La séparation de phase au sein d'un système polymère/solvant est la méthode la plus couramment utilisée pour élaborer une membrane polymère poreuse. Les principales méthodes d'inversion de phase nécessitent l'usage de solvants organiques qui génèrent des problèmes environnementaux (traitement des bains de coagulation) et sanitaires (sécurité des installations industrielles). Cette étude porte sur le développement d'une nouvelle membrane polymère poreuse à partir d'un polymère biosourcé et hydrosoluble, l'hydroxypropylcellulose (HPC), permettant de s'affranchir de l'usage de solvants organiques. La propriété de thermosensibilité de l'HPC, caractérisé par une température critique basse en solution dans l'eau (LCST) de l'ordre de 40 °C, a par ailleurs permis de développer un procédé original d'élaboration de membranes HPC par séparation de phase induite par augmentation de la température au-delà de la LCST. Ce travail vise un triple objectif : (i) déterminer la formulation idoine permettant de former une membrane poreuse insoluble dans l'eau à partir d'HPC, (ii) appréhender et comprendre les mécanismes de structuration de la matrice polymère à travers l'interaction des mécanismes interdépendants de séparation de phase par décomposition spinodale, de réticulation chimique et d'extraction du solvant par évaporation et enfin (iii) caractériser l'aptitude des membranes à la filtration d'une solution aqueuse sous pression. Le suivi en ligne de la dynamique de séparation de phase d'un système HPC/eau/réticulant ± porogène (PEG200) par microscopie optique en contraste de phase, de la réticulation par rhéologie et de l'évaporation de l'eau par thermogravimétrie a ainsi permis de mettre en évidence l'impact de la formulation et des paramètres de conduite du procédé d'inversion de phase sur les propriétés morphologiques et d'usage des membranes. La porosité membranaire et le caractère symétrique de la morphologie ont notamment été corrélés à la vitesse des phénomènes concomitants de réticulation et d'évaporation de l'eau, donc à la vitesse de montée en température du procédé TIPS-LCST. La caractérisation de la perméabilité à l'eau des membranes HPC a confirmé l'efficacité de la réticulation et la résistance structurale des membranes au cours de plusieurs filtrations continues à l'eau. En raison du caractère thermosensible de l'HPC, ces membranes ont montré une aptitude remarquable à la filtration de solutions aqueuses à température élevée (60 °C). Par ailleurs, il a été montré que la perméabilité des membranes pouvait être en partie contrôlée par la température et la pression transmembranaire appliquée. / Phase separation of polyer/solvent system is the most widespread industrial process to manufacture membranes. Large solvent quantity is usually used whatever the process, hence leading to environmental (coagulation and washing baths treatment) and health (industrial and plant safety) problems.This study focuses on the development of new porous membranes made from hydroxypropylcellulose (HPC), a water soluble polymer, avoiding the use of any organic solvent. Moreover, the thermo-sensitive character of this polymer, characterized by a Lower Critical Solution Temperature (LCST) in water of about 40 °C, enabled to design an original thermally induced phase separation process by temperature increase above the LCST. This study aims (i) to find the ideal polymer solution composition to produce insoluble HPC membrane, (ii) to approach and understand the link between phase separation mechanism by spinodal decomposition, crosslinkig reaction and water extraction by evaporation, (iii) characterize pure water permeability under pressure. On-line monitoring of phase sepration dynamics by phase contrast optical microscopy, crosslinking reaction by rheology and water evaporation by thermogravimetric analysis of the system HPC/water/cross-linking agent ± porogen (PEG200) allowed an understanding of simultaneous and related mechanisms occurring during elaboration (phase separation / cross-linking / water evaporation) and a correlation with HPC membrane morphologies and characteristics in relation with phase separation process parametres. Pure water permeability characterization demonstrated the efficiency of cross-linking and structural strength during several filtration cycles. Furthermore, it has been shown that water permeability of HPC membranes could be controlled in part by the temperature and the applied pressure.

