oleylamine, oleic acid and 1-octadecene. The reaction was conducted at 310 °C for 30 minutes and yielded the nanoparticles presented in Figure S1.

The Gd 2 O 2 S nanoparticles crystallize in the P-3m1 hexagonal space group. The anisotropic growth leads to the formation of hexagonal-shaped platelets of 7.8 ± 1.3 nm width and 1.5 ± 0.2 nm thickness, i.e. two to three unit cells thick. The S:Gd ratio is ca 0.25, hence much lower than that in the bulk phase, as reported by Ding et al. 21 and by our group 22 . This result is explained by the presence of organic ligands at the surface that stabilize [Gd 2 O 2 ] 2+ terminal layers on the {001} facets of the lamellar compounds. The sample composition is thus better described by the formula Gd 2 O 2 S 0.5 . For the sake of clarity and consistency with previous published works (by us and others), we will keep below the notation Gd 2 O 2 S that best represent the crystal structure.

Thermal stability of Gd 2 O 2 S nanoplates in inert atmosphere

The thermal stability of the Gd 2 O 2 S nanoplates was analyzed by thermogravimetric analysis (TGA) coupled with mass spectrometry in inert atmosphere (He). The results of the TGA in inert atmosphere are presented in Figure 1. Thermal decomposition of Gd 2 O 2 S nanoparticles is a complex process that can be divided into at least six steps, as shown in Figure 1B. The total mass loss between 25 °C and 1200 °C is 33.4 %. One third of the sample was lost in the form of volatile species during the heating.

The combination of mass spectrometry with the TG equipment allowed us to precisely identify the various gas species emitted during each step (Figure 2).

Remarkably, the chemical processes are well separated. For the sake of clarity, only major volatile products detected are shown in Figure 2. The loss of water and CO 2 around 100 °C is attributed to physisorbed molecules. The emission of these species is expected, considering that the condensation of acetylacetonate ligands on oleylamine during the synthesis generates H 2 O in situ and considering that the samples were isolated and washed in ambient air (Figure 1B step (i)). 31 In step (ii), the decarboxylation of oleate molecules starts and CO 2 is formed. The liberation of CO 2 alone must involve oleate groups in which the terminal functionality -(COO-) is rather labile and susceptible to decarboxylation. It seems reasonable to suppose that a certain number of COO groups are farther away from the Gd sites and thus decarboxylated at this step (free oleic acid and weakly bonded oleate), while others oleate molecules remain strongly bonded to the surface of the nanoparticles. We put forward the hypothesis that the alkyl chains of the oleate molecules decarboxylated in step (ii) remain bonded to the other oleate groups that are still coordinated to the surface gadolinium (no C x H y signature at this step). The long alkyl chains may be bonded either by weak interactions between the hydrophobic chains or covalently by reaction via the C-C double bonds.

In step (iii), a major decomposition of organic-based ligands occurs, resulting in the detection of numerous fractions related to long alkyl chains. Evidence for this is the release of a set of fragments at m/z 41, 55, 43, 69, 81 shown in Figure 3, resulting from the scission of the oleate ligands followed by hydrogen radical recombination. The main ion at m/z 30 characteristic of oleylamine is absent of the mass spectrum, which indicates that there is no significant amount of oleylamine vs. oleates on the surface of the nanoparticles. 32 nanoplates (steps (i) to (iv)).

[Gd 2 O 2 S 0.67 ] 0.67+ [Gd 2 O 2 ] 2+ [S] 2- [Gd 2 O 2.5 ] + [S] 2- [Gd 2 O 2 ] 2+ [S] 2- "Gd 2 O 2.33 S 0.67 " (neutral) Gd 2 O 2 S Gd 2 O 3 2 Gd 2 O 2 S + Gd 2 O 3 = [Gd 2 O 2 ] 2+ -H 2 O -CO -CO 2 -C x H y -CO 2 (iv)-a (iv)-b (iii) (ii) (i) (iv) -H 2 O -CO 2 -CO -CO 2 -CO 10
There are two crucial points here. First, during steps (ii) and (iii), ligands undergo important modifications: the charge balance can still be ensured by the remaining carboxylates groups.

