Pedro A Patlan-Rosales

Alexandre Krupa

A robotic control framework for 3-D quantitative ultrasound elastography

In this paper we present a novel approach to track and explore stiff tissues within 3-D ultrasound volumes acquired by a medical 3-D ultrasound probe mounted on a six degrees of freedom robotic arm. Autonomous palpation and on-line elastography process are implemented to estimate the elastic property of the tissues (strain) in a volume of interest (VoI) indicated by the user. The compression motion, required for the elastography, is performed by controlling the force applied by the ultrasound probe to the tissues. A visual servoing control for centering a rigid tissue (target) inside the field of view (FoV) of the ultrasound probe is established to always maintain the target visible. Additionally, rotations around the contact point between the tissue and the ultrasound probe are teleoperated through a haptic device handled by the user in order to allow exploration of the target surrounding areas. Results show a stable system that can be used in the future for diagnosis of diseases or tumor location.

I. INTRODUCTION

Ultrasound systems are one of the most common tools used in hospitals around the world. Besides its portability and low cost, it offers valuable information of tissues for doctors in real time. Radio frequency (RF) signals provided by an array of transducers are processed to display a bmode image. The properties of the RF signals can also be used to compute the elastic parameters of tissues, commonly known as strain values, over a region of interest (RoI). This procedure is called elastography, and it can be implemented by applying mechanical axial compression on the skin of the subject using the ultrasound probe. Elastography needs two arrays of RF signals successively measured at the pre-and post-compression states of the considered tissue to estimate its strain values along the probe axial direction.

In a previous work [START_REF] Patlan-Rosales | Automatic palpation for quantitative ultrasound elastography by visual servoing and force control[END_REF] we presented a three degrees of freedom (DoF) automatic palpation robotic system using force control and visual servoing to automatically position a 2-D US probe in a way to center on a stiff tissue based on the elastogram (map of strain values) in 2-D. Our approach was able to maintain the target in the center of the field of view (FoV) of a 2-D ultrasound probe even if perturbation motion inside the observation plane (in-plane motion) was applied to the system. However, this method did not consider motions outside the plane of observation of the 2-D probe (out-of-plane motions), as they are likely to occur in real examinations due to patient physiological alexandre.krupa@inria.fr motions. Therefore, to deal with this issue, we propose in this paper to consider the elastograms in 3-D that we obtain by controlling the 6-DoF of a 3-D ultrasound probe with the new approach presented in section II.

Strain estimation in 3-D has been explored in different ways. In recent work [START_REF] Huang | 3-D ultrasonic strain imaging based on a linear scanning system[END_REF], a method is presented using a linear probe mounted on a sliding mechanical track, estimating the elastograms in 2-D for a known number of positions resulting in an elastogram volume. In [START_REF] Rivaz | Ablation monitoring with a regularized 3D elastography technique[END_REF], a motorized 3-D probe has been used to estimate the 3-D strain map. In this previous work, a first sweep of the probe was performed to acquire the RF signals reflected by a volume of pre-compressed tissues. Then, a second probe sweep was performed to acquire the RF signals after an axial force on the consider volume of tissues was applied by the probe. The RF signals recorded before and after the tissues compression allow then to estimate the 3-D elastogram by dynamic programing. However, the procedure to obtain the elastograms in these works is performed manually, and it depends on the dexterity of the physician to move the ultrasound probe.

Recently, robotic systems have been designed to exploit the B-mode information generated by a 3D ultrasound. For example, the estimation of the deformation of an anatomical shape due to the physiological motions of a patient was presented in [START_REF] Royer | Realtime tracking of deformable target in 3d ultrasound images[END_REF] based on the tracking of a deformable target modeled as a mesh. Another work that used the ultrasound information in 3-D was presented in [START_REF] Lee | Intensity-based visual servoing for non-rigid motion compensation of soft tissue structures due to physiological motion using 4d ultrasound[END_REF] to automatically compensate the 3-D physiological motion of a subject thanks to the control of an US probe by visual servoing.

Up until now, ultrasound elastography has been used only in very few robot-assisted procedures. The surgical robot da Vinci (Intuitive Surgical Inc.) was employed to estimate the elastogram by performing an automatic palpation [START_REF] Billings | System for robot-assisted real-time laparoscopic ultrasound elastography[END_REF]. In this previous work, the motion of a laparoscopic ultrasound 2-D probe was controlled, and the elastograms, obtained after processing the acquired 2-D ultrasound data, were displayed to guide a minimal invasive surgery procedure. In a similar framework [START_REF] Schneider | Remote ultrasound palpation for robotic interventions using absolute elastography[END_REF], an external mechanical excitation was applied instead of a palpation motion with the US probe. The mechanical excitator was positioned on the skin, and the laparoscopic ultrasound probe was displaced by teleoperating the da Vinci robot. However, this system lacks in the synchronization of the excitator and the US probe. In the same field of laparoscopy, a 11-DoF snake-like robot with an integrated phased ultrasonic micro array was used to locate hard lesions by palpation motion [START_REF] Sen | Enabling technologies for natural orifice transluminal endoscopic surgery (N.O.T.E.S) using robotically guided elasticity imaging[END_REF]. However, none of these works use directly the 3-D elastograms as control feedback for robotic tasks. We propose to extend our previous work to control the 6-DoF of a 3-D ultrasound probe. In order to allow elastography exploration in the 3-D space, we propose in this paper to maintain a target visible in the center of the 3-D FoV by visual servoing. This can help the physician to track a VoI regardless of the movement of the patient. We also propose to include in our control a teleoperation control mode for the rotations of the US probe thanks to the use of a haptic device. Teleoperating the probe rotations will therefore allow the clinician to explore in detail the surrounding areas of the VoI. To our knowledge, this is the first time that volumetric elastograms are used as control feedback for a robot.

We describe in the section II the methodology used for the automatic palpation, and the automatic centering of a rigid object in the 3-D FoV. In the same section, we detail the teleoperation of the orientation of the ultrasound probe. Section III presents experimental results using an organic phantom. Then an argumentation and analysis of the results obtained from the experiments are presented in section IV.

