Predictive Models for Mixed-Matrix Membrane Performance: A Review

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Predictive Models for Mixed-Matrix Membrane Performance: A Review Hoang Vinh-Thang and Serge Kaliaguine* Department of Chemical Engineering, Laval University, Quebec, Canada to their high stability, efficiency, and ease of operation.1−17 Today some industrially relevant membrane separation processes are already implemented at commercial scale. Some others are at more or less advanced development stage, whereas some should desirably be developed. In the first category, the Monsanto hydrogen separation based on the PRISM membrane was launched in the early 1980s.18 Many such plants have been installed since that time, with one of the most significant applications being the recovery CONTENTS of hydrogen from ammonia plant purge-gas streams. Also in the 1980s, Dow produced the GENERON, the first commercial 1. Introduction A membrane for separation of nitrogen from air. Further 2. Overview of Mixed-Matrix Membranes (MMMs) B developments of this technology by Dow, Ube, and Dupont/ 3. Mixed-Matrix Membrane Morphology Models D Air Liquide resulted in a rapid expansion of the process. Over 4. Predictive Permeation Models I 4.1. Prediction Permeation Models for Ideal 10 000 such systems have already been installed worldwide. MMMs I The production of oxygen-enriched air, however, still requires 4.2. Prediction Permeation Models for Nonideal better membranes to become commercial.10 MMMs L Dried cellulose acetate membranes were first produced and 5. Model Validation: Comparison with Experimental utilized in commercial plants by Cynara, Separex, and Grace Data O Membrane Systems for the removal of carbon dioxide from 6. Methods for Avoiding Nonideal Interfacial methane in the purification of natural gas. Further developDefects AH ments of this technology involved polyimide hollow-fiber 6.1. Modifying Preparation Conditions of MMMs AH membranes by Air Liquide.19 Significant improvements of this 6.2. Modifying Interfacial Properties of Fillers technology would be required for its large-scale application to and Polymer Matrix AH purification of biogas. This resource is still underutilized mostly 6.3. Using Filler Particles and Bulk Polymers with due to the cost of current purification technologies. Some Special Properties AI research activities are conducted worldwide toward that 7. Conclusions AI objective.20 More than 100 large commercial membrane Author Information AM separation plants for vapor separations from air, with a value Corresponding Author AM of $1−5 million each, and at least 500 small systems operating Notes AM to capture vapor emissions from retail gasoline stations, Biographies AM industrial refrigerator units, and petrochemical process vents, Annex 1. Glossary of Model Designations AN with a value of $10,000−$100,000 each, have been installed by Annex 2. Nomenclature AN MTR (USA), GKSS (Europe), and NIHO Denko (Japan) since Parameters AO the early 1990s.21 Vaperma (Canada) has developed recently Greek Letters AO the SIFTEK membrane for gas-phase separation of water vapor Superscripts AO from ethanol.22 Subscripts AO The successful application of a membrane-based separation References AO process at both laboratory and industrial scale depends significantly on the appropriate chemical, mechanical, and permeation properties of membranes. The recent developments 1. INTRODUCTION of polymeric membrane materials have seemingly reached a limit in trade-off between selectivity and permeability. The Nowadays, membrane technologies are essential to a wide disadvantages of these membranes have in turn switched the range of commercial applications in petrochemical, chemical, focus of research toward mixed-matrix membranes (MMMs). food, pharmaceutical, semiconductor, biotechnological, and Logically, MMMs, where inorganic fillers are dispersed at environmental industries. While the cost of energy connanometer level in a polymer matrix, have been proposed as sumption is increasing rapidly, membrane technologies, which are especially promising for gas separation processes, are likely to play an increasingly important role in reducing the Received: September 19, 2012 environmental impact and costs of industrial processes owing © XXXX American Chemical Society

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dx.doi.org/10.1021/cr3003888 | Chem. Rev. XXXX, XXX, XXX−XXX

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Figure 1. Upper bound correlation for O2/N2 (left) and CO2/CH4 (right) separations. Reproduced with permission from ref 40. Copyright 2008 Elsevier.

The aim of the present work is to review recent scientific and technological advances in the development of theoretical models focused on the prediction of the gas separation performance of MMMs. An overview of the state of-the-art in MMMs using various kinds of fillers will be introduced. The reported MMM morphologies and theoretical prediction models are summarized and discussed. As the main objective of this review, the status, applicability, and future theoretical research for estimating the gas separation performance of MMMs are highlighted and validated. The Conclusion section summarizes the research activities and possible future research orientations.

alternative materials containing both promising selectivity benefits of the inorganic particles and economical processing capabilities of polymers. A rational choice of both inorganic and polymeric phases toward the preparation of MMMs is necessary. Theoretical predictions of the gas performance from the pure species permeation properties in these MMMs become more and more important. Up to now, different modeling attempts have been developed for the prediction of the performance of MMMs by various theoretical expressions depending on the MMM’s morphology, including ideal and nonideal MMMs.23−27 Ideal MMM morphology is a two-phase system, which consists of inorganic fillers and polymer matrix with no defects and no distortion at the filler−polymer interface. For this kind of morphology, theoretical models such as original Maxwell, Bruggeman, Bö ttcher and Higuchi, Lewis−Nielsen, Pal, Gonzo−Parentis−Gottifredi (GPG), Funk−Lloyd, Kang− Jones−Nair (KJN), and several other models have been adapted for estimating the permeation performance of MMMs. It is, however, very difficult to reach this ideal morphology because of the imperfect filler−polymer adhesion. Therefore, nonideal morphologies or three-phase systems containing organic−inorganic interface defects have been proposed. Interface defects can be classified into the following three major categories: (i) interface voids or sieves-in-a-cage, (ii) rigidified polymer layer around the inorganic fillers, and (iii) particle pore blockage. For these frequent three-phase systems, the modified-Maxwell model proposed by Koros’s group, modified Lewis−Nielsen, original and modified Felske, and modified Pal models, as well as the recent Hashemifard− Ismail−Matsuura (HIM) theory have been developed. Generally, these models have the ability to predict permeability and selectivity for most MMM morphologies, not only at low filler loading but also at higher values. However, many other additional parameters affecting the phase transport processes through MMMs, that is, the particle pore size and its distribution, particle dispersion, polymer properties, and their interactions, and operating conditions such as temperature, pressure, and gas feed composition should be considered in future MMM research.

2. OVERVIEW OF MIXED-MATRIX MEMBRANES (MMMs) During the past few decades, membrane-based separation processes have become an emerging technology, being the subject of numerous worldwide academic studies. Polymer membranes are the first and most common commercial membranes for gas separations owing to their low cost, high processability, good mechanical stability, and excellent transport properties.2−17,28−37 Both rubbery and glassy polymers, including poly(dimethyl siloxane), silicone rubber, nitrilbutadiene, ethylene−propylene and polychloroprene rubbers, cellulose acetate, polysulfone, polyethersulfone, polyamides, polyimides, polyetherimides, polypropylene, poly(vinyl chloride), poly(vinyl fluoride), and sulfonated poly(ether ether ketone), have been recognized as promising polymers for the preparation of membranes.32−39 However, a poor resistance to contaminants, low chemical and thermal stability, and a limit in the trade-off between permeability and selectivity (polymer upper bound limit) are among some of their disadvantages. Figure 1 shows the polymer upper limits for O2/N2 and CO2/ CH4 separations demonstrated by Robeson in 2008.40 On the other hand, inorganic membranes such as zeolite and carbon molecular sieve (CMS) membranes are overcoming some of the drawbacks of polymer membranes. The unique properties of microporous molecular sieves make inorganic membranes very attractive for gas separations by imparting their excellent thermal and chemical stability, good erosion resistance, and high gas flux and selectivity. Although some B


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affinity to glassy polymers, resulting in good contact at the interface. Nonetheless, several efforts to improve simultaneously the permeability and selectivity, as well as to avoid the formation of interfacial voids, in the CMS-based MMMs have been conducted.254,258 Nonporous and porous silica nanoparticles in forms such as fumed, ceramic, organosilicate, tetraethoxysilane, or colloidal silica are generally dispersed in the polymer matrix to form heterogeneous MMMs through a sol−gel process.167,186,261−314 Due to the intrinsic impermeability or weak permeability of the silica particles, the incorporation of these solids in the polymer matrix results in an improvement of both the permeability and selectivity by altering the molecular packing of the polymer chains.72,285 Some of these MMMs, which showed an increase in polymer volume without creating nonselective voids, have however enhanced permeation properties but decreased selectivity. In order to eliminate the voids, chemical modifications with silane coupling agents containing organof un c t i o n a l o r h y d r o x y l g r o up s h a v e b e e n p r o posed.259,270,277,284,290,291,294,298−303 Moreover, ordered mesoporous materials as another form of silica particles were employed as additive fillers in order to enhance the filler−polymer interaction through polymer chain penetration of the mesopores. Although no selectivity enhancement was obtained since gas diffusion in mesopores is nonselective, the incorporation of these materials in polymer matrix can bridge the polymer chains though hydrogen bonding, mainly due to their large surface area.297,315−327 To improve the selectivity, some approaches have been developed such as to incorporate micropores into the mesoporous materials,328−331 to create mesopores in pure zeolites,332−336 or to modify the interface composition by functionalizing with organic groups, that is, periodic mesoporous organosilicas (PMOs).337−340 Metal oxide nanoparticles such as Ag2O,341 MgO,342−344 and TiO2234,345−353 were also used as dispersed fillers in the fabrication of MMMs. The high specific surface area combined with the nanosize diameter of these oxides can improve the particle spatial distribution and prevent nonselective void formation at nanoparticle−polymer matrix interface. Owing to the small size of nano-oxides causing only weak selectivity effects, significant improvements in permeability with slight decreases in selectivity of the MMMs have been reported. Several other types of nonzeolitic inorganic fillers such as nonfunctionalized and functionalized activated carbon,113,167,354−361 alumina,173,174,230,362 aluminum silicate,363 Boehmite,364 C60,365,366 and microporous organic hosts367−389 have also shown an interesting potential for application as dispersed phase in MMMs. Cyclodextrin (CD), 367−371 polyamide (PA),372 polypyrrole (PPy),373 polytetrafluoroethylene (PTFE),374 ion-exchange resins,375−377 trimethylsilylglucose (TMSG),378 and polyhedral oligomeric silsesquioxane (POSS)379−389 are some examples of microporous organic host materials. As an example, CD dispersion in the poly(vinyl alcohol) (PVA)367 or poly(amide-imide) (PAI)370 polymers can increase the MMM separation performances in both permeation flux and selectivity due to the unique molecular recognition property of CD. Promisingly, the incorporation of POSS into polymers not only enhances physical and mechanical properties but also improves the thermal and oxidation resistance, as well as rheological properties, of the MMMs.384,387

inorganic membranes show gas performances well above the trade-off curve for polymers, certain aspects still require further attention such as mechanical resistance, reproducibility, longterm stability, up-scaling, and, of course, fabrication cost.2−17,41−55 Numerous efforts have been made to overcome the limitation of both polymeric and inorganic membranes. The so-called mixed-matrix membranes (MMMs), where inorganic fillers in solid, liquid, or both solid and liquid forms are dispersed in a polymer matrix, have been identified to provide a solution to the upper-bound trade-off limit of the polymeric membranes as well as the inherent brittleness obstacle of the inorganic membranes. MMMs are indeed expected to combine the potential advantages in separation performances of both inorganic and polymer membranes. Since the pioneering overview by Okumus et al.,56 several significant review reports on the prospects of MMMs have been published.26,57−75 These materials are generally classified into three main types of MMMs, including solid−polymer, liquid− polymer, and solid−liquid−polymer MMMs. Solid−polymer MMMs have received the most significant attention in the fabrication of MMMs. For this type of MMMs, both zeolitic and nonzeolitic inorganic materials have been traditionally incorporated as fillers into the polymer matrix. Zeolitic inorganic particles are conventional zeolites, AlPO and SAPO molecular sieves. These microporous materials have been considered in the fabrication of MMMs for gas separation due to their thermal stability as well as their permeation performances.76−227 Shape selectivity and specific adsorption properties of zeolitic crystals combined with the fluent processability of polymer matrix can improve the permeability and selectivity of various polymer films for gas separation of different gas pairs. During the development of zeolitic MMMs, both rubbery and glassy polymers were used as polymer matrix. The interfacial interaction between zeolitic fillers and rubbery polymers is good due to the high mobility of the polymeric chains. Despite their high mechanical stability and more desirable permeation behaviors, the MMMs fabricated using glassy polymers and zeolites have resulted in the presence of interfacial void defects. In order to resolve this adhesion problem, a wide range of surface modification has been introduced, that is, coating of a diluted solution of a highly permeable silicone rubber to eliminate the unselective gaps often occurring on the polymers,152,180 addition of a plasticizer to reduce the intrinsic gas separation performance of polymers,152,181,182 or using silane and amine coupling agents to improve both interfacial adhesion and gas selectivity by changing the surface properties of zeolites from hydrophilic to hydrophobic.94,97,108,109,121,129,145,165,166,168,180,183−197 To date, numerous reports on zeolite-based MMMs as commercial alternatives over the polymeric and inorganic membranes have been patented.228−248 On the other hand, nonzeolitic inorganic fillers have also received significant attention throughout the development of MMMs. Carbon molecular sieves (CMS), nonporous and porous silica nanoparticles, and metal oxide nanoparticles are common classes of nonzeolitic fillers. Carbon molecular sieves are carbonaceous porous solids containing relatively wide opening windows that correspond to the diffusing gas kinetic diameters. Based on their high productivity combined with excellent permeation behaviors, CMS nanoparticles with welldefined micropores offer potential opportunities in fabrication of MMMs.161,232,249−260 These porous solids display significant C


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carboxylates, phosphonates, sulfonates, cyanides, pyridine, imidazoles, gives rise to numerous opportunities for functionalization and grafting. Fortunately, the carbon subnetwork can be itself functionalized with halogeno or amino groups. Nonporous MOFs are those in which the organic ligands are filling their cavities. In contrast, the blocked channels of porous MOFs can be opened by means of thermal or chemical treatments.496−500 Although most of porous MOFs are microporous solids, some MOFs have intrinsic mesoporous channels or mesostructured pore lattices.501−503 Zeolitic imidazolate frameworks (ZIFs) are another class of MOFs with exceptional thermal, hydrothermal, and chemical stability up to 400 °C.499 The enormous number of MOFs with a broad range of sieving properties for selective gas adsorption could potentially give rise to choice of promising new dispersed fillers. The application of these organic−inorganic materials as fillers in the MMM field, however, is still in its infancy. Up to now, not more than ten kinds of MOFs have been applied for the fabrication of MMMs. The first review on the synthesis, characterization, and industrial application of these new MOFbased membranes was published by our research group in middle 2011,75 followed recently by a feature publication of Kapteijn and co-workers.493 The main drawbacks of MOFs in their application as dispersed fillers are frequently lower thermal, chemical stability, and higher production cost compared with traditional fillers such as zeolites and CMSs. Liquid−polymer MMMs are a second class of MMMs, in which the physical state of the fillers dispersed in the polymer matrix is a liquid. The existing literature on these MMMs is less developed. Poly(ethylene glycol) (PEG),504−509 phosphomolybdic acid solution (PMA),510−513 phosphotungstic acid solution (PTA),514 and some diol isomers (butanediol isomers)515 are several examples of liquid fillers in MMMs for gas separation. Because of the undesirable leakage of the liquid from the membranes, the long-term stability is still a critical drawback of these MMMs for industrial gas separation application. The third type of MMMs is the solid−liquid−polymer MMM.516,517 These hybrid MMMs are expected to combine the properties of the continuous polymer phase, the dispersed solid filler phase, and the impregnated liquid phase. Unfortunately, similar to liquid−polymer MMMs, there are too few publications on the simultaneous introduction of solid and liquid fillers into the polymer matrix. As an example of these MMMs reported by Kulprathipanja et al.,517 solid fillers such as activated carbon were impregnated with liquid fillers such as PEG. The impregnated activated carbons were then dispersed in the continuous silicone rubber polymer matrix. The presence of PEG on the surface of activated carbon and the high solubility of CO2 in PEG are yielding a relatively low permeability and an enhanced selectivity of these MMMs in CO2/N2 separation.

Despite the rapid increase in the number of publications focusing on the application of the above-mentioned conventional fillers in the fabrication of solid−polymer MMMs, in the past decade, several alternative fillers such as carbon nanotubes (CNTs), graphenes, layered silicates, and metal−organic frameworks (MOFs) have also been recognized as new filler materials with many attractive and desired properties. CNTs are novel and interesting graphitic carbon materials constructed from tube-rolled graphite sheets. Because the diameter of a nanotube ranges from a few to tens of nanometers while its length can reach up to several millimeters, these hollow cylinders, usually capped at least at one end, are considered as nearly one-dimensional structures.59,62,66,69,72,390−427 The extraordinary inherent smoothness of the potential energy surface of CNTs can allow rapid diffusion of the gas molecules through their channels, resulting in high gas permeability without changing selectivity. Unfortunately, the poor adhesion between CNT particles and polymer matrix, the agglomeration and entanglement of CNTs, and their high production cost are several drawbacks of CNTs as nanofillers for further application in the MMM domain. Surface modification by acid treatment with coupling agent containing carboxyl or hydroxyl groups is a common functionalization method to enhance the compatibility of CNTs with the polymer matrix. This acid treatment also acts as a cutting process to disentangle and disintegrate the CNTs for better dispersion in polymer matrix, as well as to open up the closed end of the CNTs to facilitate greater diffusitivity of gas molecules.391,392,398,409 Graphenes, as a new class of carbon nanomaterials, have been used as a viable and inexpensive filler substitute for CNTs in MMMs, due to their excellent in-plane mechanical, structural, thermal, and electrical properties.428−430 Another type of alternative inorganic fillers are the layered silicates that have been used for designing a new generation of MMMs.64,65,74,431−460 Among layered silicates or clay minerals, montmorillonite, hectorite, and saponite clays, as well as halloysite nanotubes, have attracted great attention for the fabrication of MMMs. Based on their relatively high cationic exchange capacity, high aspect ratio, ease of expansion, and competitive low cost and natural abundance, incorporation of layered silicates into the polymer matrix has been well-known to improve the mechanical, dimensional, thermal, and barrier permeation properties of MMMs, even at low silicate layer loading. Excellent mechanical stability of layered silicate− MMMs is obtained with intercalated and exfoliated structures of clay layers dispersed into the polymer matrix. Layered silicates are, however, usually hydrophilic. Consequently, their interactions with nonpolar polymers are not favorable but can expand upon surfactant cationic exchange. Metal−organic frameworks (MOFs) constitute a new approach to the design of solid fillers throughout the development of MMMs.75,461−495 In the past decade, this new class of crystalline and porous materials formed by selfassembly of complex subunits comprising transition metal centers connected by various polyfunctional organic ligands to form one-, two-, and three-dimensional structures have received tremendous attention. These hybrid materials have interesting properties such as framework regularity, large surface areas, high porosities, low densities, and enormous flexibility in pore size, shape, and structure. Compared with other porous materials, which are associated with only limited metals, MOFs accept almost all the cations up to tetravalent. A huge choice of organic linkers containing O or N donors, such as

3. MIXED-MATRIX MEMBRANE MORPHOLOGY MODELS A general image of an ideal MMM structure including the dispersed phase and the polymer matrix is shown in Figure 2. This theoretical two-phase morphology for the prediction of permeation of gases through MMMs has been developed and shown to represent correctly some MMMs by several researchers.23−27,200,201 While the white part represents the continuous polymer matrix phase, the small dotted squares describe the dispersed phase. The contact between the two D


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morphology is often designated as a leaky interfacial defect.180,182,193,194,518 When the adhesion between the two phases is good, the free volume in the polymer located near the filler surface is reduced. This phenomenon is known as polymer rigidification (Figure 3b). The rigidified polymer chain layer around the inorganic fillers has lower mobility in the region directly contacting the particles than that of the bulk polymer matrix.128,129,180,182,194,208,279,283 Usually, this interfacial defect may have enhanced diffusive selectivity, while the permeability is decreased.128,129,182,193,194,257,258,519 These two interfacial defects may occur when both porous and impermeable nonporous particles are dispersed in the polymer matrix. In MMMs using porous fillers, however, pore blockage in the surface region is often evoked.93,128,158,193,194,201,520 The pore blockage defect as shown in Figure 3c is often caused by the clogging or plugging of filler pores with a sorbent, solvent, contaminant, or minor component in a feed gas or polymer chains before, during, or after membrane fabrication.57,62,93,128,158,181,182,186,201,205,208 Depending on the degree of pore blockage as well as the molecular diameter of gas molecules, this defect can be classified into total or partial pore blockage. If the pores are completely plugged, the gas cannot pass through the particle fillers, and no enhancement in selectivity over the neat polymer is reached as in the case of MMMs filled with nonporous particles.138,280 When the pores are only partially blocked, depending on the gas molecular dimension as well as the degree of pore blockage, the gas permeability is generally decreased, while the selectivity is varied depending on the porous fillers. If the pore size of fillers before blockage is in the range of the gas molecular diameter, the selectivity is greatly decreased upon blockage. On the other hand, for the porous fillers with original pore size larger than the gas molecular diameter, the blockage effect may enhance selectivity.62,93,96,138,177 In fact, upon investigation of the formation and gas separation performances of MMMs using the best-established examples of porous fillers, such as microscaled zeolites and carbon molecular sieves, two more morphologies of the interfacial voids were defined.97,156,180,182,193 Sieve-in-a-cage morphology (Figure 4a) is the most representative of

Figure 2. Schematic diagram of an ideal MMM structure. Reproduced with permission from ref 26. Copyright 2010 Elsevier.

phases at the polymer−particle interface is assumed to be defect free. This case of morphology typically corresponds to the ideal Maxwell model, which was mainly derived to represent thermal or electrical conductivity in composite media. A defect-free polymer−particle contact during the introduction of dispersed fillers in a polymer matrix is, however, difficult to achieve. The incorporation of filler particles can modify the properties of the neighboring polymer phase, which affects the overall membrane separation performance. The filler encapsulation may, moreover, cause a dense packing, or dynamic conformation of the polymer chain at its interfacial surface. Thus, depending on the polymer−particle adhesion, various nanoscale structures can be observed at the interface (Figure 3).

