Benjamin Villeroy

Valérie Lalanne

Siavash Ghabezloo

Sadok Feia

Mario Sheel

Sena Hamadi

Mohamed Gerrouache

Agustin Rios

Marcelle Ammour

Ces travaux ont bénéficié d'une subvention de l'ANR dans le cadre du programme national Investissements pour l'avenir (LabEx projet MMCD

List of Figures

General introduction

Porous structures are abundantly present in Nature, particularly in honeycombs with hexagonal cells, hollow bamboo, alveoli lungs, human skin, soils, and rocks. The design of porous materials that mimic frameworks found in Nature in the micro-and nanoscale range has long been an important science topic. Porous polymers have especially received an increased interest, due to their potential to combine the properties of both porous materials and polymers. They display significant advantages, such as tunable mechanical properties, ease to be functionalized within the porous framework or at the pore surface, and particularly lower production cost in comparison with their inorganic counterparts. Additionally, porous polymer structures may permit to switch between open and closed porosity after exposure to environmental stimulation. Such particularity is generally unavailable in other porous materials. In hierarchically porous materials, the role of each porosity level is associated with a distinct mass transfer process. Biporous materials are particularly interesting for the development of sustainable materials. For instance, in scaffolds for tissue engineering, the first porosity level with pore sizes over 100 µm should allow for efficient seeding and proliferation of suitable cell lines through the material, while the second porosity level with pore sizes lower than 1 µm may permit to improve the nutrient and waste flow within the material when the larger pores are clogged at the last stage of the cell culture.

The similarities of biporous polymeric materials to natural environments or manmade materials make them ideal models for the understanding of the structures of soils, rocks, wood or lightweight cement. The effect of each porosity level on the transport properties of biporous materials is a fundamental issue. An essential factor affecting effective permeability of many porous materials is their specific microstructure. One such structure is often complex and difficult to characterize in a manner that would easily be used to correlate microstructural and transport properties of these materials. X-ray microtomography techniques have been used to visualize the 3-D internal microstructure of heterogeneous materials for more than a decade. Combining X-ray microtomography with numerical simulation enables the input of real microstructure data in the algorithms used for the computation of the effective permeability. The reconstruction of the most realistic Representative Volume Element (RVE) for the evaluation of the transport properties is challenging in the case of highly heterogeneous and multi-scale microstructures such as biporous materials.

Three main specific objectives are considered in this PhD work. The first objective is to elaborate model biporous polymeric materials with a well-defined morphology 1 by a facile methodology. The aim is to design materials which may mimic a real porous medium with a double scale of porosity. Poly (2-hydroxyethyl methacrylate) (PHEMA) has been chosen for the elaboration of the doubly porous materials, due to its chemical stability and interesting hydrophilic properties. Additionally, hydroxyl groups are available for potential further functionalization. Such a polymer is wellknown and has been studied by our teams in the last few years. The second objective is to understand the influence of biporous polymer microstructure, and in particular of the pore interconnectivity, on the transport properties. The third objective is to compare the experimental and the computational permeability.

In order to achieve such objectives, the originality of our approach lies in the development of a three-step methodology involving design, thorough characterization and modelling of polymeric materials exhibiting a bimodal porosity. Firstly, model biporous PHEMA materials will be elaborated by the double porogen templating approach, namely by simultaneously using two different porogens to generate the two porosity levels. Then, the morphological and microstructural characterization of these biporous networks in terms of pore shape, size distribution and interconnections will be performed. Finally, the computation of the transport properties will be based on the 3-D images provided by X-ray micro-tomography and synchrotron tomography. The thresholded digital images will serve as a RVE of a doubly porous solid in which a macroscopic permeability will be computed. To this purpose, a double homogenization approach is proposed since biporous materials exhibit two populations of highly different pore sizes. The first homogenization consists in computing the permeability associated with the small pores. At an intermediate scale corresponding to the large pores, a coupled Darcy/Stokes problem is considered. Finally, the macroscopic permeability incorporates both the first and second porosity levels.

The manuscript is composed of 5 chapters:

• The first chapter will include the state of the art concerning general concepts, fields of application and elaboration methods associated with porous materials. An overview of the morphological characterization techniques is also proposed.

A last part will be devoted to the experimental and computational determination of the permeability.

• The second chapter will be concerned with the elaboration of PHEMA-based doubly porous polymers via the double porogen templating approach. The effect of the macroporogen packing and size on the morphology, and the cross-linker effect on the thermo-mechanical properties will be discussed. The influence of pore interconnections on the apparent density and the swelling behavior will also be studied. Finally, the experimental determination of the permeability will be performed.

• Chapter 3 will provide a careful description of the 3-D microstructure of the previously prepared PHEMA-based materials by means of X-ray microtomography on both a laboratory setup and a synchrotron beamline, with a voxel size down to 650 nm. Observed microstructures will be discussed in connection with the elaboration route.

• Chapter 4 will be dedicated to the Fast Fourier Transform (FFT) computation of fluid flow through doubly porous polymers exhibiting non-interconnected large pores by means of a simplified approach which uses only the Darcy equation at the intermediate scale. 3-D images obtained in Chapter 3 will serve as input data.

• Chapter 5 will finally propose new FFT iterative schemes to solve the coupled Darcy-Stokes problem with applications to fluid flow in doubly porous solids with non-interconnected and interconnected large pores.

The experimental part which describes all the protocols followed in this PhD work is presented as a couple of appendices.

Chapter 1

State of the art

Regarding the main focus of investigations achieved during this PhD thesis, the literature review includes four main sections. The first one (section 1.1), presents the general concepts associated with porous materials and the various applications of this type of particular structures. Section 1.2 then provides the different methodologies so far applied for the elaboration of biporous polymeric materials. Further, section 1.3 highlights the major morphological and microstructural characterization techniques for porous materials. Finally, the last part discusses the experimental and computational determination of the permeability associated with such doubly porous materials. For conciseness purposes, the reader should be aware that this literature survey will selectively focus on particular aspects of porous materials and will not be very exhaustive on other ones.

Porous materials: classification and applications

Classification of porous networks

According to IUPAC, a porous solid is a solid with pores, i.e. cavities, channels or interstices, which are deeper than they are wide.(1) Several methods have been up to date used in the literature to classify porous networks. First, it is possible to classify the pores according to their ability to permit the flow of an external fluid. In • MC (single multiply connected domain only) where the void space (or the solid matrix) is totally interconnected,

• SC (ensemble of simply connected domain only) where the void space (or the solid matrix) is totally isolated,

• MSC (MC+SC) where a part of void space (or the solid matrix) is interconnected while the other is isolated.The combination of these 3 configurations according to whether the phase is void space or solid matrix results in different media given in Figure 1.2.The structure is called porous medium (PM) if:

1. the solid matrix and the void space are both totally interconnected (PM4), 2. one phase is totally interconnected while the other is partially interconnected (PM2 and PM3),

both phases are partially interconnected (PM1).

The medium only contains isolated pores (IP) if the void space is not interconnected and the solid matrix is totally interconnected. The medium is a fluidized bed (FB) if the void space is totally interconnected and the solid matrix is totally isolated. In the first one, the higher porosity level (fractures) influences the overall storage capacity of the media (Figure 2.a). The second case represents a medium in which the contribution of the lower porosity level is more important than the higher one (Figure 1.3b). In the third case, the two porosity levels are equally distributed (Figure 1.3c).

In the present study, our interest is focused on doubly porous materials. The concept of macropores and micropores was defined with the development of investigations on porous polymeric materials towards the end of 1950's.( 5) According to IUPAC, macropores refer to pores with a size higher than 50 nm, mesopore size ranges from 2 to 50 nm and micropores are smaller than 2 nm.However, macroporous networks usually have a broad pore size distribution ranging from 50 nm to several µm (microns). If the pore size is beyond 10 µm in hydrogels, this type of network is also called a superporous hydrogel. (6) It is also well-known that nanoscale refers to scales below 100 nm. In the context of this study but above all to avoid confusion of the reader, the pores with a size up to 10 µm are called "small pores" and represent the second porosity level, while the pores with a size higher than 10 µm are defined as "large pores" and correspond to the first porosity level.

A polymeric material with two different porosity levels can be considered as a doubly porous material. The porosity levels can be classified depending on four possible different geometrical characteristics of the considered porosity:

• interconnected large pores representing the first porosity level, and interconnected small pores corresponding to the second porosity level. In this case, fluid can flow through the material in both porosity levels; (7) • interconnected large pores in conjunction with non-interconnected small pores. In this case, the smaller porosity does not play any role in the transport properties of the materials, but it can increase the specific surface area. In addition to the interest of increased surface area, such a material introduces advantages in terms of chemical modification and surface functionalization compared to the materials with similar macroporosity only;(8)

• non-interconnected large pores and interconnected small pores. This type of structure might be interesting if the purpose is to decrease the fluid flow due to the non-interconnected large pores, or might be used in filtration applications;

• non-interconnected large pores and non-interconnected small pores. This type of material exhibiting "closed pores" does not present a particular interest as it does not allow a fluid to flow.

Applications of porous materials

Porous materials are abundant in Nature (trees, rocks, etc) and also represent an integral part of tissues of human/living beings (skin). Scientists have been inspired by these natural structures to create materials which allow the cure of sick human tissue but also as models to understand the mechanisms of fluid transport in complex structures.

For these reasons, porous materials are widely used in tissue engineering and filtration techniques. They are also used to understand fluid flow in soil mechanics and civil engineering. The presence of two porosity levels provides particularly interesting transport properties as each level has a specific role regarding fluid flow. In this section, examples of applications in which doubly porous materials play an important role are presented.

Tissue engineering

Tissue engineering has emerged as a promising research field for the repair and regeneration of diseased tissues and organs. The presence of a scaffold is essential to provide a three-dimensional template for cell adhesion and tissue ingrowth and to maintain the mechanical stability of the tissue defect. Porous natural and synthetic biodegradable polymer-based materials represent candidates of choice for skin regeneration due to their microstructure, biocompatibility, biodegradability, and low production cost. Many bio-based polymers, e.g. collagen and glycosaminoglycans, notably ensuring some role in the mechanical strength of different tissues, have the advantage of being biocompatible and biodegradable. Nevertheless, the low mechanical strength and high rates of degradation of these natural polymers generally require their formulation as composites or their chemical cross-linking to improve their intrinsic mechanical properties and reduce their degradation rates. However, these changes may cause cytotoxic effects and reduce biocompatibility.( 9) Furthermore, the versatility of chemically synthesized polymers enables the fabrication of scaffolds with different features (shapes, porosities and pore sizes, degradation rates, mechanical properties) to match the targeted specific applications. Porosity and pore size of polymeric scaffolds play a crucial role in bone formation in vitro and in vivo. Indeed, macropores (> 50 nm) permit the growth of bone cells, vascularization and nutrient delivery to the center of the regenerating tissue, while nanopores (between 5 and 50 nm) are useful for the crystallization of hydroxycarbonate apatite and cell adhesion. (10) To demonstrate the effect of pore size on cell growth, Akay et al. used polymeric foams as scaffolds for the seeding of rat osteoblasts. (11) When primary rat osteoblasts were seeded into scaffolds presenting different pore sizes, more cells were found in the scaffolds presenting small pores (40 µm). The authors also found out that cells migrated faster inside the scaffolds presenting larger pores (100 µm); however, pore size did not affect cell penetration depth or mineralization extent.

Soil mechanics and civil engineering

Auriault et al. considered fractured reservoir rocks as porous media composed of two porosity levels, as described in Figure 1.4. The first one corresponds to the pores between grains of the matrix, while the second one is associated with the fractures. (12; 13; 14) At the pore scale (in Figure 1.4a), the period Ω has a characteristic length l, the solid occupies the domain Ωs and the pores the domain Ωp, respectively. The boundary between the two domains is denoted Γ. At the fracture scale (in Figure 1.4b), the period is Ω' with a characteristic length l'. The solid and the pores occupy the domain Ω'sp and the fractures the domain Ω'f. The boundary between the two is Γ'. It should be reminded that the fractured reservoir has a characteristic length l".

Filtration

Stimuli-responsive porous hydrogel thin films have recently attracted much attention. Because of spatial constraints, swelling/shrinking of the hydrogel films results in closing/opening (or vice versa) of the pores present in the films. The major application for these materials is based on the controlled transport of molecules across a porous thin film for the controlled release of drugs, the separation and gating of molecules, biomolecules, and ions in sensors, substrates for cells. They can be also implemented as containers for enzymes or as ultrafiltration devices.( 15) Thin porous gel films with a well-defined structure and adjustable dimension of pores will be useful for molecular filters and separation membranes.( 16) Templating environmentally responsive hydrogels with sacrificial colloidal crystals allowed for the creation of porous materials with tunable optical properties. Such materials are elaborated by polymerization of monomers inside a pre-assembled colloidal crystal (monodisperse latex of silica particles) with subsequent removal of the crystal by etching or dissolution. The resulting structure consists in a highly ordered array of interconnected submicrometer-sized pores. The swelling transitions in stimuli-responsive hydrogels lead to shifts of the Bragg diffraction wavelength due to changes in the periodic lattice spacing. This property makes these materials especially useful for sensor applications.(15)

1.2 Strategies applied for designing porous polymeric materials

Porogen templating approach

The porogen templating approach is a simple and versatile strategy for the elaboration of porous polymeric materials. It consists in the exact replication of the inverse structure of the preformed templates. (17; 18) Experimentally, porogen templating procedure requires three main steps:

• infiltration of the monomer(s), crosslinker(s) and initiator onto the surfaces or into interstitial voids of the templates for the preparation of the polymeric network,

• in situ polymerization,

• final removal of the templates.

There are several conditions for an efficient templating method. First, the template should have a well-defined structure, such as controllable morphology which allow one to tune the resulting pore size. Then, the template should be easily removed after the reaction. Moreover, it should be robust enough to withstand the high interfacial energy and capillary actions that can lead to pore deformation, or even collapse, after template removal. Finally, the polymeric walls should allow the ability to incorporate designable functionalities selected for targeted applications.

Our critically selected survey is focused on materials exhibiting two porosity levels. Different templates, either organic or inorganic, liquid or solid were used to generate two levels of porosity. For example, sintered Poly(methyl methacrylate) (PMMA) beads have been used in several studies to generate macropores. (19), (20) Some authors reported on the use of inorganic macroporogenic templates, such as CaCO 3 (21; 22; 23) and NaCl particles.(24; 25) More details are given in Table 1.1 which provides a comparison between the templates used, the experimental conditions chosen, and the porosity features obtained upon polymerization and successive elimination of the two distinct types of porogens from the as-prepared doubly porous polymeric materials via the double porogen templating approach. As shown in Table 1.1 , the use of CaCO 3 as a macroporogen in conjunction with either hydroxyapatite nanoparticles or ethanol as a nanoporogen required a time-consuming and difficult extraction of the porogens. Furthermore, the main drawback when using PMMA beads as macroporogens still relied on rather time-consuming porogen extraction from the porous materials. Such materials exhibited a lower porosity ratio than that of materials prepared with NaCl particles (60 % porosity ratio vs. 77 %, respectively). Macropores of about 195 µm diameter interconnected by voids of about 65 µm were observed, while nanopores in the 360 nm range were also found. The materials arising from the PAN fibers/EtOH porogenic pair did not exhibit a very well-defined bimodal porosity as the two pore size distributions were rather close (40 µm / 10-20 µm). Finally, NaCl or PEO porogens in conjunction with a porogen solvent mixture generated the highest porosity ratios without using toxic porogenic

Thermally-induced phase separation

The thermally induced phase separation (TIPS) process is one of the most popular methods to investigate phase diagrams of different polymers and to develop microporous membranes (27; 28). The TIPS process can induce solid-liquid (S-L) and liquidliquid (L-L) phase separation. [START_REF] De Lima | Porous Polymer Structures Obtained via the TIPS Process from EVOH/PMMA/DMF Solutions[END_REF] In general, a homogeneous solution of polymer is prepared at an elevated temperature by mixing the polymer with a diluent of high boiling point, then the solution is cooled to induce phase separation, and finally a microporous structure is formed after the extraction of the solvent. Ly et al. have recently reported on this innovative concept for generating highly porous (between 85 and 90 % porosity ratio) poly(2-hydroxyethyl methacrylate) (PHEMA)-based materials via TIPS process. [START_REF] Ly | Tailoring Doubly Porous Poly(2-Hydroxyethyl Methacrylate)-Based Materials via Thermally Induced Phase Separation[END_REF] As shown in Figure 1.5, a co-solvent mixture constituted of water and 1,4-dioxane was used to solubilize linear PHEMA chains, followed by a solidification process by quenching the PHEMA/co-solvent mixture at a given temperature, and subsequent freeze-drying to produce the corresponding biporous PHEMA materials. The obtained networks exhibited two porosity levels: the first one in the 10-100 µm size range, while the second porosity level displayed pore sizes ranging from 40 nm to 800 nm. It is also possible to associate the TIPS technique with another methodology. Liu et al. have successfully created 3-D nanofibrous gelatin scaffolds by combining TIPS and the so-called porogen leaching technique. [START_REF] Liu | Phase Separation, Pore Structure, and Properties of Nanofibrous Gelatin Scaffolds[END_REF] In this particular case, the porogenic template consisted of paraffin spheres. The resulting polymeric material exhibited rather high surface areas (> 32 m 2 .g -1 ), high porosities (> 96 %), well-interconnected macropores, and nanofibrous pore wall structures. The implementation of such concomitant techniques advantageously allowed for the control of macropore shape and size by paraffin spheres, interpore connectivity by assembly conditions (time and temperature of heat treatment), pore wall morphology by phase separation and post-treatment parameters, as well as mechanical properties by polymer concentration and crosslinking density.

High Internal Phase Emulsion

High internal phase emulsions (HIPEs) are widely used in food preparation, fuels, oil recovery and cosmetics. [START_REF] Cameron | High Internal Phase Emulsion Templating as a Route to Well-Defined Porous Polymers[END_REF] One application of HIPEs that has found considerable interest in materials science consists in generating a 3-D template to elaborate highly porous structures. [START_REF] Arriaga | Monolithic Materials: Preparation, Properties and Applications[END_REF] According to most definitions, HIPEs have a volume fraction of the internal phase higher than 74 % which represents a maximum space occupied by uniform spheres. [START_REF] Weiner | Emulsions and Emulsion Technology, Part I[END_REF] The resulting porous materials, called PolyHIPEs, are formed by curing the continuous, or non-droplet, phase of the emulsion. Following solidification of the continuous phase, the emulsion droplets are embedded in the resulting material. In such a methodology, a porogenic solvent, either organic or aqueous, contributes to the formation of the macroporosity level, while interconnections between macropores, called "windows", could be considered as a second porosity level. The experimental process setup is represented in Figure 1.6. The drawbacks of this technique mainly rely on rather low mechanical strength and rather low surface area of the as-obtained porous materials. However, one such rather low surface area can be increased by proceeding to a higher degree of crosslinking (i.e. hypercrosslinking reaction). Schwab et al. reported the synthesis of polyHIPEs with bimodal pore-size distribution and very high specific surface area up to 1210 m 2 .g -1 , for instance. [START_REF] Schwab | High Surface Area PolyHIPEs with Hierarchical Pore System[END_REF] In the same topic, our team recently developed hypercrosslinked functional poly(2-hydroxyethyl methacrylate)-based frameworks exhibiting a specific surface area up to 1500 m 2 .g -1 . The immobilization of gold nanoparticles at the pore surface allowed for the preparation of hybrid materials that were successfully applied for the supported heterogeneous catalytic reduction of a nitroaromatic compound, i.e. p-nitrophenol, or a pollutant dye, i.e. Eosin Y.(36) Furthermore, Damouny et al. have described the one-pot elaboration of hydrogelfilled, shape-memory polyHIPEs based on a polyacrylate possessing crystallizable side chains.(37) Such networks exhibited closed-cell-like voids with diameter ranging from 5 to 100 µm and a 200 nm thick hydrogel coating. During the compression at 70 • C to 70 % strain, the deformation depended on whether the polyHIPE was dry or hydrated. On one hand, the crystalline domains of the polyHIPEs prevented the recovery when the temperature was below the melting point. On the other hand, the amorphous hydrogel domains prevented the full recovery in air above the melting point. Various types of nanoparticles have been incorporated into polyHIPE materials so as to enhance their mechanical properties and also to stabilize the emulsions leading to the so-called Pickering emulsions. [START_REF] Xu | Interconnected Porous Polymers with Tunable Pore Throat Size Prepared via Pickering High Internal Phase Emulsions[END_REF] For example, Toledo et al. recently demonstrated that incorporating SiO 2 nanoparticles to the HIPEs led to larger pore diameters, while the diameter of the interconnecting pore decreased.(39)

Electrospinning

During the electrospinning process, illustrated in Figure 1.7, a polymer droplet is held at the tip of a needle by surface tension. When a sufficiently high voltage is applied to the droplet, it becomes charged and an electrostatic repulsion force is generated. When the electrostatic force triggered by the high-voltage source counteracts the surface tension, an electrically charged jet of polymer solution erupts. As the jet moves forward, but before reaching the collector, the solvent evaporates and the jet solidifies into thin fibers.(40; 41) Equipment-related parameters (applied voltage, solution feeding rate, tip-to-collector distance), solution properties (solution viscosity and conductivity, polarity and volatility of solvent), and environmental conditions (temperature, humidity) are the main factors that influence the electrospinning process. By finely tuning these parameters, doubly porous materials with inter-and intra-fiber porous structures can be obtained. Han et al. prepared ultrafine porous cellulose triacetate (CTA) fibers with a solvent mixture composed of methylene chloride (MC) and ethanol (EtOH), for instance. [START_REF] Han | Ultrafine Porous Fibers Electrospun from Cellulose Triacetate[END_REF] Ultrafine porous CTA fibers electrospun with only MC had isolated circular shape pores with a narrow size distribution in the 50-100 nm range, a porosity ratio of 76 % and a BET surface area of 15 m 2 .g -1 . However, fibers electrospun with MC/EtOH (90/10 v/v) exhibited interconnected larger pores in the 200-500 nm range leading to a higher porosity ratio of 85.5 % and a specific surface area of 14 m 2 .g -1 . Using this methodology, the second level of porosity (porous fibers) was induced by phase separation resulting from the rapid evaporation of solvent during the electrospinning process. Moreover, Zheng et al. reported on the elaboration of biporous hierarchical structures from polystyrene (PS) materials exhibiting both micro-and nanoporosity. [START_REF] Zheng | Construction of Hierarchical Structures by Electrospinning or Electrospraying[END_REF] The first porosity level (micropores) included the space between fibers, beads and bead-on-string structures, while the second porosity level consisted in nanopores and nanopapilla on the surface of the beads and fibers. Solvent(s) to polymer volume ratio was the main factor influencing the first porosity level. Increasing the polymer concentration caused a morphology change from beads to fibers. Different secondary porous levels were obtained through the use of appropriate solvent(s) composition, environmental humidity, electrospinning temperature and polymer concentration. This second porosity level tends to appear when the electrospinning process is achieved in a highly humid environment. When DMF was used, nanopapilla were formed on the surface of electrosprayed beads and fibers. When THF was used, nanopores were formed on the surface of both electrospun fibers and electrosprayed fibers.

Gas foaming process

The gas foaming methodology generates porous structures by the expansion of gas (e.g., CO 2 or N 2 ). Pores are created in this process by first forming a polymer/gas solution by equilibrating polymer discs with high pressure CO 2 gas. Thermodynamic instability was then generated by decreasing the CO 2 gas pressure. Gas molecules then cluster, forming nuclei, to minimize their free energy. Dissolved gas molecules diffuse to these nuclei, thus creating macropores. Nevertheless, the gas foaming technique could only produce a monoporous network. However, gas foaming strategy can also be implemented in conjunction with a macroporogen. For instance, disks consisting of poly (D,L-lactic-co-glycolic acid) copolymer and NaCl particles were molded under compression at room temperature and subsequently allowed to equilibrate under CO 2 high pressure (800 psi). [START_REF] Harris | Open Pore Biodegradable Matrices Formed with Gas Foaming[END_REF] The polymer particles fused to form a continuous matrix in which were entrapped salt particles. The NaCl particles were subsequently leached out to yield macropores within the polymer matrix. The overall porosity and level of pore connectivity were tuned by varying the polymer/NaCl particles mass ratio and the NaCl particle size. This process, consisting of a combination of high pressure gas foaming and particulate leaching techniques, allows one to prepare polymeric matrices with a well-controlled porosity and pore structure. Additionally, this process avoids the potential drawbacks associated with the use of high temperatures and/or organic solvents in biomaterial processing.

Morphological characterization techniques of porous materials

Porous materials are commonly characterized in terms of pore size, pore size distribution, pore connectivity, porosity ratio and specific surface area. To this purpose, three main techniques are generally used: mercury intrusion porosimetry (MIP), gas sorption porosimetry, and imaging techniques, including scanning electron microscopy (SEM) and X-ray computed microtomography (µCT).

Mercury intrusion porosimetry

MIP is commonly used for the characterization of macroporous materials, giving access to key parameters such as the porosity ratio and the pore size distribution. Washburn pointed out that mercury do not naturally enter the pores in most materials and that a certain pressure is required to force it penetrating into pores. [START_REF] Winslow | Advances in Experimental Techniques for Mercury Intrusion Porosimetry[END_REF] To this purpose, the Washburn equation that relates the required pressure to the size of pore being entered was elaborated. In this equation, it is worth noticing that a cylinder was used as a geometric model of a pore to obtain Eq. (1.1) as follows:

P = (-2µ cos θ)/r (1.1)
where P stands for the pressure triggering the intrusion, µ the surface tension of mercury, θ the contact angle between the mercury and the pore wall, and r the radius of the cylindrical pore being intruded. Most of real porous materials have pore geometries that are not cylindrical. Thus, using this equation to calculate the size of a pore requires the determination of the radius of an equivalent cylindrical pore. Some caution is, however, required in data interpretation of the size distribution curves. Briefly, the method involves submerging a porous material in mercury and then applying pressure to the mercury to force it intruding the porous material. When a sufficient pressure is reached, the mercury can enter the pores of the material. The size of the intruded pores is related to the pressure causing the intrusion and the volume of the intruded pores is the volume of mercury forced into them. If this process is continued over a range of pressures, the result is a distribution of the pore volume of the material with respect to its pores sizes, i.e., a pore size distribution.

Gas sorption porosimetry

Brunauer, Emmett, and Teller (BET) developed a simple isotherm model to investigate multilayer adsorption and used this model to extract the monolayer capacity and determine the specific surface area.(46) A number of improvements of the BET model have been developed more recently but the basic BET method remains the most widely used technique for specific surface area measurements. [START_REF] Helfferich | Principles of Adsorption & Adsorption Processes[END_REF] In the BET model, each molecule in the first adsorbed layer is considered to provide one "site" for the second and subsequent layers. The molecules in the second and subsequent layers, which are in contact with other sorbate molecules rather than with the surface of the adsorbent, are considered to behave essentially as the saturated liquid while the equilibrium constant for the first layer of molecules in contact with the surface of the adsorbent is different. The resulting Eq. (1.2) for the BET equilibrium isotherm is:

P n a (P 0 -P ) = 1 n a m C + (C -1) n a m C × P P 0 (1.2)
where n a stands for the gas amount adsorbed at the relative pressure P/P 0 , n a m the monolayer capacity, and C a constant, which depends on the isotherm shape. A linear relation between P n a (P 0 -P ) and P/P 0 can be deducted from this equation so it is possible to determine n a m , thus leading to Eq. (1.3) as follows:

A(BET ) = n a m N a m (1.3)
where A(BET ) stands for the specific surface area, N the Avogadro constant, and a m the average area occupied by each adsorbed molecule in the complete monolayer (i.e. the molecular cross-sectional area). Measurements are commonly made using nitrogen as the sorbate at liquid N 2 temperature, i.e. -196 • C or 77 K, but other small molecules as Kr [START_REF] Yanazawa | Precision Evaluation in Kr Adsorption for Small BET Surface Area Measurements of Less Than 1 M2[END_REF] and CO 2 (49) may be also employed. Gas sorption porosimetry also allows for the determination of pore sizes by using the method of Barrett, Joyner, and Halenda (BJH).( 50) Such a procedure provides pore size distributions from experimental desorption isotherms using the Kelvin model of pore filling. The BJH method applies only to the mesopore and small macropore size range.

