Candice Rey

Nicolas Hengl
email: nicolas.hengl@univ-grenoble-alpes.fr

Stéphane Baup

Mohamed Karrouch

Alain Dufresne

Henda Djeridi

Rajeev Dattani

Frédéric Pignon
email: frederic.pignon@univ-grenoble-alpes.fr

Velocity, stress and concentration fields revealed by micro-PIV and SAXS within concentration polarization layers during cross-flow ultrafiltration of colloidal Laponite clay suspensions

Keywords: cross-flow ultrafiltration, micro-PIV, velocity field, shear stress, concentration polarization layer

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

field as a function of the distance z from the membrane surface with respectively 20 µm and 30 µm accuracy, within the concentration polarization layers. The results obtained show an increase of the volume fraction related to a decrease of the velocity within the layers formed during the process. The correlation of these results with the rheological behavior of the suspensions permitted to access to the calculated stress field within the concentrated layers during the cross-flow ultrafiltration process.

Three different regions near the membrane surface have been emphasized with different shear stress/shear rate behaviors. The important role of the rheological behavior and inter particle interaction in the dynamic evolution in space and time of the accumulated layers has been highlighted.

INTRODUCTION

Membrane separation processes are commonly used in several industrial applications, like bio and agro industries, waste and water treatments, and more recently biorefinery. Ultrafiltration process permits concentrating or separating colloidal suspensions, for several applications like food, pharmaceutics, cosmetics, chemical, nuclear, and automotive industries among others.

Although it presents numerous advantages, the stability of the crossflow ultrafiltration process is governed by the increase of particle concentration at the membrane surface due to simultaneous effect of the shear involved by the flow and the pressure forces generated by the transmembrane pressure (TMP). These two external forces in competition with internal colloidal forces lead to the formation of two phenomena named concentration polarization and fouling of the membrane, which decrease the filtration performance [START_REF] Romero | Global-Model of Cross-Flow Microfiltration Based on Hydrodynamic Particle Diffusion[END_REF][START_REF] Song | Theory of concentration polarization in crossflow filtration[END_REF][START_REF] Bowen | Quantitative predictive modelling of ultrafiltration processes: Colloidal science approaches[END_REF][START_REF] Elimelech | A novel approach for modeling concentration polarization in crossflow membrane filtration based on the equivalence of osmotic pressure model and filtration theory[END_REF][START_REF] Bacchin | A unifying model for concentration polarization, gel-layer formation and particle deposition in cross-flow membrane filtration of colloidal suspensions[END_REF]. This phenomenon of concentration polarization layers was evidenced in ultrafiltration processes as well as in others processes like microfiltration processes [START_REF] Rayess | Analysis of membrane fouling during cross-flow microfiltration of wine[END_REF] reverse osmosis [START_REF] Hoek | Cake-Enhanced Concentration Polarization: A New Fouling Mechanism for Salt-Rejecting Membranes[END_REF] as well as direct contact membrane distillation systems [START_REF] Lokare | Fouling in direct contact membrane distillation of produced water from unconventional gas extraction[END_REF] used for treating very high salinity produced water. In ultrafiltration processes, these concentrated layers have a thickness of tens to few hundreds micrometers depending on the filtration conditions and the inter-particles interaction within the filtrated suspension and can generate a decrease of the permeate flux of 50 %. Thus, it is of primary interest to characterize the particles organization and interactions within these accumulated layers (hundreds of micrometers thick) and to understand the mechanisms involved in the concentration phenomena during the ultrafiltration process.

Several previous works have investigated the formation of the concentrated layer during this filtration process and showed an increase of the concentration near the membrane surface in relation with the decrease of the permeation flux [START_REF] Gesan-Guiziou | Stability of latex crossflow filtration: cake properties and critical conditions of deposition[END_REF][START_REF] Espinasse | Filtration method characterizing the reversibility of colloidal fouling layers at a membrane surface: Analysis through critical flux and osmotic pressure[END_REF][START_REF] Bacchin | Colloidal surface interactions and membrane fouling: Investigations at pore scale[END_REF]. Some experimental investigations have probed the deposit formation at different length scales by means of different approaches. Specific in-situ monitoring techniques have been developed and have been reviewed by Chen et al. [START_REF] Chen | In situ monitoring techniques for concentration polarization and fouling phenomena in membrane filtration[END_REF]. To understand the mechanisms of cake formation and fouling mechanisms, some techniques are more dedicated to characterize the particle deposition, thickness of deposit and cake layer formation. In a general way, the whole of the characterization techniques used can be separated in two domains of length scales probed: at micrometric and nanometric length scales.

Firstly, at micrometric length scales some optical non invasive methods like optical, confocal or fluorescence microscopy, have been implemented by means of direct observation through or above the membrane to characterize the organization of the deposited particles [START_REF] Marselina | Characterisation of membrane fouling deposition and removal by direct observation technique[END_REF][START_REF] Huang | A new high-pressure optical membrane module for direct observation of seawater RO membrane fouling and cleaning[END_REF][START_REF] Beaufort | Use of fluorescent microorganisms to perform in vivo and in situ local characterization of microbial deposits[END_REF]. The increase of the concentration inside the concentration polarization layers has been optically investigated by holographic interferometry [START_REF] Fernández-Sempere | Measurements by holographic interferometry of concentration profiles in dead-end ultrafiltration of polyethylene glycol solutions[END_REF] during a microfiltration process. This optical method has allowed quantifying the increase of concentration in the vicinity of the microfiltration membrane before reaching a "pseudo steady state" after 20 minutes of filtration. Others techniques like, nuclear magnetic resonance (NMR) have been put in operation to investigate the phenomena during the filtration of colloidal silica suspensions using a single tubular microfiltration membrane [START_REF] Airey | An investigation of concentration polarization phenomena in membrane filltration of colloidal silica suspensions by NMR microimaging[END_REF]. The authors are able to reach the concentration polarization profile in the filtration device and show that the axial component of the flow is driven by the shear induced cross-flow filtration. Another example of concentration polarization investigation is reported by Donogh et al. [START_REF] Mc Donogh | Experimental in-situ measurement of concentration polarisation during ultra-and micro-filtration of bovine serum albumin and Dextran Blue solutions[END_REF] using two different nonintrusive techniques for different filtration processes. The authors observed a similar phenomenon for different pore sizes and a good match between literature value and measured value concerning the thickness of the concentration polarization layer in each processes for radioactively labelled protein suspensions. More recently, a new optical method using a laser sheet has been developed and validated to characterize in-situ the cake formation in a narrow channel, to obtain the mechanisms of cake growth and better understand the relationship between deposit structure and process performance [START_REF] Mendret | Dead-end ultrafiltration and backwash: dynamic characterisation of cake properties at local scale[END_REF][START_REF] Mendret | An optical method for in situ characterization of fouling during filtration[END_REF][START_REF] Mendret | In situ dynamic characterisation of fouling under different pressure conditions during dead-end filtration: Compressibility properties of particle cakes[END_REF]. New microfluidic devices in poly-dimethyl siloxane have also been developed for dead-end and cross-flow filtration, which allow direct observation of clogging dynamics in microchannels (20 µm) wide by micrometric particles (5 µm) [START_REF] Agbangla | Experimental investigation of pore clogging by microparticles: Evidence for a critical flux density of particle yielding arches and deposits[END_REF][START_REF] Sendekie | Colloidal Jamming Dynamics in Microchannel Bottlenecks[END_REF]. The results allow distinguishing different clogging behaviors according to filtration conditions and considering the different interaction (particle -particle and particle -wall) colloidal interactions. At larger length scales, direct optical observation of the surface of a single hollow fiber membrane have been performed during cross-flow filtration of monodisperse polymer model particles with warying surface charge densities [START_REF] Lorenzen | Direct observation of fouling phenomena during cross-flow filtration: Influence of particle surface charge[END_REF]. Some recent measurement by fluid dynamic gauging have investigate in-situ the thickness and cohesive strength of cake fouling layers formed during cross-flow microfiltration of a softwood kraft lignin model system [START_REF] Mattsson | The use of fluid dynamic gauging in investigating the thickness and cohesive strength of cake fouling layers formed during cross-flow microfiltration[END_REF].

