Keywords:

Improving industrial processes such as filtration of colloidal suspensions requires an in depth understanding of colloidal stability that is linked to interparticle interactions. In order to get insight into mechanisms of both deposit kinetics and its structuring with regard to the given bulk solution chemistry, a two-fold strategy has been adopted in the present work to the case of boehmite dispersions. First of all, the influence of salt concentration on the permeability of the filtration deposits obtained by pressure-driven frontal filtration was investigated and revealed a non-trivial evolution. Combining structural (SAXS) and rheological characterization on boehmite dispersions obtained by osmotic stress experiments, it appears that a possible interpretation of such changes can be linked to the interplay between particle anisotropy and Debye length resulting in a characteristic interparticle arrangement that reduces filtration fluxes associated with a more resistant deposit formation.

Lastly, the concentration profiles recorded from real-time in situ X-ray radiography tests at a resolution of 40 m have evidenced distinct changes in shape of concentration gradient depending on ionic strength. A relevant aspect arising from the above results underlies the combined/complex role of particle size, anisotropy and bulk solution chemistry in defining particle transport mechanisms and deposit structuring in the context of optimization of filtration processes.

INTRODUCTION

Porous alumina is one of the most commonly used inorganic carriers in heterogeneous catalytic processes. In typical heterogeneous catalysis, an active metal phase is dispersed over the pore surface of a solid support that acts as a catalyst carrier. The support is a key player in the heterogeneous catalytic reactions as it affects selectivity, activity and stability.

The textural properties of the support matrix, i.e., its surface area, pore structure and bulk density are of particular importance, as they control the loading of active metal sites into the support body which, in turn, influences catalytic efficiency. On one hand it is crucial to achieve a high dispersion of the active metallic species in the support. For instance, one platinum particle with a diameter of 1 nm contains less than 50 atoms and reaches high dispersion conditions close to 100% [START_REF] Geus | Production of Industrial Catalysts[END_REF]. Optimal catalytic efficiency per unit mass of metal is economically of interest and, consequently, the presence of small particles providing high dispersion is requested. As an example, for the Co/Al2O3 catalyst system investigated in [START_REF] Jacobs | Fischer-Tropsch synthesis: influence of support on the impact of co-fed water for cobalt-based catalysts[END_REF], the metal oxide-support interaction with a cobalt loading of only 25% and crystallite sizes around 11 nm was found to provide optimum reducibility and surface area [START_REF] Visconti | Alumina: A Key-Component of Structured Catalysts for Process Intensification[END_REF]. On the other hand, the porous network of the carrier also plays a key role as it controls the molecular transport of reactants by diffusion. Consequently, adjusting pore size distribution and tortuosity of the porous network in the support is of prime importance for ensuring optimal performance of the complete catalytic system.

From an industrial point of view, the production of supported alumina catalyst involves a series of individual operations, i.e. (i) alumina precursor synthesis, (ii) filtration, (iii) washing, (iv) final shaping and (v) calcination.

It has been widely documented [START_REF] Euzen | Handbook of Porous Solids[END_REF] that boehmite remains the core material in alumina catalyst carrier, leading to a highly increased microporous surface area (500 m 2 •g -1 ) after decomposition at 500C. One common technological procedure starts with boehmite precipitation from soluble aluminum salts solution, sodium aluminate being usually used as a source of Al. In such a scheme, it is crucial to add a washing procedure, as the precipitate contains excess of alkali ions.

The Ziegler procedure is another way to synthesize microporous boehmite. Originally, it was aimed at producing linear fatty alcohols using trialkyl-aluminum as intermediate species formed via oligomerization reaction of ethylene with Al metal.

( )

ROH 3 AlOOH O H 2 OR Al 2 3 + → + .
Oxidation of aluminum trialkyls leads to aluminum trialkoxides, which after hydrolysis yields boehmite and alcohol by-products [START_REF] Diblitz | Manufacturing of raw materials for the catalyst industry[END_REF]. Depending on the synthesis history of boehmite, the resulting microcrystalline structure varies from fibrous-like to lamellar and rhombic [START_REF] Busca | The surface of transitional aluminas: A critical review[END_REF].

After boehmite is formed, the resulting slurry passes through a filtration/washing process to eliminate impurities and the filtered cake is subsequently spray-dried to produce a powder suitable for handling. Spray drying is followed by a redispersion (or so-called peptization) of pre-dried boehmite powder that is needed for the subsequent shaping procedure by means of extrusion. The next step involves drying and subsequent calcination to induce the transformation of boehmite into γ-alumina. The final step in the production of supported catalysts is the activation of the support surface with catalytically active metal sites. This is generally carried out by impregnation, i.e. the porous support is contacted with a solution containing active metal salts. Adsorption under controlled conditions then leads to the formation of nano-dispersed active metal clusters. Numerous parameters are involved in the impregnation process, such as concentration of the metal precursor salt solution, contact duration, nature of the solvent, temperature, etc...

Once the supported catalyst has been obtained, it is dried and subsequently calcined in air to decompose the precursor species to the oxide state. It must be pointed out that, in addition, considerable enhancement of the mechanical resistance of the catalyst system is achieved upon calcination.

Our work is primarily focused on the mechanisms of colloidal deposit structuring upon membrane-based solid/liquid separation that controls the flow rates and, thus, the efficiency of filtration process. One of factors affecting flux declination is related to continuous retention of particles on the membrane surface resulting in the formation of highly resistant filter deposits. At large scale, this implies a small output caused by considerable time consumption and processing costs followed by additional needs for intensive washing steps in order to maximise purification from unwanted residual substances.

When filtration of colloidal slurries takes place, such a process should not be viewed as a simple sieving mechanism. The particles to be separated are retained onto the membrane surface resulting in several phenomena, such as pores fouling, formation of concentration polarisation layer, particles arches, gel formation…. All these phenomena occur at length scales where the role of interparticle interactions cannot be neglected anymore. Such interparticle interactions resulting from the combination of electrostatic and dispersion forces tremendously affect colloidal transport with regard to flow dynamics (i.e. gradient diffusion due to Brownian motion) as well as phase behavior and rheological properties with respect to growing concentration of solids in filtration deposit.

The first goal of the present study focuses on the respective role of physicochemical conditions in deposits formation mechanisms of colloidal boehmite dispersions.

To reach such a goal, the "off-line" frontal filtration tests were firstly acquired to analyze the dependence of permeability K on salt concentration and the impact of pressure difference on the filtration rate and deposit compressibility. In order to gain further insight into the mechanisms of particles deposition at a filtrating membrane surface, a dead-end filtration cell coupled with X-Ray radiography was developed to study the filtration of boehmite dispersions with contrasting and controlled physico-chemical properties. Using such a cell allows visualizing in real time deposits growth kinetics, which, in turn, provides information about particles accumulation upon densification. Using a quantitative treatment of X-ray radiographies, we are then able to derive concentration profiles as a function of the distance from the membrane and to monitor the evolution of these profiles with increasing filtration time.

In order to deepen the analysis of the mechanisms of particles deposition revealed by in situ filtration studies, the second goal is addressed to the role of colloidal interactions in defining characteristic phase behavior, rheology and structural organization in concentrated dispersions. To reach this second goal, we carried out osmotic stress experiments to obtain boehmite dispersions with various volume fractions in order to assess both rheological and structural characteristics in a homogenously concentrated state.

For ease of reading, the present manuscript is organized in 5 chapters.

Chapter 1 summarizes the literature review. Firstly, the basic principles of DLVO theory of colloidal stability will be introduced. In this regard, experimental data on anisometric systems reviewing a wide array of states and structural properties are then included. The principal mass transport model that will serve this study will be briefly presented afterwards. The key issues associated to filtration mechanisms of colloidal systems will be further reviewed for better identifying the current state of research regarding filtration phenomena.

Chapter 2 is devoted to the materials and experimental methods used in this study. This chapter describes the preparation of boehmite dispersions, the experimental filtration setups and other analytical methods and procedures.

In Chapter 3, ultrafiltration studies on the boehmite dispersions in "dead-end" mode are first presented. Their flow properties are evaluated and correlated with structural organization determined by using small angle X-ray scattering (SAXS) technique. Then, the results of filtration experiments coupled with X-Ray radiography are given. The design of these experiments followed by a consequent data discussion and analysis is therefore presented.

Chapter 4 contains the characterization of the phase behavior, structural and rheological properties of boehmite dispersions. In this chapter, we attempted to investigate the different mechanisms of particles organizations with regard to bulk solution chemistry, i.e. ionic strength, surface charge and solids content. Particular attention is given to the rheology of boehmite dispersions that is therefore followed by a structural characterization probed at a small length-scale using small angle X-Ray scattering technique. Following discussion is supported by an integrated analysis of obtained results throughout the chapters.

Chapter 5 contains a state-of-art experiment acquired on ANATOMIX Tomography beamline at synchrotron SOLEIL (Gif-sur-Yvette, France). Filtration runs were carried out on boehmite sols with addition of non-Brownian boehmite aggregates to monitor their displacement upon deposit formation at a high spatial resolution of 0.65 µm. Data treatment was implemented with the help of imageJ and discussed in the context of particle motion in non-Newtonian fluids.

CHAPTER I: Literature review I.1 Introduction: colloidal phenomena

Colloidal systems are widespread in nature and constitute an integral part of associated products and processes, ranging from food, pharmaceuticals to cleaning agents, paints or cosmetics. Simple colloidal dispersions generally refer to a two-component mixture containing the solid phase dispersed in the continuous medium. However, both phases may either co-exist in gas, liquid or solid states that results in a variety of other types of colloidal systems, for instance emulsions (unmixed liquidliquid state), mist (droplets of liquid in gaseous phase), foams (air encapsulated in liquid) or milk (solids dispersed in liquid). Colloids refer to particle sizes between 1 nm and 1 m, which is a key element leading to a list of astonishing properties. The colloidal nature of dispersions is often experimentally evidenced by their ability to scatter light or so-called opalescence that manifests itself by the appearance of a luminous cone of light directed through a colloidal dispersion (Tyndall effect). Furthermore, some dispersions are well distinguished by their long-term kinetic stability, i.e. sols containing noble particles (gold, silver ), yet upon salt addition such systems undergo irreversible flocculation that reflects the loss of thermodynamic stability. Owing to small particles size, such metastable state is closely attributed to a highly developed surface area, giving rise to a high contact area at the interface. Hence, such interaction forces could result in surface charging effects, adsorption of ions or surfactants, etc..., thus providing the basis for understanding colloidal stability. To give a sense about physical aspects of intercolloidal interactions with regard to the acting forces, the theory of Derjaguin, Landau (1941) and Verwey, Overbeek (1948) (DLVO) is reviewed next.

I.2 Interactions in colloidal systems

Interactions between colloidal particles are governed to a large degree by colloidal forces. This section focuses on colloidal interactions on the basis of the DLVO theory of colloid stability. This theory implies that potential energy of particle interactions is determined by the balance of two main contributions, the electrostatic repulsion arising from overlap of the diffuse double layers and the van der Waals attraction due to electric dipoledipole interactions.

I.2.1 Potential distribution around a charged surface: the Poisson-Boltzmann equation

Immersing particles in a liquid can lead to surface charging effects, for instance by adsorption of free ions, surfactants or polyelectrolytes. The presence of dissociated surface groups, of adsorbed species and added salt ions results in a structured ionic environment in the vicinity of charged surface that is schematically depicted in Figure 1.2. Generally, the electrical double layer is formed by the particle surface and its counterions (ions of opposite charge to the particle surface) apart from it, where they are less associated. Therefore, the particle surface comprises the thin layer of adsorbed counterions that effectively balance the surface charge of particle, namely Stern layer. The ions in Stern layer are strongly bonded and this region acts as capacitor over which the potential decays linearly. Clearly, across this region till the border with diffusive plane, the Poisson-Boltzmann equation does not hold. Accordingly, it defines the major assumption of electrical double layer concept, where only the diffuse layer is considered from which the ionic atmosphere begins to follow PB dependency (Stern layer is neglected). The layers are considered to behave as charged capacitors C1 and C2, taken to be of constant value between the surface and the inner Helmholtz plane and between the inner and outer Helmholtz planes, respectively. Adapted from [START_REF] Wood | Electrochemistry of the boehmite-water interface[END_REF].

The Stern layer is also known to be further split (for instance, in the two pK triple layer model) into two additional Helmholtz planes, as depicted in Figure 1.2. Its combined thickness is of the order of few angströms and reflects the finite size of the charged surface groups (coions) and strongly bonded counterions. The outer limit of this region is often associated with the plane of shear, beyond which the mobile ionic species of the bulk medium are present. The potential at this plane is adopted to be as a good approximation to the zeta potential that can be defined experimentally from electrophoretic mobility tests.

Thus, beyond Stern layer, the electrostatic potential  of a charged particle surrounded by the ionic medium obeys Poisson-Boltzmann equation where 0 is the ionic concentration of ions i in the bulk solution with valence z at x = 0 (infinite away from the charged surface), e0 the elementary electric charge, kB Boltzmann's constant, T the absolute temperature, r the relative permittivity of the electrolyte solution and 0 the permittivity of vacuum. Equation (1.1) contains two components: (i) the Boltzmann equation ascribing the distribution of charge density of each ionic species in the system, as Since the full Poisson-Boltzmann equation represents a nonlinear differential equation, it is usually approximated using both the Gouy-Chapmann theory that is a onedimensional simplification for electric double layer and the Debye-Hückel theory, which a special solution for spherical symmetric system given by ( )

         -   =  T k ze exp B 0 0 q , ( 1
x exp x  -    . (1.4)
The  denotes the potential at the surface;  -1 represents the characteristics decay length of the potential, namely Debye length, determined by the ionic strength and the dielectric properties of the suspending medium as (1.5)

The exact form of interaction pair potential slightly varies for surfaces or particles of different geometries. However, it always decays exponentially with distance x and the Debye length , unlike the van der Waals interaction where the decay is a power law having very different exponents depending on both geometry and nature of the material that can be approached through the Hamaker constant [START_REF] Israelachvili | Intermolecular and surface forces[END_REF].

For instance, for two similar spheres of radius a with a center-to-center separation distance x, the linear superposition of their potentials leads to the sphere-sphere interaction energy as ( ) ( ) ( )

a 2 x exp R a 4 x 2 2 0 r -  -    =  (1.6)
under assumption of weak surface potentials and small a<3.

A pure electrostatics model is often exploited in rationalizing the viscoelastic trends of electrostatically stabilized dispersions. On the basis of Equation (1.6), the rheological characterization was employed in the present work to study the effect of electrostatic screening on evolution of flow properties in boehmite gels. This will be thoroughly discussed in Chapter IV of the present manuscript.

I.2.2 Van der Waals interaction between two particles

The tendency of colloidal particles to form agglomerates and flocs through collisions due to Brownian motion is a clear indication of the presence of an attractive force. The origin of these forces purely arises from quantum mechanics: a molecule with a fluctuating instantaneous electric dipole serves as a starting point. When two molecules approach each other, the attraction arises from the local fluctuations in the polarisation due to generated electric field, which results in an induced dipole moment.

The potential energy of the van der Waals forces between two similar molecules with separation R between their centers in a vacuum is given by the well-known London's expression (1937)

( ) 6 R C R V - = , (1.7) where ( )
. 4 4 h 3 C 2 0 2 0    = (1.8)
C is the London-van der Waals constant, 0 the electronic polarisability,  the characteristic frequency of the molecule, and h Planck constant. For a Bohr atom,  holds 3.310 15 s -1 , so that h = 2.210 -18 J, which is the energy of an electron in the first Bohr radius and corresponds to the energy required to ionize the atomthe first ionization potential I.

It is worth noting that Equation (1.7) holds when the intermolecular spacing is shorter than the characteristic wavelength 2/. At larger distance V(R) the 1/R 6 dependence becomes proportional to 1/R 7 due to so-called retardation effects. This happens when two molecules are sufficiently apart from each other, so that the time taken for electric field to reach the second molecule and return back to the first one becomes comparable with the period of the fluctuating dipole 1/ itself.

Under the assumption that this interaction is additive, Equation (1.7) might be integrated for all atomes or molecules in one body with those in another one, giving the "twobody" potential. This procedure was applied for numerous geometries that might be viewed elsewhere. In the interest of brevity, our discussion is limited to the case of two parallel planar plates:

( ) , R 12 A R V 2 VdW  - = (1.9)
where A, the Hamaker constant is of the order of 10 -19 J for interactions in vacuum.

Chapter I: Literature review Equation (1.9) provides an analytical form for the long-range interaction potential decaying with distance as (1/R) 2 with Hamaker constant characterizing material properties. However, the additivity approach faces serious limitations when it comes to many-body interactions in condensed medium. Furthermore, retardation effects that are introduced by the phase shift between dipoles causes variation of Hamaker constant A with separation distance.

Given problems are fully solved in the continuum theory (Lifshitz, 1956), where objects are placed in absorbing media and considered as coupled harmonic oscillators in the fluctuating electromagnetic field. The materials are fully characterized by their bulk properties, i.e. dielectric permittivity and refractive indices.

The total van der Waals energy (VDW) between two parallel planar surfaces interacting across a medium 3 for the nonretarded Hamaker constant effects is given by ( )

2 eff VdW R 12 A R V  - =
.

(1.10) , 1975) for interaction between polystyrene plates in pure and salt water (0.1 M). From [START_REF] Russel | Colloidal dispersions[END_REF].

It is worth noting that equations of the van der Waals force for all separation distances remain complex. However, for a better understanding, the concept accounting for retardation can be illustrated schematically using effective Hamaker constant Aeff = A=0 + A>0. Accordingly, when the separation distance approaches the retardation regime, the first term A=0 always remains unchanged, as it is not subjected to retardation phenomena. However, this zero-frequency term essentially represents an electrostatic interaction with regard to charges displacement in an electrostatic field. Hence, it is affected by electrostatic screening in electrolyte solutions, typically at smaller separation distance than that of retardation A >0 . Therefore, it should be mentioned that at separations greater than 2 nm, the attraction is mostly determined by the dispersion force term only.

Contributions to electrostatic screening and retardation phenomena are well illustrated in Figure 1.3. From Figure 1.3 it follows that the van der Waals force is much reduced in a solvent medium. The effect of vacuum (not shown here) gives for Aeff (0) = 7.910 -20 J, which exceeds those in pure water Aeff (0) = 1.310 -20 J due to the larger contributions from the ultraviolet frequencies. For water, Hamaker constant Aeff (0) mostly contains the zero-frequency and infrared terms. Because of such low-frequency contribution with water, the resulting interaction potential in the presence of 0.1 M aqueous solution with  -1 = 1 nm illustrates screening of the  = 0 term due to ionic conduction, causing the potential reduction. Higher ionic strengths, for instance, 1 M NaCl with  -1 = 0.3 nm, effectively suppress the  = 0 term, causing decrease of attraction. The non-retarded  = 0 term for salts containing aqueous solution disappears for large separation distance R, reflecting retardation effects with R increase [START_REF] Israelachvili | Intermolecular and surface forces[END_REF][START_REF] Russel | Colloidal dispersions[END_REF].

I.2.3 DLVO theory of colloid stability

The framework of DLVO theory (Derjaguin, Landau, 1940 and Verwey, Overbeek, 1948) essential in understanding colloid stability has been expressed via the potential energy of interaction, as a sum of two components: (i) electrostatic repulsion arising from the overlap of diffuse double layers and (ii) van der Waals dispersion forces arising from dipoles interactions in fluctuating electromagnetic field. Schematically, the combined action of these two forces in a 1:1 electrolyte solution between two similarly charged surfaces is illustrated in Figure 1.4.

Accordingly, it follows that the VDW attraction decays quite rapidly, since it is an inverse power law function, i.e. W  -1/R n , while the double-layer interaction potential decreases slower: it decays exponentially with separation R, and typically has a range of the order of the Debye length  -1 . As a result, the VDW attraction exceeds the double-layer repulsion at a very small separation distance, whereas the double-layer repulsion is dominant at larger R. Variations in the ionic strength (as it is a measure of the concentration of charge species in a given electrolyte solution and it is inversely proportional to Debye length  -1 ), as well as pH control the range of the electrostatic repulsion, while the van der Waals attraction is dependent on particle size, medium bulk properties (dielectric permittivity and refractive indices) and particle-to-particle separation distance R. One can notice that the curves exhibit two characteristic minima and one maximum. The energy minimum at contact is known as a deep primary minimum, which is located at very close separation distances (several 100 kBT units). In more concentrated electrolyte solution appearance of secondary minimum occurs, separated by the energy barrier from the primary minimum. Electrostatic repulsion gives rise to this energy barrier, usually between 1 and 5 nm, in cases where highly charged surfaces are concerned (in dilute conditions). When approaching this barrier, the particle should have sufficient kinetic energy due to thermal motion to overcome the primary minimum resulting in aggregation. When the surface charges approaches zero or at high ionic strength, particles experience only the VDW attraction and attract each other at all positions (approaching lower dashed curve in Figure 1.4). There is a critical concentration of electrolyte, Ccrit, at which the barrier fully disappears and particles proceed towards slow aggregation, known as coagulation or flocculation [START_REF] Israelachvili | Intermolecular and surface forces[END_REF]. From [START_REF] Israelachvili | Intermolecular and surface forces[END_REF].

I.3 Colloidal interactions in practice: diversity of particular characteristics

When suspended in aqueous medium, particles usually acquire charges. Interactions are mainly established via electrostatic repulsion that counteracts particles aggregation, precipitation processes induced by natural attractive Van der Waals forces. From this point, the role of electrostatic interactions in combination with the given physico-chemical characteristics of colloidal systems is of considerable importance that might result, for instance, in formation of ordered lattice (latex spheres in water), Wigner glass transition (laponite) and causes the dispersibility of boehmite in water, similarly to clay minerals. It suggests therefore that focusing on the nature of interparticle interactions allows explaining the broad range of phase behavior, also taking into account the physico-chemical aspects (particle size and shape, surface charges, rheological behavior,…). Thus, the next section introduces a brief characterization of anisometric systems with particular focus on boehmite dispersions.

I.3.1 Structure of boehmite

Boehmite is composed of lamellar AlO6-octahedra sheets, 3 nm thick [START_REF] Mange | A microstructural investigation of nanostructured boehmite films prepared by the sol-gel route[END_REF]. The aluminum atom is coordinated by four O(1), located inside the layer, and two O(2) that are linked via hydrogen bonding with a neighboring sheet (marked as dashed lines in Figure 1.5). This provides a highly polarised, hydroxyl-functional surface, resulting in an enhanced double layer thickness, which contributes to the dispersibility of boehmite in water [START_REF] Hill | Hydrogen Atoms in Boehmite : A Single Crystal X-Ray Diffraction and Molecular Orbital Study[END_REF]. Due to its particular crystallographic structure resembling those of most studied mineral clays (i.e. smectic clays such as montmorillonite, hectorite), numerous boehmite properties are similar to those of clay minerals, although the origin of the electric charges is different.

Similarly to clay minerals, boehmite exhibits a swelling behavior in water. However, its mechanism is quite different from the one occuring in clay minerals (i.e. montmorillonite, smectite), which is closely linked to the question of interlayer water location in the crystallographic structure. Clay minerals can accommodate large quantities of water that is accompanied with lattice expansions. However, in the case of boehmite the lattice expansion was shown to be insufficient to support this as the location of water excess. Furthermore, Xray diffraction patterns [START_REF] Baker | Water content of pseudoboehmite : A new model for its structure[END_REF] also revealed an increase of water content with a decrease in crystallite size. Since the size decrease corresponds to an increased number of peripheral oxygen relatively to oxygen encapsulated in the interlayers, this suggests the main contribution of crystallite surface in accomodating such excess of water. Further studies via Controlled Transformation rate Thermal Analysis (CRTA) [START_REF] Fauchadour | Peptization mechanisms of boehmite used as precursors for catalysts[END_REF] have therefore evidenced that this water excess is likely located on the surface of boehmite crystals, distinguishing strongly linked chimisorbed water on [START_REF] Buscall | Viscoelastic properties of concentrated latices. Part 2.-Theoretical analysis[END_REF] and [001] faces from physisorbed water to surface hydroxyls on [010] faces and to water molecules of the [START_REF] Buscall | Viscoelastic properties of concentrated latices. Part 2.-Theoretical analysis[END_REF] and [001 ] faces, which is fully supported by ab initio calculations in [START_REF] Raybaud | Morphology and Surface Properties of Boehmite (γ-AlOOH): A Density Functional Theory Study[END_REF]. Therefore, these results strongly suggest some consistency with the qualitative model of Baker [START_REF] Baker | Water content of pseudoboehmite : A new model for its structure[END_REF] describing water content at the boehmite surface, in contrast to the commonly accepted practice of locating this excess water as a bonded hydrogen between the double sheets, as mentioned above.

I.3.2 Synthesis of ultradispersed boehmite particles

Colloidal boehmite dispersions are often synthesized by precipitation in aqueous solution from alumina salts. The shape of boehmite nanoparticles can be well controlled during its synthesis by varying pH conditions. Thus, after heating procedure at pH 4-5 fiber and rod structures are produced with length around 100 nm. The fibers consist of aggregated platelets (36 nm), exhibiting (100) lateral faces and (010) basal planes. The particles synthesized at pH = 6.5 look like pseudohexagonal platelets ( 155 nm) with ( 100) and ( 101) lateral faces, whereas synthesis at pH = 11.5 leads to diamond-shape particles 10-25 nm wide with (101) lateral faces, as illustrated in Figure 1.6. Boehmite particles acquire a net positive charge in aqueous suspending media until the isoelectric point is reached. The pHiep ranges between 4.5 to 9.8 depending on the nature and amount of impurities present. In the context of particle synthesis, changes in the acidity range enable to control particle shape. Around pHiep the surface energy is sufficiently high to develop surface, so it promotes flat objects in order to maximize the basal (010) plane (shape of plates or sheets). Moving away from iep region causes a strong decrease of surface energy, becoming energetically unfavorable to develop surface, so that particles shape consists of linear objects (flat rods or laths). Accordingly, the basal surfaces of boehmite particles reach their highest development and it does not evolve over large acidity range, whereas the edge surfaces, in contrast, are related to the pH values [START_REF] Bleta | An efficient route to aqueous phase synthesis of nanocrystalline γ-Al2O3 with high porosity : From stable boehmite colloids to large pore mesoporous alumina[END_REF], which defines particles organization in suspending media.