Comparison between mercury intrusion porosimetry and gas sorption porosimetry

The investigation of porosity by MIP has the advantage of being rapid. Unfortunately, it requires an important amount of sample, particularly if small pores have to be investigated. The isotherm method requires less amount but the time necessary to obtain an accurate desorption isotherm is much greater than that needed to obtain MIP data. MIP is better suited for investigating the pore volume distribution of materials presenting pores larger than 30 nm radius, while the isotherm method would cover a pore range below this value.

Imaging techniques 1.3.3.1 Scanning Electron Microscopy

Scanning Electron Microscopy (SEM) is based on interactions of electrons with a material. This imaging technique permits to obtain high-resolution images of a sample surface with a depth of field higher than that of an optical microscope. A scanning electron microscope is composed of an electron source, a high voltage device, a vacuum column, a set of electronic lenses allowing for the focus of the electron beam on a small spot on the sample surface, a deck for the samples installation, different electron detectors coupled with a signal amplifier, and a system to record and display the images. Two principles are used for the electron extraction: the thermoionic emission and the field-emission. The first one is used in traditional scanning electron microscopes. Basically, the electron source is a tungsten or lanthanum hexaboride (LaB 6 ) filament heated by Joule effect. The second system is a field-emission used in a SEM-FEG (Field Emission Gun). The application of a voltage between the cathode and the anode permits to obtain an intense electric field, thanks to the pointed shape of the metallic cathode. The SEM-FEG technology allows for a greater stability during the observation, a high spatial resolution because of smaller spot size on the sample and a better brightness when compared to the other system.( 51) The primary electrons of the beam interact with the matter underneath the spot within the so-called information volume to generate various other signals measured by the detectors, among which high energy backscattered electrons (BSE) and low energy secondary electrons (SE) are the most commonly used. 2-D images are generated when the electron beam scans the area of interest on the sample. SEM images provide highly detailed information on the surface topography (case of SE images) or chemical contrast (case of BSE images) of the sample. A wide range of magnifications can be considered by modifying the scanning parameters. In the field of porous materials, SEM has commonly been used for demonstrating pore size and morphology. The high depth of field of SEM allows in particular to observe the non-flat surface of sample, as those obtained by breaking a porous sample, without any further surface preparation, except metallization required to evacuate electric charges on non-conductive materials. However descriptions of pore interconnectivity based on SEM have most often been mostly qualitative because of the inherent limitation of SEM which is restricted to surface measurements on relatively small fields of view. Consequently, only a fraction of a sample may be viewed at once, and it may become difficult to distinguish pores from interconnections.( 52)

1.3.3.2 X-ray microtomography
X-ray computed microtomography, abbreviated as µCT, is a non-destructive method to produce fully 3-D images of the whole volume of the sample and not only of its external surface. In standard absorption contrast X-Ray tomography, the physical quantity that is mapped in 3-D is the coefficient of linear attenuation which, for single-phase porous materials as in the present study, is directly linked to the local density of matter.

The obtained data are usually represented as a series of 2-D serial sections through an object. More details about the µCT apparatus will be given in the Chapter 3. So far, many studies have reported on the investigation of the microstructure of porous materials by means of µCT.(52; 53; 54; 55) For instance, Moore et al. described an image analysis technique for quantitative investigation of poly(propylene fumarate) porous scaffold interconnectivity.( 52) Such porous materials were prepared via a particulate leaching process with 75, 80, 85, and 88 vol. % porogen fractions. Samples were scanned and resulting 3-D digital images were analyzed with an algorithm routinely used for such applications. A series of virtual, idealized scaffolds were also created so as to illustrate the algorithm's analysis approach and to demonstrate its validation. The program notably allowed for the calculation of accessible void fractions over a range of minimum connection sizes. In samples obtained experimentally, almost 100 % of the porous volume was interconnected with outside air for connections higher than or equal to 20 µm in their smallest dimension. In materials prepared from 75 vol. % of porogen, the accessible void fraction decreased down to 78 % if only those connections higher than or equal to 260 µm were considered. The relationship between accessible void fraction and connection size varied as a function of porogen content. The interconnectivity parameter described in this study might have implications for cell migration and tissue growth into scaffolds when considering tissue engineering applications. Moreover, Filmon et al. measured the pore volume and interconnectivity of PHEMA macroporous networks by means of µCT and image processing. The authors used three different macroporogens: sugar fibers, sucrose crystals, and urea beads. [START_REF] Filmon | Non-Connected versus Interconnected Macroporosity in Poly(2-Hydroxyethyl Methacrylate) Polymers. An X-Ray Microtomographic and Histomorphometric Study[END_REF] The observation of the sample consisted in using a magnification of 15 and a voxel size of 20 µm.

Transport properties of porous polymeric materials

The permeability of porous materials is a physical property describing the relative ease in which fluids can flow through materials. Permeability (k, m 2 ) is the single-phase fluid conductivity of a porous material that describes the structure regardless of sample size and fluid used. The permeability of porous materials depends on five important parameters: porosity, pore size and pore size distribution, pore interconnectivity and pore orientation.(56) Darcy's law states that permeability k can be calculated from Eq. (1.4) in which Q stands for the volumetric flow rate (ml/s), l the length of the specimen through which the fluid flows (m), µ the viscosity of the fluid (Pa.s), P the pressure difference through the sample (N/m 2 ), and A the sample cross-sectional area in the direction of the flow (m 2 ).

k = Qlµ P A (1.4)
In the field of tissue engineering, permeability of porous scaffolds plays an important role in the nutrient and waste transport within the network. There are two possible mechanisms for the transport of metabolites from cells through the scaffold i.e. diffusion and transport through capillary networks formed throughout the scaffold via angiogenesis, in the case of in vivo applications. Fluid flow, and therefore permeability, allows the transport of nutrients to and waste away from cells. The influence of fluid flow has been studied in tissues as bone,(

PGA: poly(glycolic acid); PLLA: poly(L-lactic acid); PLA: poly(lactic acid) Table 1.3: Permeability of various natural and synthetic porous polymers.

Following O'Brien's studies, collagen scaffold permeability was found to be in the 10 -13 m 2 order of magnitude. Such materials had a very high porosity equal to 99.5 %. [START_REF] O'brien | The Effect of Pore Size on Permeability and Cell Attachment in Collagen Scaffolds for Tissue Engineering[END_REF] As expected, the authors demonstrated that permeability increased significantly with pore size: k = 6.10 -4 m 2 to 1.38 × 10 -13 m 2 for 96 and 151 µm pore size, respectively. The only previous value found in the literature (Table 1.3) for a collagen material corresponded a collagen gel. [START_REF] Ramanujan | Diffusion and Convection in Collagen Gels: Implications for Transport in the Tumor Interstitium[END_REF] The values for permeability with these gels ranged from 10 -16 to 10 -15 m 2 . The collagen scaffold characterized in O'Brien's studies was 2-3 orders of magnitude more permeable than a collagen gel, an expected result if one considers the lower porosity of the gel. It is worth mentioning that the standard deviations were drastically high, suggesting the lack of reproducibility of the experiments. Scaffold permeability was also observed to decrease with increasing compressive strain.

The device used in this study, and schematically represented in Figure 1.8, permitted to apply a constant pressure on the material so as to measure the flow-rate through it. The difference between this experimental device and others in the literature relies on the mechanism by which the material is secured. Typically, such devices have used adhesives or bonding cement to hold the material sample in place.

Computation of fluid flow through porous materials

The modeling of flow through doubly porous materials requires some fundamental questions, such as the role of each porosity level on the macroscopic permeability, and the optimization of the microstructure to specific applications.

The study of the relation between the microstructural geometry and the macroscopic effective permeability of porous media is an attracting challenge in many areas such as geological science (69; 70; 71) and biomechanics. [START_REF] Daish | Estimation of Anisotropic Permeability in Trabecular Bone Based on MicroCT Imaging and Pore-Scale Fluid Dynamics Simulations[END_REF] The interactions between a free fluid flow with porous media found many applications such as fractured reservoirs (71), modeling of lung alveolar sheet [START_REF] Tang | Fluid Movement in a Channel With Permeable Walls Covered by Porous Media: A Model of Lung Alveolar Sheet[END_REF], insulation materials (75), flow through spheres or packed bed of particles [START_REF] Joseph | Lubrication of a Porous Bearing-Stokes' Solution[END_REF], convection in porous materials [START_REF] Beckermann | Natural Convection Flow and Heat Transfer Between a Fluid Layer and a Porous Layer Inside a Rectangular Enclosure[END_REF], modeling of liquid infusion into fibrous media under compaction (79), etc.

The asymptotic expansion method is widely used for the development of homogenization techniques to multiporous structures. (71; 12; 13; 14; 83; 84) The approach is applied to doubly porous materials exhibiting fractures. If the fractures are larger than pores, a double upscaling approach is used and the macroscopic permeability is A constant pressure is prescribed at infinity (far away from the spherical pore). The analytic solutions are provided inside the pore and in the infinite matrix. The authors take advantage of these close form solutions to derive estimations of the permeability of the biporous solid. Interestingly, they also identify the permeability of an equivalent porous solid in the Stokes region. Later, the results have been extended to the case of a spheroidal cavity by Rasoulzadeh et al. [START_REF] Rasoulzadeh | Effective Permeability of a Porous Medium with Spherical and Spheroidal Vug and Fracture Inclusions[END_REF] The spherical and cylindrical shapes are particular cases of the spheroid. Moreover, the case of thin cracks can be simulated by flat spheroids. The results obtained by the authors could then be applied to fractured porous media. Analytic solutions for the Stokes flow inside the spheroid are derived as a set of semi-separable solutions in terms of the ellipsoidal coordinates. Note however that the solution is given by an infinite series and no close form solutions could be delivered for the effective permeability. Another approach based on the Brinkman equation has been developed by Silva and Ginzburg.( 97) From a practical viewpoint, the Brinkman model can simply be regarded as a transmission model, bridging the limits of open to very porous media, described by Stokes and Darcy's law, respectively. Specifically, by considering the fluid as a porous medium with large permeability, the fluid/porous composite region can be treated with only the Brinkman equation, which helps circumventing the use of suitable conditions for the interface. They solve the Brinkman equation for a composite cylinder, i.e. two concentric cylinders in which the flow obeys to the Brinkman equation but with different coefficients in the core and in the coating. Close form solutions are then derived when the composite cylinder is subjected to a uniform velocity field at its boundary. By adapting the values of the Brinkman coefficients in the core and in the surrounding coating, various situations could be considered: the flow around a permeable porous cylinder or the flow in a porous solid with a cylindrical cavity. All these studies therefore considered idealized microstructure. The resolution of the coupled Darcy-Stokes for 3-D complex porous microstructures is still lacking in the literature.

Computation of fluid flow based on real µCT microstructures

A vast number of numerical studies has attempted to examine how pore-scale geometrical features (such as pores interconnections) influences the fluid flow in porous media by computing the permeability on 3-D images coming from real porous media. The state of the art permitted to notice that the Finite Element Method (FEM) and the Lattice Boltzman Method (LBM) were the most used techniques to simulate fluid flow through porous media.

For instance, Koivu et al. reported the study of transport properties of porous fibrous materials such as plastic fiber mat and hardwood. [START_REF] Koivu | Transport Properties of Heterogeneous Materials. Combining Computerised X-Ray Micro-Tomography and Direct Numerical Simulations[END_REF] The regular cubic grid used in the numerical solution was based directly on the X-ray microtomography 3-D images (with a voxel resolution of 4.84 µm) of the materials. The slices were processed by using a non-linear anisotropic diffusion filter and thresholded with a seeded region growing method. The permeability was computed by using the LBM which was based on solving the discrete Boltzmann equation instead of using the standard continuum flow equations. A simple regular grid containing the center points of the cubic unit cells (voxels of the tomographic image) was used. The major uncertainties of this method were the resolution of the tomographic image that provided the numerical grid and the small sample size provided by the imaging technique. The maximum deviation between the experimental method and the computational one was less than 30 %. This methodology permitted to determine all components of permeability tensor and was particularly useful in cases where direct measurements were not achievable. Furthermore, according to the study of Sun et al., a multiscale framework was extracted by using computational techniques from digital images of natural sandstone. [START_REF] Sun | Multiscale Method for Characterization of Porous Microstructures and Their Impact on Macroscopic Effective Permeability[END_REF] The study of the images of 6 µm resolution consisted of:

• the development of level sets to determine the 3-D medial axes of the pore structure,

• the use of a shortest-path algorithm to evaluate the geometric tortuosity,

• the evaluation of connected pore space by a growing-region algorithm. Porosities comprised between 14 % and 21 % were found. After the image processing, a 2-scale homogenization of the effective permeability using LBM for the mesoscale calculation and FEM for the macroscopic simulations was elaborated. The strategy is illustrated in Figure 1.9. The connected pore space in the sample was decomposed into simple domains for the direct mesoscale computation. Then, permeability tensors were calculated for each subdomain using LBM on the connected porosity only.

Then, each subdomain was represented geometrically by finite elements with permeability values obtained from the previous step. Finally, FEM was used to compute the effective permeability of the entire material, according to the heterogeneities implied in each subdomain. The limitation of this method was that the occluded porosity could be interpreted as connected in the computation and generate errors. The permeability is an important parameter to take into account in the study of physiological processes occurring in trabecular bones. Indeed, permeability is directly related to fluid velocity and fluid pressure in the trabecular bone pore space. Such parameters have been identified as important mechanical and biological cell stimuli. In this context, Daish et al. reported the calculation of the anisotropic permeability on bovine trabecular bone specimen by combining µCT and pore-scale fluid dynamic simulations. [START_REF] Daish | Estimation of Anisotropic Permeability in Trabecular Bone Based on MicroCT Imaging and Pore-Scale Fluid Dynamics Simulations[END_REF] µCT images had a resolution of 10 µm while a RVE of about 2 3 was proved to be suitable for a correct simulation. Mesh convergence studies showed that accurate fluid flow properties were obtained for a mesh size above 125 000 elements. Volume averaging of the pore scale fluid flow properties allowed the calculation of the apparent permeability matrix of trabecular bone specimens. The numerical results showed a good agreement with experimental data. The authors also noticed that bones sample taken from long bone generally exhibit a larger permeability in the longitudinal direction.

The fact that all coefficients of permeability matrix were different from zero indicated that bone samples are generally not harvested in the principal flow directions. The porosity was comprised between 83 % and 86 % while permeability values ranged from 0.22 to 1.45 × 10 -8 m 2 . The steps of the image analysis and computation consisted of:

• the segmentation: the contrast values needed to be converted into average fluid volume fractions in each computational cell (φ) , a cell with φ = 1 contains only fluid while a cell with φ = 0 contains only solid phase. A cell with values comprised between 0 and 1 contains both fluid and solid regions. The first segmentation of the solid-fluid interface region was performed by applying the Otsu multi-thresholding method based on the global property of the histogram. This segmentation was based on two points identified in the frequency density plots: t solid and t f luid . An optimal set of thresholds was selected by maximizing the variance between the two classes.

• the selection of the RVE: the authors investigated the porosity to select a suitable RVE. The porosity was computed for 18 different cubic RVE sizes with edge length comprised between 100 µm and 5000 µm.

• the mesh creation: an algorithm transformed each 9.87 µm edge-length voxel into a cubic volume defined by 8 nodes.

• the numerical simulations of fluid flow in trabecular bone space.

Trabecular bones have also been studied by Janc et al. who combined µCT with FEM and compared the results with homogenization techniques such as Mori-Tanaka, Self-Consistent and Incremental Scheme.(73) Finally, Piller et al. used the pore-scale Direct Numerical Simulation to compute the velocity and pressure fields in the digital pore space, reconstructed from high-resolution X-ray tomography.(102) A commercial Finite Volume Fluid dynamic solver was used to simulate incompressible laminar flow through reservoir rock and operated on voxelbased computational meshes. The methodology was validated by reproducing literature results on monodisperse periodic arrays of spheres. The hydraulic permeability of real porous media was compared with laboratory experimental measurements. The image processing consisted in using standard 3-D watershed-transform techniques. First, the distance-transform (DT) was applied to the binary volume. The map supplied each voxel of the pore space with the distance to the nearest obstacle voxel (i.e. the distance to the interface void/rock). The watershed transformation applied to the DT considers the magnitude of the DT-gradient and identifies voxels having the highest gradient magnitude as watershed lines, which represent the 3-D region boundaries.

In order to define the computational domain, each voxel corresponded to a mesh element. This strategy avoided generating the surface mesh and reduced dramatically the time required to set up the CFD analysis. Additionally, it did not require the use of sophisticated mesh generation packages.

The pore-grain interface can be identified by the list of linear indexes of the voxels lying on the border of the pore space and connected to grains according to a giving voxelneighborhood connectivity. The identification of the interface fluid voxels was required for subsequent selection of the boundary surface where the boundary conditions are to be enforced. The computational domain and the related computational mesh were trivially generated by associating each voxel to a hexahedral computational cell.

In order to overcome the limitations of µCT images resolution, Soulaine et al. proposed an original Darcy-Brinkman approach, in which a Darcy model is considered for the sub-resolution porosity, and the Stokes equation is applied to compute flow in the resolved pore-space. [START_REF] Soulaine | The Impact of Sub-Resolution Porosity of X-Ray Microtomography Images on the Permeability[END_REF] This approach allowed for quantifying the impact of this sub-voxel porosity on the pore-scale flow simulations. Indeed, some pore structures were much smaller than the image resolution, and they could not be represented explicitly in the image. The results showed that even with only 2 % of microporous regions, the sub-voxel porosity could play an important role in the flow distribution in the pore network with significant consequences on the computed permeability tensor.

In particular, some microporosity regions served as bridges across macropores, and if such bridges were not represented properly into the modeling, important connectivities could be artificially removed, and the flow capacity of the porous medium could be underestimated.

All these studies showed the importance of the image resolution to reconstruct the most realistic RVE leading to a satisfying computation of the fluid flow through porous media. Moreover, FEM is the most used method and to the best of our knowledge, no Fast Fourier Transform calculation has been realized on µCT microstructures of biporous polymers.

Conclusions

This critically selective literature survey permitted to highlight the broad diversity of applications for biporous polymeric materials. They have been notably implemented as scaffolds for tissue engineering and filtration devices for molecule separation.(10; 15; 16; 31; 66; 67; 68) Porous materials are known for their ability to permit the flow of an external fluid through the pore interconnections. In the context of tissue engineering, doubly porous materials are of particular interest as large pores permit the growth of bone cell, vascularization and nutrient delivery to the center of the regenerating tissue, while small pores are useful for the cell adhesion. Furthermore, reservoir rocks are also considered as biporous media: the first porosity level corresponds to the pores between grains of the matrix, while the second one is associated to the fractures. Porous polymeric materials could thus be considered as a model to investigate the fluid flow through soils or rocks. This chapter was also devoted to investigate the main methodologies so far implemented for designing doubly porous polymeric materials. Five major techniques have particularly attracted our attention: the double porogen templating approach, Temperature-Induced Phase Separation (TIPS), the polyHIPE approach, electrospinning, and gas foaming. The double porogen templating approach was found to be the most facile to carry out and versatile, with a fine control of the morphology and very high porosity ratio. For these reasons, this technique has been chosen to design doubly porous polymeric materials in our experimental work (Chapter 2). Additionally, the use of NaCl particles as porogen template is interesting thanks to their availability and low cost, and their easy extraction which only requires water washing. Furthermore, the complexity of biporous networks requires a careful visualization of the 3-D microstructure. In particular, pores interconnectivity, depth and shape need to be precisely investigated because of the crucial role they play regarding transport properties. The literature points out an inherent weakness of SEM, as this imaging technique is typically limited to 2-D measurements on relatively small fields of view. Consequently, only a fraction of a sample may be viewed at once, and it may become difficult to distinguish pores from interconnections.

µCT data is thus a more appropriate technique to give precise information about the 3-D microstructure. Indeed, this technique typically profits from the ability to view arbitrarily oriented slices through the three-dimensional volume represented by the data, and from the capacity to extract features of interest selectively and display their perspective views using methods of isocontouring or volume rendering. Additionally, no study has reported the investigation of doubly porous structures by means of µCT but only monoporous networks. This imaging technique, in conjunction with synchrotron tomography will be mainly used in Chapter 3 for the morphological characterization of doubly porous materials.

Finally, this bibliographic chapter demonstrated that permeability of doubly porous materials depends on porosity, pore size and pore size distribution, pore interconnectivity and pore orientation. Several studies reported on the determination of permeability (k ) of collagen or PLGA porous materials devoted to tissue engineering but not on PHEMA porous polymers. One of the objectives of Chapter 2 will be to study experimentally the transport properties of these materials. The experimental data will be compared to the computational permeability in Chapters 4 and 5.

Chapter 2

PHEMA-based doubly porous materials prepared by the double porogen templating approach

This chapter is dedicated to the preparation and physico-chemical characterization of PHEMA-based doubly porous materials via the double porogen templating approach. Sintered or non-sintered NaCl particles in conjunction with a porogenic solvent have been used as porogen agents. The obtained porous frameworks have been then characterized in terms of porosity ratio, pore size distribution, and interconnectivity between adjacent pores by means of scanning electron miscroscopy (SEM), mercury intrusion porosimetry (MIP), and gas sorption porosimetry. The next part of this chapter focuses on the investigation of the swelling behavior of PHEMA-based doubly porous materials in the presence of water. Finally, the transport properties of such biporous networks has been studied by measuring the water permeability. The microstructure of these materials will also be deeply characterized by means of X-ray microtomography in Chapter 3. The obtained images will serve as input data for the computation of fluid flow through the as-obtained porous media in Chapter 4 and Chapter 5. In the present chapter, for the sake of clarity, "interconnected networks" or "materials exhibiting interconnected large pores" refer to the materials prepared with sintered NaCl particles, while "non-interconnected networks" or "materials exhibiting non-interconnected large pores" refer to the samples prepared with non-sintered NaCl particles. Moreover, the pores generated after porogenic solvent extraction are called "small pores".

The double porogen templating approach

As previously demonstrated in different studies, NaCl particles have been successfully used to generate macroporous(1) or biporous materials through particle templating combined to a porogenic solvent after a straightforward porogen removal.(2) It should be stressed that previous reports based on the double porogen approach using NaCl as a macroporogen only concerned either non-sintered NaCl particles(3) or sintered ones(2), and no thorough investigation of the influence of nanoporogen or even macroporogen features on pore morphology was provided. Recently, Ly et al. developed a some-how smarter strategy toward doubly porous PHEMA-based networks in which NaCl particles sintered through the SPS technique were used to form a 3-D continuous porogenic template in conjunction with a second porogenic agent consisting of a suitable solvent.( 4) In one such case, the 3-D continuous NaCl particle-based template led to the first level of porosity, while the porogenic solvent ensured a lower porosity level through a phase separation process or syneresis during the polymerization step.(4; 5; 6) At this stage, it is worth mentioning that the porogenic solvent must obviously be miscible with the comonomers for the phase separation to occur during the copolymerization process. Indeed, when the polymerization is at an advanced step, the incompatibility between the solvent and the polymeric network results in the expulsion of solvent molecules from the polymer, thus leading to pores whose sizes depend notably on the nature and concentration of the solvent as well as the crosslinker concentration.( 4)The advantage of this approach illustrated in Figure 2.1, that is based on the concomitant use of two distinct porogenic agents, namely NaCl particles and a porogenic solvent (i-PrOH in the present study), lies in the independent control of each porosity level. In previous studies, Ly et al.thus showed that the use of more and more hydrophobic alcohols as porogens for the preparation of hydrophilic PHEMA-based crosslinked materials led to increasing average pore sizes for the lower porosity level without any significant modification of the upper porosity level.(4) Likewise, an increasing proportion of the porogenic solvent also led to increasing pore sizes related to the lower porosity level, while decreasing the upper to lower porosity ratio. In our study, the porogenic solvent was fixed (i.e. i-PrOH) while the effects of NaCl particles features on the doubly porous morphology were investigated. It is worth mentioning that the pore sizes obtained from MIP measurements are often underestimated when compared to SEM observations. This has already been observed notably with poly(HIPE) samples (7) or when using fused PMMA beads as porogens (5), and it could be explained by the threshold size theory that was reported in the literature concerning MIP measurements on cement-based materials. (8) The original biporous material presented an upper to lower porosity volume ratio equal to 80/20 vol. %. In this case, the resulting biporous material presented interconnected pores and thus a higher porosity ratio, i.e. 87 % compared to 80 % for the corresponding analogue prepared from non-sintered NaCl particles, due to closer particle packing resulting from SPS-mediated NaCl sintering. Samples prepared with non-sieved NaCl particles (between 50 and 500 µm size) were also analyzed by MIP. Materials obtained from non-sintered NaCl particles exhibited a porosity ratio of 89 % while materials obtained from sintered NaCl particles had a porosity ration of 95 %. It can be concluded from all these analyses that sintered NaCl particles permit to obtain polymeric materials with two distinct porosity levels but also with higher porosity ratios than those obtained from non-sintered NaCl particles.

Effect of NaCl particles size on morphology

The porosity of biporous PHEMA-based samples was carefully investigated by varying the size of NaCl particles used to generate the 3-D inorganic template of the porous materials. The choice of NaCl particles size had no significant influence on the lower porosity level as the two pore-forming mechanisms are independent, as mentioned above. The amount of porogenic solvent was kept constant, while three different fractions of sieved NaCl particles, i.e. 125-200 µm , 200-250 µm, and 250-400 µm, were used to prepare biporous PHEMA-based materials that were then characterized finely in terms of porosity features. MIP profiles shown in Figure 2.5 logically showed a correlation between an increase in the porogen size and an increase in the size of resulting pores and pore volumes. For the lowest NaCl size (i.e., 125 -200µm), a first level of porosity centered on 40 µm was observed with a 77 % porosity ratio. For the intermediate particle size (i.e., 200-250 µm), the pore size was nearly centered on 60µm with an 87% porosity ratio. Finally, biporous materials prepared with the highest porogen size (i.e., 250-400 µm) exhibited a pore size of about 70 µm and a porosity ratio equal to 88 %. Again, it is noteworthy that the porogen size was higher than the corresponding pore size. This is actually quite often observed with porous materials for which the pore size determined by MIP is generally underestimated when compared to SEM observa-tion, likely due to the threshold size theory.(7; 8) Likewise, specific surface area values increased gradually with increasing porogen sizes, while the upper to lower porosity volume ratio was nearly constant (Table 2.2). This investigation permitted to highlight that the upper porosity level can be independently controlled by varying the size of NaCl particles template, without altering significantly the lower porosity level.

Effect of crosslinker on morphology and mechanical properties

We previously demonstrated that the porogen template features could control the morphology of doubly porous materials. The effect of the length of the crosslinker on morphology, swelling and mechanical properties are now studied. Several studies have investigated the impact of crosslinking density on PHEMA swelling.(9; 10) Five different crosslinking agents from 1 to 8 ethylene oxide units (EGDMA, DEGDMA, TriEGDMA, TetraEGDMA or PEGDMA) were added to the polymerization mixture. They were chosen to provide a wide range of chain lengths and degrees of flexibility.