II. METHODS

A. Automatic palpation

We define autonomous palpation as a system to compute a strain map by compression motion with an ultrasound probe. A motorized ultrasound probe is used in this work (model 4DC7-3/40 Analogic Corporation) to obtain a volume composed by N f RF frames. Fig. 1 shows the cartesian reference frames attached to the 6-DoF robotic arm we used to move the probe. A force sensor (located at frame F s in Fig. 1) is involved in the system to achieve the compression motion. As we previously show in the 2-D case [START_REF] Patlan-Rosales | Automatic palpation for quantitative ultrasound elastography by visual servoing and force control[END_REF], oscillatory force control was used to reach the desired compression motion. This task remains the same with respect to the control law. However, the period in the oscillatory force variation changes. The acquisition of one volume takes around one second, and we need the RF signals in the states of pre-and post-compression. Therefore, a step-variation function proposed for the force changing is expressed as follows:

F d (τ ) = F max if τ is odd F min otherwise (1)
where τ ∈ Z + increases every acquisition of one volume. F min and F max are the forces applied for pre-and postcompression respectively. These force values depend in the tissue stiffness. The force/torque sensor provides the measures for the force tensor H s s expressed in the sensor frame F s . As we need to measure the force applied in the contact point frame F pc , we are taking into account the probe mass m p in the gravity force tensor H g g = [0 0 9.81m p 0 0 0] ⊤ . The tensor used to express the force measured in F pc is defined as follows [START_REF] Nadeau | Intensity-based ultrasound visual servoing: Modeling and validation with 2-d and 3-d probes[END_REF]:

H pc pc = F pc s H s s -F s g H g g
(2) where F s g and F pc s are transformation matrices from the gravity frame F g to the frame F s and from the frame F s to the frame F pc respectively.

In our case, we are aiming to control the force in the y-axis of the probe. Then, we define the feature vector to be regulated as s f = [0 1 0 0 0 0] H pc pc and the desired feature vector as

s * f = [0 F d (τ ) 0 0 0 0]. The error to minimize is e f = s f -s *
f , with an exponential decrease of e f . Therefore, the desired variation of the error is defined as ė * f = -λ f e f with λ f being the force control gain. The change of the force feature through time can be expressed as ṡf = L f v where L f is the interaction matrix that relates the force feature to the probe velocity v = [v x v y v z ω x ω y ω z ] ⊤ with the three first components corresponding to the translational velocity and the three last components to the angular velocity expressed in the contact point frame F pc (see Fig. 1). Then L f can be approximated as

L f = [0 k 0 0 0 0],
where k is an estimation of the contact stiffness. The control law for this task is defined as

v f = L + f ė * f (3)
where L + f is the Moore-Penrose pseudo-inverse of L f . In Fig. 2 we show the process to estimate the center of mass of the biggest rigid tissue. We use the volumes acquired with RF signals (RF volumes) V r and V c in the pre-and postcompression motion respectively to compute a 3-D strain volume into a VoI. We define the 3-D strain volume V s as

B. Estimation and post-processing of the 3-D elastogram

V s (i, j, k) = A s k (i, j) (4)
where i and j are the indexes of the ultrasound scanline and the sample in the scanline respectively. A s k is the strain map in 2-D estimated at the k-frame pair of V r and V c in the interval [k 0 , k n ] such that k n > k 0 . The elastograms in 2-D are approximated using the method presented in [START_REF] Patlan-Rosales | Automatic palpation for quantitative ultrasound elastography by visual servoing and force control[END_REF], which are based on the optical flow for the motion estimation between the pre-and post-compression 2-D RF frames and the least-squares strain estimator [START_REF] Kallel | A least-squares strain estimator for elastography[END_REF]. The obtained V s volume provides elastic information of the tissues inside the VoI as we show in Fig. 2, where the red color represents the soft tissue and the blue the stiff tissue. Furthermore, we propose to segment V s using a threshold between -µ and µ, to isolate the stiff tissues. We estimate µ = c s + p s max( V s ), where c s = min(V s ) + 0.5 [max(V s )min(V s )] is the central strain value, p s is the percentage of the strain values to use and ⋆ is the absolute value of (⋆). The value of p s allows to restrict the limits of the thresholding for the strain values. As we want to detect only the rigid tissues, the value of p s should be low, and we have found experimentally that p s = 0.1 (10%) gives us the best results. The thresholding provides the regions with only stiff tissue. We detect and track the biggest region in order to keep it always visible as described in subsection II-C. To compute the biggest region and its center of mass (CoM), we apply the connected components algorithm in 3-D to V s after thresholding. In our case, we use the implementation of this algorithm provided by the VTK library (Visualization Toolkit [START_REF] Schroeder | Visualization Toolkit: An Object-Oriented Approach to 3D Graphics[END_REF]) with the function vtkPolyDataConnectivityFilter to obtain the largest stiff region V sb and the vtkCenterOfMass function to calculate the center of mass (p c = (i c , j c , k c )) of V sb . We show in Fig. 3 the mesh obtained to compute the center of mass. We need to convert the value of p c in the metric coordinates p g . To do this, as we use a convex ultrasound probe, we perform a scan conversion of each point inside the RF volume to the Cartesian coordinates (see Fig. 4), s(i, j, k) → p(x, y, z), in order to obtain the metric location with respect to the Cartesian frame. We define the scan conversion using the ultrasound probe parameters as described in [START_REF] Lee | Intensity-based visual servoing for non-rigid motion compensation of soft tissue structures due to physiological motion using 4d ultrasound[END_REF]. In our case, RF data is considered instead of pre-scan images. We Every slice function of V r , f k (i, j) ∈ V r , is converted to b-mode (brightness mode), f k b (i, j), image as f k b = log{ Hilbert (f k ) } where Hilbert (⋆) is the Hilbert transform of (⋆). For visualization purposes, images are normalized from 0 to 255, and the scan conversion previously described is applied. We display one volume every two volumes due to the computational cost. The display rate is two times the period of the volume acquisition T v . We also display V sb with a mesh (using marching cubes mesh in VTK) at the same display rate as shown in Fig. 3.

C. Target-probe centering on x-z plane

Keeping a target in the field of view (FoV) of the volume of analysis is a task which requires the displacement of the ultrasound probe on the x -z plane (see Figure 5). The center of mass of the target, p g = (x g , y g , z g ) (estimated in the section II-B), comes from the automatic palpation. As we control the probe velocity in y-axis with the automatic palpation, the remaining translational velocities v x and v z can therefore be controlled by visual servoing to perform an automatic alignment of the target in the middle of the probe FoV.

The visual feature vector that we define to perform this task is s t = [x g z g ] ⊤ . In this case the desired visual feature vector to reach the centering of the object of interest in the probe FoV is directly s * t = 0 2×1 . The error is defined as e t = s ts * t . Similar to the force control designed for the automatic palpation task, an exponential decrease of the error can be obtained by defining the desired error variation as ė * t = -λ t e t with λ t being the target-probe centering control gain.

