Close
About
FAQ
Home
Collections
Login
USC Login
Register
0
Selected
Invert selection
Deselect all
Deselect all
Click here to refresh results
Click here to refresh results
USC
/
Digital Library
/
University of Southern California Dissertations and Theses
/
Experimental studies and computer simulation of the preparation of nanoporous silicon-carbide membranes by chemical-vapor infiltration/chemical-vapor deposition techniques
(USC Thesis Other)
Experimental studies and computer simulation of the preparation of nanoporous silicon-carbide membranes by chemical-vapor infiltration/chemical-vapor deposition techniques
PDF
Download
Share
Open document
Flip pages
Contact Us
Contact Us
Copy asset link
Request this asset
Transcript (if available)
Content
EXPERIMENTAL STUDIES AND COMPUTER SIMULATION OF THE PREPARATION OF NANOPOROUS SILICON-CARBIDE MEMBRANES BY CHEMICAL-VAPOR INFILTRATION/CHEMICAL-VAPOR DEPOSITION TECHNIQUES by Ryan Mourhatch A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMICAL ENGINEERING) August 2010 Copyright 2010 Ryan Mourhatch ii Dedication To my dear parents Eddie and Frances Mourhatch for believing that I could do whatever I set out to accomplish, and instilled in me the love of learning, and to my brothers Ramoun and Ramses Mourhatch for their support and encouragement. iii Acknowledgments I would like to express my deep and sincere gratitude to my supervisors, Professors Muhammad Sahimi and Theodore T. Tsotsis, for giving me the opportunity to work under their supervision. Their wide knowledge and logical way of thinking have been of great value for me, and their personal guidance has provided a good basis for the present thesis. I wish to express my warm and sincere thanks to Dr. Grace Lu, for serving on my dissertation committee. I thank her for her support and guidance. I would also want to thank Drs. Katherine S. Shing and Edward Goo for serving on my qualifying exam committee. I warmly thank Drs. Bahman Elyassi and Feng Chen for their valuable advice and friendly help. Their extensive discussions with me regarding my work were very helpful for this study. My warm thanks also go to my colleague and officemate for the last 5 years, Jocelyn Lee. I would also like to thank all my colleagues friends in our research group who have helped me in different ways: Drs. Taewook Kim, Megha Dadwhal, Aadesh Harale, Hyun Hwang, and Nafiseh Rajabbeigi, and Mr. Sarawut Thongsai and Mr. Yousef M. Hashemi I wish to thank Ms. Tina Silva for her kindness and support. Special thanks are due to the administrative staff of the Mork Family Department of Chemical Engineering and Materials Science, namely, Ms. Karen Woo, Ms. Heather Alexander, and Mr. Brendan Char for all their help and support throughout my graduate studies. iv I owe my loving thanks to my parents and brothers. Without their encouragement and understanding it would have been impossible for me to finish this work. My special gratitude is due to Henry Mourhatch, Mariet Mourhatch, Ramseen, Rameil, and Irving Betneisan for their loving support. v Table of Contents Dedication ii Acknowledgments iii Table of Contents v List of Tables vii List of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Principles of Chemical Vapor Deposition 1 1.1.1 The CVD Precursors 6 1.2 Membranes 8 1.2.1 Gas Separation using Inorganic Membranes 15 1.2.2 Membrane Reactors 23 1.2.3 Separation Mechanisms in a Porous Membrane 25 1.2.4 Inorganic Membranes Prepared using the CVD Techniques 28 1.2.5 SiC Membranes 30 1.3 Modeling 36 1.3.1 Pore Network Modeling 36 Chapter 2: Fabrication of Microporous Silicon-Carbide Membranes 41 2.1 Introduction 41 2.2 Support Preparation 41 2.3 Preparation of the Membranes 45 2.3.1 Flat-disk membranes 46 2.3.2 Tubular membranes 50 Chapter 3: A Continuum Model of Membrane Preparation 63 3.1 Introduction 63 3.2 Model Development 64 3.3 Results and Discussion 72 3.4 Summary and Conclusions 76 vi Chapter 4: A Network Model of Membrane Preparation 78 4.1 Introduction 78 4.2 Background 79 4.3 Formulation of the Model 82 4.3.1 Inside the membrane 83 4.3.2 The tube side 91 4.3.3 The shell side 92 4.4 Numerical Simulations 92 4.5 Results and Discussion 97 4.6 Extension to Binary Gaseous Mixtures 108 4.7 Summary 109 Chapter 5: Determination of the True Pore Size Distribution by Flow 110 Permporometry Experiments: An Invasion Percolation Model 5.1 Introduction 110 5.2 Flow Permporometry Experiments 113 5.3 Invasion Percolation Model of Flow Permporometry 118 5.4 Computing the Dry and Wet Curves 121 5.5 Computing the Correct Pore Size Distribution 126 5.6 Results and Discussion 129 5.7 Summary 136 Chapter 6: Fabrication of Silicon Carbide Nanotubes 138 6.1 Introduction 138 6.2 Preparation Technique 140 6.3 Characterization 142 References 148 vii List of Tables Table 1.1 Filtration through size exclusion. 9 Table 1.2 Melting points of ceramics. 14 Table 1.3 Microporous inorganic membranes for gas separation. 16 Table 1.4 CO 2 /N 2 separation using ceramic membranes. 20 Table 1.5 Hydrogen separations from gas mixtures using 22 microporous membrane. Table 2.1 The details of the conditions for all the CVD experiments. 55 Table 6.1 Summary of BET results for SiC nanotubes. 145 viii List of Figures Figure 1.1 Sequence of events during the deposition (after Pierson, 1992). 4 Figure 1.2 Schematic of various membrane types (after van Rijin, 2004). 10 Figure 1.3 Simplified schematic drawing of membrane reactor. 23 Figure 2.1 Argon permeance vs. average pressure. 45 Figure 2.2 Schematic of flat disk SiC membrane experimental setup. 47 Figure 2.3 The permeance of the inert gases vs. injection time for flat-disk 49 membranes (after Chen et al., 2008a). Figure 2.4. The He/Ar selectivity vs. time (after Chen et al., 2008a). 50 Figure 2.5 Schematic drawing of the stainless steel apparatus for 51 tubular membranes. Figure 2.6. The support for the SiC membrane before (left) and 52 after (right) glazing. Figure 2.7 Pyrolysis of the TPS. 53 Figure 2.8 Reduction in the permeance of Ar and He during deposition 55 for Experiment 1. Figure 2.9 The He/Ar separation factor vs. deposition time for Experiment 2. 56 Figure 2.10 Reduction in the permeance of Ar and He during the 57 deposition in Experiment 2. Figure 2.11 The He/Ar separation factor vs. deposition time for the 58 experiment 2. Figure 2.12 He and Ar permeance Vs. deposition time in 59 Experiment 3 (after Chen et al., 2008a) . ix Figure 2.13 The He/Ar separation factor vs. deposition time in 60 Experiment 3 (after Chen et al., 2008a). Figure 2.14 He and Ar Permeance vs. injected amount of the TPS 60 in Experiment 3. Figure 2.15 He and Ar permeance vs. temperature (after Chen et al., 2008a). 61 Figure 2.16 Indication for Knudsen flow for argon (after Chen et al., 2008a). 62 Figure 2.17 The He/Ar membrane selectivity vs. Time. 62 Figure 3.1 Dependence of the relative porosity of the tubular 73 membrane on the distance r from the support’s top surface, at several deposition times t (after Chen. et al., 2008a). Figure 3.2 The SEM image of the cross section of the SiC membrane. 