This thesis focuses on my endeavors to prepare new porous polymeric monoliths that are viable to use as supports in flow-through processes. Polymer monoliths of various porous properties and different chemical properties have been prepared utilizing the thermally induced phase separation (TIPS) phenomenon and step-growth polymerization reactions. The aim has been to find appropriate synthesis routes to produce separation supports with fully controlled chemical, physical and surface properties. This thesis includes preparation of porous monolithic materials from several non-cross-linked commodity polymers and engineering plastics by dissolution/precipitation process (i.e. TIPS). Elevated temperatures, above the upper critical solution temperature (UCST), were used to dissolve the polymers in appropriate solvents that only dissolve the polymers above this critical temperature. After dissolution, the homogeneous and clear polymer-solvent solution is thermally quenched by cooling. A porous material, of three dimensional structure, is then obtained as the temperature crosses the UCST. More than 20 organic solvents were tested to find the most compatible one that can dissolve the polymer above the UCST and precipitate it back when the temperature is lowered. The effect of using a mixture of two solvents or additives (co-porogenic polymer or surfactant) in the polymer dissolution/precipitation process have been studied more in depth for poly(vinylidine difluoride) (PVDF) polymers of two different molecular weight grades. Monolithic materials showing different pore characteristics could be obtained by varying the composition of the PVDF-solvent mixture during the dissolute­ion/precipitation process. Step-growth polymerization (often called polycondensat­ion reaction) combined with sol-gel process with the aid of porogenic polymer and block copolymer surfactant have also been used as a new route of synthesis for production of porous melamine-formaldehyde (MF) monoliths. In general, the meso- and macro-porous support materials, for which the synthesis/preparation is discussed in this thesis, are useful to a wide variety of applications in separation science and heterogeneous reactions (catalysis).

Orientador: Maria Isabel Felisberti / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-12T12:37:28Z (GMT). No. of bitstreams: 1 Lima_JulianaAristeiade_D.pdf: 4425741 bytes, checksum: e0c47d6335f66cc31a0fe6e5ac1c0457 (MD5) Previous issue date: 2008 / Resumo: Neste trabalho foram estudadas blendas de poli(etileno-co-álcool vinílico) (EVOH), um copolímero semicristalino que combina segmentos hidrofílicos e hidrofóbicos e poli(metacrilato de metila) (PMMA), um polímero amorfo e hidrofílico. As blendas de EVOH, com teor de etileno variando de 27 a 44 mol %, e PMMA foram preparadas por casting a partir de soluções em DMF, e por mistura mecânica em um mini-misturador, objetivando: i) avaliar o comportamento de fases e a morfologia das blendas, EVOH/PMMA, em função da composição das misturas, do teor de etileno nos copolímeros de EVOH e do método de preparação; ii) obter os diagramas de fases das soluções binárias, EVOH/DMF, e ternárias, EVOH/PMMA/DMF, através do processo de separação de fases induzido termicamente (TIPS). O comportamento de fases das blendas, EVOH/PMMA, foi estudado através de Calorimetria Diferencial de Varredura (DSC) e Análise Dinâmico-Mecânica (DMA). A morfologia das blendas foi investigada por Microscopia Eletrônica de Varredura (SEM). As blendas independentemente do modo de preparação e da composição são imiscíveis. Como conseqüência desta imiscibilidade, as temperaturas de fusão (Tf) e cristalização (Tc) não são afetadas pela presença de PMMA. A morfologia das blendas varia com a composição e método de preparação. As blendas obtidas pelo método mecânico são compactas, apresentando morfologia de fase dispersa em uma matriz, com inversão de fases em aproximadamente 50 % em massa de cada componente. As soluções binárias, EVOH/DMF, e ternárias, EVOH/PMMA/DMF, foram submetidas a ensaios para a determinação das temperaturas de turvamento (Tturv) por microscopia ótica, e de cristalização dinâmica (Tcd), por DSC. As soluções binárias apresentaram comportamento UCST (upper critical solution temperature), sendo que a separação de fases L-L ocorre a temperaturas superiores à separação de fases S-L. O modelo de interações binárias prevê que a curva de separação L-L para as soluções EVOH- 38/DMF esteja situada a temperaturas superiores em relação às soluções EVOH-32/DMF e indica que a separação de fases resulta da baixa afinidade entre segmentos hidrofóbicos do EVOH com os segmentos do polímero contendo hidroxila e com o solvente, DMF, concordando com os dados obtidos experimentalmente. Filmes de EVOH obtidos pela evaporação do solvente mostraram-se densos, sem a presença de poros. As soluções ternárias também apresentaram comportamento UCST, resultando em duas fases macroscópicas, F1 e F2, à temperatura ambiente. A composição destas fases foi determinada por Análise Termogravimétrica (TGA), e os resultados mostraram que uma das fases macroscópicas é rica em EVOH e outra fase é rica em PMMA. As blendas resultantes da secagem dos sistemas ternários apresentaram duas camadas: uma densa e rica em PMMA e outra porosa e rica em EVOH. A presença de PMMA foi decisiva à formação de estruturas porosas / Abstract: In this work blends of poly (ethylene-co-vinyl alcohol) (EVOH), a semicrystalline copolymer which combines hydrophobic and hydrophilic segments and poly (methyl methacrylate) (PMMA), an amorphous and hydrophilic polymer were studied. EVOH blends, with ethylene content ranging from 27 to 44 mol% and PMMA were prepared by casting from solutions in DMF, and by mixing into a mini-mixer, with the objective of: i) evaluate the phase behavior and the morphology of the blends, EVOH/PMMA, depending on the composition of mixtures, the ethylene contents in the copolymers of EVOH and the conditions of mixing; ii) obtain the phase diagrams of the binary and ternary solutions, EVOH/DMF and EVOH/PMMA/DMF, respectively, by the process of thermally induced phase separation (TIPS). The phase behavior of the blends, EVOH / PMMA, was investigated by Differential Scanning Calorimetry (DSC) and Dynamic Mechanical Analysis (DMA). The morphology of the blends was investigated by Scanning Electron Microscopy (SEM). The blends independently of the method of preparation and of the composition are immiscible. As a result of immiscibility, the melting temperature (Tm) and the crystallization temperature (Tc) are not affected by the presence of PMMA. The morphology of the blends varies with the composition and with the method of preparation. The blends produced by the mechanical method is compact, showing morphology of dispersed phase in a matrix, with inversion of phases in about 50% by weight of each component. The binary solutions, EVOH-38/DMF and EVOH-32/DMF, were submitted to experiments to determine the cloud temperature (Tcloud) by optical microscopy, and the dynamic crystallization (Tcd), by DSC. The binary solutions show UCST behavior (upper critical solution temperature), and the L-L phase separation occurs at higher temperatures than the S-L phase separation. The binary interaction model provides the L-L line to the EVOH-38/DMF, solutions which was situated at higher temperatures than the EVOH-32/DMF solutions and indicates that the phase separation results from the low affinity between the hydrophobic segments of EVOH with the segments of the polymer containing hydroxyl and the solvent, DMF, which agrees with the data obtained experimentally. Films of EVOH obtained by the solvent evaporating seemed to be dense, without the presence of pores. The ternary solutions also had UCST behavior, resulting in two macroscopic phases, F1 and F2, at room temperature. The phase compositions were determined by Thermogravimetric Analysis (TGA), and the results showed that one of the macroscopic phases is rich in EVOH and the other phase is rich in PMMA. The blends resulting from drying of ternary systems had two layers: a dense and rich in PMMA and another porous and rich in EVOH. The presence of PMMA was crucial to the formation of porous structures / Doutorado / Físico-Química / Doutor em Ciências

No description available.