Using basic kinematics, we determine the relation between the probe velocity v and the variation of the retained features as

ẋg żg = -1 0 0 0 -z g y g 0 0 -1 -y g x g 0 v (8)
The Equation ( 8) can be written as ṡt = L t v, where L t is the interaction matrix related to s t . Then, the control law for the target-probe centering can be expressed as

v t = L + t ė * t (9)

D. Teleoperation of probe orientation

The third task we propose consists in teleoperating the probe orientation in order to explore with detail the surrounding area of the target. To perform this, we define the orientation of the probe as reaching a desired orientation s * θ = [θ x d θ y d θ z d ] ⊤ from the measure feature vector s θ = [θ x θ y θ z ] ⊤ , where θ x , θ y , θ z are the measured angles of the ultrasound probe provided by the robot odometry at the current time t c . The error to minimize is defined as e θ = s θs * θ , and the desired exponential error decrease can be achieved by the desired error variation expressed as ė * θ = -λ θ e θ where λ θ is the orientation control gain. As in the previous task, we determine the interaction matrix that relates the feature vector variation, ṡθ , with the probe's velocity v as

L θ =   0 3×3 -1 0 0 0 -1 0 0 0 -1   ( 10
)
The control law for the probe orientation is then provided by

v θ = L + θ ė * θ (11)
In our case we propose to set and change on demand the desired probe orientation s * θ by manually moving the three rotational degrees of freedom of a haptic device (Haption VIRTUOSE 6D). Using the odometry of the haptic device, we obtain a reference orientation Φ init at the time t 0 and the current orientation Φ c at the current time t c when the endeffector is moved. Then, we express the desired orientation as

s * θ = R pc v (Φ c -Φ init ) ( 12
)
where R pc v the rotation matrix between haptic frame F v and the probe contact frame F pc .

E. General control law

Our goal is to combine the automatic palpation with the alignment of the probe on the target and the teleoperation of the rotation of the probe. Therefore, to achieve our goal, we enclose the previous control laws into a general control law by stacking the interaction matrices L f , L t and L θ of each task in a general interaction matrix. This latter relates the variation of the features of the three tasks to the 6-DoF probe velocity, and it is obtained as follow

L =         -1 0 0 0 -z g y g 0 k 0 0 0 0 0 0 -1 -y g x g 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1         (13)
Finally, the general control law that allows to perform simultaneously the three tasks is given by

v = L +     -λ t0 e t0 -λ f e f -λ t1 e t1 -λ θ e θ     (14)
where -λ t0 e t0 and -λ t1 e t1 represent the first and the second elements of ė * t respectively. The tasks of centering the target in the FoV and the probe orientation require low gain to converge successfully due to dependency of the time to retrieve a new RF volume in order to compute the center of mass. However, if we apply adaptive gain for these tasks, we can decrease the convergence time for these tasks. Then, adaptive gain is applied to λ t and λ θ , and it is defined as follows

λ(x) = ae -bx + c (15)
where a, b and c are constant parameters and x = s * -s is input to consider. The desired feature vector s * and the measured feature vector s are linked to the corresponding vectors in the task where the adaptive gain is applied. The parameters a, b and c are computed as follows

a = λ(0) -λ(∞) (16) b = λ(0)/a (17) c = λ(∞) (18)
where λ(0) is the gain when x = 0, λ(∞) is the gain when x = ∞ and λ(0) is the slope of λ(x) when x = 0.

III. EXPERIMENTAL RESULTS

the experiments we used a 6-DoF robot, Viper s850 (Adept Technology Inc., USA) and an Analogic Corporation 4DC7-3/40 convex ultrasound probe rigidly attached to its end-effector. The force sensor used was an ATI Gamma 65-SI FT connected to a National Instrument NI DAQmx PCI-6220 board. The acquisition of RF data was implemented using a server-client TCP/IP communication in a local network.

We used as server the SonixTouch ultrasound scanner, and as client a workstation (Intel Xeon CPU @2.1 GHz) that performs all the imaging process, control law computation and communication with the robot. The RF data from the server is sent to the client at the rate of 24 FPS (frames per second). Every volume contains 31 RF frames which means that every volume is completed in ≃ 1.3s. In the client, we developed a multi-thread program in C++, with a graphic user interface (GUI) in the main thread created to display and control the functions of the other threads, as described in Fig. 6. A shared pointer is continuously updated by the acquisition thread (frame by frame). This shared pointer is read by the RFtoBMode thread (process in charge of converting the RF volume to b-mode volume) and the Elastography thread (process to compute the 3-D elastogram in a VoI) once a volume is completed. The Display object, in the main thread, contains the functions to display three orthogonal planes (sagittal, axial, coronal) of the volume (see Fig 7). This object also allows the user to select the VoI by displacing the planes to the desired position and pressing a key of the keyboard. We perform a set of experiments on a homemade gelatin phantom with two duck gizzards inside (see Fig. 8), and we present in this paper only the evolution of one experiment, which is the base for every experiment performed. We set the values of F min = 1.5N , F max = 2.5N and λ f = 0.002. F min and F max were estimated empirically for the gelatin phantom, and they can be adapted for any other kind of tissue. The parameters in the adaptive gains are set as λ t (0) = 0.1, λ t (∞) = 0.03 , λt (0) = 0.3, λ θ (0) = 1.5, λ θ (∞) = 0.2 and λθ (0) = 2.3. The experiment begins with an initial probe position where a stiff object of interest is positioned in the 3-D US probe FoV (red point in Fig. 8left). Then, the automatic palpation with the robot is activated to perform the compression of the tissues. The centroid of the 3-D elastogram is estimated as we previously described in II-B, and the automatic centering control task is started. Next, four points delimiting the VoI are chosen using the developed GUI and displayed by small yellow spheres as shown in Fig 7 . Once the VoI is selected, the 3D elastogram is estimated for every pair of RF volumes in the same direction of acquisition. The centroid of the 3-D elastogram is estimated as we previously described in II-B, and it is sent to the automatic centering control task. The teleoperation of the probe orientation is always active, and the user can rotate the end-effector of the haptic device any time. We show in Fig. 9 the plots of the evolution of the probe velocities for one experiment. We can observe at the beginning of the experiment that the only velocity active is the v y (force control). At time t ≃ 30s the system is paused to select the VoI, and time t ≃ 75s the process continues. Then, the center of mass is computed and the velocities v x and v z start to variate, and at t ≃ 150s the probe has been automatically aligned with the stiff tissue of interest (duck gizzard) by following and exponential decrease of the visual error e t as expected. The teleoperation of the probe orientation is introduced from the time t ≃ 160s and we can see at this point the variation of ω x , ω y and ω z with eventual convergence each time a new desired probe orientation is manually indicated by the user thanks to the haptic device.

IV. DISCUSION

We observe that the process in the experiments requires fast acquisition of RF data sent by the ultrasound. However, the control of the probe is slow due to the requirement of the elastography of two volumes (pre-and post-compressed) to compute one 3-D elastogram. Therefore, if the gains for the automatic centering and for the probe orientation are set too high, these two tasks will fail due to the low volume rate acquisition. We also introduce small displacements to the gelatin when the system was running, and we observe that if the motions are slower than the time of the RF volume acquisition, then the process compensates successfully this perturbation. The main advantage of the system proposed in this work with respect to the 2-D [START_REF] Patlan-Rosales | Automatic palpation for quantitative ultrasound elastography by visual servoing and force control[END_REF] case is that the out-of-plane motions are considered. This represents a big step in the quest of medical robotic-assisted system that can help to improve medical procedures. Moreover, the proposed system allows the clinician to perform 3-D quantitative elastography by simply teleoperating the probe orientation without worrying on both the palpation motion and the target centering motion.

