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In Situ Characterization by SAXS of Concentration Polarization Layers during Cross-Flow Ultrafiltration of Laponite Dispersions F. Pignon,*,† M. Abyan,† C. David,† A. Magnin,† and M. Sztucki‡ †

Laboratoire de Rhéologie, Université Joseph Fourier - Grenoble I, Grenoble - Institut National Polytechnique, CNRS, UMR 5520, BP 53, F-38041 Grenoble Cedex 9, France ‡ European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex 9, France ABSTRACT: The structural organization inside the concentration polarization layer during cross-flow membrane separation process of Laponite colloidal dispersions has been characterized for the first time by in situ time-resolved small-angle X-ray scattering (SAXS). Thanks to the development of new “SAXS cross-flow filtration cells”, concentration profiles have been measured as a function of the distance z from the membrane surface with 50 μm accuracy and linked to the permeation flux, cross-flow, and transmembrane pressure registered simultaneously. Different rheological behaviors (thixotropic gel with a yield stress or shear thinning sol) have been explored by controlling the mutual interactions between the particles as a result on the addition of peptizer. The structural reversibility of the concentration polarization layer has been demonstrated being in agreement with permeation flux measurements. These observations were related to structure of the dispersions under flow and their osmotic pressure.

1. INTRODUCTION Membrane separation processes are used for concentrating and purifying nanoparticles dispersions. Typical industrial applications include pharmaceutical, bio- and agro-industries, as well as water and sludge treatment. The main limitation of this process is the ability to control the accumulation of matter in the socalled concentration polarization layer. An accurate understanding of the structural organization of colloids in the vicinity of the membrane in parallel with the filtration properties is of considerable interest to improve the efficiency of the filtration. Depending on the transmembrane pressure and cross-flow conditions, a transition from stable to unstable filtration performance has been identified and a limit has been proposed.1,2 The limit separates a stable domain where reversible polarization layer is formed and an unstable domain where the accumulation of matter growth leads to an incessant decrease of the permeation flux with time. Until now, these phenomena have been investigated by theoretical3−7 and modeling approaches,8−11 but no real time observation of these phenomena at the pertinent length scales under crossflow of the colloids has been completed during the filtration process. Some experimental investigations have probed the structural organization of deposits,12−16 and some others have allowed following the structural organization of colloids near the membrane surface during frontal ultrafiltration by smallangle X-ray scattering (SAXS) and small-angle neutron scattering (SANS).17−20 The focus of this paper is to characterize, in situ, at nanometer length scales, the induced structures and concen© 2011 American Chemical Society

tration profiles in the vicinity of the ultrafiltration membranes, when colloidal dispersions are simultaneously subjected to a transmembrane pressure and cross-flow over the membrane. To reach this challenge new “SAXS cross-flow filtration cells” have been developed at the “Laboratoire de Rhéologie”. These cells allowed applying small-angle X-ray scattering at the European Synchrotron Radiation Facility (ID02 beamline) for the investigation of membrane separation processes and the corresponding permeation flux measurements. The colloidal dispersions filtered are the widely studied synthetic aqueous clay Laponite, made of nanometric platelets with an average lateral dimension of 30 nm and 1 nm in thickness. The structure, phase diagram, and rheological behavior of these dispersions have been extensively studied.21−37 At moderate ionic strengths (I = 10−3 mol L−1) and pH 10, the Laponite dispersions are in the semidilute regime,26,31,35 just above the sol/gel transition and behave as shear thinning thixotropic gels. In order to study the effect of the modification of interparticle interactions on the structure of accumulated layers permeation flux and reversibility phenomena, tetrasodium diphosphate peptizer (tspp) has been added to the dispersions. This peptizer has the ability to reduce the attractive interactions between the particles and induces a dissociation of the network that is responsible of the yield stress and the gel state of the dispersions.31 Without peptizer (Cp = Received: April 22, 2011 Revised: December 8, 2011 Published: December 12, 2011 1083

dx.doi.org/10.1021/la201492z | Langmuir 2012, 28, 1083−1094

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Figure 1. (a) Schematic description and (b) picture of the SAXS cross-flow filtration cell positioned on ID02 Beamline at the ESRF. SAXS pattern and different integrations performed on it: (i) Ir radial averaged, (ii) along a sector of 30° around the vertical (Iv) and horizontal (Ih) axes, and (iii) Ic on an angular sector from 270° to 90°.

there is no eventual dissolution process of the particles, as reported before.31,35 The structure and rheological properties of these dispersions are known to evolve with time. Several works have been done to study this effect.17,26,30,31,33,35,37 Consequently, 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 kind of measurements (rheometric behavior, crossflow ultrafiltration) the results will be compared at the same tp time. In order to study the effect of a change in particle interaction on the structure and filtration performance, different contents in peptizers tetrasodium diphosphate Na4P2O7 (tspp) were added to the dispersions. Two peptizer concentrations (Cp) have been explored (0% and 6%) as percentages of the mass of dry clay. The effect of this peptizer on the structure, rheological behavior, osmotic pressure, and frontal filtration performance of the Laponite dispersions have been studied in detail in our precedent works.17,31,35 The diphosphate anion binds the positive surface charge of the edges of the platelets. This has the consequence of reducing 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. Depending on the volume fraction and aging time tp, different rheological behaviors could be reached (yield stress gel, shear thinning sol or Newtonian behavior). In this work the volume fraction and tp conditions studied have allowed to explore the behavior of a yield stress gel

0%), the dispersions behave as thixotropic gels with a yield stress. At increasing peptizer concentration the dispersions can flow as shear thinning sol or could exhibit a Newtonian behavior at low volume fractions.17 Consequently, due to the addition of this peptizer, the filtration characteristics have been evaluated as a function of interparticles interaction defining different initial states (sol or gel) of the dispersions to be filtered. The precedent works on Laponite dispersions,21−37 have studied the system under only one single solicitation as under shear or under pressure conditions. Besides the understanding of structural organizations in the concentration polarization layers, the SAXS measurements performed in situ during ultrafiltration offer the possibility to study the changes in particle organizations, particles interactions and sol/gel transition mechanisms under the mutual pressure and shear flow conditions accessible by these new developed methods.