Poly(phenylene sulfide) (PPS), an engineering thermoplastic with excellent mechanical, thermal, and chemical properties, was gelled for the first time using 1,3-diphenylacetone (DPA) as the gelation solvent in a thermally induced phase separation (TIPS) process. PPS was dissolved in DPA at high temperatures to form a homogeneous solution. The solution was cooled, initiating phase separation and eventually forming a solidified PPS network around DPA-rich domains. Evacuation of DPA from the gel network creates monolithic PPS aerogels, one of few physically crosslinked polymer aerogel systems comprised of a high-performance thermoplastic. In this work, specific properties of PPS aerogels were controlled through the manipulation of various processing parameters, such as polymer concentration, post-process annealing conditions, mode of manufacturing (casting versus additive manufacturing), dissolution temperature, and drying method. The ultimate goal was to elucidate key process-morphology-property relationships in PPS aerogels, to ultimately improve aerogel performance and applicability. The phase diagram of PPS/DPA was first elucidated to determine the phase separation mechanism of the system, which guides all future processing decisions. The phase diagram indicated that the system undergoes solid-liquid phase separation, typical for solutions with relatively favorable polymer-solvent interactions. This assignment was validated by the calculation of the Flory-Huggins interaction parameter through two independent methods - Hansen solubility parameters and fitting melting point depression data. The influence of polymer composition on PPS aerogel properties was then characterized. As polymer concentration increased, aerogel density and mechanical properties increases, and porosity decreased. The particular morphology of PPS aerogels from DPA was that of a fibrillar network, where these axialitic (pre-spherulitic) fibrils are comprised of stacks of PPS crystalline lamellae, as suggested by x-ray scattering and electron microscopy. These interconnected microstructures responded more favorably to compressive load than similar globular PEEK aerogels, highlighting the importance of aerogel microstructure on its mechanical response. Upon solvent extraction, PPS aerogels were annealed in air environments to improve their mechanical behavior. Annealing did not dramatically shrink the aerogels, nor did it appear to affect the micron-scale morphology of PPS aerogels as observed by electron microscopy. The resistance to densification of PPS aerogels was mainly a product of their interconnected fibrillar morphologies, aided by subtle microstructural changes that occurred upon annealing. Exposure to a high temperature oxidative environment (160 – 240 oC) increased the degree of crystallinity of the aerogels, and also promoted chemical crosslinking within the amorphous PPS regions, both of which may have helped to prevent severe densification. With enhanced physical and chemical crosslinking, annealed PPS aerogels displayed improved compressive properties over unannealed analogues. Additionally, the thermal conductivity of both annealed and unannealed aerogel specimens was below that of air (~ 0.026 W/mK) and did not display a dependence on polymer composition nor on annealing condition. Generally, these experiments demonstrate that annealing PPS aerogels improved their mechanical performance without negatively affecting their inherent fibrillar morphology, low density, or low thermal conductivity. To fabricate aerogels with geometric flexibility and hierarchical porosity, PPS/DPA solutions were printed through material extrusion (MEX) and TIPS using a custom-built heated extruder. In this process, solid solvated gels were first re-dissolved in a heated extruder and solutions were deposited in a layer-wise fashion onto a room-temperature substrate. The large temperature gradient between nozzle and substrate rapidly initiated phase separation, solidified the deposited layers and formed a printed part. Subsequent solvent exchange and drying created printed PPS aerogels. The morphology of printed aerogels was compositionally-dependent, where the high extrusion temperature required to dissolve highly-concentrated inks (50 wt % PPS) also destroyed self-nuclei in solution, yielding printed aerogels with spherulitic microstructures. In contrast, aerogels printed from 30 wt % solutions were deposited at lower temperatures and demonstrated fibrillar microstructures, similar to those observed in 30 wt % cast aerogel analogues. Despite these microstructural differences, all printed aerogels demonstrated densities, porosities, and crystallinities similar to their cast aerogel counterparts. However, printed aerogel mechanical properties were microstructurally-dependent, and the spherulitic 50 wt % aerogels were much more brittle compared to the fibrillar cast 50 wt % analogues. This work introduces a widely-applicable framework for printing polymer aerogels using MEX and TIPS. Intrigued by the compositional morphological dependence of the printed PPS aerogels, the dissolution temperature (Tdis), and thus the self-nuclei content, of cast PPS/DPA solutions was systematically varied to understand its influence on aerogel morphology and properties. As Tdis increased, the length and diameter of axialites increased while aerogel density and porosity were relatively unaffected. Thus, the isolated influence of axialite dimensions (analogous to pore size and pore concentration) on aerogel properties could be studied independent of density. At low relative densities (below 0.3, aerogels of 10 – 30 wt %), compressive modulus and offset yield strength tended to decrease with Tdis, due to an increase in axialite length (akin to pore size) and number of axialites (akin to number of pores). At higher relative densities (above 0.3, 40 and 50 wt %), axialitic aerogels were so dense that changes in pore dimensions did not result in systematic changes in mechanical response. All spherulitic aerogels fabricated at the highest Tdis¬ demonstrated reduced mechanical properties due to poor interspherulitic connectivity. The thermal conductivity of all aerogels increased with polymer composition but demonstrated no clear trend with Tdis. A model for thermal conductivity was used to deconvolute calculated conductivity into solid, gaseous, and radiative components to help rationalize the measured conductivity data. This work demonstrates the importance of nucleation density control in TIPS aerogel fabrication, especially at low polymer concentrations. The specific method used to dry an aerogel generally has a great influence on its microstructure and density. Vacuum or ambient drying is the most industrially-attractive technique due to low cost and low energy usage; however, it is typically the most destructive process due to high capillary forces acting on the delicate aerogel microstructure. Three drying methods, vacuum drying, freeze drying, and supercritical CO2 drying, were used to evacuate PPS gels fabricated at three PPS concentrations (10, 15, and 20 wt %). Almost all aerogel specimens displayed excellent resilience against shrinkage as a function of the drying method, besides the 10 wt % vacuum dried sample which shrunk almost 40%. While the micron-scale aerogel morphology captured by electron microscopy appeared to be unaffected by the drying method, other properties such as aerogel surface area, mesoporous volume, and mechanical properties were effectively functions of the degree of aerogel shrinkage. Aerogel thermal conductivity was low for all samples, and in particular, vacuum dried aerogels demonstrated slightly lower conductivities than other ambiently-dried aerogel systems such as silica and carbon. In general, vacuum drying appears to be industrially viable for PPS aerogels at concentrations above 10 wt %. / Doctor of Philosophy / Polymer aerogels are nanoporous solid networks of low density. These materials are used in applications such as thermal insulation, absorbance/filtration, drug delivery, biomedical scaffolds, solid state batteries, and others. One method of creating polymeric aerogels is through thermally induced phase separation (TIPS), where a polymer is first dissolved in a high boiling point solvent at a high temperature. Next, the solution is cooled, inducing phase separation and gelation. Extraction of the gelation solvent transforms the solvated gel into an aerogel. To create polymeric aerogels with good properties and wide-ranging applicability, one should use a high-performance polymer. In this work, aerogels are for the first time made from poly(phenylene sulfide) (PPS), an engineering thermoplastic with good mechanical properties, thermal stability, and chemical resistance. PPS aerogels are fabricated using TIPS over a wide compositional range, and their microstructures, physical properties, thermal properties, and compressive properties are fully characterized. To further enhance aerogel performance, the fabrication process can be optimized to precisely control the aerogel morphology and thus the resulting properties. The influence of processing variables such as the polymer concentration, the post-fabrication aerogel annealing conditions, the method of manufacturing (traditional casting versus additive manufacturing), the dissolution temperature (temperature at which the polymer dissolves in solution prior to gelation), and the drying method on the aerogel behavior is investigated. Generally, results suggest that understanding critical process-morphology-property relationships allows for precise control over the nature of PPS aerogels.

Les matériaux polymères poreux font l'objet d'intenses recherches depuis de nombreuses années et présentent certains avantages importants par rapport à leurs homologues inorganiques, comme des propriétés mécaniques modulables, une fonctionnalisation aisée et surtout un coût de production plus faible. Au cours de la dernière décennie, les matériaux à double porosité ont attiré une attention particulière de la communauté scientifique car ces matériaux offrent de nouvelles perspectives intéressantes pour l'élaboration de matériaux durables. Le rôle de chaque niveau de porosité est différent et associé à des processus de transfert de masse distincts. Les macropores (~ 100 µm) permettraient l'écoulement de macromolécules ou de cellules à travers le matériau, tandis qu'un réseau nanoporeux (10-100 nm) serait dédié au passage de molécules plus petites, agissant ainsi comme un deuxième mécanisme de transport, en particulier lorsque des macropores sont totalement obstrués. La première partie de ce travail porte sur le développement d'approches polyvalentes et efficaces pour la préparation de matériaux à double porosité biocompatibles à base de poly(méthacrylate de 2-hydroxyéthyle) (PHEMA). La première approche a reposé sur l'utilisation de deux types distincts de gabarits porogènes, à savoir un macroporogène et un nanoporogène. Pour générer la macroporosité, soit des particules de NaCl ou des billes de PMMA, pouvant être fusionnées ou non, ont été utilisées afin de contrôler la morphologie l'interconnectivité des pores. Le nanoporosité a été obtenue en utilisant diverses quantités de différents solvants