V. CONCLUSION

We present a framework to control a 3-D ultrasound probe held by a 6-DoF robotic arm in order to perform realtime 3-D quantitative ultrasound elastography. It is based on the design of a control law that simultaneously perform three tasks: autonomous palpation of tissues, probe automatic alignment on a stiff target of interest and teleoperation of the probe orientation. The experimental results demonstrated the feasibility of the proposed concept. In future work we plan to use the estimated strain volume to provide a feeling of the elasticity of the tissue to the user through the haptic device. Our system could also be used in other clinical scenarios as for example for real-time automatic estimation of tissue strain 3-D map required for robotic steering of flexible needle.

Fig. 1 .

 1 Fig. 1. Cartesian reference frames attached to the robotic arm.

Fig. 2 .

 2 Fig. 2. Proposed process to obtain the center of mass of the largest stiff region.

Fig. 3 .

 3 Fig. 3. Left image shows the volume of strains values Vs (green color means stiff tissue and orange soft tissue). Right image shows the mesh of the biggest rigid tissue, Vs b , in green.

Fig. 4 .

 4 Fig. 4. Scan conversion in 3-D required to obtain the metric coordinate.briefly recall the scan conversion formulation asx = r sin φ (5) y = [r cos φ -(r p -r m )] cos θ + (r p -r m ) (6) z = [r cos φ -(r p -r m )] sin θ (7)where r p and r m are the radii of the ultrasound probe and the motor of the probe respectively (see Fig.4right). The coordinates in the Cartesian volume are sorted as quasispherical coordinates with r as the distance from the point to the origin of the scanlines, φ as the azimuthal angle in the x-y plane and θ as the zenith angle (see Fig.4right). The quasi-spherical coordinates are computed in function of the RF coordinates as r = vs fs j + r p , φ = -0.5α l (N f -1) + α l i and θ = -0.5η(N f -1 -2k) where v s is the speed of the sound (1540 m/s), f s is the sampled frequency, α l is angle between neighboring scanlines and η is the angle of the FoV of the motor in the ultrasound probe for a motor angular step.Every slice function of V r , f k (i, j) ∈ V r , is converted to b-mode (brightness mode), f k b (i, j), image as f k b = log{ Hilbert (f k ) } where Hilbert (⋆) is the Hilbert transform of (⋆). For visualization purposes, images are normalized from 0 to 255, and the scan conversion previously described is applied. We display one volume every two volumes due to the computational cost. The display rate is two times the period of the volume acquisition T v . We also display V sb with a mesh (using marching cubes mesh in VTK) at the same display rate as shown in Fig.3.

Fig. 5 .

 5 Fig. 5. Probe motion in the space from the current position to the desired position.

Fig. 6 .

 6 Fig. 6. Short diagram of the implemented multithread program.

Fig. 7 .

 7 Fig. 7. Display of the three orthogonal planes and the VoI with the 3D elastogram.

Fig. 8 .

 8 Fig. 8. Experiment with a gelatin phantom containing two duck gizzards.

Fig. 9 .

 9 Fig. 9. Evolution through time of the reference experiment. (a) to (d) show the curves of the errors and velocities during the experiment. (e) to (g) show the pose of the ultrasound probe for centering of the object in the FoV and teleoperation of the probe orientation.

Pedro A. Patlan-Rosales is with Inria Rennes-Bretagne Atlantique, Universite de Rennes 1, campus universitaire de Beaulieu, Rennes 35042, France. pedro-alfonso.patlan-rosales@inria.fr

Alexandre Krupa is with Inria Rennes-Bretagne Atlantique, campus universitaire de Beaulieu, Rennes

35042,France.

ACKNOWLEDGMENT

This work has been jointly supported by Inria Rennes -Bretagne Atlantique and the National Council of Science and Technology of Mexico (CONACYT) under the Grant no. 242120/383765.


Feina Xu
email: feina.xu@ensem.inpl-nancy.fr

Sébastien Leclerc

Olivier Lottin

Daniel Canet

Impact of chemical treatments on the behavior of water in Nafion® NRE-212 by 1H NMR: Self-diffusion measurements and proton quantization

Keywords: NMR, self-diffusion coefficients, Nafion, chemical cleaning, counter-cations, protons transport

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

Introduction

During the past decades, a lot of renewable and clean alternative energies has been envisaged in order to replace eventually fossil fuel energy. Among them, electrochemical devices like Proton Exchange Membrane Fuel cell (PEMFC) can be considered as a means to produce competitive and clean energy, since their by-products are only water and heat. The key element of this electrochemical cell is the polymer electrolyte, which ensures both gas separation and transport of protons from the anode to the cathode. Moreover, this electrolyte must possess good (electro)chemical, mechanical and thermal stabilities.

Until now, perfluorosulfonated membranes like Nafion are the most used ionomers in PEMFC because they possess all these properties, and mostly important a high ionic conductivity when hydrated. These properties arise from their chemical structure: a hydrophobic matrix in teflon on which are grafted perfluorovinyl ether chains terminated by hydrophilic sulfonated groups. In general, the latter are compensated by cations to ensure the electroneutrality. As a result, Nafion can be in various cationic forms and the acid form -RSO 3 H is the most common. It is actually in the acid form that the ionic conductivity of Nafion is the most important, since protons can be transported via two mechanisms, vehicle and Grostthus [START_REF] Cukierman | Et tu, Grotthus! and other unfinished stories[END_REF]. The former is proton diffusion through sulfonated sites and the latter is the transport of a proton from one water molecule to its neighbor by modification of the hydrogen bond network.

Although Nafion is the most studied membrane [START_REF] Mauritz | State of understanding of Nafion[END_REF][START_REF] Hickner | Alternative polymer systems for proton exchange membranes (PEMs)[END_REF], results from literature cannot be compared without taking into account the history of the membrane. Indeed, thermal and chemical treatments on the one hand and conditioning on the other hand affect membrane swelling and performances [START_REF] Evans | Role of conditioning on water uptake and hydraulic permeability of Nafion membranes[END_REF][START_REF] Slade | Ionic conductivity of an extruded Nafion 1100 EW series of membranes[END_REF]. In addition, as the hydration state of Nafion in a running fuel cell can vary, due to gas distribution, from almost dry to fully hydrated, this parameter has to be controlled. Since water plays a crucial role on ionomer performances, it seems important to determine its influence in the transport of protons at different scales, in particular for modeling purpose. Thus, an investigation of water dynamics within Nafion should enable us to improve our understanding of the effect of water on the proton transport properties. Among all existing techniques, we have applied 1 H NMR for studying this membrane. 1 H NMR has the advantage to provide information on water dynamics in a confined medium at molecular and microscopic scales, and also, with some precautions, on the amount of water in the sample.