In the meantime, the surface becomes more accessible and thus sintering becomes easier (Figure 4).

Second, as the crystals grow and sinter, the lack of sulfur in the powder makes the formation of larger crystals of stoichiometric Gd 2 O 2 S impossible. The formation of large crystals of non-stoichiometric Gd 2 O 2 S ~0.5 is prohibited as charge balance would not be respected in this case (there are no more ligands to compensate the charge). We could also imagine that the free carbon at the surface of the crystals could limit the sintering process. However, we observe at high temperatures (> 900 °C) the coexistence of growing crystals of Gd 2 O 2 S with allotropes of Gd 2 O 3 on the XRD patterns of the annealed powders (Figure S2).

Step (iv) can be divided in two sub-steps regarding the evolution of the fragments corresponding to CO 2 and CO (Figure 2). Carbon dioxide comes from the end of the decarboxylation process. Carbon monoxide is likely the product of the reaction between the free carbon (formed by the decomposition of the ligands carbon chains in step (iii)) and CO 2 according to the Boudouard reaction. The equilibrium favors the formation of CO at high temperatures, which could explain that CO becomes the major product in (iv)-b (Figure 2B). Step (v) , from 780 to 900 °C, represents a limited mass loss induced by evolved CO, possibly formed at the previous step but adsorbed on Lewis acid sites.

Step (vi) corresponds to a loss of a small quantity of H 2 S coming from a limited degradation of the oxysulfide phase. The TG analysis associated to the structural investigations by XRD (Figure S2) indicates that the heat treatment up to 1200 °C did not significantly affect the different compositions (Gd 2 O 2 S, Gd 2 O 3 -c and Gd 2 O 3 -m) between the end of step (v) and that of step (vi).

Ligand nature, amount and coordination

The previous section showed that oleates are the most significant ligands (if not the only ones) around the nanoparticles. Their amount was estimated to be around 30 wt% of the powder of The frequency difference between the asymmetric and symmetric stretching bands of the COO group is indicative of the bonding type of the carboxylate ligand. In particular, Deacon and Phillips 34 indexed more than eighty acetate or trifluoroacetate compounds whose crystalline structures and infrared spectra were available, according to the acetate coordination type: unidentate ligand, chelating ligand, bridging bidentate ligand and monoatomic bridging ligand (Figure 5B).

For the nanoparticles studied here (Figure 5A), the difference between the two stretching modes of COO group is Δν = 112 cm -1 (ν asym = 1497 cm -1 and ν sym = 1385 cm -1 ). In comparison, the values measured on sodium oleate are Δν = 100, 118 and 139 cm -1 (separation of the symmetric band in three contributions, see Figure S3). The low value of Δν 12 (< 150 cm -1 ), close to the value of the corresponding sodium carboxylate, is typical of chelating or bridging carboxylate ligands, and rules out the monodentate ligand configuration (Figure 5B, coordination mode I). Moreover, in comparison with the numerous structures presented by Deacon and Philips, this value is one of the lowest, and suggests that chelation is favored in our case (Figure 5B, coordination mode II). 34 Geometric derivatives mixing different bonding types also exist and are not presented.

On the basis of thermogravimetric analysis, the calculated amount of ligands is around 30 wt% of the Gd 2 O 2 S nanoparticles powder. It is thus possible to estimate the ratio between A B * the quantity of oxygen coming from the oleate ligands and the exposed gadolinium of our nanoplates, considering the nanoplate geometry (Figure S1).