porogènes, générant ainsi une large gamme de distributions de tailles de pores pour ce second niveau de porosité. La seconde méthodologie a été fondée sur le procédé de séparation de phases induite thermiquement. Un mélange de co-solvants constitué de dioxane et d'eau a été utilisé pour solubiliser le PHEMA linéaire préalablement préparé, suivi par un processus de solidification par congélation du mélange de co-solvants / PHEMA, et sublimation consécutive des co-solvants pour produire les matériaux de PHEMA biporeux correspondants. Enfin, les matériaux à double porosité ont été valorisés à travers différentes réactions de fonctionnalisation en utilisant la chimie du carbonyldiimidazole, et l'immobilisation postérieure de nanoparticules d'or générées in-situ. De tels matériaux hybrides à double porosité se sont avérés être des supports catalytiques efficaces.Dans la deuxième partie, nous avons déterminé numériquement la perméabilité des matériaux à double porosité. La méthodologie a été fondée sur une approche à double changement d'échelle dans le cadre des théories d'homogénéisation périodique et sur des calculs de cellules élémentaires. Le premier changement d'échelle a consisté à déterminer une première perméabilité associée au réseau de nanopores. A cette échelle, les pores ont été saturés par un fluide visqueux obéissant aux équations de Stokes et le problème a été résolu par une approche classiques d'éléments finis ou en utilisant des techniques plus récentes à base de la transformée de Fourier rapide. À l'échelle mésoscopique, l'écoulement du fluide a obéi aux équations de Stokes dans les macropores et aux équations de Darcy dans le solide perméable. Le problème de cellules élémentaires couplant les équations de Darcy et Stokes a été résolu par la méthode des éléments finis afin de calculer la perméabilité macroscopique finale. Dans cette optique, nous avons développé une méthode fondée sur une formulation variationnelle mixte qui a été mise en œuvre en prenant différents éléments dans les domaines de solide et fluide. Divers exemples 2D et 3D sont fournis pour illustrer la précision et la capacité des méthodes numériques proposées pour calculer la perméabilité macroscopique des matériaux biporeux / Polymer-based porous materials have been the subject of intense research for many years and present some important advantages over their inorganic counterparts, such as tunable mechanical properties, ease to be functionalized, and especially lower production cost. Over the last decade, materials with dual porosity have attracted a particular attention from the scientific community, as these peculiar materials offer new interesting perspectives for engineering sustainable materials. The role of each porosity level is different and associated with distinct mass transfer processes. Macropores (~100 µm) would allow macromolecules and cells flow through the material, while a nanoporous network (10-100 nm) would be dedicated to the passage of smaller molecules, thus acting as a second transport mechanism, especially when macropores are totally clogged. The first part of this work addresses the development of versatile and effective approaches to biocompatible doubly porous poly(2-hydroxyethyl methacrylate) (PHEMA)-based materials. The first approach relied on the use of two distinct types of porogen templates, i.e. a macroporogen and a nanoporogen. To generate the macroporosity, either NaCl particles or PMMA beads that could be fused or not, were used in order to control the pore morphology and interconnectivity of the materials. The nanoporosity was obtained by using various amounts of different porogenic solvents, thus generating a wide range of pore size distributions for this second porosity level. The second methodology was based on the thermally-induced phase separation process. A co-solvent mixture constituted of dioxane and water was used to solubilize previously prepared linear PHEMA, followed by a solidification process by freezing the co-solvents/PHEMA mixture, and subsequent sublimation of the co-solvents to generate the corresponding biporous PHEMA materials. Finally, advantage of doubly porous materials was taken through different functionalization reactions using carbonyldiimidazole chemistry, and further immobilization of in-situ generated gold nanoparticles. Such hybrid doubly porous materials proved to act as efficient catalytic supports. In the second part, we numerically determined the permeability of doubly porous materials. The methodology was based on a double upscaling approach in the field of periodic homogenization theories and on unit cell calculations. The first upscaling consisted in the determination of a first permeability associated with the array of nanoscopic pores. At this scale, the pores were saturated by a viscous fluid obeying the Stokes equations and the problem was solved by means of standard Finite-Element approaches or using more recent techniques based on Fast Fourier Transform. At the mesoscopic scale, the fluid flow obeyed the Stokes equations in the macropores and the Darcy equations in the permeable solid. The unit cell problem coupling Darcy and Stokes equations was solved by the Finite Element method in order to compute the final macroscopic permeability. To this purpose, we developed a method based on a mixed variational formulation which was implemented by taking different elements in the solid and fluid regions. Various 2D and 3D examples were provided to illustrate the accuracy and the capacity of the proposed numerical methods to compute the macroscopic permeability of biporous materials

No description available.