The corresponding monoporous and doubly porous materials were elaborated and subjected to SEM, MIP, and DMA analyses. SEM micrographs (Figure 2.6) showed that monoporous materials presented a uniform porosity with a pore size around 1 µm. Thus, the length of the crosslinker seemed not to have any significant influence on the small pores sizes as this parameter is rather governed by the porogenic solvent. The average pore size was estimated by MIP analysis (Table 2.3) and a constant value of about 5 µm was determined. Moreover, it was observed that the polymerization of HEMA and DEGDMA in the presence of i-PrOH gave rise to PHEMA-based polymeric materials exhibiting a globular structure generally observed when phase separation occurs, while samples prepared with TriEGDMA, TetraEGDMA, and PEGDMA crosslinkers presented more collapsed structures in which it was more difficult to distinguish one such typical globular structure. This is more likely due to a collapse of the porous network during the drying step resulting from the increase in flexibility of the crosslinker backbone. This was confirmed by MIP results that showed a gradual decrease in the porosity ratio and the total pore volume, while increasing the crosslinker chain length. While the porosity ratio was equal to 60 % for the crosslinker short chain (i.e. EGDMA), it decreased down to 26 % for crosslinker longer chain (i.e. PEGDMA).

The fact that the pore size remained constant while the porosity ratio decreased showed that the small pores were less and less interconnected as the size of the crosslinker chain increased. Furthermore, the doubly porous materials prepared with various crosslinkers (Figure 2.7) presented very different structures depending on the crosslinking agent used during the polymerization. For crosslinkers presenting the shorter chains (i.e. EGDMA and DEGDMA), large and small pores separated by a thin polymeric wall could be easily distinguished. In the case of TriEGDMA-based materials, the polymeric envelope which separates both levels of porosity was thinner and the obtained polymer seemed to be less porous. Finally, in the case of crosslinkers presenting longer chains (i.e. TetraEGDMA and PEGDMA), collapsed structures were obtained in which it was very difficult to distinguish large and small pores. The length of the crosslinker backbone led to some elasticity of the materials, causing notably some collapse of the porous network. In the same way, Dynamic mechanical analyses (DMA) were performed on the samples in order to study the effect of the crosslinker chain on the mechanical properties associated with porous materials. DMA consists in applying a dynamic strain to the polymer following Eq.

(2.1):

γ(t) = γ 0 sin ωt (2.1)
that induces a dynamic stress. By applying a very small strain in the sample (0.1 %), the mechanical response of the material is found in the linear regime. The ratio between stress and strain is obtained from the dynamic modulus that can be decomposed into two parts: the storage modulus (G') and the loss modulus (G"), as described by Eq.

(2.2):

τ (t) = γ 0 .(G . sin ω.t + G . cos ω.t) (2.2)
A part of the applied energy is elastically restored by the polymer. This energy is associated to the storage dynamic modulus G'. The polymeric material also absorbs a part of the applied energy which is dissipated as heat; this energy is associated to the loss dynamic modulus G". The ratio between the loss and the storage moduli is called the loss angle or:

tan(δ) = G G (2.3)
which is the out-of-phasing angle between stress and strain. It represents the energy proportion dissipated as heat by the polymer in a sample.

The quantitative modulus, the α relaxation, the quantification of gelation, and the ratio of loss to storage moduli at defined temperature may be obtained using DMA.

This study is particularly focused on the effect of the crosslinker on the α relaxation temperature (T α ) and the storage modulus G' of porous polymeric materials. DMA is widely used to evaluate the effect of crosslinkers on the molecular relaxations. Oysaed et. al investigated the effect of dimethacrylate crosslinkers on the dynamic mechanical properties of multiphase acrylate systems formed by autopolymerisation of a mixture of liquid methacrylate monomers and poly(methyl methacrylate).( 12) Moreover, Cascone demonstrated that Tg values for cross-linked gelatin were higher than their uncrosslinked counterparts. (13) In this study, the storage modulus G' as a function of time, reported in This section demonstrated that the morphology and the thermo-mechanical properties of porous networks could be controlled by the crosslinker length. The use of EGDMA permitted to obtain the most porous network with a well-defined double porosity. Nevertheless, increasing the crosslinker length induced the collapse of the porous structure, along with the decrease in the T α value associated with the material.

Determination of apparent density and porosity ratio values

It has been shown that the apparent density of a porous material has an effect on its mechanical strength and on its permeability. (14) This parameter is commonly estimated in soil characterization (15; 16; 17) but also in the research field devoted to scaffolds for tissue engineering. (14; 18; 19; 20) This section is dedicated to the determination of the apparent density of the doubly porous materials by using the equilibrium mass swelling ratio (q w ) and the porous volume (V pores ). This methodology was notably used by Rohman et al. regarding investigations of poly(D,L-lactide)/poly(methyl methacrylate) (PLA/PMMA) interpenetrating polymer networks used as precursors to PMMA-based porous polymeric materials. (21) They demonstrated that the porosity ratio was related to the apparent density (cf. appendix A for experimental details). In this study, dichloromethane was selected as the solvent of choice because it is considered as a non-swelling solvent. Indeed, non-polar dichloromethane has no specific affinity with the polar PHEMA-based polymeric matrix. The solvent can thus only fill the pores in order to estimate the porous volume and, as a consequence, the apparent density. Using water would give inaccurate results because of the high swelling ratio of doubly porous PHEMA-based materials in water. Table 2.6 provides the apparent density values (d app ) and the porosity ratios by using q w , V pores and the true density of the bulk material. The cylindrical shape of the samples permitted to deduct d true from the mass and the volume of the bulk material. d true was then equal to 1.107 g.cm -3 . This density value was very close to that measured by Le Droumaguet et al. on a similar Poly(HEMA-co-EGDMA) material.(5) Indeed, the authors of this study obtained a density of 1.170 g.cm -3 by Helium pycnometry.

In this study, we chose to provide an interval of values instead of calculating a single one because of the uncertainty of measurement due to the balance (10 % uncertainty).

The determination of the apparent density and the porosity were then calculated by using the equation of distribution of uncertainties explained in appendix A.

The interconnected biporous network had a d app value equal to 0.075 (± 0.005) g.cm -3 , while the apparent density of its non-interconnected counterpart was equal to 0.15 (± 0.015) g.cm -3 range. As expected, d app value was twice higher in the case of noninterconnected networks because of a higher polymeric matrix volume fraction. Furthermore, interconnected porous networks exhibited a porosity ratio around 91%, while the value was equal to 85.5% for non-interconnected networks. We thus noticed that these values were in good agreement with the MIP measurements of materials prepared with 200 -250µm NaCl particle size.

Biporous network configuration

Apparent density Porosity ratio a (%) Porosity ratio b (%) Interconnected 0.075 ± 0.005 91.5 ± 0. The determination of the apparent density values allowed to evaluate the void proportions in these complex biporous architectures and also to understand the effect of the pore interconnectivity on the apparent density. Moreover, the similarity between porosity ratio deduced from d app and measured by MIP strongly confirmed the efficiency of the method.

Study of swelling properties in water

Several studies have been performed regarding the swelling behavior of PHEMA-based materials in the presence of water. (9; 22; 23) In this section, the swelling behavior of biporous networks was evaluated by measuring the evolution of the water uptake during 4 h. Mabilleau et al. previously investigated the effect of the crosslinker backbone length on bulk PHEMA swelling. (9) The authors highlighted the effect of the crosslinker on the water uptake. PHEMA crosslinked with PEGDA exhibited the highest value of water uptake (i.e. 56 %), while the lowest values were obtained for DVB-PHEMA and EGDMA-PHEMA (40 % and 42 %, respectively). The water uptake of polymer networks can be expressed following Eq. (2.4):

Water uptake = W w -W d W d 100 (2.4)
where W w and W d stand for the wet mass and the dry mass, respectively.

The study was performed on interconnected and non-interconnected biporous networks prepared from NaCl particle size between 200 and 250 µm. Figure 2.10 represents the water uptake of interconnected and non-interconnected biporous networks as a function of time. In both cases, two plateaux were observed that probably corresponded to the water filling of each porosity level. Indeed, it was likely to happen that water entered the large and small pores before binding with the polymer. In the case of interconnected networks, the water filling began with the channels between larger pores and then larger pores during 20 min. Then, the water uptake reached a constant value equal to 1500 % during 30 min. In a second step, water drained for 1 h through smaller pores before reaching a second plateau. At the end of the experiment, the water uptake was approximately equal to 2400 %, and the mass of the wet material was almost 24 times higher than the dry sample. In the case of non-interconnected networks, the evolution of the swelling degree was similar excepted for the second plateau which was reached less rapidly than in the case of interconnected networks (180 min vs. 120 min, respectively). This difference was likely due to the lack of interconnections between larger pores which should facilitate the draining of water. The water uptake was then lower in the case of interconnected biporous networks. Indeed, the material incorporated 1200 % of water at the end of the experiment.

Previous studies showed that a bulk PHEMA-based polymer (HEMA/EGDMA, 96/4 mol. %) incorporated 58 % of water after 2 days.(9)Moreover, Lokman et. al found that poly(HEMA-MAH) porous monoliths with 78 % porosity were able to incorporate 68 % of water.( 23) Finally, Gomes et. al showed that the water uptake of scaffolds prepared with ethylene vinyl alcohol and cellulose acetate could reach values of around 30 %. (24) For the sake of comparison, we performed this study on monoporous materials exhibiting large pores or small pores. The swelling degree as a function of time is reported in Figure 2.11. As expected, a monoporous network incorporated less water than a biporous one. We could notice that the equilibrium swelling degree was reached after 10 min in the case of a material with small pores (350 % water uptake), 40 min for a material with large interconnected pores (275 % water uptake) and 180 minutes for large non-interconnected pores (50 % water uptake). It is important to mention the presence of one single plateau in the case of monoporous samples. This observation permitted to confirm the hypothesis in which the two porosity levels were filled gradually in the case of biporous networks. In the case of materials exhibiting only small pores and samples exhibiting interconnected large pores, a constant swelling degree was reached rapidly, whereas water had more difficulties to drain through the materials with non-interconnected large pores. We could explain this phenomenon by the lack of interconnections between the larger pores. Moreover, the absence of smaller pores reduced the path of the fluid through the material. Furthermore, the thin resin formed around the material during the polymerization (due to the absence of porogenic solvent) could also slow down the passage of water. As expected, the lowest swelling degree was obtained in the case of non-interconnected monoporous network. This section highlighted the considerable capacity of biporous PHEMA materials to incorporate water compared to bulk and monoporous polymers. Indeed, these complex structures were able to incorporate until 2400 % of water in a few hours, while monoporous counterparts incorporated 300 %. In the literature, it was demonstrated that the water uptake of scaffolds and porous monoliths did not exceed 68 %. Moreover, the pore interconnectivity increased significantly the water uptake.

Experimental investigation of transport properties

The permeability of porous materials as scaffolds for tissue engineering applications controls the flow of nutrients in and waste out of the scaffolds as well as the influence of the pressure fields within the material. (25) In this study, we propose to evaluate the effect of pore interconnection and size on the permeability of biporous polymeric networks. The hydraulic conductivity K was measured by following Darcy's law in a Mariotte device at atmospheric pressure. The permeability values k (m 2 ) were then deduced and reported in Table 2.7 and Table 2.8 for biporous and monoporous networks, respectively. The measurements were realized with interconnected and non-interconnected networks prepared with NaCl particle sizes ranging from 125 to 400 µm. It was notably noticed that k was higher in the case of a biporous interconnected network (in the 10 -12 m 2 range) than for biporous non-interconnected structures (in the 10 -13 m 2 range). As expected, the permeability increased with NaCl particles size. The highest value was obtained for an interconnected network prepared with NaCl particle size between 250 µm and 400 µm. We can conclude by this study that water flows faster when larger pores are interconnected. The values for interconnected and non-interconnected networks were close and we could make the hypothesis that water flows also through smaller pores. This hypothesis was checked by measuring the permeability of a material exhibiting only small pores. Its permeability k was equal to 4.10 -13 m 2 , which was very close to the values obtained with a biporous network exhibiting non-interconnected pores. In this system, we supposed that water flows mainly through small pores since large pores are less interconnected. Permeability study on monoporous networks with large pores (Table 2.8) showed that the permeability of a monoporous and a biporous interconnected network are very close (around 10 -12 m 2 ). Interconnected large pores play thus an important role in transport properties. Otherwise, the lowest k value was obtained with a monoporous non-interconnected network (around 10 -15 m 2 ). We could explain it by supposing that the absence of small pores and interconnections between large pores slowed the fluid path. Finally, the permeability increased with NaCl particles size as in the case of biporous networks. Moreover, we compared the experimental permeability of polymers in the literature measured by following Darcy's law. We found that collagen gel permeability was around 10 -16 m 2 with a porosity of 4.5 % which is lower than our PHEMA porous polymers.(26)Interestingly, polyacrylamide gel permeability was closer to PHEMA noninterconnected monoporous network value, as it was between 4.10 -15 and 1.3.10 -14 m 2 (with 3 % porosity ratio).

NaCl particles size (µm) Permeability k 1 (10 1 Calculated from k = (Kη)/(ρg) and K = (Q.L)/(h.S) where K(m.s -1 ), η(P a.s), ρ(kg.m -3 ), g(m.s -2 ), Q(m 3 .s -1 ), L(m), h(m), S(m 2 ) stand for the hydraulic conductivity, the volume flow rate, the sample length, the hydraulic load, the section surface of the cylindrical sample, the water dynamic viscosity, water density, and the gravitational constant, respectively. The determination of water permeability of biporous PHEMA-based polymers permitted to understand the role of each porosity level in the transport properties. We could thus conclude that permeability increased with NaCl particles size and that small pores as well as interconnections between large pores had a significant effect on the fluid flow. Experimental k values will be compared to the theoretical values obtained from numerical methods in Chapters 4 and 5.

Conclusions

In this chapter, doubly porous polymeric materials with a very high porosity ratio (up to 90 %) were elaborated via a simple route named double porogen templating approach.

The main advantage of this strategy was that each porosity level could be easily tuned independently. Moreover, we demonstrated the effect of the macroporogenic agent (i.e. NaCl particles) features on the pore interconnectivity, pore size and distribution, and apparent density. The influence of the crosslinker on the thermo-mechanical properties and the morphology of these structures were also investigated. It has been shown that a high chain length led to the collapse of the biporous structure and a lower T α value.

Furthermore, this study highlighted the considerable capacity of biporous PHEMA materials to incorporate water compared to bulk and monoporous polymers, thanks to the high pore interconnectivity. Indeed, these complex structures were able to incorporate until 2400 % of water in a few hours, while the water uptake of scaffolds and porous monoliths reported in the literature did not exceed 68 %. Finally, these biporous networks permitted to evaluate their transport properties that will make them interesting model systems to understand the fluid flow through porous media as scaffolds for tissue engineering and soil mechanics.

Chapter 3

Characterization of PHEMA-based doubly porous materials microstructures by means of X-ray microtomography

In this chapter, X-ray microtomography was used to observe and characterize the 3-D microstructure of the doubly porous networks prepared via the double porogen templating approach (cf. chapter 2). The aim was to investigate the homogeneity of porosity as well as the pore shape and interconnectivity. One of the most important challenges was to optimize the imaging configuration and the experimental protocol in order to obtain the less noisy images and to characterize the two porosity levels. Indeed, the very low density and low X-ray absorption properties of the PHEMA matrix, together with the small dimensions of the microgeometries, made 3-D imaging with tools adapted for denser materials rather challenging. Basically, three different experimental protocols were used to perform the analysis. First, the samples were observed directly by using a classical laboratory X-ray microtomography technique.

After that, the samples were soaked in a contrast agent solution in order to improve the image quality. Finally, a synchrotron X-ray analysis was performed.

Introduction

X-ray microtomography (µCT) is a radiographic imaging technique that produces 3-D images of a material's internal structure.(1) Sequential contiguous images are compiled to create three-dimensional representations that can be manipulated digitally, as schematically depicted in Figure 3.1. Optimal data acquisition and interpretation require a proper selection of scanning configuration, the use of suitable X-ray sources and detectors, appropriate tuning of their parameters, and careful calibration of the apparatus. A particular attention is especially focused on origins of potential artifact and to their modes of elimination. Visualization of µCT data typically profits from the ability to view arbitrarily oriented sections through the three-dimensional volume represented by the data, and from the ability to extract features of interest selectively and display their perspective views using methods of isocontouring or volume rendering. The X-ray source can be an X-ray tube which produces a conical beam or a synchrotron that produces a parallel beam. The intensity of the transmitted beam Id is measured by the detector and compared to the intensity I 0 of the same beam in the absence of the sample. In nowadays CT devices, the detector is a 2-D matrix and the sample is rotated about a fixed rotation axis, thus generating attenuation data for a 3-D set of X-ray beams. As shown in Eq. (3.1), the ratio I d /I 0 is related to the coefficient of linear attenuation µ(x) integrated along the path of the beam:

I d = I 0 exp -µ(x)dx (3.1)
The calibration of the detector consists in measuring the signal without X-rays ("black" or "dark" image) and in the presence of X-rays but without sample ("white" or "reference" image). This calibration permits to connect the grey level of a projection to the ratio I d /I 0 in every position of the detector, and thus to the integral µ(x) dx on the corresponding path. The calibrated value of the pixel in the position (i,j) of the detector, noted p 0 (i, j), is determined by Eq. (3.2):

p 0 (i, j) = p(i, j) -p N (i, j) p B (i, j) -p N (i, j) (3.2)
where p(i, j), p N (i, j) and p B (i, j) stand for the values of the non-calibrated pixel (i,j), the black image and the white image, respectively. A projection thus corresponds to the 2-D spatial distribution of the integral of the linear attenuation coefficient on the whole source-position path of the detector (more precisely of its exponential). In the case of the laboratory µCT, the optical magnification G is defined from the distances between the source and the detector (SD) and between the source and the object (SO). G is expressed in Eq. G can thus be fitted by adapting the position of the sample and of the detector with respect to the X-ray source because of the conical shape of the beam. There is thus a continuity of the magnification on a range defined by the extremities of the sourcedetector (SD) distances and source-objet (SO) distances. For a parallel beam, the magnification is obtained thanks to the addition of an optic system, very similar to an optical microscope. It is thus possible to reach only discrete values of magnification.(2) Synchrotron sources generate however much brighter X-ray fluxes, so that images can be acquired with better signal to noise ratios, in shorter times and with higher magnifications than with laboratory setups. In order to select the most suitable energy for a successful observation of our materials, we simulated the total attenuation (µ/ρ, cm 2 .g -1 ) as a function of the photon energy by inserting the formula of the PHEMA polymer ((C 6 H 10 O 3 ) n ) and the energy range (from 10 -3 to 10 5 MeV) in the Xcom interface provided by the National Institute of Standard and Technology. Then, we deducted the global coefficient of linear attenuation of the polymer, knowing the bulk density (1.1071g.cm -3 ), the length (1 cm), and the porosity ratio (85 %) of the material. It was found that the coefficient of linear attenuation was around 0.9 for a photon energy range from 2.10 -2 MeV to 10 5 MeV, showing that the polymer absorbs a very small amount of X-ray.

Observation of doubly porous polymers without using contrast agent 3.2.1 Qualitative analysis of the µCT images

Preliminary µCT imaging was performed with a spatial resolution of 8.75 µm on doubly porous PHEMA cylindrical samples 1 cm in diameter (cf. appendix B for more experimental details). In this particular case, the NaCl porogens were not previously sieved (grain size between 50 and 500 µm). The 3-D images of samples elaborated from non-sintered (Figure 3.2a) and SPS-sintered (Figure 3.2b) NaCl particles confirmed the surface observations made by SEM (cf. chapter 2) regarding the porous network. Cross sections in Figure 3.2 show a rather homogeneous pore distribution in both samples and a porous network with large pore imprints of sizes similar to those of NaCl particles. The spatial resolution of these images was however too coarse to investigate the structure of the porous matrix in between large pores, which thus appears with an almost uniform grey level. In Figure 3.2b, some bright area could be observed on the periphery of the sample reflecting strongly attenuating constituents, which might be residual NaCl particles potentially entrapped in the material and not removed during preparation. The exploration of the 3-D image of the sample prepared from non-sintered NaCl qualitatively confirmed that the large pores were mostly isolated within the polymer matrix. The large pores morphology presented various shapes, such as pseudo-cubes, spheres, and pyramids. The matrix exhibited some large areas without pores, a few hundred micrometers in size; some thin matrix walls separating pores, or even inside pores, with a thickness close to voxel size, could however also be observed. A few neighbor pores seemed connected; it was however not clear whether they were separated or not by a thin matrix wall, because it could not be resolved by the adopted 3-D image which encompasses the whole samples. Such situations were however seldom. The 3-D µCT image of the sample obtained from the SPS-fused particles, a crosssection of which is provided in Figure 3.2, showed a microstructure with mostly pseudo-spherical pores separated by more regular and thinner matrix walls (less than 50 µm). However the 3-D exploration of these data showed that most pores seemed to be connected to their neighbors. Similar connections could be found for almost all pores in other cross-sections. Strictly speaking, this did not prove that pores were actually connected, as separating walls could have been too thin with respect to image resolution. These observations were however consistent with the expected features of the microstructure of this material. The major differences between both samples, as qualitatively demonstrated by the µCT images, laid in the interconnection between adjacent large pores, the pore shape and the matrix volume fraction. It is noteworthy that the polymer matrix proportion was obviously more important for the material displaying non-interconnected large pores. This could be explained as the consequence of the random packing of the inorganic NaCl grains used as porogens in the preparation of the template associated with the porous materials. This result was also confirmed more quantitatively by experimental measurements of the apparent density (cf. chapter 2) of samples elaborated from nonsintered NaCl particles which was comprised between 0.14 and 0.17 while materials prepared with SPS-sintered NaCl particles showed a range of values from 0.07 to 0.08 density. Such a global density measurement allows to directly compare the first level of porosity (larges pores), if one assumes the lower porosity level (small pores) to be the same in both samples, which sounds reasonable in a first approximation as both pore forming mechanisms might be considered are essentially independent.

Quantitative analysis of the µCT image

Furthermore, the sample porosity obtained from non-sintered NaCl was quantitatively evaluated from its µCT image using the following methodology: the average grey level g av of a large region of interest of the sample, made of 200 slices, was compared to the average grey level of voids g v , and of microporous matrix g m , which could be determined by a manual selection within a 2-D cross section of sufficiently large domains strictly included in the pore or matrix phase. As grey levels in µCT images reflect average X-ray attenuation coefficients of the constitutive phases in presence in the volume associated to a voxel, which in case of two-phase materials are essentially proportional to the volume fraction of constituents within voxels, a simple rule of mixture can be used to evaluate the average porosity of the first level P from these quantities, according to g av = P × g v + (1 -P ) × g m . ( 4) When applied to non-filtered images (see Figure 3.3a for a typical cross-section), this procedure led to an evaluation of P of 64 % for non-sintered samples.

A similar procedure could however not be applied efficiently to the other sample obtained from fused NaCl because the value of g m was difficult to quantify precisely, for two reasons (see Figure 3.3d): first, the matrix domains are much thinner so that it is hard to select large domains made of matrix only, and second, the grey levels within the matrix exhibited larger fluctuations, possibly because of a larger variability of the lower porosity level (as will be confirmed later), or the presence of residual traces of NaCl (as observed near the periphery of the sample). However, assuming a similar lower porosity level in both samples, the porosity of the second material could be deduced from that of the first one and from the ratios of the overall densities. In this way, a porosity of 74 % was found for SPS-sintered samples, i.e. a value very close to that found independently by means of MIP, which is 0.8 × 0.87 = 70%. Finally, let us stress that a more conventional approach to evaluate volume fractions could have been based on a segmentation of images. It would however have been less accurate and possibly biased, because of the high noise level of the images and the voxel size, which is too large with respect to many small features of the microstructure, such as wall thicknesses. To illustrate this, a filtered region of interest made of 200 slices (see Figure 3.3b for a representation of one of these) has been segmented making use of a threshold chosen such that the pore phase, defined as the voxels with a grey level below this value, represent 64 % of all voxels. The resulting image is illustrated in Figure 3.3c which shows significant differences with Figure 3.3a, especially regarding the polymer walls. This holds for images of the material obtained from non-sintered NaCl grains, with larger matrix domains; the situation is even more critical for the other one, with small matrix domains, for which conventional thresholding would lead to meaningless results. Biporous networks prepared from 125-200 µm and 200-250 µm NaCl particles were also analyzed via µCT imaging with a similar voxel size. As shown in Figure 3.4a and Figure 3.4b, the large pores dispersion was homogeneous for both samples but some areas showed larger pores (approximate diameter: 500 µm) due to NaCl particles aggregation during sintering step. For both samples, porosity ratio was again difficult to estimate accurately using either of the above presented methods, because of high noise levels, low contrast difference between the polymer matrix and voids, and too coarse voxel size with respect to microstructural features. In order to investigate more precisely the morphology of these materials, a sample prepared with 200-250 µm sintered NaCl particles was observed via µCT measurements with higher spatial resolution, namely a voxel size of 2.0 µm instead of 8.75 µm. The sample of about 3 mm in diameter, cut within the larger elaborated sample, was held in a plastic straw, which was stuck on the rotating stage. The acquisition of these images was a real challenge because of the low absorption of the polymer with respect to the energies allowed by the sources and sensors. In addition, we have almost reached the limit of the microtomograph in terms of voxel size. Finally, it was difficult to keep the sample still during the experiment because of its light weight. Despite the noise, Figure 3.5 permitted to clearly observe a porous network (corresponding to the lower porosity level) within larger matrix domains, surrounded by uniform thin walls. This thickness of the latter was about 1 voxel in the images, indicating an actual physical width lower than the voxel size. This result was in good agreement with the previous SEM micrographs (cf. Chapter 2), showing a porous structure surrounded by a homogenous polymeric shell. Furthermore, µCT image showed some isolated matrix filaments certainly formed in the thin areas near the necks between sintered grains, or in the cracks in the NaCl particles caused by the SPS sintering step. This very fine structure could not be resolved accurately within the images with a coarser voxel size, which explained the difficulty to segment the latter. To illustrate the high noise level, we quantified the grey level of pixels along a line that passes through the porous matrix by using the "plot profile" tool available on FIJI R . In Figure 3.6, X-axis represents the distance along the line while Y-axis corresponds to the pixel intensity. We could notice that the noise inside and outside the matrix is equivalent. Because of such very high noise levels, it was also difficult to segment efficiently the higher resolution images; the evaluation of volume fraction based on average grey level was also of poor accuracy because of the high image noise level. 3.3 Observation of doubly porous polymers after the incorporation of a contrast agent

Qualitative analysis of the µCT images

In the previous part, the doubly porous materials were observed directly after elaboration without any preparation. The obtained images turned out to be highly noisy and/or insufficiently resolved with respect to some microstructural features of these materials. The high level of noise is due to both the high porosity ratio (up to 90 %) and the low intrinsic absorption of the PHEMA polymer for the X-rays used in the available laboratory tomograph. Indeed, the samples appeared to be almost transparent to the used X-rays and the obtained radiographs had a poor signal to noise ratio.