2. MATERIALS AND METHODS 2.1. Laponite Dispersions. The Laponite XLG (Laporte Industry) dispersions used were prepared under high shear in a solution of distilled water (10−3 M NaCl) at 20 °C. The filtered dispersions were prepared at initial volume fraction ϕv0 = 0.01 corresponding to a mass fraction 0.0253 g/cm3 . The dispersions were then aged in closed vessels for 12−26 days before the filtration experiments. During this short aging time, the pH of the dispersions remains stable and equal to 10 and 1084


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≤ 785 nm. A temperature controlled flow-through capillary cell (diameter = 1.85 mm) was used to measure the absolute scattered intensity of dispersions at rest at T = 22 °C. It permits to measure sample and background scattering in the same position of the capillary allowing very reliable subtraction. In the SAXS cross-flow filtration cell, the X-ray beam was highly collimated (50 μm vertically × 250 μm horizontally) to reach a high spatial resolution in the vertical direction. This incident beam crosses the deposited particles allowing to probe multilevel structures and interactions as a function of the distance z from the membrane surface (Figure 1). All of the scattered intensities I(q) presented are in absolute units and correspond to the scattering of the colloidal particles only. The normalized background scattering of the different cells filled with distilled water was systematically subtracted. Several integrations of the normalized SAXS patterns were performed as shown in Figure 1. For the dispersion in flow-through capillary cell a radial averaged Ir was done. In the SAXS crossflow filtration cell the integration has been completed on angular sector from 270° to 90° Ic. This angular sector has been chosen to avoid any contribution from the membrane surface. To determine the amplitude of the anisotropy along parallel and perpendicular directions to the membrane, the scattering along the vertical (Iv) and horizontal (Ih) axes was calculated by integrating the scattered intensity along a sector of 30° around the vertical (z) and horizontal (y) axes, respectively.

without peptizer (Cp = 0%) or a shear thinning fluid (Cp = 6%). Besides, in order to calibrate the scattered intensity level as a function of volume fraction, several others dispersions at different volume fractions and peptizer contents were prepared in the same conditions. 2.2. Rheometric Measurements. The rheometric measurements were carried out with a stress-controlled rheometer (ARG2, TA Instrument) with a stainless steel cone and plate geometry (diameter 49 mm, angle 4°21′). The temperature of the samples was controlled (30 ± 1 °C) at the same range as the one reached during filtration measurements. In order to avoid interfacial effects, the surfaces of the apparatus were covered with sand-paper of 200 μm roughness. To avoid evaporation from the sample during the test, the atmosphere around it was saturated with water.28 The measurements done at a low tp time (12 to 26 days) have allowed to reach shear flow regimes for which the shear stress corresponds to volume properties and a continuous deformation of the sample between the cone and plate tools. Direct observations of the shear profile at the outer edge of the rheometer tools,28 confirmed that slip did not occur and that the samples were sheared uniformly without any localization phenomena. 2.3. “SAXS Cross-Flow Filtration” Cell and Filtration Procedure. The “SAXS cross-flow filtration cell” used for these measurements (Figure 1) is made of transparent polycarbonate and contains a flat polysulfone ultrafiltration membrane (100 kD, Pleyade Rayflow x100, Rhodia Orelis). The retentate channel is 120 mm long in tangential flow direction and the flow section is 5 mm high for 1 mm large. Pressure is applied to the rig via purified compressed air, and the retentate pressure P is measured at the entry of the cell with a pressure gauge (FP 110 FGP Sensors & Instrument). Retentate cross-flow flux is imposed (Mono pump LF series, Axflow) and measured constantly (Optiflux 6300C flowmeter, Krohne). Permeate flux J is continuously recovered in a recipient and its mass is registered with a computer every 10 s with a scale of 0.001 g of accuracy (Precisa 400M). In order to follow the structure inside the cell at different distances from the membrane during ultrafiltration process, the cell was mounted on a motorized sample stage that enables us to translate the cell across the incident beam vertically and horizontally. The analysis of the SAXS patterns registered in the cross-flow cell is the same which has already been described in the preceding work.20 Four different zones were defined by measuring the transmitted X-ray signal as a function of the distance z from the membrane surface through the SAXS crossflow filtration cell. The description of these zones has allowed to define the minimal distance z above which the scattered intensity is not influence by the X-ray beam partially crossing the polycarbonate support and the membrane. Furthermore in the results presented here, this minimal distance has been reduce to 100 μm thanks to the use of an angular motion stage which allowed to orientate the cell in order to align the surface of the membrane with the direction of the X-ray beam. 2.4. SAXS Measurement Conditions and Analysis. All X-ray scattering measurements have been performed at the ID02 High Brilliance Beamline,38 at the European Synchrotron Radiation Facility (ESRF), Grenoble, France. The wavelength of incident X-rays was λ = 0.995 Å and two sample−detector distances (1 and 10 m) were used. The SAXS measurements covered the following scattering vector range: 8.10−3 nm−1 ≤ q ≤ 5 nm−1, q = (4π/λ) sin(θ/2) and θ is the scattering angle. The corresponding (l = 2π/q) length scale range is 1.25 nm ≤ l

3. RESULTS AND DISCUSSIONS 3.1. Structure and Rheometric Behavior of Laponite Dispersions. The structure and rheometric properties of these Laponite dispersions has been extensively studied in the past using a variety of techniques including rheometry, light, X-ray, and neutron scattering.26−31 The scattering studies indicate that, for ionic strengths of 10−3 M and at high pH from 9.5 to 10, the Laponite particles attract each other in edge-to-face configurations. These attractions cause the particles to gather in microdomains, which subsequently associate to form very large fractal superaggregates, containing all the particles in the dispersion. These superaggregates form a network that controls the mechanical properties of the dispersions.26−28 A gel state is obtained when the network of connections is macroscopic. This network is destroyed by the application of sufficient strain, but it heals at rest. The addition of peptizers weakens the edge-toface attractions, and makes the healing times much slower.31 The scattered intensities of the dispersions at rest (Figure 2a) are fully in agreement with our precedent works.26,31 The calculated form factor P(q) (same calculation as in ref 26) shows that at higher q vectors (above 2.5 × 10−1 nm−1 values) the scattered intensity is not affected by the volume fraction nor by the tssp concentration (Figure 2b). At lower q vectors (below 2.5 × 10−1 nm−1) the mutual interaction between the particles induces changes in the scattered intensity, the structure factor S(q) is predominant and the scattered intensity evolves with volume fraction (Figure 3a) and CP peptizer concentration (Figure 2b). As already described,18−20 static SAXS measurements were performed in capillary to establish a calibration curve, in order to correlate the level of absolute scattered intensity to the volume fraction of the Laponite dispersions. The results of the scattered intensity as a function of volume fractions (Figure 3a) have allowed us to define the following linear relationship (Figure 3b): I (1.2 nm−1) = 0.047 ϕv (eq 1). This equation has been established in the linear part of the scattered intensity (q−2 1085


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Figure 3. (a) SAXS intensities from Laponite dispersions at rest in capillary flow-through cell as a function of volume fraction and (b) calibration curve: radial averaged absolute scattered intensity I(q) as a function of volume fraction at particular q vector of 1.2 nm−1. Cp = 0%, I = 10−3 M, pH 10, tp = 12 days, T = 22 ± 1 °C.