Figure 3. Schematic diagram of various nonideal MMM morphologies: (a) interfacial voids, (b) rigidified polymer chain layer, and (c) pore blockage. Reproduced with permission from ref 26. Copyright 2010 Elsevier.

A poor polymer−inorganic filler adhesion, polymer packing disruption in the vicinity of the dispersed particles, repulsive force between the two phases, different thermal expansion coefficients, and the elongation stress during fiber spinning are presumed to be among the causes for the formation of nonselective interfacial voids (Figure 3a). The formation of these defects allows the gases to pass and, hence, deteriorates the apparent selectivity and increases the permeability of the MMM.62,115,116,128,155,166,201−204,262 Additionally, an interfacial morphology with molecular-scale or submolecular-scale extra free volume between segments can occur, and this produces a slight decrease in selectivity below that of the pure polymer, while still showing an increase in permeability. This special

Figure 4. Representatives of interfacial voids. Reproduced with permission from ref 182. Copyright 2005 Elsevier.

voids.97,109,155,156,182 As mentioned above, this morphology results in an increased permeability with essentially no change in selectivity. To overcome this problem, the introduction of a bridging agent during the membrane fabrication in order to facilitate a good interfacial interaction was proposed. As shown in Figure 4b, a situation with bridging of the filler and polymer matrix following surface modification might be just a special E


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Figure 5. Schematic illustration of three-phase morphologies of MMMs comprising large pore size fillers: case I, ideal; case II, void (the dark-gray space surrounding the filler is the void) at low filler loading (II-A), at high filler loading with particle agglomeration (II-B), and at high filler loading with particle agglomeration combined with channeling morphology (II-C); case III, rigidification (dotted space shows the rigidified region); case IV, pore blocking (black tips shows the blocked parts); and case V, pore blocking and void. Reproduced with permission from ref 439. Copyright 2011 Elsevier.

case of the sieve-in-a-cage morphology, where the effective void thickness or high free volume region between the polymer and dispersed particles might be approximately the size of the gaseous penetrants.73,193 Recent work from our group has underlined the various effects of the filler grafting on membrane properties.185 A more detailed illustration of different types of interfacial phase was reported by Hashemifard et al. for MMMs comprising large pore size fillers.439 In this study, halloysite nanotubes (HNTs) were employed as representative of large pore size fillers to demonstrate the possible morphologies. As illustrated in Figure 5, five main morphologies including ideal (case I), void (case II), rigidified (case III), pore blocking (case IV), and agglomeration combined with pore blocking (case V) could be observed. In case II, the formation of voids between porous fillers and polymer matrix results in an increase in permeability but no significant change in selectivity at low filler loading (case II-A). However, at higher values of filler loading and poor particle spatial distribution, clusters of aggregated fillers surrounded by voids are formed (case II-B). Case II-C exhibits agglomeration combined with channeling morphology. For the latter two morphologies, especially case II-C, the selectivity of MMMs is dramatically decreased with an increase in permeability. It should be noted that for case IV, at

increasing degree of pore blockage, the permeability tends to decline, while selectivity is likely higher than that of the polymer matrix. Case V including a combination of pore blockage and channel defects is common for MMMs filled with both small and large pore size particles. It has also been shown that in MMMs containing porous fillers, the pore blocking effect is often accompanied by the polymer chain layer rigidification. The polymer rigidification defect is dominant over the pore blockage at the polymer−filler interphase. Since the rigidified defect should only affect a very thin layer of polymer in the vicinity of the filler, the pore blockage is assumed to involve a serious decrease in permeability.62,128,129,256,258 In their approach to a MMM permeation prediction model, Chung and co-workers62,128,129 proposed a schematic diagram of nonideal three-phase MMM morphology simultaneously considering both zeolite pore blockage and polymer chain rigidification. As illustrated in Figure 6, the whole MMM is assumed to be in a pseudo-threephase system. The “pseudo-insert phase” or “pseudo-dispersed phase” concept was first proposed by Koros’s research group202,208 in order to envision the combination between the insert or dispersed component and interphase. In this MMM three-phase model, the third phase including the dispersed particles and the interfacial skin with partial or F


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Figure 6. Schematic diagram of a nonideal three-phase MMM morphology. Reproduced with permission from ref 128. Copyright 2005 Elsevier.

complete pore blockage acts as the second pseudo-dispersed phase. Then, the second phase as the first pseudo-dispersed phase is constructed from this third phase encapsulated by the rigidified polymer chain layer. Finally, the continuous polymer matrix acts as the first phase. Note that this schematic diagram was developed as a modified three-phase Maxwell model. In 2008, Funk and Lloyd206 developed another representation of MMMs, including both ideal and nonideal morphologies of the zeolite-filled microporous MMMs, referred to as ZeoTIPS membranes, which are synthesized using the thermally induced phase separation (TIPS) procedure. In their report, different from a typical ideal MMM morphology including polymer and dispersed phases, an ideal ZeoTIPS structure is represented in terms of a mixture of three phase components in a parallel−series arrangement: polymer matrix, voids, and zeolite particles. As shown in Figure 7a, the permeation molecules can pass through zones I and II in parallel pathways. In zone II, the gases cross through the void− zeolite−void sequence in series channel. However, as often

reported in literature, the zeolite particles are often coated with a layer of polymer. Depending on the polymer−zeolite interaction, this interphase section has different effects on the permeation performance of MMMs. As shown in Figure 7b, a more realistic nonideal ZeoTIPS MMM is depicted including the same two zones as in ideal ZeoTIPS morphology, except in zone II, the zeolite particles are coated with a polymer layer of uniform thickness. These polymer interphases are arranged in parallel and series pathways with the void−zeolite−void sequence. In 2010, Hashemifard et al. introduced two new images of interphase MMM materials, including parallel−series and series−parallel combination morphologies, in which filler particles with body-centered cubic lattice are distributed randomly throughout the continuous polymer matrix.207 The parallel−series combination is used for the case of void and leaky interphase. As shown in Figure 8, zones I, II, and III are arranged as a series system. The penetrant gases first pass

Figure 7. (a) Ideal and (b) nonideal ZeoTIPS membrane morphologies. Reproduced with permission from ref 206. Copyright 2008 Elsevier.

Figure 8. Parallel−series combination morphology of MMMs with an interphase consisting of void and leaky defects. Reproduced with permission from ref 207. Copyright 2010 Elsevier. G


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bulk polymer and the dispersed particles is attributed to the inhibition of polymer chain mobility in compressive stress near the polymer−particle interface. In order to apply the Pal equation twice, the dispersed phase and the rigidified interphase are combined in a new phase, similar to the pseudo-insert phase concept proposed by Koros and coworkers for the modified Maxwell model.182,202,208 Very recently, to estimate permeation performance in MMMs particularly with tubular fillers, Kang et al. presented an analytical model (KJN model) with a detailed modelmorphology illustration. In their model, an ideal (defect-free) and a nonideal (defective) morphology of a composite membrane containing nanotube fillers dispersed in polymer matrix have been proposed.424 As shown in Figure 11 for the ideal morphology, the tubular fillers are oriented in a fixed direction. For a general ideal case, two possible permeation pathways are expressed in parallel combinations (Figure 11a): (i) without any tubular fillers (t0/Pm) and (ii) polymer matrix containing nf tubular fillers in series pathway (tm,nf/Pm and tf,nf/ Pf). The t/P ratio between the transport path length and the permeability is considered as the transport resistance. Unfortunately, it is difficult to estimate the total membrane resistance from all the possible resistances-in-parallel from the volume fraction of each distinct path (i.e., containing different numbers of tubular fillers (nf) and different path lengths (t0, the membrane thicknesses, or tm and tf)). Hence, the general ideal morphology is simplified to that of a MMM comprised of an average number of tubular fillers, in which polymer matrix (tm,avg/Pm) and filler (tf,avg/Pf) resistances are expressed in series (Figure 11b). Figure 11c illustrates a subcase morphology of that depicted in Figure 11b. This is a particular system with tubular fillers oriented in a fixed direction, and the total resistance of MMM is only contributed by the transport through the polymer matrix and fillers in series. The overall resistance is dominated by the one of the domain with fillers, in which the filler permeability (Pf) is much larger than that of the polymer matrix (Pm). Furthermore, in the case where the filler permeability is either much smaller than or of the same order of magnitude as that of the polymer matrix, the effective permeability of MMM (Peff) is either close to the one of the polymer matrix or significantly lower. As outlined above for all MMM morphologies, however, it is difficult to obtain a perfect defect-free MMM. For MMMs containing tubular fillers, the interfacial void surrounding the fillers owing to the incompatibility between the fillers and the polymer matrix as well as the pinholes extending from the feed to the permeate side of the composite membranes are two types of defects (Figure 12a). The so-called nonideal morphology of such MMMs having these defects is modeled as combining three pathways through three pieces of side-byside layers in parallel (Figure 12b,c): (i) a defect-free membrane with tubular fillers, (ii) a membrane comprised of polymer matrix and interfacial voids, and (iii) a pinhole. Clearly, this nonideal illustration is an extended case of the ideal morphology of MMMs containing tubular fillers. The actual contacts in the interfacial interaction between the tubular fillers and the polymer matrix as well as the blockage phenomenon of tubular pores of the fillers were not taken into account. The ideal zones, interfacial voids, and pinholes were separately considered as three independent ideal MMMs (Figure 12b).

progressively through zone III consisting of continuous polymer phase only, then diffuse through zone II consisting of continuous polymer matrix and interphase as a parallel pathway, and finally cross zone I consisting of continuous polymer matrix, interphase, and dispersed particle core as also a parallel channel. After passing zone I, the flow goes sequentially through zone II and zone III. On the other hand, the series−parallel combination is used to express the penetrant gas flow through MMMs having rigidified interface morphology (Figure 9). In this case, zones I,

Figure 9. Series-parallel combination morphology of MMMs with an interphase consisting of rigidification polymer chain defects. Reproduced with permission from ref 207. Copyright 2010 Elsevier.

II, and III are arranged as parallel channels. The gases permeate simultaneously through three zones I and II as a series pathway. The flows crossing different zones are finally recombined before leaving the MMMs. In 2011, in an attempt to modify the Pal model, Shimekit et al.27 proposed a detailed new expression of nonideal MMM morphologies to represent the presence of interfacial rigidified matrix chains. Figure 10 shows a systematic representation of a three-phase MMM system including polymer matrix, dispersed phase, and rigidified interphase. This interphase between the

Figure 10. Schematic representation of a three-phase MMM system proposed for the modified Pal model. Reproduced with permission from ref 27. Copyright 2011 Elsevier. H


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Figure 11. Ideal morphology of MMM containing tubular fillers: (a) a general ideal case with various molecular permeation paths in parallel−series combination; (b) a simplified case, in which the MMM contains an average number of tubular fillers; and (c) a subcase morphology of case b, where θ is tilt angle with respect to the bulk permeation direction of a filler in a polymer matrix; l and d are the length and outer diameter of a tubular filler, respectively. Reproduced with permission from ref 424. Copyright 2011 Elsevier.

inorganic fillers and polymer matrix with no defects and no distortion at the filler−polymer interface. However, it is very difficult to reach such ideal morphology because of the usually poor filler−polymer adhesion. Therefore, a nonideal morphology or a three-phase system containing the organic−inorganic interfacial defects was proposed. 4.1. Prediction Permeation Models for Ideal MMMs

Permeation prediction models for MMMs have been developed to predict the effective permeability of a gaseous penetrant through these MMMs as a function of continuous phase (polymer matrix) and dispersed phase (porous or nonporous particles) permeabilities, as well as volume fraction of the dispersed phase. For a given penetrant (A), the permeability coefficient (PA) can be estimated as the product of diffusivity coefficient (DA) and solubility coefficient (SA): PA = DA SA

(1)

DA and SA are usually functions of solubility and pressure. The permselectivity, αA/B, is the ideal ratio of permeability coefficients of the two components A and B: Figure 12. Nonideal morphology of MMM containing tubular fillers with (a) tubular fillers, void spaces, and pinholes, (b) three side-byside layers in parallel, defect-free composite membrane, composite membrane with void space, and pinholes, and (c) a combination of transport resistances in parallel. Reproduced with permission from ref 424. Copyright 2011 Elsevier.

αA/B =

PA DS = A A PB D BS B

(2)

Peff describes the effective steady-state permeability of a gaseous penetrant through a MMM. A minimum value of Peff is calculated when a series two-layer model (series model) is applied:

4. PREDICTIVE PERMEATION MODELS Different modeling attempts have been developed to estimate the permeation performance of MMMs by various theoretical expressions depending on the MMM’s morphology, including ideal and nonideal MMMs as mentioned above.23−27 Ideal MMM morphology is a two-phase system consisting of

Peff =

PcPd ϕcPd + ϕdPc

(3)

where, Pc is the continuous phase permeability, Pd is the dispersed phase permeability, and ϕc and ϕd are the volume fractions of continuous and dispersed phase, respectively. I


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A maximum value of Peff is reached when both phases are assumed to diffuse through a parallel two-layer membrane (parallel model): Peff = Pcϕc + Pdϕd

the separation performance of an individual phase). This expression must be modified to account for a nonideal morphology of MMMs possessing several types of interfacial defects such as interfacial voids, polymer chain rigidification, and pore blockage. The Bruggeman model, originally developed for the electric constant of particulate composites, can be also adapted to estimate the permeability of MMMs as shown:524−526

(4)

where, ϕc = (1 − ϕd). Applying the Maxwell−Wagner−Sillar model, the Peff of a MMM with a dilute dispersion of ellipsoids is given by the following expression:521,522 Peff = Pc

⎛α − 1⎞ (Pr)1/3 ⎜ ⎟ = (1 − ϕd)−1 ⎝ α − Pr ⎠

nPd + (1 − n)Pc − (1 − n)ϕd(Pc − Pd) nPd + (1 − n)Pc + nϕd(Pc − Pd)

(5)

where, Pr = Peff/Pc. This equation considers the effect of adding additional particles to a dilute suspension for a random dispersion of spherical particles. The Bruggeman model is an improvement over the basic Maxwell model for a larger range of ϕd; however, limitations similar to those of the basic Maxwell model are still obstacles in application. In addition, this model contains an implicit function that needs to be solved numerically. The Böttcher and Higuchi models, originally applied to a random dispersion of spherical particles, are, respectively, expressed as follows:527,528

Here, n is the particle shape factor. For prolate ellipsoids, where the longest axis of the ellipsoid is directed along the applied partial pressure gradient, 0 < n < 1/3. For oblate ellipsoids, that means the shortest axis of the ellipsoid is directed along the applied partial pressure gradient, 1/3 < n < 1. The limit of n = 0 leads to a parallel two-layer model and can be expressed as an arithmetic mean of the dispersed and continuous phase permeabilities: Peff = Pc(1 − ϕd) + ϕdPd

(6)

⎛ P ⎞⎛ P ⎞ ⎜1 − c ⎟⎜α + 2 eff ⎟ = 3ϕd(α − 1) Peff ⎠⎝ Pc ⎠ ⎝

On the other side, the limit of n = 1 corresponds to a series two-layer model, and eq 5 can be expressed as follows: Pd Peff = Pc Pd(1 − ϕd) + ϕdPc

(7)

Pr =

At the limit of n = 1/3, eq 5 reduces to the following equation known as the original Maxwell equation with different expressions: Peff = Pc = Pc = Pc

3ϕdβ Peff =1+ Pc 1 − ϕdβ − KH(1 − ϕd)β 2

(10)

(11)

where, KH is treated as an empirical constant and assigned a value of 0.78. β is useful to define the “reduced permeation polarizability” or a convenient measure of penetrant permeability difference between dispersed spheres and polymer matrix:

Pd + 2Pc − 2ϕd(Pc − Pd) Pd + 2Pc + ϕd(Pc − Pd) 2(1 − ϕd) + α(1 + 2ϕd)

β=

(2 + ϕd) + α(1 − ϕd) 1 + 2ϕd(α − 1)/(α + 2) 1 − ϕd(α − 1)/(α + 2)

(9)

P − Pc α−1 = d α+2 Pd + 2Pc

(12)

This parameter is bounded by −0.5 ≤ β ≤ 1. The lower and upper limits correspond to totally nonpermeable and to perfectly permeable filler particles, respectively. Note that eq 10 is a second-order algebraic expression on Peff. So, like the Bruggeman model, a trial and error procedure is needed to estimate Peff as a function of α and ϕ for the Böttcher formula. The Lewis−Nielsen model, originally proposed for an elastic modulus of particulate composites, is also possible to apply for predicting the effective permeability of MMMs:529,530

(8)

where, α is the permeability ratio, Pd/Pc. The Maxwell equation was developed in 1873, originally derived from the estimation of the dielectric properties of composite materials, and has been widely accepted as a simple and effective tool for predicting MMM permeation properties.523 The Maxwell model is strictly applicable to a dilute suspension of spherical particles at low loadings, when the volume fraction of filler particles is less than about 0.2, because of the assumption that the streamlines around particles are not affected by the presence of nearby particles. For higher values of volume fraction ϕd, the Maxwell model cannot predict the permeability of MMMs. When ϕd → ϕm, a deviation of the relative permeability Peff is more pronounced, especially for membranes with permeability ratio α → ∞. ϕm is the maximum packing volume fraction of filler particles and is usually considered to be equal to 0.64 for a random close packing of uniform spheres. ϕm is in relation with particle size distribution, particle shape, and aggregation of particles. Furthermore, this model does not consider such morphological parameters of filler as particle size distribution, particle shape, and aggregation of particles. Generally, the Maxwell model represents an ideal case of MMM morphology (no defects and no distortion on

Pr =

1 + 2ϕd(α − 1)/(α + 2) Peff = Pc 1 − ψϕd(α − 1)/(α + 2)

(13)

where: ⎛1 − ϕ ⎞ m⎟ ψ = 1 + ⎜⎜ ϕ 2 ⎟ d ϕ ⎝ m ⎠

(14)

By considering the effect of particle morphology on permeability, this model might be representing a correct behavior of permeability over the range of 0 < ϕd < ϕm. The relative permeability Peff at ϕd = ϕm is found to be, however, diverging when the permeability ratio α → ∞. Note that when ϕm → 1, the Lewis−Nielsen model reduces to the Maxwell equation. J


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K = a + bϕd1.5

A model proposed by Cussler is a form similar to the original Maxwell model applied to a dilute suspension of flake spheres531 but considering a staggered array of high aspect ratio particles. The effective permeability of an ideal MMM is then expressed as follows: Peff = Pc

1 ⎛ Pd 1−ϕ ⎞ 1 − ϕd + 1/⎜ ϕ P + 4 2 2d ⎟ α ϕd ⎠ ⎝ dc

where, the parameters a and b are functions of β: a = −0.002254 − 0.123112β + 2.93656β 2 + 1.690β 3 (20)

b = 0.0039298 − 0.803494β − 2.16207β 2 + 6.48296β 3

(15)

+ 5.27196β 4

where, α is the flake aspect ratio and ϕd is the volume fraction of flakes. A generalized Maxwell model proposed by Petropoulous522 and extended by Toy and coauthors532 to estimate the permeation properties of a binary structured composite, wherein the additive filler is randomly dispersed with sharp interfaces in a continuous polymer matrix, can be expressed as follows: ⎡ ⎢ Peff = Pc⎢1 + ⎢ ⎣

⎤ ⎥ ⎥ − ϕd ⎥ ⎦

Pd / Pc + G Pd / Pc − 1

)

(16)

where, G is a geometric factor accounting for the effect of dispersion shape. G equals 1 for long and cylindrical (elongated) particles disposed transverse to the gas flow direction. G is 2 for spherical particles or isometric aggregates. In the case of planar (laminate) particles, G tends to infinity if the dispersed particles are oriented in lamellae parallel to the gas flow direction, minimizing resistance to flow. On the other side, G tends to zero if the dispersed particles are oriented in lamellae perpendicular to the gas flow direction, maximizing impedance of flow.522 The Pal model, originally applied for thermal conductivity of particulate composites, was also adapted for prediction of permeability:533 1/3

(Pr)

−ϕ ⎛α − 1⎞ ⎛ ϕd ⎞ m ⎟⎟ ⎟ = ⎜⎜1 − ⎜ ϕm ⎠ ⎝ α − Pr ⎠ ⎝

(17)

