Gabriel Amiard Hudebine

Eric Freysz

Tunable and chirp free femtosecond signal pulses generated by a PPLN OPO pumped by Ytterbium fiber laser chirped pulses

Keywords: Optical parametric oscillators, Parametric amplification, Parametric generation, Three-waves mixing, Femtoseconde pulse, Chirped pulse, Ytterbium fiber laser, PPLN crystal

Chirp of the pulses generated in the 1.4 µm-1.8 µm spectral ranges and in a singly resonant OPO build around a 3 mm long PPLN crystal pumped at 108 MHz by 40 nJ, 1 ps chirped or 350 fs weakly chirped pump pulses centered at 1035 nm is analyzed. Whatever the chirp of the pump pulses, we achieved a pump to signal conversion efficiency > 30%. We demonstrate that by adjusting the cavity length around synchronism condition, chirped or almost chirped free (down to 80 fs) signal pulses are generated. A simple mechanism accounts for this phenomenon.

INTRODUCTION

Many different optical parametric amplifiers and oscillators (OPO) taking advantage of Periodically poled (PP) ferroelectric crystals such as LiNbO 3 , KTP, KTA have been proposed [START_REF] Kokabee | Efficient, high-power, ytterbium-fiber-laser-pumped picosecond optical parametric oscillator[END_REF][START_REF] Bhushan | Generation of High Efficiency 2 µm Laser Pulse from a Periodically Poled 5 mol % MgO-Doped LiNbO 3 Optical Parametric Oscillator[END_REF] . These crystals exhibit high effective nonlinear coefficient which under quasi phase-matching conditions (QPM) make it possible to record efficient second order nonlinear processes such as as second harmonic generation (SHG), sum-frequency generation (SFG) or difference frequency generation (DFG) in the visible and IR range. These crystals are a good mean to develop mid-infrared tunable sources. When pumped with chirped femtosecond pulses, it has been shown that chirped PP gratings impinged in the crystal can compensate for the chirp of the pump pulse [START_REF] Schober | Tunable-chirp pulse compression in quasi-phase-matched second-harmonic generation[END_REF] . This makes it possible to generated Fourier transform-limited femtosecond pulses. Hereafter, we show that when chirped pump pulses are used, one can also generate a signal pulse Fourier transform-limited without using chirped PP crystal taking advantage of velocity group delay and velocity group dispersion between the pump and signal in the PP crystal [4]. Accordingly, we show that a slight detuning of the resonator-length makes it possible to generate almost chirped free signal pulse at the expense of a reasonable decrease of the conversion efficiency. The phenomenon is investigated for weakly and largely chirped pump pulses.

EXPERIMENTAL SET-UP AND RESULTS

Set-up

A sketch of our singly resonant OPO is depicted in Fig. 1. Our linear OPO cavity is build with two plane and two concave mirrors with 10 mm radius of curvature. The mirrors CM 1,2 and M 2 transmit more than 99 % of the pump wavelength and reflect more than 99 % of light at wavelengths in between 1.4 µm and 1.8 µm. In this spectral range, mirror M 1 is reflecting 90 %. All the mirrors have very low dispersion in the 1.4 µm and 1.8 µm spectral range. The pump pulse is focused with a 20 cm focal lens through the mirror CM 2 in a 3 mm thick MgO:PP-LiNbO 3 (PPLN) crystal (MOPO1-1.0-3, from Covesion) in which five gratings with periods ranging from 29.52 to 31.59 µm are impinged. This crystal is inserted in a thermally regulated oven whose temperature can be tuned between 30 °C and 200 °C. It is also antireflection coated for both the pump and signal wave (R < 1.5 % at 1030 nm and R < 1 % in the 1.4 µm-1.8 µm range). We adjusted the cavity length of the OPO, by translating with a remote piezo-mechanic translation stage the mirror M 2 . To finely adjust their distance, the two concave mirrors are mounted on micrometric mechanic stages. The length of our singly resonant OPO's length is firstly adjusted at the repetition rate at 108 MHz. We also performed similar experiments at twice the repetition rate of the pump laser that will be not presented hereafter. The small leak through mirror M 2 is used to measure the spectra of the signal and idler wave produced by our OPO. The latter is sent in an optical spectrum analyzer (OSA205 Thorlabs) which records spectra in between 1 µm and 5.6 µm. The autocorrelation trace of the signal and pump pulses are measured with a homemade autocorrelator using a 300 µm thick BBO crystal. Our OPO is pumped by an all-normal-dispersion ytterbium fibered oscillator [START_REF] Deslandes | Picosecond to femtosecond pulses from high power self mode-locked ytterbium rod-type fiber laser[END_REF] . The linearly polarized and chirped pump pulse centered at  p =1035 nm has a duration at a spectral bandwidth of ~1.06 ps (Fig. 2a) and ~8 nm (Fig. 2b), respectively. The time bandwidth product of these pulses is ~2.2. At 108 MHz repetition rate, the averaged power delivered by this laser is ~6 W. To evaluate the impact of the pump pulse on the chirp of the signal pulses delivered by our OPO we have recompressed them using two chirped mirrors with a group delay dispersion of 10000 fs 2 at 1035 nm. The autocorrelation trace of the signal pulse after 10 reflections by these mirrors is presented in Fig. 1c. This indicates that the pump pulse, compressed down to ~380 fs, has the time bandwidth product of ~0.8 by and presents a residual chirp.

Results

Figure 3a displays the evolution of the signal power versus pump power for different gratings when the cavity length is synchronized with the repetition rate of the pump pulse. The pump threshold is P th ~300 mW whatever the used grating period. The slope is ~30 % and almost independent of used grating period and temperature of the crystal. The pump to signal + idler conversion efficiency C is ~44 %. It is defined as follow (P s +P i )/(P p -P th ) where P x refers to the power of the signal (s), idler (i); pump (p). At maximum, the depletion of the pump within our OPO is 52 %. The evolution of the signal wavelength was found to be in good agreement with the Sellmeier equations of the used for our PPLN crystal [START_REF] Umemura | Sellmeier and thermo-optic dispersion formulas for the extraordinary ray of 5 mol % MgO-doped congruent LiNbO 3 in the visible, infrared, and terahertz regions[END_REF] . In this synchronized regime, we found the duration at the half width of signal pulse (t ~1.03±0.2 ps) was about the duration at half width of the pump pulse ((t~1.06±0.2 ps) whatever the grating or the temperature of the crystal. Typical spectra and pulse duration recorded using the 30.49 µm grating at 30 °C is presented in Fig. 3b-c. According to its spectrum, we found the duration at half width of a gaussian signal pulse should be at most 0.2 ps, showing that the time-bandwidth product of the generated signal pulse is ~2.2 almost similar to the time bandwidth product of the pump pulse. To reduce the pulse duration of this signal pulse, we slightly detuned the cavity length of our OPO. Besides the change of the pulse duration, the change of the cavity length results in both a decrease of the signal intensity and a shift of the central wavelength of the signal pulse. The typical evolution of this ensemble of parameters is recorded at 30 °C when using the 30.49 µm grating presented in Fig. 4. One can notice that the shorter pulse duration is achieved when the cavity length is increased by about 6 µm. Under this condition the output power is reduced by a factor of two, but as indicated by the autocorrelation trace, the pulse duration at half width is reduced to t~236 fs (Fig. 5). This clearly indicates that the peak power of this latter signal pulse is larger. It is also interesting to mention that according to its spectrum the pulse duration at half width is at most t~156 fs indicating that under this condition the time bandwidth product of the signal pulse is reduced from ~2.2 to ~0.65, hence below the time bandwidth product of the pump pulse. The evolution of the intensity of the signal pulse delivered by the OPO when it is pumped by compressed femtosecond pump pulse is presented in Fig. 6a. It is very similar to the results presented in figure 3a. However, a linear regression of the signal intensity versus the pump intensity indicates the threshold is reduced to ~150 mW in good agreement with the increase of the peak power of the pump pulse. The spectrum of signal pulse was found to broaden. The typical evolution of the spectrum as one increase the temperature of the crystal is displayed in Fig. 6b. The autocorrelation trace of the signal pulse recorded when the used 30.4 µm grating is heated at 70 °C is displayed in Fig. 6c. As expected for shorter pump pulse the signal pulse duration is also shorter and here again about the pulse duration of the pump pulse. However, the time bandwidth product of this signal is ~0.93 slightly above the time bandwidth product of the compressed pump pulse (~0.8). This phenomenon is likely due to cross-and self-phase modulation in the PPLN crystal. We then measured the impact of the cavity length detuning on the signal pulse duration. Under this condition, the signal pulse duration and its intensity was found to change more drastically versus the cavity length. The intensity of the signal was found to decrease linearly when increasing the cavity length (0.2 W. µm -1 ). As we increased the cavity length, we also recorded a drastic change of the pulse duration. The spectrum and the pulse duration recorded as the cavity length increases (0<L<2 µm) is displayed in Fig 7 . The pulse duration is reduced down to ~80 fs at 1 µm and then increases again. It is important to stress that in between 1 µm and 2 µm, the pulses duration of the signal is almost limited by its spectrum and the time-bandwidth product is ~0.4. It is also important to indicate that while the pulse duration is reduced by a factor of ~4.7, the intensity of the signal wave is only reduced by 20 %. This clearly stress that the peak power of the generated pulse are increase by a factor of ~4. It is also worth mentioning that this behavior was recorded for all the grating whatever the temperature of the crystal.

Discussion

The impact of the cavity length detuning L on a synchronously pumped OPO has been reported by many authors [START_REF] Fallnich | Experimental investigation and numerical simulation of the influence of resonator-length detuning on the output power, pulse duration and spectral width of a cw mode-locked picosecond optical parametric oscillator[END_REF][START_REF] Khaydarov | Pulse-compression mechanism in a synchronously pumped optical parametric oscillator[END_REF][START_REF] Edelstein | Broadly tunable high repetition rate femtosecond optical parametric oscillator[END_REF][START_REF] Lefort | Generation of femtosecond pulses from order-of-magnitude pulse compression in a synchronously pumped optical parametric oscillator based on periodically poled lithium niobate[END_REF][START_REF] Seres | Nonstationary theory of synchronously pumped femtosecond optical parametric oscillators[END_REF][START_REF] Kurti | Pulse compression in a silver gallium sulfide, midinfrared, synchronously pumped optical parametric oscillator[END_REF][START_REF] Khurgin | Passive mode locking of optical parametric oscillators: an efficient technique for generating sub-picosecond pulses[END_REF][START_REF] Laporte | Analysis of cavitylength detuning in diffraction-grating narrowed picosecond optical parametric oscillators[END_REF][START_REF] Ebrahimzadeh | Design and performance of a singly resonant picosecond LiB_3O_5 optical parametric oscillator synchronously pumped by a self-mode-locked Ti:sapphire laser[END_REF] . The phenomenon has been accounted by the model proposed by Khaydarov et al. [START_REF] Khaydarov | Pulse-compression mechanism in a synchronously pumped optical parametric oscillator[END_REF] and experimentally reported by different authors [START_REF] Fallnich | Experimental investigation and numerical simulation of the influence of resonator-length detuning on the output power, pulse duration and spectral width of a cw mode-locked picosecond optical parametric oscillator[END_REF][START_REF] Cheung | Theory of a synchronously pumped optical parametric oscillator in steady-state operation[END_REF] . In agreement with our experiment, this phenomenon occurs for L >0 when group velocity of the signal pulse is larger than the group velocity of the pump pulse. One of the interesting points revealed by our experiment, not yet implemented in the available numerical simulations, is the impact of the pump chirp on the pulse compression of the signal pulse. Our experimental results clearly stresses that by slightly increasing the cavity length, indeed one can generate chirp signal pulse out of an OPO using chirp pump pulse. The compression mechanism results in a slight decrease of the conversion efficient of our OPO which is largely compensated by the increase of the peak power of the generated signal pulse. Yet, an interesting aspect will be to evaluate the chirp of the idler wave which according to the phase matching condition should carry out the chirp of pump pulse. This experimental work as well as the associated numerical simulations will be presented in a forthcoming publication.