In order to improve the images quality, we chose to incorporate a contrast agent on the samples before analysis. The so-called Lugol solution, containing I 2 and KI is widely used for µCT observations of biological tissues. (5; 6; 7; 8; 9; 10; 11) For example, Jeffrey et. al used iodine staining to describe the fibrous structure of skeletal muscle of rodents by improving the contrast between muscle and connective tissue. The authors showed that enhancement depended on the concentration of iodine solution, time in solution and specimen size. (9) In this study, six families of samples were investigated: three obtained with the SPS technique, three with a simple orbital shaking of the NaCl particles, with three different large pores size distributions. The used aqueous Lugol solution contained potassium iodide and elemental iodine (2/1 wt. %). The potassium iodide in water dissociates and adding elemental iodine promotes the formation of the tri-iodide ion, enhancing the aqueous solubility of iodine. The solution was incorporated in the porous materials by using two different protocols. In the first one (protocol 1), the samples were previously soaked in the Lugol solution, placed on a filter paper, and washed with deionized water and dried under vacuum (cf. appendix B for more experimental details). The second protocol (protocol 2) consisted in adding drops with a syringe on the materials until the material was completely colored. Moreover, each sample was weighed before the soaking and after the drying in order to estimate the quantity of Lugol solution incorporated in the material for each protocol. Table 3.1 provides the proportion of contrast agent solution in the porous samples. Table 3.1 shows that the proportion of incorporated Lugol solution in the noninterconnected networks was globally higher because these materials presented a lower porosity ratio (cf. Chapter 2).

Moreover, it could be observed that the samples (both interconnected and non-interconnected) incorporated a more constant proportion of staining agent when using protocol 1. Also, adding drops (protocol 2) did not permit a homogenous distribution of the Lugol solution into the samples. We thus selected the materials prepared with protocol 1 to perform the µCT analyses. Typical cross-sections of the 3-D images obtained from local tomography with a spatial resolution of 2 µm, of cylindrical samples with a diameter of 5 mm, cut within the larger elaborated sample, are provided in Figures 3.7, 3.8, 3.9. These images are actually crops of the full images enhancing local features of the materials. It is clearly observed that the contrast was higher and the images less noisy than for the non-labelled materials (cf. section 3.2). However, ring artifacts were observed in the center of the images. This classical artefact in µCT is due to the presence of some defective pixels in the detector, which do not behave linearly and induce a bad quantification of the attenuation of some rays. The position of a ring corresponds to the area of greatest overlap of these rays during reconstruction.

To reduce the ring artefact and the noise in general, a "Smooth" filter in conjunction with a "2-D Anisotropic Diffusion" filter (available in Fiji) were applied on a stack of 1200 slices of each sample. The Smooth filter replaces each pixel with the average of its 3 × 3 neighborhood while "2-D Anisotropic Diffusion" is able to reduce noise while preserving interfaces. Namely, this filter preserves the sharpness of edges. The operator can fix some parameters such as the number of iterations and the number of smoothings per iteration. The size of the smoothing per one complete iteration is proportional to the square root of this number. It is worth mentioning that the slices selected in this study are representative of the global microstructure of each sample. As shown in the all the figures, the Lugol solution is well distributed inside the polymeric matrix. A preliminary experiment has however, shown that Lugol solution does not necessarily bind homogeneously to the matrix leading to the formation of some agglomerates in isolated regions (red circles in Figure 3.11). On the other hand, some dark regions suggested that Lugol has not bonded. We supposed that such unequal distribution of the contrast agent was due to a lack of drying of the samples after the incorporation step. A careful observation of the microstructures revealed three different features:

• the large pores taking the pseudo-cubic shape of the salt and representing the first porosity level. Somehow, pores generated by sintered NaCl particles seemed closer to the salt shape than the pores coming from non-sintered particles.

• the polymeric matrix containing the small pores (second porosity level) generated by the porogenic solvent. The magnification of the images in Figure 3.12 showed that samples obtained with sintered templates exhibited less polymeric matrix due to the smaller space available between NaCl grains. Moreover, the pores seemed larger in the case of interconnected networks.

• the very thin polymeric walls at the interface between the porous matrix and the large pores, or in-between two large pores. Such walls exhibited the same thickness regardless the pore size as shown in Figure 3.13, revealing that the formation of the polymeric walls is a local phenomenon. One exception is for the materials prepared with 125-200 µm sintered NaCl particles that exhibited a larger amount of polymeric wall. Indeed, the smaller the particles the more are the interfaces. Less polymer is thus available to form the porous matrix. The first reason for the presence of such walls, may be the affinity of the polymer with the surface of the grains. The second reason has already been formulated while commenting the images of non-labelled materials (cf. section 3.2) and supposed that the walls were formed by the polymerization of the polymer in fissures inside NaCl grains or at the necks between grains (regions 3 in the figures). This phenomenon is specific to materials prepared from SPS sintered salt because the salt is mechanically stressed during sintering, resulting in cracks.

Finally, we could observe two different phenomena affecting the walls. The first one is the detachment of some walls in all the samples (regions 1 in the figures). The second one is the collapse of the walls because of the low mechanical properties of such materials revealed by the overall withdrawal (regions 2 in the figures). Some regions seemed however more concerned by the collapse because the matrix might be less abundant in such domains. Surprisingly, we noticed that interconnected networks obtained from 200-250 µm NaCl particle size exhibited more collapsed regions. We supposed that the detachment and the collapse of the walls were due to the vacuum drying step during the elaboration of the materials (cf. Chapter 2). Materials brittleness (especially materials obtained from non-interconnected porogen) has actually been noticed during their elaboration process. Some samples brutally broke right after the drying step.

Quantitative analysis of the µCT image

Finally, a segmentation was performed on the images in order to estimate the void fraction of the interconnected (Figure 3.14) and non-interconnected networks (Figure 3.15). The "Threshold" option of FIJI R permitted to calculate 1200 binary slices. The images showed that the geometry of the material in the thresholded images were in good agreement with the ones obtained from experiment, except for the walls in which the average thickness was higher than for the real walls. This uncertainty is probably due to the partial-volume effect that occurred when the material volume is comprised of a number of different substances and the resulting CT values represent some average of their properties. Furthermore, all material boundaries were blurred because of the limited resolution of laboratory µCT, and thus the material in any one voxel can affect CT values of surrounding voxels. Moreover, the second porosity level (small pores) was partially recognized by the thresholding algorithm. Table 3.2 showed the estimation of the void fraction obtained from the segmentation of the images. As expected, the void proportion of interconnected networks was higher than for non-interconnected networks. This result is in good agreement with MIP analysis explained in chapter 2. However, the NaCl particle size effect on the porosity was not highlighted. This is probably due the segmentation technique that modified the walls thickness. In the next section, further 3-D imaging attempts using devices with higher photon fluxes are performed to investigate more precisely the microstructure of such materials.

Investigation of microstructures by means of synchrotron tomography

The use of synchrotron radiation as an X-ray source brought important improvements to the imaging.(12; 13) Synchrotron radiation results from the bending of a highenergy electron beam due to a magnetic field. The main advantage of this technique is that the high flux allows one to resolve very subtle variations in absorptivity and therefore in internal structure. Additional advantages of synchrotron radiation include X-ray beam collimation, which simplifies the tomographic reconstruction algorithm, and the tunability of the X-ray energy to a narrow energy band. Moreover, the use of a monochromatic X-ray beam improves the accuracy of the reconstructed tomographic images by eliminating the issue of energy dependence on X-ray absorption.

In our case, synchrotron imaging is expected to provide much more accurate, less ambiguous, more spatially resolved and much less noisy images, within more reasonable scanning times.

It is important to mention that the experiments were carried out on a new and not yet optimized beamline (Anatomix, Synchrotron Soleil, Saint-Aubin, France) that did not yet include a monochromator. The experiments were therefore carried out using a wide X-ray spectrum determined by the gap of the magnets and a solid filter. The spectrum was thus reduced to approximately 10 keV providing more suitable radiation for polymer absorption compared to laboratory microtomography (sections 3.2 and 3.3). Moreover, all the samples were observed directly after the elaboration without the incorporation of a contrast agent.

The observation was performed on biporous samples prepared from sintered and nonsintered NaCl particles with sizes ranging from 125 to 400 µm (Figures 3.16, 3.17, 3.18). Monoporous materials with large interconnected and non-interconnected pores (200-250 µm) were also analyzed in a second step (Figure 3.19). As mentioned above, the synchrotron tomography permitted to observe the microstructure with 0.65 µm voxel size which is much lower than the scans made by CT (8 µm and 2 µm). Moreover, a double rotation of the materials around the axis permitted to obtain a reconstruction of a larger domain (cf. Appendix B for more experimental details). The images corresponding to these last configuration will not be presented and used in this manuscript because these experiments were run late.

Qualitative analysis of synchrotron images coming from doubly porous networks

X-ray synchrotron tomography permitted to considerably reduce the noise and the ring artifacts without image processing tools such as 2-D anisotropic filter. Moreover, the main advantage is that the use of a contrast agent was not necessary because the beam energy and the spatial resolution were more adapted. The reconstructed volumes were higher (2000 3 pixels and 4000 3 pixels vs. 1700 3 pixels for classical microtomography) as well as the number of radiographic projections (4000/8000 vs. 1440). This technique permitted also to reduce the scanning time from 9 h to few minutes (cf. appendix B for more experimental details).

As observed in the Figures 3.16, 3.17, 3.18, the conclusions that have been made for the images obtained with classical X-ray tomography are confirmed. Indeed, three different features could be observed: the large pores, the porous polymeric matrix and the thin polymeric walls. We could notice that the interconnected materials prepared with the lowest NaCl particles size (i.e. 125-200 µm) exhibited less polymeric matrix. Indeed, as the grains were smaller, these templates would have required more reactants (monomers) to form a sufficient amount of polymer which could fill all the template interstices. Furthermore, the omnipresence of the walls deformation probably induced by the drying is even more visible particularly for non-interconnected networks regardless the salt size used during the elaboration. Such observations also permitted to confirm the reliability of the use of Lugol solution, as the microstructures features are similar with and without contrast agent.

Qualitative analysis of synchrotron images coming from monoporous networks

The materials exhibiting only large pores were elaborated without the use of porogenic solvent (i.e. propan-2-ol). In Figure 3.19a corresponding to an interconnected network with large pores, the non-porous polymeric matrix could be observed and the large pores size was similar to the one obtained with biporous materials (Figure 3.17).

In the material exhibiting non-interconnected large pores (Figure 3.19b), we could observe that the NaCl particles were not completely extracted from the material. This is probably due to the resin that has been formed during the polymerization. This observation permit to explain the low permeability of these materials and their low water uptake compared to the others structures (cf. Chapter 2).

Conclusion

This chapter permitted to investigate the 3-D microstructure of doubly porous polymeric networks by means of µCT and synchrotron microtomography. The different experimental protocols and imaging devices configuration allowed a precise observation of the morphology of such materials as the pores connectivity and shape. Imaging provided also crucial information about the void/polymeric matrix interfaces which play an important role in transport properties. Three microstructural features were observed : the large pores having the NaCl particles shape, the polymeric matrix exhibiting the small pores, and thin polymeric walls. However, the quantitative analysis of the porosity was tricky because of the very thin polymeric walls and filaments which made difficult the segmentation step. Finally, some µCT and synchrotron images will be selected in the next chapters (cf. chapters 4 and 5) as input data for the computation of fluid flow through porous medium.

scale. This double homogenization is only possible when the hypothesis of scale separation is satisfied. Denoting by L the characteristic length of the macroscopic scale, that is typically a dimension of the macrostructure, the scale separation is verified if l 2 << l 1 and l 1 << L. Practically, the upscaling is possible if the ratio between two successive length-scales, l 2 /l 1 and l 1 /L remains inferior to 0.1. A second homogenization is then performed to replace it by a Darcy medium at the macroscopic scale. Due to the double homogenization process, the macroscopic permeability then accounts for both the first and the second porosity.

• First homogenization problem.

At the second porosity scale, i.e. the lower scale, the cubic unit cell has the characteristic length l 2 along each space directions. Its volume is V 2 = l 3 2 . The solid occupies the domain V s2 and the pores the domain V f 2 . The pores are fulfilled by a newtonian viscous fluid and the adherence condition is used at the frontier between the fluid and the solid. This first homogenization problem is well documented in the literature (see for instance Auriault and Sanchez-Palencia (2), Sanchez Palencia (24), Levy (14)). Note also that no slip occurs at the solid-fluid interface. The wall-slip correction is given by the Klinkenberg equation and is significant only at low Knudsen number. In the present case, the pore size is not small enough to makes these slip effects significant. More details about the homogenization of porous media with the wall-slip condition could be found in Skjetne and Auriault works (25) (see also Monchiet et al. (15) for the numerical integration of the Stokes slip-flow problem with the FFT method).

µ∆v -∇p = J ∀x ∈ V f 2 (4.1) div v = 0 ∀x ∈ V f 2 (4.2) v = 0 ∀x ∈ ∀S 2 (4.3)
In these equations, v denotes the velocity field, p is the local pressure in the unit cell, µ is the dynamic viscosity, ∆ is the Laplacian operator, ∇ is the gradient operator and div is the divergence operator. At the boundary of the unit cell, the periodic boundary conditions are used for the velocity field and the pressure. Obviously, the polymer is not periodic, but the consideration of the periodic conditions is used by the requirements of the numerical method based on Fast Fourier Transform (FFT). Note also that the periodicity conditions avoid artificial boundary effects as already observed for polycrystal microstructures (see for instance Nygards and Gudmundson (18)). Since the local pressure is periodic, the average of the gradient of pressure is

< ∇p > V = 0.
The solution of the problem linearly depends on the applied macroscopic pressure gradient J , the local velocity field can then be put into the form:

v = - 1 µ A (2) (x).J (4.4)
where A (2) (x) is the localization tensor which depends on the vector position x in the unit cell. The average of the local velocity leads to the Darcy equation: 2) .J

< v > V = - 1 µ K (
where:

K (2) =< A (2) (x) > V (4.6)
is the permeability tensor associated with the first porosity. Assuming that the material is isotropic, the permeability tensor is K (2) = k (2) I where I is the second order identity tensor.

• Second homogenization problem.

At the first porosity scale (the larger pores), the unit cell is a cube of dimension l 1 along each space directions and its volume is

V 1 = l 3 1 .
It is constituted of a porous solid which occupies the domain V s1 and the pores which occupy the domain V f 1 . The pores at this scale are also fulfilled by the same viscous fluid that at the first scale and the flow is generated by the applied constant pressure gradient J . The double homogenization of doubly porous materials has been studied in a series of papers by Auriault and Boutin.(3; 4; 5) The method is based on a double asymptotic series expansion. At the intermediate scale, the homogenization problem consists in solving a coupled Darcy-Stokes problem. However, in the present study, the normalization differs from that of the papers of Auriault and Boutin since they consider fractured porous microstructures. In their study, both the lower pores and the fractures are interconnected while in the present case, only the smaller pores are interconnected. So, the order of magnitude of the macroscopic permeability is not the same as that in the case of a fractured porous media. This is explained in the next part of this chapter. Note also that, following Auriault and Boutin, we introduce only one pressure for the macroscopic filtration law.

µ∆v -∇p = J ∀x ∈ V f 1 (4.7) div v = 0 ∀x ∈ V f 1 (4.8)
they found a value of 0.1 with the average pore size of 0.013 and 0.027 inches. Numerical studies have also been provided by Sahraoui and Kaviany for 2-D periodic structures made of cylindrical particles and found a value of the slip coefficient closed to 2.( 23) Some other theoretical support of the BJ model has been provided in the literature. For instance, Saffman used a statical treatment and found that the tangential velocity component in the porous solid could be neglected and suggested to use the following condition for the tangential velocity:( 22)

2µν.d(v (f ) ).τ = -αv (f ) .τ x ∈ S 1 (4.15)
This form has been considered in numerous works.(13; 1; 26) A tentative to demonstrate the BJS interface model has been provided by Jager and Mikelic using the asymptotic series expansion method and a scale separation.( 12) Moreover, the authors show that the slip coefficient could be determined through an auxiliary-layer type problem. However, some criticism about some hypothesis made in (12) have been formulated in the paper of Auriault who suggests that the adherence condition could certainly be the better approximation at the interface between the fluid and the solid region:( 6)

v (f ) .τ = 0 x ∈ S 1 (4.16)
The solution of the second homogenization problem also linearly depends on the applied constant pressure gradient and reads:

v = - 1 µ A (1) (x).J (4.17)
Taking the average of the velocity field v over the volume V 1 leads to:

< v > V 1 = - 1 µ K (1) .J , K (1) =< A (1) (x) > V 1 (4.18)
where K (1) is the macroscopic permeability tensor. If the porous solid is isotropic at the macroscopic scale, one has K (1) = k (1) I.

Normalization and simplification of the double homogenization problem

The normalization of the local equations has two main goals. First, we can formulate the unit cell problem independently of some fluid properties such as the dynamic viscosity. Second, it is possible to evaluate the importance of each terms in the set of equations by means of dimensionless numbers. By doing so, we can simplify the problem at the intermediate scale.

• First homogenization problem.

We consider the first set of equations (4.1)-( 4.3) and let us introduce the following change of variables:

v = |J |l 2 2 µ v, p = |J |l 2 p, J = |J |j, x = l 2 x (4.19)
where |J | denotes the norm of the macroscopic pressure gradient J and j represents the direction of J . j is a normal unit vector, |j| = 1. Variable v, p, j and x are dimensionless. Particularly, x varies in the normalized cube V 2 (a square for 2d problems) of dimension 1 along each space direction. By introducing this variables change in the set of equations (4.1)-( 4.3), we obtain the following dimensionless problem:

∆v -∇p = j ∀x ∈ V f 2 (4.20) divv = 0 ∀x ∈ V f 2 (4.21) v = 0 ∀x ∈ S 2 (4.22)
where ∆, ∇ and div are the differential operator related to the dimensionless coordinates x 1 , x 2 , x 3 . The solution v of the set of equations (4.20)-(4.22) reads:

v = -A (2) (x).j (4.23)
where A (2) is the dimensionless localization tensor of the first homogenization problem.

The average of v is:

< v > V = -K (2) .j, K (2) =< A (2) (x) > V (4.24)
Here, K (2) is the dimensionless permeability tensor that is related to K (2) by:

K (2) = l 2 2 K (2) (4.25)
for an isotropic porous solid, we have:

k (2) = l 2 2 k (2) (4.26)
• Second order homogenization problem.

We assume that the pores of the second porosity are not connected. Consequently, the macroscopic permeability is of the same magnitude that the permeability k (2) that is increased by the presence of the first porosity. It follows that the velocity at the intermediate scale is:

v = O k (2) µ J (4.27)
We use the change of variables:

v = k (2) |J | µ v, p = |J |l 1 p, J = |J |j, x = l 1 x (4.28)
Here, the space variable is normalized with the size of the unit cell at the intermediate scale, that is l 1 . The dimensionless variable x lies in the cube V 1 whose dimensions are 1 along each space directions. We introduce the above change of variables in the set of equations (4.7)-(4.13). We obtain in the fluid phase:

ε 2 ∆v -∇p = j ∀x ∈ V f 1 (4.29) divv = 0 ∀x ∈ V f 1 (4.30)
In the porous solid phase, we have:

v = -∇p -j ∀x ∈ V s1 (4.31) divv = 0 ∀x ∈ V s1 (4.32)
At the interface between the phases, the conditions are:

v (s) .ν = v (f ) .ν ∀x ∈ S 1 (4.33) 2εν.d(v (f ) ).τ = -δ(v (f ) -v (s) ).τ ∀x ∈ S 1 (4.34) 2ε 2 ν.d(v (f ) ).ν = p (f ) -p (s) ∀x ∈ S 1 (4.35)
In the dimensionless equations, we have introduced the parameter ε given by:

ε = k (2) l 2 1 = l 2 l 1 k (2) (4.36)
The computation of the dimensionless permeability for different 2-D and 3-D microstructures shows that k (2) is generally inferior to 1. For instance, in (7), the dimensionless permeability for porous media with regular and random distributions of spherical pores is approximatively comprised between 10 -2 and 10 -3 . The scale separation is satisfied if the ratio l 2 /l 1 is inferior to 0.1. Clearly, ε is a small parameter. By taking the limit ε → 0 in equations (4.20) and (4.21), we observe that:

-∇p -j = 0 ∀x ∈ V f 1 (4.37) divv = 0 x ∈ ∀V f 1 (4.38)
It is observed that equations (4.31), (4.32), (4.37) and (4.38) can be rewritten as:

v = -k(x)(∇p + j) ∀x ∈ V 2 (4.39) divv = 0 x ∈ ∀V 2 (4.40)
where the non dimensionless permeability k(x) is defined by: Regarding now the interface conditions (4.11)-(4.13), the limit → 0 leads to:

k(x) =    1 x ∈ V s1 +∞ x ∈ V f 1
v (s) .ν = v (f ) .ν ∀x ∈ S 1 (4.42) v (f ) .τ = v (s) .τ ∀x ∈ S 1 (4.43) p (f ) = p (s) ∀x ∈ S 1 (4.44)
The interface conditions involve the continuity of the pressure (4.44) and the continuity of the velocity (4.42) and (4. When the Darcy-Darcy problem is solved, the velocity field linearly depends on j:

v = -A (1) (x).j (4.45)
The average of the velocity over the volume of the unit cell is: 1) .j, K

< v > V 1 = -K (
=< A (1) (x) > V 1 (1)
Where K (1) is a dimensionless permeability. The true macroscopic permeability is:

K (1) = k (2) K (1) = l 2 2 k (2) K (1) (4.47)
and, considering an isotropic material:

k (1) = k (2) k (1) = l 2 2 k (2) k (1) (4.48)
Coefficient k (1) can be interpreted as an "amplification factor" due to the presence of the large pores. Indeed, in the absence of the large pores, the macroscopic permeability is k (2) . The presence of the macrocavity multiply this permeability by a factor k (1) . If the distributions of the large pores is not spatially isotropic, k (2) is amplified by the anisotropic tensor K (1) .

Resolution of the Darcy-Darcy problem with a FFTmethod 4.4.1 Reformulation of the problem

In this section, for the sake of simplicity, the notation of the dimensionless variables with the bar over each corresponding quantity is removed. Let us introduce the field ω by: ω = -∇p -j (4.49)

The average of the quantity ω over the volume of the unit cell is denoted W . From the above relation, and considering that the pressure field p is periodic, we deduce that:

W =< ω > V 1 = -j (4.50)
The Darcy-Darcy problem (4.39), (4.40) with equation (4.41) reads:

ω = h(x)v ∀x ∈ V 2 (4.51) div v = 0 x ∈ ∀V 2 (4.52)
with the compatibility condition:

rot(ω) = 0 (4.53)
and where h(x) is the resistivity, the inverse of the permeability, h(x) = 1/k(x). The resistivity is given by:

h(x) =    1 x ∈ V s1 0 x ∈ V f 1 (4.54)
It is observed that h(x) coincides with the characteristic function of the porous solid phase.

The effective permeability can be equivalently computed by applying W or the average of the velocity, denoted V . Indeed, if we apply the velocity V instead of W in the set of equations (4.51)-(4.53), the solution for ω linearly depends on V and:

ω = B(x).V (4.55)
and the average of ω is:

W =< ω > V 1 = H (1) .V , H (1) =< B(x) > V 1 (4.56)
in which H (1) is the macroscopic dimensionless resistivity, the inverse of K (1) .

The FFT-iterative scheme

The problem of fluid flow through an heterogeneous Darcy medium is equivalent to that of heat conduction in composites and has close similarities with the problem of elastic composites. The FFT method of Moulinec and Suquet (16; 17) and its dual form (10; 11) can be used to compute the solution of the problem at the intermediate scale.

The method is summarized below:

                           Initialization : v 0 (x) = V Iteration for i=1..Niter: ω i (x) = h(x)v i (x) ω i (ξ) = F(ω i (x)) v i+1 (ξ) = v i (ξ) -D 0 (ξ). ω i (ξ) v i+1 (x) = F -1 ( v i+1 (ξ))
Convergence test (4.57)

In the above iterative scheme, the quantities v(ξ) and ω(ξ) represent the Fourier transform of v(x) and ω(x) respectively and ξ is the wave vector. F and F -1 represent the Fourier transform and its inverse. D 0 (ξ) is the Green operator that is given, in the Fourier space, by:

D 0 (x) = 1 h 0 I - ξ ⊗ ξ |ξ| 2 (4.58)
h 0 is the resistivity of a fictitious homogeneous porous solid. The value of h 0 is adjusted in order to obtain the better convergence of the iterative scheme. Following Moulinec and Suquet ( 16), we use the value:

h 0 = h min + h max 2 (4.59)
Since h(x) takes two values, 0 and 1, that of h 0 is 1/2. The FFT method advantageously use the fact that the Green tensor is explicitly known in the Fourier space and h(x) is explicitly known in the real space. From a numerical point of view, the unit cell is decomposed in N × N × N voxels. The centers of the voxels are denoted by x n and are located at the nodes of a regular grid along the three space directions. The value of h(x) is computed at the nodes of this regular grid. As shown in Figure 4.5, when the unit cell is decomposed in voxels, for 3-D images, or with pixels, for 2-D ones, the real geometry is approximated by cubic elements (in 3-D) or squares in (2-D). A better approximation consists in using some "mixed elements". Denoting by Ω s , Ω f the volume of the porous solid and fluid in a pixel and ∆ x = l 1 /N the size of the voxel, we can define by

c s = Ω s /∆ 3 x , c f = Ω f /∆ 3
x the volume fraction of solid and fluid respectively. Since the fluid has the resistivity h(x) = 0 and the solid h(x) = 1, in a mixed voxel, the resistivity h(x) = c s is used. When real images coming from tomography are used, the position of the interface is unknown. Mixed element are determined as function of the Gray-Level. This is more detailed in the next section. At each step of the iterative scheme, the velocity v i (x n ) is known and we compute ω(x n ) at the nodes of the grid. The exact Fourier transform and its inverse are replaced by the discrete Fourier transform and its inverse which are numerically computed with the FFT algorithm. The Green operator is computed along each wave vector:

ξ n = 2π l 1 (n 1 , n 2 , n 3 ) (4.60)
where n 1 , n 2 and n 3 vary from -N/2 to N/2 -1 (N is an even number) and it is recalled that l 1 is the dimension of the cubic cell.

Thresholding of the phases

The FFT method will be applied to samples obtained by µCT. Two microstructures are considered in this chapter and are denoted by A and B. The images are provided in On the right, we provide the distribution of pixels as a function of the grayscale. On these figures, two peaks are evident. Let us denote by X the value of the grayscale and by X 1 , X 2 the values of X corresponding to the peaks. Each peak is associated to a phase, the fluid phase for the higher peak and the porous solid phase for the lower one. The reconstruction of the image for FFT computation reduces to determine the threshold below which the pixels are deemed to belong to the large pores and above which they are part of the porous solid phase. If we consider that some pixels are "mixed" or "composite", they are part of both the large pores and the porous solid phase. The introduction of two thresholds is thus needed.

Let us then introduce two characteristic values X 1 and X 2 . When the grayscale X is inferior to X 1 , then, the pixel belongs to the large pores. Alternatively, when X > X 2 , the pixel belongs to the porous phase. When X is comprised between X 1 and X 2 , the pixel is mixed. The following law is then used to determine the volume fraction of the porous solid phase (c s ) and of the fluid phase (c f ):

c s (X) =            0 X < X 1 1 X > X 2 X -X 1 X 2 -X 1 X 1 ≤ X ≤ X 2 , c f (X) = 1 -c s (X) (4.61)
The variations of c s (X) ad c f (X) with the grayscale is provided in figure 4.8. Obviously, another law could be introduced, but this choice corresponds to the simplest. The values of the two thresholds X 1 and X 2 are determined as a function of the two values at the peaks, X 1 and X 2 by:

X 1 = (1 -α)X 1 + αX 2 (4.62) X 2 = αX 1 + (1 -α)X 2 (4.63)
in which α is a coefficient which must be taken in the range [0, 0.5]. When α = 0.5, we have X 1 = X 2 = (X 1 + X 2 )/2. There is only one threshold which separates the pixels which belong to the porous solid and the fluid phase. In this case, there is no mixed pixels. When α = 0, X 1 = X 1 and X 2 = X 2 , the two thresholds are taken at the peaks and all the pixels lying between the peaks are mixed. In order to evaluate the importance in the choice of the model for c s (X) (see Eq. The method has been also applied to the microstructure B. Again, 20 slices are selected in the volume of the sample. One image of the X-ray microtomography is given in Figure 4.13. The unit cell contains 800 × 800 pixels. We can distinguish here the first porosity represented by the large pores and also the second porosity level. Compared to the previous microstructure, we can observe thin polymeric walls inside the large pores. The distribution of I s (x) (see figure 4.13) shows some dark blue pixels within the porous matrix that corresponds to the second porosity. The results show that the porous solid is almost isotropic, i.e. the off diagonal are very small compared to the diagonal components and the components 11 and 22 are close. However, we observe significant differences between these two components that will be probably attributed to the dimension of the unit cell. The convergence of the macroscopic permeability must be studied by increasing the dimension of the unit cell, this is done in the next part.