Secondly, at nanometer length scales, some ex-situ or in-situ scattering techniques, like small-angle neutron scattering (SANS) as well as small angle x-ray scattering (SAXS) have been used to probe the structural organization of colloids during the filtration processes. Thanks to the development of a filtration cell dedicated to neutron scattering, in-situ SANS measurements have allowed monitoring the in-situ development of the fouling layer inside the membrane pores [START_REF] Su | Application of Small Angle Neutron Scattering to the in Situ Study of Protein Fouling on Ceramic Membranes[END_REF]. The mechanisms of cake collapse in dead-end filtration have been studied and elucidated by filtration of aqueous dispersions of latex particles aggregated by CaCl2. Ex-situ SANS experiments on filtered cakes in different pressure conditions have allowed to show that the mechanisms of cake collapse are made of very small relative motions of the particles, which leave the local coordination of the latex particles unchanged but allow large voids to be reduced [START_REF] Antelmi | Cake Collapse in Pressure Filtration[END_REF]. In the same manner SANS and transmission electron microscopy have been linked to determine the process by which the particles move and reorganize during cake collapse [START_REF] Madeline | Restructuring of Colloidal Cakes during Dewatering[END_REF]. Combination of static light scattering , SANS and local birefringence techniques have been used to quantify the inner structure, structural orientation and organization of deposited colloid fouling layers on the membrane surface during dead-end filtration of Laponite clay dispersions [START_REF] Pignon | Structural characterisation of deposits formed during frontal filtration[END_REF]. Some developed dead-end filtration cells, simultaneously applying a transmembrane pressure and a magnetic field have allowed to simultaneously characterize by SAXS, the structure of particle deposit during ultrafiltration, manipulate the particle orientations and enhance the filtration performance [START_REF] Pignon | Small-Angle x-ray scattering studies of Femontmorillonite deposits during ultrafiltration in a magnetic field[END_REF]. On the same principle custom made ultrafiltration dead-end cells have allowed get access to the in-situ structural organization of casein micelle suspensions by SAXS and the corresponding spatial and temporal in situ concentration profile evolution during dead-end filtration [START_REF] Pignon | Structure and rheological behavior of casein micelle suspensions during ultrafiltration process[END_REF][START_REF] David | Spatial and temporal in situ evolution of the concentration profile during casein micelle ultrafiltration probed by small-angle x-ray scattering[END_REF].

On the bases of these dead-end ultrafiltration cells, dedicated to SAXS measurements, some others cross-flow ultrafiltration cells and ultrasound coupled cross-flow ultrafiltration cells have been designed to perform in-situ SAXS. Thanks to these cells, concentration profiles and structural organizations have been explored as a function of the distance z from the membrane surface, for different colloidal systems, in different cross-flow filtration / ultrasound conditions [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF][START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF][35][36].

Apart from these structural characterization of the deposited particles near the membrane surface, some measurements or numerical modelling have been done to access to the velocity fields in filtration processes [37][38][START_REF] Arndt | New Insights into Sodium Alginate Fouling of Ceramic Hollow Fiber Membranes by NMR Imaging[END_REF]. Nevertheless some important information like the velocity field near and inside the concentrated layer formed during crossflow ultrafiltration and the corresponding local shear stresses are still missing to overcome the phenomena involved during the filtration. This knowledge is necessary to develop the theoretical models and improve the understanding of stability of the ultrafiltration process and the related mechanisms responsible of the concentration polarization phenomena. The objective of this study is to use in-situ scattering techniques and develop micro-PIV investigations to get access to concentration profiles, velocity fields and deduce the corresponding calculated shear stresses inside the accumulated layer during membrane separation processes. New cross-flow ultrafiltration cells were designed to perform firstly time resolved in-situ small angle X-ray scattering (SAXS) and secondly time resolved in-situ micro particle image velocimetry (micro-PIV) at the vicinity of the membrane surface during the filtration. The SAXS measurement performed in-situ during ultrafiltration of Laponite clay suspensions allowed characterizing the structural organization at the nanometer length scales and concentration profiles measured as a function of the distance z from the membrane surface with 20 µm accuracy. These structural and concentration information's have been linked to the permeate flux, cross-flow and transmembrane pressure registered simultaneously. In the same manner, the micro-PIV measurements have allowed to establish the velocity field evolutions and corresponding shear rate as a function of the distance z with 30 µm accuracy, within the concentration polarization layers and during filtration time. These concentration and shear rate evolutions in space and time, were then correlated to the rheological behavior of the suspensions, in order to calculate the corresponding stress field evolutions within the concentration polarization and fouling layer during the cross-flow ultrafiltration process.

MATERIALS AND METHODS

Sample preparation

The clay used in this study was Laponite XLG which comes from Laporte Industries. It is a synthetic disk-shaped Laponite 30 nm in diameter and 1 nm in thickness. The particles have a density of 2.53 g.cm -3 . The suspension was prepared by mixing the Laponite powder with demineralized water at 20 °C with a fixed ionic strength of 10 -3 M NaCl and pH = 10, following the protocol used by Jin et al. [START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF]. Suspensions to be filtered were prepared at a volume fraction of 1 vol%. The suspension aged in closed bottle during 12 to 26 days before the filtration experiments. As shown before the structure and the rheological behavior evolves during the aging time and the value of the yield stress increases for increasing aging time [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF][40][41][42][43]. Changes were observed in the viscoelastic properties and structural characteristics of the gels over time. This is due, in part, to the osmotic swelling caused by repulsion between the double layers and also to the progressive organization of the particles into fractal aggregates over larger length scales as already studied in details [40]. are able to identify the presence of two distinct arrested states. While repulsion is felt almost immediately after samples are prepared, attraction, due to its anisotropic nature and to the presence of an effective repulsive barrier, develops on a much longer time scale. More recently [43] this aging dynamics was studied as a function of solid content, salt, concentration and pH, and discuss in term of the edge to face or face to face contact between the particles. Consequently, to pass over this aging phenomena, the time tp that has elapsed between the end of the preparation and the different structural or filtration investigations will always be indicated in the following. For the different kinds of measurements (rheometric behavior, cross-flow ultrafiltration) the results will be compared at the same tp time.

The interaction can be controlled by the addition of a peptizer which reduces the positive charges and change the global charge of the particles to be repulsive. The peptizer used in the present study is the tetrasodium diphosphate Na 4 P 2 O 7 (tspp) and has a molecular weight of 446 g.mol -1 . The effect of the peptizer on the rheological behavior and the structure has already been studied in previous works [START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF]40]. The reduction of the interaction leads to a change in the rheological properties from yield stress fluids to shear thinning fluids behavior with the addition of peptizer. Several suspensions were prepared for the rheometric measurements denoted L(i)-Cp(0 or 6), with i corresponding to the volume fraction content, and 0 or 6 corresponding to the Cp concentration in peptizer equal respectively to 0% or 6% of the mass fraction of dry clay.