CONCLUSIONS

We have developed a PPLN based OPO pumped at 108 MHz, 40 nJ, 1 ps chirped or 350 fs weakly chirped pump pulses centered at 1035 nm. Under optimal synchronous pumping condition signal pulse tunable from 1.4 to 1.8 µm were generated with a pump to signal conversion efficiency of ~30%. The duration of the signal pulse is then almost similar to the pump pulse. However, when the cavity is slightly detuned, shorter signal pulse are generated. Under this condition, the energy of the signal pulse is decreased by ~20 % while the peak power is increased by almost of factor 4. We also found that the time-bandwidth product of these pulses is ~0.4 indicating that the generated signal pulses are almost chirp free.

Figure 1 :

 1 Figure 1: Experimental setup of the OPO. HWP = Half-wave plate; M 1-2 = mirrors; CM 1-2 = concave mirrors. Doted arrows: signal leak and residual pump beam.

Figure 2 :

 2 Figure 2: (a) Spectrum of the pump pulse. (b) Autocorrelation trace of the uncompressed and compressed pump pulse.

Figure 3 :

 3 Figure 3: (a) Conversion efficiency of the OPO. (b-c) spectrum and autocorrelation trace of the signal pulse.

Figure 4 :Figure 5 :

 45 Figure 4: Impact of the cavity detuning on the signal average power (a) the central wavelength (b) and pulse duration (c).

Figure 6 :

 6 Figure 6: (a) Conversion efficiency, (b) evolution of the signal spectrum versus temperature, (c) typical autocorrelation trace of the signal pulse.

Figure 7 :

 7 Figure 7: Spectrum and autocorrelation trace of the signal pulse versus the cavity length detuning.


Yuemeng Feng
email: yuemeng.feng@creatis.insa-lyon.fr

Ane Etxebeste

Jean Michel Létang

David Sarrut

Voichit ¸a Maxim

Total variation and point spread function priors for MLEM reconstruction in Compton camera imaging

Keywords: Compton camera imaging, gamma ray, maximum likelihood, maximum a posteriori, total variation, point spread function

The Compton camera is a gamma ray imaging device already employed in astronomy and still in investigation for clinical domain. A key point in the imaging process is the tomographic reconstruction step. When the acquisition parameters and the a priori information are correctly accounted for, iterative algorithms are able to produce accurate images by compensating for measurement uncertainties and statistical noise. In this work we focus on the list-mode maximum likelihood expectation maximization (LM-MLEM) algorithm with smoothness a priori information expressed by the total variation norm. This type of regularization is particularly well suited for low-dose acquisitions, as it is the case in the applications foreseen for the camera. We show that the TV a priori strongly improves the images when data are acquired in ideal conditions. For realistic data, this a priori is not sufficient and deconvolution with a pre-calculated image-space kernel should also be considered.

I. INTRODUCTION

The number of photons detected by virtual elements of a Compton camera are i.i.d. Poisson random variables. Let t ij denote the probability for a photon emitted in voxel j to be detected as event i and let λ j be the mean number of photons emitted in voxel j. The contribution of the acquisition uncertainties can be modelled as a convolution in the image domain, in the sense that the data are supposed to come from projections of the real distribution λ convolved with some known kernel k. With T = (t ij ) denoting the system matrix, the data are thus observations from a multi-variate Poisson law with mean T (λ⊗k). The kernel (also called point spread function, PSF) is a low-pass filter corresponding to various physical effects that occur during the acquisition, e.g. measurement errors, attenuation and scatter. This approach has two relatively opposite effects: on the one hand it brings some smoothness to the solution, on the other hand it improves resolution. The first effect comes from the implicit relaxation of data fidelity constrains. The second is due to the deconvolution capabilities of the MLEM algorithm (known under the name Richardson-Lucy in image processing applications). However the PSF has to be carefully designed in order to capture entirely the data variability without over-corrections that would lead to Gibbs artifacts ( [START_REF] Stute | Practical considerations for image-based PSF and blobs reconstruction in PET[END_REF]). For instance, for ideal data the PSF should be chosen as a Dirac distribution.

The statistical noise and the ill-conditioned nature of the problem lead to reconstructed images that usually appear overly noisy. As a consequence, the iterations have to be stopped at some point that is difficult to anticipate and the images are usually post-smoothed. Maximum a posteriori expectation maximization (MAP-EM) algorithms have been previously investigated as they may bring information about the source distribution properties and thus regularize the solution. Often smooth priors are considered. In this study we investigate a non-smooth and spatially variant prior, the total variation norm. We show that in conjunction with the use of the image-space PSF, total variation allows to reduce the noise, improve the resolution and accelerate the convergence.

II. METHODS

A. Image-space PSF deconvolution

The logarithm of the likelihood of a measurement Y generated from the emission λ can be formulated as:

ln(L(λ|Y )) = i (- j t ij λ j + y i ln( j t ij λ j ) -ln(y i ! ))
(1) Let s = (s j ) be the sensitivity vector, containing the detection probabilities for photons emitted from voxels j. The MLEM algorithm consists in calculating, starting from some initial value λ 0 ∈ (R * + ) J , the sequence:

λ l+1 j = λ l j s j i t ij y i k t ik λ l k . (2)
It solves the minimization problem:

min{-L(λ|Y ) : λ ∈ (R + ) J }. (3)
In the list-mode version that we use in this work, the data space is highly sampled in order to have y i ∈ {0, 1}.

Image reconstruction and deconvolution can be solved simultaneously with the MLEM algorithm, with two additional convolutions. The algorithm becomes:

λ l+1 = λ l s (R ⊗ k), with R = T T 1 T ( λ l ⊗ k) . (4)
The image-space PSF method was originally introduced by Snyder et al. [START_REF] Snyder | Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography[END_REF]. In Reader et al. [START_REF] Reader | EM algorithm system modeling by image-space techniques for PET reconstruction[END_REF] the method was applied to PET imaging resolution recovery with a Gaussian model for the stationary PSF. Its use for the Compton camera imaging was originally proposed by Kim et al. [START_REF] Kim | Resolution recovery reconstruction for a Compton camera[END_REF] and was further developed by Jan et al. [START_REF] Jan | PSF reconstruction for Compton-based prompt gamma imaging[END_REF].

B. Total variation regularization

Since MLEM gives noisy results, a step forward consists in considering regularized solutions of (3) and MAP-EM algorithms. Total variation control is a common method for noise reduction, first intoduced by Rudin et al. [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF], which can remove the noise from images while preserving the edges. Adding this regularization to MLEM aims to reconstruct the image with both minimal total variation and maximal likelihood through the resolution of the minimization problem: min{-L(λ|Y ) + αG(λ) : λ ∈ (R + ) J }.

(

) 5
Here α ≥ 0 is the weight of TV regularization and G denotes the L 1 -norm of the gradient in the image:

G(λ) = Ω |∇λ(x)|dx. ( 6
)
MAP-EM formalism has already been proposed by Dempster et al. [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]. For the calculation of the MAP-EM estimator with TV prior, Panin et al. [START_REF] Panin | Total variation regulated EM algorithm [SPECT reconstruction[END_REF] proposed a modified version of the MLEM algorithm. However, this solution is unstable and restricted to small values of the smoothing parameter α. Other solutions were proposed since then (see, e.g. [START_REF] Anthoine | Some proximal methods for Poisson intensity CBCT and PET[END_REF], [START_REF] Yan | General convergent expectation maximization (EM)-type algorithms for image reconstruction[END_REF]). In [START_REF] Sawatzky | Accurate EM-TV algorithm in PET with low SNR[END_REF], Sawatzky et al. proposed a two-step scheme consisting in smoothing at each iteration the MLEM approximate solution in (2), hereafter denoted by λ l+1/2 . The smoothing step is a weighted version of the TV denoising algorithm proposed by A. Chambolle [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF]. We used a new algorithm for the denoising step proposed by Maxim et al. in [START_REF] Maxim | Tomographic reconstruction from Poisson distributed data: a fast and convergent EM-TV dual approach[END_REF], based on the dual formulation of the (M) step from the EM algorithm. The (M) step consists in solving:

λ l+1 ∈ argmin λ {H(λ) + αG(λ)} , (7)
with

H(λ) = λ -λ l+1/2 log λ, s . (8)
The method is convergent and as fast as the one proposed by Sawatzky et al.

C. Monte-Carlo simulation

Monte Carlo simulation of a cylindrical source in vacuum was performed with MEGAlib [START_REF] Zoglauer | MEGAlib-the medium energy gamma-ray astronomy library[END_REF]. The simulation geometry is described in figure 1(a). The camera is composed of silicon strip scatterers (512 strips on each face) and a segmented LaBr 3 absorber. The MLEM+TV method was evaluated with ideal data, where only Doppler broadening was accounted for. More realistic data was simulated adding Gaussian uncertainties on the measured energies with σ = 1 keV for the scatterers and σ = 11.4 keV at 511 keV for the absorber. The position errors are determined by the strip pitch (≈ 1 mm) and by the size of the crystals (0.5×0.5×4 cm 3 ). The PSF deconvolution is only tested on realistic data. The PSF FWHM is fixed to 16 mm in z direction and 8 mm in x and y directions. The reconstructed volume has (5 cm) 3 and is divided in 45 3 voxels.

III. RESULTS AND CONCLUSIONS

Results from figure 1(b) were obtained from 8 × 10 5 detected events after 200 MLEM iterations. In the first column, reconstructions for ideal data are shown, with (bottom) and without (top) total variation regularization. The results from the central column show that TV regularization smooths the image, but do not allow to distinguish the hole in the cylinder. The results improve with the use of the PSF (last column).

In this work we focus on Compton camera image reconstruction using realistic data and with low statistics. We show that in this context the image-space PSF deconvolution smooths the result and improves the resolution. TV regularization allows to further smooth the images while reinforcing the edges. We also observed that it promotes convergence. However, the deconvolution did not recover properly the hole since its size is comparable to the FWHM of the system PSF.