To be scaled up towards industrial conditions, the present study focuses on industrial boehmite powder Pural highly dispersable in aqueous media. Detailed description of its preparation mechanisms is presented in the next section.

I.3.3 Acidic peptization of boehmite

When boehmite powder comes into contact with a strong acid, two major mechanisms implying acid consumption have been reported [START_REF] Cristiani | Effect of ageing time on chemical and rheological evolution in γ-Al2O3 slurries for dip-coating[END_REF] and lead to electrostatic stabilization and final formation of a colloidal sol: (i) surface charging and (ii) dissolution reactions. The charging of boehmite surface involves protonation/deprotonation of surface oxygen atoms. Adsorption of ions occurs in the inner region of Helmholtz plane with potential-determining ions H + and OH -, adsorbing on available boehmite surface sites as

+ → + + + .
These two processes that are reported to be essential in defining the nature of the final colloidal system are referred to as acidic peptization. As soon as the sol stabilizes via acid consumption, it is characterized by a gradual increase of pH, as demonstrated by [START_REF] Cristiani | Effect of ageing time on chemical and rheological evolution in γ-Al2O3 slurries for dip-coating[END_REF] [START_REF] Cristiani | Effect of ageing time on chemical and rheological evolution in γ-Al2O3 slurries for dip-coating[END_REF]. Therefore, acid consumption due to surface charging processes is reported to be relatively small compared to dissolution reactions of alumina.

It is worth mentioning the study of [START_REF] Conroy | Importance of interlayer H bonding structure to the stability of layered minerals[END_REF] [START_REF] Conroy | Importance of interlayer H bonding structure to the stability of layered minerals[END_REF] who focused on the dissolution of boehmite under electron beam irradiation, showing the effect of hydrogen bonding on mineral solubility. By comparative analysis with gibbsite, where no dissolution was found, the results strongly suggest that the dissolution process in boehmite is likely to be governed by destabilization of the hydrogen bonds that hold the structural units together. This was therefore evidenced in the analysis of O-H stretch frequency from IR spectroscopy showing the weaker terminal O-H bond strength as compared with those for gibbsite, which, in turn, implies the stronger H bonding in the interlayer of boehmite that results in its structural stability.

The work of Fauchadour [START_REF] Fauchadour | Etude de la peptisation de la boehmite[END_REF] showed that the release of Al species in boehmite dispersions is drastically increased with higher acid content above a given critical concentration (0.035 M), as illustrated in Figure 1.7 for four different boehmite powders dispersed in HNO3. Inversely, for pH below 4, the acid consumption becomes partitioned between surface charging and drastically increased dissolution processes. One might expect that dissolution reactions result in a complex speciation of aluminium within sol formation, which is strongly influenced by the bulk chemical conditions. Studies about speciation of aluminum resulting in anomalous stability and rheological properties have been largely debated and are reviewed in the next paragraph.

I.3.4 Speciation of aluminium in boehmite dispersions

One of the earliest attempt to interpret the exceptional stability and elasticity of boehmite dispersions on the basis of interaction forces between platelets dates back to the 70s and was carried out by Ramsay and co-workers [START_REF] Ramsay | Structure and stability of concentrated boehmite sols[END_REF]. Stable boehmite dispersions were found highly elastic and thixotropic, which was assigned to a short-range interparticle repulsion, presumably, due to the presence of polymeric aluminum cations forming on the particle surface during acidic peptization. This is somehow reminiscent of the case with polynuclear alumina ions in hydrolysed aluminium salts solutions generating numerous ionic forms, as illustrated in Figure 1.8. The short range repulsion mentioned by Ramsay et al. [START_REF] Ramsay | Structure and stability of concentrated boehmite sols[END_REF] was indeed evidenced in the work of Karaman et al. [START_REF] Karaman | A comparison of the interaction forces between model alumina surfaces and their colloidal properties[END_REF] who studied the interactions between alumina coated silica spheres and flat alumina substrates over pH range 3-6.5 using atomic force microscopy (AFM). In this work the observed repulsion was assigned to natural hydration forces combined with the observed gel layer at the alumina surface. Interestingly, formation of such hydrated gel layer has been suggested by Beattie et al. [START_REF] Beattie | Anomalous aggregation behaviour of aluminium oxyhydroxides[END_REF] to be in correlation with the presence of mono-or polymeric ionic species, as for instance Al13 ion, that condenses at the alumina surface and provides the repulsive forces at small interparticle separations (< 5 nm). Its existence (Al13 with 12 edge-linked octahedral surrounding a central tetrahedron atom) was first reported by Johansson et al. [START_REF] Johansson | On the Crystal Structure of Some Basic Aluminium Salts[END_REF][START_REF] Johansson | On the Crystal Structure of a Basic Aluminium Sulfate and the Corresponding Selenate[END_REF][START_REF] Johansson | The Crystal Structures of [Al2(OH)2(H2O)8](SO4)2.2H2O and [Al2(OH)2(H2O)8](SeO4)2.2H2O[END_REF] in the 1960s, therefore supported by 27 Al NMR and SAXS studies of Akitt et al. [START_REF] Akitt | Aluminium-27 nuclear magnetic resonance studies of the hydrolysis of aluminium(III). Part 5. Slow hydrolysis using aluminium metal[END_REF][START_REF] Akitt | Multinuclear magnetic resonance studies of the hydrolysis of aluminium(III). Part 8. Base hydrolysis monitored at very high magnetic field[END_REF][START_REF] Akitt | New 27Al NM studies of the hydrolysis of the aluminum(III) cation[END_REF] in base-hydrolysed solutions of aluminum chloride and aluminum nitrate.

An extensive analysis of monomeric species behavior was carried out by Morgado et al using 27 Al nuclear magnetic resonance ( 27 Al NMR) spectroscopy, revealing some important features in the nature of the resulting boehmite colloidal sols. A narrow resonance was found characteristic of the hexaaquo ion [Al(H2O)6] 3+ and its monomeric hydrolysed species, (e.g., [Al(H2O)5(OH)] 2+ ), undergoing self hydrolysis, as similar to Al 3+ aqueous system. This resonance accounts for less than 10% of the total aluminum. The other much broader resonance was assigned to the high molecular-weight species, containing relatively symmetrical Al sites. Further detailed analysis pointed out that this resonance contributes to all aluminum present in the given sol system, that is to say, the boehmite colloidal particles. Moreover, when under certain conditions the sol/gel transition occurs, this resonance peak decreases progressively in intensity giving rise to another broad signal at about the same frequency. This latter signal was tentatively assigned to a larger developed structure (due to its reduced mobility), presumably, resulting from a gradual aggregation of the former species (colloidal boehmite particles), resulting in a gel network.

Furthermore, released Al 3+ monomers allowed to confirm partial dissolution processes during acidic boehmite peptization, as stated by Fauchadour [START_REF] Morgado | Characterization of Peptized Boehmite Systems : An27Al Nuclear Magnetic Resonance Study[END_REF]. Neither tridecameric complexes, e.g. [AlO4Al13(OH)24(H2O)12] 7+ , nor any tetrahedrally coordinated species were ever found, at least, in the investigated pH range (1.4-3.7).

I.3.5 Rheological features

Rheological studies (viscoelasticity, yield stress, viscosity, time dependence of the rheology,…) provide additional insights into resulting microstructure of colloidal dispersions for a given solution chemistry (surface charge, concentration and particle size), which, as illustrated in previous sections, is closely influenced by particle-particle interactions.

I.3.5.1 Viscosity, shear-thinning and thixotropic behavior

Rheological studies on boehmite dispersions have received particular attention. Evanko et al. [START_REF] Evanko | Influence of aqueous solution chemistry on the surface charge, viscosity and stability of concentrated alumina dispersions in water[END_REF] demonstrated the impact of interparticle interactions on the viscosity of concentrated boehmite dispersions as a function of solution pH and ionic strength. As shown in Figure 1.9, the viscosity reaches a maximum at pH 9.0, i.e. at the point of zero net proton charges (pznc) for boehmite. This is consistent with promoted particles aggregation, as surface charges are highly shielded at the pznc, thus lowering electrostatic repulsions. Furthermore, this viscosity maximum tends to decrease at elevated ionic strength. Figure 1.9. Effect of pH and ionic strength on viscosity of 15 wt.% boehmite dispersions (Shear rate of 15.8 s -1 ). Adapted from [START_REF] Evanko | Influence of aqueous solution chemistry on the surface charge, viscosity and stability of concentrated alumina dispersions in water[END_REF].

Presumably, this is due to improved particles mobility, resulting from promoted edge-to-edge particles linkage when increasing ionic strength. A similar evolution with ionic strength was found on the yield stress of kaolinite dispersions, and was interpreted as varying edges and faces configurations as a function of both pH and ionic strength [START_REF] Rand | Isoelectric point of the edge surface of kaolinite[END_REF]. Most concentrated suspensions also exhibit a shear-thinning behavior, as a result of stress-induced aggregates disruption. Typical shear-thinning behavior in concentrated boehmite dispersions is illustrated in Figure 1.10 displaying a decrease in viscosity with increasing shear rate for various volume fractions [START_REF] Bunker | The aqueous chemistry of oxides[END_REF]. Adapted from [START_REF] Pignon | Thixotropic behavior of clay dispersions : Combinations of scattering and rheometric techniques[END_REF].

More specifically, shear-induced structure orientation was experimentally demonstrated by Pignon et al. [START_REF] Pignon | Thixotropic behavior of clay dispersions : Combinations of scattering and rheometric techniques[END_REF] on laponite dispersions that has many similarities with boehmite dispersions, for instance, with regard to its small anisotropy as compared to other natural clay minerals [START_REF] Levitz | Liquid-solid transition of Laponite suspensions at very low ionic strength : Long-range electrostatic stabilisation of anisotropic colloids[END_REF]. Therefore, laponite has been largely considered as a model reference system for fundamental studies of clay colloid behavior. Results of Pignon et al. [START_REF] Pignon | Thixotropic behavior of clay dispersions : Combinations of scattering and rheometric techniques[END_REF] presented in Figure 1.11 show that changes in structure occur upon an increase in shear rate. When approaching the critical shear rate of 130 s -1 , the curve exhibits a slope change associated to modifications in orientational features. Beyond this critical value, stress increases progressively with shear rate, which is followed by the appearance of a fully isotropic scattering pattern, suggesting a structural breakdown onto smaller aggregates. At the smaller length scales (1-100 nm) investigated by SANS, no anisotropy in the scattering patterns were observed. This leads to a very important consequence that structure orientation and its disruption in shear-thinning behavior is entirely determined by the presence of macroscopic aggregates observed at large length scales (of 1 m) rather than elementary particles.

Shear-thinning is usually combined with thixotropic behavior. As a time-dependent property, thixotropy is characterized by a reduction in viscosity over time at a constant shear rate, as illustrated in Figure 1.12. Again, authors [START_REF] Weston | Connecting particle interactions to agglomerate morphology and rheology of boehmite nanocrystal suspensions[END_REF] observed the critical share rate of 100 s - 1 , at which the apparent viscosity was found to decrease substantially, by dropping 25-30%. This was therefore reinforced by small-angle scattering analysis, indicating the increasing agglomerate fractal dimension with time, thus, suggesting that the structure undergoes consolidation into denser macroscopic clusters, as in case with shear-thinning pointed before. Figure 1.12. Normalized apparent viscosity as a function of time for boehmite dispersions at pH 9. From [START_REF] Weston | Connecting particle interactions to agglomerate morphology and rheology of boehmite nanocrystal suspensions[END_REF].

I.3.5.2 Phase diagrams

Further analysis shows that various structural configurations of particles induced by bulk solution parameters (pH, ionic strength, nature of ions,…) result in a complex phase behavior, which is well-illustrated in Figure 1.13. As can be seen, rod-like boehmite dispersions exhibit two transition regions: (i) transition from sol to gel state and (ii) induced coagulation processes at a very elevated K2SO4 concentration. The boundaries of these transitions were determined from the viscosity tests. Furthermore, it is interesting to note that increasing volume fraction and ionic strength induce strong platelets interactions, which results in shear-induced birefringence, well before the sol/gel boundary is reached. This birefringence phenomena, a mosaic texture of dark and light spots with different optical axes, is a quite spectacular optical characteristic of liquid-crystalline phases as a result of parallel ordering of molecules exhibiting polarisability anisotropy. As indicated by Fukasawa et al. [START_REF] Fukasawa | Higher-order structure formation of ultrafine boehmite particles in sols, gels, and dried materials[END_REF], the tested boehmite displayed flow birefringence when particles tend to align in a flow direction, but as soon as stirring or shearing is stopped, the mosaic pattern rapidly decays through rotational Brownian motion. Shear induced birefringence was found to turn to permanent one in highly concentrated dispersion cases. This is the gel region (4) starting from 12 mM of K2SO4 concentration and boehmite volume fractions > 4 wt.% according to Figure 1.13 [START_REF] Fukasawa | Higher-order structure formation of ultrafine boehmite particles in sols, gels, and dried materials[END_REF]. Birefrigence was also found in boehmite rods dispersions of high aspect ratio (average length of 180 nm), when they were illuminated between crossed polarizers. Although, in case of shorter particles (average length of 90 nm) any form of permanent birefringent structures were found, whereas they do exhibit shear-induced flow birefringence at volume fractions higher than 1% [START_REF] Wierenga | Aqueous Dispersions of Colloidal Boehmite : Structure, Dynamics, and Yield Stress of Rod Gels[END_REF]. As opposed to boehmite dispersions, extensive studies have investigated the phase behavior of plate-like clay minerals exhibiting similar properties in some instances. Still, the phase behavior might differ substantially from one mineral to another. For instance, the phase diagram of size-fractionated Wyoming montmorillonite established the retarded sol/gel transition with an increase in particle size, as presented in Figure 1.14. Similarly, sol/gel transition has been observed on synthetic hectorite (laponite) and natural hectorite. On the contrary, few minerals as beidellite, nontronite exhibit a clear isotropic-nematic phase transition with a phase separation, which occurs below the sol/gel transition. An example of such phase diagram is presented in Figure 1.15 for nontronite aqueous suspensions with two different particles size fractions. Accordingly, authors in [START_REF] Michot | Phase diagrams of Wyoming Na-montmorillonite clay. Influence of particle anisotropy[END_REF][START_REF] Michot | Sol/gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 1. Features of the i/n transition[END_REF] demonstrated presence of the biphasic domain with the shape of the sol/gel transition, which is dramatically different from those of montmorillonite systems.

Detailed analysis showed that for ionic strength below 10 -3 M, the sol/gel zone exhibits a positive slope, which was assigned to electrostatic repulsion and might be interpreted as a colloidal Wigner glass transition. At higher ionic strength (after 10 -3 M), the slope of the sol/gel line switches on negative, which tends to, with regard to aggregation processes, join the flocculation domain at low volume fraction. Furthermore, particles of smaller size fraction display a stronger negative slope, which is expected regarding the fact that smaller particles tend to aggregate more easily.

However, the interpretation of this domain on phase diagram (ionic strength of  10 -3 M) has been widely debated. A few authors had attempted to justify the fact that shift of the sol/gel transition towards lower density region is attributed to confinement of particles in microdomains. Formation of partially oriented large-scale clusters of particles was consistent with observed shear-induced birefringence and long-term mechanical relaxation of laponite gels [START_REF] Mourchid | Phase Diagram of Colloidal Dispersions of Anisotropic Charged Particles : Equilibrium Properties, Structure, and Rheology of Laponite Suspensions[END_REF]. It must be added that large scale millimetric structures were actually observed in Wyoming montmorillonite gels using synchrotron X-ray fluorescence [START_REF] Bihannic | First Direct Visualization of Oriented Mesostructures in Clay Gels by Synchrotron-Based X-ray Fluorescence Microscopy[END_REF]. Phase diagrams of nontronite aqueous dispersions with size fraction (average length) of (A): 147 nm (anisotropy ratio 3.1) and (B): 120 nm (anisotropy ratio 2.9). From [START_REF] Michot | Phase diagrams of Wyoming Na-montmorillonite clay. Influence of particle anisotropy[END_REF][START_REF] Michot | Sol/gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 1. Features of the i/n transition[END_REF].

I.3.5.3 Viscoelasticity and yield stress

As previously seen, various anisometric colloidal systems might undergo phase transitions in the gel state with the addition of an electrolyte because of reduced electrostatic stabilization. A major rheological indication of gelled systems is the appearance of yield stress and elasticity, which defines the microstructural features and serves as a useful tool to study gelation mechanisms. For instance, the effect of progressively increasing strain amplitude 0 on the storage G and loss G moduli of boehmite dispersions at varied solid content were studied by Ramsay et al. [START_REF] Ramsay | Structure and stability of concentrated boehmite sols[END_REF]. As illustrated in Figure 1.16, the response changes markedly from viscous fluid (where G > G) to elastic solid (G > G) upon concentration increase. Therefore, a distinctive increase in G and fall of G is developed when a limit of linear elastic range is exceeded, presumably due to the onset of structural breakdown in the dispersions.

Ananthakumar et al. [START_REF] Ananthakumar | Rheology and packing characteristics of alumina extrusion using boehmite gel as a binder[END_REF] studied the effect of boehmite incorporated in alumina pastes on the rheology of final extrudes. The yield stress, determined by extrapolating the straight-line from torque rheology tests, obtained in this study is presented in Figure 1.17. The results display an improved plastic flow behavior with increasing solid content of fine boehmite acting as a binder, which is known to be an essential factor for extrusion processes. This was explained by Ananthakumar et al. as resulting from the adsorption of nano-sized boehmite at the alumina surface that creates a weakly-flocculated state of alumina-boehmite. In such a state, long range attractive interparticle Van der Waals forces and short range repulsive hydration forces co-exist, which leads to reduced frictional forces and increased degree of plasticity, thus, implying a reduction in yield stress.

Summary

This section begun with a theoretical description on the origin of interparticle forces in the context of DLVO, providing a fundamental background. Special emphasis was given in the second part of this section to experimental data on boehmite and other clay minerals systems to provide a global view on complex physical phenomena encountered in anisotropic colloidal systems. This will provide a sound basis for the next section that deals with filtration issues with regard to the role of colloidal interactions in filtration kinetics, deposit accumulation, its growth and related structural properties.

I.4 Dead-end filtration of colloidal dispersions I.4.1 Transport mechanisms across porous media

The basic principle of any liquid-solid separation process is linked to the transport of substances (particles, molecules) in a semipermeable membrane with appropriate pore size. After separation, two phases are obtained: a concentrated solid-like phase retained on the surface of membrane and a fluid stream (filtrate) driven by the applied force. In any situation involving fluid flow over the surface of solids, frictional losses lead to a pressure drop, which serves as a driving force. Darcy discovered that the driving force P per membrane thickness  and flux J are related via a proportionality factor K (Darcy, 1856), i.e.

. P K J    = (1.11)
that defines how fast the component is transported through the membrane. In other words, parameter K represents a measure of the resistance (named therefore as permeability constant) from the membrane as a transport medium, when a driving force is applied.

It must be pointed out that Equation (1.11) does not provide any information about the nature of the membrane or its relation with transport mechanism. However, as mass transport occurs through the porous media rather than the dense matrix, its structural characteristics (pore size, pore size distribution, pore dimensions, tortuosity) might be also important to describe transport mechanism more adequately. One of such transport models accounting for membrane pore geometry is given in the Hagen-Poiseuille equation via (1.12) Equation (1.12) ascribes the simplest pores geometry, which consists of cylindrical and parallel pores with radius r perpendicular to the membrane surface. The length of each of the pores is taken to be roughly equal to the membrane thickness. Comparing with the phenomenological form of Equation (1.11), permeability according to the Hagen-Poiseuille will be given as

, 8 r K 2   = (1.13)
where  is the surface porosity, which is the fractional pore area and  is the pore tortuosity (for cylindrical perpendicular pores the tortuosity approaches unity). The Hagen-Poiseuille Equation (1.13) gives a good description of transport through membranes, although very few of them have such a pore configuration in practice.

In contrast, membranes exhibiting a system of closed packed spheres are best described by the Kozeny-Carman relationship, i.e.

, P ) 1 ( S k J 2 2 KC 3    -   = (1.14)
where  is the volume fraction of pores, S is the internal surface area and kKC is the Kozeny-Carman constant, which depends on the shape of the pores and the tortuosity. Comparing Equation (1.14) with Equation (1.11) gives the well-known equation for permeability due to Kozeny as:

. ) 1 ( S k K 2 2 KC 3  -  = (1.15)
In practice, measured permeability is usually several orders of magnitude lower than that given by the above equations. This results from many factors, as porosity, particles shape and packing, particle size and distribution, concentration of the slurry being filtered, etc…Particular care should therefore be taken while working with fine particles such as colloidal dispersions: indeed, deviations from the theoretical permeability relation are expected to become significant in this case [START_REF] Mulder | Basic Principles of Membrane Technology[END_REF][START_REF] Rushton | Solid-liquid filtration and separation technology[END_REF].

I.4.2 Pressure driven membrane processes

Pressure driven membrane processes are therefore categorized by the particle size of the solute and its chemical properties determining the type of required membrane with regard to its pore size and pore distribution. Accordingly, there are four types of such processes: microfiltration, ultrafiltration, nanofiltration and reverse osmosis. Their application range is schematically illustrated in Figure 1.18. From microfiltration towards reverse osmosis, the size of the particles/molecules to be separated diminishes, which consequently leads to reduction of pores size in the membrane. This, in turn, implies that the resistance of the membranes to mass transfer increases and hence the driving force, i.e. the applied pressure has to be increased in order to obtain the same flux. Accordingly, membranes for ultrafiltration, nanofiltration and reverse osmosis membranes usually possess an asymmetric structure with a relatively dense layer (thickness 0.1-1.0 m) on its top that determines the hydraulic resistance towards transport.

The main problem encountered in a pressure driven membrane process involves certain changes in the system perfomance over processing time, causing a typical decrease of the flux. It results from several factors, such as concentration polarisation, adsorption, gel layer formation and pores fouling. Flux decline is particularly impacted by fouling, giving values for process fluxes being typically less than 5% of pure water flux. This also implies that for industrial applications, the given above Hagen-Poiseuille and Kozeny-Carman equations are irrrelevant, as they do not account for these phenomena and other models should then be considered.

Pores fouling can be partially reduced by switching the mode of filtration configuration. Basically, two major processing modes can be distuinguished, i.e. dead-end and cross-flow filtration. Schematic illustration of these modes is given in Figure 1.19. In the dead-end filtration, the feed solution flows perpendicularly to the membrane surface, as opposed to the cross-flow regime, so that the retained particles form a deposit layer on its surface, which increases over the processing time resulting in a decrease of the permeation rate. In cross-flow mode, the feed solution flows along the membrane surface, so that filter cake formation is reduced or kept at a low level. Particularly, in cross-flow mode, fouling is often accompanied by concentration polarisation phenomena. It occurs when the accumulating deposit at the membrane surface generates a diffusive flow back to the bulk solution until steady state conditions are established. In addition, it might be also combined with gel layer formation and adsorption phenomena. Consideration of above given factors results in additional resistances, which should be accounted for on the feed side to the mass transport across the membrane (Figure 1.20). Accordingly, the flux through the membrane might be rewritten with extent to the total resistance Rtot, which is the summation of all resistances due to: pore fouling Rp, adsorption Ra, gel layer formation Rg, concentration polarisation Rconc in addition to resistance of the membrane Rm considered before in Darcy's law

I.4.3 Membrane fouling

As evidenced in the previous section, the phenomenon of flux declination is very complex, and therefore very challenging to describe theoretically. As the major concern of this study, the following discussion will be restricted to membrane fouling model only. In this case, the flux through the membrane is described as the flux through two resistances in series, i.e. the cake (deposit) Rc and the membrane Rm resistances. Hence, Darcy's expression might be rewritten as

. ) R R ( P dt dV S 1 J m c m +   = =
(1.17)

The solids of comparable size or larger than the pore diameter of the membrane are retained on its surface forming a deposit layer, whose thickness increases over filtration time. If one assumes incompressible material, the deposit mass will uniformly increase with the filtrate volume giving rise to deposit resistance Rc. From mass balance, the resistance of the cake Rc can be substituted by

, S V c r R m c c c = (1.18)
where V is the filtrate volume, S the membrane surface area, rc the specific cake resistance. Physically, this term determines the 'filterability', i.e. the speed at which the filtration process occurs [START_REF] Foley | A review of factors affecting filter cake properties in dead-end microfiltration of microbial suspensions[END_REF]. The dry cake mass per filtrate volume cc can be estimated independently via the moisture ratio mR, bulk solids concentration sb (by mass fraction) and liquid density

 , s m 1 s c b R b c -  = (1.19)
Substituting Equation (1.19) in Equation (1.18), the flux can be written as

, ) R S V c r ( P dt dV S 1 J m m c c m +   = = (1.20)
which, after integration for constant pressure filtration

, dV P S R dV P S V c r dt V 0 m m V 0 2 m c c t 0      +   = (1.21) leads to . P S R V P S 2 c r V t m m 2 m c c   +   = (1.22)
A plot in the form of t/V vs. V yields a straight line with a slope In case of compressible cake filtration, both the specific resistance rc and dry cake mass per filtrate volume cc become a function of the pressure difference P and are usually given as

p 0 c c P ) p 1 ( r r  - = (1.23)
for average specific resistance c r , where p and rc0 are empirical constants (Equation for cc is derived similarly). The exponent p is usually related to the cake compressibility and the greater this value, the higher the compressibility. As a reference, p = 0 is relevant for incompressible solids, calcite yields a n value of about 0.2, for kaolinite clays n is about 0.6 and fine sludges approach unity [START_REF] Rushton | Solid-liquid filtration and separation technology[END_REF]. It seems that greater compressibility corresponds to more elongated particles shape. This might be related to the higher number of available orientations with increasing particles aspect ratio, which leads to a wider range of possible packings.

I.5 Experimental characterization of fouling mechanisms in colloidal systems

I.5.1 Effect of surface charge, pH and ionic environment

A variety of physical and chemical parameters are often involved in fouling effects. Again, interparticle interactions appear to play a significant role likewise in filtration processes. For instance, in the work of McDonogh et al. [START_REF] Mcdonogh | Charge effects in the cross-flow filtration of colloids and particulates[END_REF] dealing with the cross-flow filtration of silica colloids, the limiting fluxes were shown to significantly depend on the zeta potential of silica particles (Figure 1.22). Accordingly, the flux was increased by a factor of two between zeta potential close to 10 mV and zeta potential close to 30 mV. Such a tendency remains valid whatever the hydrodynamics as similar curves are obtained for various Reynolds numbers. The authors then concluded that particle charge might be a key parameter in controlling of either fouling or concentration polarisation phenomena that explains flux declination.