Figure 2. (a) SAXS intensities from Laponite dispersions at rest in capillary flow-through cell. (a) Form factor P(q), radial averaged scattered intensity I(q) and structure factor S(q) at 0.02 volume fraction and (b) I(q) for different peptizer (tspp) concentration, I = 10−3 M, pH 10, tp = 12 days, T = 22 ± 1 °C.

power law decay) corresponding to the form factor of the particle dispersions. Hence, this relationship is valid at increasing concentration and is not affected by the increase in mutual particle interaction, which modifies the structure factor. The same measurements on dispersions at rest with Cp = 6% tspp have been performed and give exactly the same calibration curve. This is in agreement with the result in Figure 2b showing that the form factor of the dispersions is not affected by the peptizer content. 3.2. Rheometric Behavior. The rheometric measurements were done as a function of the peptizer content and for different elapsed time tp corresponding to filtration measurements under in situ SAXS or ex situ measurements. In Figure 4 is plotted the steady state flow curve (shear stress as a function of the shear rate) for the same 0.01 volume fraction. At Cp = 0% peptizer concentration, the dispersions at tp = 12 and 26 days exhibit a stress plateau at low shear rates corresponding to a yield stress, while the stresses increase at high shear rates. This rheological behavior typically described as shear thinning gels is well depicted by a Herschel−Bulkley viscoplastic model:39 σ = σs + Kγ̇n, σ represents the shear stress, σs is the yield stress, K is the consistency, γ̇ is the shear rate, and n is the shear-thinning index. Interpolations in this model are represented by the dotted curves (Figure 4) and the fitted values given in the table in Figure 4. The model is satisfactory over the entire range of shearing explored. At Cp = 6% peptizer concentration and tp = 26 days, the flow curve of the dispersions has no stress plateau at low shear rates. A shear-thinning behavior without any yield stress is measured and well depicted by a power law σ = Kγ̇n.

Figure 4. Steady state flow curve for Laponite dispersions at different volume fractions, tp elapsed time, and Cp peptizer concentration. Dotted curves correspond to Herschel−Bulkley viscoplastic model (σ = σs + Kγ̇n) or power law (σ = Kγ̇n) fit, and corresponding calculated parameters σs yield stress, K consistency, and n shear-thinning index. T = 30 ± 1 °C, I = 10−3 M, pH 10.

3.3. Filtration Performances and Limiting Flux: Effect of Particles Interaction on the Reversibility Phenomena. Some ex situ measurements have been performed using the “SAXS cross flow cell” exactly in the same conditions (same equipment, temperature and procedures) as in situ SAXS measurements. A first set of experiments was conducted to characterize more precisely the onset of limiting flux as a function of Q cross-flow condition. The effect of elapsed time tp and peptizer content Cp has also been evaluated. The question raised here is the following: is the concentration polarization 1086


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105 Pa condition, the permeation flux is reduced to 5.2 L/m2/h at the beginning of this step certainly due to the formation of a deposit in the preceding ΔP = 0.9 × 105 Pa condition. During step A, the permeation flux increases progressively due to the disruption of this deposit until it reaches the J0 level of about 14 L/m2/h equivalent to the one at the beginning of the experiment. Consequently, the deposit formation is reversible here but depends on the competition between the time scales of disruption due to shear and the time scales of build up induced by the colloidal interactions. Several Q cross-flow conditions where investigated in order to obtain the steady state permeation flux J as a function of ΔP. The results presented in Figure 6 correspond to the filtration of

layer reversible for all filtration conditions and peptizer content? And if not, for which conditions an irreversible deposited layer is formed? To answer to these questions the following filtration procedure has been investigated: a constant cross-flow Q is maintained and successive increase and decrease in transmembrane pressure ΔP were applied. At the beginning of the experiment, a low transmembrane pressure ΔP0 = 0.3 × 105 Pa is applied which gives rise to a certain J0 permeation flux. In order to well differentiate the effect of an eventual deposit formation to a reversible concentration polarization layer, the same initial low level of 0.3 × 105 Pa is applied between each successive increase in ΔP as shown in Figure 5. Coming back to

Figure 6. Steady states J permeation flux as a function of ΔP transmembrane pressure conditions for different Q cross-flows measured during ultrafiltration of Laponite dispersions. ϕv0 = 0.01, I = 10−3 M, pH 10, T = 30 ± 1 °C: (a) Cp = 0% and tp = 12 days and (b) Cp = 6% and tp = 26 days.

Figure 5. Filtration performance of Laponite dispersions. Transmembrane pressure procedure and time dependent evolution of permeation flux at constant Q = 0.45 L/min, ϕv0 = 0.01, I = 10−3 M, pH 10, T = 30 ± 1 °C: (a) Cp = 0% and tp = 12 days and (b) Cp = 6% and tp = 26 days.

Laponite dispersions at same initial volume fraction 0.01 but for different tp and Cp conditions. The two dispersions filtered were chosen with the equivalent conditions used for rheometric measurements and filtration under SAXS (Figure 6a, CP = 0% and tp = 12 days, or Figure 6b, CP = 6% and tp = 26 days). For the dispersions without peptizer for Q = 0.60 L/min, Figure 6a shows a slow but continuous increasing J as a function of ΔP. This corresponds to the situation where a reversible concentration polarization layer is formed. The evolution of J(t) has the same features as presented in Figure 5b. Each release in pressure shows that no dense deposit layer has been formed. In contrary, at the lowest Q = 0.2 L/min and above ΔP = 0.2 × 105 Pa, the permeation flux becomes independent of the successive increases in ΔP and reaches the so-called limiting flux. Figure 5a presents the corresponding time evolution of the

ΔP0 also allowed us to identify the onset of a deposit formation; two situations are observed: if the permeation flux J at each intermediate low level (0.3 × 105 Pa) is equal to J0 as in Figure 5b no deposit formation is detected. If this permeation flux at intermediate low level 0.3 × 105 Pa is lower than J0, a deposit formation has been attained in the preceding step. Furthermore, following the evolution of permeation flux with time during the low pressure condition ΔP0 has allowed us to detect the time evolution of the erosion of a deposited layer. For example in the case presented in Figure 5a, the decrease at low ΔP0 conditions permits to erode this deposit thanks to hydrodynamic forces. At the step denoted step A between 970 and 985 min in Figure 5a, one can see that for the ΔP0 = 0.3 × 1087