Fig. 1 .

 1 Fig. 1. Images reconstructed from ideal data (first column) and realistic data (second and third column). The values of the maximum are normalized to 1.


Laurent Trotignon

Chantal Riglet-Martial

Anne Stietel

Michel Sardin

Marie-Hélène Lefèvre

Pierre Vitorge Fauré

OF MIGRATION INTERCOMPARISON

This work presents an intercomparison exercice between two geochemical migration codes, TRIO-EF (object oriented finite element code) and IMPACT (chemical engineering code). The predictions of the two codes are compared with reference experimental results obtained in a previous study of strontium transport in soil columns. This simulated geochemical system is well documented and includes ion exchange, dissolution-precipitation and complexation reactions. Solution transport is simulated by a one dimensional convection-dispersion model. The predictions of TRIO-EF and IMPACT are both in good agreement with experimental results. Slight differences are nevertheless observed between the two codes especially when concentrations discontinuities are involved such as in boundary conditions or precipitation fronts. These discrepancies between the two codes can mainly be attributed to the different discretization approaches. Further intercomparison exercices are planned for more complex geochemical systems, e.g. migration of Am, Np or Pu.

INTRODUCTION

Early predictions of radioelement migration often used a partition coefficient, so called K d , to account for the chemical reactions (sorption, ion exchange, ...) occuring during transport. The K d is generally assumed to be constant but this hypothesis is invalid in most cases. Recent predictive migration models include a more realistic description of these chemical phenomena. They yet require an experimental validation. Indeed, simple cases were analytical solutions are available are important tests for such numerically coded models, but are not enough for thorough validation and consolidation of predictions. Two main areas may be considered : i) validation of conceptual models, ii) comparison of predictions for the same conceptual model by different codes. The aim of this work is to compare, on the basis of dedicated migration experiments, the predictions of two migration codes. Reference migration experiments were designed in order to display multiple physicochemical phenomena (ion exchange, complexation and precipitation) for a well documented chemical system. Previous work (Lefèvre et al., 1993) already included IMPACT computations because this code is very convenient for modelling column experiments. The two codes used in the present exercice, TRIO-EF and IMPACT, adopt a macroscopic point of view to describe the porous medium (global dispersion coefficient, ...) but resolve the migration problem with different numerical methods. Such a comparison provides thus important information for migration codes users and developpers in the fields of time and space discretization, boundary conditions formulation, sensitivity to model parameters. In the first part of the paper, a recall of experimental results and a summary of the conceptual model are given. In the next part, a description of the two migration codes used in this exercice is made. The predictions of the two codes are then compared and discussed.

PRESENTATION OF REFERENCE EXPERIMENTS

All experimental work presented here was already discussed in previous papers (Lefèvre et al. , 1993(Lefèvre et al. , , 1994)). Among these results, two experiments were chosen as reference experiments for the intercomparison. Main results of reference experiments and conceptual model parameters chosen for the TRIO-IMPACT intercomparison are now presented.

EXPERIMENTS

The two reference experiments are described in Lefèvre et al. (1993). The soil sample (Güe sand) used for transport experiments is a clayey sand with small amounts of calcite. The column is, in a first stage, equilibrated with deionized water or a CaCl 2 solution. A definite volume of a solution containing strontium is then injected in the column. After this second stage, the composition of the feeding solution is changed in order to elute the fixed strontium. The SOLEX experiment involves ion exchange phenomena as well as dissolutionprecipitation reactions. In the IONEX experiment, dissolution and precipitation of SrCO 3 are not potent and retardation of strontium occurs only because of ion exchange phenomena. Experimental conditions are summarized in Table 1.

CHEMICAL MODEL

Calcite dissolution, strontianite precipitation or dissolution and Ca ++ /Sr ++ cationic exchange on clays have been identified as the main reactions governing strontium transport in the soil columns. In addition to these solid-solution interactions, the chemical model used in this work also includes carbonic acid equilibria and water dissociation. Pre-equilibration of the columns with deionized water leads to a pH around 9.8 that is consistent with calcitewater equilibrium in a closed system. Solutes of Ca and Sr are injected as chloride salts and Cl -is considered to be a non reactive species. Aqueous complexes of Ca ++ or Sr ++ with HCO 3 -, CO 3 --are not taken into account in this model and we verified that they were negligible by comparison with Ca ++ and HCO 3 -. Feed solutions are in thermodynamic equilibrium with atmospheric CO 2 . before injection. In the saturated porous medium there is however no longer equilibration with atmospheric CO 2 . Phases are in local thermodynamic equilibrium at a constant temperature. The speciation model is summarized in Table 2.

TRANSPORT MODEL

The transport model is based on the assumptions that the fluid flow is steady and one-dimensional ( r U is the pore velocity of the fluid (ms ) is constant. The Peclet number of the fluid flow, Pe = UL/D where L is the column length is large (~ 200) which means that the transport of aqueous species is dominantly convective. In IMPACT, the dispersion process is represented by the model of mixing cells in series. The number of mixing cells J is directly linked to Pe by the relation [START_REF] Villermaux | Génie de la réaction chimique : conception et fonctionnement des réacteurs[END_REF] : Pe = 2(J-1) (1) In TRIO, the coefficient D is defined in the dispersion operator used in the transport equation. Table 3 summarizes the main hydrodynamic parameters adopted for the computations. The pore volume, Vp, and Peclet number of the column, Pe, were determined experimentally by non reactive tracer injections (Lefèvre et al., 1993).

RESULTS OF REFERENCE EXPERIMENTS

Table 4 reports, for the two reference experiments, the measures of the dissolved concentration of Sr and Ca. The relative uncertainity on experimental results is estimated to be less than 10 %. There is no experimental data on the spatial repartition of chemical elements in the column. Such information could be obtained by the use of radioactive tracers or by other non destructive methods. Migration codes can however give representations of the column state (distribution of SrCO 3 , ...) at different moments and guide further experimental investigations.

PRESENTATION OF TRIO-EF AND IMPACT MIGRATION CODES

CODES

Both codes were run on an IBM ® RS6000 station.

TRIO-EF

TRIO-EF is an object-oriented finite element code that is developped at the CEA for the numerical modelling of thermohydraulic phenomena, atmospheric dispersion and hydrogeology. It is possible to simulate stationnary or transient fluid flow in 1-D, 2-D or 3-D space domains. Recent evolutions of TRIO-EF in the field of earth-sciences include the developpement of an algorithm coupling solute transport by diffusion or advection with chemical transformations [START_REF] Chupeau | Contribution à l'étude théorique et expérimentale du transfert des solutés en aquifère : application au traçage chimique d'un réservoir géothermique, apports de la modélisation couplée transport-géochimie[END_REF][START_REF] Abdennour-Pfiffer M | Etude hydrogéochimique et modélisation des phénomènes couplés transport-réactions liés au stockage de chaleur et/ou de déchets nucléaires dans les réservoirs géologiques[END_REF]. The user builds its own instruction set and may include, if necessary, coupling with the evolution of temperature or permeability fields during transport-chemistry calculations. The chemical module used in TRIO is derived from the geochemical code MINEQL [START_REF] Westall | A critical evaluation of recent developments in hydrogeochemical transport models of reactive multichemical components[END_REF][START_REF] Schweingruber | User's guide for extended MINEQL (EIR Version)[END_REF]. At present time, this equilibrium chemical model is able to take into account, in addition to solution complexation and dissolution-precipitation reactions, sorption and ionexchange phenomena. Temperature dependance of equilibrium constants may be taken into account for space or time dependant temperature fields. The ion strength dependance of solute activity coefficients is described by the classical Davies approximation [START_REF] Morel | Principles and applications of aquatic chemistry[END_REF]. The migration problem is solved by a two-step algorithm (Yeh and Tripathi, 1989 ;[START_REF] Abdennour-Pfiffer M | Etude hydrogéochimique et modélisation des phénomènes couplés transport-réactions liés au stockage de chaleur et/ou de déchets nucléaires dans les réservoirs géologiques[END_REF], in which equations describing transport and chemical speciation are solved iteratively between time step p-1 and time step p. At present time, reaction kinetics are not included in the model.

IMPACT

...

SPATIAL AND TEMPORAL DISCRETIZATION -BOUNDARY CONDITIONS

For TRIO-EF calculations, the column was discretized in 234 elements of equal length (dx = 0.5 mm) and the time step was kept constant (dt = 3 s). Other settings were also tried, in particular with variable time steps during calculation (from 0.6 s to 15 s) and with elements of increasing size (from 0.2 to 0.8 mm) along the column. These trials were made in order to obtain a better description of steep fronts and of boundary conditions switching at the column head. However time step increases during calculation must be progressive in order to avoid numerical artifacts as local redistribution of precipitates. This is why the first setting described above was finally found to be a simple and reasonnable compromise. The boundary condition describing the input solution was applied to the first node of the column.

For IMPACT, the number of mixing cells is directly linked to the dispersion coefficient (Eq. 1) and was determined to be J = 100. The time step is variable during computation. In particular time steps decrease strongly during the dissolution of SrCO 3 (SOLEX experiment).

RESULTS AND DISCUSSION

SOLEX EXPERIMENT (ION EXCHANGE AND DISSOLUTION-PRECIPITATION)

In this experiment, the initial injection of SrCl 2 leads to the formation of SrCO 3 at the head of the column. The subsequent injection of CaCl 2 induces the dissolution of the SrCO 3 precipitate and generates a strong Sr pulse that migrates through the column (Table 4 and Lefèvre et al., 1993). The evolution of Sr concentration at the column outlet is represented as a function of delivered volume (equivalent to time) in Fig. 1a. The two computations and experimental data are in fair agreement. Small differences between the TRIO-EF and IMPACT curves may be noted : i) the maximum Sr concentration is observed slightly earlier with TRIO-EF, ii) the maximum Sr concentration is 10 % less in TRIO-EF's prediction than for IMPACT, iii) the Sr distribution is more spread out in time for TRIO-EF. In both cases however, the ratio of restituted Sr to injected Sr at 8 Vp is comprised between 0.999 and 1. It was found that, for too large values of dx and dt, the Sr distribution predicted by TRIO-EF could be sensitive to discretization parameters, although dt and dx remained in the theoretical limits imposed by finite element discretization. Preliminary tests identified possible causes for this behaviour : i) loss of some Sr at the switching of boundary conditions when dx is large at the column head, due to finite element interpolation, ii) existence of some numerical dispersion for reactive components. Further investigations on this phenomenon are needed. At the outlet of the column, the predicted evolutions of Cl-concentration as a function of delivered volume are in very good agreement (Fig. 1b). The difference in pH curves after 3 Vp (Fig. 1b) originates from the different treatment of activity coefficients of charged species in IMPACT (no ionic strength correction) and TRIO-EF (Davies approximation for activity coefficients). A first order correction made on IMPACT simulation results accounted correctly for this pH gap. The predictions by the two codes for the distribution of chemical species at the head of the column (first mixing cell for IMPACT, second node for TRIO-EF) as a function of delivered volume are also in good agreement (Fig. 2a and2b) SrCO 3 precipitates at the head of the column during the initial injection (see Table 1) and dissolves at the eluant change at 2 Vp. Differences in produced SrCO 3 quantity and time at which precipitation starts are due to the difference in spatial discretization in the two codes. Nevertheless, the predictions of final Sr restitution (Fig. 1) are close. Fig. 3 compares the spatial distribution of chemical species in the column at a time corresponding to injection of 2.23 Vp of feed solution, i. e. just after the dissolution of SrCO 3 . At this stage, the Sr ++ pulses are in very good agreement. At final restitution (Fig. 1), larger differences between the two predictions for Sr ++ are observed. As already mentionned, this could indicate a slight numerical dispersion for reactive species in TRIO-EF.