74 The darker area near the top surface is the membrane layer. Figure 3.3 Comparison of the experimental data for the Ar permeance with 75 the results of the numerical simulations(after Chen et al., 2008a). Figure 3.4 Comparison of the experimental data for the Ar permeance with 76 the results of the numerical simulations (after Chen et al., 2008a). Figure 4.1 Schematic of the model of the tubular membrane. 82 Figure 4.2 Comparison of the experimental data for the pore size distribution 85 with the pore size distribution generated by Equation (4.1). Figure 4.3 Dependence of the gases’ permeance on the deposition time. 97 Figure 4.4 Comparison of the experimental data for the Ar permeance in the 98 membrane with the results of the numerical simulations for several values n, the order of the reaction that produces the SiC. Figure 4.5 Time-dependence of the Thiele modulus in a pore with a size 100 equal to the average of the PSD, for first- and second-order kinetics. Figure 4.6 Comparison of the computed and measured selectivity of 101 the membrane. Figure 4.7 Dependence of the average pore size on the deposition time. 103 x Figure 4.8 a,b Evolution of the pore size distribution with the time as the 104 deposition and, hence, pore shrinkage and plugging, proceed. Figure 4.9 Time-dependence of the fraction of the pores that are closed 105 to the TPS. Figure 4.10 Effect of the membrane connectivity on the Ar permeance during 106 the deposition. Figure 4.11 Dependence of the Ar permeance on the membrane’s thickness. 107 The connectivity of the pore network is Z p = 6. Figure 5.1 Schematic of the experimental apparatus. 115 Figure 5.2 Comparison of synthetic wet and dry curves with the curves 130 calculated by IP (r a =45nm). Figure 5.3 Comparison of FPP-PSD with IP PSD (r a =45nm). 131 Figure 5.4 Comparison of synthetic wet and dry curves with the curves 132 calculated by IP (r a =100 nm). Figure 5.5 Comparison of FPP-PSD with IP PSD (r a =100nm). 132 Figure 5.6 Comparison of experimental wet and dry curves with the curves 133 calculated by IP method. Figure 5.7 The progression of the convergence of the computed wet curve 134 toward the experimental one. Figure 5.8 Comparison of FPP-PSD with IP PSD (experimental data). 135 Figure 5.9 PSD computed at various stages of the simulations, using the 136 synthetic data of Figure 5.2. Figure 6.1 TEM pictures of SiC nanotubes. 142 Figure 6.2 BET Isotherm plot for SiC Nanotubes. 146 Figure 6.3 XRD analyses for SiC Nanotubes. 147 xi Abstract Silicon carbide (SiC) is a material with very attractive properties. Its excellent mechanical strength, high chemical stability and thermal conductivity, strength and abrasion resistance at high temperatures, low thermal expansion, biocompatibility, and resistance to acidic and alkali environments, have made SiC a great candidate for use as material for the preparation of high-temperature membranes. The focus of the present Thesis is to prepare SiC microporous membranes by a chemical-vapor deposition technique (CVD), using tri-isopropylsilane as the precursor and Ar or He as inert carrier gas. In the experiments we prepared membranes with a He permeance ~10 −7 mol/m 2 .s.Pa and a He/Ar ideal separation factor ~10.0. We envision the SiC membranes that we prepare to be eventually utilized in reactive separation applications with the water-gas shift and methane steam reforming reactions, where the membrane must function in the presence of high-temperature steam. A pore network model was also developed in order to describe the preparation of microporous SiC membranes by the CVD technique. The membrane's pore space was represented by a three-dimensional network of interconnected pores, in which the effective size of the pores was distributed according to a pore size distribution. The chemical reaction, the various transport processes, and the evolution of the pore sizes due to the deposition of SiC on the pores’ internal surface during the CVD process were included in the model. The Maxwell-Stefan equations were used for describing the pore- xii level transport processes, which include the Knudsen and hindered diffusion, as well as viscous flow. The effect of pore blockage was also taken into account. The simulator monitors the PSD as the membrane's structure evolves. Also computed was the carrier gas' permeance during the CVD process. Good agreement was found between the simulation results and our single-gas experimental permeation data. The results also indicate the fundamental significance of the pore blockage (i.e., the percolation effect) to the evolution of the membrane's structure. We also used the CVD technique to prepare SiC nanotubes and to characterize their properties using the BET, XRD, and TEM techniques. The SiC nanotubes have a number of potential uses as components of mixed-matrix membranes and catalyst supports, as well as for electronic and sensor applications. In the preparation of the SiC nanotubes we utilized as substrates, Anopore™ inorganic membranes (Anodisc™), which are composed of a high purity alumina matrix prepared by the electrochemical anodization of aluminum. 1 Chapter 1 Introduction 1.1 Principles of Chemical Vapor Deposition Chemical vapor deposition (CVD) has grown rapidly during the last two decades, and currently has many applications in the fabrication of semiconductors, membranes, and of various optical materials. The CVD is defined as the deposition of a solid on a heated surface as a result of a chemical reaction that takes place in the vapor phase. It belongs to the class of vapor transfer processes, which are typically atomistic in nature, meaning that the deposition species are atoms or molecules or a combination of both (Pieson, 1992). The reasons that the CVD has been constantly expanding its range of applications and continues to attract attention are as follows: • The CVD is a relatively flexible and uncomplicated technique, which can accommodate many process variations. • By using the CVD technique, it is possible to coat objects of any desired shape or size. • It is economically competitive with most other coating processes. 2 The CVD involves a number of phenomena, including flow, reaction, and mass and heat transfer, and its study involves a combination of various scientific and engineering disciplines, including thermodynamics, kinetics, fluid dynamics, mass and heat transfer, plasma physics, and chemistry. Several chemical reactions often take place during the CVD, such as thermal decomposition (pyrolysis), reduction, hydrolysis, disproportionation, oxidation, carburization, and nitridization. They occur either in series or in parallel. The net outcome of all the reactions is a reaction pathway that converts the vapor precursor into a desired solid. A key goal of the design of the CVD reactors and processes is to force the reaction to happen only where and when it is desired, typically on the surface of a pre- existing substrate. Undesired reactions must be avoided, as they may result in the formation of particles in the gas phase, which can collect on the substrate surface, coat the reactor walls, or clog the reactor exhaust openings. To achieve the selective deposition onto a substrate surface and in order to avoid the nonselective particle formation, one must appropriately control the