NMR experiments [START_REF] Stejskal | Spin-diffusion measurements: spin-echoes in the presence of a timedependent field gradient[END_REF][START_REF] Tanner | Use of the Stimulated Echo in NMR Diffusion Studies[END_REF] for measuring either relaxation times (T 1 and T 2 ) or self-diffusion of water are well known to be reliable, although the preparation and conditioning of samples may considerably affect their results [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF][START_REF] Gong | Selfdiffusion of water, ethanol and decafluoropentane in perfluorosulfonate ionomer by pulse field gradient NMR[END_REF][START_REF] Nicotera | NMR characterisation of composite polymer membranes for low-humidity PEM fuel cells[END_REF][START_REF] Macmillan | An n.m.r. investigation of the dynamical characteristics of water absorbed in Nafion[END_REF]. As 1 H NMR suffers from the presence of paramagnetic species which may entail ambiguous interpretation, some caution has to be exercised before NMR measurements, in particular about membrane cleanings. However, concerning this latter point, there is no standard protocol specifically designed prior to NMR measurements. As a result, some authors have preferred to use Nafion in its acid form as received [START_REF] Gong | Selfdiffusion of water, ethanol and decafluoropentane in perfluorosulfonate ionomer by pulse field gradient NMR[END_REF][START_REF] Nosaka | Slow morphological change of perfluorinated ionomer as deduced from 1H NMR spectra of confined water[END_REF], others have chosen to clean the membrane using either acid baths [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF][START_REF] Nicotera | NMR characterisation of composite polymer membranes for low-humidity PEM fuel cells[END_REF][START_REF] Ochi | Investigation of proton diffusion in Nafion 117 membrane by electrical conductivity and NMR[END_REF] or both acid and EDTA chelating agent baths [START_REF] Codd | Magnetic resonance microscopy of heterogeneity in polymer electrolyte membranes[END_REF].

Although Nafion was studied in acid form and in fully hydrated conditions, water self-diffusion coefficients varying from 6.110 -6 to 1.510 -5 cm 2 s -1 have been reported in the literature [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF][START_REF] Gong | Selfdiffusion of water, ethanol and decafluoropentane in perfluorosulfonate ionomer by pulse field gradient NMR[END_REF][START_REF] Nicotera | NMR characterisation of composite polymer membranes for low-humidity PEM fuel cells[END_REF][START_REF] Codd | Magnetic resonance microscopy of heterogeneity in polymer electrolyte membranes[END_REF].

In addition, the use of EDTA was suggested by Mac Millan et al. [START_REF] Macmillan | An n.m.r. investigation of the dynamical characteristics of water absorbed in Nafion[END_REF] in order to remove all paramagnetic cations within Nafion, since such species impart an additional relaxation mechanism which can become prominent. However, EDTA in acid form is not soluble in water even at a low concentration, contrary to its salted forms. This means that using EDTA in salted forms (e.g. sodium or potassium forms) provokes the exchange of counter-cations within Nafion with cations from the soaking solution.

Owing to the fact that the nature and size of cations affect Nafion hydration [START_REF] Jalani | The effect of equivalent weight, temperature, cationic forms, sorbates, and nanoinorganic additives on the sorption behavior of Nafion[END_REF], we have to check, via NMR measurements, the effects of EDTA and of counter-cations on the behavior of water in Nafion. In such a complex environment, the main question that we wish to answer is: can surrounding sodium and potassium cations? Answering those questions would allow us to select the more suitable cleaning protocol and also to have a better understanding of protons transport through Nafion in various cationic forms.

In this report, we focus on the dynamics of water within Nafion ® NRE 212, by taking into account different parameters such as the type of counter-cations, the use of a chelating agent, the way of conditioning and preparing samples. For this purpose, we have prepared five series of samples in acid, sodium and potassium forms and possibly treated with EDTA. The samples are studied over a wide range of water content varying from almost dry to fully hydrated by means of self-diffusion coefficient measurements and proton quantization through 1 H NMR. As membrane history is important for understanding the results of this study, we report with some details in the following sections the different chemical treatments of the membrane, the method for determining the drying mass and the hydration time.

Experimental Section

Materials and Sample Preparation

The materials used throughout this work were (a) Nafion ® NRE-212 from Ion Power, (b) salts with purity superior to 99%: LiCl, MgCl In order to remove possible metallic cations present within 5 mm NMR tubes (with screw cap from New Era), the latter were cleaned using a protocol similar to the one described by MacMillan [START_REF] Macmillan | An n.m.r. investigation of the dynamical characteristics of water absorbed in Nafion[END_REF]. All tubes were filled with a mixture of acid solutions (50% of H 2 SO 4 + 50% of HNO 3 , 1mol L - 1 ) for 24h. They were rinsed at least 4 times with distilled water before being immersed in an ultrasound bath in order to remove any trace of impurities. They were then soaked in a solution of EDTA-H 4 (0.015 mol L -1 ) neutralized by NaOH 1 mol L -1 for at least 24h. After that, all tubes were rinsed thoroughly with distilled water before immersing them again in an ultrasound bath.

Thereafter, they were rinsed again and dried in an oven at 60°C overnight.

The membrane Nafion ® NRE-212 is a non-reinforced dispersion-cast film and it can be considered as a new generation of Nafion ® 112. Thus, its EW (Equivalent Weight) is 1100 g equiv -1 and its mean thickness is 50 µm. The membrane sheets, as received, were cut into pieces of 1.1 x 9 cm 2 prior to preparing them for the chemical treatments. All the strips were boiled in H 2 O 2 , 3% wt for 1h in order to remove organics impurities that could come from Nafion synthesis. They were then rinsed in pure water at room temperature before being boiled again in pure water for 1h. After that, all strips were arranged in five groups for being prepared in the following forms: H + , Na + (M1), H + -Na + (M1)-H + , Na + (M2) and K + (M2).

(1) Samples in acid form: H + . The pieces of Nafion were prepared in acid form by soaking in a 2 mol L -1 of HCl solution for 2h at 80°C. They were then rinsed several times with pure water at room temperature prior to immersion in demineralized water for 2h at 80°C. Thereafter, they were acidified again in a 1 mol L -1 of HNO 3 solution for 2h at 80°C. After this last step of acidification, all samples were rinsed thoroughly in pure water at room temperature and all excess of acid within the samples were removed by cleaning twice in distilled water at 80°C for 2h.

(2) Samples in sodium form: Na + (M1) and Na + (M2). Both samples Na + (M1) and Na + (M2) were prepared according to methods 1 and 2 described below. The only difference between these two methods is the choice of EDTA salts. Method 1 makes use of a solution of EDTA-H 4 neutralized with caustic soda. This is because EDTA in the acid form (EDTA-H 4 ) is not soluble in water. Using this method, the soaking solution is generally basic with a pH around 11. Method 2 consists in soaking membranes in an EDTA-Na 2 solution, leading to a pH around 5. With these two solutions, one should be able to detect the influence of pH on self-diffusion coefficients. For both methods, the next steps are similar to the procedure of Mac Millan [START_REF] Macmillan | An n.m.r. investigation of the dynamical characteristics of water absorbed in Nafion[END_REF]. The samples were first acidified according to subsection [START_REF] Cukierman | Et tu, Grotthus! and other unfinished stories[END_REF]. Then, they were soaked in a solution at 0.015 mol L -1 for more than 3 days, either of EDTA-H 4 + NaOH (Method 1) or EDTA-Na 2 (Method 2). Note that the soaking solutions were changed at least 5 times during the complexation step and at each time a large amount of solutions was used (at least 500 mL). Finally, they were thoroughly rinsed with pure water before cleaning them twice in demineralized water at 80°C for 2h.