IONEX EXPERIMENT (ION-EXCHANGE -NO PRECIPITATION)

In this experiment precipitation of SrCO 3 does not occur and retardation of Sr migration occurs only because of ion exchange phenomena (Table 1. The agreement between TRIO-EF and IMPACT predictions and experimental data for Sr elution (Fig. 4a) is good. Slight differences can be noted between the two codes : i) the rise of the elution peak is about 0.05 Vp earlier for TRIO-EF, ii) the maximum Sr ++ eluted concentration is about 1 % less in the TRIO-EF prediction. The comparison between TRIO-EF and IMPACT predictions for the spatial Sr ++ distribution after the injection of 3 Vp of solution are shown in Fig. 4b.

CONCLUSION

This migration code intercomparison exercice involved two codes that follow very different approaches for modelling solute transport and coupling of transport with chemical transformations. The agreement between these two codes, IMPACT and TRIO-EF, for the modelling of 1-D migration phenomena involving ion-exchange, dissolution-precipitation and complexation reactions is nevertheless good. Differences in the codes predictions are often related to differences in discretization schemes and can be easily accounted for. The comparison of predictions for the migration of strongly reactive elements like Sr is fair (SOLEX experiment) or excellent (IONEX experiment) but desserves further numerical investigations. Further intercomparison exercices are prepared in relation with experimental work on more complex chemical systems involving the migration of Am, Np and Pu.

  Figures 1

Table 1

 1

	:

Table 2 :

 2

	Reactions				Equilibrium constant
	H 2 O ⇔ H + + OH -			K w =10	-14
	H 2 CO 3 ⇔ HCO 3 -+ H +		K 1 = 10	-6.3
	HCO 3 -⇔ CO 3 2-+ H +		K 2 = 10	-10.3
	CaCO 3s ⇔ Ca 2+ + CO 3 2-		K s1 =10 -8.42
	SrCO 3s ⇔ Sr 2+ + CO 3 2-		K s2 =10	-9.03
	Ca f	2+ + Sr 2+	⇔ Ca	2+ + Sr f	2+	K Sr/Ca = 1.05
						Table 3:
		diameter (mm)	length (mm)	M. sand (g)	V p (ml)			CEC (eq/l)	flow rate (ml/mn)	Pe	=	uL D
		25.4	117	94	26			0.03	1	198


Marie-Hélène Fauré

Michel Sardin

Pierre Vitorge

Release of clay particles from an unconsolidated clay-sand core: experiments and modelling

Release of clay particles from an unconsolidated clay-sand core experiments and modelling

M. Faure, M. Sardin, P.

Introduction

Particles are suspected to be involved in the transport of pollutants in soils [START_REF] Avogadro | The role of colloids in nuclear waste disposal[END_REF]. The possible phenomena involved in the control of particle transport are linked to the physical properties of the particles in solution (flocculation or dispersion according to the solution), the interaction between the particles and the porous medium (mechanical retention (filtration), adsorption (DLVO theory or sorption reaction)) and of course, the hydrodynamic properties of the porous medium ( porosity, permeability, dispersion…).

It is well known that, exchanging the salt water present in a sand core with a fresh water, reduces the permeability of the core due to clay particles migration [START_REF] Van Olphen | An introduction to clay colloid chemistry[END_REF]. This problem is of interest for petroleum engineers and agricultural researchers. They usually focus their studies on the evolution of the permeability of the sand core until plugging occurs. Few quantitative studies exist on the clay migration [START_REF] Khilar | Colloidally induced fines migration in porous media[END_REF]. These works show that the behaviour of a sand core is dependant on the type of clay, and on the nature of the salt. They show that release of particles occurs below a Critical Salt Concentration, CSC [START_REF] Khilar | The existence of a critical salt concentration for particle release[END_REF].

Here, rather than decreasing salinity to reach fresh water and clogging the porous medium by a release of a large amount of particles, the salinity decrease is stopped just below the CSC. This decrease must be large enough to generate particles, but avoids the plugging of the column. One of preliminary work has been to perform well controlled and reproducible experiments. It is needed to develop a phenomenological modelling and to test numerical code predictions. The modelling approach is to first establish, from experiments, a relationship between the salinity in the porous medium and the amount of particles released then, to develop a transport model taking into account the observed behaviours (released amount, kinetic effects) and finally, to compare the simulated and experimental results and to test the limits of the model (flow rate, pH, % of clay, column length).

This work is part of a study [START_REF] Fauré | Migration de particules et de solutés en milieu poreux. Modélisation du transport simultané de particules argileuses et de radionucléides sous l'effet d'un gradient de salinité[END_REF] on the modelling and experimental investigation of the coupled transport of radionuclides and particles. A general article has been presented earlier [START_REF] Fauré | Transport of clay particles and radioelements in a salinity gradient: experiments and simulations[END_REF], in which a particle transport model is used to simulate the coupled transport of caesium and particles in a salinity gradient. In the present contribution, we want to focus the methodology and the experimental results which have led to this model of particle transport.

Experimental set-up and method

We use a standard liquid chromatographic system. Two pumps, one fed by distilled water and the other by a NaCl solution, monitored by means of a gradient performer, allow us to feed the column at constant flow rate (usually 1ml/min) with a linearly decreasing NaCl concentration with time. At the outlet of the column, we monitor the variations of NaCl concentration with a conductimetric cell. An U.V. (280 nm) continuous detector measures the particle concentration in the effluent. On a large range of particle concentrations, the response of U.V. detector is linear. The porous medium consists in a mixture of 5% weight Foca3 bentonite clay and sand (99% SiO 2 ). A 16 mm diameter column is packed with 8 g of the mixture. The column is purged of air with CO 2 and then saturated with an upward flow of sodium chloride (0.5 M) at pH= 6.5.

Reproducible experiments: The NaCl concentration of the feed solution of the column is decreased from the initial NaCl concentration, C i , to the final, C f , then increased to C i . This cycle is repeated several times to observe the influence of these signals on the particle breakthrough curves. Progressively, the amount of particles eluted during a cycle decreases to zero, though there is still clay in the column. For a given decrease of concentration from C i to C f , the amount of particles capable of being released from the column is limited. The history of the column cannot be ignored and we will refer to reproducible experiments only for a fresh columns. Preliminary results showed that step changes in the NaCl concentration at the inlet of the column led to large perturbations (inducing possible clogging) in the column and non reproducible experiments from one fresh column to another. In a typical experiment, the NaCl concentration of the feed solution is decreased from C i to C f within a volume, V g , of solution, then the concentration of the feed solution is maintained constant at C f as long as some particles leave the column. This typical experiment is reproducible from one fresh column to another with identical physical characteristics: length (≈ 3 cm), composition (5 % clay) of porous medium and flow Fauré and all. (1994).

[NaCl] domain

Equation of P d ([NaCl]) [NaCl] > 0.16 M P d ([NaCl]) = 0 0.16 M ≥ [NaCl] ≥ 0.05 M P d ([NaCl]) = X 1 (exp([NaCl] n ) -exp(0.16 n )) 0.05 M ≥ [NaCl] ≥ 0.035 M P d ([NaCl]) = X 1 (exp(0.05 n )-exp(0.16 n )) 0.035 M ≥ [NaCl] ≥ 0.019 M P d ([NaCl]) = X 2 (exp([NaCl] n ) -exp(0.035 n )) + P d (0.05) 0.019 M ≥ [NaCl] P d ([NaCl]) = X 2 (exp(0.019 n ) -exp(0.035 n )) + P d (0.05) n=-0.6, X 1 =75 500 et X 2 =7 970
Q C Q C V J dC dt m,k-1 m,k m p m,k e,k = + + θ Φ (1)
tracer in immobile fluid zone

θ im p im,k e,k V J dC dt = Φ
(2) tracer flux between zones

Φ e,k m p m,k im,k M V J C C t = - θ (3) particles in mobile fluid zone Q P Q P V J dP dt m,k-1 m,k m p m,k e,,P,k = + + θ Φ (4)
particles in immobile fluid zone rate (1 ml/min). Important parameters are: C i , C f and V g . We limit the study with 0.015 M<C f <0.5 M to avoid plugging of the porous medium. In this case, the amount of particles leaving the column is negligible compared to the amount of clay staying in it. We shall now vary them to characterise some aspects of the particle transport. ). As long as the NaCl concentration is the same, the particle breakthrough curves are identical (column 1 , column 2 ). ) is decreased from 0.2 M to 0.015 M with V g = 336 V p .

θ im p im,k s,P,k e,P,k V J dP dt + = Φ Φ (5)

Experimental results and interpretation

Typical experiment: (see for example Figures 3 and4) At the beginning of the experiment, the salinity of the feed solution decreases and no particle are detected in the effluent. Then, the particles begin to leave the column and the concentration of particles in the effluent increases up to a maximum value and finally decreases. Let us focus on three points: the shape of the particle breakthrough curve, the time when particles begin to leave the column and the total amount of particles released.

Shape of the particle breakthrough curve: The amount of particles available for migration, during the decrease of the NaCl concentration of the feed solution, is clearly limited. However, even when the NaCl concentration decrease stops (the column is then fed with a solution at constant NaCl concentration), particles continue to leave the column. The system is not at equilibrium and we deduce that some kinetic limitations occur in the particle transport.

Measurement of the particle output threshold: We look for a NaCl critical salt concentration, CSC, below which the particles begin to be released. Several typical experiments are performed with the NaCl concentration decreasing from 0.5 M to 0.05 M and from 0.05 M to 0.015 M, and the effect of the parameter V g is studied.

Finally, for each domain of NaCl concentration variation, a threshold concentration is obtained, independently of V g : C s1 = 0.16 M and C s2 = 0.036 (Figures 3 and4). For a naturally consolidated sandstone (8%clay content, mainly kaolinite), [START_REF] Khilar | The existence of a critical salt concentration for particle release[END_REF] have determined a CSC equal to 0.07 M NaCl. This value is in the same order of magnitude as C s1 and C s2 . The difference can be attributed to the fact that particle release is dependant on clay type, the composition of sand and the structure of porous medium.