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permeation flux J. For each release in pressure at ΔP0, an important reduction of the flux pertinent to the formation of a deposit layer is registered. These macroscopic measurements will be compared in the next section with the proof of the presence or not of a deposited layer thanks to the nanoscale observation. For the dispersions with Cp = 6% peptizer concentration (Figure 6b), the evolution of the permeation flux as a function of ΔP shows the same features as the one without tspp, but the permeation flux levels are higher whatever Q or ΔP imposed. Furthermore, the limiting flux is reached at a lower Q = 0.045 L/min. Consequently, the shear thinning dispersions (Cp = 6%) have a better permeability than the yield stress dispersions (Cp = 0%). This result is well in agreement with the formation of a less concentrated polarization layer in the limiting flux regime. This decrease in concentration would originate from the reduction of mutual particle interaction with the addition of peptizer which led to a system having a lower resistance to shear-flow as shown in Figure 4. For the same cross-flow level, the hydrodynamic forces have a better ability to disrupt the network of the accumulated particles for the dispersions with peptizer than without peptizer. This hypothesis will be confirmed in the following by the in situ SAXS investigations. 3.4. In situ Structure and Concentration Profiles during Cross-Flow Filtration: SAXS Analysis in CrossFlow Filtration Cell. An example to deduce the volume fraction in the accumulated layers from the SAXS patterns acquired during the filtration, is presented in Figure 7a for ΔP = 1.2 × 105 Pa transmembrane pressure and 0.1 L/min crossflow conditions at 308 min filtration time. The absolute scattered intensity has been deduced from the integration Ic of the normalized SAXS patterns. From this curve, one can extracted the absolute scattered intensity I0 at the fixed 1.2 nm−1 q vector as a function of the distance z. Using the linear relationship (eq 1) deduced from the samples at rest, one can deduce reliably the evolution of the volume fraction as the function of the distance z from the membrane (Figure 7b). Special care has been taken on the choice of the fixed q vector equal to 1.2 nm−1 in order to be sure that this absolute scattered intensity I0 in the concentration polarization layer corresponds to the q−2 power-law decay part of the scattered intensity and is not affected by a structural evolution (increasing interactions or orientations at increasing concentration). Furthermore the particles orientations induce an anisotropic SAXS pattern. Consequently the analysis of the absolute scattered intensity in the concentration polarization layer has to be done at a q vector, which is not affected by this anisotropy. This last point is verified in Figure 8 where the integration on angular sector Ic of the oriented patterns from the cross-flow cell gives the same scattering curve (in the range corresponding to the form factor) as the radial averaged Ir of the isotropic SAXS pattern of the dispersions at rest in the capillary. 3.5. Orientation Phenomena under Shear and Pressure Forces. In the result shown in Figure 7a, the SAXS patterns exhibit an increase in scattered intensity and anisotropic level when approaching the membrane surface. At the ionic strength of 10−3 M, the mutual interaction are principally the electrostatic attractive interaction between edges and faces of the particles of opposite charge surface.31 It has been shown in previous works that at high concentration levels reached under pressure conditions (by frontal filtration17 or centrifugation35) the structural organization is anisotropic. In the accumulated layers, the particles are parallel to each other

Figure 7. (a) Variation of the scattered intensity as function of the distance z deduced form the angular sector averaged integration of the SAXS patterns measured in situ during cross-flow separation process of Laponite dispersions and (b) concentration profile and anisotropy parameter (Iv/Ih) as a function of the distance z deduced from absolute scattered intensity (Figure 7a) and calibration curve (eq 1). ΔP = 1.2 × 105 Pa and Q = 0.1 L/min at 308 min filtration time. Cp = 0%, I = 10−3 M, pH 10, tp = 12 days, T = 29 ± 1 °C.

Figure 8. Comparison of scattering intensities from Laponite dispersions at rest in capillary, radial averaged Ir (closed symbol), T = 22 ± 1 °C and in the ultrafiltration SAXS cell for different distances z, angular sector averaged Ic (open symbol), T = 29 ± 1 °C, ΔP = 1.2 × 105 Pa, and Q = 0.1 L/min at 308 min filtration time. Cp = 0%, I = 10−3 M, pH 10, tp = 12 days. 1088


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with their normal aligned along the direction of applied pressure. However, there is no regular repetition in the perpendicular direction of the applied pressure, which appears to preclude a structure made of stacked plates arranged in parallel columns. More recently,36 a detailed study of the interactions between a pair of platelets by Monte Carlo simulation has concluded that the most favorable configuration, from the free energy point of view, is that of overlapping coins, in which the platelets are aligned and slightly overlapping. In the results presented in Figure 7, panels a and b, the observation of increasing orientation is due to the mutual effect of the cross-flow and pressure conditions. First, the shear flow will orient the particles, or aggregates, and at the same time induces a breakdown of the initial particles network inducing a shear thinning effect allowing the particles to move or orientate easily in the suspension.21,27,28 Second, the transmembrane pressure that concentrates the particles and then reduces the interparticle distances allows them to get closer into contact which increases their level of orientation. It is interesting to notice that the anisotropy parameter (Iv/Ih) and the volume fraction (Figure 7b) have the same regular and continuous increase as a function of decreasing distance z to the membrane. This shows that these two parameters are linked and that the two phenomena of concentration and orientation increase near the membrane are associated and completed with each other to form a denser structure near the membrane. 3.6. Effect of Transmembrane Pressure and CrossFlow on Concentration Profiles (Cp = 0%). After the introduction of Laponite dispersion at initial ϕv0 = 0.01 in the ultrafiltration rig, several hydrodynamic parameters were explored through four steps defined in Figures 9 and 10.

Figure 10. Q cross-flow, ΔP transmembrane pressure, and corresponding J permeation flux measured simultaneously with scattering measurements (Figure 9) during cross-flow separation process of Laponite dispersions. ϕv0 = 0.01, Cp = 0%, I = 10−3 M, pH 10, tp = 12 days, T = 29 ± 1 °C.

corresponding 2D SAXS patterns are also shown for the last conditions. Figure 10 presents the simultaneously measured ultrafiltration parameters (ΔP, Q, and J). 3.6.1. Increase in Transmembrane Pressure at Constant Cross-Flow (Step 1). Initially, the concentration was equal to ϕv0 at all distances (above 100 μm) and no orientation of the SAXS pattern was detected. In step 1, for a constant cross-flow Q = 0.45 L/min, ΔP was increased step by step from (0.22 to 0.4, 0.6, 0.8, and 1.2) × 105 Pa. It does not induce any change in the concentration level above 100 μm from the membrane (Figure 9). The permeation flux increases regularly with ΔP as shown in Figure 10, which is in agreement with the fact that no deposit or cake layer is created. This corresponds to the situation for which no limiting flux is reached as shown in Figure 6a. Only a small and stable reversible polarization layer is formed. In these conditions, the hydrodynamic forces are sufficiently high to disrupt an eventual dense structural organization of the particles due to pressure forces, which tend to push and concentrate the particles toward the membrane. 3.6.2. Change in Cross-Flow at Constant Transmembrane Pressure (Steps 1 to 2). When the cross-flow is reduced from 0.45 (step 1) to 0.2 L/min (step 2), the concentration in the polarization layer starts to increase and stabilizes after a few minutes of ultrafiltration. This filtration condition corresponds to the one in Figure 6a, for which the limiting flux of about 10.4 L/m2/H is reached. As shown in Figure 9, for the same filtration conditions, the profile at time t = 178 min is a bit lower than the one at time t = 185 min, but it is the same for the two subsequent profiles at time t = 185 and 194 min, which demonstrates that a steady state value is achieved after at least 7 min under these filtration conditions. The corresponding flux measured during this step 2 decreases during time at the initialization of this step during about 7 min and stabilizes thereafter at a low level around 10.4 L/m2/H, corresponding to the same value as the limiting flux measured in Figure 6a. This decrease in permeation flux and stabilization after about 7 min is well in agreement with the stabilization of the growth of the concentration polarization layer as explained above. It is important to notice the high volume fraction reached near the membrane: at 100 μm, ϕv = 0.04 which correspond to about 9.5% in mass of dry clay. At this 0.04 volume fraction the dispersions are very strong gel with a yield stress σs of about