We based this estimation on the average mass of powder obtained per synthesis: 100 mg fractioned between 3 mg of lightly adsorbed products (including water, step (i) in Figure 2), 30 mg of oleate ligands and 67 mg of Gd 2 O 2 S x nanocrystals. Because only 40 % of the gadolinium of the nanoplates is exposed to the surface (both {100} and {001} facets), we calculated that each gadolinium atom monopolized 1.43 oxygen atoms from the oleate ligands (Figure S4). For such a value, a mix between chelation (two oxygen atoms per metallic cation) and bridging (one O per M) can be envisaged. This result is in agreement with the structures of lanthanide carboxylates (from samarium to lutetium 35 and yttrium 36 ) for which the IR signals evidenced a ratio of 2 chelating ligands for 1 bridging ligand (total: 1.6 O/Ln).

However, with this ligand quantity, the charge compensation is not reached. If we assume the Gd 2 O 2 S 0.5 formula for the nanoparticles (+ 1 total charge), they should be surrounded by 1 equiv. of oleate ligands (-1 charge per molecule) to obtain the charge balance. Here, approximately 0.72 oleate ligand per Gd 2 O 2 S surrounds the nanocrystals, which is not enough.

Nevertheless, one should remember that the oleate ligands are mainly bonded to gadolinium atoms by chelation. It means that there is not much remaining space for an additional ligand to bond a metal. Actually, the number of oleate ions represents a spatial occupation of 4.5 ligands/nm 2 , which is already a high value for such ligands (values in the literature are closer to 0.2 nm 2 for each -COO -group). 37,38 Accordingly, Anderson et al. reminded that the steric hindrance between the alkyl chains of the oleate will limit the coverage to the density of crystalline alkyl chains (4.9 chains/nm 2 ). 39,40 In this first part, we described the thermal behavior of the nanoparticles annealed under inert atmosphere. This allowed us to identify the nature and to estimate the number of organic ligands surrounding the nanoparticles. However, technological use of such nanoparticles will likely include heating devices under ambient atmosphere. To model these conditions, we performed a second analysis in flowing gas containing a mixture of helium and 20 % O 2 in volume.

Thermal behavior under oxidizing atmosphere and ligand removal

The thermal behavior of Gd 2 O 2 S nanoplates was also investigated under oxidizing atmospheres with an air-like mixture (O 2 /He with 20/80 v/v) and at intermediate reaction temperatures, in order to identify the nature of the solid formed at the most relevant steps. Under O 2 /He (20/80 v/v), the final mass loss is lower than in inert atmosphere (Figure 6). This is due to the formation of gadolinium oxysulfate Gd 2 O 2 SO 4 , detected by XRD (Figure 7).

These compounds are known to form from oxysulfides in oxidizing medium (reversible

A ii iii i 350 C 517 C B C
reaction possible in H 2 ) and were shown to be promising for thermally activated oxygen storage. 41 It is worth noting that the thermal decomposition in oxygen observed in the TGA curve (Figure 6) occurs in three steps. The first mass loss between 40 °C and 200 °C is quite similar to that obtained in inert atmosphere, and corresponds to the loss of physisorbed species H 2 O and CO 2 . The major mass loss of 25 % occurs in the second step between 250 °C and 400 °C.

The last step above 500 °C implies a moderate 4 % loss of the sample mass.

In order to identify the intermediate compounds, we performed two additional experiments for which we stopped the heating ramp at selected temperatures: 350 °C (within step (ii) in Figure 6A) and 517 °C (after the end of step (ii)). The temperature was kept constant for twenty minutes before cooling down and collecting the powder. When heated around 350 °C for 20 minutes, although the ligands have undergone considerable changes, the resulting product still corresponds to Gd 2 O 2 S nanocrystals (Figure 7B), as attested by the broadness of the XRD peaks which is similar to these of the starting material. TEM confirmed this by showing the presence of small polycrystalline aggregates as well as still isolated nanoparticles in the sample (see Figure S6). Infrared spectroscopy after this 20 min plateau confirms that the alkyl chains are not detected anymore (Figure S3).