Rheometric measurements

Rheometric measurements were carried out using a shear rate controlled rheometer (ARG2, TA Instrument) with a cone -plate geometry of 49 mm in diameter and with an angle of 4°21' in the same conditions as [START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF]. In order to avoid interfacial effect, the surface of the plate geometry has been covered with sand paper with a roughness of 200 µm. Measurements were performed at a fixed temperature of 25±1 °C. An evaporation proof system has been added on the top of the geometry to ensure the saturation of the atmosphere of the sample.

SAXS cross-flow filtration: cell and filtration procedure

A SAXS coupled to cross-flow filtration cell was designed to measure in-situ concentration profiles and structural organization of the colloidal suspensions during the time of filtration (Fig. 1 a,b). This information is obtained as a function of the distance z from the membrane surface. The cell was made of transparent polycarbonate and the membrane was a flat sheet of polyethersulfone of (100 x 4) mm (100 kD, Pleyade, Orelis environnement). The filtration cell is composed of two polycarbonate pieces with a retentate channel for the upper part and the permeate channel for the lower part. The retentate channel was 100 mm long in x cross-flow flow direction and the flow section was 7.4 mm z high for 4 mm y large. Three windows of observation were positioned at the entrance, the middle and the outlet of the cell at 43 mm of each other. These windows had a dimension of (3 x 5.5 x 0.3) mm, respectively, in width, height and thickness. The permeate channel was 100 mm long in x cross-flow flow direction and the flow section was 10 mm z high for 4 mm y large. The membrane is strengthen between these two upper and lower parts of the cell. The z c = 0 position of the filtration cell, correspond to the edge of the membrane in contact with the permeate lower part of the filtration cell.

During the filtration, the feed suspension was pumped (Mono pump LF series, Axflow) from a high pressure tank (Millipore). Pressure was applied to the rig via purified compressed air and the retentate pressure P was measured at the inlet and outlet of the cell with a pressure gauge (FP 110 FGP Sensors & Instrument). Cross flow flux was continuously measured (Optiflux 6300C flowmeter, Krohne). The permeate flux was recovered and weighed during the time with an accuracy of 0.001 g every 5 s (balance Precisa 400 M). During the filtration the temperature was monitored by a cryo thermostat (Thermo & Scientific SC, 150 A25, HAAKE) and fixed at 25 °C. Filtration procedure with several steps was performed for the Laponite suspensions as described by table 1 for each system. To emphasize the phenomenon about the formation of the concentration layer during the ultrafiltration process, only the last filtration step at 0.06 L.min -1 concerning these two kinds of suspension will be presented in detail.

In-situ SAXS cross-flow filtration: conditions and analysis

SAXS experiments were performed at the European Synchrotron Radiation Facility (ESRF, Grenoble, France), at the ID02 TRUSAXS beamline. The incident X rays beam had a wavelength of 0.1 nm and was highly collimated with a corresponding Full Width at Half Maximum (FWHM) (20 µm (FWHM) vertically and 400 µm (FWHM) horizontally) to reach a high spatial resolution in the vertical direction. The sample-to-detector distances were fixed to 2 m and 10 m. The SAXS measurements covered the following scattering vector range: 2.10 -2 nm -1 ≤ q ≤ 2 nm -1 , q = (4π/λ) sin(θ/2) and θ is the scattering angle. The corresponding (λ = 2π/q) length scale range is 3.14 nm ≤ λ ≤ 314 nm. The filtration cell was mounted on motorized stages. A rotational stage was used to align the incident beam parallel with the membrane surface. A vertical translation stage allowed moving the filtration cell in front of the beam, in order to probe the structure at different vertical position. The incident beam passed through the sample in the filtration cell and the two-dimensional scattered intensity patterns were recorded on a high-resolution CCD detector (Fig. 1 a,b).

The analysis of the SAXS patterns registered in the cross-flow cell is the same which has already been described in preceding works [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF][START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF]. Four different zones were defined by measuring the transmitted X-ray signal as a function of the distance z c through the SAXS cross flow filtration cell. The description of these zones has allowed to define the minimal distance z c above which the scattered intensity is not influence by the X-ray beam partially crossing the polycarbonate permeate channel and the membrane. The detection of this minimal z c position for which the scattered intensity can be properly registered (without influence of the polycarbonate permeate channel and the membrane) is fixed at zero (z c = 0). Thanks to a new design of the windows of observation and also with the use of the angular motion stage, the minimal distance above this z c position for obtaining exploitable SAXS data was reduce to 20 µm. To clarify the location of the zero position corresponding to the edge of the membrane surface, the description according to (Fig. 1 c) is the following: four areas are defined depending on the vertical distance in the filtration cell. A first zone Z1 correspond to the permeate channel in polycarbonate matter. The X-ray beam is totally absorbed in this region. A second area Z2 is defined as being the membrane since a transmitted signal is received. This means that the X-ray beam starts to pass through the membrane, positioned onto the permeate channel. The signal increases for increasing vertical distance. In the third area Z3 the signal received is provided by the sample but is still affected by the membrane signal. This area measures a determined thickness depending on the beam width, here 20 µm. When the scattering intensities received are constant, any influences from the membrane are detected and the area Z4 corresponding to the measure area is configured. So this area Z4 starts from 20 µm above the membrane surface in our experiments and the z c = 0 position is fixed at the bottom of the area Z3 (Fig. 1 c).

Scattering patterns were normalized to an absolute intensity scale after applying standard detector corrections and then azimuthally averaged to obtain the one-dimensional intensity profiles denoted by I(q). Before the filtration experiments the cell was filled with demineralized water and the normalized background scattering associated was systematically subtracted to the scattering of suspensions during filtration experiments [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF][START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF].

An initial investigation of 5 suspensions of known volume fractions was performed in a flow through capillary cell 2 mm in diameter. This step allowed establishing the calibration curve of the absolute intensity as a function of the volume fraction. The results of the scattered intensity as a function of volume fractions allowed us to define the following linear relationship I (q = 0.6 nm -1 ) = 0.192 φ v [START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF].

This equation has been established in the linear zone of the scattered intensity (q -2 power law decay) corresponding to the form factor of the dispersions, which indicates that it was not affected by the increasing mutual particle interaction (described by the structure factor of the dispersions) and therefore valid at high dispersion concentrations [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF]. Indeed, the size of the Laponite clay is 30 nm in diameter and 1 nm in thickness. This size corresponds to q range, which defines the form factor of the scattering curve. This form factor does not vary with particle concentration. But at smaller q vectors, (larger distances than the particle sizes) the aggregation phenomena could change the scattering intensity as a function of the particle concentration (in a q vector range which is attributed to the structure factor) [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF]. To properly extract the concentration from the scattering intensity, the analysis as to be done in the q range corresponding to the form factor, to determine the volume fraction as a function of the distance from the membrane.

Then, with the obtained calibration curve, the volume fraction of any Laponite suspension could be determined by introducing the corresponding absolute scattered intensity at a scattering vector q = 0.6 nm -1 . This method has already been used [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF][START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF][35][36] to determine the evolution of the volume fraction of the dispersions as a function of the distance z from the membrane. In other words, the concentration profiles in the accumulated layers during filtration were determined by this method. The scattering intensity for all the volume fractions reached within the concentrated particles layers at scattering vector q = 0.6 nm -1 were in the q range where the scattering intensity is not affected by neither the structure factor nor the possible anisotropy of the SAXS pattern, as already discussed in a precedent work [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF]. Following these results, in-situ SAXS measurements during Laponite clay suspensions filtrations were performed.