temperature, pressure, and time of the deposition, as well as the substrate surface specificity (for further details, see http://www.timedomaincvd.com). The conventional CVD process is characterized by complex fluid dynamical processes. The fluid (typically, one or more gases) is forced through pipes, valves and various chambers and, at the same time, is exposed to large temperature/concentration gradients as it comes into contact with the substrate where the reaction takes place. The 3 reaction is typically heterogeneous, which means that it involves a change of state, from gaseous into a solid. In some cases, the reaction may take place before the substrate is reached while the reactant is still in gas phase (gas phase precipitation). The sequence of events that lead to the deposition of solid films in the CVD is described in Figure 1.1 (the slowest step determines the rate of deposition), and is summarized as follows: • The reactant gases enter the reactor. • They diffuse through the boundary layer. • They are adsorbed on the surface of the substrate, where the surface reaction happens. • Gaseous by-products are desorbed from the surface, diffuse across the boundary layer, and are carried away by the flow of the main reactant gases. 4 Figure 1.1 Sequence of events during the deposition (after Pierson, 1992). The growth rate of the deposited layer can be limited by either the surface reaction kinetics, the mass transport, or by the gas-phase kinetics. When the surface reaction is controlling, the rate of growth is dependent upon the reactant gas-phase concentration. When mass transport is the rate limiting step, the reactor flow characteristics are also important, as they determine the transport of the reactant towards, and of the by-products away, from the surface of deposition. Several types of the CVD processes are in widespread use and are frequently referenced to in the literature. Classified by the operating pressure, they include: 5 • Atmospheric pressure CVD (APCVD). • Low-pressure CVD (LPCVD), for which the CVD processes occur at sub- atmospheric pressures. The reduced pressures typically tend to reduce unwanted gas-phase reactions and improve film uniformity. • Ultrahigh-vacuum CVD (UHVCVD) Classified on the basis of the physical characteristics of gas/vapor phase, the CVD processes are classified as: • Aerosol-assisted CVD (AACVD) • Direct liquid injection CVD (DLICVD) It is possible to control the nature of a CVD-derived structure by the proper manipulation of the various deposition parameters, including the temperature, pressure, super-saturation and, of course, the selection of the CVD precursors and reaction. For example, pressure controls the thickness of the boundary layer and, consequently, the degree of diffusion. By operating at low pressure, the diffusion process is enhanced, and surface kinetics then becomes rate-controlling. Under such conditions, the deposited structures tend to be fine-grained, which is usually a desirable situation. 6 In addition to operating under low pressures, fine-grained structures can also be obtained at low temperatures, and under high super-saturation. On the other hand, at higher temperatures, deposits tend to be more columnar, as a result of uninterrupted grain growth from the reactant source. The structure is also often dependent on the thickness of the deposit. For instance, grain sizes increase as the thickness increases and a columnar- grain structure develops. 1.1.1 The CVD Precursors The choice of the proper reactant - the precursor - is very important. The precursors fall into several general major groups, which include halides, carbonyls, metallo-organics, and hydrides. The choice of an appropriate precursor is governed by certain general characteristics that it must have, which is summarized as follows: • Stability at room temperature • Ability to react cleanly in the reaction zone • Sufficient volatility at low temperatures, so that it can be easily transported to the reaction zone without condensing in the lines. • Possibility of being produced with a very high degree of purity. • Ability to react without producing side or parasitic reactions. 7 The halides are binary compounds of halogen (fluorine, chlorine, bromine and iodine - all the resulting halides have been used in the CVD reactions) and a more electropositive element, such as a metal. Most halides are gaseous or liquid at room temperature and can be easily transported to the reaction chamber. Because of their reactivity, they readily attack most materials, which make them difficult to handle and requires equipment consisting of inert material, such as Monel or Teflon. The metal carbonyls are a large and very important group of compounds, which are used widely in the chemical industry, particularly in the preparation of heterogeneous catalysts and as precursors in the CVD and metallo-organic CVD (MOCVD). The carbonyls are relatively simple compounds and consist of only two basic components, namely, the carbonyl group and a d-group transition metal. Metal carbonyls form several complexes, but those of major interest in the MOCVD are the carbonyl halides and the carbonyl nitric oxide complexes. The use of metallo-organic compounds in the CVD was first reported in the 1960s for the deposition of indium phosphate and indium antimonite. Such compounds are molecules in which a non-transitional element (from groups IIa, IIb, IIIb, IVb, Vb and VIb) is bonded to one or more carbon atoms of an organic hydrocarbon group (mostly alkyls and acetylacetonates). The metallo-organics complement the halides and carbonyls, which are the precursors for the deposition of transition metals and their compounds. The term metallo-organic is used somewhat loosely in the CVD, since it includes compounds of such elements as silicon, phosphorous, arsenic, selenium and 8 tellurium, some of which are definitely non-metallic. The main advantage of using the metallo-organics is their lower deposition temperature. 1.2 Membranes van Rijin (2004) provided a comprehensive review of the subject In what follows we follow them and describe and discuss the issues. Membranes play an essential role not only in nature, but also in today’s modern industrial world. In the chemical industry, for example, one often encounters the problem of separating mixtures, and membranes can, in principle, carry out most of the separation and provide an economic alternative to the more conventional separation processes, such as distillation, extraction, adsorption, etc. The advantages that membranes offer compared to traditional methods include (van Rijn, 2004): (i) low energy consumption; (ii) a continuous mode of separation; (iii) scalability, and (iv) broad applicability. Based on their average pore size, one can divide membranes into four groups that are described in Table1.1 (van Rijn, 2004). 