The results clearly show a strong influence of α on the effective transport properties. Moreover, the permeability increases with α. For α = 0, there is large number of mixed pixels (see Figure 4.11), especially within the large pores and which reduce significantly the velocity compared to the case α = 0.25. When α = 0.4, there is a few number of mixed pixels compared to the case α = 0.25 and some large pores are connected (see Figure 4.11). The results show the importance of the description of the porous material in the ligament between the large pores. The distribution of the local velocity according to the direction 1 is represented according to α on In each case, we compare the distribution of c s and that of the local velocity. The value of the velocity is represented by a colors gradient which goes from the blue for low velocity to the orange-red for the highest values. We can observe that the more we increase the value of α, the more the velocity is located and high. For α = 0, we observe some red zones which correspond to the passage of the fluid along mixed pixels located between two very close pores. By increasing the values of α, i.e. for α = 0.25 and α = 0.4, the velocity is more localized since the number of mixed pixels is reduced. We also represent the distribution of local velocity for direction 2 and the distribution of the local pressure for directions 1 and 2 and α = 0.25 (4.18). In the direction 2 (a), the velocity is lower than direction 1 (4.16) and takes negative values. Pressure is generally higher and more homogeneous in the case of direction 2 (c). However, pressure takes negative values and is more located in the case of direction 1 (b) .

Computations on a cubic cell

The 3-D permeability is computed on 15 unit cells of dimension 186 × 186 × 186. One unit cell is represented in Figure 4.12. In Figure 4.23, we provide the values of the resistivity for each occurrence and the average cumulated value. The statically average values of the resistivity, its standard standard deviation and the corresponding values of permeability are provided in Table 4.9. It is observed that the components 11 and 22 are close but a small difference is however noted. Moreover, the component 33 is significantly distinct from the in-plane components (11 and 22) which reveals an anisotropy with respect to the x 3 axis. 4.9: components of the dimensionless macroscopic resistivity, its standard deviation and the components of the dimensionless macroscopic permeability in 3D case.

Experimental vs. computed permeability

In this section, we aim to estimate the macroscopic permeability for microstructure A and to compare it with the experimental data. To this end, we need to compute the permeability of the second porosity, k (2) . The latter remains undetermined since the µCT images at this scale is not available. Only k (1) , called amplification coefficient due to the presence of the larger pores, has been computed from the µCT images in section (4.6.5).

A simplified model for the microstructure at the scale of the second porosity is used.

To build this model, two parameters could be used: the diameter of the large pores and the porosity associated with the smaller pores. For microstructure A, the diameter of the pores is 1µm. We need to determine the porosity at the lower scale. To this end, denoting by V f 1 and V f 2 the volume of the larger and smaller pores respectively, we introduce the corresponding porosity f 1 and f 2 by:

f 1 = V f 1 V , f 2 = V f 2 V (4.65)
and the total porosity is:

f = f 1 + f 2 (4.66)
For the computations at the lower scale, we do not use the porosity f 2 but f 2 defined by:

f 2 = V f 2 V s1 (4.67)
where V s1 = V -V f 1 is the volume of the porous solid containing only the smaller pores. Note that f 2 is related to f 2 by the relation:

f 2 = V f 2 V V V -V f 1 = f 2 1 -f 1 (4.68)
The total porosity determined with the mercury intrusion porosimetry is f = 90%. The porosity f 1 is determined from the µCT images (see Figure 4.12) and gives f 1 = 66%. We deduce that f 2 = 34% and f 2 = 71%. A unit cell which mimic the porous microstructure at the lower scale is built by considering a centered cubic array of overlapping spherical cavities. By adjusting the radius of the spheres, we can obtain the porosity f 2 = 71%. The calculations are performed on a cubic unit cell having the dimension 1 along each space directions. The radius of the spheres corresponding to a porosity of 71% is R = 0.438. The image of the porous solid is provided on The dimensionless permeability k (2) is computed by the FFT algorithm on the dimensionless unit cell. We use 256 3 voxels. The results lead to the value 5.03 × 10 -4 . The permeability k (2) is obtained by multiplying k (2) by l 2 2 , where it is recalled that l 2 is the true dimension of the cell. Since the diameter of the smaller pores is 1µm, we deduce that the dimension l 2 is 1.14µm. The permeability k (2) is then equal to 6.55 × 10 -15 . The permeability associated with the smaller pores being now estimated, the macroscopic permeability is deduced from Eq. (4.48). The components of K (1) are taken in

Conclusions

In this chapter, we computed the macroscopic permeability of biporous polymers. Only the case of porous materials with non-interconnected pores for the first porosity has been considered. In that case, it is possible to rewrite the coupled Darcy-Stokes problem with a set of equations which only use the Darcy law. The Stokes region has then been replaced by a Darcy one with an infinite permeability. The calculations were performed for two samples (called microstructures A and B in the chapter).

The thresholding of the µCT images for the FFT computations has involved a careful investigation of the distribution of pixels as a function of the greyscale. The aim was to elaborate a model in which pixels were deemed to belong to the large pores and above which they were part of the porous solid phase. In order to obtain the most realistic thresholded images, we considered that some pixels are "mixed" or "composite". In other words, they were part of both the large pores and the porous solid phase. Interestingly, this chapter pointed out the importance of the choice of the thresholding model by using different values of a coefficient α. Indeed, the results clearly showed a strong influence of this parameter on the effective transport properties. Furthermore, it could be observed that the porosity was almost independent of the dimension of the unit cell size. The void fraction was thus found to be around 64%. The 3D computation revealed an anisotropy of the porous medium in the x 3 axis direction that could be explained by a stack default of the porogen during the material elaboration. Indeed, the NaCl particles were not sintered (see Chapter 2 for more details).

Last but not least, a comparison of the computed and experimental permeability was attempted. The limitation of a correct comparison was the lack of microstructure information concerning the second porosity level (small pores). In the chapter, we used a simplified model based on a regular array of spherical cavities in order to estimate the permeability associated with the second porosity.

Chapter 5

Computation of macroscopic permeability by using Brinkman equation

Resolution with the Brinkman equation

We consider the fluid flow problem at the intermediate scale. The unit cell is made up of a porous solid with the volume V s1 and large pores which occupy the domain V f 1 .

We still follow the notations of the last chapter. The index "1" makes reference to the scale of the first porosity (the large pores) also called intermediate scale. The smaller scale corresponds to the second porosity and we use the index "2" to make reference to that scale. The total volume of the unit cell at the intermediate scale is denoted V 1 and by S 1 we denote the interface between V s1 and V f 1 . By ∂V 1 we denote the boundary of the unit cell.

The local problem at the intermediate scale uses two different equations: the Stokes ones in the cavities and the Darcy ones in the porous solid. This constitutes a difficulty for the application of the FFT method because it has been developed for the resolution of heterogeneous problems, i.e. the problems which use the same equations for each phases but with different local properties. For instance, in the Darcy-Darcy problem, the resistivity differs from one phase to another but the Darcy equation is used anywhere in the unit cell. The simplification of the coupled Darcy/Stokes problem with the Darcy-Darcy one has been used in the last chapter in order to apply the FFT method. However, such approach is valid only for microstructures with non interconnected large pores. When the large pores are interconnected the approach leads to an infinite value of the macroscopic permeability.

To circumvent this difficulty, we propose to use the Brinkman equation within the porous solid and in the large pores:(2; 3)

µ(x)∆v -γ(x)v -∇p = J ∀x ∈ V 1 (5.1) div(v) = 0 ∀x ∈ V 1 (5.2)
123 in which γ(x) and µ(x) are two coefficients which are given by:

γ(x) =    µh (2) x ∈ V s1 0 x ∈ V f 1 µ(x) =    0 x ∈ V s1 µ x ∈ V f 1 (5.3)
In Eq. (5.1), ∇p is the local pressure gradient in which p is the local periodic pressure field. At the right of the equality in Eq. (5.1), J is the applied constant pressure gradient. Let us recall that h (1) is the resistivity of the porous matrix and µ is the dynamic viscosity of the fluid.

In the form given by Eq. (5.1), the coupled Darcy-Stokes problem falls in the category of heterogenous problems and uses two coefficients γ(x) and µ(x). Both γ(x) and µ(x) are piecewise constant functions. In the solid phase, the dynamic viscosity µ(x) is null and the Stokes term vanishes. In the fluid phase,γ(x) is null, the Darcy term vanishes and the Stokes equations are recovered. The conditions at the interface S 1 between the two phases is the continuity of the velocity field and the traction:

v (s) = v (f ) , σ (s) .ν = σ (f ) .ν ∀x ∈ S 1 (5.4)
where ν is the normal unit vector taken on S 1 , exponent (s) makes reference to the porous solid and exponent (f ) to the fluid phase, σ is the local stress tensor given by:

σ = 2µ(x)d -pI (5.5)
and where d is the strain rate tensor defined by:

d = ∇ s v = 1 2 (∇v + ∇ T v) (5.6)
that is traceless due to incompressibility. As a consequence, the deviatoric part of the stress field is denoted by s and is given by:

s = 2µ(x)d (5.7)
and the trace of the stress is:

tr(σ) = -3p (5.8)
At the boundary of the unit cell, we consider the periodicity condition for the velocity field v and the antiperiodicity of the traction σ.n (n being the normal unit vector taken on the boundary of the cell). The latter condition results form the equilibrium of the traction on the boundary between two neighboring cells. By taking the average of the local velocity v, we obtain the macroscopic velocity: 1) .J (5.9)

V =< v > V = - 1 µ K (
where K (1) represents the macroscopic permeability. By substituting the Brinkman equation to the coupled Darcy/Stokes ones, the formulation of the problem remains

Solution in real space

By adopting the notations introduced in Eq. (5.20), the solution of the inclusion problem can be also read:

d v = -   Γ 0 Ω 0 -Ω 0 G 0   q f (5.32)
where we have introduced the following Green operators:

Γ 0 ijkl = λ ξ 2 4 (Q ip P jq + Q iq P jp + P ip Q jq + P iq Q jp ), (5.33
)

Ω 0 ijk = iλ ξ 2 (Q ik n j + Q jk n i ), (5.34)
G 0 ij = λQ ij (5.35)
for any ξ = 0 and where λ is defined by:

λ = 1 µ 0 ξ 2 + γ 0 (5.36)
For ξ = 0, Γ 0 , Ω 0 and G 0 are null. In Eq. (5.32), Ω 0 is the adjoint of Ω 0 such that

d : Ω 0 .v = v. Ω 0 : d.
Let us come back to the real space. The solution of the inclusion problem is derived by taking the inverse Fourier transform of Eq. (5.32). Let us recall that the component of any quantity corresponding to ξ = 0 corresponds to its volume average over the unit cell V 1 . As a consequence, when a Green operator is applied to any pair of tensors, it generates a pair of tensors whose volume average is null. So, the average of each corresponding quantity must be added in the real space:

d v = 0 V -   Γ 0 Ω 0 -Ω 0 G 0   * q f (5.37)
where " * " denotes the convolution product. The mean value of each corresponding quantity has been added: V for the velocity field and zero for the strain rate tensor d (since the latter derivates from a periodic velocity field).

To simplify the notations, we put:

G 0 =   Γ 0 Ω 0 -Ω 0 G 0   (5.38)
such that the solution of the inclusion problem can be also read:

d v = 0 V -G 0 * q f (5.39)
and ∆ 0 = 0 for ξ = 0. We can put the solution for the stress field in the form:

s = -∆ 0 : r + Π 0 . g (5.50)
To summarize the solution for s and ω can be read:

s ω = -   ∆ 0 -Π 0 Π 0 D 0   r g (5.51)
The solution is now written in the real space. Again, when applying the inverse Fourier transform, the component of each quantity corresponding to ξ = 0 corresponds to its volume average over the unit cell V 1 must be added. It gives:

s ω = S W -   ∆ 0 -Π 0 Π 0 D 0   * r g (5.52)
where " * " denotes the convolution product.

The mean value of each corresponding quantity has been added: W for ω and S for the stress field. Since the average of the strain rate field is null, we have

S =< s > V 1 =< q > V 1 = -2µ 0 < r > V 1
that is a priori non null if we consider an arbitrary periodic eigenfield r.

In order to simplify the notations, we put:

s ω = S W -D 0 * r g (5.53)
with:

D 0 =   ∆ 0 -Π 0 Π 0 D 0   ( 5
q = 2(µ(x) -µ 0 )d, f = (γ(x) -γ 0 )v (5.55)
By considering expressions (5.55) in relation (5.39), we deduce that:

d v = 0 V -G 0 * 2(µ(x) -µ 0 )d (γ(x) -γ 0 )v (5.56)
The latter equation is an integral equation, called Lippmann-Schwinger equation, for the variables v and d. The enforcing term is the macroscopic velocity V .

Resolution with an iterative scheme: first form

Eq. (5.56) is a linear equation which can be formally put into the algebraic form:

x + Lx = b (5.57)

where x contains the unknowns:

x ≡ d v (5.58)
and L is a linear operator defined such that:

Lx ≡ G 0 * 2(µ(x) -µ 0 )d (γ(x) -γ 0 )v (5.59)
and b is given by:

b ≡ 0 V (5.60)
The inversion of the linear system (5.57) is computationally too expensive due to the presence of the convolution product in Eq. (5.56). To circumvent this difficulty, we can expand the solution along Neumann series following a methodology which was first introduced for composite conductors by Brown (4) and later by Kroner (7). The Neumann series reads:

x = b -Lb + LLb -LLLb + ... (5.61)
Each term of the series can be computed with the following recurrence relation:

x i+1 = b -Lx i (5.62)
and which starts from x 0 = b. The convergence of the Neumann series depends on the spectral radius of the linear operator L. In the context of the thermal conductivity and in the case of elasticity the conditions that lead to the convergence of the Neumann series has been well documented in the literature. (6; 9; 8; 10; 15) The conditions of convergence will be discussed in the context of the Brinkman equation in the next of this chapter. By making the analogy between Eq. (5.57) and (5.56), the following iterative scheme can be used to compute the solution of the Brinkman problem:

d i+1 v i+1 = 0 V -G 0 * 2(µ(x) -µ 0 )d i (γ(x) -γ 0 )v i (5.63)
and which is initialized with:

d 0 = 0, v 0 = V (5.64)
Eq. (5.63) is the first form of the iterative scheme. A second form is provided in the next section.

A simplification of the iterative scheme (5.63) is possible. To this end, let us consider the following property of the Green operator G 0 : for any periodic velocity field v which comply with the incompressibility and strain rate field d which derivates from v, we have:

G 0 * 2µ 0 d γ 0 v = d v -V (5.65)
The demonstration is provided in the appendix (see C.1). Since at each step of the iteration process we have d i = ∇ s v i , it follows that:

d i+1 v i+1 = d i v i -G 0 * 2µ(x)d i γ(x)v i (5.66)
that is the second form of the iterative scheme. The recurrence process is still initialized by Eq. (5.64). The iterative scheme has the same structure that one of Moulinec and Suquet and used in the last chapter for the resolution of the Darcy-Darcy problem. (13) Note that in Eq. (5.66), 2µ(x)d i is the deviatoric part of the stress field computed at iteration i and which can be denoted s i . Also, γ(x)v i can be denoted ω i . When the convergence of the iterative scheme is achieved, ω i = γ(x)v i is equal to ω the solution of the Brinkman equation (5.11)- (5.15). By taking the average of ω over the volume of the unit cell, we determine the macroscopic resistivity (and then the macroscopic permeability) following Eq. (5.19).

Third form of the iterative scheme

The solution at convergence may comply with all the equations in (5.11)-(5.15). Some equations are verified at each step of the iteration process, they are: the local relations s i = 2µ(x)d i and ω i = γ(x)v i , the compatibility between the strain rate tensor and the velocity field, d i = ∇ s v i and the incompressibility, tr(d i ) = div(v i ) = 0. Only the equilibrium div(σ i ) = ω i -W is not verified at each iteration but only at the convergence. When the convergence is achieved, which corresponds to d i+1 = d i and v i+1 = v i , we have:

G 0 s i ω i = 0 (5.67)
The latter condition is a measure of the distance from the equilibrium. Let us give the proof. Owing to Eq. (5.38) and (5.35), we deduce that:

G 0 s i ω i = -λ i E i ⊗ s ξ E i (5.68)
where E i is given by:

E i = Q.(i s i .ξ -ω i ) (5.69)
It can be noted that the quantity E i is null when equilibrium is verified. Indeed, the equilibrium is i σ i .ξ = ω i in the Fourier space. The projection of this equation onto the plane normal to the wave vector ξ is iQ. s i .ξ = Q. ω i . Note that, the pressure vanish in the latter equation when the projection along Q is considered and only the deviatoric part of the stress field remains. Owing to relation (5.68), we can simplify the iterative scheme as follows:

initialization: v 0 = V , d 0 = 0 iteration i : (a) ω i = γ(x)v i , s i = 2µ(x)d i (b) ω i = F F T (ω i ), s i = F F T (s i ) (c) E i = Q.(i s i .ξ -ω i ) (d) convergence criterion (e) v i+1 = v i + λ E i (f) d i+1 = i v i+1 ⊗ s ξ (g) v i+1 = F F T -1 ( v i+1 ), d i+1 = F F T -1 ( d i+1 )
In the third form, it is not necessary to compute and to store the components of the Green tensor. This has the advantage to reduce the computer memory because the total number of components is 45. Indeed, the Green tensors Γ 0 , Ω 0 , G 0 has 21, 18, and 6 distinct components respectively. With the third form, the components of the Green operator is not needed, only the computation of the vector E and λ is required.

The choice of the reference material

The convergence of the iterative scheme drastically depends on the choice of the reference material. The latter is given by two coefficients µ 0 and γ 0 . The reference material is a mathematical artifice without physical reality that is introduced in order to obtain the convergence of the iterative scheme. In fact, µ 0 and γ 0 could be interpreted as two preconditioner of the linear system associated with the unit cell problem. Following the methodology of Michel et al. (8) and Milton (9), the values of µ 0 and γ 0 could be determined in such a way that the spectral radius of the linear operator L is inferior to 1. Also, the values of µ 0 and γ 0 could be optimized in order to minimize the number of iterations at convergence of the iterative scheme. The optimization of the parameter consists in minimizing the spectral radus of the linear operator L. In the context of elasticity, it has been demonstrated that the optimal elastic coefficients of the reference material are defined by the average of the local elastic coefficients of the phases (for a two phase composite). Milton (9) obtained a similar result for the thermal conductivity problem.

Note that the last equation is the projection of the equilibrium equation along the projector Q to eliminate the pressure field (see section 5.3.4). As already explained in section 5.3.4, a measure of the distance from the local equilibrium can be given with the Green operator G 0 , the equilibrium being given by the condition:

G 0 s ω = 0 (5.76)
Conversely, a measure of the distance from both the incompressibility and compatibility could be given by the dual Green operator D 0 given by Eq. (5.54). Any compatible pair of tensor d and v and incompressible velocity v satisfy to the equation:

D 0 d v = 0 (5.77)
The proof is provided in section C.2. Now, we can combine both conditions (5.76) and (5.77) as follows:

αC 0 G 0 s ω + β D 0 d v = 0 (5.78)
where α and β are coefficients and C 0 is defined by:

C 0 =   2µ 0 0 0 γ 0   (5.79)
Eq. (5.78) is satisfied if v, d comply with the equilibrium and incompressibility while ω and s satisfy to the equilibrium. This is proved in section C.3.

The iterative scheme

Eq. (5.78) is the central point for the formulation of the accelerated scheme. The latter is formulated with the eigenfields q and f and reads:

q i+1 f i+1 = q i f i -αC 0 G 0 * s i w i -βD 0 d i v i (5.80)
where the values s i , w i , d i and v i are updated at each iteration as follows:

d i = 1 2(µ(x) -µ 0 ) q i , v i = 1 γ(x) -γ 0 f i , (5.81)
s i = q i + 2µ 0 d i , ω i = f i + γ 0 v i (5.82)
At the convergence of the iterative scheme, which corresponds to q i+1 = q i and f i+1 = f i , we deduce that the equilibrium, compatibility and incompressibility are recovered.

where it is recalled that:

ε = k (2) l 2 1 = l 2 l 1 k (2)
(5.93)

ε is a measure of the scale separation. When the ratio between the pore scales is very low, ε is also very small and µ(x) is null everywhere in the unit cell. The problem then reduces to the resolution of the Darcy equation with a null resistivity in the large pores. This is the approximation used in the last chapter. It is now possible to evaluate the importance of the diffusion term in the larger pores by keeping the coefficient µ(x) = ε 2 in the volume of the pores V f 1 .

The thresholding of the phase is still considered with mixed pixels, i.e. pixels which contain both the porous solid and the fluid phases. For such pixels, the coefficients are taken in the form:

γ(x) = c s γ s + c f γ f , µ(x) = c s µ s + c f µ f (5.94)
where c s and c f are the volume fraction of the porous solid and fluid in the pixel and where γ s , c f γ f are the values taken by γ(x) in the porous solid and fluid phase respectively. µ s , µ f the values taken by µ(x) in the phases. In the present case, γ s = 1 and µ s = 0 in the porous solid phase, γ f = 0 and µ f = ε 2 in the fluid phase. It follows that:

γ(x) = c s , µ(x) = c f ε 2 (5.95)
in the mixed pixels.

Effect of α

The effect of α on the macroscopic permeability is studied by giving to α different values as 0, 0.25 and 0.4 with a constant scale factor equal to 10 -2 . We now investigate the effect of the scale factor on the permeability. The computations are performed by increasing ε from 10 -5 to 1. By doing so, we increase gradually the smaller pores size.

Effect of the unit cell dimensions

The computation of the resistivity is performed for different unit cell dimensions: 186× 186 pixels, 300 × 300 pixels and 400 × 400 pixels. The thresholding of the images is made with this α = 0.25. We assume a scale factor ε=10 -4 . The 2-D dimensionless macroscopic resistivity H (1) , its standard deviation and the dimensionless macroscopic permeability K (1) are reported in Tables 5.

The dimensionless problem

We now consider the case for which the larger pores are interconnected. In that case, there is a predominant flow system through the array of large pores. As a consequence, we deduce that: By introducing this change of variable in Eqs. (5.1), we obtain the system (5.90) with the following definitions for µ(x) and γ(x):

v = O l 2
γ(x) =    ε -2 x ∈ V s1 0 x ∈ V f 1 µ(x) =    0 x ∈ V s1 1 x ∈ V f 1
(5.98)

In the mixed pixels, we use expressions given by Eq. (5.94). It follows that: Let us call X the value of the greyscale and X 1 the value of X corresponding to the peak. It is important to recall that we observed two peaks when the large pores were not connected (see Chapter 4). By introducing X 1 = αX 1 , we define by X ≤ X 1 the pixels which belongs to the fluid phase (large pores). All the pixels corresponding to X > X 1 are mixed, i.e. they belong to the fluid phase and the porous solid phase. We need a law to define the volume fraction of each phase in that mixed pixels.

c f =      1 X ≤ X 1 exp -s X -X 1 X 1 X > X 1
(5.100)

The variation of c f with the greyscale is illustrated in

Effect of coefficient s

The calculations are performed on 2-D images. 20 slices are selected along the x 3 axis. The computation is performed with s = 0.1. Firstly, we can notice that the resistivity values at each occurrence are almost equal in both directions which reveals the quasiisotropy of the microstructure C, in comparison with microstructure A. This isotropic behavior is explained by the experimental protocol applied for the elaboration of the materials (see Appendix A for experimental details). Indeed, a controlled packing of NaCl particles has permitted to obtain a well-defined porous microstructure with interconnected pores (see Chapter 2 and 3). On the other hand, the materials coming from microstructure A has been elaborated with randomly packed NaCl particles that has given rise to less ordered porous network.

Experimental vs computed permeability

The comparison between the experimental and the computed permeability consists in using the equation below:

k (2) = l 2 2 k (2)
(5.101)

where l 2 stand for the unit cell size, k (2) the computed permeability, and k (2) the experimental permeability. In the case of microstructure C, k 1 is not taken as reference because smaller pores are interconnected so the flow is not governed by the smaller pores but only by the larger pores. Let us recall that synchrotron slice size is equal to 2000 x 2000 pixels and the voxel size is equal to 0.65 µm so l 2 = 1.3.10 -3 m and k

(2) = 0.1. We can thus deduct that k (2) is equal to 1.69. 10 -7 . This value is higher than the permeability measured in Chapter 2 (in the order of 10 -12 ). The pores interconnection seems to have more impact on the transport properties when the permeability is computed. Note however that the calculations have been performed on a 2-D unit cell. The results are then probably not representative of the permeability of the 3-D microstructure.

Conclusions

In this last chapter, we attempted to determine the macroscopic permeability of biporous polymers with interconnected and non-interconnected larges pores by using Brinkman equation. The permeability of a microstructure exhibiting non-interconnected larges pores has been already computed by using Darcy-Darcy equation in Chapter 4. This time, we used an FFT iterative scheme to estimate the permeability. The results clearly showed that permeability increased with α value. In a general way, the values of permeability obtained by using Darcy-Darcy equation were widely higher than the case of the Stokes-Darcy equation. In order to investigate the effect of the scale factor ε on the macroscopic permeability, we performed the computation by increasing epsilon from 10 -5 to 1. By doing so, we increased gradually the smaller pores size while the large pores size remained constant. This study permitted then to investigate the effect of the geometry of the microstructure on the macroscopic permeability. The computation showed that the permeability of the medium decreased while the scale factor increased. Additionally, a saturation effect was reached for the lowest value of ε. In a second part, the permeability of a microstructure with interconnected large pores was computed. In that case, the fluid flow was governed by the large pores interconnections. This particularity lead us to change the thresholding strategy by introducing the s parameter which governed the porous solid fraction. The aim was to study the important role of the polymeric walls permeability in the fluid flow. By increasing the porous solid fraction of the walls, we observed an increase of the macroscopic permeability. Finally, this chapter pointed out the effect of the pore interconnections on the isotropy of the material.