Amount of particles released: Two columns are fed with a NaCl solution, the concentration of which is decreased from C i to C f1 . Then, we continue to decrease the concentration to C f2 in the first column, whereas we feed the second with a constant NaCl concentration solution. The breakthrough particle curves are identical as long as the NaCl concentrations of the feed solution are the same (Figure1). The results of several typical experiments from C i (C i =0.5 M or 0.05 M) to C f (0.05 M or 0.015 M) with different V g values are considered. The plot of the amount of particles that leave the column during an experiment, Q P , shows that Q P is only dependent on C i and C f , but not on V g . However, V g has an effect upon the concentration of the particles in the effluent of the column. Keeping C i and C f constant, the larger the required volume V g to elute the same amount of particles, the lower the concentration of particles in the effluent.

Verification of the existence of several CSC: C s1 and C s2 have been determined from distinct experiments, decreasing NaCl concentration from 0.5 M to 0.05 M, keeping the NaCl concentration constant as long as particles are eluted for the column, and decreasing it again from 0.05 to 0.015 M. To test if C s2 is not a kinetic effect due to the plateau at 0.05 M, a column is fed with a NaCl solution decreasing from 0.2 M to 0.015 M without the plateau at 0.05 M (Figure 2). We detect, as previously assumed, two peaks of particles. The particle release begins when the concentration reaches 0.16 M, then stops and resumes when the concentration drops below 0.036 M. This confirms the previous observations: two domains of NaCl concentration occur in which some particles are able to migrate in the column. The upper limits of these domains are the threshold concentrations. The ) of the feed solution is decreased from C i = 0.5 M to C f = 0.05 M with V g = 23.3 V p (■) or with V g = 107 V p (♦). lower limits, C ls1 and C ls2 , below which the particles are no longer released into solution, are 0.05 M and 0.019 M respectively. These are determined from experiments with large V g volumes to limit the kinetic effects. In each of these, it has been checked that Q p is the same for experiments with C f equal to C ls or slightly lower than C ls . This experiment, revealing the two domains for particle release, is very tricky. Indeed, we must choose a convenient intermediate value of the volume V g . If V g is too small, the two particle peaks of the two domains interfere. On the other hand, if V g is too large, the concentration of particles in the effluent will be below the detection limit of the U.V cell. This means that some sub-domains of particle output, that we have not detected, could exist. The distinct threshold concentrations might be related to the size or to the nature of the clay.

In order to model the particle release, we need an equation to account for the amount of particles available for migration. An empirical function, P d ([NaCl]), is fitted on previous experimental plots, Q p vs NaCl concentration. P d (C f )-P d (C i ) is the amount of particles that leave the column when decreasing the NaCl concentration of the feed solution from C i to C f . This relationship is given in table1.

Hydrodynamics: We have shown earlier [START_REF] Fauré | Transport of clay particles and radioelements in a salinity gradient: experiments and simulations[END_REF] that, initially, the water flow is near that of a piston flow. But immediately prior to the detection of the first eluted particles from the output of the column, preferential pathways and dead zones are created; their size increases as the NaCl concentration is decreased. This is attributed to particle transport and reordering inside the column. ) of the feed solution is decreased from C i = 0.05 M to C f = 0.02 M and V g = 37.5 V p (■). In the other experiment, the NaCl concentration of the feed solution is decreased from C i = 0.05 M to C f = 0.02 M with V g = 80 V p then NaCl is decreased to 0.01 M with V g = 254 V p (♦).

Modelling the transport of particles in a salinity gradient

Hydrodynamics modelling: To model the hydrodynamics of the porous medium in the different situations, we used the classical description of onedimensional convective-dispersive solute transport with dead end pore volume [START_REF] Coats | Dead end pore volume and dispersion in porous media[END_REF]. A mobile water zone, in which the transport is controlled by convection and dispersion, is assumed to diffuse through the immobile water zone located in dead end pore volume. This description is applied to the discrete model of mixing cells in series [START_REF] Villermaux | Deformation of chromatographic peaks under the influence of mass transfer phenomena[END_REF]. The column, of porous volume V p , is represented by J identical cells in series. Equivalence between this discrete approach and the classical continuous model has been shown under these conditions [START_REF] Sardin | Modelling the nonequilibrium transport of linearly interacting solutes in porous media: a review[END_REF], the relationship between J and the axial dispersion coefficient, D, being Pe=uL/D=2(J-1) (Pe= Peclet number, L= length of column, u= interstitial velocity). Each cell contains a mobile fluid zone of volume θ M V p /J and an immobile fluid zone of volume θ im V p /J. The mass balance equations in the k th cell, for a tracer, are listed in table 3. C m,k and C im,k are the concentrations of the tracer in the mobile and the immobile fluid zones. Φ e,k is the flux between the mobile and the immobile fluid zones. As a first approximation, this flux is assumed to be controlled by a linear first order kinetic law (table 2, equation (3)) and t M is a characteristic mass transfer time. According to the nature of the mass transfer resistance (film model, internal diffusion in the aggregates…), the expression of t M can be calculated from the main characteristics of the grain packing [START_REF] Sardin | Modelling the nonequilibrium transport of linearly interacting solutes in porous media: a review[END_REF].

Particle transport modelling: We apply the description of the tracer transport to the particles by adding a term for the generation of particles, P d ([NaCl]). As for the solute, a characteristic mass transfer time for the particles, t Mp , is introduced. No other term of kinetic limitations is added. We first want to verify if t Mp accurately takes ) and typical experiments with small and large V g values. The NaCl concentration ( ) of the feed solution is decreased from C i = 0.05 M to C f = 0.02 M and V g = 37.5 V p (■). In the other experiment, the NaCl concentration of the feed solution is decreased from C i = 0.05 M to C f = 0.02 M with V g = 80 V p then NaCl is decreased to 0.01 M with V g = 254 V p (♦). into account all the kinetic limitations found earlier in the experimental study of particle transport. We assume that the immobile fluid zone is in contact with the porous medium that produces particles according to the NaCl concentration. The mass balance equations for the particles, the flux between the mobile and the immobile fluid zones and the source term of particles from the porous medium to the immobile fluid zone are in table 2 (equation ( 4) to ( 7)).

Determination of the parameters and model verifications: Equations ( 1), ( 2) and (3) describe the NaCl transport, considered as a tracer, equations (4) to (7) describe the transport of particles. The parameters (V p , J, θ im , t M ) are determined from experimental measurements of Residence Time Distribution [START_REF] Fauré | Migration de particules et de solutés en milieu poreux. Modélisation du transport simultané de particules argileuses et de radionucléides sous l'effet d'un gradient de salinité[END_REF]. The hydrodynamic characteristics change continuously as NaCl concentration is decreased, but we only know these values for a few NaCl concentrations. So, during simulation, we keep these parameters constant. For each domain of destabilisation, the hydrodynamic characteristics of the porous medium used are in Table 3. To determine t Mp (Table 3), we fit it with two breakthrough curves in each domain. Then, the model is tested on a set of a dozen of experiments of clay destabilisation, of which four are shown here (Figures 3 and4). These figures are representative of the two NaCl concentration domains studied and of the values of V g used. For typical experiments from 0.5 M to 0.05 M, or from 0.05 M to 0.015 M, the agreement between experiments and simulation is good when V g is not too large (< 150 V p ). When V g increases, the simulation anticipates the particle breakthrough curve. These differences are possibly due to the approximation on the invariance of the porous medium hydrodynamic characteristics. During the simulation, we keep θ im , t M , t Mp constant, whereas actually, they increase. The increase of θ im and t Mp involves a retardation of the particle migration. An other reason could be the insufficient knowledge of the P d ([NaCl]) function.

Meaning of the characteristic parameters for the particles: P d , t Mp

To simulate the transport of particles in a porous medium, a filtration term is often introduced. We do not use this approach. This term of filtration is implicitly enclosed in the P d function. We have made typical experiments with a long column (V p =11.8 ml instead of 3 ml usually). The amount of particles (referred to the mass of porous medium) that leave this column is less than that from a column of smaller length. It would be interesting to get a flocculation curve of clay particles, that is, for a given time, the rate of flocculation of the clay as a function of the NaCl concentration.

t Mp is the characteristic mass transfer time. Additional experiments with different flow rates and a constant V g , show that the particle concentration in the effluent increases as flow rate decreases (the total amount of particles that leave the column being constant). The flux between the mobile and the immobile water zones in a mixing cell is then probably in diffusive mode.

Conclusion

The transport of clay particles, caused by a decrease of the salinity of the feed solution, in a unconsolidated sand core is modelled by coupling a non equilibrium transport model with a term of particles generation, P d ([NaCl]). The hydrodynamic properties of the porous medium vary as soon as the particles are in movement inside the column. The kinetic limits of particle transport are simply taken into account by the hydrodynamic model. P d ([NaCl]) is linked to the properties of flocculation of the clay particles and of the filtration in the porous medium. This model is tested on a set of a dozen of experiments.

Figure 1 :

 1 Figure 1: Effect of the final concentration, C f , on the amount of particles leaving the column. The NaCl concentration of the feed solution is decreased to 0.001 M in column 1 (), while in column 2, the NaCl concentration is just decreased from 0.1 M to 0.025 M (). As long as the NaCl concentration is the same, the particle breakthrough curves are identical (column 1 , column 2 ).

Figure 2 :

 2 Figure 2: Particle elution profile ( ) from a column where the NaCl concentration of the feed solution () is decreased from 0.2 M to 0.015 M with V g = 336 V p .

Figure 3 :

 3 Figure 3: Comparison of particle release between simulation () and typical experiments with small and large V g values. The NaCl concentration ( ) of the feed solution is decreased from C i = 0.5 M to C f = 0.05 M with V g = 23.3 V p (■) or with V g = 107 V p (♦).

Figure 3 Figure 4 :

 34 Figure 3 (details): Comparison of particle release between simulation () and typical experiments with small and large V g values. The NaCl concentration ( ) of the feed solution is decreased from C i = 0.5 M to C f = 0.05 M with V g = 23.3 V p (■) or with V g = 107 V p (♦).

Figure 4

 4 Figure 4 (details): Comparison of particle release between simulation () and typical experiments with small and large V g values. The NaCl concentration ( ) of the feed solution is decreased from C i = 0.05 M to C f = 0.02 M and V g = 37.5 V p (■). In the other experiment, the NaCl concentration of the feed solution is decreased from C i = 0.05 M to C f = 0.02 M with V g = 80 V p then NaCl is decreased to 0.01 M with V g = 254 V p (♦).