9 Table 1.1 Filtration through size exclusion. Note, that a microporous membrane is defined by the International Union of Pure and Applied Chemistry (IUPAC) as one with a pore diameter < 2.0 nm. Such membranes are also called nanoporous, because their pore diameters are in the nanometer range. However, for consistency with the IUPAC nomenclature, the term microporous is used in this Thesis. The structure of the membrane plays an important role in its performance. Membranes can be either symmetric or asymmetric. In symmetric membranes the structural properties of the membrane are constant along its thickness. Asymmetric membranes are composed of a thin selective layer, which is placed on a strong support layer that provides the mechanical strength. A schematic of several types of membranes is shown in Figure 1.2. 10 Symmetric Asymmetric Figure 1.2 Schematic of various membrane types (after van Rijin, 2004). Inorganic (ceramic and metal) and polymeric membranes are the two most common types of membranes, each with its own advantages and disadvantages. In the last two decades membrane-based processes have become important new unit operations in the chemical and petroleum industry; in gas separations where they are utilized in such processes as nitrogen production, refinery hydrogen recovery, the production of CO 2 for enhanced oil recovery, sour-gas treatment, organic solvent recovery, air dehydration, oxygen enrichment, and ammonia purge gas purification, among others. Virtually all gas- separation processes currently utilize polymeric membranes, which is due to their low cost and sufficient gas selectivity for most important applications. A downside of such membranes is that they do not perform well under harsh environments (for example, at high-temperature and pressure, or in the presence of corrosive gases), and their stability is suspect under such conditions. Due to their superior physical and chemical properties 11 under harsh conditions, inorganic membranes have been attracting considerable attention during the last decade. However, Pd-alloy membranes are the only of such membranes that are currently used for industrial gas separations (Yan et al., 1996). Most ceramic membranes have qn asymmetric structure with a thin porous layer (typically 1 to 5 µm thick) and a thicker porous support, which is made of sintered ceramic particles (alumina, titania, zirconia, and SiC). Experience has shown that the preferred shape of the ceramic particles constituting the ceramic membrane support is rod-like, since flat discs are brittle. The advantages of ceramic membranes are that they are thermally stable, mechanically strong, and very resistant chemically, as previously noted. Mesoporous alumina membranes (and many types of other inorganic membranes), for example, utilize α-alumina tubes as supports with a pore size of to 110-180 nm (the support tubes also have important liquid-phase applications such as microfiltration (MF) or ultrafiltration (UF) membranes). In order to form mesoporous membranes, a γ-alumina layer is applied on the top of the α-alumina support by means of dip-coating a boehmite (γ-AlOOH) sol (Yoldas, 1975), which subsequently solidifies through drying, yielding a porous γ-alumina film, which is further calcined through heating for several hours at high temperatures (up to 700 °C). The duration and temperature of the calcination treatment determine the final pore size which, for γ-alumina, is typically 3 - 7 nm. Each dip-coating and subsequent controlled drying step forms a layer with a thickness of around 0.5 µm (Sea et al., 2001; see also Lin et al., 1994; Guizard et al., 1994). 12 Microporous membranes need additional layers. Various methods are available for forming these films. For example, CVD is a common technique for preparing microporous silica membranes using various precursors (e.g., tetraethoxysilane (TEOS) and methyltriehoxysilane (MTEOS)). During the CVD process the silica forms within the pores of the γ-alumina layer. Silica microporous membranes are also fabricated by the sol-gel deposition methods (Raman et al., 1995). The advantages of ceramic membranes over their polymeric counterparts include (the latter membranes have a key advantage which is their significantly lower cost): • Ceramics do not typically adsorb large amounts of water or other organic vapors, and if they do they do not swell. Swelling is a common problem with many polymeric materials, impacting membrane selectivity and overall performance. • They are thermally stable. The membranes allow processes to be run at high temperatures (see also below), which is beneficial for many applications, such as the filtration of viscous fluids like oils, because their viscosity decreases. • They are wear-resistant and physically hard. This allows operating under cross- flow conditions, which is beneficial for the removal of particles from solutions or for the removal of the cake layers formed on the membranes during such operations, without the danger of damaging the membrane. 13 • They are (particularly silicon carbide and titania) resistant to various corrosive chemicals, which makes it easy to filter solutions that contain the chemical, as well as to clean the membranes without fear that they will be destroyed. There is a host of materials that is currently used for fabricating porous ceramic membranes, but the majority of them are made of metal oxides. Common oxides utilized include Al 2 O 3 , ZrO 2 , TiO 2 , and SiO 2 , and their mixtures. Other materials used include various zeolites, SiC, silicon and aluminum nitrides, and microporous carbons. As noted above, inorganic materials generally show superior chemical and thermal stability relative to polymeric materials. For example, polymers can be applied over a temperature range from room temperature to 300 °C (though they are most commonly applied at temperatures < 100 °C), while most of the aforementioned ceramics operate in a much broader range of temperatures, see Table 1.2 below that shows their melting points. As previously noted, the high temperature resistance makes such materials very attractive for gas separation in combination with a chemical reaction. 14 Ceramic Melting Point (°C) Alumina Al 2 O 3 2050 Zirconia ZrO 2 2770 Titania TiO 2 1605 Silicon Carbide SiC 2500 Table 1.2 Melting points of ceramics. Most common polymeric membrane materials have, in addition, limited stability towards acidic and alkaline solutions, as well as organic liquids. The inorganic materials are superior in that regard. This is important in liquid-phase UF and MF processes where, as a result, inorganic membranes have found most of their applications. In such processes fouling is an important challenge, since it leads to a drastic decrease of the flux so that frequent cleaning is, therefore, necessary. Inorganic membranes offer an important advantage in that they can tolerate much harsher cleaning agents than their polymeric counterparts. 15 1.2.1 Gas Separation using Inorganic Membranes Bernardo et al. (2009) provided a comprehensive review of the subject. In what follows we follow them and describe and discuss the issues. Table 1.3 summarizes some key studies on gas separation applications (other than those involving hydrogen and CO 2 separations, see below) of microporous inorganic membranes during the last decade or so. Microporous inorganic membranes are also finding applications in liquid/vapor applications (e.g., microporous zeolite membranes for anhydrous ethanol recovery applications, see Lin et al., 2000, 2001, 2003). Two important gas separation applications that have recently been the focus of many studies are hydrogen purification and carbon dioxide removal. The first application relates to the use of H 2 as an alternate clean fuel (see, for example, Asaeda et al., 2001; Gavalas et al., 1989; Kanezashi et al., 2005, 2006; Lee et al., 2004; Sushil et al., 2006), while the second application is driven by the need to capture and sequester CO 2 (Bredesen et al., 2004). 