Figure 9. 2D-SAXS patterns and concentration profiles versus distance z deduced from in situ SAXS during cross-flow separation process of Laponite dispersions. ϕv0 = 0.01, Cp = 0%, I = 10−3 M, pH 10, tp = 12 days, T = 29 ± 1 °C.

Spatial and temporal evolution of volume fraction deduced from the SAXS measurements are presented in Figure 9 as a function of the height z from the membrane surface. The 1089


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4000 Pa and osmotic pressure of about 12 000 Pa.35 This σ0 level has been calculated from the power law σs ∝ ϕv3 and data published in ref 26. This result is the first experimental evidence that the limiting flux described in the literature1,40 is reached consecutively to the formation of a layer composed of a highly concentrated and strong colloidal gel. 3.6.3. Reversibility of the Concentration Polarization Layer (Steps 2 to 3). The next point investigated is the question if the highly concentrated deposition layer formed during step 2 is reversible or not. In step 3 for the same 1.2 × 105 Pa pressure condition as step 2, the cross-flow is increased to the same level 0.45 L/min as in step 1. This change in the hydrodynamic forces under a constant pressure implies a disturbance of the highly concentrated layer formed during step 2 and the corresponding permeation flux reaches the same level as the one at the end of step 1. In consequence, the reversibility of this concentrated polarization layer has been revealed. Finally in step 4, the cross-flow is reduced to a lower level of 0.1 L/min, which induces a further increase of the concentration and orientation of the particles near the membrane. At z = 100 μm the volume fraction is 0.055 that corresponds to a mass fraction of about 12.8% in mass of dry clay. This gel has a yield stress of about 6400 Pa and osmotic pressure of about 22 000 Pa. The deduced volume fraction from the scattered intensity is only accessible above 100 μm from the membrane surface. This is partially due to the limited spatial resolution of the X-ray beam in the vertical direction and in part to the scattering coming from the membrane surface and its support. However, by extending the concentration profile to the membrane surface in first approximation as a linear growth (presented in Figure 9) or as an exponential one, the volume fraction at the membrane surface ϕvm can be evaluated. In this filtration condition, the extended volume fraction reaches the value of about ϕvm = 0.08 (0.17 g/mL). From previous work35 for a volume fraction of 0.08, the osmotic pressure level of Laponite dispersions is slightly above 1 × 105 Pa. Comparing these 2 orders of magnitude, the osmotic pressure level is in the same order of the transmembrane pressure (1.2 × 105 Pa) applied in this experiment. In the following, a more detailed analysis is presented in order to compare the effects of pressure applied to the dispersion by osmotic stress with those applied by transmembrane pressure during cross-flow ultrafiltration. The application of the osmotic pressure model3−7 is also discussed. 3.6.4. Successive Increase and Decrease in Transmembrane Pressure. The next investigation on this concentrated polarization layer followed the reversibility of this layer by successive decreasing ΔP steps and then increasing ΔP steps at the same constant cross-flow 0.1 L/min as step 4 (Figures 11 and 12). To clarify the interpretation of the graph, each filtration condition giving rise to a concentration profile has been labeled with a number in Figure 11. The corresponding permeation fluxes measured simultaneously are indicated similarly in Figure 12. Furthermore, the level of the mean permeation flux reached along each ΔP step has been reported in Figure 12b as a function of this increasing and decreasing cycle of ΔP. Starting from point 1, the decrease in transmembrane pressure at constant Q implies a continuous disruption of the concentrated polarization layer. Simultaneously, the mean permeation flux measured for each step decreases gradually (Figure 12). It is interesting to notice that at the point 2 and more particularly at the point 3 no permanent regime of the permeation flux is reached but the

Figure 11. Reversibility features of the concentration profile versus distance z deduced from in situ SAXS during a procedure of successive decreasing and increasing transmembrane pressure at a constant crossflow rate on Laponite dispersions. ϕv0 = 0.01, Cp = 0%, I = 10−3 M, pH 10, tp = 12 days, T = 29 ± 1 °C.

Figure 12. (a) Q cross-flow, ΔP transmembrane pressure, and corresponding J permeation flux measured simultaneously with scattering measurements (Figure 11) during a procedure of successive decreasing and increasing transmembrane pressure at a constant crossflow rate of Laponite dispersions. (b) Mean level of permeation flux deduced from panel a. The label numbers correspond to the same measurements as the one in Figure 11. ϕv0 = 0.01, Cp = 0%, I = 10−3 M, pH 10, tp = 12 days, T = 29 ± 1 °C.

permeation flux is increasing during the step. This could be interpreted by the fact that at this low transmembrane pressure the concentration polarization layer is continuously eroded by 1090