The zeta potential dependence on filtration rate was indeed observed in numerous works. For instance, Wiesner et al. [START_REF] Wiesner | Membrane Filtration of Coagulated Suspensions[END_REF], [START_REF] Waite | Colloidal Fouling of Ultrafiltration Membranes: Impact of Aggregate Structure and Size[END_REF] carried out dead-end filtration tests on flocculated humic acid solutions at various dosage of aluminum coagulant. A minimum in specific cake resistance (meaning the highest filtration rate) was reached for suspensions with zeta potential close to 0 as shown in Figure 1.23, independently of the aluminum dosage. For higher zeta potential values, the specific cake resistance increases tremendously, which was assigned by the authors to the presence of less aggregated particles. Indeed, in such conditions particles size was found comparable with the size of membrane pores, giving rise to pore blockage and fouling. Similar trends were observed by [START_REF] Hlavacek | Simple Relationships among Zeta Potential, Particle Size Distribution, and Cake Specific Resistance for Colloid Suspensions Coagulated with Ferric Chloride[END_REF] in the case of kaolinite suspensions coagulated with various doses of ferric chloride. For certain coagulant doses the filterability of suspensions was improved while overdosing caused aggregate breakage, thereby increasing resistance to filtration. The results were therefore complemented by TEM analysis distinguishing variations in particles packing mode. The cake structure without ferric chloride displayed a dense packing of randomly oriented particles, whereas in the presence of ferric chloride, particles were found to be preferably associated in stacks forming large channels in between. Authors suggested that such a difference in packing modes comes from reduced surface charge asymmetry between edges and faces of kaolinite platelets caused by fixation of iron ions that bring positive charges onto the negatively charged faces, which promotes faceto-face interactions and, thus, allows for the stacking of particles on top of each other. This implies an improved filterability due to the presence of large channels associated with tortuosity increase. In other works, e.g. Bacchin et al. [START_REF] Bacchin | Influence of surface interaction on transfer during colloid ultrafiltration[END_REF] demonstrated the influence of ionic strength on the deposition kinetics of bentonite clay dispersions. An observed maximum in specific cake resistance, which was followed by a strong decrease afterwards (illustrated in Figure 1.24), was explained by authors due to proximity of critical concentration point for coagulation (ccc). Their data are particularly pertinent to our results displayed and discussed in Chapter IV of present manuscript. As surface charge is highly dependent on solution pH, it therefore influences filtration kinetics. Studies on microbial suspensions (C. glutamicum) in the work of Ohmori and Glatz [START_REF] Ohmori | Effects of pH and ionic strength on microfiltration of C. glutamicum[END_REF], for instance, revealed that pH adjustment from 6.6 to 2.0 at constant ionic strength leads to a five-fold increase of the filtration flux (meaning the rc decreases). When ionic strength was uncontrolled such a rise was even more dramatic. Additional zeta potential measurements suggested once again that increasing specific resistance was associated with elevated surface charge, leading to reduced aggregation and, thus, to a more resistant deposit (Figure 1.25).

I.5.2 Effect of particles shape and aspect ratio

In Section I.2, it was shown that particles with asymmetric shape might undergo orientational ordering, resulting in optical (birefringence) or rheological (i.e. shear-thinning) features. Hence, it is evident that particles orientation with varied shape and aspect ratio might be likewise important in filtration of anisotropic particles.

Indeed, reorientation of plate-like particles in filter cakes was observed in several studies. For instance, Tanaka et al. [START_REF] Tanaka | Filtration characteristics and structure of cake in crossflow filtration of bacterial suspension[END_REF] found orientational ordering of rod-shaped bacteria cells with the flow direction when switching from dead-end to cross-flow mode (Figure 1.26). Further estimations of specific cake resistance showed that such arrangement leads to a more compact deposit layer than those formed in dead-end filtration, which reduces filtration rate.

In work of Perdigon-Aller et al. [START_REF] Perdigon-Aller | Preferred orientation in filtercakes of kaolinite[END_REF] using neutron diffraction technique, weakly aggregated and deflocculated kaolinite particles were found also aligned parallel to the filter membrane in cross-flow filtration. Both systems displayed significantly different permeability. Thus, it might be suggested that particles orientation is not the only parameter controlling permeability, but its correlation with colloidal interactions must also be taken into account. The strong influence of particle/cell morphology on the filtration properties of microbial suspensions was found in work of McCarthy et al. [START_REF] Mccarthy | Effect of cell morphology on dead-end filtration of the dimorphic yeast Kluyveromyces marxianus var. marxianus NRRLy2415[END_REF]. The shape of microorganisms, ranging from yeast-like to filamentous, was controlled by altering growth conditions. The tendency for cake compressibility to increase as the cells become more elongated (increasing mean aspect ratio) was observed. This results is consistent with the wider range of voids displayed by beds containing long cylinders due to wider possibilities of cells packing, as mentioned previously in Section I.3.3.

I.5.3 Effect of particle size distribution

Size distribution has also been shown to affect filtration performance, as well as particles packing in filtration deposits. For instance, filtration tests were conducted on a bimodal mixture containing micron sized gibbsite and nanometer ranged boehmite particles by Bruinsma et al. [START_REF] Bruinsma | Rheological and Solid-Liquid Separation Properties of Bimodal Suspensions of Colloidal Gibbsite and Boehmite[END_REF]. This work demonstrated the effect of reducing permeability in the presence of boehmite nanoparticles that therefore influences the rheological properties of dispersions by enabling the solids content to be increased without affecting viscosity. Such an effect was attributed by the authors to the small fraction of nanoparticles adsorbed at the surface of larger particles, hence providing steric repulsion. This was accompanied by a continuous drop in the filter cake permeability with increasing boehmite fraction. This was presumably due to the fact that boehmite can fill the voids between micron-sized gibbsite particles resulting into a denser filter deposit. Therefore, at higher addition of boehmite content its contribution begins prevailing the suspension behavior and, thus, contributing to a more shear-thinning behavior.

Work of Shi and Zhang [START_REF] Shi | Compaction and Sintering Behavior of Bimodal Alumina Powder Suspensions by Pressure Filtration[END_REF] also revealed better flowability of alumina dispersions when mixing fine and coarse particles. Their studies were aimed at investigating the compaction and sintering behavior of alumina powders that undergo pressure-driven filtration. Again, at the optimum fraction of fines, bimodal suspensions were found to display the lowest viscosity and the resulting pressure-filtered compacts attained the highest compaction density, which is in line with the results and discussion mentioned previously by Bruinsma et al. [START_REF] Bruinsma | Rheological and Solid-Liquid Separation Properties of Bimodal Suspensions of Colloidal Gibbsite and Boehmite[END_REF].

I.5.4 Filtration studies in situ

Despite visible progress in the research on liquid/solid separation of colloidal slurries, as illustrated by the above given review, it is apparent that off-line characterization methods are not sufficient to investigate in depth dynamic processes. For instance the schemes of deposit growth or particles transport mechanisms can be evidenced via in situ approach coupled with non-invasive characterization techniques, thus, providing a more detailed picture of filtration process.

Few studies were attempted to assess a direct visualization of clogging dynamics in filtration using numerous imaging or scattering techniques. The group of Bacchin [START_REF] Liot | Pore cross-talk in colloidal filtration[END_REF][START_REF] Agbangla | Experimental investigation of pore clogging by microparticles : Evidence for a critical flux density of particle yielding arches and deposits[END_REF] used a poly-dimethylsiloxane (PDMS) microfluidic device coupled with a confocal microscopy to investigate the clogging of microchannels by hard sphere systems (polydisperse polymersomes, latex particles). Different patterns of forming cake structure were observed and discussed with regard to applied operating conditions (i.e. hydrodynamics, particles volume fraction, physico-chemical surface interactions, …). However, as typical size of microchannels in PDMS cell usually does not overcome 5-20 m, this approach is unable to reproduce ultra-or nano-filtration processes with membrane pore sizes below 1 m.

Scattering techniques (static light scattering (SLS), small-angle X-ray scattering (SAXS), small-angle neutron scattering (SANS)) are the most reported for monitoring filtration processes down to the length scale of a few nm. The works of Pignon et al. (in frontal [START_REF] Pignon | Structural characterisation of deposits formed during frontal filtration[END_REF], cross-flow [START_REF] Pignon | In situ characterization by SAXS of concentration polarization layers during cross-flow ultrafiltration of Laponite dispersions[END_REF] ultrafiltration modes and centrifugation [START_REF] Martin | Osmotic compression and expansion of highly ordered clay dispersions[END_REF]) have extensively demonstrated the anisotropic organization of platelet particles close to the membrane where high concentration levels are reached. By combining filtration tests in situ with X-ray scattering and birefringence techniques, for the first time structural changes upon deposit formation were distinguished at length-scales of elementary particles up to a large-scale organization. Notably, increasing particles orientation was found closely associated to an increasing concentration of the laponite dispersions forming a denser structure near the membrane. These works further suggested that particles alignment is strongly favored under pressure-induced flow environment.

Characterization of deposit growth within dead-filtration was carried out in situ also using an optical laser sensor [START_REF] Mendret | In situ dynamic characterisation of fouling under different pressure conditions during dead-end filtration: Compressibility properties of particle cakes[END_REF]. The method is based on the use of a laser beam focused tangentially to the surface of a membrane. Due to growing deposit, the laser beam will be deflected from its original position. The measurement of such deflection followed by a calibration procedure enables the cake thickness to be evaluated at a resolution of several m. Accordingly, two different stages in the deposit growth were detected by Mendret et al. [START_REF] Mendret | In situ dynamic characterisation of fouling under different pressure conditions during dead-end filtration: Compressibility properties of particle cakes[END_REF]. The first layers were found thin but highly resistant to mass transfer whatever the applied pressure, indicating the occurrence of concentration polarisation phenomena. In contrast, the pressure effect become explicit in the top layers, when a certain quantity of deposited clay content is reached, resulting in the formation of thick layers displaying compressible character.

The effects of flow on the aggregation and fragmentation of boehmite particle were studied by Krzysko et al. [START_REF] Krzysko | Correlating inter-particle forces and particle shape to shear-induced aggregation/fragmentation and rheology for dilute anisotropic particle suspensions : A complementary study via capillary rheometry and in-situ small and ultra-small angle X-ray scattering[END_REF] using a combined capillary rheometry with small-angle X-ray scattering (SAXS), ultra-small-angle X-ray scattering (USAXS) and wide-angle X-ray scattering (WAXS) techniques in order to quantify boehmite particle behavior at length scales of 6-10 000 Å. In this work, boehmite dispersions were exposed to a pressure-driven pipe flow (Poiseuille flow) at varied Peclet number (Pe) providing an estimate of the hydrodynamic shearing on the boehmite particles. Besides formation of multi-scale fractal aggregates authors evidenced in situ shear-thinning accompanied by fragmentation of agglomerates (shift of the gyration radius Rg towards smaller values as the shearing force increases), therefore confirming its (shear-thinning) orientation-dependent nature rising from the non-spherical plate-like shape of the boehmite particles.

One might notice from the above given theoretical models and review on experimental data that permeability factor K is the most operating parameter in characterization of filtration kinetics. In contrast, porosity is less relevant as it is hard to estimate correctly in "offline mode". In this regard, Nuclear Magnetic Resonance imaging (NMR) was successfully exploited in a direct estimation of porosity evolution within filtration cakes in situ at a resolution of 65 µm [START_REF] Hieke | Analysis of the Porosity of Filter Cakes Obtained by Filtration of Colloidal Suspensions[END_REF]. A typical NMR test measures the relaxation processes of protons when interacting with external magnetic field. It is referred to the signal decay, which is dependent on the solid content, and, thus, if calibration is carried out, it allows to generate data on the pore distribution profile within a filter cake, similarly to SAXSbased concentration profiles (Pignon et al., [START_REF] Pignon | In situ characterization by SAXS of concentration polarization layers during cross-flow ultrafiltration of Laponite dispersions[END_REF]).

I.6 Summary

This chapter first introduced the origin of interparticle forces with regard to DLVO concept, which was followed by experimental data on anisotropic colloidal systems showing diverse structural, rheological and optical properties.

Next part of the chapter was focused on the basic principles of membrane separation processes and the transport related models that will be further used in the present study. A critical analysis of literature dealing with filtration processes of anisotropic colloidal systems was then carried out, allowing to identify a key role of colloidal interactions through a given bulk chemistry, such as surface charges, size and size distribution, pH, ionic strength, etc., in filtration rate and particles accumulation within filtration. Furthermore, a few studies show that particles anisotropy and aspect ratio affect filtration performances as well due to particles alignment, which alters the deposit resistance. Accordingly, it seems interesting to study the filterability of boehmite suspensions via a thorough analysis of particle size, shape and, in particularly, via a control of interparticle interactions provided by a control of the physicochemical properties.

In the last part of the present chapter, it was shown that significant progress in understanding fouling mechanisms within filtration has been achieved due to state-of-the art in situ filtration approach. It then appears promising for the present work to employ an in situ characterization with non-invasive observation techniques. Finally, in the following chapters, the preparation of boehmite dispersions as well as experimental filtration set-ups and other analytical methods and procedures will be introduced.

CHAPTER II: Materials and methods

II.1 Product information

The colloidal suspension considered in the present study is a high purity boehmite (AlOOH) Pural SB containing 45 ppm of Fe, 450 ppm of Ti, <40 ppm of S, <50 ppm of Cl, <20 ppm of Na and 0.2 wt.% of C provided by Sasol group (Germany) [START_REF] Fauchadour | Etude de la peptisation[END_REF]. Technical data provided by Sasol report a crystallite size of 5 nm for boehmite SB without any information about the shape of the particles [START_REF] Gallois | Etude des propriétés physico-chimiques de suspensions de boehmite : Application aux supports catalytiques[END_REF]. Nevertheless, primary characteristics of SB powder had been widely ascribed previously [START_REF] Fauchadour | Etude de la peptisation[END_REF], [START_REF] Mange | A microstructural investigation of nanostructured boehmite films prepared by the sol-gel route[END_REF]. Thermogravimetric measurements indicate an excess of water 0.45H2O when compared to the well crystallized form of alumina hydroxide AlOOH. The particles are characterized by a density of 3.035 gcm -3 and a high surface area of 300 m 2 g -1 .

II.2 Samples preparation

II.2.1 Peptization of boehmite

The boehmite dispersions used in this study were prepared using an acid peptization procedure, as already discussed in Chapter I.2.1. In this dispersion process, the large m boehmite powder agglomerates are broken down into a mixture of smaller aggregates ( 20-200 nm) and elementary crystallites (10 nm). These small aggregates are reported to be often comprised of elementary crystallites stacked along the b axis. The initial state of boehmite agglomerates dispersion scales with the amount of acid introduced: higher acid contents provide a higher amount of well-dispersed small particles, whereas lower acid additions that are not high enough to ensure surface charging result in the occurrence of large aggregates and a lesser fraction of colloids (Figure 2.1). In the present work, we focused on boehmite dispersions with two different size distributions: a well-dispersed sol in the absence of coarse particles, that will be referred to as "colloidal" dispersion, and a mixture of colloids with a fraction of coarser particles, that will be referred to as "bimodal mixture" (in studies on synchrotron (Chapter V)). Accordingly, colloidal boehmite sols are obtained by peptizing the starting boehmite powder at a total solid concentration of 143 gL -1 in an aqueous solution containing a high HNO3 content of 0.186 molL -1 . The bimodal mixture is prepared using a lower nitric acid content (0.074 molL -1 ) with the same solid concentration. The mixing is carried out under continuous stirring at 700 rpm for 5-10 min. To ensure suspension uniformity, a mechanical dispersion step using Ultraturax ® rotor-stator mixing is applied afterwards at 15 000 rpm for 10-15 min. In a typical peptization procedure, any non complexing, usually monovalent acid (in the present work HNO3) serves as a peptizing agent and, as mentioned previously, it charges the surface and also leads to dissolution reactions. Thus, the resulting sol is a complex mixture of well-dispersed primary particles, aggregates, agglomerates of particles and soluble aluminum species.

II.2.2 Dialysis

The use of dialysis procedure allows removing dissolved aluminum species after peptization and, thus, lowers the ionic strength. A schematic illustration of dialysis is given in Figure 2.2. Unlike pressure driven filtration or reverse osmosis, purification by dialysis is governed by ions transport across a semi-permeable membrane resulting from a concentration difference as a driving force. The initial dispersion is placed in a dialysis tube made of regenerated cellulose at a cut off in molecular mass chosen to ensure the transport of ions and weakly adsorbed molecules only. Since the concentration is higher inside the dialysis bag, salts and ionic species are washed out into the buffer reservoir on the outside, until an equilibrium concentration is reached. The procedure is simple and does not require any external forces to be applied. It has a few drawbacks though: it is time consuming and requires large volumes of liquid, as well as frequent changes of buffer reservoir in order to speed up the purification process. In the present work, prior to dialysis, the colloidal sol obtained at the higher acid/alumina molar ratio was centrifuged at 4 000 g during 4 hours in order to eliminate remaining agglomerates. Both the resulting supernatant of colloidal system after centrifugation (pH 1.3) and bimodal mixture (pH 4.2) were placed in a porous cellulose membrane with a cut off of 6-8 kDa for dialysis against a buffer reservoir at pH 5.0-5.1. The buffer solution was renewed 1-2 times per week to accelerate the process. The pH of the reservoir was fixed by adding a few drops of nitric acid (6.8 wt.% HNO3). The conductivity of the added solution is around 2-3 S•cm -1 and the conductivity in the reservoir was then monitored. Renewal by fresh reservoir solution is stopped once the conductivity of reservoir displays an equilibrium value of 5 S•cm -1 . Typical total time for this procedure is around 2 months. The volume fraction of the resulting boehmite dispersions after dialysis is  = 0.4 vol.% (= 12.7 g•L -1 ) and the minimum ionic strength  810 -5 M.

In order to vary the ionic strength of the suspensions, various amounts of KNO3 were added into the buffer reservoir at concentrations of 0.0182 g•L -1 , 0.0809 g•L -1 , 0.196 g•L -1 and 1.110 g•L -1 that corresponds to ionic strengths of 1.810 -4 M, 810 -4 M, 1.9410 -3 M and 110 -2 M, respectively according to

 =  = n 1 i 2 i i z c 2 1 I , (2.1)
where ci is the molar concentration of ions and zi the number of charges on the ion.

II.2.3 Osmotic stress

The principle of osmotic stress is based on a controlled removal of water from the system of interest using an external polymer solution of known osmotic pressure separated by semipermeable membrane (Figure 2.3). The method has been largely applied to a wide set of charged rodlike (e.g. tobacco mosaic virus) or plateletlike (e.g. nontronite) colloids that could form ordered arrays during osmotic equilibration, thus, enabling to assess the structural features of the system. Chapter II: Materials and methods A high molecular weight polymer solution at a set concentration exerts osmotic stress on the dispersion. It is accompanied by volume changes of dispersion resulting from water and ions diffusion from the dispersion to the external polymer solution that results in a more concentrated dispersion with respect to its original state. At chemical equilibrium, the osmotic pressure will be balanced on both sides of the membrane and, thus, the osmotic pressure of the dispersion corresponds to that of the polymer in the reservoir bath. If in addition, the final  of the dispersion is monitored and supported by X-ray scattering analysis, one gets the dependence of the osmotic pressure over volume fraction, i.e. its equation of state, which provides insight into the nature of interparticle interactions.

PolyEthylenGlycol (PEG) with a molar mass (MW) of 20 000 provided by Sigma-Aldrich, Germany was used as a stressing polymer. The required osmotic stress was estimated from the polynomial expression relating pressure  (in dyn•cm -2 ) to PEG concentration (in wt.%): (2.2)

( ) ( )
PEG solutions at osmotic pressures in the range 5 000-100 000 Pa were prepared by dissolving the polymer in the last bath reservoir for the given ionic strength (I1 = 810 -5 M, I2 = 810 -4 M and I3 = 110 -2 M) in order to preserve the same conductivity and pH conditions. As dialysis, osmotic stress is a thermodynamic phenomenon, and it therefore requires about 2 weeks for equilibrium to be reached. Afterwards, the solid content of the osmotically stressed concentrated dispersions is evaluated by mass balance after drying at 140°C during 24h.

II.3 Sols characterization

II.3.1 Particle size and shape

The morphological parameters of boehmite particles were determined via transmission electron microscopy (TEM) using JEOL JEM-2100F microscope operating at 200 kV. Samples for microscopy analysis were prepared by diluting the given boehmite dispersion in ultrapure water with a dilution factor 20 000. A drop of solution was dried in ambient conditions on a carbon grid prior to being introduced into the vacuum chamber of TEM. From the TEM micrographs, the average length with a standard deviation was defined based on statistical analysis of around 50 particles.

Depending upon the particle size distribution, two methods were used to acquire size distribution. Analysis of bimodal mixtures containing micron range aggregates was conducted on Malvern Mastersizer 2000 with a Hydro 2000SM wet dispersion unit. The sample was run at 1300 rpm of stirrer speed. The refractive and absorption indices of the particles were selected at 1.66 and 0.01, respectively. The refractive index of the medium (water) is 1.33.

Optical transparency of diluted boehmite sols after dialysis ( = 0.4 vol.%) enables performing size distribution tests via Dynamic Light Scattering (DLS). These measurements have been carried out on a DLS setup of Cordouan Technologies. The incident beam of a He-Ne laser ( = 656 nm) passes through the sample in a 10 mm Hellma quartz cuvette. The scattered beam is collected at an angle of 90° by an Avalanche PhotoDiode (APD) detector and used after to build an intensity autocorrelation function G2(Q,t) (with wave vector Q and time t) to describe fluctuations in scattered light. G2(Q,t) is related to the modulus of the normalized field autocorrelation function g1(Q,t) by the Siegert equation

) t , Q ( g ) t , Q ( G 2 1 2  +  = , (2.3)
where  is a baseline and  is the coherence factor. The time dependence of g1(Q,t) is essentially linked to the dynamical properties of the given dispersion. In case of Brownian monodisperse systems, it is described by a single exponential decay where  is the decay rate

) t exp( ) t , Q ( g 1  - = (2.4)
For a polydisperse system g1(Q,t) is written as a sum of exponential decaying functions as

) t exp( A ) t , Q ( g i i i 1  - =  . (2.5)
From the sum of exponential decaying functions, a standard Levenberg-Marquardt is used to adjust the weight of each exponential decaying function. If the studied dispersion is composed by Brownian particles, the self-diffusion coefficient can be further defined via

2 i i Q D = 
, which enables determining the hydrodynamic radius a of the hard sphereequivalent using Stokes-Einstein equation

a 6 T k D B i  = , (2.6)
where kB is the Boltzmann constant and  the dynamic viscosity of the continuous phase.

II.3.2 Volume fraction, pH and -potential

Solid content of boehmite in sols  was estimated by measuring mass after drying at 140°C according to the following equation

m p m p p m m m   + =  , (2.7)
where ρp density of boehmite equals 3.035 g•cm -3 , ρm of dispersion medium is 1 g•cm -3 . The pH values were monitored using a pH meter from Mettler Toledo, SevenExcellence.

The zeta potential  was determined from electrophoretic mobility measurements using Zetasizer Nano-ZC (Malvern) equipped with a He/Ne laser operating at 633 nm. Measurements were performed in a folded capillary cell containing two electrodes. When a voltage is applied, the particles start moving toward the electrode of opposite charge. As particles are illuminated by a laser crossing the capillary cell, the intensity of scattered light will fluctuate at a frequency that is proportional to the particle velocity (Doppler shift). By measuring particle velocity at multiple voltages, one can determine the zeta potential using Smoluchowski's approximation. The values of boehmite density and mean refractive index are 3.035 g•cm -3 and 1.6577 [Sahama et al., (1973)], respectively, whereas refractive index of water (1.330), its density (1 g•cm -3 ) and its dynamic viscosity (1.002 Pas, at 20°C) were used for calculations of the zeta potential. Measurements were carried out without any sample dilution (boehmite concentration of 12.7 g•L -1 ). The dispersions were equilibrated for 1 min prior to each measurement. Analysis was repeated 3 times, each measurement cycle was based on over 50-100 runs and averaged.

II.3.3 Ionic composition of the aqueous phase

The composition of the aqueous phase in the boehmite sols was analyzed in the following way. Assuming the equilbrium state inside the dialysis bag and in the buffer reservoir, the later was used for the analysis of iron, nitrate, potassium and aluminum ions without any sample dilution. The measurements were performed at the Ultra Traces Analysis Aquitaine (UT2A) laboratory of the University of Pau using inductively coupled plasma emission spectrometry (ICP-AES) to trace iron, pottasium, aluminium and the Dionex ICS-3000 ion chromatography system for nitrate determination.

II.3.4 Rheometric studies

Rheology is the science that deals with the deformation and flow of materials, both solids and liquids. Let us consider a simple shear flow, as schematically depicted in Figure 2.4. The material is confined between two parallel plates separated by a distance h, where the top plate moves with velocity  in the x direction and the bottom plate is stationary. At sufficiently low velocities, thereby in laminar regime, the fluid is assumed to flow everywhere parallel to the plates. The local velocities x will therefore vary linearly across the gap h, resulting in a gradient dx/dy of x across the gap, namely the "shear rate", which is defined by h dy

d x  =  =   . ( 2

.8)

A force Fxy, which is applied tangentially to the top plate and is proportional to the surface area of the plates, leads to a shear stress xy that induces a flow in the x direction. This shear stress is transmitted from one plate to another and acts on each fluid element in between. The kinematic parameter that determines the level of internal stresses in this case is the velocity gradient or shear rate. The simplest example of a rheological constitutive In contrast, for a Hookean ideal elastic solid a shear stress applied to the surface y = h results in instantaneous deformation xy, followed by an elastic recoil as soon as the load is removed, similarly to a spring model

  =  G xy xy , ( 2.10)
where G* is referred to as the "rigidity shear modulus". In reality, most materials combine properties of Newtonian viscous fluids and Hookean elastic solids, thus, the term "viscoelasticity". While purely elastic materials store energy as they are being deformed and purely viscous materials dissipate this energy, in viscoelastic materials both responses manifest themselves at different time scales of the experiment in relation to a natural time of the material. One approach to measure viscoelasticity is to use forced oscillation methods, also known as dynamic measurements. In a typical dynamic test, the response of the sample to a small amplitude oscillatory shear is measured. As the sample structure is only slightly perturbed with respect to that at rest without a complete destruction, oscillatory measurements is a commonly used method for correlating elastic effects with the structural characteristics of colloidal dispersions.