Table 1. Definition of Volume Fractions for the Funk−Lloyd Model206

Using the differential effective medium approach considering the packing difficulty of filler particles, this model also considers the effect of morphology as a function of the maximum packing volume fraction, ϕm, like the Lewis−Nielsen model. The Pal model, however, like the Bruggeman equation, should be solved numerically. Moreover, when ϕm → 1, the Pal model reduces to the Bruggeman model. Applied to an ideal morphology including a two-phase system (i.e., compatibility between fillers and polymer matrix with no defects and no distortion), the Pal model covers a wide range of ϕ (0 < ϕ < ϕm). Based on the percolation theory, in which a simple power law can describe the relation between composite permeability and filler concentration,24 and the hard-sphere fluid model proposed by Chiew and Glandt,534 Gonzo and coauthors proposed an extension of the original Maxwell model in terms of ϕd, namely, the Gonzo−Parentis−Gottifredi (GPG) model:24 Pr =

Peff = 1 + 3βϕd + Kϕd 2 + O(ϕd)3 Pc

(21)

In these equations, β is a convenient measure of penetrant permeability difference between the two phases. Like the Böttcher and Higuchi models, this parameter is bounded by −0.5 ≤ β ≤ 1. The second term represents the interaction between particles and polymer matrix, and the third term implies the interaction between these filler particles. At low values of particle loading (ϕd ≪ 1), GPG equations give the same results as the original Maxwell model. Compared with the original Maxwell model, the particle−polymer interfacial interaction is an additional issue, which is further considered in the GPG model. In 2008, Funk and Lloyd reported a prediction model for microporous zeolite-filled MMMs, namely, ZeoTIPS membranes.206 This Funk−Lloyd model takes into account the zeolite loading as well as the ratio of void volume to polymer volume. In a nonideal ZeoTIPS membrane, a polymer layer of uniform thickness often coats the dispersed zeolites at the interfacial surface. The interfacial interaction between zeolite particles and polymer layer is assumed to be a good defect-free contact. Thus, path II contains voids, polymer layers, and a region III in series arrangement (see section 3 and Figure 7 for definitions). Region III includes the zeolites and polymer layers in parallel combination. Based on this parallel−series model construction, the volume fractions are defined as listed in Table 1. Note that the volume fractions of continuous polymer and voids ignoring the zeolite portion in the entire MMM are denoted as ϕ*c and ϕ*v , respectively. Factor ξ refers to the

(1 + G)ϕd

(

(19)

volume fractions

definition Entire MMMs

zeolites

ϕdo

polymer layers

ϕvo = (1 − ϕdo)ϕv*

(22)

continuous polymer

ϕco

(23)

= (1 −

ϕdo)ϕc*

Path II zeolites

polymer layers

voids

ϕdII = ϕcII = ϕvII =

ϕdo ϕdo

+ ϕvo + ξϕco

(24)

ϕdo +

ξϕco ϕvo +

ξϕco

(25)

ϕdo +

ϕvo ϕvo +

ξϕco

(26)

Region III

(18)

Here, K and O are the needed corrections of Maxwell expression. The coefficient K is a function not only of β but also of ϕ: K

zeolites

ϕdIII =

polymer layers

ϕcIII

ϕdII ϕdII

1

+ 2 ϕcII

1 = ϕcII 2

(27)

(28)


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fraction of total polymer-coated zeolite particles in the entire MMM. Then, the permeability through region III (PIII), path II (PII), and entire MMMs (Ptotal) can be expressed as follows: P III = ϕdIIIPd + ϕcIIIPc

(1 − ϕvII − 12 ϕdII)PvPc + ϕvIIPcP II + 12 ϕcIIPvP III

= (1 − ξ)ϕcoPc + [1 − (1 − ξ)ϕco]P II

(30)

(31)

Peff,v Pc

ϕd + ϕv + ϕp

Peff,v

Pp (35)

Pv + 5Pc − 5(Pc − Pv)ϕv Pv + 5Pc + (Pc − Pv)ϕv

(36)

1 RT

32r 2RT 9πM

(37)

where, DKn is the Knudsen diffusivity, SIG is the solubility coefficient, r is the average diameter of the pinhole or void space, and M is the molecular weight of the transported molecule. 4.2. Prediction Permeation Models for Nonideal MMMs

For a nonideal MMM, interfacial defects affect the membrane performance and should be considered in prediction models. As mentioned above, interfacial defects can be classified into the following three major categories: (i) interfacial voids or sievesin-a-cage, (ii) rigidified polymer layer around the inorganic fillers, and (iii) particle pore blockage. These defects are generally formed at the interfacial region between inorganic fillers and polymer matrix, namely, as an interphase. The presence of an interphase was first hypothesized in a report of Erdem-Şenatalar et al.199 based on the effective medium theory (EMT).536 In this report, two additional parameters, namely, the permeability (Pdi) and the volume fraction (ϕdi) of an inorganic filler−interphase composite medium, are introduced. Since inorganic fillers and interphase are in series arrangement, the overall permeability can be estimated as follows:

−1 ⎡⎛ ⎞ ⎛ ⎞ ⎤ Pc ⎜ cos θ 1 ⎢⎜1 − ⎥ ⎟ ⎟ ϕf + ⎜ ϕ 1 ⎢⎜ Pd ⎝ cos θ + 1 sin θ ⎟⎠ d ⎥⎦ cos θ + α sin θ ⎟⎠ ⎣⎝ α

(33)

where, α = l/d is the aspect ratio of tubular fillers and θ is the filler orientation angle with respect to the membrane transport direction, varying from 0 to π/2 radians. For a completely random distribution of filler orientations, the KJN model can be rewritten as:

∫0

=

Pv = Pp = SIGDKn =

Peff = Pc

⎤−1 Pc dθ ⎥ Peff, θ ⎥⎦

ϕd + ϕv + ϕp

(32)

Clearly, the Funk−Lloyd model represents an enhanced approach to the permeation prediction for an ideal two-phase MMM. This model discriminates the participation of the polymer layer coating the zeolite particles from the neat polymer matrix. However, the interfacial contact between the zeolitic and polymer phases is not discussed yet. The Kang−Jones−Nair (KJN) model was specifically derived for an ideal composite membrane with tubular fillers having a fixed orientation that possesses perfectly anisotropic 1D transport properties. It can be expressed as follows:424

π /2

ϕp

ϕv

where, Pv is the permeability in the void space. The permeation mechanism in the void as well as in the pinholes can be considered as the Knudsen diffusion.209 Therefore, the permeabilities of voids (Pv) and pinholes (Pp) can be expressed as:

PvPd ϕdIIPv + ϕvIIPd

Peff,d +

where, Peff,d and Peff,v are the effective permeabilities of an ideal MMM composed of filler/polymer and void/polymer, respectively. Peff,d is deduced from eq 34. ϕd, ϕv, and ϕp are the volume fractions of the imaginary membranes composed of filler/polymer, void/polymer, and pinhole, respectively. Peff,v can be predicted by the Hamilton−Crosser model, assuming that the void space is cylindrical and the diffusion in this space is isotropic:535

where, Pv, Pc, and Pd are the permeabilities of voids, continuous polymer, and zeolite particles, respectively. For an ideal ZeoTIPS membrane, there is a free polymer layer coating around zeolite particles (Figure 7a). Corresponding to a zero thickness of polymer layer (ξ = 0), eq 30, which expresses the permeability through path II (PII), reduces to:

Peff π⎡ = ⎢ Pc 2 ⎢⎣

ϕd + ϕv + ϕp

(29)

Ptotal = ϕIPc + ϕIIP II

P II =

ϕd

+

PvPcP III

II

P =

Peff =

ϕdi(Peff − Pdi) Pdi + 2Peff

+

ϕd(Peff + Pc) Pc + 2Peff

=0

(38)

where Pdi = PiPd/(Piϕd + Pdϕi) and ϕdi = ϕd + ϕi. Pi and ϕi denote the permeability and the volume fraction of interphase, respectively. ϕi is related to the filler particle radius (r) and the thickness of interphase (t) for spherical particles by the following equation:

(34)

Specifically, for a MMM consisting of tubular fillers, the incompatibility between the fillers and the polymer matrix can often create an interfacial void space surrounding the fillers. Moreover, the polymer membrane may contain intrinsic pinholes extending from the feed to the permeate side of the membrane. Considering these two defects, the ideal KJN model can be extended to estimate the permeation performance of a MMM composite, which is assumed to contain separately three idealized membranes in parallel arrangement. Hence, the effective permeability (Peff) becomes:

⎛ 3t 3t 2 t3 ⎞ ϕi = ϕd⎜ + 2 + 3 ⎟ ⎝r r r ⎠

(39)

For a cubic particle, r is replaced by half of the length of cube edge. For other particle shapes, the particle radius (r) should be rationally derived. L


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Table 2. Modified Maxwell Model with Different Interfacial Defects182 interface defects voids or sieve-in-a-cage parameters

PI, void permeability PI = DS ⎛ dg ⎞ Da = DKn⎜1 − ⎟ 2r ⎠ ⎝

Sa =

pseudodispersed phase components

model for the pseudodispersed phase permeability

particle pore blockage

β, chain immobilization factor PI = PC/β

β′, permeability reduction factor PI = PC/β′

lI, thickness of rigidified region fillers and surrounding polymer of reduced permeability

lϕ, thickness of the reduced permeability region fillers (including bulk particle and region of reduced permeability)

2 dg ⎞ 1 ⎛ ⎜1 − ⎟ RT ⎝ 2r ⎠

lϕ, void thickness fillers and surrounding voids

⎡ P + 2P − 2ϕ(P − P ) ⎤ d I d s I ⎥ Peff = PI⎢ ⎢⎣ Pd + 2PI + ϕs(PI − Pd) ⎥⎦ ϕs =

model for the overall MMMs

matrix rigidification

ϕd ϕd + ϕi

=

rd 3 (rd + l i)3

,

li = lϕ , lI , or lϕ′

⎡ P + 2P − 2(ϕ + ϕ )(P − P ) ⎤ eff c c eff d I ⎥ P3MM = Pc⎢ ⎢⎣ Peff + 2Pc + (ϕd + ϕI)(Pc − Peff ) ⎥⎦

(40) (41)

volume fractions of pseudodispersed phase, and the influence of particle size distribution, particle shape, and aggregation of particles on the permeation performance of MMMs is not to be considered. As an enhancement of the Mahajan and Koros’s approach, Li et al.129 modified the basic Maxwell model to simultaneously consider both polymer chain rigidification and particle pore blockage. The basic Maxwell model was applied three times to obtain the final overall permeability. In this modeling, two pseudodispersed phases are defined: (i) the first pseudophase includes the dispersed particles and the interfacial skin affected by the partial pore blockage as the continuous phase; and then (ii) this phase is considered as dispersed phase combined with the rigidified polymer region as continuous phase to form the second pseudophase. Equations for this modification are detailed in Table 3. The Felske model, originally developed for thermal conductivity of composites of core−shell particles (core particle covered with an interfacial layer) as well as for permeability measurement, can be written as follows:537

In this method, the relationship between interphase thickness and filler particle size as well as that between interphase and polymer matrix have been taken into consideration. A prediction model for a series arrangement between two components can, however, only predict the minimum bound of the expected effective permeability of MMMs. Mahajan and Koros193,194,202 carried out a similar work on the influence of interfacial effects on the separation performance of MMMs containing a poor filler−polymer adhesion. As mentioned in section 3, such poor bonding results in interfacial voids or in “sieve-in-a-cage”, forming an interphase between dispersed phase and polymer matrix. In their model, a dispersed phase is encapsulated by an interphase, creating a pseudodispersed phase.180 Based on this assumption, a three-phase system including polymer matrix, inorganic fillers, and interphase can be described as an idealized two-phase system containing continuous polymer matrix and pseudodispersed phase. The permeation prediction model for the overall permeability of MMMs is obtained by applying the Maxwell equation twice, as summarized in Table 2. As shown in Table 2, this approach in predicting the separation performance of MMMs can be readily extended to a more complex morphology, in which interfacial void, polymer chain rigidification and particle pore blockage are considered as interfacial defects. For a MMM morphology containing an interfacial void, the void permeability (PI) is calculated as a product of the Knudsen diffusivity (D) through a pore with the same diameter as the void and the solubility coefficient (S) of gas in the void. On the other hand, for a MMM morphology holding a polymer chain rigidification or a particle pore blockage defect, the permeability of interphase is reduced by chain immobilization factor (β) or permeability reduction factor (β′), respectively. In addition, the interfacial thickness must be fitted to the experimental data. By application of this modified Maxwell model, the separation performances of nonideal MMMs have been estimated quantitatively considering these interfacial defects as reported elsewhere.62,128,182,208 However, like the original Maxwell model, this modified model is only valid for low

Pr =

2(1 − ϕd) + (1 + 2ϕd)β /γ Peff = Pc (2 + ϕ) + (1 − ϕd)β /γ

(47)

where, β and γ are given by: β = (2 + δ 3)λdm − 2(1 − δ 3)λIm

(48)

γ = (1 + δ 3) − (1 − δ 3)λdI

(49)

where, δ is the ratio of outer radius of interfacial shell to a core radius. In this model, ϕd is the volume fraction of core−shell particles. For consideration of a single core−shell particle surrounded by a matrix material in the same volumetric proportion as in the whole composite, the ϕd value is divided by the combined volume fractions of core−shell particle and its surrounded matrix material. P d , P I , and P m are the permeabilities of filler core particle, interfacial shell, and polymer matrix, respectively. λdm, λIm, and λdI are the permeability ratios Pd/Pm, PI/Pm, and Pd/PI, respectively. M


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Table 3. Modified Maxwell Model Extended by Li et al.129 param Pblo ϕblo ϕps1

Pps1 Prig ϕrig ϕps2

Pps2

P3MM

Table 4. Definition of Volume Fractions for the Hashemifard−Ismail−Matsuura Model207

definition

volume fractions

permeability of the interfacial skin affected by the partial pore blockage volume fraction of the interfacial skin in the entire MMMs volume fraction of the dispersed particles in the first pseudodispersed phase: ϕd ϕps1 = ϕd + ϕblo (42)

permeability of the rigidification region volume fraction of the rigidification region in the entire MMMs volume fraction of the first pseudodispersed phase in the second pseudodispersed phase: ϕd + ϕblo ϕps2 = ϕd + ϕblo + ϕrig (44)

(51)

2/3 1/3

3 ϕd′θ

zone II

ϕII = 2

zone III

ϕIII = 1 − ϕI − ϕII

dispersed particles

ϕdI = πϕd′2

interphase

ϕiI = 4 3

continuous polymer

ϕmI = 1 − ϕdI − ϕiI

(52) (53)

(54)

⎛ ⎛ 3 ⎞2 2 2 ⎜ ⎟ πϕ′ ⎜θ + ⎝2⎠ d ⎝

3

2 ⎞ θ⎟ 3 ⎠

interphase continuous polymer

ϕiII =

3

(55)

(56)

Zone II ⎡ ⎛3⎞ ⎛ 3 ⎞2 2 2 ⎜ ⎟ πϕ′ ⎢ 3 ⎜ ⎟ + 4θ 2 + 4 3 ⎝ 2 ⎠ d ⎢⎣ ⎝ 2 ⎠ 3 2

permeability of the second pseudodispersed phase: ⎡ P + 2P − 2ϕ (P − P ) ⎤ blo rig ps1 ps2 rig ⎥ Pps2 = Prig ⎢ ⎢⎣ Pps1 + 2Prig + ϕps2(Prig − Pps1) ⎥⎦ (45)

1 + 2ϕd(β − γ )/(β + 2γ ) Peff = Pc 1 − ϕdψ (β − γ )/(β + 2γ )

ϕd πϕd′2

Zone I

ϕmII = 1 − ϕiII

⎤ θ⎥ ⎥⎦

(57)

(58)

Where

ϕd′ =

3

ϕd π

θ=

t dp

dl =

3

2 dp 3

(59)

(46)

for definitions). As outlined in Table 4, ϕd is the volume fraction of the dispersed phase in the entire MMM, t is the thickness of interphase, and dp and dl are the diameter and height of dispersed particles, respectively. Based on the above definition, the overall permeability can be expressed as:

The permeation predictions obtained from this model are similar to those from the modified Maxwell model proposed by Koros and co-workers.182 When δ = 1, corresponding to an absence of interfacial layer, this model reduces to the original Maxwell model. Unfortunately, this model has the same limitations as those of the modified Maxwell model. The predictions are valid only when the volume fraction of core− shell particles is small (typically below 0.2). The computational procedure needed for the Felske model is, however, simpler than that applied for the modified Maxwell model. Accounting for a composite morphology with varying packing density of particles, the Felske model was further modified by Pal25 as follows: Pr =

ϕI =

zone I

permeability of the first pseudodispersed phase: ⎡ P + 2P − 2ϕ (P − P ) ⎤ d blo d ps1 blo ⎥ Pps1 = Pblo⎢ ⎢⎣ Pd + 2Pblo + ϕps1(Pblo − Pd) ⎥⎦ (43)

permeability of the overall MMMs: ⎡ P + 2P − 2(ϕ + ϕ + ϕ )(P − P ) ⎤ ps2 c c ps2 d blo rig ⎥ P3MM = Pc⎢ ⎢⎣ Pps2 + 2Pc + (ϕd + ϕblo + ϕrig )(Pc − Pps2) ⎥⎦

definition Entire MMMs

⎡ ⎛ ⎞ 1 Pr = ⎢1 + ϕII⎜⎜ − 1⎟⎟ u ⎢⎣ ⎝ ϕiII(λ i − 1) + 1 ⎠ −u ⎛ ⎞⎤ 1 + ϕI⎜⎜ − 1⎟⎟⎥ u u ⎝ ϕdI(λd − 1) + ϕiI(λ i − 1) + 1 ⎠⎥⎦

(60)

where, λi (Pi/Pc) and λd (Pd/Pc) are the ratios of interphase permeability and dispersed filler permeability to the permeability of continuous polymer matrix, respectively. Regarding the exponent parameter u, u = 1 is for the parallel−series combination, while u = −1 is for series−parallel pathway. The Hashemifard−Ismail−Matsuura (HIM) model is capable of predicting both permeability and selectivity through a nonideal MMM over a wide range of filler loadings. As expected, the HIM model should be able to give a true trend of transport properties through a MMM having different interfacial defects, including voids and leaky and rigidified polymer chain. In addition, the HIM model can also be applied for an ideal MMM morphology, in which the existence of interphase is ignored. In this case, ϕII = ϕiI = ϕiII = 0. Then, eq 60 for the overall permeability reduces to:

(50)

where, β and γ are given by eqs 48 and 49, respectively, and ψ is expressed by eq 14 in terms of ϕm (maximum packing volume fraction of core−shell particles). When ϕm = 1, this modified model reduces to the original Felske model, while when δ = 1, eq 45 reduces to eq 14 for the Lewis−Nielsen model. Also, when ϕm = 1 and δ = 1, the modified Felske model returns to the original Maxwell model. As expected, this model showed an improved permeation prediction at the limit ϕ → ϕm. Similar limitations mentioned above for the original Maxwell, Lewis− Nielsen, and Felske models, however, still apply to the modified Felske model. Recently, Hashemifard et al.207 developed a new model based on the flow pathways of penetrant gas through MMMs in both series and parallel channels. Based on the two proposed morphologies for a nonideal MMM, the volume fractions of three zones, as well as those of each component phase in zones I and II, are assumed as summarized in Table 4 (see section 3

−u ⎡ ⎛ ⎞⎤ 1 Pr = ⎢1 + ϕI⎜⎜ − 1⎟⎟⎥ u ⎢⎣ ⎝ ϕdI(λd − 1) + 1 ⎠⎥⎦

(61)

Very recently, Shimekit et al. modified the Pal model to take into account the interfacial rigidified polymer chain effect on the separation performance of MMMs.27 Applying for a threeN


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phase morphology including filler particles, polymer matrix, and rigidified interfacial layers, a functional relationship between relative permeability (Pr) and other variables can be expressed as follows: Pr = f (δ , λIm , λdI , ϕd , ϕm , ϕs)

Additionally, there are other prediction models for estimating the permeation properties of MMMs available in literature including those of Cheng and Vachon,538 Agari and Uno,539 and te Hennepe,540 as well as the effective medium theory (EMT).23,536 Moreover, the nonequilibrium lattice fluid (NELF) model for estimating the solubility and diffusivity of gases in MMMs,267,541−544 as well as the Maxwell−Stefan simulation for predicting the diffusive performance of MMMs containing selective flakes,545−547 has also been reported.