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the shear flow forces, which imply an increase in permeation flux. At point 4, the pressure is fully released and the measured pressure in Figure 12b corresponds to the one imposed by the pump. The volume fraction measured is the one as the initial volume fraction ϕv0. At this point, the hydrodynamic forces will be certainly sufficiently high to mostly remove all the concentrated layers near the membrane. In the increasing ΔP part of the experiment, in point 5, a small layer is again built and the permeation flux reaches a permanent regime with a level higher than in the point 3. Consequently for two identical filtration conditions in points 3 and 5 (0.2 × 105 Pa Δp and 0.1 L/min Q), two kinds of concentration polarization layers and corresponding permeation flux are reached. The concentration profiles in point 3 exhibits a higher concentration near the membrane during the decreasing step than in point 5 in the increasing step. That is to say that when the pressure is completely released no more layers of concentrated particles are accumulated at the membrane surface and the successive increase in pressure induces the formation of a new layer less concentrated than the one which has been formed during the decreasing branch in ΔP. This change in concentration near the membrane during these two identical filtration conditions is due to the different history in preceding pressure conditions and is directly linked to the thixotropic behavior of the dispersions. This result emphasizes the well-known effect that the way the operating conditions are performed could have a great importance on the filtration performance. 3.7. Effect of tspp Content on Concentration Profiles and Permeation Flux. The same kind of measurements were done on Laponite dispersions with 6% tspp at the same initial volume fraction ϕv0 in order to explore the effect of change in mutual particle interactions on the structural organization of the polarization layers. In Figure 13, the same filtration condition (ΔP = 1.2 × 105 Pa, Q = 0.2 L/min) has been explored for dispersions with and without peptizer content at the same physicochemical and tp conditions as the one presented in Figures 4−6. This filtration condition has been compared with two reference conditions corresponding to a pressure release and an equivalent cross-flow of Q = 0.2 or 0.1 L/min. The low transmembrane pressure level measured (ΔP = 0.03 × 105 Pa) is only due to the cross-flow. For this reference condition, the volume fraction profile is flat at the same level as the initial ϕv0 volume fraction filtered and the anisotropy level Iv/Ih of about 1.1. Under the application of the same transmembrane pressure ΔP = 1.2 × 105 Pa and cross-flow conditions Q = 0.2 L/min, one can see that with 6% tspp the concentration polarization layer at distances z above 100 μm is thinner, has a lower concentration level and a lower level of orientation (Iv/Ih), than without tspp. This observation is well in accordance with the permeation flux measurements Figure 6, which shows a higher permeation flux (J = 15.2 L/m2/h) with tspp than without tspp (J = 10.5 L/m2/h). The precedent work35 on these systems has shown that the osmotic pressure (ϕv) is the same with and without peptizer at the volume fractions reached in these measurements. Therefore, even if the volume fraction reached at the membrane surface is at the same level with and without tspp, the variation of the concentration as a function of the distance z from the membrane will be modified by the equilibrium between the interaction forces and hydrodynamic forces. In others words, the differences in concentration profile with and without tspp will be mainly due to hydrodynamic forces and not to a change in internal pressure within the accumulated layers. Consequently, this result

Figure 13. Comparison of concentration profiles versus distance z, deduced from in situ SAXS during cross-flow separation process for Laponite dispersions with and without peptizer content. ϕv0 = 0.01, I = 10−3 M, pH 10, T = 29 ± 1 °C.

illustrates the importance of the rheological behavior and structural organization of the interacting colloidal particles on the filtration performances. In the case of 6% tspp the dispersion behaves like a shear thinning fluid without yield stress and has less attractive interparticle forces. Consequently, the hydrodynamics forces originating from the cross-flow have a higher capability to disrupt the accumulated layers with 6% tspp than in the case with 0% tspp which has a yield stress and stronger attractive inter particle forces. The consequence is that with 6% peptizer the polarization layer is more permeable due to a thinner accumulation of colloidal particles, less concentrated and less oriented, which increases the filtration performance. These results illustrate the performance of this nanometric approach to emphasize the effect of particle interactions on both the concentration polarization layer and filtration performance of colloidal dispersions. 3.8. Osmotic Pressure Model: Effect of Shear and Kinetic of Consolidation. The osmotic pressure model can be expressed by the following equations and considerations:41−44 In the case of fixed bed of stationary particles, the particles impart a drag force on the flowing solvent. Thus, the solvent encounters a pressure drop when traversing the particle bed, which is equal to the cumulative drag force of the stationary particles. Let us consider the osmotic pressure viewpoint for permeate transport across a concentration polarization boundary layer and a membrane. The permeate flux J can be expressed in two equivalent forms namely41 1091


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J=

ΔP − ΔΠ m ηR m

(1)

J=

ΔP − ΔΠb η(R m + R cp)

(2)

where ΔP is the transmembrane pressure, ΔΠm = Πm − Πp denote the osmotic pressure difference between the membrane surface and permeate, ΔΠb = Πb − Πp denote the osmotic pressure difference between the feed bulk and permeate, η0 the dynamic viscosity of the solvent, and Rm and Rcp the resistance of the membrane and the polarized layer to the solvent flow, respectively. ΔP = Pm − Pp in eq 1 and Pb − Pp in eq 2. With Pb, Pm, and Pp being the pressure at the bulk solution, the membrane surface and the permeate, respectively. As mentioned,41,42 the solvent flow is subject to a hydrodynamic boundary layer resistance but the energy dissipation in this layer is exactly compensated by the decrease in the chemical potential of the solvent due to its concentration gradient. Consequently, there is no pressure drop across the polarization layer. In others words, any pressure applied to the polarization layer from the bulk solution is transmitted undiminished to the membrane surface. This implies in our case that ΔP is identical in both eqs 1 and 2 as Pm = Pb. The transmembrane pressure applied is given by the difference between the atmospheric pressure in the permeate and the pressure applied to the bulk solution, resulting in ΔP = Pm. The question addressed here is the following: could we use the osmotic pressure evolution Π(ϕv) deduced from osmotic stress experiments to deduce a permeation flux J thanks to the osmotic pressure model (eq 1 or 2) or do we have to take into account others effects due to shear flow and time dependent structural organization in the value of Πm? Others related questions are: Is the volume fraction reached at the membrane surface the one for which the osmotic pressure Π(ϕv) of the system at rest equilibrates the trans-membrane pressure ΔP(ϕ v )? Are this volume fraction and the structural organization influenced by the cross-flow filtration conditions (shear-flow)? Are there any differences in the structural organization of the particles subjected simultaneously to a shear flow and uniaxial transmembrane pressure (in cross-flow ultrafiltration) and the particles subjected only to the isotropic osmotic pressure of osmotic stress experiments? Figure 14 compares the osmotic pressure Π(ϕv) of Laponite dispersions deduced from osmotic stress35 and the evolution of the transmembrane pressure ΔP(ϕv) = Pm(ϕv) as a function of the extended volume fraction at the membrane surface ϕvm. These extended volume fractions ϕvm have been deduced from the concentration profiles coming from the SAXS experiments during cross-flow ultrafiltration conditions. The evaluation has an accuracy of about 20% due to uncertainties in the evaluation (linear or exponential trend of the extension of the data, steady or unsteady state reached for the different profiles, and growth of the polarization layer along the canal). The first interesting point is that the extended volume fraction reached at the surface level during cross-flow ultrafiltration process is in the order of magnitude of the one measured by osmotic stress. This tells us that the osmotic pressure model, which is extensively used in modeling approach, is a good way to take into account the internal pressure of the colloidal dispersions as compared to the transmembrane pressure applied. Another interesting point is that the evolution of Π(ϕv) or ΔP(ϕv) follows the same