Oscillatory shear flow can also be studied in the parallel plate geometry of Figure 2.4. In this case, instead of a steady velocity, the top plate executes a sinusoidal motion that generates a time-dependent sinusoidal strain (t) as

) t sin( ) t ( 0   =  , (2.11)
where 0 is the maximum deformation,  the angular velocity ( = 2f, f being the frequency) and t is time. Substituting Equation (2.11) into Hooke's law Equation (2.10), a stress function will be rewritten as

) t sin( G ) t ( 0   =   .
(2.12)

Similarly, substituting Equation (2.11) into Newton's law and since dt / d =   the time derivative of the sinusoidal strain function (t) results in the cosinusoidal function, the stress function will then be given by

) t cos( ) t ( 0    =   .
(2.13)

As seen, the stress function is now shifted by 90° with respect to the strain. Hence, in the oscillatory flow, the difference between elasticity and viscosity is expressed by a specific value of the phase shift between stress and strain. In general, viscoelastic materials are characterized by a phase shift angle 0 <  < 90°. Given in Equation (2.10) the modulus can be decomposed into two components, G' and G" portions defined as

   +  =  ) G i G ( . (2.14)
The in-phase or "real" part G' defines the elastic component of the stress, namely the storage modulus. The out-of-phase or "imaginary" part G" (shifted by 90°) is the loss modulus, representing the viscous part. It is worth noting that proportionality between stress and strain given by Equation (2.14) only applies in the linear region, i.e. at sufficiently low strains or stress, the so-called linear viscoelastic range (LVE range), as depicted in Figure 2.5. When plotting both moduli G' and G" as a function of stress or strain amplitude (the frequency is at a constant value), there is a critical deformation value where curves begin to deviate noticeably from the linearity that defines the limit of the LVE region. After this test, the time-dependent viscoelastic behavior is investigated by means of a frequency sweep, since frequency is the inverse of time. Short-term behavior is simulated by rapid motion, i.e. high frequencies and long term, respectively, by slow motion, i.e. low frequencies. Thus, when both moduli are plotted as a function of frequency, G' > G" in the range of high frequencies and G' < G" when exposed to very slow motion, presenting good flow properties [START_REF] Moreno | Encyclopedia of Materials: Science and Technology[END_REF][START_REF] Mewis | Colloidal suspension rheology[END_REF].

In the present work, rheological measurements were performed using a Modular Compact rheometer, MCR 302 from Anton Paar instruments equipped with parallel plates geometry (Figure 2.6). Two plates with a crosshatched surface were used to avoid wall slip during measurements. The gap between plates was adjusted by controlling the normal force between 2 and 10 N depending on the volume fraction of the samples. Prior to the measurements, the rest interval at the start of the test was set to enable sample relaxation from accumulated internal stresses after gap setting. Measurements were held at fixed temperature (20 °C). Firstly, oscillatory strain controlled amplitude sweeps were conducted to characterize the onset of yielding mechanisms and determine the LVE range. Next, the elastic G and viscous G moduli were measured again in an oscillation frequency sweep in the range of 0.01-100 rad•s -1 . A strain of  = 0.01 % was chosen that corresponds to the LVE region, where the storage moduli display a plateau. Steady-state viscosity curves were also recorded under controlled shear rate from 110 -5 to 0.1 s -1 with a measuring point duration of 150 s (this measured point duration was determined on the basis of the evolution of the shear rate viscosity as a function of time ) t (   ).

II.3.5 Small angle X-ray scattering (SAXS)

Elastic collisions of X-rays with the electron cloud in the matter result in scattering processes within a small angle of 0.1-10 °C around the incident beam (Thompson scattering). Since the dimensions of macromolecules and particles are relatively large compare to the wavelength of the incident X-ray radiation (the range of 0.01-10 nm), SAXS technique enabling to probe submicroscopic structural and morphological characteristics over very small distances. A schematic SAXS set-up is shown in Figure 2.7. The sample is illuminated by an incident X-ray beam. The source and detector are sufficiently far from the origin that the incident and scattered X-rays are viewed as plane waves. This is the so-called far-field limit associated with the Fraunhoffer diffraction.

The scattered intensity is recorded as a function of the so-called scattering angle 2. If the momenta of incident and scattered waves are given by wavevectors ki and ks, respectively, a wave scattered by a nucleus in the sample at a point r produce a phase shift with respect to the incident beam by a phase-factor exp (iQ•r), where Q = kiks is the momentum transferred in the scattering process. The scattered intensity I is represented as a function of momentum transfer Q

   = = sin 4 Q Q  , (2.15)
holding the dimension of a reciprocal length, where  is the radiation wavelength and 2 is the elastic scattering angle. For a single scattering process the amplitude produced by the scattering of all the electrons with their corresponding phase shift exp (-iQ•r) inside volume Vp is

  -   = p V 3 r d ) r iQ exp( ) Q ( A , (2.16)
where  = p -m represents the electronic density difference between the particle and the medium. It is worth noting that the scattering amplitude A(Q) in Equation (2.16) is a three-dimensional Fourier transform of the electron density distribution within the object. Since

I(Q) = A(Q)A(Q)* = |A(Q)| 2
, the square root of measured I(Q) function provides only the modulus of the scattering amplitude and not its complex number. This is referred to the wellknown phase problem in crystallography and is the characteristic in X-ray structure analysis when the phase information is getting lost while obtaining the intensity as an absolute square of the amplitudes.

If one considers an assembly of identical particles of volume Vp, the total scattering intensity can be presented by the product of a single particle property, the form factor P(Q) and the structure factor S(Q), which reflects the local structure based on the interactions between the individual particles as following

    factor Structure factor Form 2 p ). Q ( S ) Q ( P ) Q ( A V 1 ) Q ( I  = =
(2.17)

In the absence of interactions (in dilute systems) the structure factor S(Q) approaches the unity 1 and the scattering function at very small Q yields the well-known Guinier form where Rg is the radius of gyration of the particle defined as the root-mean squared distance from the particle's center of gravity. The simplest example of form factor is given for a sphere with a radius of gyration Rg equals to R 3/5 of radius R as follows

) R Q 3 1 exp( V ) Q ( I 2 g 2 2 p 2 -    , ( 2
2 3 ) QR ( ) QR cos( QR ) QR sin( 3 ) Q ( P       - = . ( 2

.19)

At very small Q, the form factor P(Q) is independent of particle shape, however, it becomes a function of the particle morphology and its size at large Q. This is the so-called Porod regime.

In case of the uniform sphere, cos 2 (QR) oscillates rapidly as a function of Q with an average value of ½. The intensity might be then rewritten by replacing the volume Vp with the surface Sp of a sphere

p 4 2 S Q 2 ) Q ( I    = , ( 2.20)
The scattered intensity is then proportional to the surface area of the sphere and inversely proportional to Q to the power 4. Form factor for disk-and rod-like particles shape is also illustrated in Figure 2.8. As shown, significant differences in form factors appear following a power law  Q -4 , Q -2 and Q -1 for 3, 2 and 1 particle dimensions, respectively. The slope of the plot of ln I(Q) vs. ln Q is usually used to represents the interface and fractal dimension of the scattering objects.

The above considerations are valid for well-separated systems, where the interactions between suspended particles are negligible. Upon further increasing particle concentration, the interference pattern will therefore comprise contributions from neighboring particles, and the structure factor in Equation (2.17) becomes important in the total scattering intensity. The structure factor describes the arrangement of single particles proceeding either ordering or aggregation processes and allows calculating the interactions in the system. Large aggregates mostly show their presence through the appearance of interaction peak in scattering intensity at small Q values, the position of which is related to the main first neighbor distance between particles, as shown in Figure 2.9. Also, as might be viewed from Figure 2.9, that for large distances the structure factor approaches unity in order to preserve the fact that at large Q only the inner structure of the particle is visible and not its arrangement in space.

II.3.5.1 SAXS acquisition and data treatment

Firstly, Small angle X-ray scattering (SAXS) measurements were carried out on boehmite dispersions of various solids content and ionic strength at room temperature (at 20 °C) and without vacuum insolation on beamline SWING at synchrotron SOLEIL. Samples were conditioned between two 50 µm Kapton foils. Beam energy was fixed at 12 keV ( ≈ 1 Å) and two sample to detector distances (6.2 m and 0.50 m) yielding a Q range between 1•10 - 3 and 1 Å -1 . The beam size on the sample was approximately 400 µm wide and 100 µm high. Two-dimensional scattering patterns (typical acquisition times around 1 s) were recorded on an Eiger detector. The curves of scattering intensity versus scattering vector Q, were deduced from azimuthal angular integration of the SAXS patterns. Curves are provided in arbitrary intensity as the thickness of the various samples was not controlled precisely.

Therefore, the scattering spectra were similarly obtained on filtration deposits at IFPEN by conditioning in Kapton capillaries and sealing both sides to prevent sample evaporation. SAXS acquisitions were acquired at room temperature (at 20°C) and without vacuum insolation. To investigate isotropic-to-anisotropic transition along the filtration deposits, the azimuthal profiles were deduced from the 2D scattering patterns. A Cu rotating anode (Rigaku MM07 X-Ray generator) is used to provide a monochromatic parallel beam with a wavelength of 1.54 Å. The sample to detector distance was 1.4 m, providing a range of scattering vector of 0.005  Q  0.25 Å -1 . Two pairs of crossed slits provide a beam size of 0.2×0.2 mm 2 on the sample. To investigate isotropic-to-anisotropic transition along the filtration deposits, the azimuthal profiles were deduced from the 2D scattering patterns in the following range: 0.01 Q  0.03 Å -1 .

II.4

Dead-end filtration studies on boehmite dispersions

II.4.1 Filtration unit for ex situ tests and data acquisition

Filtration tests were acquired on two types of boehmite dispersions: (i) purely colloidal dispersion and (ii) bimodal mixture. A scheme of the pressure driven frontal filtration unit used in the present study is shown in Figure 2.10. The filtration cell contains Polyethersulfone (PES) ultrafiltration membranes (25 mm in diameter ) (Millipore, Amicon Bioseparations) with a nominal cut off of 500 kDa. Its top layer is comprised of porous organic polymer (PES), which is fixed on a relatively dense support layer resulting in an average membrane thickness of 200 m. Prior to filtration, these membranes were rinsed in distilled water during at least one hour, changing water 2-3 times. Filtration runs were carried out in constant pressure operating mode. The filtration cell (Amicon, Model 8010) was filled with 10 mL of suspension for each experiment and the pressures, obtained from using nitrogen, were set in the range between 1 and 3 bar. The outgoing filtrate is collected in a container, placed on a balance connected to a computer that automatically monitors the filtrate weight every 30 s.

II.4.2 Determination of the deposits permeability and compressibility

A conventional method for estimating the permeability of the deposits consists of monitoring the filtrate volume V as a function of filtration time t at constant pressure. In the case of constant pressure filtration, Equation (1.22) ascribing mass transfer through a porous medium (derived from Darcy's law, see Section I.3.3) was applied to estimate the deposits permeability. This method consists of following data treatment protocol:

1. Registered filtrate weight was converted to volume values using density correlations in order to plot filtrate time over volume t/V, s•m -3 versus volume filtrate collected V, m 3 . 2. The corresponding graph has a straight line dependency, where the slope of a plot of t/V versus V enables estimating the values of specific cake resistance rc. It must be a pointed out that medium resistance Rm is negligible compared with the high levels of estimated rc (of the order 10 15 m•kg -1 ) and, thus, the second term in Equation (1.22) can be ignored. 3. The dry cake mass per filtrate volume cc was estimated independently via the moisture ratio mR, bulk solids concentration sb (by mass fraction) and liquid density , according to Equation (1.19). The solid content of recovered concentrated dispersions is evaluated by a mass balance after drying at 140°C during 24 h. 4. The cake permeability K is related to the specific cake resistance rc according to the following equation

c p c c r 1 K  = , ( 2.21)
where p is the solids density (3.035 g•cm -3 in the case of boehmite) and cc the solid concentration in the filtration cake. 5. The average value for the compressibility factor p of boehmite dispersion is extracted from the linear asymptote of an ln (rc) vs. ln (P) plot, as the slope according to Equation (1.23). The main principle is illustrated in Figure 2.11. A sample is placed on a rotating stage between an X-ray source and a detector. The beam emitted by the X-ray source through the sample is captured by a flat panel detector covered by a crystal scintillation counter, which, first, converts X-ray photons into visible light and transform the latter ones into electrical signal by means of a photodetector. The recorded image gives intensity distribution proportionally to the attenuated X-rays passed through the object, so the 3D spatial distribution is projected into a 2D-image (radiography). In case if X-ray tomography test is carried out, a three-dimensional microstructure is then gathered by acquiring a series of such 2D projection images when setting in a full rotation of sample's axis of 180° followed by data processing using a tomographic reconstruction algorithm.

II.5 X-

Traversing a sample, the beam intensity experiences an exponential attenuation, which is therefore proportional to the linear attenuation coefficient (z) at point z according to Beer-Lambert law where I1 and I0 are the intensity of the transmitted and incident beams, respectively and l is the sample thickness. For monochromatic X-ray beams, Equation (2.22) simplifies to

       - =  dz ) z ( exp I I l 0 0 1 , (2.22)
  l exp I I 0 1  - = . (2.23)
In laboratory systems, beams are usually polychromatic. Still, the same principle of image formation is applied in this case apart from the fact of summing of all attenuation coefficients that vary with X-ray energy. The linear attenuation coefficient  is therefore dependent on the energy E (voltage or wavelength) of X-ray source, the physical density  and the atomic composition Z of medium as follows

3 4 E Z    . (2.24)
Thus, for a high quality absorption-contrast image depending on the material thickness or density, the X-ray energy can be controlled for better image optimization by adapting the acceleration voltage of the X-ray tube. In lab-based X-ray sources, it is varied typically in the range of 30-130 kV, providing more contrast with low energy photons in studies on low density materials, while high energy is required for highly dense samples [START_REF] Ortona | Reticulated SIC Foam X-Ray CT, Meshing, and Simulation[END_REF].

Besides attenuation processes, propagating through a sample X-rays display also Fresnel diffraction effects forming fringes at the object edges on the corresponding image, as a result of interference between passed through different parts of sample beams. This is socalled inline phase contrast, which increases with a long propagation distances beyond the sample, as depicted in Figure 2.12.

Typical X-ray laboratory sources usually generate a cone beam geometry. Due to this, the final scanning resolution will be determined by the position of the object with regard to X-ray source and detector according to magnification SOD SDD = .

(2.25)

Figure 2.12. Simulated phase-contrast images of a 80 m polymer sphere imaged with 30 keV parallel-beam X-rays at three different propagation distances. The fringes at the edge of the sphere increase in visibility and width as the propagation distance increases. In the last image at 250 mm multiple fringes are beginning to merge. The sphere would be too weakly absorbing to be visible in an absorption-contrast image at zero propagation distance. From [START_REF] Meyendorf | Handbook of advanced nondestructive evaluation[END_REF].

The closer the object is located to the source (a source-object distance (SOD) decreases), the higher is the magnification and, therefore, the smaller the voxel size vs. The voxel size vs is defined as

ion magnificat p v s s = (2.26)
with ps the detector pixel size. Each voxel represents a gray value corresponding to the effective X-ray attenuation coefficient respectively to the position in the volumetric data. For higher beam brilliance that have better optical properties, synchrotron radiation offers significant advantage due to its parallel beam geometry. Synchrotrons exploit charged particles accelerated to relativistic speeds, as the source of spontaneous X-ray radiation. Thanks to a parallel monochromatic beam configuration, an exact and quantitative reconstruction enables accurate and detailed structural investigation without geometrical constraints and beam hardening phenomena, as compared to laboratory X-ray sources. Therefore, the use of highly energetic X-ray fluxes allow to reduce significantly acquisition time while maintaining a high signal to noise ratio. Smaller beam sizes enable therefore to acquire highly contrasted images with to submicronic spatial resolution.

II.5.2 X-Ray tomography in membrane separation studies

The use of X-Ray tomography is relatively new in the membrane filtration separation field. So far, only one study carried out by Frank et al. [START_REF] Frank | Visualization of concentration fields in hemodialyzers by computed tomography[END_REF] used X-ray computed tomography for visualizing the concentration field in a hemodialysis module. To reinforce contrast, sodium iodide was added into water. The resulting voxel size was about 5 mm in thickness possessing a cross-sectional area of 0.1875×0.1875 mm. This means that particles size or deposit thickness smaller than the given voxel could not be observed directly, which illustrates the first limitation of X-ray tomography. Also, images were recorded by the authors every 5 minutes requiring 1-2 s per each projection, and results revealed that the concentrations are heterogeneous in the module, indicating that the fibers and flow repartition are not well optimized. However, particular attention should be taken here, as deposition processes might be well faster than acquisition time of experiment, which affects the precision of the final reconstruction. Thus, it implies that observation in real time by X-ray tomography is only possible for slow enough evolving phenomena.

However, it is possible to gain shorter acquisition times in monitoring dynamic processes by acquiring only radiographies. Yeo et al. [START_REF] Yeo | Non-invasive observation of external and internal deposition during membrane filtration by X-ray microimaging (XMI)[END_REF] applied this principle of X-ray radiography (X-ray macroimaging (XMI)) in their studies to follow the deposition of ferric hydroxide particles inside the lumen of a hollow fiber membrane. Coupled with synchrotron radiation, the resolution of 1 m in magnitude was reached, which demonstrated its remarkable capacity in the real-time observation of fouling mechanisms. Although, lack of quantitative data treatment on the characterization of deposited layers have been noted, which requires additional study. Thereby, in our work, both a lab-based X-ray tomographer and Xray microtomography with synchrotron source were applied to acquire in situ filtration tests of boehmite dispersions for the following characterization of deposit growth with controlled physico-chemical dispersion conditions. The design of these experiments is presented in the following sections.

II.5.3 Filtration cell for X-Ray radiography imaging

Lab-based X-ray tomograph (EasyTOM RX Solutions Instrument) was used in filtration studies on colloidal boehmite dispersions. For these purposes, a newly developed filtration cell was used to acquire in situ 2D radiography scans of deposit growth with a relevant resolution. The design of this cell was adapted in order to minimize artifacts in the zone of interest (close to the membrane surface) that could occur when using the preceding filtration cell, as shown in Figure 2.13. The cell is built entirely of polyether ether ketone (PEEK), a polymer that is radiolucent and resistant to X-rays. The internal diameter is 22 mm with a 30 mL capacity. The membrane holder is located in the center of the cell containing an elastomeric O-ring along its edges (Figure 2.14). To ensure membrane sealing, an additional cylinder is placed afterwards, which maintains the contact with the membrane after the cap is screwed onto the cell body. Boehmite dispersions 20 mL in volume were inserted before closing the cell. Transmembrane pressure is then applied at the top of the cell via a connection with nitrogen gas supply. Thin flexible plastic hoses of around 1 mm in diameter were used to interconnect gas supply at the top and filtrate output on membrane holder through the central channel of the cell. Filtration runs were registered during the first 16 filtration hours at an applied pressure of P = 1 bar.

2D radiography scans (2 304×2 304 pixels) were acquired using EasyTOM RX Solutions Instrument on a 110×110 mm 2 Hamamatsu flat detector at an applied tube voltage of 70 kV and a current of 200 A. The source-cell distance was fixed at 4 mm and the sourcedetector distance at 179.3 mm, which yields an effective pixel size of 40 μm. A metal Cufilter (0.1 mm) was used to minimize beam hardening artifacts. The following data treatment was performed using ImageJ software.

II.5.4 Synchrotron X-ray tomography

In situ filtration runs were carried out at the beamline ANATOMIX, at Synchrotron SOLEIL (Gif-sur-Yvette, France) a beamline dedicated to full-field radiography and tomography in the hard X-ray range. A scheme of the beamline is given in Figure 2. [START_REF] Bleta | An efficient route to aqueous phase synthesis of nanocrystalline γ-Al2O3 with high porosity : From stable boehmite colloids to large pore mesoporous alumina[END_REF]. ANATOMIX uses a cryo-cooled undulator. The beamline optics contains diaphragms defining the opening aperture of the beamline, attenuators, focusing mirrors, transfocator and a monochromator. The attenuators use a CVD diamond plate, thickness 600 m, coated with 20 m of Au, as a filtering element to reduce heat load on the transfocator and other downstream elements. The beamline provides a monochromatic beam from a double-crystal Si(111) monochromator (DCM, 28.5 keV).

An indirect conversion detector (revolver-type optics) was used. In such configuration, X-ray photons collide with a 22 m thick lutetium aluminium garnet (LuAG) scintillation foil, where they are then converted to visible light. A mirror redirects vertically the visible light path towards a very high-resolution microscope objective with magnification of ×10 mounted on a CMOS-based digital camera (Hamamatsu Orca Flash 4.0 V2), which results in an effective pixel size of 0.65 m.

The functional principle of filtration cell remained unchanged (Figure 2.17), however its size was reduced down to 140 mm (including peripheral pieces) in order to obtain a better resolution, which is imposed by the minimum sample to detector distance (Fresnel diffraction) and with regard to the geometry of the detector mechanics. Consequently, the membrane positioning of  = 10 mm was shifted towards the bottom of the cell in order to gain a total capacity for gas and dispersion of around 3.4 mL. The total wall thickness was reduced down to 2 mm in order to enhance the transmission signal. The filtration runs were carried out with a dispersion volume of 1 mL providing an optimal filtration time of 1.5-2 hours and resulting in  1 mm of deposit thickness. The applied pressure P of 1, 2 and 3 bar was used. The membrane conditioning and pre-treatment procedures, except its reduced diameter, remained unchanged. The cell was fixed on an additional support providing the necessary height relative to the X-ray beam and the detector with an adapter plate mounted on the standard rotation stage RT500S (LAB Motion Systems, Heverlee, Belgium) (Figure 2.17). Tomography scans were conducted at 10 min interval. In case of bimodal mixtures, a long exposure time of 200 ms was chosen with an overall projections number of 2 000 for the better visualization of particles agglomerates. The user interface for experiment control is based on Spyc, an iPython-based command line. The PyHST2 reconstruction software was used, based on a python script for parametrization of the reconstruction. The radiographies at a given rotation angle were then collected using ImageJ macros in order to derive concentration profiles of growing deposits, similarly to data treatment carried out on the laboratory X-ray tomograph at IFPEN. Similarly, absorption data for 5 reference samples with pre-determined volume fraction of boehmite and water were measured for the quantification of absorption profiles along the growing deposit by means of accumulated volume fraction.

CHAPTER III: Frontal filtration studies on boehmite dispersions

The present chapter is focused on boehmite dispersions when subjected to flow using pressure driven frontal filtration. Firstly, particle morphology, size distribution, dispersions composition and physico-chemical parameters of interest are defined. In a second step, "offline" filtration tests are acquired and the structural organization of filtration deposits is determined by using small angle X-ray scattering (SAXS). In parallel, we attempted to assess detailed growth kinetics of the deposits by X-ray radiography using a specific filtration cell adaptable to an X-ray tomograph.

III.1 Size and shape of boehmite particles

Representative TEM micrographs of the studied boehmite particles are displayed in Figure 3.1. Boehmite appears as small aggregates of slightly isometric platelets with an estimated average length of 9.5  0.4 nm, a value that concurs with the one deduced from Xray diffraction and small angle X-ray studies [START_REF] Mange | A microstructural investigation of nanostructured boehmite films prepared by the sol-gel route[END_REF] (with average length and thickness of 10 nm and 3 nm, respectively).

The size distribution evaluated via dynamic light scattering (DLS) measurements is illustrated in Figure 3.2. The studied boehmite sols represent a complex mixture of welldispersed primary particles (a small peak around 6 nm), aggregates and agglomerates of particles that can be observed by peaks located at 50 nm and 147 nm. The signal-to-noise ratio from residue is rather high, which provides confidence in the obtained results.

III.2 Ionic composition of the aqueous phase

The interactions of particles are highly affected by the ionic environment that directly influences the magnitude of electrostatic repulsion forces. Thus, ionic composition of dialyzed boehmite sols was examined to enable an accurate determination of ionic strength. The composition of this aqueous phase is defined by (i) dissociation of added salts and (ii) dissolution reactions of alumina species (arising from peptization procedure), whose amount by the end of dialysis becomes negligible. Accordingly, ionic strength can be determined by the concentration of nitrate   -3

NO or potassium   + K ions and the pH given in the exponent, which determines its (ionic strength) order of magnitude (in M).

    (

)

pH Al 3 10 Al z NO 2 1 I - - + + = (3.1)
The concentration of H + ions determined by pH 5.0-5.1 in the buffer dialysis reservoir is 10 -5 mol•L -1 . The concentration of Al ions is found to be 7•10 -5 mol•L -1 , which results in the lowest ionic strength considered in the present study I1 = 8•10 -5 mol•L -1 . Increase of ionic strength in this "mother" solution was therefore attained by adding KNO3 into the buffer reservoir at a given pH. The expression for ionic strength can be rewritten as

      ( ) pH Al 3 10 Al z K NO 2 1 I - + - + + + = . ( 3.2)
The detected concentration of ions for intermediate ionic strength is found to be 7•10 -4 mol•L - 1 for K + ions and 7.5•10 -4 mol•L -1 of

- 3
NO ions, resulting in an experimental value of 8•10 -4 mol•L -1 (ionic strength I2 hereinafter). For the highest ionic strength considered in this study the measured concentration of ions is 1•10 -2 mol•L -1 for K + and -3

NO ions, leading to a value of 1•10 -2 mol•L -1 (ionic strength I3 hereinafter). These three ionic strengths that are the "main" focus in the present work will be constantly mentioned throughout the following chapters. In some sections of this chapter two additional ionic strengths of 1.810 -4 M and 1.9310 -3 M also appear, which will be therefore detailed in the text.