In-situ velocity field measurements during cross-flow ultrafiltration process

Some relevant precedent work have proposed a methodology to predict pressure and velocity field in hollow fiber network of a filtration module [START_REF] Arndt | New Insights into Sodium Alginate Fouling of Ceramic Hollow Fiber Membranes by NMR Imaging[END_REF]. The results were focalized on the effect of geometry of the module and the membranes, the membrane permeability as well as the operating filtration conditions. The effects of shear forces induced by crossflow velocities on ex situ dead end filtration formed alginate fouling layer, were studied by crossflow experiments using multi slice multi echo imaging (MSME). The flow velocity measurements were performed to gain more insight into the hydrodynamics in the fouled membranes [START_REF] Arndt | New Insights into Sodium Alginate Fouling of Ceramic Hollow Fiber Membranes by NMR Imaging[END_REF]. Nuclear magnetic resonance imaging (MRI) combined with MSME flow velocity measurements allowed further characterization of alginate layer structure and was used to elucidate the influence of Ca 2+ on the fouling layer structure for alginate filtration within ceramic hollow fiber membranes. However, until now not any direct in-situ measurements of the flow field within the accumulated layers near the membrane surface during cross flow ultrafiltration, have been reported. Consequently, in this goal, a new cell was designed using Plexiglas (PMMA) to allow a complete visualization of the flow field along the feed channel. Micro-PIV technique was used to quantify the velocity field near the membrane surface during the filtration of Laponite suspensions. The dimensions of the feed channel were the same as for the cell used for SAXS filtration coupled experiments, The retentate channel was 100 mm long in x cross-flow flow direction and the flow section was 7.4 mm z high for 4 mm y large. The design of the cell allows performing micro-PIV measurements all along the length of the channel in the cross-flow flow directions (Fig. 1d). Then velocity profiles have been characterized in the same conditions and at the same x positions as those used for SAXS experiments. Micro-PIV measurements were performed with high-resolution camera synchronized with a pulsed laser. The laser used was a Nd:YAG pulsed with a wavelength of 532 nm and a power of 50 mJ/pulses (New Wave Solo). A specific lens was used to generate the thickness of the laser sheet depending on the flow and geometric characteristics. The thickness of this laser sheet was 1 mm. The laser sheet has been positioned in the middle of the canal in the y direction. Although the camera has a resolution of 2657 x 4000 pixels (Hisense 11M) with a cadency of 15 Hz, a visualization area of 2657 x 2000 pixels was used to optimize the storage size of the pictures. Polystyrene fluorescent particles 5 µm in diameter (microParticle GmBh), with an absorbance and emission wavelength of 532 nm and 640 nm, respectively, were included in the filtrated Laponite suspension. Fluorescent particles were used in order to inhibit reflection phenomenon near the wall and surface membrane. The choice of tracers was the most suitable for our experiments and the Stokes number was 8.6 x 10 -4 . The concentration in polystyrene fluorescent particles has been optimized to avoid any change in filtration performance or rheological behavior of the Laponite suspensions, and simultaneously permit suitable signals detected by the camera to get reliable correlation functions and velocity measurements.

The optical device was composed of a Canon 65 mm MPE lens linked to a filter (LaVision) with a range of 545 to 800 nm. The device is parameterized in order to the field of view be focused in the middle of the channel thickness in the y direction, corresponding to the laser sheet position. PIV processing requires a careful selection of interrogation window size to take into account robustness and accuracy for the calculation of the velocity especially in regions of high gradients. Adaptive PIV approach leads to obtain the optimal local interrogation window size and shape based on flow gradients and image quality. The interrogation window size is varied according to the local flow gradient and the correlation value. Couples of images have been taken during the filtration time with a time laps between the first and the second image of couples corresponding of 150 µs in average, taking into account the order of magnitude of the velocity (about 0.06 m.s -1 ) to a particles displacement of 8 pixels on average, according to literature about that parameter [38]. Taking into account the order of magnitude of the velocity (about 0.06 m.s -1 ) the time lap chosen corresponds to 150 µs in average. This time must be small enough to obtain a valid measure of the velocity and a certain length of displacement in the window of measure is required to quantify the velocity field. If the time laps are too high, particles will not be detected by the software from one picture to the other, and if the time laps are too small, the particles will not move enough and any displacement will be detected.

Commercial software "DynamicStudio" (Dantec) was used to extract the mean velocity fields from the recorded image frames. The processing principle is based on cross-correlation analysis of the particleimage patterns in small sub-domains between two successive image frames. To convert the pixel space into physical space, a calibration was performed using a specific target. Adaptive PIV processing requires a careful selection of interrogation window size to take into account robustness and accuracy for the calculation of the velocity especially in regions of high gradients. Adaptive PIV approach leads to obtain the optimal local interrogation window size and shape based on flow gradients and image quality. The interrogation window size is varied according to the local flow gradient and the correlation value.

The number of image frames (200 pairs of images) allowed a good convergence for statistical values of the velocity field. Due to high spatial-resolution of the camera, the measurements of the mean velocity components allowed an accurate quantification of the velocity gradients and the strain rate tensor each 30 µm of distance above the membrane surface. An example of velocity profile obtained by this method is presented in Fig. 1e.

RESULTS AND DISCUSSIONS

Rheometric measurements

From rheometric measurements, the flow curves (viscosity and shear stress as a function of the shear rate) are presented in Figure 2 as a function of volume fraction and tspp content. The rheological behavior of the Laponite suspensions without tspp from 1 vol% to 3 vol% exhibits a shear thinning behavior: the viscosity decreases while the shear stress increases with an increasing shear rate. The yield stress behavior of the suspensions above a certain volume fraction is emphasis on (Fig. 2b) at low shear rates 0.1 s -1 , the stresses reach a plateau value corresponding to the yield stress of the suspensions. When the peptizer is added to the suspensions interparticular interactions are turned to globally repulsive via linking to positive charges on the surface of the particles [40]. As discussed before, the effect of this peptizer on the structure, rheological behavior, osmotic pressure and dead-end filtration performance of the Laponite dispersions have been studied in details in our precedent works [START_REF] Pignon | Structural characterisation of deposits formed during frontal filtration[END_REF][START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF]40]. The diphosphate anion binds the positive surface charge of the edges of the platelets. This has the consequence to reduce the strength of edge-face and edge-edge attractions between the particles. The result of this reduction in attractive forces between the particles is a partial disruption of the network. As a consequence, the yield stresses measured for the same volume fractions are reduced when the peptizer is added to the suspensions (Fig. 2b). For example, at 2 vol% the yield stress without peptizer (L2-Cp0) is equal to 105 Pa and with peptizer (L2-CP6) the yield stress is reduced to

Method for calculation of the shear stress profile within the accumulated layers, during cross-flow ultrafiltration

In the aim to calculate the stresses reach within the concentrated layers during the cross-flow ultrafiltration process, an analysis of the rheometric flow curves (Fig. 2) has been made. This will allow relating the shear rate to the shear stress by rheological laws. From the shear rate measured by micro-PIV within the concentrated layers during the cross-flow ultrafiltration process, these equations will allow to calculate the corresponding shear stress within the concentrated layers. The linear regression of the experimental rheometric data (Fig. 2b) gives power law (eq. 1) or Herschel-Buckley (eq. 2) laws depending on the volume fraction and peptizer content:

φ = φ . φ
(eq 1)

φ = φ . φ + φ (eq 2)
Where φ is the volume fraction (vol%), τ φ is the shear stress (Pa), the shear rate (s -1 ), K (φ) is the consistency, n(φ) the shear thinning index, and τ φ is the yield stress.