16 Membrane material Synthesis method Permeation data available Reference Polymeric sol-gel He, H 2 , CO 2 , CH 4 , C 3 H 6 Uhlhorn, et al. 1989 Colloidal/Polymeric sol-gel He, Ar, N 2 , CH 4 , C 2 H 4 C 2 H 6 , C 3 H 6 , C 3 H 8 Asaeda, et al. 2006 Polymeric sol-gel He, H 2 , CO 2 , N2, CH 4 , C 3 H 6 , i-C 4 H 10 Krishna and Wesselingh, 1997 Surfactant template sol-gel He, N 2 , CO 2 , CH 4 , SF 6 Raman, 1995 Colloidal sol-gel H 2 , N 2 , CO 2 , CH 4 , C 2 H 6 , C 3 H 8 Asaeda, et al. 2002 Polymeric sol-gel H 2 , C 3 H 8 Sato, et al. 2007 SiO 2 Polymeric sol-gel H 2 , N 2 Li, et al. 2001 Table 1.3 Microporous inorganic membranes for gas separation. 17 Table 1.3: continued. Membrane material Synthesis method Permeation data available Reference Polymeric, colloidal sol-gel N 2 , CO 2 Yoshioka, et al. 2001 CVD N 2 , CO 2 Cuffe, et al. 2006 Thermal decomposition of TEOS H 2 , N 2 , CO, CO 2 Brunetti, et al. 2007 Koresh, and Soffer 1983 Jones, and Koros 1994 Hatori, et al. 1992 Carbon Pyrolysis O 2 , N 2 Suda, et al. 1995 18 Table 1.3: Continued. SiC Pyrolysis He, Ar, N 2 , H 2 , CO 2 , CH 4 , Elyassi, et al. 2007 Membrane material Synthesis method Permeation data available Reference SiC CVD/CVI He, Ar, N 2 , H 2 , CO 2 , CH 4 , Ciora, et al. 2004 SiO 2 -ZrO 2 Colloidal sol gel He, H 2 , N 2 , O 2 , Tsuru, et al.1998 ZrO 2 Polymeric sol-gel H 2 Gu, et al. 2001 TiO 2 Polymeric sol-gel N 2 , CO 2 Abidi, et al. 2006 SiO 2 -ZSM-5 Hydrothermal He, H 2 , CO 2 , O 2 , SF 6, Bonhomme, et al. MFI Hydrothermal H 2 , Propane Julbe, et al. 1994 ZSM-5 Hydrothermal N 2 , CO 2 Shin, et al. 2005 Zeolite T Hydrothermal He, H 2 , CO 2 , O 2 , N 2, CH 4 , C 3 H 8 Cui, et al. 2004 19 Hydrogen separations require molecular-sieving materials with high selectivity (Kanezashi et al., 2005, 2006). For CO 2 separations high selectivity can be attained not only by molecular sieving, but also based on the interaction of the permeating molecules with the pore surface. This is true as well with other condensable gases. For example, zeolite-MFI membranes have also been utilized for the separation of difficult mixtures, such as n/i-C 4 H 10 , where one of the components in the mixture condenses within the pores and blocks passage for the other component, thus improving the separation factor of the membrane (Motuzas, 2007). Table 1.4 lists some of the key studies on the use of inorganic membranes for CO 2 separation from various gas mixtures (see Bernardo, Drioli, and Golemme, 2009, for more details), which represents an industrial application where polymeric membranes find common use at lower temperatures. The interest in inorganic membrane arises from the potential of performing the same separation at high temperatures and in the presence of other flue-gas components (e.g., hydrogen sulfide and tar), thus avoiding the high energy costs of cooling down the effluent, and heating it up again for further processing. 20 Membrane Temperature (K) ΔP (atm) S.F. CO 2 /N 2 Mechanism Reference γ-Alumina 293 0.73 4.50 Knudsen Abidi, et al., 2006 Silica ZSM-5 298 1.09 2.95 Surface Diffusion Bonhomme, et al., 2003 Zeolite FAU 473 0.86 4.60 Surface Diffusion Gu, et al., 2001 Zeolite-FAU/NaY 313 1 20 Selective adsorption +surface diffusion Kusakabe, et al., 1998 Si-NH 2 298 1 80 Molecular sieveing Xomeritakis, et al., 2007 Table 1.4 CO 2 /N 2 separation using ceramic membranes 21 Table 1.5 lists some of the key studies on the use of microporous membranes for hydrogen separation. Several types of membranes have been utilized. Silica membranes prepared by the CVD- chemical vapor infiltration (CVI) technique were the earliest to be used. They exhibit high permselectivity towards hydrogen, but generally low permeation rates, since they tend to consist of nearly dense silica plugs inside the pores of the supporting structure (Benes et al., 2000). Sol-gel processing allows for fine-tuning of the microporous silica layer through the careful choice of the coating solution concentration and temperature, and the coating, drying, and sintering procedures (Kitao et al., 1990; Lee et al., 2004; Verkerk et al., 2001). Silica and silica-zirconia membranes have been successfully utilized for hydrogen separation in steam reforming reactors at 500 °C (Kanezashi, 2006). More recently, metal-doped silica membranes have also been developed with high flux and separation factors, and good stability in steam (Kanezashi, 2005). Zeolite membranes, when used for hydrogen separation, should be carefully prepared in order to avoid intercrystalline pores through which the small hydrogen molecule could pass. When pinhole-free zeolite (e.g., ZSM-5) films are prepared, they usually exhibit high separation factors towards hydrogen (Li et al., 2002). Aluminophosphate zeolite materials have also been recently investigated (Sklari et al., 2007) whereby the calcination temperature of the membrane is the most important factor that controls the process performance as the layer is grown in-situ on the support. 22 Membrane Temp. (K) ΔP (atm) Mixture S.F. Reference SiO 2 473 3 H 2 /C 3 H 6 156 Uhlhorn et al, 1991 SiO 2 543 3 H 2 /N 2 185 Kitao, et al.1990 SiO 2 473 3 H 2 /CH 4 30 Lee, et al. 2004 SiO 2 423 1 H 2 /N 2 76 Verkerk, 2001 H 2 /N 2 19 SiO 2 /Stain less steel 564 4 H 2 /CO 2 35 Brinetti, 2007 Ni-doped SiO 2 773 0.90 H 2 /N 2 400 Kanezashi, 2006 Zeolite LTA 293 0.20 H 2 /C 3 H 8 9.17 Li, et al. 2006 Al 100 P 60 O z 303 2 H 2 /C 3 H 8 15.5 Sklari, 2007 Table 1.5 Hydrogen separations from gas mixtures using microporous membrane. 23 1.2.2 Membrane Reactors As noted on more than one occasion so far, inorganic membranes exhibit a distinct advantage in applications in membrane reactors (MR). A schematic of a simple such reactor is shown in Figure 1.3 (Fogler, 2006). Figure 1.3 Simplified schematic drawing of a membrane reactor. Membrane reactors combine reaction with separation in one unit with the goal of increasing conversion. In the conventional MR one of the products of the reaction is removed from the reactor through the membrane, which for equilibrium-limited reactions leads to enhanced conversions according to the Le Chatelier’s Principle. Catalytic MR are applied to a broad range of reactions, including dehydrogenation and dehydration reactions. The membrane may itself be catalytic (e.g., zeolites and Pd), or rendered catalytic by impregnation with active components. Most often than not, though, the membrane is inert and does not participate in the reaction directly, but simply acts as a 24 barrier to the reactants and to some of the products. The catalytic action is provided instead by catalyst pellets placed in the reactor in contact with the membrane. One area where catalytic