III.3 Filtration studies ex situ: deposits permeability

As far as the industrial purpose is concerned, separating colloidal particles from the liquid by filtration is of great importance for the global operation to be efficient. As detailed in the literature review of present manuscript (Chapter I.4), any membrane-based liquid-solid separation operation involves a continuous two-phase separation process: a fluid stream passing through a semipermeable membrane and a concentrated solid phase retained on its surface. Thus, in our studies such mass transport mechanism across porous media, to a first approximation, can be ascribed by a rising resistance parameter due to membrane fouling model (see Chapter I.4.3). As a reminder, medium resistance Rm was found negligible compared with the high levels of estimated rc (of the order 10 15 m•kg -1 ) and, thus, Rm was omitted.

In this section, frontal filtration tests on boehmite dispersions were conducted in order to assess the dependence of permeability K on salt concentration and the impact of pressure difference on the filtration rate. A scheme of the filtration unit for ex situ studies has been displayed previously in Figure 2.10 (Chapter II). The slope after linear fitting is used to calculate K. Boehmite dispersions at a volume fraction of  0.4 vol.%, a pH of 6.0-6.2 and ionic strengths of 8•10 -5 M, 8•10 -4 M and 1•10 -2 M were investigated. The filtrated volume of dispersions was fixed at 10 mL. The permeate was examined using DLS, where negligible quantity of particles was detected. This confirms the choice of the filtration membrane, whose nominal cut off roughly corresponds to 40 nm in pores diameter.

Figure 3.3 displays the evolution of the filtrate mass with filtration time at applied constant pressures of 1, 2 and 3 bar. Such profiles are indicative of both filtration kinetics and cake formation followed by its growth. From Figure 3.3 it appears that the runs are reproducible, which enables using such plots for assessing the role of various parameters on filtration. In order to evaluate cake permeability, these profiles can be plotted as t/V versus V (Figure 3.4). Clear tendency can be deduced from these curves: increasing the applied pressure leads to lower slopes. This indicates an acceleration of filtration when a stronger force is applied to push suspensions towards the porous membrane. Based on fitting (Equation 1.22) to experimental data (Figure 3.5), the K values evaluated from a series of constant pressure filtration measurements are of the order of 10 -18 m 2 , values that are somehow close to those obtained in filtration studies on protein and laponite systems [START_REF] Loginov | Centrifugal ultrafiltration for determination of filter cake properties of colloids[END_REF], bentonite [START_REF] Bacchin | Influence of surface interaction on transfer during colloid ultrafiltration[END_REF], aggregated silica [START_REF] Madeline | Filtration de dispersions agrégées de silice colloïdale : structure, consolidation et perméabilité de dépôts[END_REF]. In the given range of K, our samples can be ascribed as highly impervious. Deposits formed at the lowest ionic strength (I1 = 810 -5 M) are characterized as the most permeable, whereas the lowest filtration rate is obtained for moderate ionic strength (I2 = 810 -4 M), displaying a two-fold decrease as compared to I1 = 810 -5 M. Such tendency remains unchanged whatever the operating pressure. Still, the drop in permeability at moderate ionic strength becomes less and less marked with increasing P. Such an evolution with pressure can be assigned to deposit compressibility. The average value for the compressibility factor of boehmite dispersion is extracted from the linear asymptote of a ln () vs. ln (P) plot, as in such a representation the slope should provide the value of n according to α = α 0 • (1 -n) • (ΔP) n . The parameter n thus determined is equal to 0.4. As a reference, n = 0 is relevant for incompressible solids, calcite yields a n value of about 0.2, and for kaolin clays it approaches 0.6. Classically, higher compressibility factors are associated with a higher aspect ratio of particles. This could be explained by the fact that for anisotropic materials orientational diversity leads to a wider range of possible particles packing [START_REF] Foley | A review of factors affecting filter cake properties in dead-end microfiltration of microbial suspensions[END_REF]. Clearly, the permeability drastically declines for an ionic strength I2=810 -4 M. Similar tendencies have been observed for bentonite dispersions in [START_REF] Bacchin | Influence of surface interaction on transfer during colloid ultrafiltration[END_REF], where authors suggested that the minimum in permeability might be correlated to the proximity of critical coagulation concentration (ccc). However, this explanation is not relevant in our case as we are at least one order of magnitude below this region (ccc for boehmite lies between 0.1 M and 0.15 M, according to [START_REF] Fauchadour | Etude de la peptisation de la boehmite[END_REF]). This suggests that the main cause of the minima lies elsewhere. Another possible mechanism could be put forward here: effect of surface charges on deposit formation, as explicitly discussed in Chapter I of present manuscript. This issue is addressed in the next section.

III.4 The boehmitesolution interface

Electrophoretic mobility tests are commonly used to measure  potential values that provide information about the pH evolution of particle surface charges. This knowledge provides useful insights regarding electrostatic stabilization of particles via their double layers as well as interparticle interactions. Referring to Chapter I of the present manuscript, the  potential defines the electric potential at the plane d between Stern layer of counter-ions and diffuse layer in the bulk solution. From Figure 1.2 it follows that the electrical potential (x) decays with distance from the particle surface, consequently affecting the ions distribution across the diffuse layer and the bulk medium. To remind, the reciprocal thickness  -1 of the diffuse layer is referred to as the Debye screening length, which holds the linkage with ionic strength of solution via ( )

I 304 . 0 nm 1 =  - , ( 3.3)
for a symmetric monovalent electrolyte (at 25°C). The above given equation shows that the Debye length decreases with increasing ionic strength, so that consequently double layer repulsion is expected to be enhanced in dispersions at low ionic strengths. Studies on electrical double layer properties of boehmite have been poorly disclosed, apart from few works dated back to the 80's. In particular, Jiratova [START_REF] Jirátová | Influence Of Aluminium Hydroxide Peptization On Physical Properties Of Alumina Extrudates, in Preparation of Catalysts III -Scientific Bases for the Preparation of Heterogeneous Catalysts[END_REF] carried out zeta measurements of modified alumina with reported isoelectric point values varying between 4.5 and 9.8 depending on the type and level of impurities present. The work of Drouin et al. [START_REF] Drouin | Rheology and structure of peptized boehmite pastes[END_REF] reported the isoelectric point of boehmite at pH 8.6 using NaNO3 supporting electrolyte. Wood et al. [START_REF] Wood | Electrochemistry of the boehmite-water interface[END_REF] carried out an extensive analysis by simultaneously assessing surface charge and zeta-potential. This enables a precise description of the solid-aqueous interface of boehmite via a triple layer site-binding approach, as previously mentioned in Chapter I of the present manuscript. Accordingly, Stern layer is further split into two additional Helmholtz planes, where the inner one closest to the particle surface accounts for surface hydration reactions. Therefore, authors observed discrepancies between the point of zero charges (pzc) (pHpzc = 8.5) and the iep (pHiep = 9.1) that was assigned to different affinities of the KNO3 electrolyte containing cations and anions for the boehmite surface.

As opposed to commonly used potentiometric titration experiments, the pH of boehmite dispersions in our experiments was reached by performing dialysis against a buffer reservoir at a corresponding pH. For the lowest ionic strength considered in present study (I = 8•10 -5 M), by the end of dialysis it is defined by the concentration of nitrate   for acid and basic range, respectively. Thus, using such a preparation method, corrections for proton release by dissolved species in high solubility regions are not required, as it is the case with titration curves. In our case, for the zeta measurements of boehmite dispersions the pH is constrained by the range of 5-10 due to increasing concentration of potential-determining H + and OH¯ ions that alters the order of magnitude of I in highly acidic and alkaline conditions. Three additional ionic strengths of 810 -4 M, 1.9310 -3 M and 110 -2 M of the suspensions were therefore adjusted to a "mother" solution at a given pH by adding KNO3 into the buffer reservoir. The conductivity is seen consequently rising with addition of salt concentration due to an enhanced number of mobile charge carriers (co-ions K + and counterions NO3¯). The expression for ionic strength is rewritten as

      ( ) pH Al 3 10 Al z K NO 2 1 I - + - + + + = (3.6)
and

      ( ) pOH Al 3 10 Al z K NO 2 1 I - + - + + + = (3.7)
for acid and alkaline range, respectively. Two important parameters characterizing the pH-dependent charging behavior of particles are: (a) the isoelectric point (iep) indicating that the potential at the slipping plane (namely, the zeta potential) is zero; and (b) the point of zero charge (pzc) when the surface charge caused by the potential-determining ions (H + and OH¯ ions) is zero. In the simplest case, if there is no any specific adsorption of ions, the pHiep then coincides with the point of zero charge (pHpzc). According to Figure 3.6 showing distribution of zeta potential as a function of pH for three different ionic strengths, the iep for boehmite lies around pH 9.6, which is in good agreement with other studies on alumina [START_REF] Wood | Electrochemistry of the boehmite-water interface[END_REF][START_REF] Fauchadour | Etude de la peptisation de la boehmite[END_REF][START_REF] Evanko | Influence of aqueous solution chemistry on the surface charge, viscosity and stability of concentrated alumina dispersions in water[END_REF]. Since the octahedral stoichiometry of boehmite (Al-OH) comprises hydroxyl-functional groups at the basal plane, the edge surface acquires a net positive charge over a wide range of pH (below pHiep) as a result of protonation-deprotonation reactions. From Figure 3.6 it therefore follows that decreasing pH and increasing ionic strength leads to lower zeta potential. Intuitively, this can be well explained in terms of reducing electrostatic potential resulting from double layer compression due to increasing ions concentration forming a cloud around the particle. It is worth noting that flocculation starts appearing above pH 7 and rises progressively when approaching the pH of the iep. Consequently, electrophoretic measurements were difficult to handle for the highest ionic strength (I = 110 -2 M) due to increasing flocs formation. Since the zeta potential is calculated from the electrophoretic mobility of particles, displacement of particles aggregates is more relevant at high ionic strengths. Consequently, the measuring zeta potential values in this case were found significantly overstated in comparison with that of lower ionic strengths and were not presented for clarity.

Additional filtration tests were therefore conducted as a function of pH (for ionic strength I = 810 -5 M) (Figure 3.7). Accordingly, the permeability displays a six-sevenfold increase when pH rises from 5 to 9.6 corresponding to the iep. This is also coherent with the progressively decreasing zeta potential. Indeed, changes in particles packing mode was experimentally observed by Hlavacek et al. [START_REF] Hlavacek | Simple Relationships among Zeta Potential, Particle Size Distribution, and Cake Specific Resistance for Colloid Suspensions Coagulated with Ferric Chloride[END_REF] in the case of coagulated kaolinite dispersions filtered in cross-flow microfiltration. Their results demonstrated that particles tend to preferably align in stacks, leaving large channels in between, when approaching nearly zero surface charges. This implies an improved filtration operation (a more permeable deposit), since charge destabilization invokes aggregation processes, as evidenced in the present work by flocs appearance and more turbid dispersions near the iep. With regard to permeability curves displayed in Figure 3.5 and zeta measurements detailed above, the following question might be asked: does the evolution of permeability as a function of ionic strength result from a corresponding decrease of interaction potential? According to zeta measurements acquired for ionic strengths of 810 -5 M, 810 -4 M and 1.9310 -3 M (Figure 3.6), surface charges diminish proportionally upon salt addition (at a fixed pH of boehmite sols in the range of 6.1-6.2) from 60 mV to 40 mV for I = 810 -5 M down to I = 1.9310 -3 M, respectively. In the meantime, for one given ionic strength (I = 810 -5 M, Figure 3.7) it was shown that the zeta potential decreases from 60 mV to 40 mV with a concomitant six-fold increase in permeability, which arises from promoted flocculation state due to sol destabilization close to the iep region. Thus, one might expect a denser particles packing and, hence, a reduced permeability K when also lowering the ionic strength with associated surface charge reduction. However, since zeta potential was seen decreasing proportionally with increasing ionic strength, this fact does not seem to provide a viable explanation for the nonlinear dependence of permeability K with ionic strength (Figure 3.5). Lastly, the permeability drop observed for an ionic strength of I = 810 -4 M could alos be associated to orientational effects. The works of Pignon et al. (in frontal [START_REF] Pignon | Structural characterisation of deposits formed during frontal filtration[END_REF], crossflow [START_REF] Pignon | In situ characterization by SAXS of concentration polarization layers during cross-flow ultrafiltration of Laponite dispersions[END_REF] ultrafiltration modes and centrifugation [START_REF] Martin | Osmotic compression and expansion of highly ordered clay dispersions[END_REF]) have extensively demonstrated an anisotropic organization of platelet particles next to the membrane where high concentration levels are reached. By combining filtration tests in situ with X-ray scattering and birefringence techniques, structural changes upon deposit formation were distinguished at length-scales ranging from that of elementary particles up to supra-particular organization. It was also shown that anisotropic arrangement of particles (at moderate ionic strengths 10 -3 M) is strongly favored under pressure-induced flow environment. Though the effect of ionic strength was not addressed in the studies, stronger particles orientation was closely associated to increasing concentrations of laponite dispersions forming a denser structure near the membrane. This shows that it is highly relevant to characterize the structure of the deposits by scattering techniques in order to assess the correlation between microstructure of the filtration deposits and flow related properties with regard to filtration performance. Such a characterization is the main focus of the next section.

III.5 Structural organization of filtration deposits

In order to obtain structural features along the filtration deposits at various ionic strengths, ex situ Small Angle X-ray Scattering (SAXS) tests were acquired in capillaries with 1 mm diameter, as schematically displayed in Figure 3.8. Both sides of the capillaries were sealed in order to prevent sample drying within acquisition. The sample-to-detector distance was 1.4 m, providing a range of scattering vector of 0.01  Q  0.03 Å -1 . Figure 3.9 displays the azimuthal intensity profiles of the filtration deposit acquired along the capillary for three different ionic strengths. Azimuthal intensity profiles were extracted by integrating sectors of the image over azimuthal angles along the principle directions of the 2D SAXS pattern, as defined by its ellipsoidal orientation. The periodic evolution can be fitted using a Maier-Saupe type orientation function according to

)) ( cos m exp( A y ) ( f 0 2 0  -  + =  , (3.8)
where A is a background term, B is an intensity multiplicative term, m is an orientation parameter,  is the azimuthal angle (in degree °) and 0 is related to the orientation angle (in degree °).

In the present work, we consider the second-order Legendre polynomial S given by ( )

1 cos 3 2 1 S 2 -  = , (3.9)
where  2 cos indicates the average. It follows from Figure 3.9 that, for the two lowest ionic strengths, near the filter, the particles are slightly more oriented, as seen by the occurrence of deeper oscillations. For these two IS values, away from the membrane, the particles tend to orient in an increasingly random manner causing the reduced oscillations. The resulting order parameter S is displayed in Figure 3.10 together with the corresponding concentration profiles obtained by exploiting transmission data of SAXS for each ionic strength. Using Equation (2.23) with absorption coefficients of 84.07 cm -1 and 9.82 cm -1 for boehmite boehmite and water water, respectively, For all ionic strengths and distance for the membrane, the parameter S remains rather low with maximal values around 0.07 1 . Such values are significantly different from those obtained for instance on kaolinite dispersions where values of 0.3 to 0.6 were measured [START_REF] Perdigon-Aller | Preferred orientation in filtercakes of kaolinite[END_REF] when high flow rates were used under static and under cross-flow conditions. The fact that small values of S are observed on boehmite dispersions could be explained by various reasons. (i) boehmite particles display anisotropy much smaller than those of clay mineral clays, which could lead to low particles ordering. (ii) orientational arrangement could be highly limited by the chosen operational filtration conditions. Indeed, for the experimental conditions chosen the Reynolds number Re can be calculated according to

fluid fluid L Re      = , (3.11)
where fluid is the initial fluid velocity (estimated to be here of the order of 2•10 -6 ÷ 2•10 -7 m•s - 1 at P = 1 bar), L is a characteristic dimension of the flow geometry, fluid is the kinematic viscosity of the fluid (taken  1 mPa•s as for water which is appropriate for such low viscosities slurries. The values of Reynolds number thus obtained vary between 0.006 and 0.06, i.e. values << 1 that implies slow flows.

1 Filtration test at the highest ionic strength (I3 = 1 10 -2 M) as compared to others was conducted up until dewatering processes followed by the deposit compaction. Hence, the concentration profile and the ordering parameter S reach their maximum level. Accordingly, the filtration fluxes developed in the chosen operational filtration conditions could be insufficient to induce significant particles alignment under flow. (iii) the spatial scale associated to the explored scattering vectors Q corresponds to intraparticle dimensions. At such short length scales, particles display weak intercorrelations with their neighbors whatever the ionic strength.

Still, the variations in ordering deduced from this SAXS analysis are not high enough to explain the permeability drop observed for the intermediate ionic strength (Figure 3.5). It may be conceivable that this is linked to the spatial scale investigated and USAXS measurements or optical birefringence measurements could certainly be interesting to perform to explore higher spatial scales. In contrast with the orientational order parameter S, significant evolution of volume fractions with ionic strength can be observed. Still, before exploiting such differences, it is important to check if the volume fractions deduced by SAXS are relevant and not affected by the sampling procedure used. To get better insight into this question, after SAXS characterization, the kapton tubes with filtration cakes were analyzed using a lab-based X-ray tomograph at IFPEN. sample appears rather homogeneous. To go further in the analysis, it is possible to measure more precisely the volume fraction by analyzing a series of boehmite samples of various known concentrations. Using such a procedure a calibration curve of the gray levels can be obtained (Figure 3.12). Figure 3.13 displays a comparison between the concentration profiles deduced from SAXS and X-ray tomography. Both methods yield nearly identical results and therefore, the evolution of volume fraction as a function of distance along the deposit appears reliable. Still, some uncertainties associated to the preparation procedure remain and the image thus obtained does not lead to any kinetic information. For this reason, we also used the X-Ray tomograph to carry out an in situ characterization of the filtration deposit during its formation by designing a new dead-end filtration cell. The results of such runs are presented in the next section.

III.6 Monitoring concentration profiles by X-Ray radiography imaging

In situ filtration tests have been performed on colloidal boehmite dispersions at three ionic strengths (I1 = 810 -5 M, I2 = 810 -4 M and I3 = 110 -2 M) using a lab-based X-ray radiograph at IFPEN. With the designed filtration cell (see Figure 2.14), the initial volume of dispersion was increased to 20 mL for visualizing thick deposit layers. Filtration runs carried out with an applied pressure P = 1 bar were monitored during the first 16 hours of filtration The height resolution was estimated around 40 m. For such resolution single agglomerates could not be resolved, and the main purpose of these experiments was then follow deposit growth kinetics at large-scale. In order to determine the concentration profiles within deposit layers, the images obtained by the experiment were further processed. Figure 3.14 displays some typical radiographies obtained along a filtration run. The gray level on each pixel on the image corresponds to the average intensity transmitted across the object. Accordingly, the bottom of the filtration cell appears darker as the volume fraction is higher close to the filtration membrane. Using a calibration curve for the used geometry (Figure 3.15) is then possible to extract 2D plots of the evolution of volume fraction as a function of distance from the membrane surface z for the first sixteen hours of filtration.

In view of the cylindrical shape of the filtration cell, volume fraction calculations were performed only in the middle of the filtration cell, where the thickness is almost constant. As an example, Figure 3.16 displays a plot showing the time evolution of the concentration profiles obtained for a filtration run carried out at an ionic strength of I1 = 8•10 -5 M. The shape of the profile clearly evolves with time. For high filtration times, far from the membrane, a sharp drop corresponding to the air-liquid interface can be observed. As it is of no interest for the analysis we wish to perform, it is not shown here. Visually, the shape of the concentration gradient appears to display three regimes: (i) next to the membrane surface the

Gray level normalized

Chapter III: Frontal filtration studies on boehmite dispersions slope of the concentration is rather high, it then smoothens at higher distance, and becomes steeper again before reaching the volume fraction of the initial slurry. The first regime could indicate an increased compression in the lowest layers, resulting in a more compact and resistant deposit. Interestingly, other in situ filtration studies using Magnetic Resonance Imaging (MRI) (Hieke et al. [START_REF] Hieke | Analysis of the Porosity of Filter Cakes Obtained by Filtration of Colloidal Suspensions[END_REF]) or optical laser sensor (Mendret et al. [START_REF] Mendret | In situ dynamic characterisation of fouling under different pressure conditions during dead-end filtration: Compressibility properties of particle cakes[END_REF]) have reported similar trends, that were convincingly assigned to concentration polarisation phenomena. The interpretation of the second regime is less clear. Figure 3.17 presents a comparison of the concentration profiles obtained for three different ionic strengths, i.e. I1 = 8•10 -5 M, I2 = 8•10 -4 M and I3 = 1•10 -2 M. Clearly, ionic strength has a strong influence on the shape of the concentration profiles. This is particularly striking for the highest ionic strength for which the concentration profile is almost linear. As mentioned previously, electrophoretic measurements and visual observations (Figure 3.18) strongly suggest that, for such an ionic strength, aggregation processes may well take place. It may then tentatively proposed that the featureless concentration profile obtained for I3 = 1•10 -2 M, could be linked to the presence of an at least partially aggregated structure. Further experiments on other systems shoud be performed for consolidating such an assumption.

The situation is obviously different for the filtration run carried out at the intermediate ionic strength (I2 = 8•10 -4 M). For such a system, aggregation phenomena do not take place but still, the concentration profile is significantly different. Furthermore, such physicochemical conditions lead to significantly lower filtration rates. Interestingly, filtration studies of Pignon et al. [START_REF] Pignon | Structural characterisation of deposits formed during frontal filtration[END_REF] also evidenced that changes in deposit structuring with evolving physicochemical conditions induce significant modifications of both filtration fluxes and resulting permeability. In particular, they showed that the presence of a connected gel leads to higher

steep drop slight fold level of suspension permeability and increased filtration fluxes due to the existence of a more porous structure as compared to the case of a suspension in the liquid state. The structure and interparticle interactions (that depend on ionic strength) then appear to play a major role on particle transport mechanisms upon filtration. It is then clearly relevant to investigate in more details the properties of boehmite dispersions at equilibrium. This can be achieved through osmotic stress experiments that will be detailed in the next chapter.

Figure 3.17. Concentration profiles as a function of the distance z from the membrane surface deduced from in situ radiography tests during pressure-driven dead-end filtration of boehmite dispersions at varied ionic strength. Filtration conditions: V = 20 mL; P = 1 bar; ultrafiltration membrane PES, 500 kDa, surface area 3.14 cm 2 . 1 time step : increment 60 ×10 s  10 min.

III.7 Summary

In the present chapter, we analyzed the flow properties of anisometric colloidal boehmite dispersions subjected to pressure-driven ultrafiltration runs. Permeability data from ex situ filtration tests revealed significant differences in filtration behavior depending on the ionic strength of boehmite dispersions. In particular, an unexpected drop in cake permeability was observed for an intermediate range of ionic strength. This was tentatively assigned in our work to induced orientational ordering, as illustrated in other filtration studies on plate-like systems [START_REF] Tanaka | Filtration characteristics and structure of cake in crossflow filtration of bacterial suspension[END_REF][START_REF] Perdigon-Aller | Preferred orientation in filtercakes of kaolinite[END_REF][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]. However, SAXS structural analysis on filtration deposits evidenced only moderate alignment at intraparticular levels irrespectively of ionic strength. Alignment at the scale probed by SAXS does then not provide any explanation for the evolution of permeability with salt addition.

In parallel, the use of X-Ray tomography showed promising prospects for the direct observation of deposit growth kinetics. The concentration profiles obtained from real-time in situ X-ray radiography filtration tests, confirmed the influence of ionic strength on filtration by evidencing significant different shapes depending on salt addition. This confirms the importance of particle-particle interactions in deposit structuring in the context of filtration processes.

CHAPTER IV: Properties of osmotically densified boehmite dispersions: microstructure and particles organization

The present chapter is focused on boehmite dispersions equilibrated by osmotic stress experiments to investigate in depth the properties of these dispersions when submitted to a uniform compression. The phase diagram, rheology and structure of boehmite dispersions are studied with regard to bulk solution chemistry, i.e. solids content and suspensions ionic strength. In a first step, boehmite dispersions were equilibrated at fixed ionic strength by osmotic stress experiments to yield the equation of state (osmotic pressure curves vs. solid fraction) and phase behavior as a function of both solid fraction and ionic strength. The rheological features of the suspensions (in particular yielding and viscoelasticity) were then measured in various areas of the previously determined phase diagram. In parallel the structure of the suspensions was investigated by small angle X-Ray scattering (SAXS). Based on such information, we attempted to rationalize high frequency elastic moduli on the basis of electrostatic effects in the systems and took into account the anisotropic nature of boehmite particles by using effective volume fractions. In a last step, electrostatic interaction potentials between boehmite particles have been derived.

IV.1 Osmotic pressure curves and phase diagram

The osmotic pressure versus volume fraction curves and phase diagram both provide valuable information about the role of interparticle interactions on the resulting structure of particles assembly. Figure 4.1 shows the results obtained by osmotic equilibration of dilute boehmite dispersions against PEG solutions of fixed ionic strength at pH = 5.0-5.2. As expected, upon salt addition, the induced electrostatic screening results in a lower osmotic pressure. Such an effect is observed for colloidal dispersions whatever the particle shape. In contrast, the presence of a plateau for medium volume fractions and osmotic pressure that is observed in the present case appears to be typical of some plate-like particles. Indeed, it was observed for Na-Wyoming montmorillonite [START_REF] Michot | Phase diagrams of Wyoming Na-montmorillonite clay. Influence of particle anisotropy[END_REF] and laponite [START_REF] Mourchid | Phase Diagram of Colloidal Dispersions of Anisotropic Charged Particles : Equilibrium Properties, Structure, and Rheology of Laponite Suspensions[END_REF], whereas no plateau was reported in the case of nontronite [START_REF] Michot | Sol/gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 1. Features of the i/n transition[END_REF] and beidellite [START_REF] Paineau | Liquid-crystalline nematic phase in aqueous suspensions of a disk-shaped natural beidellite clay[END_REF] (at least for ionic strengths lower than 10 -3 M).