(3) Samples in acid form and treated with EDTA: H + -Na + (M1)-H + . The samples H + -Na + (M1)-H + were prepared in the same way as samples Na + (M1). They were then acidified with a solution of HNO 3 at 1 mol L -1 , 80°C for 2h and rinsed twice in pure water at 80°C for 2h. At each step, samples were thoroughly rinsed in distilled water at room temperature.

(4) Samples in potassium form: K + (M2). The set up was the same as for the samples Na + (M2), with EDTA-K 2 in place of EDTA-Na 2 .

Following the above described treatments, all samples were dried in an oven overnight at 60°C before being introduced in a rolled form at the bottom of NMR tubes. To prevent a possible humidification of the rolled membranes within NMR tubes during the insertion step, they were dried again at 60°C overnight prior to equilibration at a desired humidity. After that, the dry mass of each piece of membrane was weighed (typical weight for a sample is around 0.1g).

In this work, we had recourse to two methods (dubbed "vapor phase" and "liquid phase") for equilibrating samples. The first one (vapor phase) makes use of water vapor pressure P/P 0 (P 0 : atmospheric pressure) of a hygroscopic solution for hydrating Nafion at room temperature. Each sample in its NMR tube is placed in a sealed jar containing the saturated salt solution. In order to cover a large range of water relative humidity -11, 32, 53, 75 and 98% respectively, we have used the following salts: LiCl, MgCl 2 , Mg(NO 3 ),H 2 O, NaCl and KClO 3 . Thus, knowing the relative humidity imposed by the hygrostatic solutions, we were able to prepare a series of samples at different water contents. These water contents can be converted into λ parameters by means of the sorption isotherm curves. This parameter is defined as the number of water molecules per number of SO 3 -cation pairs and is calculated according to (1) [START_REF] Saito | Mechanims of ion and water transport in perflurosulfonated ionomer membranes for fuel cells[END_REF]:

m(dry) m M EW λ O H O H 2 2    (1)
With M H2O = 18 g mol -1 O H 2 m = m(wet sample) -m(dry) m(dry) = mass of the sample obtained at 60°C. and where EW + is the Nafion Equivalent Weight (EW) corrected for the exchanged cation C + as expressed below.

1) (M EW EW C      (2)  C
M is the molar mass of the cation C + .

Since conditioning samples in vapor phase limits λ to 14 due to Schroeder's paradox [START_REF] Mauritz | State of understanding of Nafion[END_REF][START_REF] Bass | Hydration of Nafion and Dowex in liquid and vapor environment: Shroeder's paradox and microstructure[END_REF], we had to resort to the second method (liquid phase) which consists in swelling membranes with liquid water so as to go beyond this limit. For obtaining a given water content, the excess of liquid was removed from hydrated samples by condensing water at the upper part of the NMR tube with the help of a water bath at 70°C. Water droplets were progressively removed with a soft paper until the desired value of λ is reached.

Chemical characterizations

The thermo-gravimetric analysis was performed using a TGA 2950 High Resolution from TA Instruments. The samples were heated in nitrogen atmosphere from 25°C to 600°C with a scanning rate of 10 °C min -1 . In order to distinguish between free and non-free water in the Nafion matrix, all samples were immersed in liquid water overnight prior to the analysis.

The Infra-red spectra were recorded using an FTIR-ATR from Bruker, equipped with a ZnSe crystal. Before recording an FTIR-ATR spectrum, all samples were dried in an oven at 60°C overnight in order to get rid of water absorption bands.

NMR measurements

Diffusion experiments were performed at ambient temperature on a Bruker Avance DRX-600 spectrometer equipped with a cryogenic probe. The pulse-gradient stimulated echo sequence (PGSTE) method was applied throughout this work, with bipolar sine-shaped gradient pulses [START_REF] Stejskal | Spin-diffusion measurements: spin-echoes in the presence of a timedependent field gradient[END_REF][START_REF] Tanner | Use of the Stimulated Echo in NMR Diffusion Studies[END_REF].

As longitudinal (T 1 ) and transverse (T 2 ) relaxation times vary with the water content in the polymer, the duration of each sine-shaped gradient pulse (/2) and of the diffusion interval (Δ) were chosen according to the relaxation time values: a range of 5 to 10 ms for δ and a range of 60 to 150 ms for Δ. The gradient strength, with a maximum of 50 G cm -1 , was incremented in 32 steps between 2 % and 95 % of this maximum value, each step corresponding to a separate experiment.

Standard proton spectra have been obtained at room temperature on a homemade 200 MHz spectrometer equipped with a 4.7 Tesla Oxford magnet. The same experimental set-up was maintained throughout all experiments. The recycle time (between two consecutive experiments) has been set to five times the longest T 1 (or longer). Moreover, all samples correspond to the same membrane roll height, matched to the receiver coil height. These precautions ensure quantitative comparison.

Characterization of samples prior to NMR measurements

As NMR results depend strongly on the history of the sample, they have to be characterized prior to NMR measurements by the drying temperature, the possible presence of EDTA and the equilibration time of membranes in rolled form.

Choice of the drying temperature

Before hydrating samples, drying mass must be determined in order to calculate the water content or the number of H 2 O molecules per sulfonated groups. A number of methods has been reported for getting a dry mass of Nafion, like drying over P 2 O 5 at room temperature [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF], drying under vacuum for a long period at different pressure (for example 10 -5 torr or 1 torr) either at room temperature [START_REF] Bunce | Proton NMR method for the quantitative determination of the water content of the polymeric fluorosulfonic acid Nafion-H[END_REF] or at a temperature higher than 105°C [START_REF] Bunce | Proton NMR method for the quantitative determination of the water content of the polymeric fluorosulfonic acid Nafion-H[END_REF]. Other authors have mixed both methods by drying theirs samples under vacuum at 50°C in presence of P 2 O 5 [START_REF] Zhang | An NMR study of the state of ions and diffusion in perfluorosulfonate ionomer[END_REF]. One molecule of water per sulfonated group was found by drying under vacuum at room temperature, whereas a value of λ close to zero was calculated by dehydrating Nafion under vaccum at 105°C or over P 2 O 5 at room temperature. However, Yeo et al. [START_REF] Yeo | Physical properties and supermolecular structure of perfluorinated ion-containing (nafion) polymers[END_REF] have demonstrated that there is a residual water of 0.2 wt. % even after drying Nafion under vacuum at 170°C. According to literature data, it seems that reaching an "absolute" drying mass is difficult and depends on experimenters and mostly on the homogeneity of the polymer.