(62)

where, δ is the ratio of outer radius of rigidified interfacial matrix chain layer (ri) to a core radius (rd) and is estimated by the least-squares method. Generally, ri is assumed to be half the distance between the filler particles and the polymer. In this equation, λIm and λdI are the ratios of permeability between interfacial rigidified polymer chain layer and polymer matrix and between dispersed phase and interfacial rigidified polymer chain layer, respectively. ϕs is the volume fraction of the dispersed phase in the combined region between dispersed phase and rigidified interfacial polymer chain layer. ϕs could be estimated by ϕs =

ϕd ϕd + ϕi

=

5. MODEL VALIDATION: COMPARISON WITH EXPERIMENTAL DATA In 1997, Bouma et al.521 reported an investigation on the influence of solid fillers dispersed in a polymer matrix on permeability. The MMMs were prepared via the solvent-casting method with ethyl acetate as a solvent, using polyvinylidene fluoride−hexafluoropropene (PVDF) as a polymer matrix. Instead of the commonly used solid fillers like zeolites and CMS, a nematic liquid crystalline mixture E7, consisting of 51 wt % 4′-n-pentyl-4-cyano-biphenyl, 25 wt % 4′-n-heptyl-4cyano-biphenyl, 16 wt % 4′-n-octoxy-4-cyano-biphenyl, and 8 wt % 4′-n-pentyl-4-cyanotriphenyl, was used. The loading of E7 was varied between 0.5 and 25 wt %. Both the original Maxwell and Bruggeman models were applied to predict the O2 permeability of these co-PVDF/E7 MMMs. From the fitting results summarized in Table 5, the author pointed out that the

rd 3 rd 3 + ri 3

(63)

where, ϕi is the volume fraction of the interfacial rigidified polymer chain layer. Then, the relative permeability of species in this combined region can be written in modifying the original Pal equation533 (eq 17) by replacing α and ϕd with λdI and ϕs, respectively: 1/3

(Pr)

−ϕ ⎛ λdI − 1 ⎞ ⎛ ϕs ⎞ m ⎟⎟ ⎜ ⎟ = ⎜⎜1 − ϕm ⎠ ⎝ λdI − Pr ⎠ ⎝

Table 5. O2 Permeability Predicted by the Original Maxwell and Bruggeman Models through co-PVDF/E7MMMs at 40 °Ca

(64)

Replacing Pr by Peff*/PI in eq 64, the effective permeability of combined phase (Peff*) can be calculated: ⎛ Peff * ⎞ ⎟ ⎜ ⎝ PI ⎠

1/3

⎛ λdI − 1 ⎞ ⎛ ϕ⎞ ⎜ ⎟ = ⎜⎜1 − s ⎟⎟ ϕm ⎠ ⎝ λdI − Peff */PI ⎠ ⎝

Pd/Pc predicted

−ϕm

(65)

In this equation, the permeability of rigidified interface layer PI = Pm/β, where Pm is the measured permeability of polymer matrix, and β is the polymer matrix rigidification or chain immobilization factor. A value of β = 3 is usually assumed because it is the approximate value with typical gas penetrants in semicrystalline polymers.181,208,258 Using this calculated value of Peff*, the ratio of permeability between the combined phase and polymer matrix (λeff*m) is estimated:

λeff * m = Peff */PI

a

(66)

ri 3 R m3

(67)

where, Rm is the distance from the center of the particles to boundary of the polymer surface. Finally, the relative permeability of the whole MMM can be estimated by replacing λdI and ϕs with λeff*m and ϕz in eq 64, respectively: −ϕ ⎛λ ϕ⎞ m − 1⎞ ⎛ (Pr)1/3 ⎜ eff * m ⎟ = ⎜⎜1 − z ⎟⎟ ϕm ⎠ ⎝ λeff * m − Pr ⎠ ⎝

Peff/Pc, exp

Maxwell

Bruggeman

1.03 1.30 1.31 1.45 1.70 1.71 2.03 2.35

2.2 24.2 7.5 12.1 28.0 10.0 19.7 16.0

2.2 15.6 6.6 9.1 13.2 7.5 10.4 9.1

Reproduced with permission from Ref 521. Copyright 1997 Elsevier.

Maxwell equation is an analytically correct method, which is applicable at low filler loadings, while the Bruggeman model much better describes the membrane performance of MMM at higher filler concentration. Zimmerman et al.57 investigated the O2/N2 separation using Ultem and Udel polyimide membranes filled with different types of microporous solids such as zeolite 4A and carbon molecular sieve (CMS) particles. Using the original Maxwell model and effective medium theory (EMT), predictions of MMM separation performance indicated that significant improvements in the theoretical O2 permeability and O2/N2 selectivity can be achievable by polymer matrix selection or filler−polymer defect elimination. In 1999, Mahajan et al.58,200 also studied the O2/N2 separation through a series of MMM containing zeolite 4A and CMS particles dispersed in a Matrimid polymer. The results pointed out that the membrane performance predicted by the unmodified Maxwell model showed a slight reduction in permeability, combined with a large increase in selectivity as a

The combined volume fraction of the filler phase and the interfacial rigidified polymer matrix in the whole MMM is defined as:

ϕz = ϕd +

ϕf 0.035 0.103 0.137 0.166 0.210 0.255 0.296 0.373

(68) O


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incorporated into a poly(4-methyl-2-pentyne) polymer (PMP), were estimated and compared with the experimental data for N2 and O2 obtained from constant pressure/variable volume method.269 However, an increase in gas permeability with increasing silica loading was observed for the experimental results, while this performance predicted by the original Maxwell model was in an opposite trend (Figure 13). The existence of a large free volume in PMP/fumed silica (TS-530) MMMs might explain this behavior.

function of particle loading up to 90 vol %. The experimental permeabilities of oxygen and nitrogen at 35 °C were, however, found to be higher than those of the predicted results, with no improvement in selectivity. The interfacial voids between filler particles and Matrimid polymer matrix were designated as the cause of this deviation. Later in 2000, Mahajan and Koros prepared another series of MMMs using zeolite 4A crystals as fillers and a poly(vinylacetate) (PVAc), which is more compatible with zeolite than polyimide, as polymer matrix.201 By priming of the zeolite particles via adsorbing a layer of polymer before dispersion in order to have a good contact between PVAc polymer and zeolite 4A fillers, MMMs were synthesized with high fractional volumes of filler up to 40 vol %. As summarized in Table 6, the Table 6. Experimental Data and Predictions by the Original Maxwell Model of O2/N2 Separation through PVAc−Zeolite 4A MMMs at 35 °Ca ϕ (vol %)

PO2 exp (Barrer)

PO2 pred (Barrer)

PN2 pred (Barrer)

(PO2/PN2) exp

(PO2/PN2) pred

0 15 25 40

0.5 0.45 0.4 0.28−0.35

0.5 0.53 0.55 0.55

0.0847 0.0724 0.0648 0.0538

5.9 7.3−7.6 8.3−8.5 9.7−10.4

5.9 7.4 8.7 10.9

Figure 13. Pure-gas nitrogen permeability of PMP/fumed silica MMMs as a function of filler content. Reproduced with permission from ref 269. Copyright 2002 Elsevier.

a

Reproduced with permission from ref 201. Copyright 2000 American Chemical Society. αO2 = 1.54; βO2 = 0.1525; αN2 = 0.246; βN2 = −0.336.

In 2003, Vu et al. reported a series of works on the comparison between the experimental permeation data and the predicted values estimated by the Bruggeman and original Maxwell models for CO 2 /CH 4 and O 2 /N 2 separations.198,257,258 In these papers, CMS/polymer MMMs were synthesized via a flat-sheet solution casting method using carbon molecular sieves (CMS) dispersed within two different polymer matrices: Matrimid 5218 and Ultem 1000. The loading of CMS particles was varied from 10 to 50 vol %. For CMSbased MMMs prepared with Ultem 1000 polymer, the Bruggeman model predicts better permeability and permselectivity than the original Maxwell model. This was attributed to the fact that the Bruggeman model appears to account for disrupted flow patterns around the CMS particles under high loading condition (Figures 14 and 15). For Matrimid/CMS MMMs, both the Bruggeman and original Maxwell models provide a fairly poor prediction compared with the experimental permeation data. The permeability and selectivity predicted by the Bruggeman

O2 experimental results are in reasonable agreement with the predictions of the theoretical Maxwell equation. However, as the loading of filler increases, the deviation becomes more pronounced. Similar results were also observed in two other publications subsequently published by this research group.193,194 Using the original Maxwell model to study the O2/N2 separation performance through two series of MMMs synthesized with intermediate and high loadings of zeolite A particles (up to 40 vol %) embedded in 2,2′-BAPB + BPADA (BAPB = 1,4-bis(4-amino-phenoxy)benzene and BPADA = 4,4′-bisphenol A dianhydride) and Ultem polymer matrix, respectively, the deviations between the predicted results and experimental data are illustrated in Table 7. Also applying the original Maxwell model, the permeation properties of a series of MMMs prepared from a hydrophobic silica type (TS-530), with silica contents up to 25 vol %, Table 7. O2/N2 Separation Performance of MMMs Consisting of Zeolites A Dispersed in 2,2′-BAPB + BPADA and Ultem Polymers at 35 °C: Experimental Data vs Predictions by the Original Maxwell Modela O2 permeability (Barrer)

MMMs polymer 2,2′-BAPBBPADA

Ultem

O2/N2 permselectivity

loading (vol %)

exp

pred

exp

pred

20

0.47

0.55

9.4−9.6

9.4

30 40 15 35

0.4 0.37 0.38 9.7

0.57 0.6 0.42 9.7

10.6−10.8 12.4−12.5 0.28 12.85

10.8 12.6 0.49 13

Figure 14. Theoretical predictions by the original Maxwell and Bruggeman models for Ultem− and Matrimid−CMS MMMs with 10−50 vol % zeolite loading. Reproduced with permission from ref 258. Copyright 2003 Elsevier.

a

Reproduced with permission from refs 193 and 194. Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA. P


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Figure 15. Comparison between the experimental permeation data and model predictions (original Maxwell and Bruggeman models) for the CO2/ CH4 and O2/N2 separations through Ultem−CMS MMMs. Reproduced with permission from ref 258. Copyright 2003 Elsevier.

Figure 16. Comparison between the experimental permeation data and model predictions (original Maxwell and Bruggeman models) for the CO2/ CH4 and O2/N2 separations through Matrimid−CMS MMMs. Reproduced with permission from ref 258. Copyright 2003 Elsevier.

model are especially higher than those of the experimental data. In addition, these predicted values are also higher than those estimated by the original Maxwell model (Figure 16). Owing to the more rigid and less flexible nature of the Matrimid polymer than that of the Ultem polymer, the matrix rigidification effect occurring at the region near the CMS surface may not be significant for the Ultem/CMS MMMs. Moreover, both the Bruggeman and original Maxwell models are generally known not to take this effect into account. In order to account for the influence of the interfacial rigidified polymer chain defect in the prediction process of membrane performance, these authors also applied the modified Maxwell model to predict the permeation properties in CO2/CH4 separation through Matrimid/CMS MMMs. In this theoretical modeling, the authors assumed that the CMS particles are spherical (n = 1/3), and the chain immobilization factor (β) is 3, corresponding to a radius (or thickness) of the rigidified region of l = 0.075 μm. When the Maxwell equation is applied twice, a significant improvement in both CO 2 permeability and selectivity predictions compared with those obtained from the orginal Maxwell model was observed (Table 8 and Figure 17). As a further study of the deviation between experimental values and predicted permeation by the original Maxwell model, Koros and co-workers208 proposed an extension of the original Maxwell model to analyze gas permeation properties with the hypothesis of an inhibited contraction or a polymer chain rigidification at the interphase. A three-phase MMM morphology was demonstrated with an assumption that the inserted phase is an associated interphase, having a finite distance from the zeolite surface to the polymer matrix. Within this interphase, the polymer chain mobility is reduced. The inhibited contraction affects only the interphase thickness. This

Figure 17. Comparison between the experimental permeation data and predictions by the modified (three-phase) Maxwell model for the CO2/CH4 separation through Matrimid 5218/CMS MMMs at 35 °C. Reproduced with permission from ref 258. Copyright 2003 Elsevier.

“reduced mobility” modified Maxwell model was applied to estimate the membrane performance of PVAc/zeolite 4A MMMs,201 as well as that of a series of MMMs containing zeolite 4A crystals dispersed in BAPB−BPADA and Ultem polymer matrix.194 Figures 18 and 19 show a comparative illustration between the experimental data and prediction results estimated using the modified Maxwell model, together with the calculated values obtained from the unmodified original Maxwell model. As shown in both figures, the permeation values estimated by the “reduced mobility” modified Maxwell model are closer to the observed permeabilities with increasing zeolite loading than those predicted by the unmodified Maxwell model. These authors pointed out that the inhibited segmental motion due to surface attachment and matrix contraction at the interface during preparation of MMMs is a critical key of this observation. Q


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Table 8. Comparison between the Experimental Data and Predictions by the Modified (Three-Phase) and Nonmodified Maxwell Model for CO2/CH4 Separation through Matrimid−CMS MMMsa CO2 permeability (Barrer) Matrimid MMMs ϕ (vol % CMS) 17 19 33 36 a

CO2/CH4 permselectivity

exp

Maxwell model predictions

modified Maxwell model predictions

10.3 10.6 11.5 12.6

13.0 13.3 16.5 17.2

11.0 11.2 12.1 12.3

exp

Maxwell model predictions

modified Maxwell model predictions

44.4 46.7 47.5 51.7

47.7 49.3 63.1 66.2

44.2 45.4 55.3 57.5

Reproduced with permission from ref 258. Copyright 2003 Elsevier.

assumed to make no effect on permeability. For the closed model, the zeolites showed a very strong affinity for water, blocking the pores of these molecular sieves. The latter case is called the equilibrium adsorption model, in which the permeability of the zeolites is proportional to the fraction of the pore structure not occupied with water. As listed in Table 9, Table 9. Experimental Data and Predictions by the Open, Equilibrium Adsorption, and Closed Models for O2/N2 Separation through Zeolite 4A/PVAc MMMs at 25 °Ca permeability (Barrer) relative humidity (%)

zeolite fraction (vol %)

open model

equilibrium adsorption model

closed model

experimental data

1.5 10

40 25

0.50 0.51

0.30 0.30

0.20 0.27

0.33 0.28

a

Reproduced with permission from ref 138. Copyright 2003 WileyVCH Verlag GmbH & Co. KGaA.

Figure 18. Comparison between the experimental data and predictions by the original Maxwell and reduced mobility modified Maxwell models for the O2/N2 separation through zeolite 4A/PVAc MMMs at 35 °C. Reproduced with permission from ref 208. Copyright 2004 Wiley-VCH Verlag GmbH & Co. KGaA.

for a MMM containing zeolite 4A crystals dispersed in PVAc polymer at 25 °C, the equilibrium adsorption model predictions are most closely parallel to the experimental data. Similar observation was also reported in the case of a MMM synthesized using PVAc polymer matrix filled with hydrophobic SSZ-13 particles. Erden-Senatalar et al.23 reported an interesting study on the applicability of several conventional models for estimating the gas performance of zeolite−polymer MMMs. The series, parallel, original Maxwell, effective medium theory (EMT), and te Hennepe models were employed to investigate the effects of different types of polymer matrix and zeolites, as well as varied zeolite loadings, in providing an accurate prediction of membrane performance. The authors used the experimental permeability data reported in literature for various zeolite− polymer MMMs containing silicalite, 13X, 4A, and 5A type zeolites dispersed in PDMS (polydimethylsiloxane), Udel, PES (polyethersulfone), EPDM (ethylene propylene diene monomer), NBR (nitrile butadiene rubber), PEI (polyethyleneimine), CA (cellulose acetate), and TPX (poly(4-methyl-1pentene)) polymers to evaluate these investigated models. The results indicate that these models taking into account the influence of zeolite and polymer permeabilities, as well as zeolite loadings, might not be sufficient to describe the permeation performance of MMMs. The interfacial interaction occurring at the zeolite−polymer interface was hypothesized to play a critical role. An additional phase, designated as the interphase, should be taken into consideration to explain this phenomenon. It may be deduced that the relatively lower zeolite permeability values estimated by various models are an indication of a more significant contribution of the interphase

Figure 19. Comparison between the experimental data and predictions by the original Maxwell and reduced mobility modified Maxwell models for the O2/N2 separation at 35 °C: (a) zeolite 4A/ BAPB−BPADA MMMs (β = 3.0 and l = 0.66 μm); (b) zeolite 4A/ Ultem MMMs (β = 2.6 and l = 0.88 μm). Reproduced with permission from ref 208. Copyright 2004 Wiley-VCH Verlag GmbH & Co. KGaA.

Moore et al.138 studied the effect of humidified feeds on the permeability of oxygen through a series of MMMs consisting of hydrophilic 4A or hydrophobic SSZ-13 zeolites dispersed in a poly(vinyl acetate) polymer (PVAc). Three morphological models were considered to describe the influence of both zeolite properties and humidified feeds on the oxygen permeability. For the open model, the humidified feeds are R


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in decreasing the effective permeability values. This may be due to either lower interphase/polymer permeability ratios or higher interphase thickness. The relationship between interphase thickness and zeolite particle size as well as that between interphase and polymer matrix should be carefully investigated for different ranges of zeolite particle size, as well as for different types of zeolites and polymers. In 2004, Jeong et al.443 fabricated a new type of MMM, namely, polymer/selective-flake MMMs, using porous layered aluminophosphates (AlPOs) as fillers and hexafluoroisopropylidene−dipthalic anhydride−(hexafluoroisopropylidene)−8%dianiline-diaminobenzoic acid (6FDA−6FpDA−8%-DABA) as polyimide polymer matrix. The synthesized AlPO/PI MMMs with up to 10 wt % layered aluminophosphate loadings displayed a substantial enhancement in both O2/N2 and CO2/ CH4 selectivities. As shown in Figure 20, the estimated O2

Figure 21. Comparison between the experimental data and predictions by the original Maxwell and different modified Maxwell models for the O2 permeability through PES/zeolite 4A MMMs. Reproduced with permission from ref 128. Copyright 2005 Elsevier.

Figure 22. Comparison between the experimental data and predictions by the original Maxwell and modified Maxwell models for the O2/N2 selectivity through PES/zeolite 4A MMMs. Reproduced with permission from ref 128. Copyright 2005 Elsevier.

Taking into account the effect of polymer chain rigidification in modeling, the Maxwell equation was applied twice under several assumptions that these zeolites A have a cubic morphology, the chain immobilization factor for O2 (β) is 3, and the thickness of the rigidified region is r = 0.30 μm. As shown in Figures 21 and 22, the O2/N2 separation predictions by the modified Maxwell model, however, are still far from the experimental data, although they are much closer than those predicted by the original Maxwell model. It is possible that the partial pore blockage of zeolites by polymer chains might be considered as another effect on the gas permeation performance of MMMs. Therefore, the authors continued to investigate this interfacial effect by representing the whole morphology of PES−zeolite A membrane as a pseudo-three-phase composite and applied the Maxwell equation three times. For the O2 permeation, the average decline factor of gas permeability nears the zeolite skin due to the partial pore blockage (β′) and the average thickness of this influenced region (r′) are assumed to be 250 and 0.01 μm, respectively. For the N2 case, both β and β′ parameters are different from those for O2 and are assumed to be 4 and 10, respectively. Hence, both predicted O2 permeability and O2/N2 selectivity provided by the three-time modified Maxwell model are in better agreement with experimental data (Figures 21 and 22). A similar prediction result considering both chain rigidification and partial pore blockage effects was also reported by Kulprathipanja’s group.129 As shown in Figure 23, the predictions by the three-time modified Maxwell model for the O2/N2 separation performance through PES/zeolite 4A MMMs functionalized with amine group is in very good agreement with experimental data compared with those estimated by the original Maxwell model. Here, the average particle size of the used zeolite 4A is 1.5 μm; then, r and r′ parameters are assumed to be 0.12 and 0.007 μm, respectively.

Figure 20. O2 permeability and O2/N2 selectivity estimated by the Cussler model through layered AlPO/6FDA−6FpDA−DABA MMMs: M1 (5 wt % AlPO), M2, and M3 (10 wt % AlPO). Reproduced with permission from ref 443. Copyright 2004 American Chemical Society.

permeability and O2/N2 selectivity obtained from the semiempirical effective medium model developed by Cussler are comparable to those of the experimental data reported for two MMMs composed of zeolite 4A particles dispersed in polysulfone (PSF)170 and PVAc181 polymers. In Li et al.’s report,128 the effects of polymer chain rigidification and zeolite pore blockage on the O 2 /N 2 separation permeation predictions by the original and modified Maxwell models were studied. Upon a modified solutioncasting procedure at high temperature close to the glass transition temperatures (Tg) of polyethersulfone (PES) polymer, a series of MMMs was fabricated using PES polymer as polymer matrix and three different types of zeolite A (3A, 4A, and 5A) with different pore sizes and zeolite loadings up to 50 wt %. The obtained results showed that the original Maxwell model fails to predict the gas permeation performance of these MMMs. This is expected because the original Maxwell model does not discuss these two interfacial effects. For example, for PES−zeolite 4A MMMs, the experimental data of O 2 permeability exhibit an opposite trend compared with the values predicted by the original Maxwell model (Figure 21). Unfortunately, this theorical model also predicts a much higher magnitude of O2/N2 selectivity than that of the experimental results (Figure 22). S


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Table 10. Experimental and Estimated Permeability Values Obtained from the GPG Model for Different Gases through Zeolite 13X/Polysulfone MMMsa permeability (Barrer) MMMs

CO2 exp

CO2 est.

He exp

He est.

H2 exp

H2 est.