The presence of such a plateau might indicate the coexistence of two phases. Indeed, if a phase transition occurs, the osmotic pressure is expected to be equal in both phases which would lead to the appearance of a plateau at the volume fractions corresponding to both phases. This was for instance evidenced in the study of Pages et al. [START_REF] Page | Osmotic pressure and phase boundary determination of multiphase systems by analytical ultracentrifugation[END_REF] who suggested that the plateau observed in osmotic pressure curves for an ionic strength of 10 -3 M can be assigned to the formation of a gel that is kinetically trapped during osmotic stress compression and equilibration. Such a micro-phase separated state was also experimentally evidenced on laponite dispersions using ultra-small angle X-ray scattering experiments (USAXS), displaying a Q -3 dependence at low angles [START_REF] Saunders | A Small-Angle X-ray Scattering Study of the Structure of Aqueous Laponite Dispersions[END_REF]. The formation of this latter phase was assigned to the onset of an isotropic/nematic phase transition (Onsager transition), in agreement with numerical predictions for the phase diagram of charged disk-like colloids [START_REF] Jabbari-Farouji | On phase behavior and dynamical signatures of charged colloidal platelets[END_REF].

To further investigate the system, it is relevant to plot a phase diagram obtained on the basis of rheological measurements that will be detailed later in this chapter. Basically at this point, the aim is to distinguish between a free-flowing sol phase and a non-flowing (gellike) phase. A gel-like state refers to its mechanical property that defines a stiffness of solid material. Rheologically, it displays the typical dominance of the storage modulus G over the loss modulus G over the tested frequencies (0.01-100 rad•s -1 ) and in the linear viscoelastic (LVE) range). On this basis, the resulting phase diagram is plotted in Figure 4.2. The dotted line in this diagram corresponds to the sol/gel transition defined as an average between two distinct points separating the two states. Osmotic pressure with a step of 10 Pa was imposed to ensure the accuracy window accounting for an error of 11-12 %. It must be pointed out that the evolution of the transition line with ionic strength is far from being trivial. Indeed, the sol/gel transition line first exhibits a positive slope up to 810 -4 M, indicating a system dominated by electrostatic repulsion whereas in the high ionic strength region, i.e. from 810 -4 M to 110 -2 M, the slope is reversed. By interpolation, it can be foreseen that this line will join the flocculation line located at high ionic strength (between 0.1 M and 0.15 M, according to [START_REF] Fauchadour | Etude de la peptisation de la boehmite[END_REF]). Similar features with a "nose" in the phase diagram have already been observed in other charged anisotropic systems, e.g. synthetic compound laponite [START_REF] Martin | Dissociation of thixotropic clay gels[END_REF] or natural clays such as nontronite [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF] or beidellite [START_REF] Paineau | Liquid-crystalline nematic phase in aqueous suspensions of a disk-shaped natural beidellite clay[END_REF]. It must be pointed out that the sol/gel transition in Figure 4.2 coincides with the onset of the plateau in the osmotic pressure curves up to I2 = 8•10 -4 M, as observed for laponite [START_REF] Mourchid | On Viscoelastic, Birefringent, and Swelling Properties of Laponite Clay Suspensions : Revisited Phase Diagram[END_REF]. In contrast, such correlation significantly diverges for ionic strength of I3 = 1•10 -2 M. It

I 1 = 810 -5 M I 2 = 810 -4 M I 3 = 110 -2 M
Chapter IV: Properties of osmotically densified boehmite dispersions: microstructure and particles organization 83 is tempting to attribute this discrepancy to aggregation processes that become apparent at such high concentrations, and it will be discussed in more details later in the text. As already mentioned, the shape of the phase diagram displayed in Figure 4.2 is close to the one observed for laponite. In this latter case, the evolution of the gelation mechanism with ionic strength has been highly debated and remains somehow controversial. At very low ionic strength (in the range of the phase diagram with a positive slope) Levitz et al. [START_REF] Levitz | Liquid-solid transition of Laponite suspensions at very low ionic strength : Long-range electrostatic stabilisation of anisotropic colloids[END_REF] have shown that the sol/gel transition could be interpreted as a repulsive glass transition (Wigner glass transition) of charged spherical like objects. According to Wigner predictions, this arises from the long-range Coulombic interactions that block the particles at distances from each other that are much higher than the particle diameter resulting in a solidlike behavior for very small densities as compared to ordinary solids. Along similar lines, based on light scattering tests, Bonn et al. [START_REF] Bonn | Aging of a colloidal "Wigner" glass[END_REF] rationalized the dynamics of laponite dispersions at I  10 -4 M (i.e. just at the "nose" of the transition line) as corresponding to a glassy-like state (combining features of crystals and fluid-like structure), where particles are caged/locked in dynamically arrested metastable states formed by their neighbors. For a symmetric monovalent electrolyte, such as KNO3, the Debye length  -1 evolves with ionic strength as ( )

I 304 . 0 nm 1 =  - . ( 4.1)
In the case of boehmite particles, considering the size of the platelets, i.e. 10 nm, at the lowest ionic strength used in the present work (8•10 -5 M), the corresponding Debye length (≈ 34 nm) is much higher than the average particle size. The particles can then be assimilated to repulsive spheres with a radius of 34 nm forming a gel, once their ionic clouds start

Sol Gel

Flocs interpenetrating each other. Numerical simulations carried out by Jabbari-Farouji et al. [START_REF] Jabbari-Farouji | On phase behavior and dynamical signatures of charged colloidal platelets[END_REF] suggest that particles immobilized by strong repulsive double-layer interaction (= in the limit of low ionic strengths) can indeed reach a solid-like state with ordered regions. These results are in accordance with data on high-aspect ratio boehmite rods [START_REF] Wierenga | Aqueous Dispersions of Colloidal Boehmite : Structure, Dynamics, and Yield Stress of Rod Gels[END_REF] revealing the existence of glassy state that is formed way before the isotropic-nematic (I-N) transition occurs. Moreover, neither the glassy state nor the phase separation was relevant for dispersions containing less anisotropic particles, similarly to our results, suggesting that these processes are presumably driven by excluded volume effects.

In the limit of high ionic strengths, i.e. I3 = 110 -2 M, the Debye length is reduced down to 3 nm in thickness, electrostatic repulsions are strongly screened and Van der Waals forces start playing a major role. Even if the structure of the gel formed in such regions of the phase diagram remains debated, attractive forces likely start playing a role. Still, two possible mechanisms were put forward either: (i) a jamming transition with regard to closely packed clusters stabilized by long-range electrostatic repulsions due to interacting double layers or (ii) a percolation threshold of small stacks in a linked three-dimensional structure governed by attraction between edges and faces of the platelets, resulting in the so-called "house of cards" configuration. The formation of objects resulting from attraction appears likely in the case of the boehmite suspensions investigated in the present work as (i) suspensions in this region of the phase diagram appear slightly more turbid than those at lower ionic strength, which could well indicate the occurrence of aggregation processes; (ii) these suspensions evolve with time in the long term. Indeed, gel-like behavior appears with time for volume fractions as low as 0.8 % but the sample can be reversibly transformed back to sol after gentle shaking. Theoretical works (i.e. [START_REF] Delhorme | Monte Carlo simulations of a clay inspired model suspension : The role of rim charge[END_REF]) associated with numerical simulations on model clay dispersions suggest various modes of association between platelets when sufficiently high salinity is reached.

In the intermediate region of the phase diagram (810 -4 M) the Debye length  -1 is comparable to the length of platelet ( 10 nm). In this situation, particle anisotropy starts playing a role and long-range interaction with anisotropic short-range excluded interactions both participate in a complex way to the sol/gel transition. Norrish [START_REF] Norrish | The swelling of montmorillonite[END_REF] proposed in the 1950's that gel formation results from the mutual repulsion of particles resulting from interactions between their double layers. Evidence for such processes was provided for instance, by viscosity tests of Evanko et al. [START_REF] Evanko | Influence of aqueous solution chemistry on the surface charge, viscosity and stability of concentrated alumina dispersions in water[END_REF] on concentrated alumina dispersions or by shear stresses and shear moduli measurements on montmorillonite dispersions acquired by Rand et al. [START_REF] Rand | Investigation into the existence of edge-face coagulated structures in Na-montmorillonite suspensions[END_REF].

On the basis of phase diagram, the folded zones observed previously (in situ filtration studies, Chapter III) on the concentration profiles can therefore be interpreted. For clarity reasons, it seems important to recall them in this section for the following discussion (Figure 4.3). As seen in Figure 4.3, the broad folded zone on the given profiles may be tentatively assigned to the liquid/gel transition located differently for the three ionic strengths investigated. Again, in the limit of high ionic strengths, the featureless concentration gradient might be explained by the fact that early gelation at rest occurs in a range of sufficiently low volume fractions that is close to that of the initial slurry. Alongside visual observations (Figure 3.18) and electrophoretic measurements mentioned previously (Chapter III), it is evident to speculate that the presence of aggregated structure induces significant modification on the deposit formation resulting in a distinct concentration profile as compared to the case where repulsive contribution prevails (ionic strength of 810 -5 M and 810 -4 M).

Moreover, our results are in agreement with the fact of the structural equivalence of osmotic and mechanical (by means of pressure-driven filtration) modes of consolidation/concentration mechanisms, as demonstrated by Miller et al. [START_REF] Miller | Comparison of the Compressive Yield Response of Aggregated Suspensions : Pressure Filtration, Centrifugation, and Osmotic Consolidation[END_REF] on kaolinite and alumina suspensions displaying similar volume fraction profiles.

If what we claim is true, this is the first experimental evidence of the time resolved sol/gel transition recorded upon increasing solid concentration in the growing deposits during frontal membrane separation process. In any case, the osmotic pressure curves and the sol/gel transition of boehmite dispersions appear to result from a subtle balance between electrostatic interactions and excluded volume effects. This suggests that the physical origin of gelation could lie in the structural diversity with regard to different configurations of particles assemblies. Distance z, (m)

I 3 = 110 -2 M
Chapter IV: Properties of osmotically densified boehmite dispersions: microstructure and particles organization 86 Accordingly, it is worthwhile carrying out a structural analysis of the given dispersions by SAXS.

IV.2 Structure and particle organization

The structure of dispersions in various zones of the phase diagram was examined by small-angle X-ray scattering in the Q range of 110 -3 -1 Å -1 . The total scattering intensity I(Q) per unit particle volume V is usually expressed as

) Q ( S ) Q ( P V n A ) Q ( I 2 e 2 e   = , (4.2)
where Q is the scattering vector (with

   = sin 4 Q
), Ae is the amplitude scattered by one electron, ne is the number of electrons,  2 is the electronic density difference between particle and solvent, P(Q) is the form factor and S(Q) the structure factor. In very dilute dispersions, particle interactions are minimized and in this case, S(Q) is close to unity and the scattering pattern then corresponds to the form factor P(Q). A curve corresponding to such a situation ( = 0.2 vol.%) is shown in Figure 4.4. At short distances (high Q), the asymptotic decrease in intensity follows a Q -4 power law, which is described as the Porod law region dependent on the interfacial area of the solid-liquid interface. At intermediate scattering vector, the intensity decreases as  Q -2 over a narrow range of Q, suggesting the presence of flat objects characterized by low anisotropy [START_REF] Mange | A microstructural investigation of nanostructured boehmite films prepared by the sol-gel route[END_REF]. A similar dependence, but rather more pronounced was constantly reported on several mineral clays, such as lath-shaped nontronite [START_REF] Michot | Sol/gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 1. Features of the i/n transition[END_REF] or disk-shaped laponite, which is consistent with scattering expected from the scattering objects displaying bidimensional nature. Interestingly, whatever the ionic strength, at low Q, the intensity keeps on rising, which reveals the presence in the dispersion of a few primary aggregates that dominate the signal at low Q. SAXS patterns were also acquired on boehmite gels concentrated by osmotic stress (also presented in Figure 4.4). In such cases, the structure factor S(Q) cannot be neglected and can be experimentally estimated by dividing the intensity obtained for the concentrated dispersion I(Q)conc by the intensity measured for a dilute non-interacting dispersion I(Q)dil (volume fraction of 0.2 % in the present study), that is to say the single-particle function. The measured intensities were divided by the known concentration in both cases in order to exclude the effect of different particle concentrations as

f ) Q ( I ) Q ( I ) Q ( S conc dil dil conc   = . (4.3)
Therefore, Equation (4.3) contains an adjustable parameter to account for the effects linked to the non-constant thickness of the gels analyzed by SAXS. An example of derived S(Q) is shown in Figure 4.5 for boehmite gels prepared at an ionic strength of 8•10 -4 M. For this particular ionic strength, the sol/gel transition occurs when  = 2.69 vol.% .The positive values of S(Q) regardless of concentration and ionic strength imply a repulsive nature of interparticle interactions in the given boehmite dispersions. Furthermore, the curves display at low Q the presence of a correlation peak. The position of this peak shifts to high Q with volume fraction whereas at high Q, S(Q) approaches unity. The maximum observed is rather broad, which can be linked to the fact that the investigated samples are gels and consequently, rather disordered in terms of particles positions. Still, the The correlation distances decrease with increasing ionic strength. This is expected and reflects the screening effects of the double layer that results in a shorter distance between particles. At low volume fractions, all curves display a scaling exponent of -1/3, which is a typical of an isotropic state. A similar behavior was observed on several clay minerals, such as beidellite and montmorillonite [START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 1. Structure and electrostatic interactions[END_REF], or nontronite [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF]. At higher volume fractions, the curves for the I1 = 810 -5 M and I2 = 810 -4 M scale as  -0.96 and  -0.8 , respectively, suggesting a local lamellar ordering of the platelets. In such a case, the swelling law should be d* = t/, where t is the thickness of individual platelets. The calculation yields a value of 3 nm, which is coherent with the known thickness of a single boehmite sheet deduced from diffraction patterns [START_REF] Mange | A microstructural investigation of nanostructured boehmite films prepared by the sol-gel route[END_REF]. This then suggests that some boehmite layers are exfoliated in the gels. The transition between both swelling regimes that could indicate the position of a potential isotropic/nematic phase transition is located at volume fractions ≈ 8 % and 6 % for I1 = 810 -5 M and I2 = 810 -4 M, respectively, i.e far in the gel region according to the phase diagram of Figure 4.2. Consequently, the I/N transition, if present is totally hindered by the sol/gel transition in the boehmite suspensions used in the present study. Notably, several studies on clay mineral have demonstrated that the orderdisorder transition does not display a trivial dependence with increasing repulsion. For instance, occurrence in the gel phase of flow-induced birefrigence reminiscent of nematic texture has been reported on laponite [START_REF] Mourchid | On Viscoelastic, Birefringent, and Swelling Properties of Laponite Clay Suspensions : Revisited Phase Diagram[END_REF] at lower volume fractions when lowering the ionic strength (in the range of 10 -2 M-10 -4 M). Also, aqueous dispersions of gibbsite platelets have revealed that the I-N interface (not hindered by gelation, as in our case) decreases with ionic strength and increases with particle concentration [START_REF] Van Der Beek | Nematic ordering vs. gelation in suspensions of charged platelets[END_REF]. Another example is the phase diagram of beidellite displaying the similar sol/gel transition line as in our study. However, the narrow regions of a two-phase system and the nematic phase were additionally observed by authors [START_REF] Paineau | Liquid-crystalline nematic phase in aqueous suspensions of a disk-shaped natural beidellite clay[END_REF] showing dependence on the particle size only, whereas effect of ionic strength was found negligible. Similar trends have been therefore reported on the phase diagram of nontronite [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF].

Referring again to the concentration profiles from in situ filtration studies, as displayed in Figure 4.3, one may try to correlate the steep rise of concentration next to the membrane to the potential isotropic/nematic phase transition. In view of limited data points (Figure 4.6), the change in slope between two regimes likely appears to be gradual and is difficult to predict accurately. At the same time, exploring the range of higher volume fractions becomes highly impaired due to increasing gels stiffness. Clearly, additional experiments, namely static and dynamic birefringence, should be carried out for consolidating such an assumption.

IV.3 Rheological characterization

The use of oscillatory rheology provides detailed information about the material structure and further illustrates the substantial effect of bulk solution chemistry on interparticles forces. All the above obtained osmotically stressed samples of varied volume fractions and ionic strengths are examined via oscillatory strain controlled measurements and will be discussed in the following sections.

IV.3.1 Yielding behavior of boehmite gels

Yielding is a crucial rheological property providing information about gel strength. It is associated to the structural network and can be viewed as the critical force required to break down the structure so that the sample starts flowing. In the present work, oscillatory amplitude sweeps were carried out by applying increasing oscillatory strains  at a constant frequency of 1 Hz. Some typical curves are displayed in Figure 4.7. All curves exhibit G values significantly higher than G at low strain values, indicating that, in this region, dispersions have a solid-like behavior. G and G curves remain constant, as long as  remains below a certain limiting value lin. This range is referred to as the linear viscoelastic (LVE) range. Linearity is lost for  ≥ lin, Whatever the ionic strength, the critical strain decreases with increasing volume fraction, which demonstrates that the breakage of gel network is mainly governed by a critical deformation.

Interestingly, when the storage moduli G' start decreasing at the end of the LVE range, the loss moduli display a characteristic peak which position changes with volume fraction. This peak is significantly weakened and broadened for the highest ionic strength (I3 = 1•10 -2 M). A similar behavior was reported for kaolinite (Jogun and Zukoski [95,[START_REF] Jogun | Rheology and microstructure of dense suspensions of plateshaped colloidal particles[END_REF]; Bossard et al., [START_REF] Bossard | Linear and nonlinear viscoelastic behavior of very concentrated plate-like kaolin suspensions[END_REF]), laponite (Ramsay [START_REF] Ramsay | Colloidal properties of synthetic hectorite clay dispersions[END_REF]) and montmorillonite (Paineau et al. [START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 2. Rheological characterization[END_REF]) suspensions. Still, the physical origin of such nonlinear G" peak at intermediate shear strain amplitudes remains ill-understood. In their study, Bossard et al., [START_REF] Bossard | Linear and nonlinear viscoelastic behavior of very concentrated plate-like kaolin suspensions[END_REF] experimentally demonstrated that such an evolution of G" can be interpreted as resulting from the coupled increase in the excluded volume of platelets and electrostatic interactions for decreasing ionic strength. Evolution of moduli G and G" from the amplitude sweeps for boehmite dispersions at ionic strengths (from top to bottom) of 810 -5 M, 810 -4 M and 110 -2 M. For clarity, not all experimental curves are presented. The yield strain was determined via the maximum in G" (marked in red).

Various methods can be used for determining the yield strain. Some authors define it as corresponding to the point at which G' and G" are equal whereas other authors use the The values for low and intermediate ionic strength exhibit a power law dependence with an exponent that varies with ionic strength, the minimal exponent (-0.96) being observed for an ionic strength of 8•10 -4 M. The fact that exponents vary with ionic strength contrasts with what was reported for other more anisotropic systems such as nontronite [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF], montmorillonite and beidellite [START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 2. Rheological characterization[END_REF] where constant exponents were observed whatever the particle size and ionic strength. The differences obtained in the present study could suggest changes in the suspension structure with ionic strength that need to be analyzed in more depth. Still, in view of the uncertainty associated to the determination of both the volume fraction of the sol/gel transition and c the differences that we observe could also be in the error bars.

IV.3.2 Elasticity of boehmite gels

The presence of an electric double layer also contributes to the material's elasticity. In order to obtain additional information on the suspensions, dynamic moduli were also measured as a function of angular frequency, presented in Figure 4.9. As can be seen, all curves exhibit the dominance of G over the G over the chosen range of frequencies, which is typical for a gel-like state. Moreover, when samples are exposed to a faster oscillatory motion  ( is linked to the inverse characteristic relaxation time ), elastic moduli tend to rise progressively until a terminal high-frequency plateau is reached. Similarly to yield strain, the evolution of such high frequency plateau for different volume fractions and ionic strengths was examined via a power law scaling, as presented in Figure 4.10. All data can be grouped in a single master curve following a power law with a ( -sol/gel) 1.9 dependence. Notably, data points are more scattered at low volume fractions, which is related to the accuracy of the determination of the critical volume fraction corresponding to the sol/gel transition. The data obtained here are similar to those obtained on

, (rads -1 )

Chapter IV: Properties of osmotically densified boehmite dispersions: microstructure and particles organization 93 various clay minerals (laponite [START_REF] Mourchid | Phase Diagram of Colloidal Dispersions of Anisotropic Charged Particles : Equilibrium Properties, Structure, and Rheology of Laponite Suspensions[END_REF], nontronite [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF], montmorillonite and beidellite [START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 2. Rheological characterization[END_REF]) where an exponent of 2.35 was determined.

IV.3.3 Rheological determination of the interaction potential energy

As shown in different publications, the elastic modulus can be used to obtain information about electrostatic interactions in various systems characterized by an elastic response to small deformations, e.g. lattices [START_REF] Buscall | Viscoelastic properties of concentrated latices. Part 2.-Theoretical analysis[END_REF], aqueous clay dispersions [START_REF] Baravian | Rheological Determination of Interaction Potential Energy for Aqueous Clay Suspensions[END_REF], sterically stabilized [START_REF] Bergenholtz | Colloidal Charge Determination in Concentrated Liquid Dispersions Using Torsional Resonance Oscillation[END_REF] or weakly flocculated systems [START_REF] Goodwin | The elasticity of weakly flocculated suspensions[END_REF]. Buscall et al. [START_REF] Buscall | Viscoelastic properties of concentrated latices. Part 2.-Theoretical analysis[END_REF] were the first ones to propose a formula linking the high frequency modulus of a slightly deformed lattice with face-centered cubic symmetry (fcc) and the interaction potential that writes

         =  2 2 max dR d R 5 N ' G . (4.5)
where  is the interaction potential, R the center-to-center separation distance between particles, N the coordination number (assuming a f.c.c. packing configuration N = 12) and max the maximal packing density (max = 0.74).

In the case of a repulsive system, neglecting attractive interactions, and in the limit of weak surface potentials and small a<3 the repulsive potential (R) can be written as where a is the sphere radius, r the relative permittivity of the electrolyte solution, 0 the permittivity in the vacuum and  the diffuse layer potential defined from -potential measurements. After its differentiation twice and combining with Equation (4.5), one then obtains the resulting expression for elastic modulus

  ) a 2 R ( exp R a 4 ) R (
( )   a 2 R exp R 5 2 R 2 R a N 4 ' G 4 2 2 2 2 0 max -  -         +  +     =  . ( 4.7)
Increasing concentration results in shorter distances between particles and, for spherical objects, the relation between R, a and  can be written as

3 max a 2 R   = , (4.8)
Since elasticity is linked to a Brownian contribution that acts between particles like elastic springs to restore the microstructure, it is best investigated when the suspension is probed at high frequencies . Therefore, the G' value that must be considered in the above described analysis is the high frequency limit of G'.

In the case of boehmite suspensions, we applied such a procedure to low and intermediate ionic strength, as for the highest ionic strength (I3 = 1•10 -2 M), van der Waals attraction is most likely involved, which impairs the analysis.

Two additional points must be considered. (i) charged colloidal particles cannot be considered as hard spheres due to the presence of double layers and (ii) boehmite particles are not spheres. Indeed, the SAXS analysis presented above clearly shows the existence of anisotropic features for boehmite suspensions. Consequently, some additional calculations must be performed before being able to apply Equation (4.7) to boehmite dispersions.

Let us first introduce an effective volume fraction eff defined as

3 eff ) a 1 (   +  =  , (4.9)
that depends on the range of electrostatic repulsion through a and in which  is a scaling parameter accounting for the overlap of double layers. Whereas a is known,  must be determined. To do this, we start with the classical dependence of viscosity on volume fraction described for hard spheres by the Maron-Pierce-Quemada equation, as

2 * f 0 r 1 -           - =   =  , (4.10)
where r is the relative viscosity of the suspension defined as the ratio of its viscosity to the viscosity of the suspending medium and * is the maximum packing fraction. According to this equation, for increasing volume fraction, the viscosity of hard spheres dispersions should follow a power law with an exponent of -2 and finally diverge at maximum packing *. Figure 4.11 presents the evolution of the viscosity of boehmite suspensions obtained for various volume fractions at low and intermediate ionic strengths. The extrapolation of data at zero relative viscosity provides estimates of * of 9.6 vol.% and 13.6 vol.% for ionic strengths of 8•10 -5 M and 8•10 -4 M, respectively (see Figure 4.12). Further assuming that for particles such as the boehmite used in our study, the maximum volume fraction can be assimilated to *eff = 1 [START_REF] Baravian | Rheological Determination of Interaction Potential Energy for Aqueous Clay Suspensions[END_REF], it is then easy to determine the parameter  in Equation (4.9). Furthermore, for non-spherical particles, the evolution of the interparticle distance R as a function of volume fraction cannot be described by Equation (4.8) and it was shown recently [START_REF] Baravian | An effective geometrical approach to the structure of colloidal suspensions of very anisometric particles[END_REF] that the following expression applies well to repulsive anisometric particles

n * eff eff 3 / 1 * eff eff a 2 R           +           = . (4.11)
The term in the denominator in Equation (4.11) provides the harmonic average between the isotropic state at low volume fractions and the organized state in the limit of high concentrations. The parameter n can be directly obtained from SAXS data and in our case, values of 0.96 and 0.8 were chosen for ionic strengths of 8•10 -5 M and 8•10 -4 M, respectively. Figure 4.13 compares the experimental data obtained for G' with the predictions made on the basis of Equation (4.5). An increase in ionic strength leads to a decrease of G', a trend that was already observed for PBMA systems [START_REF] Bergenholtz | Colloidal Charge Determination in Concentrated Liquid Dispersions Using Torsional Resonance Oscillation[END_REF] and polystyrene lattices [START_REF] Benzing | The viscoelastic properties of ordered latices : Experiments[END_REF]. For calculating G', we used -potential data acquired for dilute suspensions at pH = 6 previously (see Figure 3.6, Chapter III) and as a consequence, in our fitting exercise, the only remaining adjustable parameter is the number of neighbors N. This appears reasonable as for systems such as boehmite dispersions, defining a priori the number of neighbors is far from being trivial. Using radii of 30 nm and 20 nm for I1 = 810 -5 M and I2 = 810 -4 M, respectively, a reasonable fit of G' data is obtained (Figure 4.13). Therefore, such values of a appear in close accordance with DLS measurements, acquired earlier in Chapter III (see Figure 3.2). It must be pointed out though that such values are average values whereas the actual particles likely display significant polydispersity [START_REF] Morin | Préparation d'alumine à porosité contrôlée : étude de l'interaction de la boehmite dans des solvants et des propriétés fonctionnelles des matériaux résultants[END_REF].

Remarkably, for the lowest ionic strength it can be seen that the zeta values are slightly over predicted in comparison with experimental  data that results in increasing discrepancies at higher volume fractions. This may be well a consequence of the constraints of the rheological model applicable in the limit of weak surface potentials ( < 50 mV).