the MR show good potential is in hydrogen production. Hydrogen is mainly produced industrially by the steam reforming reaction (SMR) of hydrocarbons, especially methane which is a key component natural gas. The steam reforming of methane is described by the following reactions (Sugawara et al., 2006): CH 4 + H 2 O CO + 3H 2 ΔH=206 KJ/mol (1.1) CO + H 2 O CO 2 + H 2 ΔH=-41 KJ/mol (1.2) Membrane reactors have been previously applied to both reactions (Sanchez and Tsotsis, 2002) utilizing mostly metal (Pd and its alloys) membranes. Using such noble metal membranes is dictated by the fact that, other available membranes (such as polymeric ones) cannot withstand the high temperatures required, are sensitive to steam (e.g., SiO2 membranes), or are not permselective enough towards H2 (e.g., Al2O3 or zeolite membranes) under the prevailing high temperature, pressure, and steam conditions. Noble metal membranes perform well (Sanchez and Tsotsis, 2002). They are, however, costly, susceptible to deactivation from coke and sulfur impurities, and must be handled with great care to avoid embrittlement and mechanical failure. The SiC membranes, the topic of this Thesis, exhibit good potential for the SMR and WGS/MR applications. 25 1.2.3 Separation Mechanisms in a Porous Membrane Baker (2004) provided a comprehensive review of the subject. In this section we summarize their paper and describe and discuss the various issues. The transport through membranes is expressed in terms of the “permeance” or the pressure normalized flux, that is, the transport flux per unit of trans-membrane pressure gradient (kmol/m 2 .s.kPa). The data are also described in terms of the permeability, which is equal to the permeance multiplied by the membrane’s thickness (mol.m/m 2 .s.Pa). The driving force for transport through microporous membranes is the chemical potential gradient (Baker, 2004; Bowen et al., 2004; Bredesen et al., 2004; Koriabkina et al., 2005; Koukou et al., 1999). So, although we often think of the flux being proportional to the pressure gradient, most commonly that is not the case. A better way to fit the experimental data is through an alternative, semi-empirical expression (Burggraaf et al., 1991, 1996). (1.3) the exponent b, expressing the dependence of permeance on the dimensionless pressure P d (=P/P ref - the reference pressure P ref is chosen as a typical or average pressure) is a measure of the extent of departure from Fickian transport (e.g., b=1 for completely viscous flow, and b=0 for Knudsen diffusion). The pre-factor is temperature- dependent, particularly for activated transport through microporous membranes. The selectivity of a membrane, α, is defined as: 26 (1.4) where C P and C F are the mass concentrations of component i or j in the permeate and the feed mixture, respectively (Burggraaf et al., 1991,1996). In addition to convective (viscous) flow, several different types of diffusion processes participate in transport in porous membranes, namely, bulk, Knudsen, configurational, and surface diffusion. In some cases, molecules can move through a membrane by more than one mechanism. Separation of a certain species in a mixture through the membrane may come as a result of different rates of component transport through the membrane. In some cases the membrane pores are so small that they only allow one species to go through, while excluding all other species in the mixture. This is known as molecular sieving. We also described above the phenomenon of pore blockage through molecular condensation, which also is highly effective in preventing transport of all other molecules in the mixture. Knudsen diffusion prevails in pores with sizes that are significantly smaller than the molecular mean free-path. The Knudsen number (K n ), defined as the ratio of the mean free path of the gas molecules to the pore radius, determines whether Knudsen diffusion prevails. This is typically the case for Knudsen numbers larger than 10. Selectivity is 27 limited during Knudsen transport with permeance ratios being inversely proportional to the square root of the ratio of the molecular weights of the gasses involved. For even larger K n (where the radius of the pores is close to the size of the molecule), the configurational or hindered diffusion prevails. The selectivity under such conditions is much higher, depending strongly on the pore size distribution, temperature, pressure and the interactions between the gases being separated and the membrane surfaces. Under such conditions, molecular sieving or pore condensation often becomes the determining factors. For pores with small K n (for microporous membranes, small Knudsen numbers are associated with pinholes and defects) bulk diffusion, and for extremely large pores viscous flow, which is non-selective, prevail significantly impacting membrane performance characteristics. Surface diffusion often occurs in parallel with Knudsen and configurational diffusion. In fact, in pores with dimensions close to the molecular size it is often difficult to separate hindered and surface diffusion. During surface diffusion, gas molecules are adsorbed on the pore walls of the membrane and subsequently migrate along the surface. Surface diffusion tends to increases the permeability of the components adsorbing more strongly to the membrane pores. At the same time, the effective pore diameter is reduced. Consequently, transport of the non-adsorbing components is reduced and selectivity is high. For low temperatures and/or high enough pressures capillary condensation may occur, whereby a condensed phase fully (or partially) fills the membrane pores. If the pores are completely filled with the condensed phase, only the species soluble in the condensed phase can permeate through the membrane. Fluxes and selectivities are 28 generally high under such conditions. The appearance of capillary condensation strongly depends on gas composition and pore size in addition to temperature and pressure. 1.2.4 Inorganic Membranes Prepared using the CVD Techniques Li (2007) provided a comprehensive review of the subject, which is summarized in what follows. As already noted, there have been several prior efforts in preparing inorganic microporous membranes using the CVD techniques. Gavalas et al. (1989) were the first to report the preparation a hydrogen-permselective silica membrane on the surface of a porous Vycor® glass tube, using the CVD technique. Smith et al. (1994) used the TEOS as the precursor to prepare H 2 -permselective SiO 2 membranes. Kusakabe et al. (1994, 1995) and Yan et al. (1996) produced silica membranes by thermal decomposition of the TEOS by evacuating the reactant through the porous support, which led to the plugging of the macropores. The same group, in a later study, used a different precursor, phenyl-substituted ethoxysilicanes, to produce membranes with improved properties (Morooka et al., 1997). Nomura et al. (2005) prepared silica membrane by the counter-diffusion CVD technique, with an excellent H 2 /N 2 permeance ratio of > 800. The membrane also exhibited good steam stability with the H 2 /N 2 permeance ratio, staying stable for 82 hours under 76 kPa of steam