ϕ (vol %)/β 0 10 20

−0.2 6.5 6.1 6.1

−0.2 6.5 6.1 5.8

0.05 12.4 12.8 12.5

0.05 12.4 12.6 12.8

0.22 13.2 14.4 14.7

0.22 13.2 14.1 15.0

a

Table is based on data from ref 104. Reproduced with permission from ref 24. Copyright 2006 Elsevier.

listed in Table 11, together with predicted values. Again, the predicted values of permeability showed a reasonable agreement with the experimental data. Table 11. Experimental and Predicted Permeabilities Obtained from the GPG Model for 1,1,1-Trichloroethane/ Water Separation through Activated Carbon/Polyamide MMMsa

Figure 23. Comparison between the experimental data and predictions by the original Maxwell and modified Maxwell models through PES/zeolite 4A-NH2 MMMs for (a) O2 permeability and (b) O2/N2 selectivity. Reproduced with permission from ref 129. Copyright 2006 Elsevier.

The other modeling parameters are similar to those reported in their previously work.128 Both O2 permeability and O2/N2 selectivity of MMMs synthesized using zeolite 4A particles functionalized with a silane agent (e.g., (3-aminopropyl)diethoxymethyl silane, APDEMS) are higher than those of the PES/unmodified zeolite A MMMs at a same zeolite loading of 20 wt %. This effect can be explained by the fact that the presence of APDEMS agent introduces a distance of around 5− 9 Å between polymer chains and zeolite surface, thus reducing the extent of blockage occurring in the pore of zeolites by these polymer chains. In 2006, Gonzo et al.24 developed an extended form of the basic Maxwell model, namely, the Gonzo−Parentis−Gottifredi (GPG) model, based on the hard-sphere fluid model proposed by Chiew and Glandt,534 to estimate the effective permeation performance of gases and liquids through several types of MMMs prepared from different polymer matrices as continuous phase and organic (polymer or liquid crystal mixture) or inorganic (zeolites and activated carbons) compounds as fillers at a relatively high filler loading. First, this GPG model was tested for the gas permeation data measured via the constant volume technique at room temperature through a series of MMMs containing zeolite 13X incorporated in a Udel P-1700 polysulfone polymer. The zeolite−polymer MMMs were fabricated by the melt extrusion process.104 The 13X zeolites, 2−8 μm in size, were distributed uniformly in a polymer matrix (Udel P-1700 polysulfone) with three different loadings, 10, 20, and 40 vol %. As expected, the predicted values obtained from this GPG model were agreeing very well with the experimental permeation data for CO2, He, and H2 separation through these MMMs. For the polysulfone membranes filled with 0, 10, and 20 vol % zeolite 13X crystals, the maximum deviation was below 5%.24,104 Second, the experimental permeation data of the 1,1,1trichloroethane transport through polyether-block-polyamide membranes with incorporated activated carbon particles reported by Ji et al.113 were also used to validate this GPG model. The reduced experimental permeabilities (Peff/Pc) as a function of activated carbon loadings (0, 8, and 15 vol %) are

ϕ (vol %)

Peff/Pc, exp

Peff/Pc predicted

0 8 15

1.00 1.358 1.925

1.00 1.27 1.57

a


With application of the GPG model, the experimental data obtained from the pervaporation of acetic acid from 50 wt % water solution over silicalite/PDMS MMM composites with 0, 20, and 40 wt % (0, 17, and 33 vol %) silicalite loadings were also used for modeling.24,213 As expected, a very good agreement between the experimental data and estimated values was also reported (Figure 24).

Figure 24. Relative pervaporation rate calculated by the GPG model for acetic acid separation from 50% water solution through silicalite/ PDMS MMMs. Figure is based on data from ref 215. Reproduced with permission from ref 24. Copyright 2006 Elsevier.

Using the experimental data of the separation of n-pentane from i-pentane through a series of polydimethylsiloxane (PDMS) membranes filled with several types of zeolites such as ZSM-5, 4A and 5A particles,158 a fairly good agreement between the experimental data and predicted values was observed, except for the point located at volume fraction ϕ = 0.57 near the region where the zeolite particles almost touch one another (Figure 25). As revealed by SEM analysis, the interphase region connecting the bulk polymer to dispersed T


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the authors, but again, the relative CO2 and CH4 permeabilities and CO2/CH4 selectivity predicted by the GPG model are reasonable at the other volume fraction values compared with the experimental data. Pechar et al.146 synthesized a series of MMMs consisting of 20 wt % zeolite L powders dispersed in a glassy 6FDA−6FpDA polyimide (PI), as well as in a block copolymer blend composed of 6FDA−6FpDA and amine-terminated polydimethylsiloxane (PDMS) made by using a bulk imidization technique. As shown in Table 14, the predicted effective Figure 25. n-Pentane relative permeability estimated by the GPG model for n-pentane/i-pentane separation through zeolite 5A/PDMS MMMs. Figure is based on data from ref 158. Reproduced with permission from ref 24. Copyright 2006 Elsevier.

Table 14. Comparison between the Effective Gas Solubility Coefficients Predicted by the Original Maxwell and Series Models and Those Measured by the Time-Lag Method for All Gases through MMMs Consisting of 20 wt % Zeolites L Dispersed in 41 wt % PDMS/PI Copolymera

zeolite particles and the absence of voids nearby the zeolite− polymer interface were related to this observation. The experimental permeation data of He, O2, N2, CH4, and CO2 separations through a series of zeolite-filled MMMs containing a glassy polyimide polymer filled with modified zeolite (ZSM-2) particles were also modeled by this extended Chiew−Glandt equation.144 The experimental data were obtained via the time-lag procedure through MMMs filled with 20 wt % (16 vol %) ZSM-2 particles. As summarized in Table 12, a fair agreement between the experimental data and estimated relative permeabilities for all these gases was demonstrated.

solubility coefficient in cm3 (STP)/(cm3 (zeolite) 1.01325 × 105 Pa)

a

Peff/Pc, exp

Peff/Pc, predicted

α

β

He CO2 O2 N2 CH4

0.871 0.726 1.259 1.237 0.904

0.876 0.77 1.26 1.235 0.898

0.36 0.013 4 3.45 0.46

−0.27 −0.49 0.5 0.45 −0.22

a


Finally, the experimental data and permeability predictions through polysulfone membranes filled with carbon black at three different values of volume fraction ϕ (1.4%, 3.5%, and 7.1%) were also compared by using the GPG model.260 In fact, at ϕ = 3.5% (see Table 13), the CH4 experimental permeability exhibits a maximum value, while that for CO2 shows a minimum. This observation is difficult to explain, as stated by Table 13. Experimental and Predicted Values Obtained from the GPG Model for CO2/CH4 Separation through Coated Carbon Black/Polysulfone MMMsa Peff/Pc

permselectivity

ϕ (%)

CO2, exp

CO2, pred

CH4, exp

CH4, pred

CO2/CH4, exp

CO2/CH4, pred

0 1.4 3.5 7.1

1 0.885 0.798 0.872

1 0.98 0.95 0.90

1 1.03 2.27 1.162

1 1.03 1.08 1.17

41.0 35.3 14.4 30.8

41.0 38.9 36.1 31.5

CO2

O2

N2

CH4

2.87 2.19 2.29

0.43 0.33 0.18

0.203 0.15 0.12

0.712 0.53 0.39


solubility coefficients obtained from the original Maxwell and series models are somewhat higher than those measured by the time-lag method for all examined gases such as CO2, O2, N2, and CH4. One possible explanation for this observation might be related to the pore blockage effect of zeolites by the flexible PDMS polymer. Applying the basic Maxwell model to the experimental permeation data measured by the constant-volume method for the CO2/CH4 separation through zeolite SSZ-13/polymer MMMs, Hillock et al.180 reported an investigation of the effect of zeolite surface modification on permeation performances. The pore surfaces were first modified by priming zeolite SSZ-13 particles in a pure 3:2 6FDA−DAM/DABA copolyimide and its specific form modified by cross-linking with 1,3-propane diol. This provides a resistant to CO2 swelling, designated as propane diol monoester cross-linked (PDMC) polymer. Second, these pore surfaces were also functionalized with an APDMES (γ-aminopropyl-dimethylethoxy silane) silane coupling agent. As expected, the cross-linked polymer−zeolite membranes containing 15 wt % SSZ-13 crystals dispersed into a PDMC polymer matrix displayed an increase in both CO2/CH4 selectivity and CO2 permeability compared with the pure PDMC membrane. As listed in Table 15, for the MMM composites containing zeolites SSZ-13 primed with a pure PDMC polymer, the experimental CO2 permeability is lower than that estimated by the original Maxwell model, which is related to the rigidified polymer chain effect after priming these zeolites into an unmodified PDMC polymer. On the other hand, upon priming with a modified PDMC polymer, the experimental gas performance is approximately equal to that predicted by the original Maxwell model. The cross-linking reaction was presumed to have a significant effect at the interfacial region, resulting in a good adhesion between zeolites and polymer matrix. Exceptionally, the experimental CO2 permeability of the MMM filled with zeolites SSZ-13 silanated with APDMES was significantly improved, while the CO2/CH4 selectivity was almost unchanged. The formation of a “sieve-ina-cage” morphology of MMM after incorporing zeolites SSZ-13

Table 12. Experimental and Estimated Values Obtained from the GPG Model for Different Gases through 16 vol % Zeolite ZSM-2/6FDA−6FpDA−DABA MMMsa gases

method Maxwell model series model experimental

a

Table is based on data from ref 260. Reproduced with permission from ref 24. Copyright 2006 Elsevier. U


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with different gases studied (He, O2, N2, CH4, and CO2).144 Figure 26 (set 2 and 3) compares the permeation values estimated by the modified Felske model for CO2/CH4 and O2/ N2 separation, respectively, with the experimental data reported by Vu et al.258 for CMS/Matrimid 5218 MMMs. Figure 26 (set 4 and 5) displays the same comparison, in particular, for O2/N2 separation through zeolite 4A/BAPD−BPADA membrane193 (BAPD, 4,4′-bis(4-aminophenoxy)-2,2′-dimethylbiphenyl) and zeolite 4A/poly(vinyl acetate) composite,201 respectively. For all MMMs, the zeolite loading values are varied up to 40 wt %. As illustrated in Figure 26, a good agreement between model predictions and experimental data is observed for almost all reported cases of zeolite−polymer MMM composites. By considering the existence of an interfacial rigidified layer surrounding the zeolite particles, the modified Felske model seems to be a powerful technique to estimate the gas separation performance of MMMs at high filler loadings, surpassing the drawback of the original Maxwell model, which is only useful for an ideal morphology of MMM with perfect filler−polymer interaction. In 2009, Liu et al. carried out an investigation on the butane isomer transport through MFI/6FDA−DAM MMMs synthesized from 6FDA−DAM polymer matrix filled with up to 45 wt % zeolite MFI crystals (DAM, diaminomesitylene).192 Before incorporation into the polymer matrix, the MFI crystals were modified via a two-step Grignard treatment (GT) that results in Mg(OH)2 whiskers at the zeolite surface. As summarized in Table 17, the experimental permeation values of nC4/iC4 separation through the MFI/6FDA−DAM MMMs filled with up to 35 wt % Grignard-treated MFI particles are very close to those predicted by the original Maxwell model. This observation thereby supports the presence of a defect-free interface between the MFI particles and 6FDA−DAM polymer matrix. At higher zeolite loadings (i.e., 45 wt %), however, due to the higher nC4 hydrocarbon permeability of MFI zeolites compared with the polymer, the mismatch between the permeabilities of zeolites MFI and 6FDA−DAM polymer matrix might be the main reason that causes a slight increase in both nC4 permeability and nC4/iC4 selectivity. In 2010, Ahn et al.263 reported a study of the gas transport of O2, N2, and CO2 through PIM-1/silica MMMs. PIM-1 microporous polymer, containing the contorted spirobisindane unit, can be easily prepared with high molecular weight polymer and possesses a combination of outstanding permeability with moderate selectivity, especially for O2/N2 and CO2/CH4 pairs, which overcomes the upper bound trade-off proposed by Robeson.40,548 The loading of silica nanoparticles was limited to a maximum of 24 vol %, because PIM-1/silica membranes prepared with more than 30 vol % silica loading were brittle. The authors pointed out that the gas permeabilities are generally increased with silica volume fraction, which is in contrast to the predictions by the original Maxwell model. As shown in Figure 27, a similar behavior is also observed with the permeation predictions by the Bruggeman model for O2 gas. The existence of interfacial voids at the silica−PIM-1 interface or the effect of nanoparticle-induced disruption of polymer chain packing is barely distinguishable. Keskin and Sholl reported a comparative molecular simulation study of the gas mixture separation through a series of IRMOF-1/Matrimid MMMs by applying both the original Maxwell and Bruggeman models.472 IRMOF-1 is a metal− organic framework (MOF) material having the MOF-5 structure. Using experimental permeation data of several gas

Table 15. CO2/CH4 Separation Properties of PDMC/SSZ13 Membranes Primed with Unmodified PDMC Polymer, PDMC Polymer Modified with 1,3-Propane Diol, and Zeolites SSZ-13 Silanated with APDMESa experimental PDMC/SSZ-13 MMMs cross-linked PDMC primed with unmodified PDMC primed with modified PDMC silanated with APDMES a

Maxwell model

PCO2 (Barrer)

αCO2/CH2

PCO2 (Barrer)

αCO2/CH2

57.5 ± 2.9

37.1 ± 0.7

67.0 ± 3.4

49.6 ± 0.7

56.5 ± 2.8

43.8 ± 0.7

66.0 ± 3.3

41.0 ± 0.7

88.6 ± 4.4

41.9 ± 0.7


silanated with APDMES into a PDMC polymer would be a reasonable explanation for this enhancement. The basic Maxwell model predictions were again well matched with the O2 and N2 single-component experimental data reported by Bae and coauthors for a series of Ultem/MFI MMMs prepared using a solution-casting technique.80 In this report, all membranes made with untreated MFI crystals were found to contain some voids at the zeolite/polymer interface. On the other hand, Ultem polyimide membranes synthesized with up to 35 wt % solvothermally treated zeolites MFI displayed a highly uniform distribution of MFI crystals in the polymer without interfacial voids, likely having a desired ideal morphology. For example, the experimental O2 and N2 singlecomponent permeabilities through MMMs fabricated with 20 wt % solvothermally treated MFI crystals were similar to those predicted by the original Maxwell model, which is often accurate in predicting the gas performance of a MMM with an ideal morphology (Table 16). Table 16. Pure-Component Gas Permeation Properties of Pure Ultem Membrane and 20 wt % MFI/Ultem MMMs at 35 °C and 4.5 atm Upstream Pressurea membrane pure Ultem membrane untreated Ultem/MFI MMM treated MFI/Ultem MMM Maxwell model prediction for treated MFI/Ultem MMM

O2 permeability (Barrer)

N2 permeability (Barrer)

O2/N2 selectivity

0.43 ± 0.01 0.43 ± 0.02

0.055 ± 0.002 0.056 ± 0.003

7.8 ± 0.2 7.5 ± 0.7

0.35 ± 0.01 0.35

0.041 ± 0.002 0.044

8.5 ± 0.6 7.8

a

Reproduced with permission from ref 80. Copyright 2009 American Chemical Society.

In 2008, Pal proposed a modification of the Felske model for estimating the gas effective permeability of MMMs.25 This new model takes into account the presence of interfacial layer (shell) at the surface of core filler particles. In order to evaluate this modified Felske model, the author carried out a comparative study between the permeation predictions and experimental data reported in literature for several MMM systems. Figure 26 (set 1) shows a comparative result for ZSM2/6FDA−6FpDA−DABA MMMs at the volume fraction of 0.16. The permeability ratio, λdm, was varied from 0.0132 to 4 V


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Figure 26. Comparison between predictions by the modified Felske model and permeation experimental data. Experimental data taken from (set 1) Pechar et al.,144 (set 2 and set 3) Vu et al.,256 (set 4) Mahajan and Koros,211 and (set 5) Mahajan and Koros.193 Reproduced with permission from ref 25. Copyright 2008 Elsevier.

Table 17. nC4/iC4 Separation Performance of MMM Containing Grignard-Treated MFI Particles Dispersed in 6FDA−DAM Polymer at 100 °C, Upstream Pressure of 25 psi, and Downstream at Vacuuma PnC4 (Barrer) αnC4/iC4 a

method

35 wt % GT, uncalcined

25 wt % GT, uncalcined

pure polymer

25 wt % GT, calcined

35 wt % GT, calcined

experimental model experimental model

2.7 ± 0.2 2.5 23 ± 2 21

3.0 ± 0.2 2.8 23 ± 2 21

3.7 ± 0.2

6.4 ± 0.2 6.2 22 ± 2 21

7.8 ± 0.2 7.7 23 ± 2 21

21 ± 2


mixtures such as H2/CH4, CO2/CH4, CH4/N2, H2/CO2, and H2/N2 taken from Perez et al.,478,479 these authors showed a slight agreement between predictions by these two models and experimental data. For example, both model predictions are

slightly overestimating the experimental permeabilities of CH4, N2, CO2, and H2 gases. The Bruggeman model predicted higher permeability values than the original Maxwell model, especially at high MOF loadings. Additionally, the ideal selectivities of gas W


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Figure 27. N2 (■), O2 (●), and CO2 (○) gas transport properties of PIM-1/silica MMMs measured at 25 °C and Δp of 50 psig after being soaked in methanol and dried at 100 °C in a vacuum oven; the dotted line represents the predictions by the Bruggeman model for O2. Reproduced with permission from ref 263. Copyright 2010 Elsevier.

pairs calculated from the original Maxwell model are slightly lower than the experimental values. Hudiono et al.107 demonstrated an enhancement in CO2 separation performance through three-component mixedmatrix membranes synthesized using RTIL organic salts and SAPO-34 crystals dispersed in a poly(RTIL) polymer. RTILs, 1-ethyl-3-methyl imidazolium bis(trifluoromethyl-sulfonyl)amide (emim [Tf2N]), are molten organic salts at ambient temperature and pressure and have unique physicochemical properties (e.g., negligible vapor pressure, nonflammable, and high ionic conductivity). An incorporation of RTIL can both increase the permeability of the membrane and facilitate the interfacial interaction between the poly(RTIL) (polymer) and zeolites (inorganic particles). In this report, three types of membrane composites, designated as RTIL/SAPO-34, poly(RTIL)/SAPO-34, and poly(RTIL)/RTIL/SAPO-34 MMMs, were synthesized. For RTIL/SAPO-34 composite, an increase in the permeability of CO2, CH4, and N2 was reported when 10 wt % SAPO-34 was added. The selectivity of CO2/CH4 was increased, while the selectivity of CO2/N2 remained constant. For poly(RTIL)/SAPO-34 system, an addition of 10 wt % SAPO-34 in poly(RTIL) resulted in an improvement in the permeability of all measured gases compared with the neat poly(RTIL). The permselectivity of CO2/N2 was increased, while the permselectivity of CO2/CH4 was decreased. The presence of a styrene group on the poly(RTIL) polymer may cause larger void spaces between the polymer chains, particularly at the organic−inorganic interface, which results in a decrease of CO2/CH4 selectivity. For three-component poly(RTIL)/RTIL/SAPO-34 membrane, the permeabilities of CO2, CH4, and N2, as well as both CO2/CH4 and CO2/N2 selectivities, were increased compared with the neat poly(RTIL)/RTIL composite. The increase in selectivity indicates that interfacial defects between poly(RTIL) polymer and zeolite particles are not present. This deduction could be confirmed by using the original Maxwell model, which is typically used to predict the gas performance of an ideal MMM. As expected, the experimental CO2 permeabilities and CO2/ CH4 selectivities were in good agreement with predicted values (Figure 28). One can explain this result by the fact that the RTILs may react as an interfacial agent to adhere the inorganic particles to the polymer matrix, producing a membrane without interfacial defects. Basu et al. reported a study on the synthesis of two MMMs comprising Cu3(BTC)2 metal−organic framework (MOF)

Figure 28. Robeson plots of the experimental data and predictions by the original Maxwell model for 10 wt % SAPO-34 particles in poly(RTIL), poly(RTIL)−emim[Tf2N], and emim[Tf2N] composites compared with those of these neat polymers for (a) CO2/CH4 performance and (b) CO2/N2 performance. Reproduced with permission from ref 107. Copyright 2010 Elsevier.

crystals embedded in Matrimid or Matrimid−polysulfone polymers via phase inversion under mechanical stirring.463 In the synthesis, up to 30 wt % Cu3(BTC)2 microparticles were dispersed in a mixture of N-methylpyrrolidinone (NMP) and dioxolane before introduction into these polymers. An improvement in thermal and mechanical properties with increasing Cu3(BTC)2 loading, confirmed by TGA, dynamic mechanical storage, and Young’s modulus, tensile strength, and break at elongation test results, was observed for both MOFfilled MMMs. The CO2/CH4 and CO2/N2 separation performances of Cu3(BTC)2-filled MMMs were higher than those of unfilled polyimide membranes and increased with increasing MOF loading. Furthermore, these experimental data were in accordance with the predictions by the original Maxwell model (Figure 29), suggesting a defect-free morphology of Cu3(BTC)2/polymer MMMs with good compatibility at the filler/polymer interface. Gavle et al.99 reported a quantitative analysis applying the Cussler model for a series of MMMs synthesized with oriented flakes containing up to 10 wt % disaggregated microporous titanosilicate (JDF-L1) particles dispersed in a carboxyl group containing 6FDA−4MPD/6FDA−DABA copolyimide polymer (molar ratio of 4:1). As summarized in Table 18 for the separation of H2/CH4 mixture, at low filler loadings, the Cussler’s model predictions are somewhat lower than those of the experimental data. The main reason is that in these oriented JDF-L1/copolyimide MMMs, the orientation of JDF-L1 sheets is not perfect, and some aggregates of JDF-L1 parts still exist. Interestingly, this model seems to be more precise to describe the CH4 permeation at higher filler loadings, because these microporous JDF-L1 particles are not permeable to CH4. In consequence, the Po/P values calculated from the Cussler model are in good accordance with the experimental data. X