General conclusion and perspectives

This PhD work developed an original three-step approach toward the multiscale investigation of model polymeric materials with a bimodal porosity through the combination of their design and synthesis, their thorough microstructural characterization, and the computation of their macroscopic permeability. In the first part of the study, PHEMA-based biporous networks were prepared via a facile route referred to as the double porogen templating approach. MIP measurements showed that the first porosity level generated by NaCl particles (sintered or not) with different sizes exhibited pores of about 70 µm size, while the second porosity level corresponded to a porous network of about 1 µm pore size. Such complex frameworks were thoroughly analyzed in terms of morphological and physico-chemical characterizations. The effect of the macroporogenic agent features (i.e. NaCl particles size and packing) on the pore interconnectivity, pore size and distribution were investigated. We found that the Spark Plasma Sintering of the particles permitted to obtain materials with a well-defined bimodal porosity and increased the pore interconnectivity, the porosity ratio and the specific surface area. An interesting advantage of the double porogen templating approach is the control of the first porosity level without changing the features of the second one. Furthermore, a particular interest has been focused on the effect of large pore interconnectivity that gave to the biporous networks a considerable capacity to incorporate water compared to their bulk and monoporous counterparts. The influence of the cross-linker on the thermo-mechanical properties and the morphology of these structures was also investigated. It has been shown that a dimethacrylate with a long ethylene glycol spacer length led to the collapse of the biporous structure and a low T α value. Finally, it was found that the large pore sizes (first porosity level) and the interconnectivity had a significant effect on the experimental permeability k.

The second part of the study was devoted to the investigation of the 3-D microstructure of doubly porous polymeric networks by means of laboratory and synchrotron X-ray microtomography. The low absorption properties of the polymer and the high porosity of the network made their observation difficult. During the three years of this PhD, the effort was focused on obtaining the least noisy images in order to reconstruct a realistic Representative Volume Element (RVE) for the computation of the transport properties. Different experimental protocols such as the incorporation of a contrast agent and the optimization of X-ray energies were thus tested. High-resolution images with voxel sizes down to 650 nm were eventually obtained and revealed three different microstructural features: the large pores having the NaCl particles shape, the polymeric matrix exhibiting the small pores and thin polymeric walls. We noticed the collapse of the walls that probably occurred during the vacuum drying of the newly obtained materials. Furthermore, the image processing pointed out the difficulties encountered during the segmentation precisely because of the thickness of these walls that did not allow for a precise distinction between the matrix and the void space. The last part of this work was concerned with the development of computational tools to estimate the macroscopic permeability of biporous polymers. We first considered the case of microstructures with non-interconnected large pores derived from µCT analyses. In this case, we simplified the coupled Darcy-Stokes problem with a set of equations which only use the Darcy law. The Stokes region was then replaced by a Darcy one with an infinite permeability. The thresholding of the µCT images for the FFT computations involved a careful investigation of the distribution of pixels as a function of the greyscale. A model in which pixels were deemed to belong to the large pores and above which they were part of the porous solid phase was elaborated. The results clearly showed a strong influence of the thresholding strategy parameters on the effective transport properties. The 3-D computation revealed an anisotropy of the porous medium in the x 3 axis direction that could be explained by a stack default of the macroporogen during the material elaboration. A comparison of the computed and experimental permeability was attempted. The limitation of a correct comparison was the lack of microstructure information concerning the second porosity level. To circumvent this limitation, a model based on a regular array of spherical cavities was used to mimic the microstructure related to the second porosity, thus enabling to obtain a good agreement between experimental and computed macroscopic permeability values.

The second approach proposed new FFT based algorithms to solve the coupled Darcy-Stokes problem on microstructures exhibiting non-interconnected and interconnected large pores. This approach allowed us to understand the effect of the microstructure features on the effective permeability. Last but not least, the important role of the permeability associated with the polymeric walls in the fluid flow was highlighted in the case of interconnected large pores. By increasing the porous fraction of the walls, we observed an increase in the macroscopic permeability. Finally, this chapter pointed out the effect of the pore interconnectivity on the isotropy of the material. In order to extend the previous studies, we propose the investigation of the mechanical behavior under compression of PHEMA-based doubly porous materials prepared with different dimethacrylate crosslinkers (i.e. EGDMA, DEGDMA, TriEGDMA, TetraEGDMA, PEGDMA). Indeed, the DMA apparatus did not permit reliable measurements on doubly porous materials. Compressive mechanical testing could be thus more adapted to such highly porous materials. Moreover, SEM and MIP results showed that the material's morphology was strongly influenced by the crosslinking agent. It could be interesting to explore the effect of the crosslinker chain length on the swelling behavior of the materials since only EGDMAbased frameworks have been investigated so far. Furthermore, the collapse phenomenon observed in the 3-D microstructures could be better understood by monitoring the evolution of the material's morphology during drying by radiography. We may also scan a sample before and after the drying process by means of synchrotron tomography. Additionally, the characterization of the second porosity level could be helpful for the computation of the macroscopic permeability since no clear microtomography image of the porous matrix has been obtained so far. An observation by means of X-ray microscopy with an effective spatial resolution lower than 100 nm and a voxel size A potassium iodide aqueous solution (I 2 /KI), the so-called Lugol solution, was used in order to improve the low X-ray absorption contrast provided by highly porous polymeric materials. 1 wt. % iodine (I 2 ) and 2 wt. % potassium iodide (KI) were dissolved in 100 ml distilled water. This Lugol solution was then stirred overnight on an orbital shaking plate at room temperature. The porous samples were then soaked in the Lugol solution for 1 min. A second protocol consisted in adding solution drops on the sample. The Lugol impregnated porous materials were placed on a filter paper and washed with deionized water to remove the excess of staining agent and prevent surface saturation. Finally, the labelled materials were dried under vacuum and submitted to microtomography analysis. Because the available beamline computer resources did not yet allow to reconstruct the images during the experiment, this operation was performed later at laboratoire Navier using a specifically developed procedure. The radiographic projections were normalized using the dark and flat images by using a FIJI Macro developed by the team. The normalized projections were converted into appropriate format and 3-D images were then reconstructed using the X-act software. Flat images were either used as acquired on the beamline, or from the averaging of the whole set of radiographs. The latter solution provided often better results. More experimental details are reported in Table B. 3.

1. 1 1 . 1

 111 Schematic cross-section of a porous solid. . . . . . . . . . . . . . . . . . 1.2 Classification of porous media related to void space and solid matrix connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Representation of the classification of porous media based on the storage capacity of the void and fractures: (a) fractures storage is more important, (b) smaller porosity storage is more important, and (c) the two porosity levels are equally distributed. . . . . . . . . . . . . . . . . . . 1.4 Representation of the bimodal porosity in reservoir rocks: (a) periodic cell of the microporous medium, (b) periodic cell of the fractured medium. 1.5 Schematic representation of the experimental process applied for the elaboration of doubly porous polymers via Thermally Induced Phase Separation process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Schematic representation of the experimental setup implemented for the preparation of porous polymers via the HIPE technique. . . . . . . . . 1.7 Schematic representation of the experimental setup required for the electrospinning process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Schematic representation of the experimental device implemented for measuring the permeability of collagen scaffolds. . . . . . . . . . . . . . 1.9 Two-scale homogenization of permeability using LBM and FEM. . . . . 1.10 Representation of the fluid flow in the x-direction (a), in the y-direction (b) and in the z-direction (c). . . . . . . . . . . . . . . . . . . . . . . . 1.11 Grey-scale values histogram for the entire 3-D sample of soil aggregate.2.1 Representative scheme for the double porogen templating approach. . . 2.2 Molecular structures of the functional monomer, i.e. HEMA, and of the ethylene glycol-based dimethacrylate crosslinkers used in this study. . . 2.3 MIP profiles of non-interconnected (on the left) and interconnected (on the right) doubly porous networks. . . . . . . . . . . . . . . . . . . . . 2.4 SEM micrographs of non-interconnected (A & B) and interconnected porous networks (C & D). . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 (a) Pore size distribution profiles and (b) cumulative intrusion volumes of interconnected biporous networks in function of NaCl particles size as determined by MIP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 SEM micrographs of monoporous networks with small pores prepared with a) DEGDMA, b) TriEGDMA , c) TetraEGDMA, d) PEGDMA. . iii 2.7 SEM micrographs of biporous networks prepared with a) DEGDMA, b) TriEGDMA, c) TetraEGDMA, d) PEGDMA. . . . . . . . . . . . . . . 2.8 Pore size distribution and cumulative intrusion profiles of a) DEGDMAbased biporous network and b) its monoporous counterpart, c) TriEGDMAbased biporous network and d) its monoporous counterpart. . . . . . . 2.9 Storage modulus G' of DEGDMA, TriEGDMA, TetraEGDMA, and PEGDMAbased monoporous materials as a function of temperature. . . . . . . . 2.10 Water uptake of non-interconnected and interconnected biporous networks as a function of time (*P=porosity ratio). . . . . . . . . . . . . . 2.11 Water uptake of monoporous networks with interconnected and noninterconnected large pores and small pores as a function of time (*P=porosity ratio). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Schematic illustration of µCT acquisition and reconstruction processes. 3.2 µCT images of biporous networks prepared with non-sintered (a) and sintered (b) NaCl particles.Scale bars represent 500 µm. . . . . . . . . . 3.3 µCT images of porous PHEMA-based networks:(i) exhibiting non-interconnected large pores before (a) and after (b) image processing, and corresponding image after segmentation (c). (ii) exhibiting interconnected large pores (d).Scale bars represent 500 µm. . . . . . . . . . . . . . . . . . . . . . . 3.4 µCT images of biporous networks prepared with sintered 125-200 µm(a) and 200-250 µm (b) NaCl particles. . . . . . . . . . . . . . . . . . . 3.5 µCT image (left, voxel size = 2 µm) and SEM image (right) of a biporous network prepared with 200-250 µm NaCl particles. . . . . . . . . . . . 3.6 Representation of the grey level of pixels along a line that passes through the porous matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 µCT images of labelled doubly porous networks prepared with 125-200 µm sintered (a) and non-sintered NaCl particles (b) (zoom x 150 % ). . 3.8 µCT images of labelled doubly porous networks prepared with 200-250 µm sintered (c) and non-sintered NaCl particles (d) (zoom x 150 %). . 3.9 µCT images of labelled doubly porous networks prepared with 250-400 µm sintered (e) and non-sintered NaCl particles (f) (zoom x 150 % ). . 3.10 Doubly porous networks prepared with 125-200 µm NaCl particles before (a) and after (b) image processing . . . . . . . . . . . . . . . . . . . . . 3.11 Doubly porous networks prepared with 250-400 µm NaCl particles and labelled with a Lugol solution. . . . . . . . . . . . . . . . . . . . . . . . 3.12 Images of the porous polymeric matrix of an interconnected network (on the left) and a non-interconnected network (on the right) (x 300). . . . 3.13 Images of the polymeric walls of non-interconnected networks prepared with 125-200 µm (a), 200-250 µm (b), and 250-400 µm NaCl particles (c) (x 300). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 µCT images before (a, c, e) and after (b, d, f) segmentation of doubly porous networks prepared with sintered 125-200 µm (a, b), 200-250 µm (c, d), and 250-400 µm (e, f) NaCl particles. . . . . . . . . . . . . . . . iv 3.15 µCT images before (a, c, e) and after (b, d, f) segmentation of doubly porous networks prepared with non-sintered 125-200 µm (a, b), 200-250 µm (c, d) , and 250-400 µm (e, f) NaCl particles. . . . . . . . . . . . . 3.16 Synchrotron images of doubly porous networks prepared with 125-200 µm sintered (a) and non-sintered NaCl particles (b). . . . . . . . . . . . 3.17 Synchrotron images of doubly porous networks prepared with 200-250 µm sintered (a) and non-sintered NaCl particles (b). . . . . . . . . . . . 3.18 Synchrotron images of doubly porous networks prepared with 250-400 µm sintered (a) and non-sintered NaCl particles (b). . . . . . . . . . . . 3.19 Synchrotron images of monoporous networks prepared with 200-250 µm sintered (a) and non-sintered NaCl particles (b). . . . . . . . . . . . . . 4.1 Double homogenization approach of the doubly-porous material. . . . . 4.2 Stokes-flow problem at the lower scale. . . . . . . . . . . . . . . . . . . 4.3 Darcy-Stokes flow problem at the intermediate scale. . . . . . . . . . . 4.4 Normal and the tangential unit vectors acting on S 1 . . . . . . . . . . . 4.5 Discretization of the unit cell with pixels (2d case). . . . . . . . . . . . 4.6 X-ray microtomography of the sample A (on the left) and sample B (on the right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 X-ray microtomography (on the left) and number of pixels as function of the grayscale (on the right). . . . . . . . . . . . . . . . . . . . . . . . 4.8 Variation of c s and c f with the grayscale X. . . . . . . . . . . . . . . . 4.9 Thresholding of the phase from the grayscale distribution. . . . . . . . 4.10 X-ray microtomography image (at the left) and distribution of I s (x) (at the right) for α = 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 (a) X-ray microtomography image and distribution of c f for α = 0, for α = 0.25, for α = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Distribution of I s in a cubic unit cell. . . . . . . . . . . . . . . . . . . . 4.13 X-ray microtomography (on the left), distribution of I s (x) (on the right). 4.14 Resistivity at each occurrences and average cumulated value of the resistivity for α = 0.25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Distribution of c s (on the left) and distribution of the local velocity for α = 0 and direction 1 (on the right). . . . . . . . . . . . . . . . . . . . 4.16 Distribution of c s (on the left) and distribution of the local velocity for α = 0.25 and direction 1 (on the right). . . . . . . . . . . . . . . . . . . 4.17 Distribution of c s (on the left) and distribution of the local velocity for α = 0.4 and direction 1(on the right). . . . . . . . . . . . . . . . . . . . 4.18 Distribution of the local velocity for α = 0.25 and direction 2(a), distribution of the local pressure for direction 1 (b) and direction 2 (c). . . . 4.19 X-ray microtomography images with the resolution (a) 186 × 186, (b) 300 × 300, (c) 400 × 400. . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Principle of the reduction of the image by a factor Q. . . . . . . . . . . 4.21 X-ray microtomography, distribution of I s (x) with the factor Q = 1, Q = 2, Q = 3 etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 4.22 Dimensionless macroscopic resistivity as a function of the reduction factor. (a) component 11, (b) components 12, (c) component 21, (d) component (22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.23 3-D resistivity, standard deviation and permeability of a cubic unit cell 4.24 Unit cell of the porous solid at the lower scale. Model obtained with a centered cubic array of overlapping spherical cavities. . . . . . . . . . . 5.1 X-ray microtomography of the microstructure A . . . . . . . . . . . . . 5.2 Resistivity at each occurrence and average cumulated value of the resistivity for α = 0.25 with the components 11, 12, 21, 22 . . . . . . . . . . 5.3 Resistivity as a function of the scale factor with the component 11, 12, 21, 22 (α = 0.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Synchrotron tomography image of microstructure C. . . . . . . . . . . . 5.5 Synchrotron tomography images (on the left) and number of pixels as a function of the greyscale (on the right) for microstructure C. . . . . . . 5.6 Thresholding of the phases as a function of the greyscale. . . . . . . . . 5.7 Representation of c f as a function of X. . . . . . . . . . . . . . . . . . . 5.8 Representation of c f as a function of X. . . . . . . . . . . . . . . . . . 5.9 Thresholded images as a function of s value. . . . . . . . . . . . . . . . 5.10 Representation of the dimensionless resistivity as a function of the occurrence (s=0.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Representation of the dimensionless resistivity of microstructure C as a function of s value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Picture of the cell containing a sample used to determine the water permeability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Picture of the X-ray microtomography device. . . . . . . . . . . . . . . B.2 Picture of the sample used for the large scale investigation. . . . . . . . B.3 Picture of the sample used for the small scale investigation. . . . . . . . B.4 Picture of the synchrotron X-ray tomography equipment equipment. . . vi List of Tables 1.1 Examples of porogen templates, experimental conditions and porous features of doubly porous polymeric materials obtained through the porogen templating approach. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Major advantages and limitations of the techniques implemented for the preparation of doubly porous polymers. . . . . . . . . . . . . . . . . . . 1.3 Permeability of various natural and synthetic porous polymers. . . . . . 2.1 Porous features of biporous PHEMA-based materials with non-interconnected and interconnected porous networks. . . . . . . . . . . . . . . . . . . . 2.2 Porous features of biporous networks in function of NaCl particles size. 2.3 Porous features of monoporous PHEMA-based materials as determined by MIP by varying the cross-linker structure. . . . . . . . . . . . . . . . 2.4 Porous features of biporous PHEMA-based material as determined by MIP by varying the cross-linker structure . . . . . . . . . . . . . . . . . 2.5 Glass transition temperature of monoporous materials prepared with DEGDMA, TriEGDMA, TetraEGDMA, and PEGDMA. . . . . . . . . 2.6 Apparent densities and porosity ratios of interconnected and non-interconnected biporous networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Permeability k of interconnected and non-interconnected biporous networks as a function of NaCl particles size. . . . . . . . . . . . . . . . . 2.8 Permeability k of interconnected and non-interconnected monoporous networks (large pores) as a function of NaCl particles size. . . . . . . . 3.1 Percentage of Lugol solution as a function of the sample and the protocol. 3.2 Void proportion as a function of the sample. . . . . . . . . . . . . . . . 4.1 Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 2-D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (186 × 186 unit cell). . . . . . . . vii 4.5 2-D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (300 × 300 unit cell). . . . . . . . 4.6 2-D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (400 × 400 unit cell). . . . . . . . 4.7 Effect of the unit cell on the porosity ratio of a porous polymeric material with non interconnected large pores. . . . . . . . . . . . . . . . . . . . 4.8 In plane components of the dimensionless average of the resistivity and its standard deviation, components of the dimensionless permeability as function of the reduction factor Q. . . . . . . . . . . . . . . . . . . . . 4.9 components of the dimensionless macroscopic resistivity, its standard deviation and the components of the dimensionless macroscopic permeability in 3D case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Components of the macroscopic permeability. . . . . . . . . . . . . . . 5.1 Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α=0 and ε=10 -2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0.25 and ε=10 -2 . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0.4 and ε=10 -2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Dimensionless resistivity, standard deviation and permeability in function of the scale factor for the components 11, 12, 21 and 22 (α = 0.25) 5.5 2D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (186 × 186 unit cell) . . . . . . . . 5.6 2D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (300 × 300 unit cell) . . . . . . . . 5.7 2D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (400 × 400 unit cell) . . . . . . . . 5.8 Dimensionless resistivity, standard deviation and permeability as a function of s for the components 11, 12, 21 and 22. . . . . . . . . . . . . . . B.1 Experimental conditions for the µCT observation of samples without using contrast agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Experimental conditions for the µCT observation of samples by using a contrast agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Experimental conditions of the synchrotron tomography observation of the samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Porous materials: classification and applications . . . . . . . . . . . . . 1.1.1 Classification of porous networks . . . . . . . . . . . . . . . . . 1.1.2 Applications of porous materials . . . . . . . . . . . . . . . . . . 1.2 Strategies applied for designing porous polymeric materials . . . . . . . 1.2.1 Porogen templating approach . . . . . . . . . . . . . . . . . . . 1.2.2 Thermally-induced phase separation . . . . . . . . . . . . . . . 1.2.3 High Internal Phase Emulsion . . . . . . . . . . . . . . . . . . . 1.2.4 Electrospinning . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Gas foaming process . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Morphological characterization techniques of porous materials . . . . . 1.3.1 Mercury intrusion porosimetry . . . . . . . . . . . . . . . . . . . 1.3.2 Gas sorption porosimetry . . . . . . . . . . . . . . . . . . . . . 1.3.3 Imaging techniques . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Transport properties of porous polymeric materials . . . . . . . . . . . 1.5 Computation of fluid flow through porous materials . . . . . . . . . . . 1.5.1 Computation of permeability of doubly porous media based on simplified unit cells . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Computation of fluid flow based on real µCT microstructures . 1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 ,

 11 region (a) describes a first category of closed pores which are totally isolated from their neighbors. They influence macroscopic properties as mechanical strength, thermal conductivity and bulk density but are inactive in some processes as fluid flow and gas adsorption. Furthermore, pores exhibiting a continuous channel of communication with the external surface of the solid, such as (b), (c), (d), (e) and (f ) in Figure1.1, are called open pores. Some of them can be open only at one end (such as (b) and (f )); they are then described as blind pores. Others may be open at two ends (through pores), such as pores (e) for instance. Pores can also be classified according to their morphological shape: cylindrical (either open (c) or blind (f )), ink-bottle shaped (b), funnel shaped (d) or slit-shaped. The roughness of the external surface, represented around (g), is close to the porosity concept but still different. To make the distinction, a convenient and simple convention is to consider that a rough surface is not porous unless it has irregularities that are deeper than they are wide.

Figure 1 . 1 :

 11 Figure 1.1: Schematic cross-section of a porous solid.

Figure 1 . 2 :

 12 Figure 1.2: Classification of porous media related to void space and solid matrix connectivity.

Figure 1 . 3 :

 13 Figure 1.3: Representation of the classification of porous media based on the storage capacity of the void and fractures: (a) fractures storage is more important, (b) smaller porosity storage is more important, and (c) the two porosity levels are equally distributed.

Figure 1 . 4 :

 14 Figure 1.4: Representation of the bimodal porosity in reservoir rocks: (a) periodic cell of the microporous medium, (b) periodic cell of the fractured medium.

Figure 1 . 5 :

 15 Figure 1.5: Schematic representation of the experimental process applied for the elaboration of doubly porous polymers via Thermally Induced Phase Separation process.

Figure 1 . 6 :

 16 Figure 1.6: Schematic representation of the experimental setup implemented for the preparation of porous polymers via the HIPE technique.

Figure 1 . 7 :

 17 Figure 1.7: Schematic representation of the experimental setup required for the electrospinning process.

Figure 1 . 8 :

 18 Figure 1.8: Schematic representation of the experimental device implemented for measuring the permeability of collagen scaffolds.

  computed by solving the Darcy/Stokes coupled problem at the intermediate scale. The Stokes equations are applied to the fluid flow in the fractures while the Darcy equation is associated to the fluid flow in the porous matrix. The computation of the permeability in monoporous microstructures is often performed by using FEM (85; 86; 87; 88), and more recently, a new strategy based on the Fast Fourier Transform has been also developed by Monchiet et. al..(89; 90) Such approach is the extension of an original method applied to elastic composites (91; 92; 93) and reduced the computer memory occupancy, compared to the FEM. It is worth noticing that the modelling of fluid flow in bi-porous solid encounter more difficulties due to the resolution of the coupled Darcy-Stokes equation at the intermediate scale. The discretization with FEM has been the subject of many researches in the literature with different strategies.(94; 95; 79) Its application to 3-D porous microstructures has been recently provided for the determination of the permeability of bi-porous polymers.(80) 1.5.1 Computation of permeability of doubly porous media based on simplified unit cells The determination of the permeability of doubly porous medium requires the resolution at the intermediate scale of the Darcy-Stokes coupled problem with the appropriate interface conditions at the frontier between the Stokes region and the Darcy region. Generally, the Beaver-Joseph-Saffman model is used at the interface.(81; 82) The first analytic solution has been provided by Markov considering a cylindrical or spherical pore fulfilled by a viscous fluid and embedded in an infinite porous medium.et al.(98) The fluid flow is described by the Darcy equation in the infinite solid and by the Stokes equation in the cylindrical/spherical pore.

Figure 1 . 9 :

 19 Figure 1.9: Two-scale homogenization of permeability using LBM and FEM.

Figure 1 . 10 :

 110 Figure 1.10: Representation of the fluid flow in the x-direction (a), in the y-direction (b) and in the z-direction (c).

Figure 1 .

 1 Figure 1.11: Grey-scale values histogram for the entire 3-D sample of soil aggregate.

Figure 2 . 1 :

 21 Figure 2.1: Representative scheme for the double porogen templating approach.

Figure 2 . 2 :

 22 Figure 2.2: Molecular structures of the functional monomer, i.e. HEMA, and of the ethylene glycol-based dimethacrylate crosslinkers used in this study.

2. 2

 2 Morphological characterization 2.2.1 Effect of NaCl particles packing on morphologyThe present study is specifically focused on studying the effect of NaCl particles packing on biporous PHEMA-based networks by comparing sintered vs. non-sintered NaCl particles with a size range comprised between 200 and 250 µm. The aim of this study relies on understanding the effect of NaCl particles packing on the pore interconnectivity and size distribution. The polymerization mixture was constituted of a fixed HEMA/EGDMA molar ratio of 70/30 mol. % in conjunction with 80 vol. % i-PrOH with respect to the comonomer total volume. The MIP profiles and SEM micrographs of the resulting doubly porous PHEMA-based materials are respectively shown in Figure 2.3 and Figure 2.4, while their main porous features are gathered in Table2.1. The biporous structure prepared from non-sintered NaCl particles (Figure2.3(left)) exhibited an ill-defined dual porosity, along with pore sizes with a first porosity level centered around 200 µm and a lower porosity level centered around 6 µm. In sharp contrast, doubly porous networks prepared from SPS-sintered NaCl particles presented a well-defined bimodal porosity with a first pore size distribution of about 60 µm and a second one around 1.5 µm, as observed in Figure 2.3(right).

Figure 2 . 3 :

 23 Figure 2.3: MIP profiles of non-interconnected (on the left) and interconnected (on the right) doubly porous networks.

Figure 2 . 4 :

 24 Figure 2.4: SEM micrographs of non-interconnected (A & B) and interconnected porous networks (C & D).

Figure 2 . 5 :

 25 Figure 2.5: (a) Pore size distribution profiles and (b) cumulative intrusion volumes of interconnected biporous networks in function of NaCl particles size as determined by MIP.

Figure 2 .

 2 8a and Figure 2.8b correspond to the MIP profiles of DEGDMA-based biporous structure and its monoporous counterpart, respectively.

Figure 2 .

 2 8c and Figure 2.8d illustrate TriEGDMA-based biporous structure and its monoporous counterpart, respectively.The collapse of the porous structure was confirmed in both cases, as the second porosity level was not observed in MIP profile of the biporous material. Indeed, for DEGDMA-based material, the peak centered on 5 µm corresponding to the lower porosity level (Figure 2.8b) totally disappeared in the biporous material (Figure 2.8a), while a new peak attributed to the upper porosity level was observed around 55 µm. The same morphology was observed for TriEGDMA-based materials (Figure 2.8c and Figure 2.8 d).

Figure 2 . 6 :Table 2 . 3 :

 2623 Figure 2.6: SEM micrographs of monoporous networks with small pores prepared with a) DEGDMA, b) TriEGDMA , c) TetraEGDMA, d) PEGDMA.

Figure 2 . 7 : 4 :Figure 2 . 8 :

 27428 Figure 2.7: SEM micrographs of biporous networks prepared with a) DEGDMA, b) TriEGDMA, c) TetraEGDMA, d) PEGDMA.

Figure 2 . 9 ,

 29 was determined on monoporous materials prepared with DEGDMA, TriEGDMA, TetraEGDMA and PEGDMA crosslinkers. The mechanical response of the samples as they were deformed under periodic stress was measured. The polymers were cooled down to -140 • C and heated up to 150 • C at 3 • C.min -1 . The frequency of the applied stress was fixed at 1 Hz and the deformation was equal to 0.1 %. Monoporous EGDMA-based materials and biporous samples had not been analysed because of their high friability (biporous materials prepared with EGDMA, DEGDMA, TriEGDMA) or their softness (biporous materials prepared with TetraEGDMA and PEGDMA). Their mechanical properties were then below the detection threshold of the apparatus. A good alternative would have been to work with bigger samples but the equipment was not adapted. As it can be observed in Figure2.9, G' reached a peak before drastically decreasing. The maximum shear storage of PEGDA-based materials was around 25 MPa and decreased down to 15 MPa with the chain length.

Figure 2 . 9 : 5 :

 295 Figure 2.9: Storage modulus G' of DEGDMA, TriEGDMA, TetraEGDMA, and PEGDMA-based monoporous materials as a function of temperature.

Figure 2 . 10 :

 210 Figure 2.10: Water uptake of non-interconnected and interconnected biporous networks as a function of time (*P=porosity ratio).

Figure 2 . 11 :

 211 Figure 2.11: Water uptake of monoporous networks with interconnected and noninterconnected large pores and small pores as a function of time (*P=porosity ratio).