Further tests of the values obtained by fitting G' values can be obtained by comparing the evolution with volume fractions of distance R obtained by SAXS with those derived from applying Equation (4.11) as displayed in Figure 4.14. The results agree fairly well, values calculated from Equation (4.11) being higher than experimental data by a few %. Considering the complexity of the system under investigation, such a result is truly convincing. The systematic slight overestimation of calculated values could be due to an overestimation of the effective volume fraction that could be related to the choice that was made to fix * eff  at 1. according to Equation (4.6). The results of such a calculation are displayed in Figure 4.15. For all the regimes investigated in the present study, the electrostatic potential is higher than the energy kBT of thermal fluctuations by at least a factor of 20. The potential decays rapidly with increasing surface separation and increasing ionic strength decreases the electrostatic, which concurs with the electrical double layer concept. As this model was successfully applied to montmorillonite disk-like particles [START_REF] Baravian | Rheological Determination of Interaction Potential Energy for Aqueous Clay Suspensions[END_REF], the results presented in our study strongly suggest that the methodology used here could really be fruitful for analyzing correlations between electrostatic effects and viscoelastic properties in colloidal suspensions of anisometric particles.

IV.4 Summary

In present chapter we investigated in depth the gelation mechanisms, viscoelastic properties and structure of boehmite dispersions obtained by the use of osmotic stress experiments that enables to reconcile the filtration studies of Chapter III. In terms of phase diagram distinct ranges of interactions modulated by salt addition have been revealed: the sol/gel transition line displays a positive slope at low ionic strength with a negative reverse at higher ionic strength. The range of such interactions has been compared with the corresponding Debye length,  -1 , of the supporting electrolyte suggesting an anisotropy effects of boehmite platelets. Moreover, the observed phase transition most likely allows to identify the folded zones on the concentration profiles deduced from in situ X-ray radiography tests previously from Chapter III. Structural analysis by small angle X-ray scattering has therefore evidenced local layered ordering far above the sol/gel transition, likewise on different clay minerals ( [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF], [START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 1. Structure and electrostatic interactions[END_REF]). This suggests that the I/N transition is likely to be hindered by gelation, a feature that was also observed on numerous charged anisotropic systems such as montmorillonite [START_REF] Michot | Phase diagrams of Wyoming Na-montmorillonite clay. Influence of particle anisotropy[END_REF] or laponite [START_REF] Mourchid | Phase Diagram of Colloidal Dispersions of Anisotropic Charged Particles : Equilibrium Properties, Structure, and Rheology of Laponite Suspensions[END_REF].

Therefore, the gelation mechanism has been established to contribute to both the elasticity and yielding arising from the coupled increase in interactions between the diffuse/ionic double layers of boehmite platelets and the excluded volume effects, leading to a weakening of gels strength and rigidity.

Lastly, by coupling rheological measurements with structural information obtained by SAXS measurements, we were able to extend the approach initially developed by Buscall et al. [START_REF] Buscall | Viscoelastic properties of concentrated latices. Part 2.-Theoretical analysis[END_REF] to the case of boehmite platelets despite displaying some polydispersity. An effective volume scaling accounting for excluded volume effects due to induced rotational motion of particles allows capturing interactions over a wide range of particle concentrations and in the limit of moderate ionic strengths before aggregation processes comes into effect. We could thus rationalize rheological data on the basis of the electrostatic interactions in the system, which provides a coherent picture of viscoelastic properties with regard to increasing volume fraction and ionic strength.

CHAPTER V: PARTICLE MOTION IN A YIELD STRESS ENVIRONMENT

The focus of this chapter is an experimental study on the monitoring by synchrotronbased X-ray tomography of flow-induced particles motions upon continuous deposit growth during frontal filtration. To reach this goal, non-Brownian boehmite aggregates are introduced into colloidal boehmite sols to enable particles visualization. One of the aims of these experiments is to evidence a correlation between particles displacement and the rheological properties of the surrounding colloidal media.

V.1 Bimodal mixture: in situ dead-end filtration by using synchrotron radiation

The following filtration tests are focused on boehmite sols to which of non-Brownian aggregates were added to yield the so-called "bimodal mixture" referred to as BM in the following. The displacement of these aggregates was monitored to try establishing some correlation between particles motion and local rheological properties inside the growing deposit. These experiments were performed at the ANATOMIX Tomography beamline at synchrotron SOLEIL (Gif-sur-Yvette, France), a beamline dedicated to full-field radiography and tomography in the hard X-ray range. Thanks to a reduced cell size design, as depicted previously in Figure 2.17, a high spatial resolution of 0.65 µm could be reached. It is important to point out that the presence of non-Brownian particles marginally affects the permeability K when compared with purely colloidal boehmite (Annex 1). This allows reconciling the following analysis with respect to all the previously obtained results on colloidal boehmite sols.

In situ filtration runs on BM were carried out at three ionic strengths I1 = 8•10 -5 M, I2 = 8•10 -4 M and I3 = 1•10 -2 M. The procedure of suspensions preparation remains the same that used for purely colloidal boehmite sols. Filtration runs were acquired using a dispersion volume of 1 mL providing an optimal filtration time of 1.5-2 hours and resulting in 1 mm of deposit thickness. At first, data treatment was processed in the X-ray imaging mode. Figure 5.4 displays the resulting concentration profiles obtained for three ionic strengths as a function of the distance z from the membrane surface. As in the case of colloids presented previously (see Figure 3.17, Chapter III), we can also observe a gradient in deposit formation characterized by folded zones propagating upon filtration time. It should also be mentioned that the phase diagram of bimodal mixture strongly resembles the one observed for colloids (Annex 2), apart from the fact that the sol/gel line for BM is slightly shifted towards lower volume fractions, likely due to the broaden size population with regard to micron aggregates.

The tomographic reconstruction obtained from the recorded projection images allows distinguishing the particles aggregates whose motion within the growing deposit can be tracked. Such a strategy is somehow similar to that used in the trajectory analysis of labeled dyes used in fluorescent microscopy. In the acquired image sequences (Figure 5.5), aggregates appear as bright spots on a dark background representing the colloidal medium. One acquisition made of 2 000 projections with an acquisition time of 200 ms accounts for  7 min, with a pixel size of 0.65 m. Particles that move faster than the given acquisition time of experiment are detrimental to the precision of the final reconstruction and are viewed as blurry spots on the images. With continuous deposit growth characterized by an increased viscoelastic behavior (when a gel is formed), particles speed diminishes, resulting in wellresolved particles positioning. Thus, by extraction of the spots coordinates for a given time sequence where particles are detectable, their trajectories can be obtained to yield information about aggregates displacement velocity within the growing and densifying filtration deposit. The subsequent image analysis was conducted using the plug-in Trackmate available in ImageJ software for 2D particles tracking. Before this analysis, the images were re-sliced in a vertical plane. Particles used for tracking were selected by setting an intensity threshold. Their coordinates were subsequently used for calculating the corresponding trajectories by

%)

Gray level normalized assuming a linear vertical motion of particles. This implies that chaotic particles displacement is fully excluded, which is likely relevant to the case of microaggregates, since they do not undergo Brownian motion. In that case, if the particle remains in the same plane (tomographic slice) of the observation, the track which is built between two consecutive frames can be truly considered as belonging to the same particle (Figure 5.5). This needs to be further confirmed by performing a three-dimensional particles tracking (work yet to be done) to ensure that the pattern is relevant throughout the entire volume of the filtration cell. From this analysis of trajectories (Figure 5.5), the average velocity was determined and plotted as a function of distance z from the membrane surface, as illustrated in Figure 5.6. One can distinguish the velocity profile characterized by a plateau of nearly zero velocity close to the membrane surface that extends progressively over larger z distances upon increasing filtration time. If one takes particle velocity, for instance, at a fixed distance z of 150 m from the filter and compares with the associated concentration profile, it is obvious that the particles velocity continuously diminishes, as expected, in response to increasing The particles motion in a colloidal environment exhibiting visco-elasticity, non-Newtonian behavior (i.e. shear-thinning, shear-thicknening), yield stress or time dependent rheology… is a complex subject. To deepen knowledge into this phenomenon, a starting point should be made first by focusing on a simplified case: the slow (at low Re number, where inertial effects are negligible) particles settling in a Newtonian fluid under the gravitational force at the Stokes velocity , (5.1) where d is the particle diameter,  = p -f the difference between the density of the particle and that of the fluid,  the fluid apparent viscosity and g the acceleration due to gravity. For Newtonian liquids the apparent viscosity  remains unchanged. For a boehmite aggregate with a diameter of 4 µm diameter settling in the feed suspension (  10 -3 Pa‧s), the Stokes velocity is 20 µm/s which 3 orders of magnitude higher than the measured velocity. Such velocity values were not measurable with tomography settings. In systems displaying shear-thinning properties, by definition this parameter becomes sensitive to the shear rates , thereby the particle velocity in Equation (5.1) results in a more complex function of dependent parameters. For yield stress fluids the apparent viscosity goes to infinity as the shear rate tends to zero, and the terminal velocity may approach zero, until the critical drag force is overcome to initiate motion of particles. In this context, coexisting static-flowing regions along the filtration deposit might stem from the existence of a yield stress. When the stress is smaller than a certain threshold of critical stress within the deposit, the material is in the regime of a strong gel. By contrast, in the situation when the yield stress is overcome, the material experiences a plastic flow, therefore implying the fact that the shear rate continuously varies within the sample. Indeed, by monitoring the slope of the velocity profile as a function of filtration time, one might deduce the equivalence of vertical velocity gradient 2 within specified region of the deposit 2 Note, however, that the true shear rates are related to the slope of the tangential velocity profile via the definition of shear rate as , where x is velocity in the direction of the flow across the channel diameter y, as depicted in (marked in yellow in Figure 5.8). The corresponding results for each of the tested ionic strengths are displayed in Figure 5.9. As seen, the "shear rate" progressively decreases in response to increasing volume fraction in a specified region of the deposit. Remarkably, this provides the flow properties of boehmite dispersions in the range of sufficiently low deformation rates around 10 -5 s -1 , which establishes the correlation with the low shear viscosity curves, i.e. the limit of a viscosity plateau and the onset of shear-thinning flow behavior (the shear viscosity is a decreasing function of shear rate, as seen in Figure 3.18, Chapter IV).

Based on estimated experimental values of particles velocity deduced above, it is therefore interesting to deduce the viscosity range and compare it with experimental data. The combination of forces exerted on a particle of diameter d can be expressed by equating the gravitational Fg, buoyant Fb, driving Fdriving and drag due to gel deposit Fgel forces as where liq is the liquid viscosity, liq the liquid velocity, p -f the difference in density between the particle and the medium, g the acceleration due to gravity and p the particle velocity (He et al. [START_REF] He | Particle movement in non-Newtonian slurries: the effect of yield stress on dense medium separation[END_REF]).

              
It is expected that wall effects contribute to the drag force when the sheared region around the moving particle extends to the walls, i.e. in the case of confined particles flow.

Experimental studies [START_REF] Atapattu | Wall effect for spheres falling at small reynolds number in a viscoplastic medium[END_REF] on viscoplastic materials suggest that the extent of the sheared region constitutes four times the particle radius and tends to decrease once the diameter of channel exceeds the size of such sheared cavity around the particle. This indicates that in our studies, where particle-to-filtration cell diameter ratio is small enough, wall effects should be relatively unimportant.

If one consider the case of an ideally Bingham (plastic) fluid, the apparent viscosity app due is expressed as There are several possible reasons of such discrepancy. (i) Firstly, the fact that the velocity profiles are deduced locally by estimating the displacement of each single particle, whereas rheological properties are averaged over the whole sample volume. (ii) Secondly, the observed discrepancy could be the result of the deposit compression during filtration. (iii) Calculations of driving force accounts for water viscosity, whereas the dispersion viscosity becomes dependent on volume fraction. (iii) And lastly, it is most likely that other rheological parameters than viscosity and yield stress are involved that does not allow to ascribe correctly the particles motion. It is interesting to note that the correlation between particle motion and properties of yield stress media is not a new subject and it has been studied both experimentally and numerically by a number of research groups. The most extensive studies have been carried out on polymer-based solutions, e.g. Carbopol gels by Atapattu et al. (1990 [108] and 1995 [START_REF] Atapattu | Creeping sphere motion in Herschel-Bulkley fluids : Flow field and drag[END_REF]), Jossic and Magnin [START_REF] Jossic | Drag and stability of objects in a yield stress fluid[END_REF], Tabuteau et al. [START_REF] Tabuteau | Drag force on a sphere in steady motion through a yield-stress fluid[END_REF], whereas only a few on colloidal suspensions, i.e. on laponite (Ferroir et al. [START_REF] Ferroir | Motion of a solid object through a pasty (thixotropic) fluid[END_REF]) and bentonite clay suspensions (Chafe and Bruyn [START_REF] Chafe | Drag and relaxation in a bentonite clay suspension[END_REF]). However, in both cases the flow around a settling sphere has been evidenced to show a breakage in a fore-aft symmetry, which is found to be typical for non-Newtonian fluids. For instance, Putz et al. [START_REF] Putz | Settling of an isolated spherical particle in a yield stress shear thinning fluid[END_REF] attempted to visualize the flow patterns around a non-Brownian sphere moving in a yield stress fluid using Particle Image Velocimetry (PIV). Their findings highlight a strong asymmetry between the upward and downward flow directions (Figure 5.10), which has a striking consistency with the oscillatory flow in rheometer with increasing and decreasing stress curves that corresponds to the downstream and upstream velocity fields, respectively. Moreover, the yield surface calculated from velocity pattern displayed likewise an asymmetric shape propagated strongly along its major axis in the upstream direction. In the downstream region the fluid relaxes and the applied stresses decay rapidly, which is consistent with the decreasing stress part of the flow curve from oscillatory shear stress measurements. Such asymmetry from Figure 5.10 indirectly suggests the coexistence of zones with sheared fluid and unsheared solid state distributed unequally upon the particle movement in the non-Newtonian medium. The shear stresses occurring upon the particle motion are first translated to a fluid up to the limits of the fluid cavity, where it has the magnitude of the yield stress, and then through a solid-like state in which it decays rapidly.

The motion in the direction opposite to the sphere was named "negative wake", and its physical origin still remains a subject of ongoing discussions. Its existence has been repeatedly reported in Carbopol solutions (Putz et al. [START_REF] Putz | Settling of an isolated spherical particle in a yield stress shear thinning fluid[END_REF]), as well as in colloidal laponite dispersions (Gueslin et al. [START_REF] Gueslin | Flow induced by a sphere settling in an aging yield-stress fluid[END_REF]). It was largely attributed either to a competition between viscoelastic and extensional effects in the wake of the sphere or the balance of extensional and normal stresses difference. Depending on the values of both effects, different situations can be encountered, namely an "extended wake" (the fluid velocity continuously decreases away from the sphere) or a "negative wake, when the elastic recoil prevails producing an upward flow. Its worth noting that most numerical and experimental studies are focused on the yield stress property described by pure Bingham or Herschel-Bulkley model fluids, suggesting that the flow is confined within an envelope around the sphere, where a yield surface separates the fluidized (flow, yielded) and solid-like (no flow, unyielded) regions. However, the appearance of fore-aft symmetry and negative wake, which is inconsistent with the numerical studies based on simplistic models (Beris et al. [START_REF] Beris | Creeping motion of a sphere through a Bingham plastic[END_REF]) suggests that other rheological properties than yield stress might be also involved and should be taken onto account to reach a better understanding of the complex flow behavior in non-Newtonian environments.

V.2 Summary

Particles settling upon pressure-induced dead-end filtration has been studied on ANATOMIX Tomography beamline at synchrotron SOLEIL. Our results demonstrated interesting prospects in mapping the trajectories of non-Brownian particles displacing in a colloidal media. We also showed that the analysis of particles displacement enables to describe the flow properties of boehmite dispersions in the range of sufficiently low deformation rates, which provides information about local rheological changes. In the future prospects it would certainly be interesting to expand this analysis by monitoring velocity fields in conjunction with additional rheological measurements in order to deepen the understanding of particles flow in a yield-stress medium. More specifically, selection of larger settling particles will allow to measure the instantaneous velocity (and not the averaged one as in the present study). Therefore, additional studies on the displacement of non-Brownian particles in a gel with constant rheological properties (without any concentration gradient) could be beneficial.

GENERAL CONCLUSIONS AND PERSPECTIVES

The present study focused on the role of interparticle interactions in the membrane separation process of boehmite dispersions by means of pressure-driven ultrafiltration. The results obtained can be summarized as follows:

Firstly, the respective role of physico-chemical conditions in filtration mechanisms has been investigated at different length scales.

At a macro-scale, the influence of ionic strength on the filtration rate of boehmite dispersions has been established, suggesting that its adjustment might be one possible way for optimizing filtration processes. More specifically, it appears that filtration at intermediate ionic strengths leads to the lowest filtration rates associated with a more resistant deposit with minimal permeability. Two related mechanisms can be proposed to explain such an effect: (i) a role of surface charge that has repeatedly shown to influence particles transport mechanisms [START_REF] Mcdonogh | Charge effects in the cross-flow filtration of colloids and particulates[END_REF], [START_REF] Wiesner | Membrane Filtration of Coagulated Suspensions[END_REF], [START_REF] Waite | Colloidal Fouling of Ultrafiltration Membranes: Impact of Aggregate Structure and Size[END_REF], [START_REF] Hlavacek | Simple Relationships among Zeta Potential, Particle Size Distribution, and Cake Specific Resistance for Colloid Suspensions Coagulated with Ferric Chloride[END_REF], [START_REF] Ohmori | Effects of pH and ionic strength on microfiltration of C. glutamicum[END_REF], and (ii) an influence of particle anisotropy that has also been shown to affect filtration performance [START_REF] Perdigon-Aller | Preferred orientation in filtercakes of kaolinite[END_REF], [START_REF] Mccarthy | Effect of cell morphology on dead-end filtration of the dimorphic yeast Kluyveromyces marxianus var. marxianus NRRLy2415[END_REF], [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]. In the former case, the pHdependent behavior of boehmite particles at fixed ionic strength evidenced a sixfold increase of permeability arising from promoted flocculation in proximity of the iep region (determined at pHiep = 9.1 for boehmite in the present study). Thus, one may expect denser particles packing and, hence, reduced permeability K when lowering the ionic strength. Still, the evolution of zeta potential with salt addition, does not seem to provide a viable explanation for the nonlinear dependence of K on ionic strength.

On the other hand, considering the particle size of boehmite, changes in actual anisotropy could be linked to variations of the Debye length with salt concentration. Still, a weak alignment at intraparticle levels has been distinguished by SAXS analysis irrespectively of ionic strength, which contrasts with other works on platelike systems [START_REF] Tanaka | Filtration characteristics and structure of cake in crossflow filtration of bacterial suspension[END_REF][START_REF] Perdigon-Aller | Preferred orientation in filtercakes of kaolinite[END_REF][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]. This calls for more extended studies at larger-scale, for instance using optical techniques. In addition, it would certainly be fruitful in future work to combine scattering experiments with rheological ones, to analyze in greater depth the orientation of particles under flow.

In parallel, the use of X-Ray tomography has illustrated its promising prospects for the direct observation of deposit growth kinetics. The concentration profiles deduced from real-time in situ X-ray radiography filtration tests revealed nontrivial concentration gradients that are significantly different depending on the used ionic strength. This confirms the crucial role of particle-particle interactions in filtration processes, as such interactions define the structuration mechanisms of deposit formation.

For this reason, in the present study, we further focused on an in depth analysis of the phase diagram, structure and rheology of boehmite slurries prepared by osmotic stress, i.e. in true equilibrium conditions.

The detailed study of the phase diagram, rheology and structure of industrial boehmite slurries provided a consistent picture of the impact of interparticle interactions on gelation mechanisms, viscoelastic effects and structural features. In terms of phase diagram distinct ranges of interactions modulated by salt addition were revealed. Notably, the sol/gel transition line exhibits a change of slope from positive to negative when increasing ionic strength, a behavior similar to that observed for other plate-like charged colloids (i.e. laponite [START_REF] Levitz | Liquid-solid transition of Laponite suspensions at very low ionic strength : Long-range electrostatic stabilisation of anisotropic colloids[END_REF] or nontronite [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF]). This slope inversion occurs at an ionic strength value (I = 8•10 -4 M), for which the corresponding Debye length becomes comparable to the platelet size. Moreover, this explains the shape of the concentration profiles deduced from in situ filtration studies that provide direct evidence for the occurrence of a sol/gel transition in growing deposits during frontal membrane separation process.

Additional structural analysis by SAXS further evidenced local nematic ordering far above the sol/gel transition, as previously observed for various clay minerals ( [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF], [START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 1. Structure and electrostatic interactions[END_REF]). Still no isotropic/nematic transition was observed for boehmite, in contrast with what was found for aqueous dispersions of gibbsite [START_REF] Mourad | Sol-gel transitions and liquid crystal phase transitions in concentrated aqueous suspensions of colloidal gibbsite platelets[END_REF] or nontronite [START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF]. The sol/gel transition appears to hinder the I/N transition, a feature that was also observed on numerous anisotropic systems such as montmorillonite [START_REF] Michot | Phase diagrams of Wyoming Na-montmorillonite clay. Influence of particle anisotropy[END_REF] or laponite [START_REF] Mourchid | Phase Diagram of Colloidal Dispersions of Anisotropic Charged Particles : Equilibrium Properties, Structure, and Rheology of Laponite Suspensions[END_REF]. The features revealed in the present work may help in rationalizing these different situations on the basis of particle anisotropy and ionic strength. Lastly, by coupling rheological measurements with structural information obtained by SAXS measurements, we were able to extend the approach initially developed by Buscall et al. [START_REF] Buscall | Viscoelastic properties of concentrated latices. Part 2.-Theoretical analysis[END_REF] to the case of boehmite platelets displaying moderate polydispersity. An effective volume scaling accounting for excluded volume effects due to induced orientational and rotational motion of particles allows capturing interactions over a wide range of particle concentrations and in the limit of moderate ionic strengths before aggregation processes comes into effect. We could thus rationalize rheological data on the basis of electrostatic interactions in the system, which provides a coherent picture of viscoelastic properties with regard to increasing volume fraction and ionic strength.

The use of the model developed here should be further examined on experimental data of other anisotropic systems to reinforce its validity. This could also be completed by taking into account electroviscous effects that were neglected in the present approach [START_REF] Quemada | Energy of interaction in colloids and its implications in rheological modeling[END_REF]. Additional static and dynamic birefringence measurements could also provide clearer information about the local nematic ordering of platelets and future work could certainly focus on coupling rheological experiments with scattering measurements. This would strengthen the analysis of flow-induced orientation which might lead to changes in processing condition upon filtration of such "complex" colloidal slurries.

Lastly, filtration studies on ANATOMIX Tomography beamline at synchrotron SOLEIL also holds out interesting prospects in mapping of non-Brownian particles trajectories displacing in a colloidal media. Such an analysis reveals the flow properties of boehmite dispersions in the range of sufficiently low deformation rates and therefore establishes some correlation with rheological changes in yield stress fluids. As a next step, it would be worthwhile to carry out Particle Image Velocimetry tests with additional rheological characterization to extend/develop studies regarding the complex particle flow behavior in non-Newtonian environments.

ANNEX 1

Comparative permeability of filtration deposits for colloidal and bimodal mixture of boehmite dispersions containing non-Brownian particles

Figure A1 compares the permeability of filtration deposits resulting from boehmite dispersions of different size distributions. The curves in gray corresponding to colloids are plotted here for illustrative purposes. Globally, the shape of the curves remains similar, even if permeability is slightly shifted towards higher values in the presence of large particles. Since the contribution of colloids prevails on that of coarse particles (11 vol.% in the tested dispersions), the effect of ionic strength, a feature crucial for colloidal particles, also remains valid for bimodal mixture.

Figure 1 .

 1 Figure 1. Schematic representation of an industrial production of supported alumina catalysts. Adapted from [L. Rouleau, IFPEN, unpublished].

Figure 1 . 2 .

 12 Figure 1.2. Potential distribution of ions around charged surface.The layers are considered to behave as charged capacitors C1 and C2, taken to be of constant value between the surface and the inner Helmholtz plane and between the inner and outer Helmholtz planes, respectively. Adapted from[START_REF] Wood | Electrochemistry of the boehmite-water interface[END_REF].

. 2 )

 2 and (ii) the Poisson equation relating the variation of electrostatic potential  to the dielectric properties of the suspending medium and charge density q,

Figure 1 . 3 .

 13 Figure 1.3. Estimated values for the Hamaker constant Aeff (Parsegian, 1975) for interaction between polystyrene plates in pure and salt water (0.1 M). From[START_REF] Russel | Colloidal dispersions[END_REF].

Figure 1 . 4 .

 14 Figure 1.4. Interaction energy as a function of distance between similarly charged surfaces immersed in a 1:1 electrolyte solution.From[START_REF] Israelachvili | Intermolecular and surface forces[END_REF].

Figure 1 . 5 .

 15 Figure 1.5. Unit cell-diagram of the boehmite crystal structure representing a lamellar structure: double sheets of octahedral oxygen layers with Al atom at the center. Dashed lines indicate possible hydrogen bond interactions between nearest donor oxygen atoms. From [11].

Figure 1 . 6 .

 16 Figure 1.6. Electron micrographs of boehmite nanoparticles synthesized at (a) pH = 4.5, (b) pH = 6.5, (c) pH = 12 and schemes of particles as deduced from modelling and simulation of the X-ray diffraction patterns. From [15].

Figure 1 . 7 .

 17 Figure 1.7. Amount of aluminum released in solution as a function of initial HNO3 concentration. Adapted from [18].Boehmite surface charging involves as well simultaneous dissolution of Al species via

Figure 1 . 8 .

 18 Figure 1.8. Example of aluminium speciation in aqueous solutions as a function of pH for 10 -4 M total Al 3+ . Adapted from [21].

Figure 1 .

 1 Figure 1.10. Shear-thinning behavior of concentrated boehmite dispersions. Adapted from [33].

Figure 1 .