From the fitted laws, the expressions of K and n and τ as a function of the volume fraction are deduced (eq. 3 to 8).

For Laponite suspensions with peptizer, the expressions for K, n and φ are the following:

φ = × φ . (eq. 3) φ = × φ² - × φ + . (eq. 4) φ = . × × φ .
(eq. 5)

For Laponite suspensions without peptizer, the expressions for K, n and φ are the following:

φ = = .
(eq. 6)

φ = = .
(eq. 7)

= . × × φ .

(eq. 8)

From the velocity profiles (V(z)) obtained by micro-PIV the following expression (eq. 9) gives access to a calculation of the corresponding shear rate:

= ! "
(eq. 9)

With z the distance from the membrane surface (m), and V the measured velocity (m.s -1 ).

For each filtration condition during time and distance z from the membrane surface, it is then possible to calculate the shear stress within the concentration polarization (eq. 10), thanks to the knowledge of i) the value of the volume fraction measured by SAXS, ii) the value of the calculated shear rate deduced from the micro-PIV (eq. 9) and iii) the rheological laws (φ) (eq. 3 to 8). It is important to notice here that one assumption is made which consider that rheological behavior of concentration polarization and gel layers of Laponite in the cross-flow experiment is equivalent to those of bulk Laponite suspensions having same particle concentration. It will be difficult to verify this assumption as any direct measurements with a specific probe, introduce in the flow, will modify the measuring stresses.

φ, = φ . φ + φ, (eq. 10)

For Laponite suspensions without peptizer, the rheological behavior is shear thinning only for 1 vol%.

For higher volume fractions, the expression of the shear stress (eq. 10) can be replace only by the expression of the yield stress corresponding to this kind of suspension, because n (φ>1) = 0 and γ z & φ'( = 0 . So for φ > 1 vol%, (eq. 10) equals (eq. 8). 3a). In a second step, under the same pressure conditions the cross-flow has been reduced to 0.06 L.min -1 in order to emphasis the phenomena in a regime for which critical conditions (limiting flux) are reached as shown previously in details in previous works [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF]. The corresponding concentration profiles (Fig. 3b) deduced from the SAXS measurements (Fig. 3c andd) show a continual increase in concentration as a function of z and filtration time. On (Fig. 3 c) are plotted the scattering intensities I(q) as a function of the q vector for different distances registered in-situ inside the crossflow ultrafiltration cell as a function of distance z, at filtration time t = 94 min of step 2. This curve show the increase of the whole scattering intensity curve at decreasing distances z toward the membrane in relation with the increase of particle concentration. The corresponding Kratky plot representation (q²*I(q)) is plotted in (Fig. 3d). Kratky plot is another way to represent scattered intensities, emphasizing the variation of scattering intensities (slopes and local maximum) as a function of q vectors. Peaks can be observed at a certain q max wave vector which correspond to a maximum of scattering intensity. This q max position correspond to a mean inter-particle distance d = 2π/q max [START_REF] Pignon | Structural characterisation of deposits formed during frontal filtration[END_REF][START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF] Fig. 3d show that the q max position is increasing at decreasing z distances which emphasizes the fact that inter-particle distances d decreases as concentration increase toward the membrane surface.

In-situ

b) Lowering of the membrane. In a third step, to explore the relaxation phenomenon during time and the dynamic of the change in structural organization and concentration evolution, the cross-flow was stopped and the pressure was released. In this arrested state, the SAXS measurements were continuously registered as a function of the distance z c and time (Fig. 4 a). Thanks to this relaxation step, a specificity of our experimental setup was emphasized. In our experimental set-up the membrane is not supported by any porous rigid media, which has induced an effect of lowering of the membrane in the permeate channel due to the effect of transmembrane pressure. Fig. 4 a compares the scattering intensities inside the accumulated layers just before and after the time when the cross-flow was stopped and the pressure was released. A sudden jump in the scattering intensity in the upper distances has been highlighted. This sudden increase of the corresponding calculated concentration profile is due to the fact that without transmembrane pressure, the membrane has now the ability to destress and recover its initial flat form, which allowed now to let appear in front of the beam from z c = 0 position, the whole concentration profile from its surface. In this relaxed configuration the z c = 0 position correspond to the distance z from the membrane surface as the membrane is not lowered anymore. Many measurements on different samples have allowed checking this phenomenon of lowering of the membrane under pressure and de-stress of the membrane when the transmembrane pressure is released. All the measurements gives the same sudden change of the concentration profile which is shift 350 µm above the z c = 0 position. In order to take into account this lowering of the membrane, all the results presented thereafter, under transmembrane pressure, will be plotted as a function of the distance z from the membrane surface named z (µm) in the graphs, where z = z c + 350 µm.

The consequence of this observation is twofold: firstly it has allowed us to evidence that the first layers deposited near the membrane surface during filtration reach higher concentration levels and better orientated states than initially detected (Fig. 3). Secondly, it has allowed us to define the effective z distance from the membrane surface (z = z c + 350 µm). In order to take into account this , for the Laponite suspension without peptizer. The concentrated layer is thicker and the volume fraction reaches a higher value than for Laponite without peptizer. As for a suspension of Laponite with peptizer, the SAXS patterns after 81 min of filtration (Fig. 5b) show an increase in the orientation for closer distances from the membrane surface during the process. It is worth noting that the concentrated layer is thicker for the suspension without peptizer due the ease of the particles to create aggregates. This observation confirms the role played by the interparticular interactions and the rheological behavior in the build up of the concentrated layers.

Velocity field deduced from the in-situ micro-PIV

A first set of experiments was conducted to characterize the permeation flux evolutions as a function of cross-flow conditions at increasing transmembrane pressures. The goal is to evaluate if the limiting flux is reached or not. In this first experiment, a constant cross-flow is maintained and an increasing transmembrane pressure was applied for the dispersions with and without peptizer. For each transmembrane pressure applied, the permeation flux was recorded until reaching a steady state value after an average value of 25 minutes. The steady state permeation flux J is plotted in Figure 6 for the different cross-flow conditions investigated in this article. For all these conditions the limiting flux is reached for a higher permeation flux of about 6 L.h -1 .m -2 for the higher cross-flow conditions (Q = 0.3 L.min -1 ) compared to about 2.5 L.h -1 .m -2 for the lower cross-flow conditions (Q = 0.06 L.min -1 ). In the following some measurements of the shear velocity field have been performed for all these cross-flow conditions, and give the same features. In order to keep concise presentation, only the results at (Q = 0.06 L.min -1 ) are presented. The same phenomena are observed during the filtration step 1 with a flow rate of Q = 0.3 L.min -1 for the Laponite suspension with peptizer, and the filtration step 1 and 2 for the Laponite suspension without peptizer. The velocity profiles show the reduction of the velocity near the membrane surface for increasing time of filtration and show the formation of the concentrated layers during the time of filtration and as a function of the flow rate. Indeed, when the flow rate is decreased, the velocity profiles show a higher area with any measurement of flow than for a high flow rate.

Suspensions of Laponite with peptizer:

Because of previous filtration steps to highlight the effect of the hydrodynamic condition on the formation of the concentrated layers, the velocity profile at the beginning of the filtration step 2 at Q = 0.06 L.min -1 , does not show any velocity at the surface of the membrane and an area of no velocity of 700 µm above the membrane surface (Fig. are lower than the one in the more concentrated stagnant layer. This emphasis the fact that the most important resistance inducing a flux reduction is due to the stagnant concentrated layer near the membrane surface. This result agrees with SAXS experiment results which show an increasing volume fraction near the membrane surface for increasing filtration times. We can conclude that the reduction of the velocity is linked to the increase in volume fraction, which conduct to an arrested state of the suspensions in the most concentrated layers near the membrane surface.