Figure 14. Comparison of osmotic pressure Π(ϕv) of Laponite dispersions deduced from osmotic stress35 and the evolution of the transmembrane pressure ΔP(ϕv) = Pm(ϕv) as a function of the extended volume fraction at the membrane surface ϕvm for different cross-flow and Cp conditions. I = 10−3 M, pH 10, T = 29 ± 1 °C.

power law increase as ϕv2. However under the same pressure, the extended volume fraction of the Laponite dispersions reached at the membrane surface during cross-flow ultrafiltration is lower of about 40% than the volume fraction of the dispersions measured under osmotic stress. Different explanations and hypothesis could be proposed to explain this difference. First, the extrapolation at the membrane surface from the concentration profile will not follow the same linear or exponential trend but a specific phenomenon at the membrane interface will change drastically the growth of the concentration near the membrane. For example specific interaction as electrostatic attractions between the membrane surface and the particles, will increase the volume fraction in the closest layers to the membrane. Second hypothesis: under cross-flow filtration, specific orientation of the particles or shear-induced structure could form a denser structural organization. This less porous oriented structure will reduce the ability of the applied pressure during cross-flow filtration to evacuate the water from the accumulated layers in comparison to the pressure effect during osmotic stress where no shear flow would induce a specific orientation. The consequence is that (for the same applied pressure) the volume fraction reached at the surface membrane during cross-flow filtration is lower than the one reached under osmotic stress. That is to say that the equation of state of the dispersions is influenced by the shear flow history during the growth of the concentration polarization layer. This hypothesis is comforted by the data in Figure 14 for the different cross-flow levels applied. One can see that at increasing cross-flow the extended volume fraction is lowered. Furthermore, this effect of orientation is also enhanced by the different ways the pressure acts on the particles for the two kinds of applied pressure method. In filtration process, the pressure applies a uniaxial strength to the particles which leads to an increasing orientation of the particles at increasing volume fractions (Figure 7b). As a consequence, in the cross−flow filtration, a part of the compression energy is dissipated in the work to rotate the particles and to disrupt some interaction. In contrary, in the osmotic stress method, the 1092


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layer can be deduced

pressure applies an isotropic force on the particles which push the water away independently of the orientation of the particles. Thus, the energy lost to orientate the particles is much lower in osmotic stress method than in cross-flow filtration. Consequently, for the same applied pressure a higher volume fraction can be reached under osmotic stress method. Third hypothesis, the kinetics of consolidation will affect the level of concentration reached near the membrane. The time scales of consolidation in ultrafiltration process are of the order of a few minutes or few hours, while in osmotic stress, the time scales of equilibrium is of about 3 weeks. Furthermore, the dispersions have thixotropic behavior and it has been shown in previous works that the time scales of build up after a disruption by shear will be of the order of few hours. Consequently, in ultrafiltration process the time scale of the strength applied to the particles to concentrate them is in the same order of the time scale of build up to reach an equilibrium structural organization. In the case of the osmotic stress, the time scale during which the pressure strength is applied is highly longer than the time scale of build up of the dispersions. Furthermore, this method implies less disruption of the particles interactions or organization as the dispersions are at rest during all the application of pressure. Effect of tspp. In the case of dispersion with Cp = 6% tspp, for the same filtration conditions, the extended volume fraction is lower than without tspp. As shown on (Figure 13), the concentration polarization layer is thinner, has a lower concentration and a lower level of orientation than without tspp. As the osmotic pressure is the same with and without tspp, the other explanation of this reduction of concentration level with tspp will be mainly due to hydrodynamic forces and not to a change in internal pressure within the accumulated layers. This effect supports the second hypothesis: the shear flow can mainly be responsible for the difference in volume fractions between filtration and osmotic stress methods. The two last hypotheses (loss of energy in orientation of the particles due to shear flow and the effect of kinetic of consolidation) are in accordance with previous results.35 In this work the authors compare the Π(ϕv) curves obtained by osmotic stress experiment or biaxial filtration or ultracentrifugation. The two last methods lead to a lower volume fraction than osmotic stress for the same pressure applied. Another analysis could be done if in a first approach we do not take into account the effect of shear or kinetic of consolidation. This analysis will allow to determine an approximation or the resistance Rcp of the concentration polarization layer and its evolution with transmembrane pressure. Denoting the osmotic pressure difference across the polarization layer as Δ∏CP = ∏b − ∏m, another expression of the permeation flux is42

J=

−ΔΠCP ηR CP

R CP =

−ΔΠCP Π − Πb = m ηJ ηJ

(4)

From our data we have access to the values of permeation flux J, the viscosity of the solvent (water) is taken to 1 × 10−3 Pa s, and from the evaluation value of the osmotic pressure in the bulk solution and the one at the membrane surface, we get the order of magnitude of the evolution of Rcp for different transmembrane pressure at Q = 0.1 L/min (Figure 15). This

Figure 15. Rcp resistance of the polarization layer and Πm(ϕvm) osmotic pressure at the membrane surface as a function of permeation flux for different transmembrane pressure ΔP at Q = 0.1 L/min. Cp = 0%, I = 10−3 M, pH 10, tp = 12 days, T = 29 ± 1 °C.

order of magnitude of the Rcp deduced from our evaluation of volume fraction at the membrane surface, is well in accordance with precedent measurements44−46 on colloidal dispersions and comforts the usefulness of the osmotic pressure model. In conclusion of this analysis of Laponite dispersions and filtration conditions (ΔP and Q) explored, the volume fraction predicted at the membrane surface has the same order of magnitude than the one reached under osmotic stress. But a lower volume fraction of about 40% less is estimated in crossflow ultrafiltration than the one measured in osmotic stress. In consequence, the use of osmotic pressure model will be a good way to describe the permeation flux through the membrane and simulate the concentration polarization layer growth of colloidal systems. But with the most probable hypothesis responsible for this difference and deduced from our analysis, it will be necessary to add some adjustments in theoretical and numerical modeling of cross-flow ultrafiltration processes. For example, to take into account the effect of shear flow on the orientation of the particles and the change in the kinetics of the consolidation, one should introduce a rheological constitutive equation which describes the evolution of the viscosity of the suspension as a function of the volume fraction, shear rate, and distance z to the membrane. This constitutive equation needs also to contain a structural parameter taking into account the time scales of break down and build-up of these thixotropic suspensions also dependent on the volume fraction.