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monoester cross-linkable polyimide (PDMC) polymer, Ward and Koros171,172 applied the three-phase (3MM) Maxwell model described by Mahajan and Koros201 to analyze the effect of interfacial interaction between dispersed zeolites and polymer matrix on the membrane performance. As listed in Table 19, the CO2 gas permeability of 25 wt % uncalcined SSZTable 19. Comparison between the Experimental Pure Gas Permeance and Predictions by the Basic Maxwell and ThreePhase (3MM) Maxwell Models through 25 wt % Uncalcined SSZ-13/PDMC MMMs at 35 °C and ∼65 psiaa Figure 29. Comparison between the experimental data and predictions by the original Maxwell model for CO2, N2, and CH4 permeances through Matrimid/Cu3(BTC)2 MMMs with different MOF loadings. Reproduced with permission from ref 463. Copyright 2010 Elsevier.

experimental basic Maxwell prediction 3MM Maxwell prediction (Pbased) 3MM Maxwell prediction (αbased)

Table 18. H2/CH4 Separation Performance of JDF-L1 Copolyimide MMMs: Comparison between the Experimental Data and Cussler’s Calculated Resultsa

a

filler loading (wt %)

volume fraction (θ)

cal Po/P

exp Po/P

5 8 10

0.051 0.067 0.099

2.0 2.7 4.9

2.8 3.5 4.5

PCO2 (Barrer)

method

a

155 48.5 155 49.3, 82.3

αCO2/CH2 25.5 36.4 36.4 25.5

estimated void size (nm)

26 0.3, 1.0


13/PDMC MMMs is 220% higher than that predicted by the basic Maxwell model. In contrast, the CO2/CH4 selectivity is 30% lower. The reduction of selectivity might suggest the existence of a very small amount of interfacial voids at the surface between the two phases, whose dimensions approach those of the penetrant molecules. By the 3MM model, the size of interfacial voids estimated based on the permeability data was 26 nm, while that calculated based on the selectivity values was 0.3 or 1 nm. It is possible that some interfacial phenomena other than interfacial void defect might be responsible for the discrepancy between the permeability- and selectivity-based void size estimates. The presence of an interphase containing rigidified polymer chain defects or a dilated interphase of lower polymer density could further support this discrepancy. In addition, SSZ-13 crystals are also capable of causing localized packing disruption in a polymer chain of PDMC matrix. Fortunately, the gas transport properties are more accurately predicted by the 3MM Maxwell model. Lately, Ge et al.393 reported an investigation on the CO2/N2 separation performance of carbon nanotubes (CNTs)/poly(ether sulfone) (PES) MMMs synthesized via the phase transition method followed by a compression molding. Two different modification procedures were used to functionalize the CNTs particles. Carboxyl-modified CNTs with oxidized carboxyl groups on the outer walls were prepared by an acid oxidation using a mixture of concentrated H2SO4/HNO3. Ru-


In 2011, Sadeghi et al.289 applied the modified Higuchi model to predict the permeability of polyether-based polyurethane−silica MMMs prepared via a solution blending and casting method with silica contents of 2.5, 5, 10, and 20 wt % for CO2, CH4, N2, and O2 separations. Polyether-based polyurethane was synthesized upon a two-step polymerization of poly(propylene glycol)/hexamethylene-diisocyanate/1,4-butanediol mixture at a molar ratio of 1:3:2. Silica nanoparticles were prepared via a sol−gel method under hydrolysis of tetraethoxysilane (TEOS). As a principal result, a slight decrease in permeability of all CO2, CH4, N2, and O2 gases, and a significant enhancement in CO2/N2, CO2/CH4, and O2/ N2 selectivities was observed, owing to the better molecular sieve property of nanocomposite membrane with increasing silica loading. The authors also modeled these permeability data using the modified Higuchi model. As expected, the experimental data for all gases were in good agreement with the predicted values (Figure 30). As a continuation of Hillock et al.’s study180 on a dense film MMM consisting of SSZ-13 zeolites dispersed in a propanediol

Figure 30. Comparison between the experimental permeability data and predictions by the modified Higuchi model through polyurethane/silica membranes for (a) pure CO2 and (b) N2, CH4, and O2 gases. Reproduced with permission from ref 289. Copyright 2011 Elsevier. Y


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On the other hand, the generalized Maxwell model gives better predictions at low CNT loadings ( modified Lewis− Nielsen model> modified Maxwell model > Felske model. Generally, the modified forms of Maxwell and Lewis−Nielsen models are much more accurate in estimating the gas properties of MMMs than the unmodified models. Note that, among these models, the Felske model shows the best accordance with experimental data, very close to that of the modified Maxwell model. Based on their results, in overall view, all the existing models considering an ideal MMM morphology showed a relatively high deviation from the experimental data. In order to overcome this drawback, the same research group proposed a new theoretical prediction model, namely, the Hashemifard− Ismail−Matsuura (HIM) model, which gives a reasonable value

Figure 31. Experimental and predicted CO2 permeability through nanocomposite membranes synthesized with various modified CNTs. Reproduced with permission from ref 393. Copyright 2011 American Chemical Society. Z


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which are derived for ideal MMMs. On the other hand, for the models considering the interfacial rigidified polymer chain defect such as the Felske and modified Felske models, the modeling lines are closer to the experimental data. Hence, the author concluded that among the studied models, the modified Felske model is the best tool to estimate the CO2 permeation property of Matrimid/CMS membranes. Considering several critical parameters to evaluate the studied models such as the absolute average relative error (% AARE) and standard deviation (σ), the authors pointed out that %AARE and σ values estimated by the models among the first group decrease in the following order: Pal model > Lewis− Nielsen model > original Maxwell model > Bruggeman model (Table 21). For the second group including Felske and modified Felske models, this order is follows: Felske model > modified Felske model. It is possible that owing to the consideration of only several experimental points, the predictions obtained by these models are in good agreement with the theoretical results reported by Hashemifard et al.,209 which were calculated from a larger number of experimental points. In 2011, the same research group carried out a modification of the basic Pal model by additionally considering the rigidified interfacial interaction between zeolites and polymer matrix to estimate the gas performance of MMMs.27 The estimated CO2 and CH4 permeabilities as well as CO2/CH4 selectivity of Matrimid−CMS MMMs are summarized in Table 22 and compared with the experimental data reported by Vu and coauthors.257,258 As an overall observation, the authors pointed out that the predictions by the modified Pal model are also in good accordance with the experimental data. These authors also carried out a comparison of the predictive ability of various existing models such as the original and modified Maxwell, Bruggeman, Lewis−Nielsen, Pal, Felske, and modified Felske models as well as the modified Pal model using the experimental data reported by Vu et al.257,258 for Matrimid/ CMS composites. As shown in Figure 35, the modified Pal model predicts more accurately both relative CO2 permeability and CO2/CH4 selectivity than other existing models. Moreover, in the case of theoretical models derived for an ideal MMM morphology, the estimated absolute average relative error (% AARE) is in the following order: Pal model > Lewis−Nielsen model > Bruggeman model > original Maxwell model. On the other hand, for the other models considering an interfacial rigidified interaction, this order is modified Maxwell model > original Felske model > modified Felske model > modified Pal model. In addition, the authors also tried to test the modified Pal model with experimental data of O2/N2 separation through several types of MMMs such as BAPB−BPADA/zeolite

Figure 32. Bar diagram exhibiting the standard deviation (σ) and percentage of average absolute relative error (%AARE) predicted by several studied models for a set of the experimental O2, N2, CO2 and CH4 permeabilities through various MMMs. Reproduced with permission from ref 209. Copyright 2010 Elsevier.

of both gas permeability and selectivity, approximating the experimental results.207 Considering not only the presence of an interphase thickness around filler particles, but also the polymer chain rigidification, interfacial voids, and leaky defects, the HIM model was applied to seven sets of experimental data reported in literature and specified in Table 20. The predicted gas performances are depicted in Figure 33. As shown in Figure 33, for all studied sets, a high degree of agreement between the model predictions and published experimental data was observed. The results also revealed that this model is capable of predicting the gas performance of MMMs over a large range of filler volume fraction of (ϕ up to 40%). However, this model still does not account for the influence of pore filler blocking, as well as particle size distribution, particle shape, and aggregation of particles. In an interesting publication, Shimekit et al. compared and evaluated various well-known gas permeation theoretical models such as the original Maxwell, Bruggeman, Lewis− Nielsen, Pal, and original and modified Felske models using the experimental data of relative permeability of the CO2/CH4 separation through Matrimid−CMS MMMs.210 Modeling the experimental data reported by Mahajan and Koros193,194 and Vu and coauthors,257,258 the estimated values of CO2 relative permeability are plotted in Figure 34 as a function of the volume fraction of CMS particles and compared with the experimental data. Here, the solid lines representing permeation predictions are classified into two groups. The first group, showing a huge deviation between the experimental data and predictions, includes the predicted lines obtained from the original Maxwell, Bruggeman, Lewis−Nielsen, and Pal models, Table 20. Specification of Cases 1−7a

a

case no.

NDPb

1 2 3 4 5 6 7

2 2 6 4 8 6 8

filler loading (ϕ) 0.20 0.20 0.20, 0.15, 0.17, 0.17, 0.18,

0.30, 0.35 0.19, 0.19, 0.28,

0.40 0.33, 0.36 0.33, 0.36 0.38, 0.48

filler type

polymer matrix

gases

θ

morphology

ref

A A A A CMS CMS A

Matrimid Matrimid BAPB−BPADA Ultem Matrimid Matrimid polyether-sulfone

O2/N2 O2/N2 O2/N2 O2/N2 CO2/CH4 O2/N2 O2/N2

0.10 0.0003 0.10 0.10 0.10 0.10 0.10

void leaky rigidified rigidified ideal rigidified, close to ideal rigidified

193 193 194 194 257 257 128

Reproduced with permission from ref 207. Copyright 2010 Elsevier. bNumber of data points. AA


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Figure 33. HIM model predictions of relative permeability and selectivity versus experimental data. Reproduced with permission from ref 207. Copyright 2010 Elsevier.

4A,193,194 PVAc/zeolite 4A,201 and Matrimid/CMS.257,258 As a general result, a satisfactory agreement was then obtained. Hashemifard et al. recently reported an investigation on several expected morphologies for nanocomposite membranes consisting of haloysite nanotubes (HNTs) and polyethyleneimine (PEI) polymer.439 Haloysite (a member of the clay family) nanotubes possess a highly unusual meso/macroscopic

structure with large pore size, resulting from the wrapping of clay layers around themselves to form hollow cylinders.453 The loading of HNTs was varied from 1 to 5 vol %. As mentioned in section 3, there are five different cases of HNT−PEI composite morphology. Case I represents an ideal morphology, in which a uniform distribution of HNTs along with proper adhesion between these particles and PEI polymer is achieved. This AB


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Figure 34. Overall comparison among the predictions by several selected models and the experimental data from Mahajan and Koros193,194 and Vu et al.257,258 for the CO2 relative permeability through Matrimid/CMS MMMs. Reproduced with permission from ref 210. Copyright 2010 Asian Network for Scientific Information.

Figure 35. Comparison between the predictions by different available models and those by the modified Pal model using experimental data reported by Vu et al.257,258 for (a) CO2 relative permeability and (b) CO2/CH4 relative selectivity. Reproduced with permission from ref 27. Copyright 2011 Elsevier.

Table 21. Comparison of the Standard Deviation (σ) and Absolute Average Relative Error (%AARE) Obtained from the Studied Theoretical Models Using the Experimental Data from Mahajan and Koros193,194 and Vu et al.257,258 for CO2/CH4 separation through Matrimid/CMS MMMsa theoretical models

σ

%AARE

Maxwell model Bruggeman model Lewis−Nielsen model Pal model Felske model modified Felske model

31.74 23.51 33.78 41.12 12.75 5.86

32.06 23.75 34.13 41.53 12.88 5.92

HNTs loadings was observed as illustrated in Figure 36 (case II-A). The clusters of aggregated HNTs surrounded by voids are frequently observed with increasing HNT loading. As shown in the shadowed region A of Figure 36 (case II), the predictions by the HIM model are also in accordance with the experimental data reported by Reid et al.322 for CO2/CH4 and O2/N2 separations. As long as the voids do not form channels from one side of the skin layer to the other side, no effect on membrane selectivity was reported (refers to the shadowed region, case II-B). Case II-C exhibits an agglomeration combined with channeling morphology, which often results from a poor particle distribution at higher HNT loadings. Again, the predictions plotted in the shadowed regions B and C are in good agreement with the experimental data reported by Boroglu and Gurkaynak for O2/N2,184 Reid et al. for O2/N2,322 and Hashemifard et al.440 for CO2/CH4 and those carried out by Mustafa et al.401 and Hashemifard et al.440 for CO2/CH4. In the case of MMMs including dispersed particles coated by a rigidified polymer layer, a rigidified morphology of HNT−PEI MMMs was defined (case III). This polymer layer at the rigidified interface usually causes a reduction of permeability,

a

Reproduced with permission from ref 210. Copyright 2010 Asian Network for Scientific Information.

morphology is likely observed at low HNT loadings. As shown in Figure 36 (case I), there is good agreement between the predictions by the HIM model and several experimental data reported by Yong et al. for O2/N2,177 Cong et al. for CO2/ N2,392 Kim et al. for O2/N2,397 and Hashemifard et al.440 for CO2/CH4 (shadowed region). Case II exhibits a void morphology, which is developed in the case of a poor adhesion between HNTs and the polymer matrix. An increase of permeability without significant change in selectivity at low

Table 22. Comparison between the Experimental Data and Predictions by the Modified Pal Model through Matrimid/CMS MMMsa CO2 permeability (Pr)

a

CH4 permeability (Pr)

CO2/CH4 selectivity (αr)

volume fraction

exp

present model

%AARE

exp

present model

%AARE

exp

present model

%AARE

0.00 0.17 0.19 0.33 0.36

1.00 1.03 1.06 1.15 1.26

1.00 1.04 1.06 1.13 1.25

0.00 0.97 0.00 1.74 0.79

1.00 0.83 0.81 0.86 0.87

1.00 0.84 0.82 0.88 0.86

0.00 0.83 1.65 1.64 0.89

1.00 1.24 1.31 1.34 1.45

1.00 1.24 1.29 1.28 1.45

0.00 0.00 1.53 4.48 0.00

Reproduced with permission from ref 27. Copyright 2011 Elsevier. AC


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Figure 36. HIM model relationship between the MMM morphologies and transport properties through HNT/PEI MMMs. Experimental data reported by (1) Yong et al.,177 (2) Cong et al.,392 (3) Kim et al.,397 (4) Hashemifard et al.,440 (5) Reid et al.,322 (6) Boroglu and Gurkaynak,184 (7) Mustafa et al.,401 (8) Zornoza et al.,297 (9) Zornoza et al.,197 (10) Süer et al.,156 (11) Li et al.,129 and (12) Chung et al.365 Reproduced with permission from ref 439. Copyright 2011 Elsevier.

C, d < 4 Å, which is less than the light gas kinetic diameters. If the pore blockage is not severe, most likely, both permeability and selectivity are increased. As shown in the shadowed region of case IV-A, the experimental data reported by Li et al.129 for O2/N2 are somewhat lower than those predicted by the HIM model. In case IV-B, permeability tends to decline, while selectivity is likely higher than that of the neat polymer. When pore blocking is more effective, a dramatic decline in permeability combined with a slight increase or no significant change in selectivity was observed in the shadowed region of case IV-C. The experimental data reported by Chung et al.365 for O2/N2 and CO2/CH4 separation are well plotted in this region but are still lower, however, than the model predictions.

while selectivity is increased. Interestingly, compared with case I, case III yields an improvement in both permeability and selectivity, owing to the large pore size of HNT. As illustrated in Figure 36 (case III), the prediction results obtained from the HIM model and experimental data reported by Süer et al. for O2/N2,156 Zornoza et al. for O2/N2 and CO2/CH4,197 and Zornoza et al. for CO2/N2,297 are in good accordance as represented in the shadowed region of case III. Pore-blocking morphology of HNT−PEI MMMs was obtained when the pores of HNT particles were filled with the polymer chains to different degrees (case IV). Because the degree of pore blockage often depends on the pore diameter (d) of filler, case IV was divided into three subcases: (i) case IVA, d > 9 Å; (ii) case IV-B, d in the range of 4−9 Å; (iii) case IVAD


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Finally, the last case (case V) is a combination of pore blocking and void morphologies. In this case, a reduction in both permeability and selectivity caused by the pore blockage and Knudsen diffusion through the interfacial void, respectively, are illustrated in Figure 36 (case V). Only one experimental point reported by Hashemifard et al.440 for CO2/CH4 is well plotted in this figure. This behavior is commonly observed for MMMs filled with both small and large pore size particles. Using the practical methodology developed by Li et al.,549 Yang et al.482 further studied the gas performance of MMMs containing fully dispersed nanosize ZIF-7 in polybenzimidazole (PBI) polymer. Interestingly, by mixing the as-synthesized ZIF7 nanoparticles without a traditional drying process with PBI polymer, a series of MMMs with ZIF-7 loading as high as 50 wt %, high transparency and mechanical flexibility, together with enhanced H2 permeability and ideal H2/CO2 permselectivity surpassing both neat PBI and ZIF-7 membranes, was prepared. Based on the experimental results from such advanced instrumental analyses as dynamic light scattering (DLS), transmission electron microscopy (TEM), field emission scanning electron microscopy (FESEM), thermogravimetric analysis (TGA), Fourier transform infrared spectroscopy (FTIR), X-ray diffraction (XRD), and positron annihilation lifetime spectroscopy (PALS), the authors explained that the superior gas performance of ZIF-7/PBI membrane is a combined result of a good interfacial interaction between the ZIF-7 particles and the PBI polymer, minimal nonselective voids, rigidified PBI chains around ZIF-7 particles, and the creation of more free volume and slightly larger pores as subnano-interphase channels for gas transport. Moreover, in order to affirm this explanation, the authors applied the original Maxwell model, which is normally used to predict the gas permeation performance of an ideal MMM at low filler loadings. As expected for such complex ZIF-7/PBI system, the obtained predictions of H2 permeability were somewhat lower than the experimental values. This trend becomes more significant with increasing ZIF-7 loadings. Similar behavior was also reported with regard to CO2 permeability; however the deviation was slightly less. Recently, Erucar and Keskin used various theoretical permeation models including the original and modified Maxwell, Bruggeman, Lewis−Nielsen, Pal, and original and modified Felske models to predict the gas performance of MOF-based MMMs.467 The authors first compared the obtained predictions by these models with the experimental data available in literature for both pure CO2 and CH4 permeance through pure Matrimid and IRMOF-1/Matrimid membranes479 and for mixed-gas permeabilities of CO2/CH4 mixture through CuBTC/Matrimid MMMs.463 As shown in Figure 37a,b, the Pal, Bruggeman, Lewis−Nielsen, and original Maxwell models predict overestimated values of both CO2 and CH4 permeability compared with the experimental data reported by Perez and coauthors.467 The modified Maxwell, Felske, and modified Felske models display better predictions relative to the other models. In addition, regarding the percentage average absolute relative error (%AARE) values estimated from these models from this CO 2 and CH4 permeation data, among the prediction models derived for an ideal MMM morphology, the %AARE values are in the following order: Pal model > Bruggeman model > Lewis− Nielsen model > original Maxwell model. On the other hand, for the permeation models considering nonideal interfacial interactions of MMM, this order is modified Maxwell model >

Figure 37. Comparison between the experimental data and predictions by different theoretical models for pure-gas permeabilities of (a) CO2 and (b) CH4 through IRMOF-1/Matrimid MMMs and (c) mixed-gas permeabilities of a CO2/CH4 35:65 mixture through CuBTC/Matrimid MMMs (1 GPU = 1 Barrer/μm). Reproduced with permission from ref 467. Copyright 2011 American Chemical Society.