Suspensions of Laponite without peptizer: in the same way, for the last filtration step3 with a low flow rate of Q = 0.06 L.min -1 (Fig. 7 c) the same phenomenon is observed. The region of no velocity has grown for increasing filtration times, even if the permeate flow J has reached a permanent state (Fig 7 d). At the end of the filtration, a concentrated layer of 2100 µm (higher than the one reported with tspp) is formed at the membrane surface. flow rate, corresponding to a limiting flux previously described (Tab. 1 and Fig. 6).

Shear rate fields within accumulated layers during cross-flow ultrafiltration

These shear rate profiles exhibit three different evolutions as a function of the distance z from the membrane surface. Firstly, a zero value corresponding to the area of no velocity near the membrane surface is observed as shown in Fig. 7. Secondly, an increase in the shear rate as a function of z is reported (Fig 8a andb) which corresponds to the increase in the velocity until reaching a maximum.

Thirdly a decrease in the shear rate until reaching a lower value corresponding to an area where the velocity field reaches middle of the cell in z position is noted. For higher filtration times, the observed phenomena are the same with a difference in the thickness of the area of no flow; a change in the maximum value and position of the shear rate, which is in accordance with the fact that the concentrated layer becomes thicker for increasing filtration times and decreasing flow rate, in agreement with the previous SAXS and micro-PIV results.

For the suspension with peptizer, Figure 8 a shows a high shear rate of about 140 s -1 near 700 µm above the membrane surface at the beginning of the filtration step (t= 0 min), which corresponds to the shear of the suspension at the initial filtered concentration (φ = 1 vol %). Indeed, at 700 µm above the membrane surface, fig. 5a shows for time t = 0 min of this filtration step, that the concentration is the initial value of the suspension, which means that any increase in concentration is detected at this distance on z direction. The shear properties of Laponite suspension correspond to a shear thinning behavior with a shear thinning index n = 0.4 according to eq. 4. After 60 and 90 min filtration, the maximum value of the shear rate decreases and is positioned at increasing z levels. This evolution is in accordance with the formation of a higher concentrated and thicker layer during the filtration process as shown in Figures 5 and7.

For the suspension without peptizer, for the same filtration conditions the maximum of the shear rate profile is positioned at higher z position (1500 µm above the membrane surface) than with peptizer, and the shear rate maximum value (60 s -1 ) is lower than with peptizer. This is in accordance with the fact that the filtration of the Laponite suspension without peptizer forms a thicker and more concentrated layer than without peptizer, as shown in Figures 5 b and7 c. Nevertheless, at increasing filtration times the maximum value of the shear rate increases, contrary to the case with peptizer. This emphasises a more complex phenomenon certainly related to the more complex thixotropic rheological of the Laponite dispersions without peptizer. One way to better understand these evolutions is to link the shear rate, velocity and concentration profiles to the rheological behavior of the suspensions as a function of their volume fraction.

Shear stress fields within accumulated layers during cross-flow ultrafiltration

To understand the concentration polarization phenomena, we will take advantage of the combined results obtained by SAXS and micro-PIV on the same suspensions in the same filtration procedures which have allowed to get access to the velocity shear rate and concentration profiles as a function of the distance z from the membrane surface. Combining all these results with the rheological behavior of the suspensions, we had the opportunity to calculate the shear stresses inside the concentrated layers as described in part 3.2 of this article. The only assumption made is that the rheological behavior of concentration polarization and gel layers of Laponite in the cross-flow experiment is equivalent to those determined by rheometric measurements for Laponite suspensions having the same particle concentration. suspension of Laponite without peptizer has a higher value than for a suspension of Laponite with peptizer in accordance with the change in the rheological behavior due to the modification in interparticular interactions. Starting from the same volume fraction of 1 vol % and following the same filtration procedure in the same conditions, the changes observed in the concentration, velocity and stresses profiles in the concentrated layers for the suspensions with and without peptizer, are linked to the change in the rheological behavior. One of the principal change in this behavior is the sol/gel volume fraction transition. Without peptizer the sol/ gel transition is around 1 vol%, while with peptizer, the sol/gel transition is between 1 and 2 vol%. This has a huge consequence on the way the particles concentrate near the membrane surface submitted simultaneously to shear, pressure and colloidal forces. Another interesting point is that as already shown in previous papers [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF][START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF] the osmotic pressure of the Laponite suspensions with and without peptizer follows the same dependence with the volume fraction, same equation of state. The differences emphasized in these results for the suspensions with and without peptizer reinforce the precedent conclusions [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF], namely that the pertinent parameter which control the filtration performance is the hydrodynamic/rheological behavior of the suspension and even though the osmotic pressure evolution is an important parameter of the filtration, it is not the most important parameter in this kind of colloidal suspensions.

Concentrated layers behavior: relationship with rheological behavior of the suspensions

In this section we will analyze the effect of the rheological behavior on the formation of the concentrated layer and the influence on the hydrodynamic field, structural organization and the stress field applied near the membrane surface. From the whole data and calculations obtained in this article, the whole accumulation phenomena is on the order of 100 minutes, this small difference of 5 minutes in recording times will negligibly affect the calculation of the shear stress. Consequently, the conclusions on the whole concentration phenomena and trend of the calculated shear tress as a function of z or filtration time, will not be affected by this difference of recording times.

These combined results and calculations allow to highlight three different regions (A, B, C) from the membrane surface to the feed channel, with different specific behaviors in terms of rheological and filtration accumulation. In the first region called A which expends up to 500 µm, the velocity and shear rate are close to zero, corresponding to a suspension at rest. This means that the calculated stress, which corresponds to the yield stress, is not high enough to give rise to a flow inside the concentrated layer. This region matches with the fouling layer, with a decreasing stress while the volume fraction decreases, in agreement with the rheometry measurements (Fig. 10 In the region B the maximum shear rate obtained in this concentration polarization layer is higher at the end of the filtration than at the beginning (almost 80 s -1 instead of 60 s -1 at the beginning) because the velocity profiles associated show more brutal changes for increasing filtration times. Indeed, the thickness of the concentrated layer increases for increasing filtration times and the mean flow velocity is more important for larger distances from the membrane surface. Therefore, during the time of filtration when the particles leave the accumulated layer to be introduced in the flow, the velocity that they will adopted will be more important on shorter distances during the time of filtration.

The change between region A and B is not only related to the static yield stress of the suspension but also to the shear history of the suspension of this thixotropic system. Consequently, for the two different filtration times the limit between these two regions A and B is pushed towards higher z values, at increasing filtration times. At 0 min filtration time this limit is at z = 1030 µm and shifts to z = 1400 µm at t = 35 min until z = 1700 µm at t = 81 min. It is then necessary to take into account the complex relationships between the restructuring level of the Laponite suspensions and the stresses applied by the upper flowing layers which are higher at increasing z distances.

Another interesting point is that in this region B, the maximum value of the shear rate reached without peptizer, is lower than for a suspension of Laponite with peptizer, in agreement with the fact that for the same volume fraction 1 vol%, the viscosity of the suspension without peptizer is higher than the one with peptizer. The required stress to involve a flow at the interface between the fouling and the concentration polarization layers is weaker for a Laponite suspension without peptizer than when the peptizer is added. This is mostly explained by the difference of the sol/gel volume fraction transition that depends on the presence of the peptizer. When the peptizer is mixed to the suspension, the sol/gel volume fraction transition is higher than without peptizer and according to the rheometric measurements, the yield stress is more important for increasing volume fractions.