(3)

As shown in Figure 14, we can evaluate an ideal value of Πm(ϕvm) extracted from the osmotic pressure curve determined by osmotic stress experiment knowing the extended volume fraction ϕvm. Thus we make the hypothesis that the osmotic pressure at the membrane surface is not affected by filtration conditions (combined shear flow and pressure on a short time scale). A value of the osmotic stress in the bulk Πb(ϕ0 = 0.01) = 600 Pa has been extracted from the previous results.23,29 From eq 3 an approximation of the Rcp resistance of the polarization

4. CONCLUSIONS For the first time, concentration profiles and structural organization in the dynamic polarization concentration layer have been characterized at the pertinent nanometer length scale. These results have been obtained from the analysis of 1093


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(15) Chen, J. C.; Li, Q.; Elimelech, M. Adv. Colloid Interface Sci. 2004, 107 (2−3), 83−108. (16) Madeline, J. B; Meireles, M.; Bourgerette, C.; Botet, R.; Schweins, R.; Cabane, B. Langmuir 2007, 23, 1645−1658. (17) Pignon, F.; Magnin, A.; Piau, J. M.; Cabane, B.; Aimar, P.; Meireles, M.; Lindner, P. J. Membr. Sci. 2000, 174 (2), 189−204. (18) Pignon, F.; Alemdar, A.; Magnin, A.; Narayanan, T. Langmuir 2003, 19 (21), 8638−8645. (19) Pignon, F.; Belina, G.; Narayanan, T.; Paubel, X.; Magnin, A.; Gésan-Guiziou, G. J. Chem. Phys. 2004, 121 (16), 8138−8146. (20) David, C.; Pignon, F.; Narayanan, T.; Sztucki, M.; GésanGuiziou, G.; Magnin, A. Langmuir 2008, 24, 4523−4529. (21) Ramsay, J. D. F.; Swanton, S. W.; Bunce, J. J. Chem. Soc. Faraday Trans. 1990, 86, 3919−3926. (22) Ramsay, J. D. F.; Lindner, P. J. Chem. Soc. Faraday Trans. 1993, 89, 4207−4214. (23) Mourchid, A.; Delville, A.; Lambard, J.; Lécolier, E.; Levitz, P. Langmuir 1995, 11, 1942−1950. (24) Gabriel, J. C. P.; Sanchez, C.; Davidson, P. J. Phys. Chem. 1996, 100, 11139−11143. (25) Kroon, M.; Wegdam, G. H.; Sprik, R. Phys. Rev. E 1996, 54, 6541−6549. (26) Pignon, F.; Magnin, A.; Piau, J. M.; Cabane, B.; Lindner, P.; Diat, O. Phys. Rev. E 1997, 56, 3281−3289. (27) Pignon, F.; Magnin, A.; Piau, J. M. Phys. Rev. Lett. 1997, 79, 4689−4692. (28) Pignon, F.; Magnin, A.; Piau, J. M. J. Rheol. 1998, 42, 1349− 1373. (29) Mourchid, A.; Lécolier, E.; Van Damme, H.; Levitz, P. Langmuir 1998, 14 (17), 4718−4723. (30) Cocard, S.; Tassin, J. F.; T.Nicolai, T. J. Rheol. 2000, 44, 585− 594. (31) Martin, C.; Pignon, F.; Magnin, A.; Piau, J. M.; Lindner, P.; Cabane, B. Phys. Rev. E 2002, 66, 021401. (32) Balnois, E.; S. Durand-Vida, S.; Levitz, P. Langmuir 2003, 19, 6633−6637. (33) Ruzicka, B.; Zulian, L.; Ruocco., G. Phys. Rev. Lett. 2004, 93, 258301. (34) P. Mongondry, P.; J.F. Tassin, J. F.; T. Nicolai, T. J. Colloid Interface Sci. 2005, 283, 397−405. (35) Martin, C.; Pignon, F.; Magnin, A.; Meireles, M.; Lelièvre, V.; Lindner, P.; Cabane, B. Langmuir 2006, 22 (9), 4065−4075. (36) Jönsson, B.; Labbez, C.; Cabane, B. Langmuir 2008, 24, 11406− 11413. (37) Ruzicka, B.; Zulian, L.; Zaccarelli, E.; Angelini, R.; Sztucki, M.; Moussaïd, A.; Ruocco, G. Phys. Rev. Lett. 2010, 104, 085701. (38) Narayanan, T.; Diat, O.; Bosecke, P. Nucl. Instrum. Methods Phys. Res. Sect. A 2001, 467−468 (2), 1005−1009. (39) Macosko, C. W. Rheology: principles measurements and applications; Wiley: New York. (40) Bacchin, P.; Aimar, P.; Field, R. W. J. Membr. Sci. 2006, 281 (1− 2), 42−69. (41) Wijmans, J. G.; Nakao, S.; Van den Berg, J. W. A.; Troelstra, F. R.; Smolders, C. A. J. Membr. Sci. 1985, 22, 117−135. (42) Elimelech, M.; Bhattacharjee, S. J. Membr. Sci. 1998, 145, 223− 241. (43) Bhattacharjee, S.; Kim, A. S.; Elimelech, M. J. Colloid Interface Sci. 1999, 212, 81−99. (44) Bessiere, Y.; Abidine, N.; Bacchin, P. J. Membr. Sci. 2005, 264, 37−47. (45) Grandison, A. S.; Youravong, W.; Lewis, M. J. Lait 2000, 80, 165−174. (46) Jimenez-Lopez, A J E.; Leconte, N.; Garnier-Lambrouin, F.; Bouchoux, A.; Rousseau, F.; Gésan-Guiziou, G. J. Membr. Sci. 2008, 325, 887−894.

SAXS patterns measured both as a function of filtration time and spatially resolved as a function of the distance from the membrane. Measurements were recorded with a resolution of 50 μm in growth direction of the accumulated colloids starting at 100 μm from the membrane surface. The continuous increase or decrease in concentration within the accumulated layers has been linked to the variation of the permeation flux for different cross-flow and transmembrane pressure conditions. The reversibility of this accumulation and the effect of change in interaction between particles on the concentration mechanisms inside the polarization layers have been highlighted for the first time at the nanometer pertinent length scales. Essential experimental data have been collected and they will contribute to a large improvement in theoretical and numerical modeling of cross-flow filtration processes. For example the determination of the concentration profiles offers the possibility to discuss the validity of the osmotic pressure model, in terms of the effect of shear flow and kinetic of consolidation on the concentration polarization layer.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

ACKNOWLEDGMENTS We sincerely thank Theyencheri Narayanan (ESRF, Grenoble) for his kind help in scattering experiments and fruitful discussions. We thank also Jacques Gorini (ESRF, Grenoble), Mohamed Karrouch, Didier Blésès, Frédéric Hugenell, and Hélène Galliard (Laboratoire de Rhéologie) for technical assistance. We thank Nicolas Hengl for constructive discussions. We gratefully acknowledge the ESRF for the SC 2409 beam time allocation and the Institute for Engineering and Systems Sciences of the CNRS for the financial support of the postdoctoral position of M. Abyan.

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REFERENCES

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