Table 1 :

 1 Empirical function P d ([NaCl]): amount of particles available to migrate as a function of NaCl concentration. P d is in (D.O. unity)/(Kg of porous medium). This function is plotted with experimental points in

Table 2 :

 2 Mass transfer equations of NaCl solute and of particles used in the model.

	tracer in mobile fluid zone

Table 3 :

 3 Values of hydrodynamic parameters used to simulate particle release in the two domains of NaCl concentration.

		Domain 1	Domain 2
		[NaCl] 0.16 M to 0.05 M	[NaCl] 0.035 M to 0.019 M
	J	51	15
	θ im t M t Mp	0.125 0.05 τ 12 τ	0.285 0.09 τ 20 τ


Chantal Riglet-Martial

Pierre Vitorge

Véronique Calmon

Electrochemical Characterisation of the Ce(IV) limiting carbonate complex

Keywords: cerium (III/IV), actinides (III/IV), carbonate, complex, activity coefficients, cyclic voltametry

The stoichiometry and the thermodynamic formation constant of the limiting complex of Ce(IV) were determined at 19.3 ± 1.0°C by using cyclic voltametry technique at a hanging mercury drop working electrode in concentrated bicarbonate/carbonate medium. The Ce(IV/III) redox potential was measured at pH varying from 9.3 to 10.6 and [ ] CO 3 2varying from 1.0 to 1.5 M by performing a CO 3 2-titration with CO 2 gas. The ionic strength and junction potential effects were taken into account for the potentiometric calibrations and measurements. Quantitative interpretation of the variations of the formal potential E IV/III showed that no polymerisation took place during the redox reaction, and that two CO 3 2-ligands, but no OH ligand, were exchanged. As the accepted stoichiometry for the limiting complex of Ce(III) is Ce CO ( ) 3 4 5-, the Ce(IV) species is Ce CO ( ) 3 6 8-. In a 3.06 molal Na + carbonate/bicarbonate medium (Ionic strength = 4.33 mol.kg -1 ), E IV III / 0 = 0.161 ± 0.008 V/SHE (in molal units) was measured. This value, combined with the published Ce CO ( ) 3 4 4-formation constant and the (re-evaluated) (Ce 4+ /Ce 3+ ) standard potential, is used to calculate the Ce CO ( ) 3 6 8-formation constant log 10 ( b 6 IV ) = 42.2± 0.5 (defined in molal concentration except for Ce 4+ in activity : see table 1) in the same medium. The values of E IV III / 0 and log 10 ( b 6 IV ) are ionic strength dependant, e.g. E IV III / 0 = 0.182 ± 0.009 V/SHE and log 10 ( b 6 IV ) = 41.8 ± 0.5 in molar units in a 2.67 M NaClO 4 medium (Na + molality = molal ionic strength = 3.06 mol.kg -1 ). The possible formation of Ce CO ( ) 3 5 6-is discussed under the experimental conditions used, log 10 ( b 5 IV ) £ 41.8 ± 0.5 (in molal units).

Introduction

The actinide chemistry in reducing aqueous solutions is important for many waste disposal issues. In a recent bibliographic review [1], we pointed out contradictory interpretations for the speciation of actinide(IV) elements in bicarbonate/carbonate media, and showed that the limiting complex can be a starting point to determine aqueous speciation in environmental conditions. From redox measurements involving the known U(VI), Np(V), Pu(VI) and Am(III) limiting carbonate complexes, the stoichiometry An CO ( ) 3 5

6-has been proposed for the actinide(IV) limiting carbonate complex ( [2,[START_REF] Grenthe | NEA-TDB Chemical thermodynamics of uranium[END_REF] for U(IV), [START_REF] Delmau | [END_REF] for Np(IV), [START_REF] Capdevila | [END_REF] for Pu(IV) and [START_REF] Bourges | [END_REF]7,[START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF] for Am(IV)). This stoichiometry has been confirmed from the published redox data and the stability of the limiting complex has been accurately determined for uranium only. Recently, this stoichiometry has been confirmed for the Pu(IV) limiting carbonate complex from EXAFS measurements [START_REF] Clark | communication T3-A1 and[END_REF]. Surprisingly, among analogue elements, both Ce CO ( ) 3 5

6-and Ce CO ( )

3 6
8- [START_REF] Dervin | [END_REF][START_REF] Dervin | [END_REF]12,13,14,15] have been proposed for the Ce(IV) limiting complex. As the chemistry of Ce(III) [16] and Am(III) [7,[START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF] in carbonate medium is well known, the reversible (or quasi-reversible) M(IV/III) redox couple may be used to study the chemistry of Ce(IV) and Am(IV) in concentrated bicarbonate/carbonate media. Convincing evidence, based on potentiometric measurements, of the esacarbonato cerium(IV) stoichiometry was proposed in reference [15] in a 8 molal Na + medium and in a carbonate concentration range from 2.084 to 4.00 mol/kg. But since the authors used an Na + specific electrode as a reference electrode, they had to correct their data for the Na + activity variations to determine the number of carbonate ions exchanged during the redox reaction. In the present work, the stoichiometry of the Ce(IV) limiting carbonate complex was checked by measuring the Ce(IV/III) formal potential in 3.06 m Na + solutions of much lower sodium carbonate concentrations (1.00 to 1.45 mol/kg) using cyclic voltametry with a different electrochemical cell and a different methodology in order to avoid the Na + activity coefficient corrections. The liquid junction potentials were measured and taken into account in the treatment of the data. ) + e(i, X) m X for an ion i of charge z i in a X medium, where X is an ion of charge opposite to that of i [START_REF] Grenthe | NEA-TDB Chemical thermodynamics of uranium[END_REF][START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF][START_REF] Grenthe | Modelling in Aquatic Chemisty[END_REF]. Numerical values and definitions used are given in table 1.

Experimental

The voltamograms were recorded between +0.240 and -0.020 V/SHE at a scanning speed of 0.002 V/s. The upper limit of +240 V/SHE was imposed by the oxidation wall of mercury. Ten millilitres of the c 0 M Na 2 CO 3 , (I-3c 0 ) M NaHCO 3 test solution was first added to a thermostated (at 19.3 ± 1.0°C) electrochemical cell (of type Tacussel RM06-C + CRSR) and then deoxygenated with argon (HP45 Carboxyque) for one hour.

After deoxygenation, a voltamogram of the electrolyte was recorded in order to verify the absence of any electroactive impurities.

In order to determine whether the redox reaction under study involved a polymer or not, a first set of experiments was first carried out with a cerium total concentration [Ce] t varying from 2.5 10 -4 to 2 10 -3 M. Two compositions of test solutions P1 (1.42 M Na 2 CO 3 , 0.13 M NaHCO 3 , pH = 10.36) and P2 (1.33 M Na 2 CO 3 , 0.37 M NaHCO 3 , pH = 9.86) were investigated. Five determinations of the E 1/2 potential (the mean value of the oxidation and reduction peak potentials) were performed for each Ce concentration. In order to determine the stoichiometry of the Ce(IV) complex, a second set of experiments was made as follows : [Ce] t was kept constants but pH and [ ] CO 3 2-were varied simultaneously by stepwise acidification of the initial (P1 + 10 -3 M Ce) test solution by means of CO 2 (Air Products, quality 4.5). The gas was previously passed through a NaClO 4 solution of same molar ionic strength as that of P1. At each step of acidification, the test solution was left for stabilisation with an argon cover at the surface until constant pH (variation < 0.05 pH unit during 10 minutes) was obtained. The voltamogram of the solution was then recorded and the pH was checked again. Five determinations of the E 1/2 potential were performed for each pH step. an electrode of same type as REF. Differences higher than 0.2 mV were never observed. A linear variation of the junction potentials versus pH (with a slope of about 1 mV/pH unit) was found in the pH range under study. The measured E 1/2 potentials of the Ce(IV/III) couple in carbonate media were corrected for the junction potentials calculated from the experimental regression straight line at the pH of the corresponding test solutions. The pH was measured with a Radiometer XC161 combined glass electrode connected to a Tacussel ISIS 20000 pH-meter. The Ag/AgCl reference electrode compartment was filled with a 0.02 M NaCl, (I-0.02) M NaClO 4 solution of same molar ionic strength I as that of the initial NaHCO 3 /Na 2 CO 3 test solution. As the acidification by CO 2 gas involved a gradual variation of the ionic strength of the test solution, the glass electrode was standardised in activity units (pH = -log 10 a H + and not -log 10 [H + ]) by means of two NaHCO 3 /Na 2 CO 3 buffer solutions of same Na + concentration (3.0 M) as that of the initial NaHCO 3 /Na 2 CO 3 test solution. The theoretical pH of these buffer solutions were calculated on the basis on their chemical compositions, determined by acid-base titration, and by using the equilibrium constants and SIT coefficients from table 1. The precision of the pH measurement for the buffer solutions was about 0.05 pH unit. In addition, the accuracy of the pH electrode response was controlled by means of a standardisation between pH 7 to 12. The pH range from 8.2 to 12 was checked by a titration of a 1 M NaHCO 3 solution with a 1 M NaOH solution, while the pH range under 8.2 was controlled by measuring the pH of a 1 M NaHCO 3 solution in equilibrium with CO 2 gas. The slope of the glass electrode (figure 1) was found to be the theoretical one (58.03 mV/pH unit at 19.3°C) within a precision of 0.05 pH units in the pH range between 7.5 to 10. Above pH=10, a systematic error varying between 0.05 to 0.6 pH unit was observed. Because of the limited solubility of NaHCO 3 (about 1 M), the control of the accuracy of the pH electrode could not be carried out with concentrated carbonate solutions (about 1.5 M) similar to that used for the electrochemical study of cerium.

Treatment of the data

Preliminary experiments showed that the interval between the oxidation and reduction peaks of Ce(IV/III) increased with the potential scanning rate, which was an indication of a quasi-reversible redox system. For reversible or quasi-reversible systems, the cyclic voltametry curves are theoretically symmetrical around the (constant) half-wave potential E 1/2 , whatever the potential scanning rate. This feature was observed for scanning rates varying from 0.002 to 0.008 V.s -1 (at higher scanning rates, the shift of the oxidation peak towards the oxidation wall of mercury was such that the oxidation peak potential could not be measured any more). In addition, the current intensity of the oxidation and reduction peaks varied linearly as a function of the square root of the scanning rate, with similar absolute values of the slope during oxidation (+0. 16) and in reduction (-0.11). We then assumed that the Ce(III) and Ce(IV) diffusion coefficients were similar and that the experimentally measured E 1/2 potential could be used as an estimate of the formal redox potential E IV/III of the Ce(IV/III) couple. The speciation at varying pH and [ ] CO 3 2-was calculated from the experimental conditions ( c Na CO 2 3 , c NaHCO 3 , pH) using equations (1 to 8) and the definitions and numerical values listed in table 1. The Na 2 CO 3 and NaHCO 3 initial (molal) total concentrations are denoted m Na CO 2 3 and m NaHCO 3 . Since m H + could always be neglected, the approximation g H + = 1 was used in the mass balance equation ( 4) and in the definition of the ionic strength I (Table 1 system indicate that the limiting carbonate complex of Ce(III), of stoichiometry Ce CO ( )

m Na + = (2 m Na CO 2 3 + m NaHCO 3 ) 1 2 1 2 3 2 3 2 + + - m m HCO H O NaHCO H O M M / / (2)
m Na + + m H + = 2 m CO 3 2-+ m HCO 3 -+ m OH - (4) K' m1 = K m1 / K me = m m m CO HCO OH 3 3 - - - (3)
3 4
5-, is the major soluble species in media of [ ] CO 3 2-> 1.0 M [16]. The electrochemical equilibrium between Ce(III) and Ce(IV), and its equilibrium potential are :

Ce CO ( )

3 4 5-+ ( j n -4) CO 3 2-+ i n H 2 O Û 1 n Ce OH CO n i j n i ( ) ( ) 3 4 -+ i n H + + e - ( 9
)
In cyclic voltametry, the half-wave potential E 1/2 corresponds to equal concentrations of the two oxidation states at the working electrode. As the difference between the diffusion coefficients is usually negligible (as we shall verify that the Ce(IV) complex is mononuclear, this approximation is certainly valid), E 1/2 is assumed to be equal to E IV/III (equation 11).