( 3 . 3 )

 33 as the ratio SD/SO:

The 3 -

 3 D data set of integrated attenuations is converted into the 3-D map of local attenuation coefficients by a numerical algorithm which performs this "inverse Radon transform. Various algorithms have been proposed, among which the "filtered back projection" (FBP), extended from parallel configuration to conical one by means of the Feldkamp, Davis & Kress (FDK) extension is the most popular one.(3) After this step, a volumetric image whose elementary volumes are called voxels (3-D pixels) is obtained. Each voxel is associated with a grayscale that is related to the local attenuation coefficient.

Figure 3 . 1 :

 31 Figure 3.1: Schematic illustration of µCT acquisition and reconstruction processes.

Figure 3 . 2 :

 32 Figure 3.2: µCT images of biporous networks prepared with non-sintered (a) and sintered (b) NaCl particles.Scale bars represent 500 µm.

Figure 3 . 3 :

 33 Figure 3.3: µCT images of porous PHEMA-based networks:(i) exhibiting noninterconnected large pores before (a) and after (b) image processing, and corresponding image after segmentation (c). (ii) exhibiting interconnected large pores (d).Scale bars represent 500 µm.

Figure 3 . 4 :

 34 Figure 3.4: µCT images of biporous networks prepared with sintered 125-200 µm (a) and 200-250 µm (b) NaCl particles.

Figure 3 . 5 :

 35 Figure 3.5: µCT image (left, voxel size = 2 µm) and SEM image (right) of a biporous network prepared with 200-250 µm NaCl particles.

Figure 3 . 6 :

 36 Figure 3.6: Representation of the grey level of pixels along a line that passes through the porous matrix.

Table 3 . 1 :

 31 Percentage of Lugol solution as a function of the sample and the protocol.

Figure 3 . 7 :

 37 Figure 3.7: µCT images of labelled doubly porous networks prepared with 125-200 µm sintered (a) and non-sintered NaCl particles (b) (zoom x 150 % ).

Figure 3 . 8 :

 38 Figure 3.8: µCT images of labelled doubly porous networks prepared with 200-250 µm sintered (c) and non-sintered NaCl particles (d) (zoom x 150 %).

Figure 3 . 9 :

 39 Figure 3.9: µCT images of labelled doubly porous networks prepared with 250-400 µm sintered (e) and non-sintered NaCl particles (f) (zoom x 150 % ).

Figure 3 .

 3 10 showed an example of two images of an interconnected network (125-200 µm NaCl particle size) before (Figure3.10a) and after (Figure3.10b) image processing. We could notice that the noise was highly reduced after the processing.

Figure 3 . 10 :

 310 Figure 3.10: Doubly porous networks prepared with 125-200 µm NaCl particles before (a) and after (b) image processing .

Figure 3 . 11 :

 311 Figure 3.11: Doubly porous networks prepared with 250-400 µm NaCl particles and labelled with a Lugol solution.

Figure 3 . 12 :

 312 Figure 3.12: Images of the porous polymeric matrix of an interconnected network (on the left) and a non-interconnected network (on the right) (x 300).

Figure 3 . 13 :

 313 Figure 3.13: Images of the polymeric walls of non-interconnected networks prepared with 125-200 µm (a), 200-250 µm (b), and 250-400 µm NaCl particles (c) (x 300).

76 Table 3 . 2 :

 7632 µm non-interconnected NaCl particles 75 200-250 µm interconnected NaCl particles 89 200-250 µm non-interconnected NaCl particles 67 250-400 µm interconnected NaCl particles 83 250-400 µm non-interconnected NaCl particles Void proportion as a function of the sample.

Figure 3 . 14 :

 314 Figure 3.14: µCT images before (a, c, e) and after (b, d, f) segmentation of doubly porous networks prepared with sintered 125-200 µm (a, b), 200-250 µm (c, d), and 250-400 µm (e, f) NaCl particles.

Figure 3 . 15 :

 315 Figure 3.15: µCT images before (a, c, e) and after (b, d, f) segmentation of doubly porous networks prepared with non-sintered 125-200 µm (a, b), 200-250 µm (c, d) , and 250-400 µm (e, f) NaCl particles.

Figure 3 . 16 :

 316 Figure 3.16: Synchrotron images of doubly porous networks prepared with 125-200 µm sintered (a) and non-sintered NaCl particles (b).

Figure 3 . 17 :

 317 Figure 3.17: Synchrotron images of doubly porous networks prepared with 200-250 µm sintered (a) and non-sintered NaCl particles (b).

Figure 3 . 18 :

 318 Figure 3.18: Synchrotron images of doubly porous networks prepared with 250-400 µm sintered (a) and non-sintered NaCl particles (b).

Figure 3 . 19 :

 319 Figure 3.19: Synchrotron images of monoporous networks prepared with 200-250 µm sintered (a) and non-sintered NaCl particles (b).

Figure 4 . 1 : 4 . 1 .

 4141 Figure 4.1: Double homogenization approach of the doubly-porous material.

Figure 4 . 2 :

 42 Figure 4.2: Stokes-flow problem at the lower scale.

Figure 4 . 3 :

 43 Figure 4.3: Darcy-Stokes flow problem at the intermediate scale.

  (3; 4; 5) Indeed, in some works (Barenblatt et al. (8), Warren and Root (27), Wilson and Aifantis (28)) the authors provide a macroscopic description which uses two pressures, a pressure for the smaller pores and a pressure for the larger ones. The asymptotic series expansion method developed in Auriault studies proved that only one pressure is needed for the macroscopic filtration law.(3; 4; 5) The coupled Darcy-Stokes problem uses the Stokes equation in the larger pores:

(4. 41 )

 41 The Stokes phase is then replaced by a Darcy one having the permeability k = +∞. The set of equations (4.39), (4.40) with equation (4.41) is called Darcy-Darcy problem. It replaces the initial Darcy-Stokes problem.

  43) (both the normal and tangential components). When the Darcy-Stokes problem is replaced by the equivalent Darcy-Darcy problem only the conditions (4.44) and (4.42) are satisfied. No conditions for the tangential velocity are involved at the frontier between two Darcy media. The consideration of the Darcy-Darcy problem instead of the Darcy-Stokes equation is motivated by the fact that conventional numerical methods based on Fast Fourier Transform could be used. No FFT iterative schemes has been yet formulated to compute the solution of the Darcy-Stokes problem. This is the subject of the next chapter.

Figure 4 . 5 :

 45 Figure 4.5: Discretization of the unit cell with pixels (2d case).

Figure 4 . 6 .

 46 For the calculations, squared and cubic unit cell are extracted.

Figure 4 . 6 :

 46 Figure 4.6: X-ray microtomography of the sample A (on the left) and sample B (on the right).

Figure 4 . 7 :

 47 Figure 4.7: X-ray microtomography (on the left) and number of pixels as function of the grayscale (on the right).

Figure 4 . 8 :

 48 Figure 4.8: Variation of c s and c f with the grayscale X.

Figure 4 . 9 :

 49 Figure 4.9: Thresholding of the phase from the grayscale distribution.

Figure 4 .

 4 Figure 4.10: X-ray microtomography image (at the left) and distribution of I s (x) (at the right) for α = 0.25.

( 4 . 5 )

 45 ), the computations will be performed for different values of α, namely, α = 0, α = 0.25 and α = 0.4. Coefficient α can carry on to 0.5, however, at this value, there is percolation of the large pores and the FFT schemes is not applicable since the series diverges. This is due to the simplification of the initial Darcy-Stokes problem by a Darcy-Darcy one with the value +∞ for the permeability in the large pores. The Darcy-Darcy approximation is only valid when the fluid phase is not interconnected. However, with α = 0.5, some open channels are observed between the large pores. When α = 0, one can observe in Figure4.11 some mixed pixels within the volume of the large pores while they should be confined next to the interface. This suggests that the value α = 0 is not pertinent. Still in Figure4.11, when α = 0.4, there is no mixed pixels inside the large pores but not at the interface too. Finally, the intermediate value α = 0.25 appears to lead to the more realistic distribution of c f . This value has been hereafter used for the computation of the effective permeability. Note that one computation is however provided in the next with the values α = 0 and α = 0.4 just to evaluate the influence of this parameter on the macroscopic permeability.

Figure 4 .

 4 Figure 4.11: (a) X-ray microtomography image and distribution of c f for α = 0, for α = 0.25, for α = 0.5.

Figure 4 . 12 :

 412 Figure 4.12: Distribution of I s in a cubic unit cell.

Figure 4 .

 4 Figure 4.13: X-ray microtomography (on the left), distribution of I s (x) (on the right).

Figure 4 . 14 :

 414 Figure 4.14: Resistivity at each occurrences and average cumulated value of the resistivity for α = 0.25.

Table 4 . 1 , 4 . 2

 4142 and 4.3 give the value of the ensemble average (or cumulated average) dimensionless macroscopic resistivity for α = 0, α = 0.25 and α = 0.4 respectively. Let us recall that the FFT algorithm delivers the values of the resistivity. All given in these tables are the cumulated average values obtained from Eq. (4.64). The components of the dimensionless macroscopic permeability are then determined by inverting the statistic average values of dimensionless macroscopic resistivity. The direction 1 is horizontal and direction 2 is vertical. The components 12 and 21 are different while the resistivity is a symmetric two order tensor. In fact, component 12, is determined by computing the component 2 of the macroscopic velocity field due to an applied pressure gradient in direction 1. Conversely, the component 21 is determined by computing the component 2 of the macroscopic velocity due to an applied pressure gradient in direction 1. Since the macroscopic velocity is determined with a truncated Neumann series, it results to two different values for the off-diagonal components.

Figure 4 .

 4 15 (α = 0), Figure 4.16 (α = 0.25) and Figure 4.17

Figure 4 . 15 :

 415 Figure 4.15: Distribution of c s (on the left) and distribution of the local velocity for α = 0 and direction 1 (on the right).

Figure 4 . 16 :

 416 Figure 4.16: Distribution of c s (on the left) and distribution of the local velocity for α = 0.25 and direction 1 (on the right).

Figure 4 . 17 :

 417 Figure 4.17: Distribution of c s (on the left) and distribution of the local velocity for α = 0.4 and direction 1(on the right).

Figure 4 . 18 :

 418 Figure 4.18: Distribution of the local velocity for α = 0.25 and direction 2(a), distribution of the local pressure for direction 1 (b) and direction 2 (c).

4. 6 . 3

 63 Effect of the unit cell dimensionNow, we investigate the effect of the size of the unit cell on the effective permeability. The computations are first performed on unit cells of the first bi-porous material generated by non-sieved NaCl particles. Three unit cells with 186 × 186 pixels, 300 × 300 pixels and 400 × 400 pixels are considered and given in Figure 4.19.

Figure 4 .

 4 Figure 4.19: X-ray microtomography images with the resolution (a) 186 × 186, (b) 300 × 300, (c) 400 × 400.

4. 6 . 4

 64 Effect of the image resolutionIn order to reduce the time computations and the memory requirement, the computations can be performed by reducing the resolution of the image. The latter are obtained by replacing Q × Q pixels by only one pixel (seeFigure 4.20). By doing so, the total number of pixels is reduced by a factor Q 2 . The value of I s (x) in the equivalent pixel is obtained by its average value over the Q × Q pixels.

Figure 4 . 20 :

 420 Figure 4.20: Principle of the reduction of the image by a factor Q.

Figure 4 .

 4 Figure 4.21: X-ray microtomography, distribution of I s (x) with the factor Q = 1, Q = 2, Q = 3 etc.

Figure 4 .

 4 Figure 4.22 shows the components of the dimensionless macroscopic permeability as function of the reduction factor. It is observed that the resistivity components increase with the reduction factor but no convergence is observed. All the components of the resistivity and permeability are reported inTable 4.8.

Figure 4 . 22 :

 422 Figure 4.22: Dimensionless macroscopic resistivity as a function of the reduction factor. (a) component 11, (b) components 12, (c) component 21, (d) component (22).

Figure 4 .

 4 Figure 4.23: 3-D resistivity, standard deviation and permeability of a cubic unit cell

Figure 4 . 24 .

 424 The spherical pores at the corner are connected with the pore at the center of the cube along the diagonals.

Figure 4 . 24 :

 424 Figure 4.24: Unit cell of the porous solid at the lower scale. Model obtained with a centered cubic array of overlapping spherical cavities.

.54) 5 . 3

 53 Resolution with a basic FFT based iterative scheme5.3.1 Integral equationThe Brinkman problem (5.1) can be put into the form (5.11)-(5.15) by considering the following expression for the pair of eigentensors:

Figure 5 . 2

 52 provides the value of the dimensionless resistivity for each occurrence and the average cumulated value of the resistivity for α = 0.25. Let us recall that α = 0.25 corresponds to the value for which the thresholded image is the most realistic. The numerical results for α = 0, 0.25 and 0.4 are reported in Tables 5.1, 5.2, 5.3. Direction 1 is horizontal and direction 2 is vertical. The results clearly show that the permeability increases with α. The components 11 and 22 are very close regardless α value. The components 12 and 21 are also very similar and close to zero. We can notice that the component 21 takes a value less than zero for α = 0.25 and α = 0.4. In a general way, the values of permeability obtained by darcy-darcy solution (see chapter 4) are widely higher than the case of the Stokes-Darcy solution.

Figure 5 . 2 :

 52 Figure 5.2: Resistivity at each occurrence and average cumulated value of the resistivity for α = 0.25 with the components 11, 12, 21, 22

Figure 5 . 3

 53 illustrates the dimensionless resistivity as function of the scale factor with the components 11, 12, 21 and 22. The curves show that the resistivity increases with the scale factor value in the case of the components 11 and 22. Moreover, we can observe a saturation of the resistivity when the scale factor is less than 10 -5 . The limit ε → 0 can be also determined by considering the Darcy equation for both phases with a null resistivity in the large pores. These limits are 0.0752 (for the component 11) and 0.0609 (for the component 22) and correspond in fact to the resistivity computed in the last chapter.

Figure 5 . 3 :

 53 Figure 5.3: Resistivity as a function of the scale factor with the component 11, 12, 21, 22 (α = 0.25)

5. 6 5 . 6 . 1

 6561 Application to a biporous polymer with interconnected large pores The biporous materialThe FFT method is now applied to a biporous material with interconnected large pores. A synchrotron image (see Chapter 3) is provided in Figure5.4. The samples are composed of interconnected large pores coming from 250 -400µm NaCl particles. In this case, the medium is highly porous and composed of very thin polymeric walls separating the fluid and the solid phases. We thus need to build a new thresholding strategy because the morphology is different from microstructures A and B. The aim is to study the role of the polymeric walls porosity upon the macroscopic permeability. The images have the dimension 2000 × 2000. In order to reduce the memory computation requirement, the images are reduced by a factor Q = 8 following the methodology already depicted in chapter 4, section 4.6.4.

Figure 5 . 4 :

 54 Figure 5.4: Synchrotron tomography image of microstructure C.

  change of variable must then be used in the Brinkman equations:v = l 2 2 |J | µ v, p = |J |l 2 p, J = |J |j, x = l 2 x(5.97)

γ

  (x) = c s ε -2 , µ(x) = c f (5.99)5.6.3 Thresholding of the phases

Figure 5 . 5

 55 Figure5.5 shows 3 different slices (on the left) and the related distributions of pixels as a function of the greyscale (on the right). On these figures, one single peak is observable. Let us call X the value of the greyscale and X 1 the value of X corresponding to the peak. It is important to recall that we observed two peaks when the large pores were not connected (see Chapter 4).

Figure 5 . 5 :

 55 Figure 5.5: Synchrotron tomography images (on the left) and number of pixels as a function of the greyscale (on the right) for microstructure C.

Figure 5 . 6 :

 56 Figure 5.6: Thresholding of the phases as a function of the greyscale.

Figure 5 . 7 .

 57 If X is inferior to X 1 , c f is equal to 1, meaning that the pixel only contains fluid. If X is superior to X 1 , the pixel is mixed. For large values of the parameter s, the exponential decay is very rapid and a majority of mixed pixels have a volume fraction of fluid close to zero and they essentially contain the porous solid. Conversely, if the value of the parameter s is chosen very small, the exponential decay is very low and the mixed pixels contain a large proportion of fluid.

Figure 5 . 7 :

 57 Figure 5.7: Representation of c f as a function of X.

Figure 5 . 8 :

 58 Figure 5.8: Representation of c f as a function of X.

Figure 5 . 9

 59 Figure 5.9 shows the thresholded images as a function of parameter s. It can be observed that the thresholded image with s = 0.1 is very close to the real synchrotron slice. Indeed, the distinction between the polymeric matrix (dark blue) and the pores (yellow) is quite clear. By increasing the value of s, the volume fraction of fluid in the mixed pixel increases and the pixels are light blue.

Figure 5 . 9 :

 59 Figure 5.9: Thresholded images as a function of s value.

Figure 5 .

 5 10 represents the dimensionless resistivity as a function of the occurrence.

Figure 5 . 10 :

 510 Figure 5.10: Representation of the dimensionless resistivity as a function of the occurrence (s=0.1).

Figure 5 .

 5 Figure 5.11 shows the dimensionless resistivity as a function of s value. For the components 11 and 22, we can observe a gradual decrease of the resistivity from 9 to 8 (corresponding to s=0.1 and s=1, respectively) and from 8 to 4 (corresponding to s=1 and s=10).

Figure 5 . 11 :

 511 Figure 5.11: Representation of the dimensionless resistivity of microstructure C as a function of s value.

Figure A. 1 :

 1 Figure A.1: Picture of the cell containing a sample used to determine the water permeability.

Figure B. 3 : 1 :

 31 Figure B.3: Picture of the sample used for the small scale investigation.

Figure B. 4 :

 4 Figure B.4: Picture of the synchrotron X-ray tomography equipment equipment.

  2 PHEMA-based doubly porous materials prepared by the double porogen templating approach 2.1 The double porogen templating approach . . . . . . . . . . . . . . . . . 2.2 Morphological characterization . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Effect of NaCl particles packing on morphology . . . . . . . . . 2.2.2 Effect of NaCl particles size on morphology . . . . . . . . . . . 2.2.3 Effect of crosslinker on morphology and mechanical properties . 2.3 Determination of apparent density and porosity ratio values . . . . . . 2.4 Study of swelling properties in water . . . . . . . . . . . . . . . . . . . 2.5 Experimental investigation of transport properties . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 3 Characterization of PHEMA-based doubly porous materials microstructures by means of X-ray microtomography 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Observation of doubly porous polymers without using contrast agent . 3.2.1 Qualitative analysis of the µCT images . . . . . . . . . . . . . . 3.2.2 Quantitative analysis of the µCT image . . . . . . . . . . . . . . 3.3 Observation of doubly porous polymers after the incorporation of a contrast agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Qualitative analysis of the µCT images . . . . . . . . . . . . . . 3.3.2 Quantitative analysis of the µCT image . . . . . . . . . . . . . . 3.4 Investigation of microstructures by means of synchrotron tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Qualitative analysis of synchrotron images coming from doubly porous networks . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Qualitative analysis of synchrotron images coming from monoporous networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Computation of macroscopic permeability by using a simplified equivalent Darcy medium 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Equations of flow in doubly porous materials . . . . . . . . . . . . . . . 4.2.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . 4.3 Normalization and simplification of the double homogenization problem 4.4 Resolution of the Darcy-Darcy problem with a FFT-method . . . . . . 4.4.1 Reformulation of the problem . . . . . . . . . . . . . . . . . . . 4.4.2 The FFT-iterative scheme . . . . . . . . . . . . . . . . . . . . . 4.5 Thresholding of the phases . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Principle of the method . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Effect of α on the macroscopic permeability . . . . . . . . . . . 4.6.3 Effect of the unit cell dimension . . . . . . . . . . . . . . . . . . 4.6.4 Effect of the image resolution . . . . . . . . . . . . . . . . . . . 4.6.5 Computations on a cubic cell . . . . . . . . . . . . . . . . . . . 4.6.6 Experimental vs. computed permeability . . . . . . . . . . . . . 4.7

Table 1 .

 1 2 presents the major advantages and limitations of the five methodologies above-mentioned for the engineering of doubly porous polymers.

	Methodology	Major advantage(s)		Major limitation(s)
	double porogen templating	easy to carry out, pre-	destruction of the template
	approach	cise	control	of	over	which has to be easy to re-
		meso/macropores	struc-	move
		ture
	thermally-induced phase sep-	can form an intrinsically in-	difficult to find a suitable sol-
	aration	terconnected polymer net-	vent to dissolve the crystal-
		work in one simple step		lized polymers
	high internal phase emulsion 3-D highly interconnected	weak mechanical strength,
		porous structures, very high	low specific surface area,poor
		porosity (74-95 %)			mechanical resistance during
						flow-through processes
	electrospinning	very simple setup, samples	difficult to produce uniform
		often uniform and do not re-	mats with a fiber diameter
		quire expensive purification,	lower than 50 nm
		very high open porosity cou-
		pled with remarkable specific
		surface area, good structural
		mechanical properties
	gas foaming	no organic solvent			yields a mostly closed pore
						structure, generates a mono-
						porous network if not com-
						bined to particle leaching
	Table 1.2: Major advantages and limitations of the techniques implemented for the
	preparation of doubly porous polymers.

Table 2 .

 2

	Porous characteristics	Non-	Interconnected
		interconnected	network
		network
	Large pore size (µm) a	200	60
	Small pore size (µm) a	6	1.5
	Upper to lower porosity ratio (vol.%) a	n.d. c	80/20
	Total pore volume (mL.g -1 ) a	6.5	6.0
	Porosity ratio (%) a	80	87
	Specific surface area (m 2 .g -1 ) b	80	105
	a Values as determined by MIP
	b Values as determined by nitrogen sorption measurements (BET method)
	c This ratio could not be determined properly

1: Porous features of biporous PHEMA-based materials with noninterconnected and interconnected porous networks.

Table 2 .

 2 2: Porous features of biporous networks in function of NaCl particles size.

	Upper to lower porosity ratio a (vol. %) 83/17	80/20	77/23
	Total pore volume a (mL.g -1 )	4.5	6	10.7
	Porosity ratio a ( %)	77	87	88
	Specific surface area (m 2 .g -1 ) b	95	105	177

a Values as determined by MIP b Values as determined by nitrogen sorption measurements (BET method)

  Price et al. demonstrated that the repetition of ethylene oxyde units (i.e. -(CH 2 -CH 2 -O)-) in EGDMA, TetraEGDA and PEGDA increased the flexibility of the crosslinking chain.(11) MIP results (Table2.4) were in good agreement with MEB observations as the porosity ratio decreased dramatically from 90 % (in the case of EGDMA-based network) to 67 % (DEGDMA-based network) before reaching an almost constant value comprised between 54 and 57 %. Furthermore, the pore size seemed not to depend on the crosslinker length.

Table 2 . 8

 28 -12 m 2 )

	Permeability k 1 (10 -13 m 2 )

: Permeability k of interconnected and non-interconnected monoporous networks (large pores) as a function of NaCl particles size.

Table 4 .

 4 2: Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0.25

	Component	11	12	21	22
	Resistivity	0.1490 -0.0018 -0.0018 0.1375
	Standard deviation 0.0319 0.0132 0.0133 0.0266
	Permeability	6.710	0.08	0.08	7.271
	Table 4.1: Components of the dimensionless macroscopic resistivity, standard deviation
	and components of the dimensionless macroscopic permeability for α = 0
	Component	11	12	21	22
	Resistivity	0.0764 0.0010 0.0011 0.0663
	Standard deviation 0.0280 0.0141 0.0140 0.0344
	Permeability	13.09 -0.1974 -0.2172 15.07
	Component	11	12	21	22
	Resistivity	0.0367 0.0019 0.0020 0.0373
	Standard deviation 0.0297 0.0090 0.0088 0.0381
	Permeability	27.20 -1.392 -1.465 26.78

Table 4 .

 4 3: Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0.4

Table 4 .

 4 4: 2-D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (186 × 186 unit cell).

	Component	11	12	21	22
	Resistivity	0.0603 -0.0007 -0.001 0.0893
	Standard deviation 0.0245 0.0074 0.0076 0.1381
	Permeability	16.58 -0.1300 0.1857 11.20
	Table 4.5: 2-D Resistivity, standard deviation and permeability of a porous material
	with non interconnected large pores (300 × 300 unit cell).

Table 4 .

 4 6: 2-D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (400 × 400 unit cell).

	Size	Average porosity Standard deviation
	186 × 186	0.64	0.05
	300 × 300	0.66	0.03
	400 × 400	0.64	0.05

Table 4 .

 4 7: Effect of the unit cell on the porosity ratio of a porous polymeric material with non interconnected large pores.

Table 4 .

 4 ). 8: In plane components of the dimensionless average of the resistivity and its standard deviation, components of the dimensionless permeability as function of the reduction factor Q.

	Q Component Resistivity Standard deviation Permeability
	11	0.0451	0.0198	22.13
	12	-0.0044	0.0063	2.777
	21	-0.0040	0.0065	2.530
	22	0.0361	0.0176	27.66
	11	0.0494	0.0210	20.20
	12	-0.0040	0.0064	2.018
	21	-0.0040	0.0064	2,054
	22	0.0396	0.0187	25.20
	11	0.0513	0.0219	19.49
	12	-0.0033	0.0066	1.619
	21	-0.0027	0.0069	1.585
	22	0.0410	0.0192	24.39
	11	0.0524	0.0223	19.07
	12	-0.0030	0.0070	1.219
	21	-0.0030	0.0070	1.219
	22	0.0420	0.0195	23.77
	11	0.0550	0.0217	18.18
	12	-0.0028	0.0069	1.188
	21	-0.0028	0.0069	1.158
	22	0.0452	0.0176	22.09

Table 4 .

 4 9. The components of the macroscopic permeability are reported on Table4.10. It is observed that the components 11 and 22 are very closed to the experimental measure of the permeability that is 10 -13 . 1.06 × 10 -13

	12	-7.20 × 10 -15
	13	-1.11 × 10 -15
	21	-8.96 × 10 -15
	22	1.10 × 10 -13
	23	-6.36 × 10 -15
	31	-1.50 × 10 -15
	32	-6.15 × 10 -15
	33	7.65 × 10 -14

Table 4 .

 4 10: Components of the macroscopic permeability.

Table 5 . 1 :

 51 Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α=0 and ε=10 -2

	Component	11	12	21	22
	Resistivity	0.1835 -0.00057 -0.00055 0.1642
	Standard deviation 0.0291 0.0130	0.0130 0.0335
	Permeability	5.4468 0.0497	-0.0431 6.0886
	Table 5.2: Components of the dimensionless macroscopic resistivity, standard devia-
	tion and components of the dimensionless macroscopic permeability for α = 0.25 and
	ε=10 -2
		140

Table 5 .

 5 3: Components of the dimensionless macroscopic resistivity, standard deviation and components of the dimensionless macroscopic permeability for α = 0.4 and ε=10 -2

	5.5.2.2 Effect of ε

Table 5 .

 5 5, 5.6 and 5.7. The results show no significant influence of the unit cell dimension on the values of the effective permeability. 5: 2D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (186 × 186 unit cell)

	component	11	12	21	22
	resistivity	0.0802 -0.0046 -0.0023 0.0668
	standard deviation 0.0490 0.0122 0.0115 0.0586
	permeability	12.4633 0.8603 0.4301 14.9503
	component	11	12	21	22
	resistivity	0.0673 -0.0007 -0.0009	0.062
	standard deviation 0.0204 0.0079 0.0078 0.0212
	permeability	14.8588 0.1677 0.2157 16.1290

Table 5 .