These observations emphasize the fact that for the same filtration conditions, same cross-flow and same transmembrane pressure, the stresses inside the concentration polarizations layers adjust their values in order to find an equilibrium between the external forces imposed by the cross-flow (which tend to disrupt the structure) and the resistance forces (which tend to build up the structure related to internal forces of the suspension). In the particular case of this thixotropic colloidal suspension these internal colloidal forces are a complex combination of the colloidal interaction between the particles, the history of the previous shear and pressure forces applied, which could have disrupt the structure of the suspensions and weaken the interparticle forces as a function of time. These time-dependent evolutions (break-down and build-up) of the structure are highly dependent on the volume fraction of the particles and also on the modification of the surface forces between the particles (with peptizer or not) as already shown [START_REF] Pignon | In situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions[END_REF] Consequently during the filtration process, under a constant cross-flow and transmembrane pressure, starting from an initial volume fraction, the final equilibrium concentration polarization layer reached is the consequence of the way the particles (starting from a certain

  33 Pa (respectively open and closed square symbols in Fig 2b).

  Concentration profiles during cross-flow ultrafiltration: SAXS analysis, relaxation phenomena and lowering of the membrane a) SAXS Analysis. To investigate the accumulation phenomena during cross-flow ultrafiltration, different filtration condition steps have been explored. In a first step, under a transmembrane pressure of 1.1 x 10 5 Pa a cross-flow of 0.3 L.min -1 has been applied and the structural organization and concentration profiles deduced by SAXS have been recorded during time (Fig.

3 . 4 Fig 5 a

 345 Fig 5 a shows the results of accumulation during step 2 at a flow rate of 0.06 L.min -1 , for a Laponite

  7 a). The corresponding permeation flux measured in-situ (Fig.7 b) continuously decreases and stabilizes around 2.5 L.h -1 .m 2 at time t = 22 min until the end of the experiment. For increasing filtration times, PIV results show an increasing thickness of the area of no velocity near the membrane surface. At the end of this filtration step, any velocity is detected over a distance of about 900 µm above the membrane instead of 700 µm at the beginning of this step. This is important to notice that the velocity profile starts to stabilize since 90 minutes of filtration whereas 22 minutes are enough to show a stabilization of the permeate flux. the stabilization of the upper flowing layers (from time t = 22 min to t = 140 min) will affect not to much the permeation flux as the corresponding changes in concentration

Figures 8 a

 8 Figures 8 a) and b) show shear rate profiles calculated from velocity PIV measurements during the time of filtration of a suspension of L1-Cp6 (Fig. 8 a) and for a suspension of Laponite without peptizer L1-CP0 (Fig. 8 b) during, respectively, the step 2 and the step 3 of filtration at 0.06 L.min -1

Figure 9 a

 9 Figure 9 a shows the shear stress profiles calculated for increasing filtration times as a function of the distance z from the membrane surface of a Laponite suspension with peptizer during the last step 3 of filtration at Q = 0.06 L.min -1 . This result shows that the stress near 350 µm above the membrane increases during the filtration time due to the formation of the concentrated layer. For the 3 different times of filtration presented, the shear stress profile has the same behavior: a first area where the stress value decreases in link with firstly the decrease in volume fraction at increasing z and secondly in link with the area of no flow measured by micro-PIV (Fig. 7 a) and (Fig. 8 a). The values of shear stress calculated for this area correspond to the yield stress value. Figure 10 a locate the calculated shear stress in the concentrated layers during the filtration, on the shear stress/shear rate flow curve of the suspension as a function of the volume fraction. The stresses reach in this decreasing part of the shear stress (Fig.9a) are located in the yield stress domain of the flow curve of the suspension. After this first area of decreasing stress, the evolution of the shear stress profile reaches a plateau with a constant value for increasing filtration times. In this area above z = 1000 µm, the volume fraction of the suspension reaches the initial filtered suspension 1 vol% and the shear rate stabilizes towards its lower value.

Figure 9 b

 9 Figure 9 b shows the calculated shear stress profiles during the last filtration step3 for a suspension of Laponite without peptizer. The shear stress profiles show the same behavior during the time of filtration for L1-Cp0 suspensions and L1-Cp6 suspensions. Nevertheless, near the membrane surface, the stress has a higher value than with peptizer (from 500 Pa to 8000 Pa depending on the filtration time) and decreases until reaching a plateau at a constant value around 10 Pa, for the 3 different filtration times. As previously seen for a suspension of Laponite with peptizer, the shear stress measured near the membrane surface (before the plateau) corresponds to the stresses obtained when no flow is measured by micro-PIV in the concentrated layer, and thus corresponds to the yield stresses. Figure10 b shows the location of the stress calculated from the velocity profiles in the concentrated layer, on shear stress/shear rate flow curve of the suspension without pepetizer as a function of the volume fraction measured by rheometric measurements. It appears that the stress obtained for a

Figures 11 and 12

 12 Figures[START_REF] Bacchin | Colloidal surface interactions and membrane fouling: Investigations at pore scale[END_REF] and 12 summarize and compare on the same figure the volume fraction (red empty circles curve), calculated shear stress (black full circles curve) and calculated shear rate (blue full triangles curve) as a function of the distance z from the membrane surface. For each suspension with peptizer (Fig.11) and without peptizer (Fig.12) these profiles have been plotted for three different filtration times, in the same cross-flow and transmembrane pressure conditions. The calculations have been done from the SAXS and micro-PIV measurements performed mostly at the same filtration time. For example for the results presented in Fig.11, the calculations are done from the volume fractions deduced from SAXS experiments (Fig.5a) at filtration time (t= 0 min, t = 59 min and t = 94 min) and the calculated shear rates and shear stress from the micro-PIV measurements (Fig.7 a) at the related filtration times (t = 0 min, t= 60 min, t = 90 min). These two set of data from SAXS and micro-PIV experiments are recorded at namely the same filtration times. The largest difference of recording times between the SAXS and micro-PIV data is 5 minutes. As the phenomena of accumulations are slower:

  Laponite suspensions with peptizer (L1-Cp6): Figures 11 a), b) and c) exhibit the three different regions (A, B, C) at different times of filtration, respectively t = 0 min, t = 60 min and t = 90 min. At the beginning of step 2 of filtration at Q = 0.06 L.min -1 , the volume fraction is the same as the volume fraction measured at the end of the previous filtration step 1 at Q = 0.3 L.min -1 , as shown in Figures 3 a) and b).