Similarly, if Ce IV polynuclear species are formed, we have :

(m III ) 1/2 = n (m IV ) 1/2 = m Ce / 2 (14
) where m Ce is the Ce total molal concentration. The introduction of equation ( 14) into the Nernst relation [START_REF] Dervin | [END_REF] gives :

E 1/2 = E IV/III + A[(1 - 1 n ) log 10 2 - 1 n log 10 n] -(1 - 1 n ) A log 10 m Ce ( 15
)
The plot of the experimental data E

Results and discussion

As [Ce] t had no influence on E 1/2 (figure 2), the Ce(IV) complex is mononuclear (that is n = 1). The experimental data were then interpreted with the simplified equation ( 12a

The effects of the ionic strength variations during the experiments were estimated to be relatively small : among all the chemical conditions investigated during the redox measurements, the Debye-Hückel term D(I m ) varied from 0.255 to 0.259 and the Na + molal concentration varied from 3.03 to 3.10 mol/kg. This induced a maximum variation of log ( )

10 3 2 g CO -
of 0.015 (from -1.280 to -1.265) which corresponds to a variation of the potential of less than 1mV (which is negligible). In addition, all the attempts to fit the experimental data according to equations ( 12) and ( 13) by taking into account the variations of the activity coefficients (by using the SIT formula) proved to be unsuccessful because the ionic strength changes were not large enough. Hence, all activity coefficients and the E IV III / 0 (I) potential, could be considered as constant. The small ionic strength variations were neglected in the qualitative slope analysis (figures 3 and 4) but not for the following calculations. All the experimental data could be interpreted with the simplified equation (12a) where the i and (j -4) coefficients had clearly to be set to 0 and 2 respectively (figures 3 and 4). Nevertheless, other sets of (i, j) coefficients were tried e.g. the Ce CO ( ) 3 5

6-species which gave poorer standard deviation and some systematic deviations, while species such CeHCO CO consistent with the first estimation, the uncertainty was increased and k m IV 6 was considered as a maximum possible value (table 1). The Ce(IV/III) redox potential shift between carbonate and non complexing media is determined by the ratio of the Ce(III) and Ce(IV) complexing constants. To avoid ionic strength correction we used the b m III m I 4 ( ) and b m IV m I 6 ( ) definitions given in table 1. For these constants, as for E IV III / 0 (I ) m , the numerical values are valid for a given ionic strength and a given Na + concentration (here 3.06 mol.kg -1 ), whatever the major anion in the aqueous solution. The thermodynamical cycle is calculated in the following way using the definitions given in table 1.

This last formula was used to estimate the formation constant of the Ce(IV) limiting carbonate complex (table 1). Cyclic voltametry was successfully used at high ionic strength but some difficulties arose at lower ionic strength, probably due to precipitation when the Ce(IV) limiting carbonate complex is dissociated. This work confirms that Ce(IV) forms the Ce CO ( ) <2M. This result corroborates the conclusions of Salvatore et al. [15] obtained at higher ionic strength, but the interpretation of our data is more straightforward since our method avoided any Na + activity coefficient correction. However, our methodology did not allow the determination of the Na + content of the Ce CO ( ) 3 5

6-complex, as suggested in reference [15].

The estimation of the Ce CO ( )

3 5
6-formation constant value (table 1) is one order of magnitude or slightly higher than for actinide(IV) [1]. The chemical form, Ce CO ( ) 2-in which the 6 bidentate groups define a nearly regular icosahedron [START_REF] Greenwood | Chemistry of the Elements[END_REF][START_REF] Cotton | Lanthanides and Actinides[END_REF] (In that respect, it seems that the size of the metal is not necessarily a dominant factor [START_REF] Greenwood | Chemistry of the Elements[END_REF]). In addition, the same icosahedral coordination geometry was found for the Th CO ( )

3 6
8-anion in the solid state structure of tuliokite Na 6 BaTh(CO 3 ) 6 .6H 2 O [14]. Further information about the molecular and solid state structure of the limiting carbonate complex of Ce(IV) could be obtained by using XRD and EXAFS spectroscopy techniques, which have already been used successfully to clarify the structure of the limiting complex Pu CO ( ) 3 5

6- [START_REF] Clark | communication T3-A1 and[END_REF]. The ratio b 6 IV /b 4 III of 28.6 calculated in the present work at I m = 4.33 mol.kg -1 seems consistent with the reported estimation of 25 (using very approximate ionic strength corrections) at I m = 12 mol.kg -1 [15] : the difference is in the order of magnitude of the activity coefficient (or ion-pairing, as proposed in [15]) corrections. As for Ce(IV), no evidence of any mixed OH --CO 3 2-or polynuclear Ce(IV) complex was found. For quantitative comparison with actinides(IV), further work could consist in studying the Ce CO ( )

3 6 8-dissociation into Ce CO ( ) 3 5
6-(probably), and performing a similar investigation on the Am(IV)/Am(III) system [START_REF] Bourges | [END_REF] with proper junction potential measurements and I corrections [7,[START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF]. Similar electrochemical studies referring to the Np(IV) [START_REF] Delmau | [END_REF] and Pu(IV) [START_REF] Capdevila | [END_REF][START_REF] Capdevila | [END_REF] limiting carbonate complexes would be more tedious as those species are involved in irreversible redox couples only.

Table 1. Physical and chemical parameters used or determined in this study.

For a pure electrolyte : log 10 2 a H O » r m , p = m / c and log 10 p » q m ; where c and m denote the molar (mol.l -1 or M) and molal (mol.kg -1 ) concentrations of the salt in solution. When the electrolyte is a mixture of two salts (Na 2 CO 3 , NaHCO 3 ), the approximations log 10 p » q 1 m 1 + q 2 m 2 and log 10 2 a H O » r 1 m 1 + r 2 m 2 are used, where q x and r x are the q and r parameters for the pure salt x, and m x is the molal concentration of the salt x in the mixture. p values (l.kg -1 ) are taken from [START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF]. The q and r values are calculated in this work. The log 10 2 a H O values (to obtain r) are calculated from Pitzer parameters [START_REF] Pitzer | Activity Coefficients in Electrolyte Solutions , 2 nd edition[END_REF]. c i , mi, a i and g i denote respectively, the molar (mol.l -1 or M) concentration, the molal (mol.kg -1 ) concentration, the molal activity and the activity coefficient of the species i : a i = m i g i and p = m i / c i . The coefficients g i . are calculated by using the SIT formula [START_REF] Grenthe | NEA-TDB Chemical thermodynamics of uranium[END_REF][START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF]1]. The SIT coefficients e( , )

M z z M A A (kg.mol -1
) are taken from [START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF] ; the uncertainty is increased for the SIT coefficients which are estimated by analogy.

Electrolyte X q X r X e(i,j) 6-. b estimated from [START_REF] Pitzer | Activity Coefficients in Electrolyte Solutions , 2 nd edition[END_REF] by analogy with La and Pr, 0.47 ± 0.03 is tabulated in [START_REF] Grenthe | NEA-TDB Chemical thermodynamics of uranium[END_REF][START_REF] Silva | NEA-TDB Chemical thermodynamics of americium[END_REF] c Estimated by analogy with U 4+ .

j = ClO 4 - j = OH - j = CO 3 2- j = HCO 3 - j = Cl - j = Ce CO ( ) 3 4 5
Table 2. Thermodynamic equilibrium constants and redox potentials used or determined in this work. The equilibrium constants are defined in molal units (as in the text, and when nothing else is stated for numerical values). Subscript m is used for molal units (only when needed in order to make the difference with molar units).

I m = 1 2 z m i i 2 å
is molal ionic strength. A = R T ln10 / F, typically 0.05916 and 0.05803 V at 25 and 19.3°C. IV and III denote the Ce(IV) and Ce(III) limiting carbonate complexes. For each constant, the first value is usually the numerical datum taken from the corresponding reference, while values at other ionic strengths are calculated in this work by means of the SIT formula [START_REF] Grenthe | NEA-TDB Chemical thermodynamics of uranium[END_REF] ) is calculated with equation ( 16) by using the necessary auxiliary data at about 25°C (the temperature corrections were assumed to be less than uncertainty).

Name Definition Numerical value Reference K mw (I m )

= m OH -a H + log 10 [K w (0)] = -14.001 ± 0.14 (1) [17] page 371

K m1 (I m ) = m a m CO H HCO 3 2 3 - + - log 10 [K 1 (0)] = -10
.329 ± 0.020 (1) [8]

E AgCl Ag / 0 (I ) m » E AgCl Ag / 0 (0) -A log 10 a Cl - E AgCl Ag / 0 (0) » 0.
2263 (1) at 19.3°C [START_REF] Bard | Standard Potential in Aqueous Solutions[END_REF] E Ce Ce

+ + 4 3 0 / (I) » E 1/2 in acidic media of ionic strength I E Ce Ce + + 4 3 0 / (I) = 1
.72 ± 0.02 (2) 1.821 ± 0.023 (1) at I = 0 ± 0.1 (3) at I m = 3.5

11.67 ± 0.98 (1) at I = 0 13.51 ± 0.16 (4) at I m = 3.06

13.61 ± 0.16 (5) at I m = 4.33

E IV III / 0 (I ) m » E 1/2 + 2 A log 10 3 2 m CO - E I IV III m / 0 ( ) =
0.182 ± 0.009 (4) at I m = 3.06 0.161 ± 0.008 (5) IV m I = 41.8 ± 0.5 (4) at I m = 3.06 42.2 ± 0.5 (5) at I m = 4.33 £ 0.03 ± 0.47 (4) at I m = 3.06 0.03 ± 0.19 (5) at I m = 4.33 ± 0.5 (4) at I m = 3.06 41.8 ± 0.5 (5) at I m = 4.33 this work (1) Standard value (at 0 ionic strength).

( Still our E 0 value calculated at I=0 is consistent with the value selected in [15]. (3) 3 M NaClO 4 . (4) 2.67 M NaClO 4 which corresponds to Na + molality = I m = 3.06 mol.kg -1 .

(5) Na 2 CO 3 / NaHCO 3 media as used in the present work ; Na + molality = 3.06 ± 0.04 mol.kg -1 and I m = 4.33 ± 0.37 mol.kg -1 .