 1 Figure 1.11. SLS scattering patterns and shear stress response of Laponite dispersion as a function of shear rate: volume fraction of 0.56 wt.%, pH = 9.5, [NaCl] = 10 -3 M, gelation time = 150 days. Open circles: cone-plate configuration (roughness  200 m); filled squares: plate-plate configuration (roughness  0.05 m).Adapted from[START_REF] Pignon | Thixotropic behavior of clay dispersions : Combinations of scattering and rheometric techniques[END_REF].

Figure 1 .

 1 Figure 1.13. Rheological phase diagram of boehmite dispersion: region (1) sol without birefringence by flow; region (2) sol showing birefringence by flow; region (3) gel showing birefringence by flow; region (4) gel showing permanent birefrigence. From [37].

Figure 1 .

 1 Figure 1.14. Rheological phase diagrams of size-fractionated Wyoming montmorillonite with average size (A): 490130 nm, (B): 295100 nm and (C): 7550 nm, showing the delayed sol/gel transition with particle size increase. Adapted from [39].

Figure 1 .

 1 Figure 1.15.Phase diagrams of nontronite aqueous dispersions with size fraction (average length) of (A): 147 nm (anisotropy ratio 3.1) and (B): 120 nm (anisotropy ratio 2.9). From[START_REF] Michot | Phase diagrams of Wyoming Na-montmorillonite clay. Influence of particle anisotropy[END_REF][START_REF] Michot | Sol/gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 1. Features of the i/n transition[END_REF].

Figure 1 .

 1 Figure 1.16. Dynamic shear moduli, G (open symbols), G (solid symbols) of boehmite dispersions ([HNO3]/[AlOOH] = 2•10 -2 ) as a function of strain amplitudes 0. From [20].

Figure 1 . 17 .

 117 Figure 1.17. Yield stress of alumina-boehmite paste as a function of boehmite content. Adapted from [43].

Figure 1 . 18 .

 118 Figure 1.18. Classification of pressure driven membrane processes according to membrane pore and characteristic object sizes.

Figure 1 . 19 .

 119 Figure 1.19. Schematic representation of dead-end and cross-flow filtration.

Figure 1 . 20 .

 120 Figure 1.20. Types of resistance towards mass transport across a porous membrane in pressure driven processes. Adapted from [44].

  procedure for determining the specific resistance rc, and, thus, the permeability K (Figure1.21).

Figure 1 .

 1 Figure 1.21. A plot of the experimental data points of t/V as a function of V for calculation of the slope of Equation (1.23). From [45].

Figure 1 .

 1 Figure 1.22. Cross-flow fluxes of monodisperse silica colloids as a function of zeta potential  with Reynolds numbers Re of 78, 140 and 200. P = 100 kPa. Adapted from [47].

Figure 1 . 23 .

 123 Figure 1.23. Relative cake resistance as a function of zeta potential for solutions of humic acid coagulated with 10 -3.6 M (in triangles) and 10 -4 M (in circles). Adapted from [48].

Figure 1 . 24 .

 124 Figure 1.24. Specific cake resistance rc of bentonite dispersions vs. salt concentration. Adapted from [51].

Figure 1 . 25 .

 125 Figure 1.25. The profile of zeta potential  as a function of pH. The  was measured at either constant conductivity (in open triangles) or constant ionic strength (in open squares). (open circles) specific cake resistance rc as a function of pH. Ionic strength was adjusted to 0.1 molkg -1 by adding NaCl for each slurry. The rc as a function of pH without maintenance of constant ionic strength is shown here. Adapted from [52].

Figure 1 .

 1 Figure 1.26. SEM cross sections of cake after 15 min of filtration formed in (a) dead-end (transmembrane pressure, 49 kPa) and (b) cross-flow (circulation flow rate, 30 cm 3 •s -1 ; transmembrane pressure, 49 kPa; channel depth, 2.5 ram; cell concentration, 5 kg•m 3 ) filtrations of Bacillus subtilis. From [53].

Figure 2 . 1 .

 21 Figure 2.1. Scheme of boehmite peptization at high and low acid content.

Figure 2 . 2 .

 22 Figure 2.2. Schematic illustration of the dialysis bag technique.

Figure 2 . 3 .

 23 Figure 2.3. Schematic illustration of the osmotic stress technique.

  equation is Newton's law representing ideal viscous flow behavior. It expresses the intrinsic relation between the applied stresses and the velocity gradient as dy the proportionality constant, *, the viscosity that represents the internal friction, i.e. the resistance to flow.

Figure 2 . 4 .

 24 Figure 2.4. Shear flow between two parallel plates separated by a distance h. The upper plate moves with a velocity  in the x direction.

Figure 2 . 5 .

 25 Figure 2.5. Typical stress amplitude sweep on an aqueous alumina slurry with 55 vol.% solids showing the variation of G', G" and  at a frequency of 1 Hz. From [69].

Figure 2 . 6 .

 26 Figure 2.6. View on the rheometric acquisition in a parallel plates configuration.

Figure 2 . 7 .

 27 Figure 2.7. A schematic representation of a SAXS experiment showing the scattering of a monochromatic X-ray beam by a twoelectron system. The scattered waves produced from assembly electrons within the sample volume result in the interference pattern at the detector.

. 18 )Figure 2 . 8 .

 1828 Figure 2.8. Illustration of the form factor P(Q) dependence on particle dimensionality plotted as a function of the wavevector transfer Q and the radius of gyration Rg. From [71].

Figure 2 . 9 .

 29 Figure 2.9. Illustration of the form factor P(Q), the structure factor S(Q) and their product S(Q)P(Q) for core-shell micelle showing the effect at low Q of the structure factor on the measured intensity. The lowering of the intensity traduces repulsive interactions and the peak position is related to the mean distance between micelles. From [72].

Figure 2 .

 2 Figure 2.10. (a) Schematic diagram of filtration apparatus and data acquisition for ex situ tests, (b) photo of filtration station.

Ray tomography II. 5 . 1

 51 Principle of X-Ray computed tomography X-ray computed tomography is an imaging technique that has been widely used to non-destructively explore internal structure of objects (e.g. cracks, defects, voids,…) or enable direct observation of time-evolving processes (for instance, charging-discharging cycles of Li-batteries (Taiwo et al., 2017) or intergranular corrosion propagation of alloys (Zhao et al., 2003)) with sub-micron resolution.

Figure 2 . 11 .

 211 Figure 2.11. Principle of cone-beam X-ray computed tomography (top view). Geometrical magnification is determined by the sourceobject distance (SOD) and source-detector distance (SDD). f is the focal spot size, ps is the detector pixel. Adapted from [73].

Figure 2 . 13 .

 213 Figure 2.13. Schematic diagram of filtration cell (left) and its corresponding radiography image (right) showing the presence of artifacts in the zone of deposit accumulation due to the various types of component pieces of the cell.

Figure 2 . 14 .

 214 Figure 2.14. Schematic diagram of the filtration unit developed for the investigation of filtration deposit by X-ray radiography at IFPEN (J. Triadou).

Figure 2 . 15 .

 215 Figure 2.15. Configuration of X-ray tomograph inside the acquisition chamber (IFPEN).

Figure 2 . 16 .

 216 Figure 2.16. Schematic of ANATOMIX beamline. The source is a cryo-cooled undulator (on the left). Starting from undulator to the right, the other elements are (a) diaphragm, (b) vertical slit, (c) primary slits, attenuators (are not shown), then two focusing mirrors, transfocator (beryllium lenses), (b) vertical slit, (d) secondary slits, attenuators (are not shown), double-crystal monochromator Si (111) (DCM), (e) guard slits, followed by experimental hutch. The distances are given relatively to the undulator center.

Figure 2 .

 2 Figure 2.17. Left: Experimental set-up of filtration on microtomography station at ANATOMIX; right: schematic diagram of the corresponding filtration unit developed at IFPEN for synchrotron measurements.

Figure 3 . 1 .

 31 Figure 3.1. Top: TEM images of peptized boehmite dispersion. Initial conditions 143 g•L -1 in an aqueous solution containing 0.186 mol•L -1 of nitric acid. Dilution factor 20 000 on a carbon grid. Bottom: Particle size (length) distribution histogram.

Figure 3 . 2 .

 32 Figure 3.2. (Top) Hydrodynamic radii obtained from SBL fit of DLS correlogram (not shown) and (bottom) residue for colloidal boehmite sols peptized with a high acid content.

Figure 3 . 3 .

 33 Figure 3.3. Example of a V vs. t plot for boehmite dispersion at an ionic strength of I = 1.810 -4 M. Filtration conditions: V = 10 mL; P = 1 -3 bar; ultrafiltration membrane PES, 500 kDa, surface area 4.91 cm 2 .

Figure 3 . 4 .

 34 Figure 3.4. Example of a t/V vs. V plot for boehmite dispersion at an ionic strength of I=1.810 -4 M. Filtration conditions: V = 10 mL; P = 1 -3 bar; ultrafiltration membrane PES, 500 kDa, surface area 4.91 cm 2 .

Figure 3 . 5 .

 35 Figure 3.5. Cake permeability K versus ionic strength I for colloidal boehmite dispersions. Filtration conditions: V = 10 mL; P = 1 -3 bar; solids density 3035 kgm -3 ; liquid viscosity 0.89 mPas; ultrafiltration membrane PES, 500 kDa, surface area 4.91 cm 2 .

  the pH given in the exponent, which determines its (ionic strength) order of magnitude (in M).

Figure 3 . 6 .

 36 Figure 3.6. Evolution of pH and zeta potential for boehmite dispersions with varied ionic strength.

Figure 3 . 7 .

 37 Figure 3.7. Evolution of permeability K (in brown circles) as a function of dispersions pH at fixed ionic strength of 810 -5 M. The corresponding zeta potential (in blue squares) is re-plotted for clarity.

Figure 3 . 8 .

 38 Figure 3.8. Schematic diagram of samples preparation for SAXS acquisition. Red arrow displays the scanning direction within the filtration deposit.

Figure 3 . 9 .

 39 Figure 3.9. Intensity evolution of the first Bragg peak as a function of the azimuthal angle along capillaries filled with filtration deposits at varied ionic strength. The solid line is the fit used to determine the orientation parameter. Filtration conditions: V = 15 mL; P = 1 bar; ultrafiltration membrane PES, 500 kDa, surface area 3.14 cm 2 .

2 ; 71 I 7 ;Figure 3 . 10 .

 2717310 Figure 3.10. Concentration (black symbols) and order parameter S (red symbols) profiles of filtration deposits obtained at varied ionic strength of boehmite dispersions.

Figure 3 . 11 .

 311 Figure 3.11. Profile plots of the filtration deposit (I = 1•10 -2 M) sampled in the kapton tube. The bottom of capillary is sealed with modeling clay. Tomographic scans were acquired at a resolution of 2.47 m displaying homogeneous sample distribution along the tube walls.

Figure 3 . 12 .

 312 Figure 3.12. Gray value data quantification after tomographic scans of reference samples: water and boehmite dispersions with volume fraction  of 1.6 vol.%, 3.0 vol.%, 4.6 vol.% and 10.5 vol.%.

Figure 3 . 13 .

 313 Figure 3.13. Comparative concentration profiles obtained from SAXS and CT on the filtration deposit (I = 1•10 -2 M) sampled in the kapton tube.

Figure 3 .

 3 Figure 3.11 shows an example of a tomographic scan acquired at a resolution of 2.47 m for a deposit formed for an ionic strength of 10 -2 M. The gray level distribution on this

Figure 3 .

 3 Figure 3.14. 2D radiographies of boehmite dispersions under 1 bar of applied pressure for filtration time of 280 min, 330 min, 400 min and 460 min. t = 400 min t = 460 min

Figure 3 . 15 .

 315 Figure 3.15. Gray value data quantification after 2D radiography scans of water reference and boehmite dispersions with volume fractions  of 1.5 vol.% and 4.73 vol.%.

Figure 3 . 16 .

 316 Figure 3.16. Example of concentration profile versus distance z from the membrane deduced from in situ radiography tests during pressure-driven dead-end filtration of boehmite dispersions at ionic strength of 8•10 -5 M. Filtration conditions: V = 20 mL; P = 1 bar; ultrafiltration membrane PES, 500 kDa, surface area 3.14 cm 2 . 1 time step : increment 60 ×10 s  10 min.

Figure 3 . 18 .

 318 Figure 3.18. Appearance of boehmite dispersions from left to right: at ionic strength of I1 = 8•10 -5 M, I2 = 8•10 -4 M and I3 = 1•10 -2 M, respectively.

Figure 4 . 1 .

 41 Figure 4.1. Equation of state for boehmite dispersions at various ionic strengths.

Figure 4 . 2 .

 42 Figure 4.2. Rheological phase diagram for boehmite dispersions showing the ">" shape of the sol/gel transition. Boundaries between the sol and gel phases were determined using dynamic oscillation shearing in LVE range.

Figure 4 . 3 .

 43 Figure 4.3. Concentration profiles as a function of the distance z from the membrane surface deduced from in situ radiography tests during pressure-driven dead-end filtration of boehmite dispersions at varied ionic strength. Filtration conditions: V = 20 mL; P = 1 bar; ultrafiltration membrane PES, 500 kDa, surface area 3.14 cm 2 . 1 time step: increment 60 ×10 s  10 min. Red lines correspond to the sol/gel transition defined from osmotic stress experiments.

Figure 4 . 4 .

 44 Figure 4.4. Example of small-angle X-ray spectra for boehmite dispersions at different volume fractions (ionic strength I2 = 810 -4 M).

Figure 4 . 5 .

 45 Figure 4.5. Structure factor S(Q) for boehmite dispersions at I2 = 810 -4 M and at different volume fractions.

Figure 4 . 6 .

 46 Figure 4.6. Evolution of the interparticle distances as a function of volume fraction of boehmite dispersions at various ionic strengths.

Figure 4 .

 4 Figure 4.6 displays the evolution with volume fraction of the correlation distances d*.Various trends can be deduced from this plot. (i) The correlation distances decrease with increasing ionic strength. This is expected and reflects the screening effects of the double layer that results in a shorter distance between particles. At low volume fractions, all curves display a scaling exponent of -1/3, which is a typical of an isotropic state. A similar behavior was observed on several clay minerals, such as beidellite and montmorillonite[START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 1. Structure and electrostatic interactions[END_REF], or nontronite[START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF]. At higher volume fractions, the curves for the I1 = 810 -5 M and I2 = 810 -4 M scale as  -0.96 and  -0.8 , respectively, suggesting a local lamellar ordering of the platelets. In such a case, the swelling law should be d* = t/, where t is the thickness of individual platelets. The calculation yields a value of 3 nm, which is coherent with the known thickness of a single boehmite sheet deduced from diffraction patterns[START_REF] Mange | A microstructural investigation of nanostructured boehmite films prepared by the sol-gel route[END_REF]. This then suggests that some boehmite layers are exfoliated in the gels. The transition between both swelling regimes that could indicate the position of a potential isotropic/nematic phase transition is located at volume fractions ≈ 8 % and 6 % for I1 = 810 -5 M and I2 = 810 -4 M, respectively, i.e far in the gel region according to the phase diagram of Figure4.2. Consequently, the I/N transition, if present is totally hindered by the sol/gel transition in the boehmite suspensions used in the present study. Notably, several studies on clay mineral have demonstrated that the orderdisorder transition does not display a trivial dependence with increasing repulsion. For instance, occurrence in the gel phase of flow-induced birefrigence reminiscent of nematic texture has been reported on laponite[START_REF] Mourchid | On Viscoelastic, Birefringent, and Swelling Properties of Laponite Clay Suspensions : Revisited Phase Diagram[END_REF] at lower volume fractions when lowering the ionic strength (in the range of 10 -2 M-10 -4 M). Also, aqueous dispersions of gibbsite platelets have

Figure 4 . 7 .

 47 Figure 4.7.Evolution of moduli G and G" from the amplitude sweeps for boehmite dispersions at ionic strengths (from top to bottom) of 810 -5 M, 810 -4 M and 110 -2 M. For clarity, not all experimental curves are presented. The yield strain was determined via the maximum in G" (marked in red).

I 3 = 110 - 2 M

 32 Chapter IV: Properties of osmotically densified boehmite dispersions: microstructure and particles organization 91 onset of non-linearity. This latter method is used in situations where the G/G" cross-over is hardly accessible due to wall slip effects. In the present work, we chose to use the position of the maximum in G" (arrow in Figure4.7) in line with the study by Pham et al.[START_REF] Pham | Yielding behavior of repulsion-and attraction-dominated colloidal glasses[END_REF].

Figure 4 . 8 .

 48 Figure 4.8. Evolution of onset of yielding c as a function of reduced volume fraction ( -sol/gel) obtained from strain amplitude sweeps for boehmite dispersions at ionic strength of 810 -5 M, 810 - 4 M and 110 -2 M.

Figure 4 .

 4 Figure 4.8 presents a log-log plot of the peak location c as a function of the reduced volume fraction ( -sol/gel). The values for low and intermediate ionic strength exhibit a power law dependence with an exponent that varies with ionic strength, the minimal exponent (-0.96) being observed for an ionic strength of 8•10 -4 M. The fact that exponents vary with ionic strength contrasts with what was reported for other more anisotropic systems such as nontronite[START_REF] Michot | Sol-gel and isotropic/nematic transitions in aqueous suspensions of natural nontronite clay. Influence of particle anisotropy. 2. Gel structure and mechanical properties[END_REF], montmorillonite and beidellite[START_REF] Paineau | Aqueous suspensions of natural swelling clay minerals. 2. Rheological characterization[END_REF] where constant exponents were observed whatever the particle size and ionic strength. The differences obtained in the present study could suggest changes in the suspension structure with ionic strength that need to be analyzed in more depth. Still, in view of the uncertainty associated to the determination of both the volume fraction of the sol/gel transition and c the differences that we observe could also be in the error bars.

Figure 4 . 9 .

 49 Figure 4.9. Moduli G (closed symbols) and G (open symbols) from frequency sweep measurements for boehmite dispersions at ionic strengths (from top to bottom) of 810 -5 M, 810 -4 M and 110 - 2 M. For clarity, not all experimental curves are presented.

Figure 4 . 10 .

 410 Figure 4.10. High frequency elastic modulus G as a function of reduced volume fraction ( -sol/gel) for boehmite dispersions at ionic strength of 810 -5 M, 810 -4 M and 110 -2 M.

Figure 4 .

 4 Figure 4.11. Zero-shear viscosity curves for colloidal boehmite dispersions at ionic strengths of 810 -5 M and 810 -4 M (steady shear flow measurements).

Figure 4 . 12 .

 412 Figure 4.12. Determination of maximum packing fraction for colloidal boehmite dispersions at ionic strengths of 810 -5 M and 810 -4 M.

97 Figure 4 . 13 .

 97413 Figure 4.13. High frequency elastic modulus as a function of the volume fraction for boehmite gels at different ionic strengths. The solid lines corresponds to the fitting as discussed in the text according to Equation (4.7) with effective volume fraction eff given by Equation (4.9). The effective maximum packing * eff  is taken 1. The zeta potentials are  = 100 mV for I1 = 810 -5 M and  = 50 mV for I2 = 810 -4 M. The reciprocal Debye length  -1 calculated from Equation (4.1) is set 34 nm and 10.75 nm for I1 = 810 -5 M and for I2 = 810 -4 M, respectively. N is taken as an adjustable parameter, N = 10-20.

Figure 4 . 14 .

 414 Figure 4.14. Evolution of the interparticle distance R with volume fraction of boehmite gels. Considering the satisfactory fit between experimental and calculated values displayed in Figure 4.14, Equation (4.11) can be used for determining the interaction potential

Figure 4 . 15 .

 415 Figure 4.15. Estimated electrostatic potential for boehmite dispersions at ionic strengths of 810 -5 M and 810 -4 M from Equation (4.6), using values of interparticle distance R determined from calculations, Equation (4.11), experimentally measured zeta potentials  and values of a as mentioned in the text.

Figure 5 . 1 .

 51 Figure 5.1. Illustration of Tyndall effect in boehmite sols differing on particle size population.The associated size distribution analysis reveals a relatively narrow particle size distribution by volume with a D50 (50% median) around 3.34 m in the range 1-10 m. Noncolloidal particles contribute to  11 wt.% of the total volume of dispersion. The comparative

Figure 5 . 2 .

 52 Figure 5.2. Examples of 2D radiographies acquired upon filtration of boehmite dispersions; I3 = 110 -2 M, P = 1 bar. V = 1 mL; ultrafiltration membrane PES, 500 kDa, surface area 0.5 cm 2 .

Figure 5 .Figure 5 . 3 .

 553 Figure 5.2 represents an example of 2D-scans acquired at different filtration times. In order to extract concentration profiles, a calibration function was obtained by acquiring the transmission signal of 4 reference boehmite samples with known volume fraction (Figure 5.3).

Figure 5 . 4 .

 54 Figure 5.4. Concentration profiles as a function of the distance z from the membrane surface deduced from in situ radiography tests during pressure-driven dead-end filtration of boehmite dispersions at varied ionic strength. Filtration conditions: V = 1 mL; P = 1 bar; ultrafiltration membrane PES, 500 kDa, surface area 3.14 cm 2 . 1 time step : increment 60 ×10 s  10 min.

Figure 5 . 7 .

 57 Figure 5.7. Comparison of concentration and velocity profiles as a function of time at a distance z = 150 m from the membrane for ionic strengths I1 = 810 -5 M, I2 = 810 -4 M and I3 = 110 -2 M.

Figure 5 . 9 .

 59 Figure 5.9. Equivalent shear rates deduced from Figure 5.8 and plotted as a function of filtration time t for I1 = 810 -5 M, I2 = 810 -4 M and I3 = 110 -2 M. The corresponding concentration profile deduced previously from in situ radiography tests is presented.

Figure 2 . 4 ,

 24 Chapter II.

  c is the yield stress, pl the plastic viscosity and the shear rate   . Since the characteristic shear rate   around the settling particle scales with the particle velocity as p/d, parameters: the yield stress c and the plastic viscosity pl. When the particle velocity p = 0, the yield stress in Equation (5.5) approaches a value of  5.5•10 -3 Pa. This value is far below yield stress commonly measured for concentrated boehmite suspensions (which is typically of hundreds of Pa).

Figure 5 . 10 .

 510 Figure 5.10. The calculation of yield surface around a sphere from the PIV velocity fields for large applied stresses. Particle moves from right to left. From [109].

Figure A1 .Figure A2- 1 .

 A11 Figure A1. Cake permeability K versus ionic strength I for colloidal (in gray) and bimodal mixture (in green) boehmite dispersions. Filtration conditions: V = 10 mL; P = 1 -3 bar; solids density 3035 kg•m -3 ; liquid viscosity 0.89 mPa•s; ultrafiltration membrane PES, 500 kDa, surface area 4.91 cm 2 .

Figure A2- 2 .

 2 Figure A2-2. Rheological phase diagram for boehmite dispersions showing the ">" shape of the sol/gel transition.

Sol Gel Flocs NOMENCLATURE

	Rg 	Gyration radius Characteristic frequency (for a Bohr atom 2.2•10 -18 J)	m J
	Rh q Latin symbols Hydrodynamic radius Electric charge density A Hamaker constant Rm Hydraulic resistance of membrane m Solvent density Af Proportionality factor Aeff Effective Hamaker constant Rp Filtration resistance due to pore fouling RT Total resistance p Density of dispersed phase 0 Ion concentration in the bulk solution A(Q) Scattering amplitude a (Hydrodynamic) particle radius C London-van der Waals constant cc Dry cake mass per filtrate volume D Diffusion coefficient S(Q) Structure factor sb Mass fraction of solids in suspension  Shear stress S Surface area  Pore tortuosity Sm Effective surface of membrane  Velocity t Time  Volume fraction T Temperature  Angular frequency	m C m -3 m -1 kg m -3 J m -1 kg m -3 m -1 kg m -3 J m -1 m -kg m -3 m 2 s -1 -N m -2 --m 2 m s -1 m 2 % s °C rad s -1
	E e0 V  Vcake ζ	Energy Volume Electrostatic potential Elementary electric charge (1.6•10 -19 C) Filter cake volume Zeta-potential	J V m 3 C m 3 V
	G* G' Vp G" vs Vvdw	Shear modulus Storage modulus Volume of dispersed phase Loss modulus Voxel size Van der Waals dispersion potential	N m -2 m 3 N m -2 m N m -2 V
	G2(Q,t) w	Intensity autocorrelation function mass of solids deposited per unit area	a.u. kg m -2
	g1(Q,t) X1	Normalized field autocorrelation function Baseline of the correlation function	a.u. -
	h X2	Planck constant Intercept of the correlation function	J s -
	h Z	Gap between two parallel plates Atomic number	m -
	I z	Ionic strength Distance from the membrane surface	M m
	I1	Transmitted intensity	a.u.
	I0	Incident intensity	a.u.
	J Greek symbols Volumetric filtration flux	m 3 m -2 s -1
	K 	Cake permeability Baseline constant	m 2 -
	kB 0	Boltzmann constant (1.38•10 -23 J K -1 ) Electronic polarisability	m 2 kg s -2 K -1 C m 2 V -1
	kKC 	Kozeny-Carman constant Coherence factor	--
	 	Membrane thickness Decay rate	m -
	l 	Sample thickness Shear strain	m %
	mm  	Mass of dispersion (solvent) Shear rate	kg s -1
	mR P	Moisture ratio (wet to dry cake mass ratio) Pressure	-N m -2
	mp 	Mass of dispersed phase Porosity	kg %
	ps r	Detector pixel size Dielectric constant	-
	P(Q) 0	Form factor Permittivity of vacuum (8.85•10 -12 F.m -1 )	-F m -1
	p Q * R  -1 Ra  Re 0 Rc  rc 	Empirical constant related to the cake compressibility Wave vector Complex viscosity Interparticle separation distance Debye screening length Filtration resistance due to adsorption Laser wavelength Reynolds number Vacuum wavelength of the incident light Filtration resistance of the cake (deposit) Osmotic pressure of solution Specific cake resistance Elastic scattering angle	-Pa s m -1 m m m m -1 m -Pa m -1 m kg -1 °
	Rconc µ	Filtration resistance due concentration polarisation Dynamic viscosity of filtrate	m -1 Pa s
	Rg (z)	Filtration resistance due to gel layer formation Attenuation coefficient	m -1 -

Chapter I: Literature review

Chapter II: Materials and methods

Chapter III: Frontal filtration studies on boehmite dispersions

volume fraction until reaching nearly zero values that therefore corresponds to the characteristic plateau on the concentration profile, as presented in Figure 5.7.