  a). At the beginning of the second region called B, the shear rate and corresponding calculated shear stress start to increase until reaching a maximum value of 7 Pa when the shear rate maximum is about 135 s -1 . In this region B the shear flow forces are sufficiently high, compare to colloidal interactions and pressure forces, to initiate the flow of the suspension. This region B can be assimilated to the concentration polarization layer. The third region called C appears when the shear rate starts to decrease until reaching a minimum value. In this region C, the volume fraction is constant and equal to the initial volume fraction 1 vol%., the shear rate decreases with the corresponding shear stress until reaching the conditions where the suspension is circulating in the feed channel. This region C is out of the concentration phenomena and corresponds to the flowing layer above the concentration polarization layer (region B).After 60 and 90 mins of filtration corresponding to Figures11 b) and c), the same phenomena are observed but over a higher z distance range due to accumulation of particles. The values of the calculated shear stresses near the membrane surface increases when increasing the volume fraction for fixed distances depending on the filtration time. Near 350 µm above the membrane surface, the stress reaches 200 Pa after 60 min of filtration (Fig.11 b) and 300 Pa after 90 min of filtration (Fig.11 c) in the fouling layer (region A). When the shear rate increases (region B), the required stress to involve a flow is about 40 Pa at 60 min and 90 min of filtration. This critical stress value to initiate the flow has the same value because this phenomenon appears when the volume fraction is between 1 and 2 vol%, corresponding to the sol/gel volume fraction transition for a suspension of Laponite with peptizer. The value of the shear rate and shear stress far away from the membrane (z = 2000 µm), are the same at time t = 60 min and t = 80 min which shows that this region of the flowing suspensions is not influenced anymore by the concentrated lower layers.Laponite suspensions without peptizer (L1-Cp0):In the case of Laponite suspensions without peptizer the three regions A, B and C are also well identified but extend on higher z distances from the membrane surface (Fig.12 a). In the first region A corresponding to the fooling layer, the calculated stress decreases as a function of z as a consequence of the decrease in volume fraction, and the stress corresponds to the yield stress of the suspensions. When the filtration time increases (Fig.12b and 12 c), the thickness of the fouling layer (region A) increases within higher values of shear stress due to the increase of volume fraction during the time of filtration. The stress level can reach high values at time t = 80 min of filtration, as high as 5000 Pa for a corresponding volume fraction of 20 vol% at z = 350 µm. It is interesting to underline that in this region A, the volume fraction decreases until reaching the initial volume fraction of 1 vol% at z = 1150 µm for filtration time t = 35 min (Fig. 12 b) and at z = 1360 µm for filtration time t = 80 min (Fig. 12 c), then the stress becomes flat and corresponds to the yield stress 5 Pa of the Laponite suspensions.

  structuring state) start to accumulate and concentrate during time. It has been previously shown that the concentration of the particles within the accumulated layers follow an exponential trend as a function of distance z from the membrane surface as well as a function of filtration time[START_REF] Jin | Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS[END_REF]. Each layer of the Laponite suspension at each volume fraction has a certain "destructuring" or restructuring time. It is then a complex combination of this restructuring time (at each sublayer with a given volume fraction) in competition with the destructuring time imposed by the cross-flow, which at the end, result in a certain stabilized concentration polarization layer. These time-and spatial-dependent phenomena are well emphasized in Figures11 and 12, for which in the same filtration conditions (cross-flow and transmembrane pressure) the two different suspensions (with and without peptizer) in terms of interparticle interactions and consequently rheological behavior, give rise to different equilibrium concentration layers. Nevertheless, it is important to claim that the mechanisms involved for these two different suspensions are the same with the existence of the three same behaviors associated to the three regions A, B and C.CONCLUSIONSThis work combines in-situ SAXS, in-situ micro-PIV and rheological measurement to bring a better understanding of the time and spatial-dependent behavior of the concentrated layer near the membrane surface during the ultrafiltration process of Laponite suspensions. The structural and kinematic evolutions of the concentrated layers have been linked to the rheological behavior evolutions of the filtered suspensions.The results of SAXS experiments performed in-situ during the filtration process have allowed highlighting the exponential increase of the volume fraction as a function of the distance z from the membrane surface as well as a function of the filtration time. The obtained results revealed the importance of the level and the type of electrostatic interactions between the particles in the mechanism of formation of the concentrated layers. The Laponite suspensions without peptiser, namely in attractive particle interactions, give rise at equilibrium to a thicker and more concentrated layers than the suspensions with peptisant in repulsive particles interactions.The original micro-PIV characterization performed within the accumulated layers during the crossflow ultrafiltration of colloids allowed giving access to the velocity field as a function of the distance z from the membrane surface. Thanks to rheometric measurements, the rheological behaviors were investigated for each concentration reached in the accumulated layers and revealed by in-situ SAXS experiments. The integration of the in-situ shear rate fields obtained from the velocity field by micro-PIV into the rheological laws gives access to the calculated in-situ stress field of the concentrated layers.These measurements allowed revealing certain mechanisms of flow in link with the rheological behavior evolutions of the suspensions, induced by the simultaneous effect of velocity and pressure at the vicinity of the membrane surface during the filtration process. It thus was highlighted several specific flow regions with well-identified behaviors. Firstly, a region A has been identified at the nearest vicinity to the membrane surface, for which there is no velocity, corresponding to a suspension at rest. Secondly a region B for which the shear rate and corresponding calculated shear stress start to increase until reaching a maximum. In this region the shear flow forces are sufficiently high compare to colloidal interactions and pressure forces, to initiate the flow of the suspension. This region B can be assimilated as the concentration polarization layer. These two first regions A and B had a thickness of about a few hundreds of micrometer. Their thickness evolves according to the time of filtration and the operating conditions, the history of shear flow involved and the physico-chemical conditions of the suspensions. This original information has led to the stress level required to involve a flow between regions A and B during the filtration process and for different physico-chemical conditions of the suspensions, which highlights the important role played by the rheological behavior in the formation of the concentrated layers. Lastly, a third region C is defined in which the suspension reaches the conditions where the suspension is circulating in the feed channel without being affected by any concentration phenomenon. The concentration of the suspension is constant and corresponds to the initial filtered concentration, with a decreasing shear rate until reaching the maximum velocity value between 2000 and 2500 µm above the membrane towards the center of the cell.From a fundamental point of view, obtaining the stress levels inside the concentrated layers, related to the concentration profile, the cross-flow and the rheological behavior of the filtered suspensions, improves the understanding of the phenomena involved in the formation of the concentrated layers and fouling phenomena. From an industrial point of view, this work allows to improve the efficiency of ultrafiltration since it gives a better understanding of the formation of these phenomena.

FiguresFig. 1 :

 1 Figures

Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :Fig 9 :

 23456789 Fig. 2: Steady-state flow curves of Laponite dispersions (10 -3 M NaCl, pH = 10) at different volume fractions and different peptizer concentrations. T = 25 ± 1 °C, tp = 12 days. a) viscosity and b) corresponding shear stress as a function of the imposed shear rate.

Fig. 11 : 39 Fig. 12 :

 113912 Fig. 11: Behavior regions A, B and C in the concentrated layer formed from Laponite suspensions with peptizer as a function of the distance z from the membrane surface (with the lowering correction) during cross-flow ultrafiltration, for the last step of filtration, Step 2, Q = 0.06 L.min -1 . At filtration time a) t = 0 min, b) t = 60 min, c) t = 90 min. L1-Cp6, tp = 12 days, TMP = 1.1 x10 5 Pa, T = 25 ± 1 °C, middle window.

ACKNOWLEDGEMENTS

We would like to thank Theyencheri Narayanan and Michael Sztucki (ESRF, Grenoble) for their kind help in scattering experiments and fruitful discussions, Jean-Jacques Lasserre (Dantec Dynamics) for his help and productive discussions, William Chèvremont for his helps in stress calculations. Mohamed Karrouch, Eric Faivre, Hélène Galliard, Didier Blésès and Frédéric Hugenell (Laboratoire Rhéologie et Procédés, Grenoble) and Jacques Gorini (ESRF, Grenoble) for technical assistance. We gratefully acknowledge the ESRF for the SC 4177 beam time allocation. This work was supported by the Labex TEC 21 (Investissements d'Avenir -grant agreement n°ANR-11-LABX-0030). LRP and LGP2 are part of PolyNat Carnot Institut (Investissements d'Avenir -grant agreement no. ANR-11-CARN-030-01) and of program (ANR-15-IDEX-02).