Thermo-gravimetric analysis enables us, on the one hand, to apprehend the decomposition of hydrated Nafion through a range of temperature (25-600°C) and, on the other hand, to select the temperature for drying Nafion. Data from thermo-gravimetric analysis (Figure 1) show four regions of Nafion degradation. The first two regions can be described as the release of two kinds of water: free water released between 25°C and 60°C, non-free water released between 60°C and 260°C.

Non-free water is strongly bound to the ionomer matrix and cannot be released at low temperature.

By increasing temperature, sulfonated groups disappear from 260°C to 360°C and perfluoroether side chains from 360°C up to more than 600°C [START_REF] Lavorgna | Hybridization of Nafion membranes with an acid functionalised polysiloxane: Effect of morphology on water sorption and proton conductivity[END_REF]. From data in Figure 1, we notice that 60°C is high enough to remove almost all free water, the remaining amount (non-free water) being not totally negligible. Indeed, depending on the sample, the latter part represents only 1.8 to 2.1 wt %, thus leading to a λ value which lies between 1.1 and 1.3. Owing to Nafion inhomogeneity, we shall retain λ = 1.2 for all samples which have been dried at 60°C. Thus, for all results given later, the non-free water will be taken into account for the calculation of λ.

Effect of EDTA on Nafion ® membrane

EDTA-H 4-is a large anion which cannot penetrate into Nafion because of the repulsion from SO 3 groupings. However, it may interact with the membrane surface leading to a possible poisoning of the membrane. In order to check this eventuality, FTIR-ATR of Na + (M1) and H + membranes (the latter being devoid of EDTA) were compared. As shown in Figure 2, both spectra are quite similar except in the 1600-1800 cm -1 region and exhibit features due to vibrations of C-F (1202 and 1144 cm -1 ), SO 3 -(1052 cm -1 ) and COC (980 cm -1 ) groups. The peak around 1730 cm -1 observed in the spectrum of sample H + confirms the acid form of the membrane (HOH deformation in oxonium ions), while this band shifts to 1650 cm -1 for the Na + (M1) sample, this latter frequency corresponding to the deformation of water [START_REF] Xu | Chemical modification of perfluorosulfonated membrane with pyrrole for fuel cell application: preparation, characterisation and methanol transports[END_REF]. As no characteristic band of C=O in the carboxylate COO -grouping (expected around 1800 cm -1 ) is visible, we can conclude that EDTA is not present in the sample Na + (M1).

Equilibration time

Samples were conditioned in 5 mm NMR tubes at room temperature with different relative humidity (RH -%). The water content of the samples H + , Na + (M1), Na + (M2) and K + (M2) was determined by weighing after 3 weeks and after 3 months (Figure 3). The comparison of these two curves show that membranes in rolls form require quite a long time to be hydrated properly if placed in NMR tubes. 3 weeks for conditioning the membranes are actually not enough to reach a fully hydrated membrane, in particular at 98% RH. By increasing the equilibration time, we can notice that results in term of water uptake for the acid form are in good agreement with the literature [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF][START_REF] Jalani | The effect of equivalent weight, temperature, cationic forms, sorbates, and nanoinorganic additives on the sorption behavior of Nafion[END_REF]. Although, as reported by Zawodzinski et al. [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF], the equilibration time for Nafion 117 in flat form is around 4 days, it turns out that, for our samples, a much longer time is necessary (hence the three months to make sure that full hydration has been reached). It was indeed reported that several weeks are necessary for hydrating samples in rolled form inserted into 10 mm diameter NMR tubes [START_REF] Perrin | Water dynamics in ionomer membranes by field-cycling NMR relaxometry[END_REF]. This is not surprising since water must penetrate by the membrane thickness rather than by its surface. Even though this method is time consuming, samples in rolled form lead to a better NMR signal/noise ratio, in particular with thin membranes. In addition, from the isotherm curves given in Figure 3(b), it is confirmed that water uptake is related to cation solvation number [START_REF] Mauritz | State of understanding of Nafion[END_REF][START_REF] Jalani | The effect of equivalent weight, temperature, cationic forms, sorbates, and nanoinorganic additives on the sorption behavior of Nafion[END_REF]. Indeed, the solvation number is 51, 41 and 32 for H + , Na + and K + , respectively [START_REF] Jalani | The effect of equivalent weight, temperature, cationic forms, sorbates, and nanoinorganic additives on the sorption behavior of Nafion[END_REF]. Moreover, it seems that the pH of the soaking solutions does not affect the sorption of Nafion in sodium form, regardless of the sample preparation (Na + (M1) and Na + (M2)). Thus, hydration of Nafion depends essentially on the type of cation.

Results and Discussion

Water self-diffusion coefficient measurements

The water self-diffusion coefficient (D) of each sample was measured by the PGSTE method (see the experimental section) [START_REF] Stejskal | Spin-diffusion measurements: spin-echoes in the presence of a timedependent field gradient[END_REF][START_REF] Tanner | Use of the Stimulated Echo in NMR Diffusion Studies[END_REF]. The results shown in Figure 4 exhibit two major trends. The first one, as expected, is that water self-diffusion coefficients depend on the membrane hydration. It actually varies by two orders of magnitude between the lower and the higher water content (λ) for all series studied here. Our results with Nafion in acid form (samples H + and H + -Na + (M1)-H + ) can be compared to those of literature [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF][START_REF] Ochi | Investigation of proton diffusion in Nafion 117 membrane by electrical conductivity and NMR[END_REF]. However the literature water self-diffusion coefficients are scattered, in particular at the lower and higher  values and therefore seem to depend to a large extent on the sample preparation. For example, Nicotera et al [START_REF] Nicotera | NMR characterisation of composite polymer membranes for low-humidity PEM fuel cells[END_REF] reported an increase of D at 20°C from 0.5 to 1.5  10 -5 cm 2 s -1 for extruded Nafion 112 with a water content ranging from 14% to 37% (obtained by weighing). These samples were prepared by immersion in distilled water for 2h to 2 days at room temperature. After removing water droplets on their surface, membranes were inserted into 5 mm NMR Pyrex tube and hermetically sealed. For the sake of comparison with our samples in acid form H + for the same water uptake (28% or λ = 18), we find a D value of 6.210 -6 cm 2 s -1 instead of 1.310 -5 cm 2 s -1 [START_REF] Nicotera | NMR characterisation of composite polymer membranes for low-humidity PEM fuel cells[END_REF]. By contrast, at λ = 2, a value of 610 -7 cm 2 s -1 was reported for a flat Nafion 117 [START_REF] Zawodzinski | Determination of water diffusion coefficients in perfluorosulfonate ionomeric membranes[END_REF] while a value of 810 -8 cm 2 s -1 is obtained here. This comparison emphasizes that both sample preparation and the method of insertion into the NMR tube (in flat or rolled form) have a considerable impact on water uptake and also on the self-diffusion coefficient.