at 773 K (the H 2 /H 2 O permeance ratio was ~300 at 773 K). The TEM showed that the deposited silica layer was inside the γ-alumina layer in the substrate, indicating that silica deposition is controlled by counter-diffusion of the precursors. The same group impregnated Rh or Ni catalyst on the porous alumina substrate before depositing a silica film by the counter-diffusion CVD method, using the 29 TMOS and O 2 . As a result, a composite catalytic membrane containing a hydrogen permselective silica layer and a catalyst layer was obtained, which was used in the steam reforming reaction to extract hydrogen (Gopalakrishnan et al., 2005). The membrane was reported to have hydrothermal stability for more than 80 hr at 773 K. They also prepared multi-membrane modules that displayed an activated transport for hydrogen, while effectively filtering larger molecules, such as N 2 , CH 4 and CO 2 . Hydrogen permeance increased from 1.57×10 -10 mol m -2 s -1 Pa -1 at 373 K to 4.38×10 -8 mol m -2 s -1 Pa -1 at 873 K, while the permeances for N 2 , CH 4 and CO 2 remained below 10 -11 mol m -2 s -1 Pa -1 , implying that the selectivity with H 2 of was over 2000. Nomura et al. (2005) also prepared silica membranes that had a H 2 /N2 selectivity of over 1000, using a counter- diffusion CVD deposition method at 873 K, on porous γ-alumina substrates with the TMOS and O 2 as the reactants. They showed that, increasing the deposition temperature resulted in increased activation energies for permeation through the membrane, with H 2 permeance at 873 K being 1.57×10 -7 mol m -2 s -1 Pa -1 . They also studied the steam stability of the membrane for 21 hr, during which they observed no reduction in H 2 /N 2 selectivity. Itoh et al. (2005) prepared Pd membranes on tubular α-alumina supports by forced-flow CVD through applying a vacuum on one side of the substrate and using palladium-diacetate [(CH 3 COO) 2 Pd] as a CVD precursor. Due to the pressure difference applied between the outside and the inside of the support tube, the chemical vapors enter into the porous support where they decomposed. The Pd membrane was as thin as 24 µm and had a H 2 /N 2 selectivity of 5000. 30 Gopalakrishnan et al. (2008) investigated the performance of high flux CVD silica membranes for the separation of gas mixtures containing H 2 and CO 2 at various temperatures. The membranes were prepared by counter-diffusion CVD method, using tetraethyl orthosilicate (TEOS) and O 2 as reactants. They reported H 2 /CO 2 and H 2 /N 2 selectivities of 36 and 57 at 400 °C, respectively. Koutsonikolas et al. (2009) used a low- temperature (300 °C) CVI technique to modify commercial sol–gel silica membranes, in order to close the meso- or micropores and to repair defects often formed in their top sol– gel layer. The pore size of the CVI-modified silica membranes decreased to less than 0.55 nm, which significantly improved the selectivities for the various gas pairs (H 2 /N 2 : 91, H 2 /CO 2 : 50, He/N 2 : 244 at 300 °C). 1.2.5 SiC Membranes A number of research groups have also studied the preparation of the SiC membranes. Sea et al. (1996), for example, prepared SiC membranes on macroporous α- alumina tubular substrates by the CVD method using tri-isopropylsilane (TPS) as the precursor at 700-800 o C under forced cross-flow. After the deposition process the membranes were heat-treated in deoxidized argon at 1000 °C for 1 hr. The resulting membranes permeated gases mainly through a Knudsen diffusion mechanism at temperatures of 50-400 °C, except for steam. The membranes were tested for the separation of H 2 /H 2 O mixtures at temperatures between 200 and 400 °C, and exhibited a H 2 /H 2 O permselectivity in the range of 3-5. When tested for a mixture of H 2 -H 2 O-HBr, 31 the H 2 permeance of the same membranes was (5-6)×10 -7 mol m -2 s -1 Pa -1 at 50-400 °C (Kusakabe et al., 1998). Lee and Tsai (1998) used the LPCVD method (at temperature of 800 °C with SiH 4 , C 2 H 2 , and Ar mixtures) to prepare asymmetric mesoporous SiC membranes on the surface of Al 2 O 3 -doped SiC macroporous supports. They prepared two asymmetric SiC membranes: one with its pore size reduced from 297 nm to 14 nm, with a 93% reduction in the permeance, and a second one with a reduction in the pore size from 670 nm to 44 nm, with a 95% reduction in permeance. Hong and Lai (1999) used a hot-wall CVD reactor with the SiH 4 /C 2 H 2 /NH 3 reaction system. SiC-Si 3 N 4 nanoparticles were formed in the gas phase at 1323 K, and were deposited on and within the pore structure of a macroporous α-alumina support disk. The deposition resulted in a decrease of the average membrane pore size from 0.3 to 0.21 µm, and the permeability decreased by almost 20%. The authors estimated the decrease to be less than what was expected from dense layer deposition on the support pore walls Lai and Hong, (1999). Takeda et al. (2001) prepared SiC membranes by the CVI in asymmetric γ-Al 2 O 3 membrane tubes with the SiH 2 Cl 2 -C 2 H 2 and H 2 reaction system, for a period of time at temperatures 800-900 °C, followed by an evacuation period. The membrane support tube was subjected to a number of the CVI and evacuation cycles, each reducing its H 2 permeance through the membrane. The final membranes had a H 2 permeance of ~1×10 −8 mol m -2 s -1 Pa -1 at 350 °C, with H 2 /N 2 selectivity of 3.36 (Knudsen ideal selectivity of 3.74). 32 Kleps and Angelescu (2001) studied the correlation between the SiC film preparation conditions and their properties. They showed, based on the deposition conditions of liquid precursors using the LPCVD, that different structure and composition of amorphous α-SiC and silicon-carbonitride form. Films deposited under kinetically- controlled conditions reveal a stable structure and composition. They were shown to form an interface that is very reactive with metallic layers, an aspect that must be taken into consideration in the SiC membrane fabrication. Krawiec et al. (2004) prepared the SiC membranes using the CVI with dimethyldichlorosilane (DDS) as the precursor and hydrogen as the carrier gas. They reported that the CVI process could be used to coat the mesoporous matrix with a nanosized coating on the inner or outer surface, by adjusting the deposition temperature. In a related study, Park et al. (2004) developed a simple and economic process for the preparation of mesoporous SiC ceramics with high surface area. They infiltrated polycarbosilane as a SiC precursor into a sacrificial silica sphere template, which was etched-off by aqueous HF after pyrolysis in Ar at temperatures in the range 1000-1400 °C. The mesoporous SiC, which was prepared from 20-30 nm aqueous silica sol, exhibited a high surface area. The USC group has used several methods to prepare the SiC membranes applicable to gas separation, as described below. Suwanmethanond et al. (2000) prepared mesoporous and macroporous supports by pressureless sintering of SiC powders under Ar, and characterized them by measuring their porosity, pore size distribution, surface characteristics, and structure, as well as their transport properties, using N 2 and He as the test gases. Various powders were utilized, as were