Felske model > modified Felske model. Figure 37c shows the mixed-gas permeabilities predicted by the original Maxwell and modified Felske models using the experimental data reported by Basu et al.463 for a CO2/CH4 35:65 mixture through CuBTC/Matrimid MMMs at 35 °C and 10 bar. The predictions by these two models were again in better agreement with the experimental data than those estimated by the other models. On the basis of these results, the authors concluded that the original Maxwell and modified Felske models provide the best prediction trend among the studied models based on the ideal and interfacial morphology concepts, respectively. After identifying several promising MOF/polymer combinations among the 80 different types of new MOF-based MMMs AE


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that possess high CO2 permeation properties, as a subsequent work, these authors then used these two permeation models to predict the membrane performance of ten different MOF particles filled in eight different polymers for CO2/CH4 separation. As a feature result, the orginal Maxwell model generally gave more optimistic results than the modified Felske model. One can conclude that the combination of atomic simulations and theoretical permeation models is a potential tool to select an appropriate MOF as dispersed filler to obtain MMMs with extraordinary properties for CO2/CH4 separation. As a continuation of the above-reported work for CO2/CH4 separation, Erucar and Keskin recently extended the use of these two models to examine the challenge of selecting MOFs as filler particles in high-performance MMMs for H2/CH4 separation.491 In this work, the authors carried out an estimation of membrane performance of 119 new MOFbased MMMs, composed of 17 different MOFs and 7 different polymers. The promising application of these two models was again confirmed by comparing the predictions with the available experimental data for pure gas permeabilities of H2 and CH4 through four different MMMs: Cu-BTC/PSF and CuBTC/PDMS,465 IRMOF-1/Matrimid,479 and Cu-BPY-HFS/ Matrimid.485 As expected, a good agreement between experimental data and theoretical predictions for all MMMs was also observed (Figure 38). The modified Felske model shows better predictions than the original Maxwell model, especially at high loadings of filler particles. Before predicting the gas performance of a new MOF-based MMMs using both original Maxwell and modified Felske models, the authors categorized 17 studied MOFs into five different groups based on their common characteristics in H2/CH4 separation: (i) high H 2 permeability and low H 2 selectivity; (ii) high H 2 permeability and high H2 selectivity; (iii) low H2 permeability and high H2 selectivity; (iv) low H2 permeability and low H2 selectivity; (v) extraordinarily high H2 selectivity and low H2 permeability. The seven polymers used in this study to make MOF/polymer pairs are sulfonated polyimide BAPHFDS(H), polyimide 6FDA−mMPD, polyimide 6FDA−DDBT, Hyflon, Teflon (AF-2400), poly(trimethylsilylpropyne-co-phenylpropyne) (PTMSP-co-PP), and poly(trimethyl-silylpropyne) (PTMSP). Among these polymers, the sulfonated polyimide shows high H2 selectivity but very low H2 permeability; whereas the last two represent polymers having almost no H2 selectivity but high H2 permeability. The other polymers have moderate H2 selectivity and permeability. Similar to the predictive results for CO2/CH4 separation mentioned above also by these authors,467 the original Maxwell model often produced higher estimated values than the modified Felske model. Teflon, PTMSP-co-PP, and PTMSP polymers seem to be good candidates to pair with all studied groups of MOF, resulting in a MOF/polymer membrane with high performance for H2/ CH4 separation, for which an enhancement in both H2 permeability and H2/CH4 selectivity is more pronounced. Moreover, pairing a MOF crystal that has an extraordinarily high H2 selectivity with three of the studied polyimides may yield MOF/polymer membranes with interesting estimated properties above the Robeson’s upper bound trade-off for H2/ CH4 separation.40 In early 2012, Zhang et al. studied the separation of C3H6 and C3H8 through a series of MMM fabricated from 6FDA− DAM polyimide and zeolitic imidazolate framework (ZIF-8) particles with high ZIF-8 loadings (28.7 wt % and 48.0 wt %).484 As shown in Figure 39, for a MMM filled with 28.7 wt %

Figure 38. Comparison between the experimental measurements and theoretical predictions by the original Maxwell and modified Felske models through CuBTC/PDMS,465 Cu-BPY-HFS/Matrimid,465 CuBTC/PSF,479 and IRMOF-1/Matrimid485 MMMs for gas permeabilities of (a) H2 and (b) CH4. Reproduced with permission from ref 491. Copyright 2012 Elsevier.

Figure 39. Exprerimental data and predictions by the original Maxwell model of C3H6/C3H8 separation through ZIF-8/6FDA−DAM MMMs. Reproduced with permission from ref 484. Copyright 2012 Elsevier.

ZIF-8 particles, the C3H6 permeability and C3H6/C3H8 ideal selectivity predicted by the original Maxwell model are in good accordance with experimental results. However, at higher filler loadings (i.e., 48.0 wt % ZIF-8), the predictions are less satisfactory. This observation is not surprising since the original AF


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Maxwell model is well-known not to be capable of predicting permeation performance of a MMM with high filler loading. Similar modeling work on MOF-based MMMs was recently reported by Atci and Keskin on the transport of single component gases (CH4, CO2, and H2) and binary gas mixtures (H2/CO2 and CO2/CH4) through ZIF/polymer (ZIF-90 and unfabricated ZIF-65) composites.487 The authors used the original Maxwell model to predict both gas permeability and selectivity through several ZIF-90/polymer and ZIF-65/ polymer membranes composed of various polymers including Matrimid, Ultem, and 1.1-GFDA−DMA polyimides as well as liquid crystalline polyester, polyaniline, and poly(trimethylsilylpropyne) (PTMSP) polymers (Figure 40). Again, the model predictions are in good agreement with the experimental data at low volume fraction (i.e., ZIF loading is below 20 wt %). For example, Figure 40a shows the experimental data reported by Bae et al.462 for pure polymer membranes (Matrimid and Ultem), pure ZIF-90 membrane, and ZIF-90/polymer composite membranes, together with the model predictions. Interestingly, the theoretical predictions are close to the experimental data of ZIF-90/polymer composites using both Matrimid and Ultem polyimides. Figure 40b,c shows the predicted H2/CO2 selectivity and H2 permeability of MOFbased MMMs consisting of ZIF-90 or unfabricated ZIF-65 crystals dispersed in several polymer matrices including liquid crystalline polyester, polyaniline, 1.1-GFDA−DMA polyimide, and poly(trimethylsilypropyne) (PTMSP). As the volume fraction of ZIF-65 or ZIF-90 increases, the predicted H2 permeabilities are significantly improved in the case of ZIFbased MMMs prepared with liquid crystalline polyester, polyaniline, and polyimide polymers, while the H2/CO2 selectivities estimated using the original Maxwell model are almost unchanged. In contrast, both the predicted H2 permeability and H2/CO2 selectivity of PTMSP membrane are increased when ZIFs are used as filler particles, because the H2 permeabilities of pure PTMSP polymer and ZIFs crystals are very close. Hussain and König synthesized a series of MMMs comprised of up to 66 wt % amine functionalized ZSM-5 particles dispersed in a polydimethylsiloxane (PDMS) polymer to study CO2 separation from gas mixtures, focusing on the high solubility of CO2 in ZSM-5/PDMS MMMs for separating a CO2/N2 mixture.225 The permeability of single gases (CO2, and N2), as well as the permeance of a CO2/N2 50:50 mixture, was significantly increased with increasing zeolite loading, while the CO2/N2 selectivity was slightly enhanced. As shown in Figure 41, the experimental data follow the theoretical predictions by the original Maxwell model up to 30 wt % ZSM-5 loading but afterward start deviating. These experimental values are even higher than those predicted by the parallel maximum model after 56 wt % ZSM-5 loading. It might be explained by the existence of an interfacial gap or void between the zeolite and polymer phases at high ZSM-5 loadings. The membrane performance of this “sieve-in-a-cage” morphology may be better predicted by applying the modified Maxwell model for a threephase system including interfacial voids.193,194,202 Kang et al. recently reported an investigation on the fabrication, detailed characterization, and molecular transport properties of nanocomposite membranes containing high volume fractions (up to 40 vol %) of individually dispersed aluminosilicate single-walled nanotubes (SWNTs) in poly(vinyl alcohol) (PVA).410 PVA/SWNT MMMs were synthesized from two different starting sources of SWNT including

Figure 40. Comparison between the experimental data and theoretical predictions by the original Maxwell model through ZIF-90/Matrimid and ZIF-90/Ultem composite membranes for (a) CO 2 /CH 4 separation and H2/CO2 separation with different types of polymers embedded with (b) ZIF-90 and (c) ZIF-65 crystals. The volume fractions are varied from 0.1 to 0.5. The solid line represents the present upper bound trade-off established for H2/CO2 separations. Reproduced with permission from ref 487. Copyright 2012 American Chemical Society. AG


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Figure 42. Comparison between the water permeability and water/ ethanol selectivity obtained from pervaporation experiments (■), predictions by the KJN model (○), and by the Hamilton−Crosser (HC) model (△) through PVA/SWNT membranes prepared from SWNT gels. Reproduced with permission from ref 410. Copyright 2012 American Chemical Society.

Figure 41. Comparison between the experimental data and predictions by the parallel, series, and original Maxwell models for CO2/N2 separation through ZSM-5/PDMS MMMs. Reproduced with permission from ref 225. Copyright 2012 Wiley-VCH Verlag GmbH & Co. KGaA.

interfacial defects at the interphase of MMMs. As mentioned in the review of Aroon and coauthors,26 there are several reasons that cause different interfacial defects and nonideal morphologies in MMMs, including polymer−inorganic incompatibility, stresses that arise due to solvent evaporation during membrane formation, poor polymer−particle adhesion, polymer packing disruption in the vicinity of the inorganic phase, repulsive forces between continuous and dispersed phases, and different thermal expansion coefficients for polymer and particle. For avoiding these interfacial defects and fabricating defect-free MMMs, the following methods have been reported:

powders and gels. Combined with the pure PVA polymer, the composite membranes prepared with SWNT powders displayed the presence of large SWNT agglomerates. On the other hand, membranes made of SWNT gels showed a uniform dispersion of individual SWNTs in the PVA matrix with a random distribution of orientations. The transport performance of these PVA/SWNT membranes was investigated experimentally via ethanol/water mixture pervaporation and then computationally by the Hamilton−Crosser (HC) and Kang− Jones−Nair (KJN) models. For example, the PVA/SWNT membranes prepared from SWNT gels possessed a monotonic increase in water permeability and a decrease in selectivity with increasing SWNT volume fraction. On the other hand, the PVA/SWNT membranes synthesized with SWNT powders showed no significant improvement in both water permeability and water/ethanol selectivity at low volume fractions. An abrupt increase in water permeability combined with a dramatic drop of water/ethanol selectivity at higher volume fractions was, however, observed. This result was explained by the occurrence of defects such as interfacial voids in the membranes made from aggregated SWNT powders. Figure 42 shows a comparison of water permeability and water/ethanol selectivity obtained from pervaporation experiments (solid squares), predictions by the KJN model (open circles), and predictions by the HC model (open triangles) for PVA/SWNT membranes prepared from SWNT gels. The KJN model predictions are in good accordance with the experimental data, except at a volume fraction of 0.4, whereas the HC model considerably overestimates them. Both the KJN and HC models predict higher water/ethanol selectivity than the experimental observations. A pronounced deviation of both permeability and selectivity estimated by the KJN model is observed at a volume fraction of 0.4. It is possible that the presence of large quantities of SWNT fillers dispersed in the PVA polymer matrix might create more interfacial defects between the PVA polymer and the outer surfaces of the SWNTs.

6.1. Modifying Preparation Conditions of MMMs

(1) Glass-transition temperatures (Tg) of polymer matrix give information about filler−polymer interaction and the rigidity of the polymer matrix. Casting at temperatures above Tg,90,97,100 using polymers with low Tg,88,152,181,259,359 fabricating the MMMs at temperatures close to the Tg,128,129,193,194 and annealing already formed MMMs at temperatures above Tg130,181,186,494,495,555 are several techniques that can maintain matrix flexibility during membrane formation to maximize stress relaxation. (2) Casting on a dense liquid surface with a high surface tension can allow the polymer solution to freely contract after casting, and thus the stresses can be distributed uniformly in all directions, resulting in interfacial void-free MMMs, with a likely rigidified polymer layer around the particles.182 (3) Preparation of membranes using melt processing can maintain polymer flexibility during membrane preparation. This method could result in void-free MMMs; however, it is not a feasible process for industrially relevant membranes.104,182 6.2. Modifying Interfacial Properties of Fillers and Polymer Matrix

(1) Incorporation of a plasticizer into the polymer solution can decrease the polymer Tg and thus maintain polymer chain mobility and flexibility during membrane fabrication. There are several types of plasticizer such as RDP Fyroflex, dibutyl phthalate, 4-hydroxy benzophenone,181 and polyethylene glycol.516,517 (2) Adding low molecular-weight additives (LMWAs) to the membrane formulation may act as a compatibilizer or a third component. Among different LMWAs, Table 23 lists several examples of compatibilizer used to improve the compatibility between fillers and polymer matrix.

6. METHODS FOR AVOIDING NONIDEAL INTERFACIAL DEFECTS It is clear that predictive modeling is a useful and practical technique to detect, identify, and confirm the existence of AH


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Table 23. Compatibilizers To Improve the Compatibility between Fillers and Polymer Matrix compatibilizer

abbreviation

maleic anhydride/styrene 2,4,6-triaminopyrimidine, 2,4,6-pyrimidinetriamine

MA/ST TAP

poly(vinyl pyrrolidone) kollidone 15

PVP K-15

N-(p-carboxyphenyl)maleimide polystyrene-b-poly(hydroxyl ethyl acrylate) p-nitroaniline

pCPM PS-b-PHEA

chemical structure C4H7N5

C6H6N2O2

magnesium hydroxide 2-hydroxy-5-methyl aniline

HMA

room-temperature ionic liquid β-cyclodextrin aminopropylisooctyl polyhedral oligomeric silsesquioxane, octaammonium polyhedral oligomeric silsesquioxane

RTIL β-CD AP-POSS, OAPOSS

(3) Coating the surface of the filler particles with a dilute polymer dope is known as the priming method. The dilute polymers are the same as the bulk polymers already used for the preparation of MMMs.152,180,257,318,358,440,461,495 This method can reduce stress at the polymer−particle interface, minimize agglomeration of the particles, and promote interfacial interaction between the two components. (4) Modification of filler particles using a sizing technique could increase the strength of the interphase by introducing more chemical reactive surface sites or more surface area for adhesion, resulting in a significant reduction of voids or gaps around the particles.251,254,257 (5) In minimization of zeolite−solvent/nonsolvent interaction, especially for use of the modified zeolite in asymmetric membranes, the zeolite particles have undergone a vigorous surface modification to replace their hydroxyl groups with methyl groups.109 Therefore, the obtained hydrophobic surface can suppress the zeolite particles from acting as nucleating agents, resulting in void-free MMMs. (6) Surface modification of filler particles using coupling agents is a most frequent technique to improve the polymer− filler adhesion in MMMs. Pore blockage by coupling agents is, however, an important limiting factor for the usefulness of this method. Table 24 summaries several coupling agents that have been used in the preparation of MMMs.

Mg(OH)2 C7H9NO

filler/polymer montmorillonite/polypropylene (PP)452 4A/, 5A/, 13X/, or Y/Matrimid polyimide (PI)177 CMS/polysulfone (PSF),254 Al2O3/ 6FDA-6FpDA,362 silica/polybenzimidazole (PBI),310 4A/Matrimid polyimide (PI)163 4A/poly(bisphenol A)carbonate (PC)152 MFI/poly(etherimide) Ultem80,183 SAPO-34/polyethersulfone (PES)190,553,554 SAPO-34/poly(RTIL)107,108 MWCNTs/poly(vinyl alcohol)418 montmorillonite/polyamide 12 (PA 12)450

intrinsic or modified functional groups of organic linkers can act as bonding agents.

7. CONCLUSIONS Depending on the interaction between the continuous and dispersed phases, two groups of theoretical models are written. In the so-called ideal interface morphology, the interfacial layer contains no defects and no distortion and its thickness is considered zero. The original Maxwell, Bruggeman, Böttcher and Higuchi, Lewis−Nielsen, Pal, Gonzo−Parentis−Gottifredi (GPG), Funk−Lloyd, and Kang−Jones−Nair (KJN) models are some examples in this group. These models originate from description of electrical and thermal conductivity of particulate composites, and they are valid at low volume fraction. At higher values of filler loading, the estimated results are only in good accordance with the experimental data when an excellent interfacial contact between dispersed particles and continuous polymer is achieved. Numerous research groups have performed many examples of using the original Maxwell and Bruggeman models to predict the permeation behavior of several kinds of MMMs. However, most of the results revealed that these models often failed in predicting permeation performance due to the frequent presence of an interfacial layer between the two phases.93,107,144,155,208,258,269,278,280,294,378,393 In order to resolve this drawback, the GPG, Funk−Lloyd, and KJN models were proposed to specifically take into account the interfacial void between the dispersed filler and polymer matrix. However, even if these models have a potential to extend the predictive ability at higher loading of filler particles, favorable predictions are only reached for MMMs in which morphology could be represented as combining the presence of voids with an ideal filler/polymer interaction. In addition, the effect of interfacial voids on permeation properties of MMMs has also been considered in both the Funk−Lloyd and KJN models. The formation of an interfacial layer surrounding a filler particle during the fabrication of MMMs, however, often occurs. Poor adhesion, mobility of polymer chains, and pore blockage by polymer matrix are some critical phenomena that are often observed when dispersing filler particles into a polymer phase. The trend in the past decade has then shifted toward modeling these interfacial defects. Various models considering these defects, including the modified Maxwell, modified Lewis−Nielsen, Felske and modified Felske, modified

6.3. Using Filler Particles and Bulk Polymers with Special Properties

(1) When some copolymers such as polyimide siloxane are used, siloxane segments can enhance the interfacial polymer− filler contacts, and thus void-free MMMs can be achieved.389 (2) Hydrophobic material can be used for both inorganic and organic phase. For example, due to the hydrophobic character of both the zeolite Nu-6(2) and the polysulfone polymer, interfacial macroscopic void-free MMMs can be prepared.123 (3) Using large pore-size zeolites or mesoporous materials for MMMs could potentially offset the negative effects of partial pore blockage and polymer chain rigidification on permeability, however not favoring selectivity.129,324,325,490,550 (4) Using metal−organic frameworks (MOFs) with or without functionalization modification as fillers is a potential technique to enhance interfacial contacts between the two components of MMMs.75,462,481−486,488,489,495,549,557−559 In this case, the AI


APTMS, APTrMOS, AmPS GPTMS MPTMS, MrPS TMOPMA, MtPS CDMS CDMOS CDMPS, DMPSCl CDMVS, CDVS CTMS, TMCS DMDCS DOCS, CDOS

3-aminopropyl-trimethoxysilane or 3-(trimethoxysilyl)propylamine or γ-aminopropyl-triethoxysilane

chlorotrimethyl silane or Trimethylchlorosilane or Trimethylsilyl chloride dimethyldichlorosilane or dichloro(dimethyl)silane or dichlorodimethylsilane dimethyloctadecylchlorosilane chlorodimethyloctadecylsilane or chlorooctadecyldimethylsilane or octadecylchlorodimethylsilane or octadecyldimethylchlorosilane or dimethyl(octadecyl)silyl chloride or dimethyl(octyldecyl)silyl chloride glycerinedimethacrylaturethanetriethoxysilane hexadecyltrimethoxysilane or trimethoxyhexadecylsilane methyltrimethoxysilane or trimethoxy(methyl)silane octadecyldimethylmethoxysilane or dimethylmethoxyoctadecylsilane or dimethyloctadecylmethoxysilane octadecyltrichlorosilane or trichlorooctadecylsilane octadecyltrimethoxysilane or trimethoxy(octadecyl)silane propyltrimethoxysilane or trimethoxy(propyl)silane N-octyltriethoxysilane or triethoxy(octyl)silane N-β-aminoethyl-γ-aminopropyltrimethoxy silane styryl amine functional silane triethoxysilane vinyltrimethoxysilane or trimethoxysilyl)ethylene or ethenyltrimethoxysilane or trimethoxy(vinyl)silane 3-methylpyridine or 3-picoline N-vinylpyrrolidone or 1-vinyl-2-pyrrolidinone p-phenylenediamine diethanolamine or 2,2′-iminodiethanol or bis(2-hydroxyethyl)amine methylmagnesium bromide

chlorodimethylvinyl silane or dimethylvinylchlorosilane or vinyldimethylchlorosilane

chlorodimethyl silane or dimethylchlorosilane chlorodimethyloctyl silane or dimethyloctylchlorosilane or octyldimethylchlorosilane chlorodimethylphenyl silane or dimethylphenylchlorosilane or dimethylphenylsilylchloride or phenyldimethylchlorosilane

AJ

TES VTMS 3-MPy NVP PPD DEA Grignard reagent

GUS HTOS MTOS ODMOS OTS OTOS PTOS OTES AEAPTMS

APTES, APTEOS

3-aminopropyl-triethoxysilane or 3-triethoxysilylpropylamine or Dynasylan Ameo (DA) silane

3-glycidoxypropyltrimethoxysilane or trimethoxy[3-(2-oxiranylmethoxy)propyl]silane 3-mercaptopropyltrimethoxysilane or 3-(trimethoxysilyl)-1-propanethiol 3-trimethoxysilyl propylmethacrylate or 3-methacryloxypropyl trimethoxyl silane