In comparison, the same thermal treatment at 517 °C led to nanoscaled powders majorly consisting of gadolinium oxide. For higher temperatures, XRD indicates that the oxysulfide phase completely disappears as a result of the oxidizing treatment. It may form an amorphous intermediate first, while part of the product crystallizes into cubic gadolinium oxide (Gd 2 O 3cubic). Then the amorphous part crystallizes into gadolinium oxysulfate (Figure 7A). In this section, we demonstrated that the inorganic crystal of Gd 2 O 2 S is stable up to around 350 °C under oxidizing atmosphere. The nanoscale is also preserved at this temperature. It is thus possible to remove the surface ligand at this temperature without degrading the inorganic core of the nanoparticles. The above study enabled us to better characterize the surface species present on the nanocrystals. Our analysis rules out the possibility of a significant amount of amide ligands derived from oleylamine chains and shows that only oleate groups cover the nanoparticle surface. The oleate groups are coordinated at gadolinium surface atoms in both chelation and bridging modes.

In the present study, we demonstrated that the hydrophobic oleate chains are readily removed by a mild treatment at 350 °C in oxygen, which does not alter the nanoplate properties. This smooth process may be useful in view of the practical applications of these nanomaterials as thin films for lightening devices or post-functionalization with hydrophilic ligands for biomedical applications.

Figure 1 :

 1 Figure 1: (A) Thermogravimetric analysis of Gd 2 O 2 S nanoplates performed in inert

Figure 2 :

 2 Figure 2: Thermal evolution of the major fragments detected in the gas phase by mass

Figure 3 :

 3 Figure 3: (A) Oleic acid mass spectrum with m/z between 20 and 130, from the NIST database. 33 The inset displays the total spectrum. (B) Thermogramm of Gd 2 O 2 S nanoplates

Figure 4 :

 4 Figure 4: Proposed mechanism of the thermal decomposition during the annealing of Gd 2 O 2 S

  At the same time, oxygen from the ligands necessarily feeds the surface of the nanoparticles to maintain charge balance. The carboxylate groups coordinated on the surface form C-O-Gd bonds. Upon heating, a number of the C-O bonds break, leaving excess oxygen on the surface (Figure4). [Gd 2 O 2 ] 2+ terminal layers become [Gd 2 O 2.5 ] + (red layers in Figure4) while the ligands decompose. The average formula for the resulting nanoplate turns from [Gd 2 O 2 S 0.67 ] 0.67+ to neutral Gd 2 O 2.33 S 0.67 . In parallel, the phase separation and growth lead to large crystals of Gd 2 O 2 S and Gd 2 O 3 .

Gd 2 O 2 S 1 (

 221 nanoplates. Infrared spectroscopy was then employed to determine the chemical bonds present in the nanopowders. A representative IR spectrum of Gd 2 O 2 S nanoplates is displayed in Figure5. The main absorption bands are due to the oleate ligands: 2959 cm -CH 3 asymmetric elongation); 2923 cm -1 (CH 2 asymmetric elongation); 2853 cm -1 (CH 2 symmetric elongation); 1497 cm -1 (COO asymmetric stretching); 1385 cm -1 (COO symmetric stretching).

Figure 5 :

 5 Figure 5: (A) FTIR spectrum of Gd 2 O 2 S nanoplates collected in ATR mode. The peak at 850 cm -1 is under investigation. (B) Four coordination modes between a carboxylate ligand

Figure 6 :

 6 Figure 6: Thermal behavior of Gd 2 O 2 S nanoplates under oxidizing atmosphere. Evolution of

Figure 7 :

 7 Figure 7: Powder XRD patterns of annealed Gd 2 O 2 S nanoplates under O 2 :He (20/80 v/v), (A)

Conclusion

  Many analytical techniques have been employed to monitor the thermal stability of Gd 2 O 2 S nanoplates stabilized by organic ligands. Our results demonstrate that the thermal stability critically depends upon the surrounding atmosphere. In inert atmosphere, the degradation of the nanoplates follows a multi-step process associated with a variety of mechanisms (dehydration, decarboxylation, ligand degradation, decarbonation/decarboxylation and sintering) that lead to distinct mass losses. In oxidizing atmosphere, the various reactions occur simultaneously at lower temperatures. The sulfur-defective structure of the nanocrystals triggers off the formation of gadolinium oxide crystals as well as oxysulfide or oxysulfate species depending on the atmosphere.