 5 6: 2D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (300 × 300 unit cell)

	component	11	12	21	22
	resistivity	0.1075 0.0003 0.0007 0.1154
	standard deviation 0.1072 0.0121	0.012 0.1045
	permeability	9.3023 -0.0241 -0.0564 8.6655

Table 5 .

 5 7: 2D Resistivity, standard deviation and permeability of a porous material with non interconnected large pores (400 × 400 unit cell)

Table 5 .

 5 8: Dimensionless resistivity, standard deviation and permeability as a function of s for the components 11, 12, 21 and 22.

	s value Component Resistivity Standard deviation Permeability
		11	9.0743	0.8236	0.1102
	0.1	12	-0.0064	0.0190	0
		21	-0.0064	0.0190	0
		22	9.0747	0.8139	0.1101
		11	8.7437	0.7787	0.1143
	0.5	12	-0.0053	0.0160	0
		21	-0.0053	0.0160	0
		22	8.7443	0.7699	0.1143
		11	8.3508	0.7274	0.1197
	1	12	-0.0044	0.0137	0
		21	-0.0044	0.0137	0
		22	8.3516	0.7192	0.1197
		11	4.45	0.3586	0.2247
	10	12	-0.0003	0.0030	0.0003
		21	-0.0003	0.0030	0
		22	4.4511	0.3566	0.2246
		11	2.8856	0.2569	0.3465
	20	12	0	0.0013	0
		21	0	0.0013	0
		22	2.8864	0.2563	0.3464
		11	2.1395	0.2063	0.4673
	30	12	0	0.0007	0
		21	0	0.0007	0
		22	2.1400	0.2060	0.4672

Acknowledgements

). Il s'inscrit dans le cadre d'une collaboration de recherche entre l'équipe "Systèmes Complexes Polymères (SPC)" de l'Institut de Chimie et des Matériaux de de doctorat de 3 ans. Ensuite, je remercie les membres du jury d'avoir accepté de juger mon travail : Pr Jannick Duchet-Rumeau (rapporteur), Pr Christian Geindreau (rapporteur), Pr Helène Dumontet (examinateur), Dr Patrick Kekicheff (examinateur) d'avoir accepté de juger mon travail. Mes remerciements à Dr Latroche (ICMPE), Pr Valérie Langlois (SPC), Pr Salah Naili (MSME) et , son énergie positive inébranlable, ses encouragements ainsi que la confiance qu'il m'a accordée durant ces 3 années de thèse ; Dr Vincent Monchiet pour sa disponibilité, pour tout le temps qu'il m'a accordé et pour son aide précieuse; Dr Michel Bornert pour ses bons conseils et pour tout ce qu'il m'a appris, Dr Benjamin Le Droumaguet pour nos riches échanges, et sa bienveillance. Egalement, mes remerciements à tous ceux qui m'ont aidé de près ou de loin à mener à bien mes travaux de thèse. Un merci particulier à Patrick Aimedieu pour

Chapter 4

Computation of macroscopic permeability by using a simplified equivalent Darcy medium

Introduction

In this chapter, we determined the macroscopic permeability of the doubly porous polymers coming from µCT images. In this study, we only consider microstructures with interconnected smaller pores but non interconnected larger pores. The methodology is based on a double homogenization approach. The first homogenization consists in computing the permeability associated with the second porosity. At an intermediate scale, that of the first porosity, we have to solve a coupled Darcy/Stokes problem. Because the larger pores are not interconnected, it is possible to simplify the Darcy/Stokes problem by a Darcy/Darcy one that is obtained by replacing the Stokes phase by a Darcy one with an infinite permeability. Advantageously, we can employ standard FFT code to solve the Darcy/Darcy problem. The method is applied to 2-D and 3-D microstructures. A methodology is provided for the thresholding of the phase which uses composite pixels at the interface between the fluid and porous solid phases. Finally, the permeability is computed for various occurrences and the statistic average values are provided and discussed.

Equations of flow in doubly porous materials 4.2.1 Statement of the problem

Consider a Representative Volume Element (RVE) of a doubly porous solid. Both the first and the second porosity are fulfilled by a newtonian viscous fluid with the dynamic viscosity µ. The material contains two scales : l 1 and l 2 which are characteristic of the first and the second porosity. The second porosity is related to the smaller pores while the first porosity refers to the larger pores. Due to the presence of three scales, that of the first and second porosity and the macroscopic scale, a double homogenization approach could be used to determine the mass transfer properties at the macroscopic Within the porous solid, the fluid flow obeys to the Darcy law with the incompressibility condition: v = -k (2) µ (∇p + J ) ∀x ∈ V s1 (4.9) div v = 0 ∀x ∈ V s1 (4.10)

In the above equation, k (2) is the isotropic permeability computed from the first homogenization problem. At the interface between the Stokes and the Darcy media, the Beavers-Joseph model ( 9) is often considered:

v (s) .ν = v (f ) .ν ∀x ∈ S 3 (4.11) 2µν.d(v (f ) ).τ = -α(v (f ) -v (s) ).τ ∀x ∈ S 3 (4.12) 2µν.d(v (f ) ).ν = p (f ) -p (s) ∀x ∈ S 3 (4.13) in which p (f ) , v (f ) and p (s) , v (s) represent the pressure and velocity fields taken at each side of the interface S 1 , exponent (s) makes reference to the porous solid and exponent (f ) to the fluid phase (i.e. the first porosity). Also, ν and τ represent the normal and the tangential unit vectors acting on S 1 (see Figure 4.4). In equation (4.12), α is a coefficient of the BJ model, d(v (f ) ) represents the strain rate tensor computed in the fluid region. These interface conditions represent: the continuity of the mass flux (4.11), the discontinuity for the tangential velocity (4.12) and the continuity for the normal traction (4.13). At the interface S 1 , the normal component of the velocity field is then continuous but the tangential component and the pressure are discontinuous. The following expression is used in the literature for coefficient α:

in which δ is dimensionless coefficient called slip coefficient and is characteristic of the porous solid. This coefficient has been determined experimentally by Beavers and Joseph for Nickel foam metal and found 0.78, 1.45 and 4.0 for porous microstructures having average pore size of 0.016, 0.034 and 0.045 inches respectively.(9)For Aloxite 4.6 Numerical results

Principle of the method

The macroscopic permeability of the polymeric bi-porous solid K (1) or resistivity H

is computed with the FFT method. Note that the permeability k (2) is unfortunately unknown since the resolution of the X-ray tomograph is not sufficient. The latter can be only estimated. However, by performing the calculations at the intermediate scale, we determine the dimensionless macroscopic permeability on 2-D and 3-D unit cells. We apply the procedure of ensemble averaging (see, for instance Sab (21)). Let i be a given occurrence, an estimation of the ensemble averaging of a quantity x is given by the cumulated average procedure:

where P represents the total number of occurrences.

Effect of α on the macroscopic permeability

We first investigate the effect of parameter α which determine the thresholding of the phases. The numerical results are performed on 20 2-D images of dimension 186 × 186 illustrated in Figure 4.7. In Figure 4.14, we provide the values of the resistivity for each occurrences. We also compute the average cumulated value following equation (4.64).

unchanged excepted for the interface. Indeed, the interface model of Beaver-Joseph-Saffman (BJS) has been replaced by the interface condition (5.4) of the Brinkman model. (1; 16) Particularly, with the latter model, the tangential component of the velocity field remains continuous across the interface while its is discontinuous with the BJS model. Imperfect interface model could be used to account for a discontinuity of the tangential velocity field, however, we assume the continuity of the fields for simplicity. The consideration of discontinuous fields with the FFT method is possible but involves extended discretization spaces.( 12)

Mathematic preliminaries

New FFT algorithms are derived in this chapter to solve the Brinkman equation. The cornerstone of the FFT method is the formulation of the local problem into an integral equation called Lippmann-Schwinger equation which uses Green operators. This integral equation is established by taking advantage of the solution of an auxiliary problem: the inclusion problem in the sense of Eshelby. (5) In this section, we derive the close form solution of the inclusion problem and we introduce the corresponding Green operators.

Reformulation of the Brinkman problem

Let us first introduce the vector ω defined by:

The Brinkman problem is rewritten into the following form:

with W = -J and the periodicity condition at the boundary of the unit cell.

The set of equations is constituted of:

• Eq. (5.11): an equilibrium equation for the stress field σ and ω,

• Eqs. (5.12) and Eq. (5.13): an uncoupled local linear relation between the pair of tensors (σ, ω) and the pair (d, v),

• Eq. (5.14): a compatibility relation between the strain rate tensor and the velocity,

• Eq. (5.15): the incompressibility condition.

By taking the average of the first equation in (5.11) over the volume V 1 , we obtain:

By making use of the divergence theorem, the first term in the above equation can be transformed into an integral over the boundary of the cell that is null due to the antiperiodicity of the traction. We deduce that:

Then, W represents the average of the quantity ω over the volume V 1 . The average of the local velocity reads: 1) .W (5.18) or equivalently: (1) .V (5.19) where H (1) is the macroscopic resistivity.

The relation between the pair of tensors (s, ω) and (d, v) can be put into the form:

Inclusion problem

Solution in Fourier space

Consider a homogeneous medium obeying to the Brinkman equation with the coefficients µ 0 and γ 0 . We are interested with the following inclusion problem in the sense of Eshelby (5) (see also the book of Mura ( 14)): the homogeneous porous solid is subjected to the following pair of periodic eigenfields q (with tr(q) = 0) and f :

The solution of the above linear system can be explicitly derived in the Fourier space. The Fourier transform of any quantity f is denoted f :

By applying the Fourier transform to Eqs. (5.21), we obtain ∀ξ = 0:

where the notation v ⊗ s ξ represents the symmetrized tensorial product:

By eliminating successively the strain rate tensor d, the stress σ and ω in the set of Eq. (5.23), we obtain the following equation for the velocity field v and the pressure p:

Owing to the incompressibility, v.ξ = 0, we deduce that the pressure reads:

and the solution for the velocity field is:

in which P and Q are the two order projectors defined ∀ξ = 0 by:

and are null for ξ = 0. These projectors have the properties:

The solution for the strain rate tensor is:

(5.31)

Dual form

The solution of the inclusion problem is now provided for the stress field σ and for the field ω. Considering the third equation in (5.23) with Eq. (5.32), we obtain:

Considering now the following expressions for the eigenfields (q, f ):

where (r, g) are the dual eigenfields, it gives:

Introducing the Green operators D 0 , defined ∀ξ = 0 by:

and D 0 = 0 for ξ = 0. Let us also denote Π 0 defined ∀ξ = 0 by:

and Π 0 = 0 for ξ = 0. Eq. (5.42) can then be read:

Considering now the stress field in Eq. (5.23) and let us replace the strain rate field given in Eq. (5.32), we obtain:

q and f are replaced by the expression in Eq. (5.41), it gives:

where K is the fourth order projector for the deviatoric two order tensors defined by:

and where I is the fourth order identity tensors.

Let us introduce the Green operator ∆ 0 defined ∀ξ = 0 by:

It is possible to extend the demonstration in the case of the Brinkman equation (not provided here). Indeed, we obtain an equivalent expression for µ 0 and γ 0 :

where the index "s" and "f " make reference to the corresponding quantity taken in the solid and fluid phase respectively.

Formulation of an accelerated scheme

The convergence of the FFT iterative schemes drastically depends on the contrast between the phases. For instance, in the case of elasticity, the contrast represents the ratio between the rigidity of the matrix and that of the inclusions. When the inclusion are highly more rigid than the matrix, the rate of convergence of the iterative schemes is strongly reduced. In the present problem, two contrasts are considered. A first one is given by the ratio between the resistivity of the phases while the second is given by the ratio between the dynamic viscosity of the phases. In the present problem, the properties of the phase are highly contrasted. In order to improve the convergence of the iterative scheme, more sophisticated algorithms could be used. Two class of improved algorithms have been used in the context of elasticity:the accelerated schemes ( 6),( 8),( 10)) and the conjugate gradient. (( 17)) In the present chapter, we propose to extend the accelerated scheme of Monchiet and Bonnet to deal with the Brinkman equation.( 10)

Preliminaries

Let us first come back to Eq. (5.55). The strain rate tensor d and the velocity field v can be expressed in term of the eigenfields q and f as follows:

The deviatoric part of the stress, s, and vector ω are then given by:

We search the fields v, d, ω and s in the form given by (5.71) and (5.72) and which comply with the compatibility equation:

the incompressibility:

and the equilibrium:

Note however that the average of v and ω is uncontrolled in that iterative scheme. An additional condition can be applied to fix the value of the average of the velocity:

with this additional term, the condition < v i > V = V is retrieved at the convergence. δ is a coefficient which, as α and β, would be adjusted in order to obtain the better rate of convergence. Let us observe that the accelerated scheme use both the Green operators G 0 and D 0 which each of them depend on 45 components. A total number of 90 components must be computed and stored for the implementation of this scheme. As for the basic one, we propose to simplify the iterative scheme in order to avoid the storage of the Green operator and to make the algorithm computationally less expensive.

It is interesting to observe that:

in which d i is the deviatoric part of d i which results form the product between d i and the fourth order tensor K which appears in the expression of ∆ 0 (see Eq. (5.49)). Also, the term < v i > V is subtracted to v because the Green operators G 0 and D 0 are null for ξ = 0, so when they are applied to the pair of tensors (s i , ω i ) and (d i , v i ) the result is two pairs of tensors which have a null volume average. As a consequence, we can put the iterative scheme in the form: [START_REF] Barrere | Détermination numérique de la perméabilité en milieu poreux périodique tridimensionnel[END_REF] with:

The product between the Green operator G 0 and the pair of tensors (a i , b i ) can be simplified, owing to Eq. (5.68). Note also that the deviatoric part d i can be replaced by the strain rate tensor d i if the first term of the series, q 0 is also deviatoric. Indeed, in the recurrence relation (5.81), if the eigenfield q i is deviatoric, also d i is deviatoric. The convolution product between C 0 G 0 and the pair of tensors (a i , b i ) also provides a deviatoric two order tensor, as a consequence, in the recurrence relation (5.85), the eigenfield q i+1 is also deviatoric. If the first term of the series, q 0 , is deviatoric, all the terms of the series q i and all the strain rates d i are also deviatoric. In the present case, the iterative scheme is initialized with q 0 = 0 and f 0 = 0. In a trivial way, q 0 = 0 is deviatoric.

The different steps of the numerical integration of the accelerated scheme are summarized below: initialization:

5.4.3

The choice of the reference material. Choice of α, β, δ.

The accelerated scheme uses five preconditioners : µ 0 , γ 0 , α, β and δ. The choice of such coefficients has been already studied in the context of elasticity (10) and thermal conductivity. (11) It has been shown that the values of α, β and δ must satisfy:

In practice, we put β = δ = -α. The better rate of convergence is generally observed for a value α close to 2. Note also that since the analysis convergence indicates that the value α = 2 is prohibited, the convergence is still observed at this value. In our computation the value α = 2 is considered.

Regarding the coefficients of the reference material, following the studies conducted in the case of elasticity and thermal conductivity, we use the following negative coefficients:

The values of µ 0 and γ 0 must be optimized numerically to obtain the better rate of convergence.

5.5

Application to a biporous solid with non-interconnected large pores

The biporous material

In this section, we apply the basic FFT based iterative scheme in order to determine the macroscopic permeability of the biporous polymers with non-interconnected large pores. We consider the images coming from microstructure A (non sieved NaCl particles and non interconnected large pores) illustrated in Figure 5.1. We use the 20 X-ray microtomography images coming from chapter 4.

Figure 5.1: X-ray microtomography of the microstructure A

Dimensionless problem

Following chapter 4, the computations are performed on a dimensionless problem at the intermediate scale. The microstructure of the second porosity being unknown, the corresponding permeability can be only estimated. The latter is still assumed to be isotropic and is denoted k (2) following the notations used in the last chapter. The computations at the intermediate scale allow the determination of the amplification factor K (1) . Following the procedure already used in the last chapter, we apply the variable change:

The Brinkman equation then becomes:

in which the quantities µ(x) and h(x) are given by:

around 30 nm could permit a better characterization of the porous matrix. This imaging technique that is based on Fresnel lens is under development in Anatomix beamline (Synchrotron SOLEIL). Last but not least, this work paves the way to future developments in the FFT-based computation strategy. The estimation of the 3-D permeability by using Brinkman's equation should be more investigated. The extension of the computation strategy to the other microstructures obtained by synchrotron tomography could be instructive.

For instance, we could evaluate the effect of the pore size on the fluid flow.

The determination of the effective elastic properties of the biporous polymer could be also investigated by means of FFT based iterative schemes.

Finally, the development of the FFT algorithms allows various other applications such as the prediction of the transport properties of fractured rocks.

Appendix A

A.1 Materials . For the sake of comparison, NaCl particles were also used without applying any sintering procedure. 2 g of particles were weighted in a glass vial and then gently stirred over an orbital shaking plate in order to ensure homogeneous particle packing. HEMA was used as the functional monomer, and 5 crosslinking agents were assessed: EGDMA, DEGDMA, TriEGDMA, TetraEGDMA, and PEGDMA. In a second step, HEMA and each one of the crosslinkers were mixed in a 70/30 molar ratio in the presence of DMPA (2 wt. % with respect to the total comonomers content) and 80 vol. % propan-2-ol (with respect to the total comonomers volume). The mixture was added to the fused 3-D NaCl particle template (either sintered or not), and the polymerization was conducted in a UV oven for 4 h at 365 nm. Once the polymerization completed, non-sintered NaCl particles were removed by extraction with deionized water for 3 days (water was changed once a day), while sintered NaCl particles could be more easily removed in a single day. Upon porogen extraction, the samples were washed abundantly with water, and dried at room temperature under vacuum. The samples were analyzed by gravimetry so as to confirm the completion of the porogenic agents extraction (NaCl macroparticles and solvent)(1).

A.2.2 Preparation of monoporous PHEMA-based materials

Monoporous PHEMA-based materials were prepared following the same procedure as that used for biporous materials. In order to generate only the upper porosity level (large pores), HEMA and the considered crosslinker were mixed in presence of DMPA without using propan-2-ol. Similarly, no NaCl template was used in the experimental procedure to generate only the lower porosity level (small pores).

A.3 Characterization of PHEMA-based porous materials A.3.1 Mercury intrusion porosimetry

Porosity ratios, pore volumes, and pore size distributions of the polymeric materials were determined by mercury intrusion porosimetry (MIP) using a Micromeritics Au-toPore IV 9500 porosimeter. The determination of the porosity features was based on the Washburn equation between the applied pressure (from 1.03 to 206.8 MPa) and the pore size into which mercury intruded.

A.3.2 Gas sorption porosimetry

The nitrogen sorption measurements were carried out at 77 K with a Quantachrome Autosorb iQ analyzer. The specific surface area values for the investigated porous polymers were quantified using the BET method at relative pressure (P/P 0 ) values ranging from 0.05 to 0.3.

A.3.3 Scanning electron microscopy

SEM observation of the materials was performed on a MERLIN microscope from Zeiss equipped with InLens and SE2 detectors using a low accelerating tension (5-10 kV) with a diaphragm aperture of 30 µm. The samples were first cryofractured and coated with a 4 nm layer of palladium/platinum alloy in a Cressington 208 HR sputter-coater. Energy-dispersive X-ray spectroscopy (EDX) was performed using a SSD X-Max detector of 50 mm 2 from Oxford Instruments (127 eV for the Kα of Mn).

A.3.4 Dynamic mechanical analyses

Samples of 2 mm thickness and 10 mm diameter were used to characterize the thermomechanical properties. A Q800 TA Instruments apparatus was used to characterize these samples using the shear method in which the cantilever was sandwiched between two identical samples. The experiment was carried out in a closed furnace; the samples were cooled down to -140 • C and heated up to 150 • C at 3 • C.min -1 . The frequency of the applied stress was fixed at 1 Hz and the deformation was equal to 0.1 %.

A.4 Swelling experiments of PHEMA-based porous materials

PHEMA-based materials were immersed in distilled water and then deposited on a filter paper in order to eliminate the residual water. The materials were then weighted every 10 min for 4 h, and the water content was calculated following Eq. (A.1) (2):

Where W w and W d stand for for the wet mass and the dry mass of the sample, respectively.

A.5 Determination of apparent density and porosity ratio values of PHEMA-based porous materials

The apparent density and the porosity ratio were calculated by following a strategy developed by Rohman et al.The equilibrium mass swelling ratio, q w , was evaluated by using Eq. (A.2) where m d and m w represent the residual mass of the materials after vacuum drying and their wet mass, respectively (3):

Dichloromethane was considered as a non-swelling solvent for PHEMA-based materials.

The solvent uptake could be then attributed to the filling of pores within the porous polymer. The pore volume V pores was deducted from Eq. (A.3), and the apparent density, d app , was obtained from Eq. (A.4):

where d s and d true represent the solvent density (i.e. d s = d dichloromethane = 1.33 g.cm -3 ), and the true density of the PHEMA polymer. d true was calculated by measuring the density of a bulk PHEMA material (i.e. without pores). The non-porous material was weighted and its volume was attributed to that of a cylinder. Finally, the porosity ratio values (P) were determined by using Eq. (A.5)

Experimentally, this study was performed on 5 similar materials exhibiting interconnected and non-interconnected networks prepared with 200-250 µm size NaCl particles. The samples were first immersed in dichloromethane until obtaining a constant mass corresponding to m w in Eq. (A.2). The materials were then dried overnight under vacuum and weighted a second time to determine md. The results were provided by taking into account the distribution of uncertainties provided in Eq. (A.6) . The uncertainty of measurement due to the balance was equal to 10 %:

where f corresponds to a function representing the product of variables named x i (m w and d in this case).

A.6 Determination of transport properties

Each polymer sample was submitted to water permeability measurements by using a constant load permeameter following the Mariotte principle at constant atmospheric pressure. Permeability tests were performed in a triaxial cell on a water saturated sample under isotropic loading. The cell was connected to a closed column full of water in which a smaller column was installed. The sample was previously saturated with water. Before measuring permeability, the cylinders of 10 mm diameter were enrobed with a Wood alloy (55 % Bi, 15 % Pb, 15 % Sn, 15 %) at 70

where L(m) and S(m 2 ) stand for the sample length and the section surface of the cylindrical sample respectively, h for the hydraulic load (m), K (m.s -1 ) for the hydraulic conductivity, η (Pa.s) for the water dynamic viscosity, g (m.s -2 ) for the gravitational constant, and ρ (kg.m -3 ) for water density, respectively.

Appendix B B.1 Classical X-ray microtomography imaging

The computed X-ray microtomography investigations were performed using an Ultratom microtomograph system manufactured by RX Solutions illustrated in The choice of the experimental parameters depends on the addition of the contrast agent that made the material more absorbent and implied a higher energy level.

B.2 Synchrotron X-ray tomography imaging

The synchrotron X-ray tomography investigations were performed at Synchrotron Soleil (Saint Aubin, France) on the ANATOMIX beamline with a parallel beam and a pink beam with an energy range from 5 to 25 keV, resulting from the use of a 12.5 mm gap on the ondulator and filtering by Au (Figure B.4). Note that the Anatomix beamline is the latest beamline of Synchrotron, constructed in the context of the EquipEx project NanoImagesX. Our experiment run in April 2018 was the first official user-experiment on this line, which was still in a temporary, not optimized, configuration. A local tomography was performed with a 10 x optics in absorption contrast mode and a Hammamatsu Orca Flash 4.0 V2 camera (2048 × 2048 pixels, 6.5 µm), associated with a scintillator 500 µm in thickness. The cylindrical material of about 1 cm in diameter was stuck on a PMMA sample holder inserted on the rotating stage (see figure ??). Radiographic scans using 2000 to 8000 projections were run together with the acquisition of "dark" (i.e. without beam) and "flat" images (i.e. without sample), averaged over 50 frames. Because the setup was not yet optimized, only the central part of the radiographs were perfectly focused, on a height of about 1000 pixels. During these experiments, 2 acquisition modes were carried out. In the first one, the sample is scanned over 180 A total of 6 materials were investigated this way : 3 samples prepared with sintered NaCl particles and 3 samples obtained from non-sintered NaCl particles.Fractions of 125-200 µm, 200-250 µm or 250-400 µm NaCl Additional experiments were also run on two monoporous materials: one with only small pores and the other with only large pores. A last experiment was realized on a materials containing 200-250 µm NaCl particles before and after the extraction of the porogen agent. For these experiments, 4000 radiographic projections were collected after one rotation of 360 • . Note finally that because these experiments were run late, only a small part of the obtained images could be used during this PhD work.

Appendix C C.1 Proof of property 1

We aim to simplify the product:

for any pair of compatible field d and v, i.e. related in the Fourier space by:

To this end, we replace the green operator G 0 by the expression given in Eq. (5.38).

It follows that:

Considering the definitions provided in Eq. (5.35) and after various simplifications, we obtain:

Considering additionally the incompressibility, v.ξ = 0, we deduce that

Coming back to the real space, we finally obtain:

where the volume average of each quantities has been eliminated because the Green operator is null for ξ = 0. So, the macroscopic velocity V has been subtracted to v.

The strain rate tensor d has a null volume average since it derivates from a periodic velocity field.

C.2 Proof of property 2

Let us introduce the two quantities M and B defined by :

The incompressibility corresponds to B = 0 and the compatibility to M = 0. If we replace D 0 by (5.54) in which the Green tensor ∆ 0 , Π 0 and D 0 are given by (5.49), (5.44) and (5.43) respectively, we obtain after some elementary algebraic manipulations:

At this stage, it is easy to show that the latter expression is null if M = 0 and B = 0.

C.3 Proof of property 3

Considering Eq. (5.68) and (C.7), we deduce that the condition (5.78) leads to the following linear system:

where M and B are given by Eq. (C.6) and E is given by (5.69) (and by omitting the index i in E).

Owing to relation E.n = 0 and Q.n = 0 in the second equation in (C.8), we deduce that B = 0 and

so that:

Introducing this result in the first equation in (C.8), we find (after some elementary algebraic manipulations) 2βµ 0 M = 0. This prove the property because, by reporting the condition M = 0 in Eq. (C.9), we obtain E = 0.

Abstract

Biporous polymeric materials are widely used in many applications as three-dimensional scaffolds for tissue engineering. They are also model systems for the understanding of many natural or man made materials, such as soils, rocks, wood or lightweight cement. The role of each porosity level on the transport properties of such porous frameworks is crucial. In this context, an original approach combining the design, thorough physico-chemical characterization and modelling of polymeric materials exhibiting two levels of porosity was developed. Model biporous poly (2-hydroxyethyl methacrylate) (PHEMA) materials were prepared by using two types of templates as porogens: sieved NaCl particles, sintered or not, to generate the first level of porosity, and a solvent, propan-2-ol, to induce the second level. Porous networks with small pore sizes of about 1µm were obtained, while larger pores had an average size of about 80µm, as evaluated from mercury intrusion porosimetry. A careful X-ray computed microtomography analysis of the 3-D microstructure and pore interconnectivity of the elaborated porous materials was also performed, using both laboratory and synchrotron facilities. The optimization of experimental parameters (X-Ray energies, use of contrast agent) permitted to obtain high-resolution images with voxel sizes down to 650 nm. In order to investigate the relationship between the microstructure and the transport properties, data extracted from 3-D images were selected to compute a fluid flow through biporous porous media, first in 2-D and then in 3-D. Such networks exhibiting at least three scales of heterogeneity, namely the characteristic scales of the two porosity levels and the macroscopic scale, the permeability was computed in the framework of homogenization with a double upscaling. The numerical resolution of fluid flow in the selected unit cells was performed with iterative schemes, which make use of the Fast Fourier Transform. The numerical evaluations of the effective permeability were compared to experimental measurements. Keywords: biporous polymeric materials, X-ray microtomography, numerical simulation by Fast Fourier Transform.