APDEMS

abbreviation

3-aminopropyl-diethoxymethylsilane or 3-(diethoxymethylsilyl)propylamine or 3-aminopropyl-methyl-diethoxysilane

coupling agent

Table 24. Coupling Agents for Surface Modification of Filler Particles

C3H10O3Si H2CCHSi(OCH3)3 3-CH3C5H4N C6H9NO C6H4(NH2)2 HN(CH2CH2OH)2 CH3BrMg

H3C(CH2)15Si(OCH3)3 C4H12O3Si C21H46OSi C18H37Cl3Si C21H46O3Si C6H16O3Si C14H32O3Si C8H22N2O2Si

(CH3)3SiCl C2H6Cl2Si C20H43ClSi

H2CCHSi(CH3)2Cl

C9H20O5Si HS(CH2)3Si(OCH3)3 H2CC(CH3) CO2(CH2)3Si(OCH3)3 (CH3)2SiHCl CH3(CH2)7Si(CH3)2Cl C6H5Si(CH3)2Cl

H2N(CH2)3Si(OCH3)3

H2N(CH2)3Si(OC2H5)3

CH3Si(OC2H5)2(CH2)3NH2

chemical structure

filler

beta140 beta137 beta137 beta137 beta,137 beta137 beta137 beta,137 ZSM-5191 silicalite-1,97 silica,299 HNTs440,454 silicalite-197 silica292 silica,306 Al2O3362 SSZ-13171,172 montmorillonite457 CMS253 LTA94 SSZ-13,109 4A,155 MFI192

silica,167 carbon black167 MCM-41,324 MCM-48,325 CNTs,551 beta137

silica305

silica302 silica296 silica296

PWA and PMA552 silica277,292,300 4A,185 silica300

94,129

3A, 4A,129 5A,129 SSZ13, FAU/EMT,187 silica291 L,145 silica,186 4A,189 silicalite-1,221 MCM41,318 CNTs,401,551 MWCNTs,408 PWA, and PMA552 silicalite-1,97 ZSM-5,166 4A,184 13X,184 silica,273,291,299,300 MCM-41324

silicalite-1,115 109,180

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Figure 43. Comparison between predicted and experimental literature data for permeability of (a) CO2 and (c) O2 and ideal selectivity of (b) CO2/ CH4 and (d) O2/N2 through a variety of MMMs. Data are taken from refs 24, 25, 27, 57, 58, 80, 107, 128, 129, 171, 172, 180, 193, 194, 200, 201, 207, 208, 210, 225, 257, 258, 260, 263, 289, 393, 443, 463, 465, 467, 472, 479, 485, and 491.

packing density of filler particles. Note that the latter factor is in relation to particle size distribution, particle shape, and particle aggregation; however, the participation of these parameters in these models is still not explicit. The Hashemifard−Ismail−Matsuura (HIM) model has attracted great interest rooted from the ability to cover all known MMM morphologies and introducing as a parameter the thickness of the interphase surrounding the filler particles. An excellent agreement between model predictions and experimental data has been observed for almost all studied MMM systems.207,439 Because in this model there are four independent variables, including the volume fraction ϕ, the interfacial thickness ratio θ = t/dp, and the two permeability ratios λi = Pi/Pc and λd = Pd/Pc, the relative permeability, Pr, of

Pal, and Hashemifard−Ismail−Matsuura (HIM) models, comprise the second group. Consequently, more realistic descriptions of the membrane properties were gained. Among these models, the modified Maxwell and modified Felske approaches were frequently used to predict the gas performance of MMMs having a complex three-phase morphology. A good agreement between predictions by these two models and experimental data was generally established.128,129,180,208,467 The modified Maxwell model proposed by Majahan and Koros193,194,202 and extended by Li et al.62,128,129 simultaneously takes into account the interfacial voids, polymer chain rigidification, and pore blockage, while the modified Felske model developed from Lewis−Nielsen and Felske models considers these nonideal interfacial defects as well as the AK


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a given MMM is realistically described. The ideal morphology is represented by setting θ = 0 and suppressing λi. The rigidified interfacial layer case is obtained at nonzero θ values and λi < 1. The leaky MMM or void in MMM situations correspond to high λi values. This versatility makes the HIM model an easy technique in realistically predicting the MMM performance. This model, however, does not account for the pore filler blocking (known to be a significant effect), as well as particle size distribution, particle shape, and aggregation of filler particles. Figure 43 shows a comparison between predicted and experimental literature values of permeability of CO2 and O2, as well as ideal selectivity of CO2/CH4 and O2/N2, through a variety of MMMs. The selection of experimental data was made to cover all data that had been compared with models in the literature. As a general observation, it may be stated that the HIM, modified Felske, and modified Pal models yield the most accurate fitting of these experimental data. Numerous researchers have already pointed out that the permeation properties of MMMs are strong functions of particle shape and size, particle pore size, pore size distribution, particle sedimentation and agglomeration, as well as operating conditions, that is, temperature, pressure, and feed composition. Up to now, the particle shape, size and size distribution, and agglomeration effects on the transport behavior of MMMs were only illustrated using the maximum packing volume factor. However, this factor is usually fixed as equal to 0.64. As the particle size decreases, at constant volume fraction ϕ, the total external area and number of filler particles increases. Therefore, the mass transfer resistances at the interfacial layer become more significant. Particle pore size is obviously also an important factor to influence the gas performance of MMMs. The literature results indicate that the permeability increases with increasing particle pore size.128,129,141,157,212 Particle agglomeration, sedimentation, or migration of filler particles to the surface during the fabrication of MMMs often occurs, especially at high particle loading. The significant difference in physical properties and density between filler particles and polymers is well-known to cause the sedimentation of fillers. As a consequence, the formation of inhomogeneous suspension of fillers in polymers is thought to be responsible for interfacial defects among the particles or between the particle fillers and polymer matrix.76,88,174,184,213,262,323,359,384,396,398,404,439,440,461,482 Although literature studies on the dependence of permeabilities and selectivities of MMMs on operating conditions such as temperature, pressure, and feed composition are limited, it is appropriate to state that these parameters are other important factors that affect the diffusion and adsorption mechanism of gas species through MMMs. The gas permeabilities are generally enhanced with increasing temperature. The permeation rate is enhanced as the feed concentration of the preferential selective component is increased, but selectivity is decreased due to high swelling of the polymer matrix.82,103,141,212,218,219,417,489 Furthermore, as the permeate pressure increases, the gas permeability gradually decreases, while the selectivity factor shows an opposite trend.18,121,141,180,308,486,495 A noticeable exception was observed in a very recent publication of our group.495 Although the CO2 permeabilities of both neat 6FO polyimide and MIL53/polyimide membranes were found to decrease with increasing feed pressure (Figure 44A), interestingly, a significant enhancement in CO2/CH4 separation factor through

Figure 44. (A) CO2 permeability and (B) separation factor (α*) for gas mixture CO2/CH4 = 50:50 as a function of feed pressure through 6FO polymer and 6FO−MIL-NH2-25% MMMs at 308 K and (C) CO2 adsorption isotherm of pure Al−MIL-53-NH2 taken from ref 486. Reproduced with permission from ref 495. Copyright 2012 American Chemical Society.

a 6FO polyimide membrane containing 25 wt % aminefunctionalized MIL-53 was observed over a pressure range from 150 to 300 psi (Figure 44B), while that of neat polyimide was decreased. In this range of CO2 feed pressure, a large increase in CO2 adsorption capacity was also observed (Figure 44C).486 This behavior can be explained by both strong interactions of CO2 molecules with the amine groups located at the aminefunctionalized MIL-53 walls and a breathing effect of the MIL53 lattice, which allows a higher adsorbed CO2 content, does not improve CO2 diffusivity, but excludes CH4 transport from amine-functionalized MIL-53 cages, resulting in a strong decrease in CH4 permeability. Just like with modeling of any other physicochemical process, a reflection about the proper usage of MMM models is necessary. The mere prediction of a large scale gas phase separation process is one possible objective. Another objective is optimization of that process operating conditions, and yet another goal could be to optimize the MMM design and preparation procedure. In the latter case, the choice of targeted optimal membrane performance is to be defined first. If the ideal filler grafting in the polymer phase is targeted, then deviation of relative permeabilities from ideal model predictions has a diagnostic value. The preparation procedure may then be modified until the resulting membrane behaves as predicted. In this regard, the recent calculations performed by Erucar and Keskin are AL


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especially instructive.467,491 These authors have, for example, compared on one hand the predictions of H2/CH4 selectivities and H2 permeabilities and on the other hand the predictions of CO2/CH4 selectivities and CO2 permeabilities using in each case the Maxwell and the modified Felske models. The MMMs encompassed seven different polymers and up to 17 MOFs. From these calculations, introducing the MOF filler may involve increasing selectivity at constant (or almost constant) permeability or increasing permeability at constant (or almost constant) selectivity. This of course depends essentially on the relative position of the neat polymer membrane and the filler on a selectivity vs permeability chart. Comparing the predictions of the two models indicates that in all modeled situations, the ideal interface represented by the Maxwell model is more effective than the nonideal contact described by the modified Felske model. It may be also be remembered that a model is based on a necessary mental simplification of the physical world and that multiple confrontations of the hypotheses and conclusions of a model with experimental confirmations is always necessary. For example, any defective interface detected by analysis of experimental membrane properties should be confirmed by other experimental techniques such as electron microscopy. As a general issue mathematical modeling of MMM properties is based on a reflection on the nature of defects at particle/polymer interface. It is remarkable that the current developments are entirely based on experimental results obtained with flat membranes. Large-scale membrane separation processes, however, mostly implement hollow fiber bundles. The preparation of MMM-based hollow fibers is still in its infancy, but several preparation procedures have already been proposed.489 These methods involve modifications of the phase inversion procedure initially proposed by Loeb and Sourirajan,561 which produces a thin overlayer of dense polymer. In the making of MMM hollow fibers, it is thus expected that lateral shrinkage stresses could be accommodated by shrinkage of the nascent fiber as solvent rapidly evaporates in the air gap. This may in turn help to preserve interfacial adhesion integrity since unrelaxed transverse stresses should be responsible for adhesion failure at particle/polymer interface. Such effects may result in a significant asset for MMM hollow fibers.

Biographies

Vinh-Thang Hoang was born in 1971 in Hanoi, Vietnam. He received his B.S. degree in Organic and Petrochemical Technology from Hanoi University of Science and Technology (Hanoi, Vietnam) in 1993. From 1993 to 2001, he joined the laboratory of Prof. Huu-Phu Nguyen at Institute of Chemistry (Vietnam Academy of Sciences and Technology, Hanoi, Vietnam) and worked on a range of applied catalysis topics. After receiving a M.Sc. degree in Physico- and Theoretical Chemistry at the same institute in 2001, he moved to the laboratory of Prof. Serge Kaliaguine in the Department of Chemical Engineering at Laval University (Quebec, Canada), where he focused his graduate research on exploring applications of hierarchical zeolites and mesoporous materials in adsorption and diffusion fields. After graduating in 2005 with a Ph.D. degree in Chemical Engineering under the guidance of Profs. Serge Kaliaguine, Trong-On Do, and Mladen Eić, he continued to work at the same laboratory as a Postdoctoral Fellow. He is now a Research Staff member of the Chemical Engineering Department at Laval University. His research interests are focused on experimental and modeling investigations of zeolites, mesoporous materials, and metal−organic frameworks in catalysis, adsorption−diffusion, membrane separation, and electrochemistry.

Professor S. Kaliaguine has worked for more than 35 years in catalysis and surface science research. He has published over 400 papers in refereed journals, and the quality of his research has resulted in several prestigious awards. These include Le prix Urgel-Archambaut (ACFAS), The Canadian Catalysis Award (CIC), The Catalysis Lectureship Award (CCF), and The Century of Achievements Award (CSChE). Professor Kaliaguine has chaired several national and international meetings. He was chairman of the Catalysis Division of the Chemical Institute of Canada, a director of the Canadian Society for Chemical Engineering, and President of the International Mesostructured Materials Association. From 2003 to 2008, he was

AUTHOR INFORMATION Corresponding Author

*Phone: 1-418-656-2708. Fax: 1-418-656-3810. E-mail: Serge. [email protected]. Notes

The authors declare no competing financial interest. AM


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Felske

developed by Felske for thermal conductivity of composites of core−shell particles, ref 537 modified Felske a modification of the Felske model proposed by Pal for a composite morphology with packing density of particles, ref 25 Hashemifard−Ismail−Matsuura developed by Hashemifard, Ismail, and Matsuura for MMMs based on the flow pathways of penetrant gas in both series and parallel channels, ref 207 modified Pal a modification of the Pal model by Shimekit and coauthors in order to take into account interfacial rigidified polymer chain defect, ref 27

the holder of an NSERC Chair for Industrial Nanomaterials (Adsorbents, Catalysts and Membranes).

ANNEX 1. GLOSSARY OF MODEL DESIGNATIONS series parallel Maxwell−Wagner−Sillar Maxwell

Bruggeman

Böttcher Higuchi Lewis−Nielsen

Cussler

generalized Maxwell

Pal Gonzo−Parentis−Gottifredi

Funk−Lloyd Kang−Jones−Nair

modified Maxwell

two-layer MMM model with series combination, ref 24 two-layer MMM model with parallel combination, ref 24 developed for a MMM with dilute dispersion of ellipsoids, refs 521, 522 developed by Maxwell for a MMM with dilute suspension of spherical particles at low loadings, ref 523 developed by Bruggeman for the electric constant of particulate composites, refs 524−526 developed by Böttcher for a random dispersion of spherical particles, ref 527 developed by Higuchi for a random dispersion of spherical particles, ref 528 developed by Lewis and Nielsen for an elastic modulus of particulate composites, refs 529, 530 proposed by Cussler with a form similar to the original Maxwell model applied to a dilute suspension of flake spheres, ref 531 a generalization of the original Maxwell model proposed by Petropoulous and extended by Toy and coauthors for binary structured composites, refs 522, 532 developed by Pal for thermal conductivity of particulate composites, ref 533 an extension of the original Maxwell model in terms of ϕd proposed by Gonzo, Parentis, and Gottifredi, ref 24 developed by Funk and Lloyd for microporous zeolite-filled ZeoTIPS membranes, ref 206 developed by Kang, Jones, and Nair for an ideal composite membrane with tubular fillers, ref 424 a modification of the original Maxwell model proposed by Mahajan and Koros and extended by Li and coauthors in order to take into account interfacial defects, refs 129, 193, 194, 202

ANNEX 2. NOMENCLATURE 4MPD 2,3,5,6-tetramethyl-1,4-phenylene diamine 6FDA hexafluoroisopropylidene-dipthalic anhydride 6FpDA hexafluoroisopropylidene-dianiline BAPB 1,4-bis(4-amino-phenoxy)benzene or 2,2′-bis(4-aminophenoxy)biphenyl BAPD 4,4′-bis(4-aminophenoxy)-2,2′-dimethylbiphenyl BPADA 4,4′-bisphenol A dianhydride CA cellulose acetate CD cyclodextrin CMS carbon molecular sieve CNT carbon nanotube DABA 3,5-diaminobenzoic acid DAM diaminomesitylene DAPHFDS 2,2-bis-4-(4-aminophenoxy)phenyl hexafluoro propane disulfonic acid DDBT 3,7-diamino-2,8(6)-dimethyldibenzothiophene sulfone DLS dynamic light scattering EMT effective medium theory EPDM ethylene propylene diene monomer FESEM field emission scanning electron microscope FTIR Fourier transform infrared spectroscopy GPG Gonzo−Parentis−Gottifredi model HC Hamilton−Crosser model HIM Hashemifard−Ismail−Matsuura model HNT halloysite nanotubes KJN Kang−Jones−Nair model MMM mixed-matrix membrane MOF metal−organic framework NBR nitrile butadiene rubber NLF nonequilibrium lattice fluid PA polyamide PAI polyamide-imide PALS positron annihilation lifetime spectroscopy PDMC propanediol monoester cross-linkable PDMS polydimethylsiloxane PEG poly(ethylene glycol) PEI polyethyleneimine PES polyethersulfone PMA phosphomolybdic acid AN


Chemical Reviews PMO PMP POSS PPy PSF PTA PTFE PTMSP PTMSP-co-PP PVA PVAc PVDF RTIL SWNT TEM TEOS TGA TIPS TMSG TPX XRD ZIF

Review

cal calculated pred predicted

periodic mesoporous organosilica poly(4-methyl-2-pentyne) polymer polyhedral oligomeric silsesquioxane polypyrrole polysulfone phosphotungstic acid polytetrafluoroethylene polytrimethylsilylpropyne poly(trimethylsilylpropyne-co-phenylpropyne) poly(vinyl alcohol) poly(vinylacetate) polyvinylidene fluoride-hexafluoropropene 1-ethyl-3-methyl imidazolium bis(trifluoromethylsulfonyl)amide (emim [Tf2N]) single-walled nanotube transmission electron microscope tetraethoxysilane thermogravimetric analysis thermally induced phase separation trimethylsilyl-glucose poly(4-methyl-1-pentene) X-ray diffraction zeolitic imidazolate framework

Subscripts

avg average blo blockage c continuous phase d dispersed phase or filler/polymer eff effective f filler i or I interphase Kn Knudsen l height of dispersed particle m matrix m maximum p pinhole p diameter ps1 first pseudodispersed phase ps2 second pseudodispersed phase r relative rig rigidified polymer region v void or void/polymer

REFERENCES

Parameters

%AARE %ARE %RE d D G K l M n n O P R r r S t T

(1) Winston Ho, W.S.; Sirkar, K. K. Membrane Handbook; Chapman & Hall: New York, London, 1992. (2) Koros, W. J.; Fleming, G. K. J. Membr. Sci. 1993, 83, 1. (3) Matsuura, T. Synthetic Membranes and Membrane Separation Processes; CRC Press: Boca Raton, FL, 1994. (4) Paul, D. R.; Yampolskii, Y. Polymeric Gas Separation Membranes: CRC Press: Boca Raton, FL, 1994. (5) Mulder, M. Basic Principle of Membrane Technology, 2nd ed.; Kluwer Academic Publisher: Dordrecht/Boston/London, 1996. (6) Singh, R. CHEMTECH 1998, 28, 33. (7) Koros, W. J.; Mahajan, R. J. Membr. Sci. 2000, 175, 181. (8) Pandey, P.; Chauhan, R. S. Prog. Polym. Sci. 2001, 26, 853. (9) Baker, R. W. Ind. Eng. Chem. Res. 2002, 41, 1393. (10) Baker, R. W. Membrane Technology and Applications: John Wiley and Sons Ltd: Chichester, 2004. (11) Bredesen, R.; Jordal, K.; Bolland, O. Chem. Eng. Process. 2004, 43, 1129. (12) Smitha, B.; Suhanya, D.; Sridhar, S.; Ramakrishna, M. J. Membr. Sci. 2004, 241, 1. (13) Lin, H.; Freeman, B. D. J. Mol. Struct. 2005, 739, 57. (14) Javaid, A. Chem. Eng. J. 2005, 112, 219. (15) Nunes, S. P.; Peinemann, K.-V. In Membrane Technology in the Chemical Industry, 2nd ed.; Nunes, S. P., Peinemann, K.-V., Eds.; WileyVCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2006; p 53. (16) Bernardo, P.; Drioli, E.; Golemme, G. Ind. Eng. Chem. Res. 2009, 48, 4638. (17) Yampolskii, Y.; Freeman, B. Membrane Gas Separation; John Wiley and Sons Ltd: West Sussex, U.K., 2010. (18) Henis, J. M. S.; Tripodi, M. K. Sep. Sci. Technol. 1980, 15, 1059. (19) Kulkarni, S. S.; Hasse, D. J. U.S. Patent 7,776,137, 2010. (20) Basu, S.; Khan, A. L.; Cano-Odena, A.; Liu, C.; Vankelecom, I. F. J. Chem. Soc. Rev. 2010, 39, 750. (21) Ohlrogge, K.; Stürken, K. In Membrane Technology in the Chemical Industry, 2nd ed.; Nunes, S. P., Peinemann, K.-V., Eds.; WileyVCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2006; p 93. (22) Cranford, R.; Roy, C. U.S. Patent 7,556,677, 2009. (23) Erdem-Şenatalar, A.; Tatlier, M.; Tantekin-Ersolmaz, Ş. B. Chem. Eng. Commun. 2003, 190, 677. (24) Gonzo, E. E.; Parentis, M. L.; Gottifredi, J. C. J. Membr. Sci. 2006, 277, 46. (25) Pal, R. J. Colloid Interface Sci. 2008, 317, 191. (26) Aroon, M. A.; Ismail, A. F.; Matsuura, T.; Montazer-Rahmati, M. M. Sep. Purif. Technol. 2010, 75, 229.

percentage of average absolute relative error percentage of absolute relative errors percentage of relative error outer diameter or diameter diffusivity geometric factor accounting for the effect of dispersion shape needed correction of Maxwell expression length molecular weight particle shape factor number needed correction of Maxwell expression permeability gas constant average diameter filler or particle radius solubility interphase thickness or membrane thickness temperature

Greek Letters

α α α α β β′ δ θ λ ξ σ ϕ

permselectivity or selectivity permeability ratio flake aspect ratio aspect ratio of tubular fillers, α = l/d reduced permeation polarizability or chain immobilization factor permeability reduction factor ratio of outer radius of interfacial shell to a core radius orientation angle permeability ratio thickness of polymer layer standard deviation volume fraction or filler loading

Superscripts

exp est

experimental estimated AO


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