Acknowledgements

This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE led by Sorbonne Université. Sorbonne Université, CNRS, and the Collège de France are acknowledged for financial support. We acknowledge the National Center for Electron Microscopy (UCM) for assistance with the generation of electron microscopy data.

Experimental section

Synthesis of Gd 2 O 2 S nanoparticles

Gadolinium oxysulfide nanoparticles were prepared under inert atmosphere in a mixture of organic solvents following a previous report. 22 Oleylamine (OM; technical grade, 70 %), oleic acid (OA; technical grade, 90 %), sulfur (S 8 ; ≥ 99.5 %) and sodium oleate (Na(oleate); ≥ 99 %) were purchased from Sigma-Aldrich. 1-Octadecene (ODE; technical grade, 90 %) was purchased from Acros Organics. Gadolinium acetylacetonate hydrate (Gd(acac) 3 •xH 2 O; 99.9 %) was purchased from Strem Chemicals. All products were used as received without further purification.

In a typical synthesis of Gd 2 O 2 S, Gd(acac) 3 •xH 2 O (0.50 mmol), S 8 (0.032 mmol), Na(oleate) (0.50 mmol), OM (17 mmol), OA (2.5 mmol) and ODE (32.5 mmol) were placed in a 100 mL three-neck flask at room temperature. The yellow solution was heated to 120 °C under vacuum for 20 min to remove water and other impurities with low boiling points. The mixture was then heated to 310 °C and stirred at this temperature for 30 min under purified N 2 . The transparent solution gradually became turbid starting from 280 °C. Then the mixture was left to cool to room temperature under N 2 . The nanoparticles were isolated by centrifugation using ethanol and washed at least three times using a THF/ethanol (1/5) mixture to remove the remaining reagents and organic matter.

Thermogravimetric analysis coupled with mass spectrometry (TGA-MS)

Thermogravimetry analysis (TGA) (Netzsch STA449F3 Jupiter apparatus) coupled with a quadrupole mass spectrometry (MS) (Aëolos QMS403D, 70 eV, electron ionisation) via a heated capillary system was used to monitor the decomposition of samples during the annealing process and to analyze the evolved gaseous species. Before each experiment, the TGA system was first evacuated and then flushed with the same ultrahigh purity gas which was used for the thermal treatment. The experiments were carried out under dynamic inert or 20 reactive gas: He (99.999 % purity) or 20 % O 2 in He (99.999 % purity) at a flow rate of 50 cm 3 /min. The samples were heated in Al 2 O 3 crucibles up to 900 °C or 1200 °C with a heating rate at 2 °C/min.

X-ray Powder Diffraction (XRD)

The different X-ray diffraction patterns of dry powders were measured on a Bruker D8 diffractometer using Cu Kα radiation at 1.5406 Å. Typical diffractograms were collected with steps of 0.05 ° and a scanning speed of 5 s/point. The backgrounds of the patterns were subtracted using the EVA software.

Fourier Transform Infrared Spectroscopy (FTIR)

Infrared spectra were collected on a Spectrum 400 (PERKINELMER) spectrometer. The dry sample (1 to 3 mg) was deposited on the attenuated total reflectance (ATR) crystal.

Transmittance was measured between 4000 cm -1 and 550 cm -1 with steps of 0.5 cm -1 .

Table of Content

Capping ligands play an important role in the chemistry of nanoparticles and their biomedical applications. We show that the thermal stability of oleate-covered Gd 2 O 2 S x nanocrystals is limited and strongly depends on the annealing atmosphere (inert or oxidizing). Annealing the nanoparticles in air enables removing the ligands without altering the nanocrystals structure. This work establishes the range of conditions for their practical use and provides an access to ligand-free nanoparticles.