The second trend is that, for a given water content, protons from samples in acid form (the series H + and H + -Na + (M1)-H + ) diffuse more rapidly than those from samples treated with EDTA salts (Na and K). It can be noticed that this behavior is more pronounced when λ is lower than 11.

These experimental observations can be rationalized on the following basis: when the countercation is H + , proton hopping occurs via two mechanisms, vehicle (using sulfonated sites) and Grotthuss (using water network); this feature does not evidently exist when the counter-cation is Na + or K + . Indeed, translational diffusion is favored by proton hopping and this explains why, for a given water content (λ), diffusion coefficients of water in the acid form are systematically higher (by an order of magnitude at low λ) than in sodium or potassium forms. In fact, a similar explanation has been proposed for explaining the evolution of the water self-diffusion coefficients of Nafion 117 in acid and sodium forms, equilibrated in a fully hydrated state [START_REF] Saito | Mechanims of ion and water transport in perflurosulfonated ionomer membranes for fuel cells[END_REF]. Now, in the case of Nafion in acid form, the very small values found for self-diffusion coefficients at low λ can be explained by the ability for a water molecule (or an oxonium ion) to jump from one sulfonated site to another (vehicle mechanism) and/or to proton exchange between water molecules. This latter possibility will become more important at higher λ values, in addition to normal molecular diffusion of water. This explains the increase of D with λ. Concerning water diffusion in Nafion in sodium and potassium forms at low water content, it can be supposed that water is transported by solvation sites, considering that sulfonated groupings make a pair with Na + and K + cations. For a given self-diffusion coefficient of water, Nafion is more hydrated in sodium form than in acid form: for instance, for D ~ 8.10 -8 cm 2 s -1 , λ = 3.5 in the sodium form and drops to 2 in the acid form.

Moreover, from the λ values of Figure 4, it is seen that (i) sodium and potassium behave similarly, (ii) pH has no significantly effect (Na + (M1) and Na + (M2)), (iii) in the acid form, there is no difference between the two treatments (with or without EDTA).

Water quantization from proton NMR spectra

In order to make sure that diffusion measurements are meaningful, it appears interesting to measure the amount of protons in a given sample. For that purpose, the signal intensity of standard Results of NMR measurements and those obtained by the weighing procedure described above are gathered in Figure 5. These results can be summarized by the ratio of the number of protons determined by NMR to the number of protons as obtained by weighing (that is the protons from water molecules). In table 1, we have reported the average of these ratios for the two ways of sample equilibration used in this work and for the different forms (acid, sodium and potassium). At low water content for Na + and K + forms, the uncertainty of NMR measurements renders meaningless the comparison between the two methods (NMR and weighing). Still for Na + and K + forms, at higher water content, we do not observe any significant variation. Conversely, in the acid form, NMR measurements provide systematically higher values. Evidently, this increase is due to the counter-cation H + which is not detected by the weighing method. These counter-cations are in fact chemically equivalent to the oxonium ion and are probably involved in the increase of the proton hopping process, thus in the increase of the diffusion coefficient. Notice that more H + cations seem to appear for "Vapor phase" samples than for "Liquid phase" samples. This can be understood on the following basis: as we are dealing with a ratio and as the amount of H + is constant, the ratio is necessarily larger at small water amount.

Impact of counter-cations on the self-diffusion measurements

Owing to the fact (i) that the number of solvation between sodium and potassium is different; (ii) that the difference between cations is not visible in the usual representation of the water selfdiffusion coefficient vs. water content cannot make, we propose in Figure 6 a different representation (diffusion coefficient vs. relative humidity) which highlights the importance of counter-cations. In contrast with Figure 4, we clearly observe here a specific behavior for each counter-cation. This behavior is simply related to the affinity of cations for water (solvation number) and to the increase of the number of protons involved in the diffusion process.

Conclusion

Water translational diffusion in cation exchange membranes is one of the key parameters as far as the properties of these membranes are concerned. Here we have dealt with the influence of chemical treatments to which Nafion membranes can be subjected. If an EDTA salt is used, this induces an exchange of cations inside the membrane; Na + or K + (according to the nature of the EDTA salt) are substituted to protons of the acid form. Owing to well established properties (confirmed here in Figure 3), the water uptake depends strongly on the nature of the counter-cation. Indeed the main results of this study can be formulated as follows:

1) Diffusion is the same for Na + and K + forms for a given water content.

2) At a given water content, diffusion is faster in the acid form as compared with diffusion in sodium or potassium forms. This feature is noticeably accentuated at low water contents.

3) The quantity of protons has been measured by NMR and compared with the quantity of water obtained by weighing. As H + cations are not seen by the latter method, the NMR method is fully justified in order to emphasize the importance of H + as far as measured diffusions are concerned.

The interpretation of all these data rests on the ability for a water molecule to jump from one site to other and/or for a proton to exchange with another neighboring proton, in addition to the classical molecular diffusion at high water content. The lability of water in the acid form (oxonium ion) is more effective than in Na + or K + forms.

Finally, it turns out that the original representation of figure 6 summarizes well the essential findings of this study and, moreover, emphasizes the importance of counter-cations.

Tables Table 1 Average ratio of the number of protons determined by NMR and weighing for the 5 Series (see text). -0.20 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00

Samples

dm/dT

Weight loss dm/dT

Figure 1 .

 1 Figure 1. Thermo-gravimetric analysis (TGA) of a sample of Nafion ® NRE-212 in acid form.

Figure 2 .

 2 Figure 2. ATR-FTIR spectra of Nafion: (a) H + sample, (b) Na + (M1) sample.

Figure 3 .

 3 Figure 3. Water content of Nafion ® NRE 212 expressed by λ after 3 weeks (a) and 3 months (b) of

Figure 4 .

 4 Figure 4. Evolution of the water self-diffusion coefficients of Nafion of the five different pre-

Figure 5 .

 5 Figure 5. Water content determined by NMR (dots) and weighing (grey) for acid, sodium and

Figure 6 .

 6 Figure 6. Evolution of the water self-diffusion coefficients of Nafion with relative humidity (RH

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Figure 5 .

 5 Figure 5.

1 )Figure 6 .

 16 Figure 6.

  H spectra should be the proper tool. The NMR signal includes protons from H 2 O, H + , H 3 O + , aqueous complexes of water and possibly OH -(from the other form of water), which usually are in rapid exchange. The intensity of the NMR signal was deduced from the integral of each spectrum, eventually corrected by means of a calibration curve. The latter was obtained with a series of H 2 O/D 2 O samples containing a known amount of H 2 O. Furthermore, the liquid contained in these samples has the same height as the membranes in the samples under investigation so that the measurement should provide a reliable value for the water content. Standard precautions regarding quantization by NMR were taken (repetition time between scans, identical spectrometer settings…).

1

H NMR spectroscopy make the difference between water in acid form, in sodium and potassium forms in a confined medium like Nafion? Moreover, can 1 H-NMR techniques distinguish water