Footnotes : CO 3 ] and [NaHCO 3 ]) and the pH values measured while the acidic titration with CO 2 gas, the speciation is calculated by solving the system of equations (1 to 8) with the parameters listed in tables 1 and 2. [i] and m i denote the molar (mol.l -1 or M) and molal (mol.kg -1 ) concentrations of the species i. The glass electrode is calibrated in activity unit, with buffers at the same molar ionic strength as that of the initial test solutions. The potential E 1/2 , mean value of the oxidation and reduction peaks, is measured on the voltamograms. It is then corrected for the junction potential E j (mV), and the reference electrode potential E REF (mV/SHE) (calculated with the parameters of table 1 (or 0.38) mol.kg -1 and pH = 10.36 (or 9.86) respectively, T = 19.3 ± 1.0°C, A = 58.03 mV/log 10 unit. This figure shows that the measured redox potential E IV/III is independent of the Ce total concentration. Since in these conditions the Ce(III) major aqueous species is mononuclear [16], the Ce(IV) species is also mononuclear (equation 15). Theoretical curves are plotted for n = 1 (solid line) and n = 2 (dotted line), where n is the Ce stoichiometric coefficient in the Ce(IV) aqueous species (for n > 2 the deviation between the experimental data and the theoretical lines are even greater than for n = 2).

  For both sets of experiments (with test solutions of general initial composition c 0 M Na 2 CO 3 , (I-3c 0 ) M NaHCO 3 ), the junction potential arising between the test solution and the REF electrode was measured by means of the cell REF || 0.02 M NaCl, c 0 M Na 2 CO 3 , (I-0.02-3c 0 ) M NaHCO 3 || REF', where REF = Ag/AgCl | 0.02 M NaCl, (I-0.02) M NaClO 4 |, REF' is an Ag/AgCl wire, || denotes a liquid-liquid junction (capillary extension) and | denotes liquid-solid contact. The variations of the junction potential as a function of pH were investigated by stepwise acidification of an initial 0.02 M NaCl, c 0 M Na 2 CO 3 , (I-0.02-3c 0 ) M NaHCO 3 solution with CO 2 gas, and measuring the electromotive force of the cell rapidly after stabilisation (to avoid the damage to the REF' electrode in direct contact with carbonate ions). After each measurement, the REF electrode was controlled by means of the cell REF' || 0.02 M NaCl, (I-0.02) M NaClO 4 || REF 0 , where REF 0 is

D

  10 K me (I m ) = log 10 K me (0) + D(I m ) -e(Na + ,OH -) m Na + + log 10 2 a H O (7) log 10 K m1 (I m ) = log 10 K m1 (0) + 3 D(I m ) -([e(Na + , CO 3 2-) -e(Na + , HCO 3 -)]) m Na + (8) Previous studies of the Ce III CO H O ( ) --

  1/2 as a function of log 10 m Ce gives (equation 15) a straight line with slope (1-1/n) which we shall use for the determination of the stoichiometric coefficient n of the Ce(IV) complex. For n = 1, E 1/2 = E IV/III (equation 15). When n is known, equation 12 and the data (E IV/III , m CO 3 2-, pH) give the stoichiometric coefficients i and j, and the formal potential E IV III / 0 (I) at ionic strength I.

  theoretical curves completely erroneous when compared with experimental data. Finally the following interpretation is proposed : = 0.155 3 ± 0.002 3 V/SHE (where the uncertainty is 1.96 times the standard deviation) was fitted. Assuming partial dissociation of the Ce(IV) limiting complex did not significantly improve the fitting, which gave log 10 the Ce(IV) Carbonate Limiting Complex. Ch. Riglet-Martial, P. Vitorge, V. Calmon 5/10

) 1 M

 1 HClO 4 ; this value may be in error since the hydrolysis correction does not seem to be validated,

Fig. 1 .Fig. 2 . 13 Electochemical

 1213 Fig. 1. Glass electrode calibration. The calibration was checked by two titrations of a 1 M NaHCO 3 solution with a 1 M NaOH solution at T = 19.3 ± 1.0°C. The theoretical pH (-log 10 a H + and not -log 10 [H + ]) values were calculated by using the equilibrium constants and SIT coefficients listed in tables 1 and 2. The observed deviation at pH>10 was taken into account in the treatment of the experimental data of the Ce(IV)/Ce(III) redox potential. No deviation (of the glass electrode slope) was observed at pH<8.2 where NaHCO 3 /CO 2(g) buffers were used for the pH calibration.

Fig. 3 . 1 , 8 -

 318 Fig. 3.Influence of the CO 3 2-concentration on the Ce(IV/III) formal potential in concentrated carbonate/bicarbonate media. [Na + ] = 3.06 mol.kg -1 , 1.00 < [ CO 3 2-] < 1.45 mol.kg -1 , [Ce] t = 0.001 M, 9.2 < pH < 10.6, T = 19.3 ± 1.0°C, A = 58.03 mV/log 10 unit. This figure shows that, within experimental uncertainties, the measured data falls into a single straight line (dotted line), with a slope corresponding to the theoretical one assuming that two CO 3 2-ions are exchanged during the redox reaction.Since in these conditions the Ce(III) major species is Ce CO ( ) 3 45-[16], the Ce(IV) species is

  All bicarbonate/carbonate solutions, prepared from crystallised NaHCO 3 (Merck, p.a.) and concentrated NaOH (Merck, p.a., d=1.35), were checked by acid-base titration using H 2 SO 4 (Merck Titrisol). A 0.1 M Ce(III) stock solution was prepared from crystallised Ce(ClO 4 ) 3 ,6H 2 O (Johnson Mattey, purity > 99.9%). The test solutions were obtained by adding an aliquot of the Ce stock solution to the previous sodium bicarbonate/carbonate aqueous solutions. The solutions used in the reference electrodes were prepared from crystallised NaCl and NaClO 4 (both Merck, p.a.). The response of the glass electrode was checked by using commercial buffer solutions (Merck Titrisol) of pH 2, 7, 9 and 12. Demineralized water delivered by a Millipore Milli-Q plus purifier was used for all the dilutions. The redox potential of the Ce(IV/III) couple was measured by cyclic voltametry, using a classical three electrode device -including an Ag/AgCl reference electrode (Radiometer XM630/D8), a hanging mercury drop electrode (HMDE Metrhom E410) as a working electrode and a platinum wire (Radiometer Pt11/CMP) as a counter electrode -connected to an electrochemical analyser (Radiometer Voltalab 32) including a programmable interface with the Voltamaster 2 software. In order to avoid the possible damage of the reference electrode by the carbonate ions, it was isolated from the test solution by means of a capillary extension (Radiometer FDL2/CMP) filled with a 0.02 M NaCl, (I-0.02) M NaClO 4 solution, of same ionic strength I as that of the test solution. The potential of this electrode (noted REF) was checked once a day. Differences higher than 0.2 mV were never observed. The formal potential of the REF electrode was calculated from Nernst law: E REF = E AgCl Ag

/ 0 + A log 10 (a Cl -). Activity coefficients were estimated by applying the SIT formula : log 10 (g i ) =z i 2 D(I m

  -

	NaHCO 3	0.01249	-0.0148	i = Na +	0.04	-0.08	0.00	0.03	-0.20
			± 0.0015		± 0.01	± 0.03	± 0.02	± 0.01	± 0.25 a
	Na 2 CO 3	0.00426	-0.0136	i = Ce 3+	0.51
			± 0.0002		± 0.04 b
	M H2O = 0.0180 kg.mol. -1	i = Ce 4+	0.76
					± 0.20 c
	a Estimated by analogy with U CO (	) 3 4 4 -and U CO (	) 3 4

  by using the e values listed in table1. The E IV III

	0	/	(4.33m) (V/SHE) value was measured in this work in a
	0.46 m NaHCO 3 1.29 m Na 2 CO 3 aqueous solution at 19.3°C. b m IV 6 4 33 m ( .

Table 3 : Ce(IV/III) formal potential measured in concentrated carbonate/bicarbonate media

 3 From the Na 2 CO 3 and NaHCO 3 initial concentrations (noted [Na 2

  as indicated in the text) to give E IV/III (mV/SHE).

		Initial conditions		Measurements		Values calculated from initial conditions
	[Na 2 CO 3 ]	[NaHCO 3 ]	[Ce] t	E j	pH	I m	m Na +	m HCO 3 -	m CO 2 -	m OH -	E REF	E IV/III
	1.468	0.084	0.001	12.5	10.552	4.564	3.071	0.085	1.493	4.88E-04	329.8	136.1
	1.468	0.084	0.001	12.3	10.276	4.531	3.073	0.157	1.458	2.58E-04		138.3
	1.420	0.120	0.001	13.0	10.388	4.487	3.032	0.122	1.455	3.35E-04	329.8	137.7
	1.420	0.120	0.001	12.3	9.766	4.334	3.041	0.457	1.292	7.97E-05	329.8	143.9
	1.420	0.120	0.001	12.1	9.563	4.235	3.047	0.673	1.187	4.98E-05	329.8	147.3
	1.420	0.120	0.001	11.9	9.388	4.125	3.054	0.912	1.071	3.32E-05	329.8	150.9
	1.420	0.120	0.001	11.8	9.294	4.057	3.058	1.059	0.999	2.67E-05	329.8	153.4
	1.422	0.127	0.001	13.8	10.361	4.473	3.026	0.130	1.448	3.15E-04	329.8	138.1
	1.422	0.127	0.001	13.3	9.844	4.354	3.033	0.391	1.321	9.55E-05	329.8	141.5
	1.422	0.127	0.001	13.1	9.616	4.253	3.039	0.609	1.215	5.64E-05	329.8	144.4
	1.422	0.127	0.001	12.9	9.440	4.150	3.045	0.835	1.105	3.75E-05	329.8	149.0
	1.422	0.127	0.001	12.8	9.329	4.073	3.049	1.002	1.024	2.90E-05	329.8	152.6
	1.422	0.127	0.001	12.8	9.257	4.019	3.053	1.119	0.967	2.45E-05	329.8	155.7
	1.462	0.087	0.001	14.4	10.536	4.551	3.063	0.088	1.487	4.70E-04	329.8	134.6
	1.462	0.087	0.001	14.1	9.943	4.445	3.070	0.320	1.375	1.20E-04	329.8	138.7
	1.462	0.087	0.001	13.9	9.671	4.338	3.076	0.552	1.262	6.38E-05	329.8	143.0
	1.462	0.087	0.001	13.8	9.494	4.241	3.082	0.764	1.159	4.24E-05	329.8	147.7
	1.462	0.087	0.001	13.7	9.305	4.111	3.090	1.047	1.022	2.73E-05	329.8	153.4
	1.326	0.369	2.49E-04 to	13.3	9.864	4.452	3.094	0.378	1.358	9.95E-05	329.8	141.2
			1.96E-03
	1.422	0.127	2.49E-04 to	13.8	10.361	4.473	3.026	0.130	1.448	3.15E-04	329.7	137.0
			1.96E-03
	1.484	0.078	0.001	13.2	10.584	4.608	3.099	0.079	1.509	5.23E-04	329.7	134.5

--(Figure 3), the formula of the the Ce(IV) complex is finally Ce CO ( )

8-, which is in agreement with the interpretation of the figure 3.