four different sintering aids, namely, 33 Al 2 O 3 , B 4 C, carbon black, and phenolic resin, either by themselves or in combination in order to test their ability for preparing good quality substrates. It was found that the porosity, pore size distribution, and transport characteristics of the resulting SiC-sintered substrates depend on the nature of the original powder, and the type and molar ratio of the sintering aid utilized. Fukushima et al. (2008) studied the corrosion effect of high-temperature water vapor on porous SiC with and without additives, and examined the corrosion resistance of porous membrane supports for hydrogen production by steam modification of methane. They found that the alumina-doped support has weight gain of 1.3 mg/cm 2 , while the undoped support indicated a weight gain of 0.7 mg/cm 2 . In the alumina-doped support, pore growth was observed due to the coalescence among oxidized fine particles. In contrast, the pore size of the supports without alumina was slightly reduced, due to thin silica layer formed on the SiC particle. Maddocks and Harris (2010) prepared micoporous SiC by the pyrolysis of the preceramic polymer, polycarbosilane (PCS), with and without the addition of an inert filler. They showed that hydrosilylation crosslinking of the PCS with divinylbenzene prior to pyrolysis appears to have little influence on the development of micro- and mesoporosity. They reported the maximum micropore volumes of 0.28 cm 3 g −1 for non- crosslinked PCS and 0.25, 0.33 and 0.32 cm 3 g −1 for the PCS crosslinked with 2, 6 and 10 wt.% DVB, respectively. Moreover, the micropore volumes decreased under 34 hydrothermal conditions to 0.03 cm 3 g −1 for non-crosslinked, and to 0 cm 3 g −1 for the crosslinked PCS. The USC group, in collaboration with Media and Process Technology, Inc., (M&P) prepared microporous SiC membranes using various methods (Ciora et al., 2004). They used two different precursors, namely, the TPS and 1,3-disilabutane (DSB), to prepare the SiC membrane by the CVD/CVI. In the deposition of the TPS, they used the M&P commercial γ-alumina membranes (prepared by sol-gel deposition on macroporous α-alumina substrates) with an average pore size of 40 Å. The substrates were calcined for 4 hr in air at 1000 °C. The treatment enlarges the pore size of the membrane to ~100 Å, but further treatment at this temperature leaves the pore size unaffected. The deposition experiments were done in temperatures between 700-750 °C for 2-20 min. Following the deposition, the membranes were further annealed at 1000 °C for 2 hr for further conversion to SiC, and then finally activated at 700-750 °C in the presence of steam to eliminate any excess carbon. The membranes prepared by the method are mostly amorphous. The SiC mostly deposited inside the γ-alumina layer forming a film ~3µ thick. The SiC membranes prepared by this method have He permeance of 8.06×10 −8 to 1.72×10 −6 mol m -2 s -1 Pa -1 and a selectivity for H 2 /N 2 , ranges from 4 to > 100. The He permeance increases with temperature, which is indicative of activated diffusion, whereas the permeance of gases with larger kinetic diameters (N 2 , CH 4 , CO 2 , CO, and H 2 O) decreases with temperature, indicative of Knudsen flow of the gases. Experiments with steam demonstrated good stability at 500 °C for over 1000 hr. The USC group has also prepared defect-free nanoporous membranes using the CVD/CVI of the DSB, with a 2-20 35 min deposition time at temperatures in the range 650-750 °C, and a 2 hr post-deposition annealing at the same temperatures. The permeance of He reached 3.5×10 −7 mol m -2 s -1 Pa -1 , with a He/N 2 selectivity of ~55 at 550 °C. The DSB membranes are thermally stable, but they fail the steam stability test. Ciora et al. (2004) also reported on the preparation of SiC microporous membranes by conventional dip-coating techniques, using a novel pre-ceramic polymeric precursor allylhydridopolycarbosilane (AHPCS), a partially allyl substituted hydridopolycarbosilane (HPCS). The earlier membranes were stable in air-treatment at 450 °C, but proved unstable in high-temperature steam. Elyassi et al. (2007) reported on the continuation of the USC efforts to prepare the SiC microporous membranes by the pyrolysis of thin AHPCS films. They coated the films, using a combination of slip- casting and dip-coating techniques, on tubular SiC macroporous supports. They reported a H 2 /CH 4 selectivity in the range 29-78, but most importantly that the new membranes were hydrothermally stable. Elyassi et al. (2008) further improved the SiC preparation method through the use of sacrificial interlayers. The technique involves periodic and alternate coatings of polystyrene sacrificial interlayers and the SiC pre-ceramic layers on the top of slip-casted tubular SiC supports. Membranes prepared by this technique exhibit single-gas ideal separation factors of He and H 2 over Ar in the range 176–465 and 101– 258, respectively. The mixed-gas experiments with the same membranes indicate separation factors as high as 117 for an equimolar H 2 /CH 4 mixture. 36 1.3 Modeling 1.3.1 Pore Network Modeling A key goal of the studies reported in this Thesis is to develop a better understanding of the processes involved during the preparation of the SiC microporous membranes, and of the important characteristics of the materials that determine their properties. A key tool in our efforts is the use of pore network models. Our efforts are part of a broader effort by the USC group that aims to understand equilibrium and non- equilibrium properties of fluids and their mixtures in confined media. Examples of such media include micro- and mesoporous materials, such as catalysts, adsorbents, and synthetic and natural (biological) membranes (Pinnavaia, 1995). For practical applications, it is important to understand how mixtures transport through the pore space of such materials. To develop such understanding, one must have an accurate model of the pore space, as well as to be able to model correctly the mechanisms by which molecules transport through that space. As previously noted, the emphasis in this Thesis is on SiC microporous membranes that have been under active investigation by the USC group for some time now, both experimentally and by computer simulations, for the separation of mixtures into their constituent components, and as potential sensors that can detect trace amounts of certain chemical compounds (Suwanmethanond, et al., 2000; Ciora, et al., 2004; Elyassi et al., 2007, 2008). Though not explicitly discussed here, the same modeling techniques are also applicable to a 37 different class of microporous membranes, namely, carbon molecular-sieve (CMS) membranes that have also been previously investigated by the USC group (Sedigh et al., 1998, 1999, 2000), and are currently being commercialized by the M&P. Optimizing the performance of the SiC membranes, and in particular their selectivity, entails developing deeper understanding of how the transport of mixtures through them is affected by the morphology of the pore space, and in particular their pore size distribution and pore connectivity. There are various approaches to modeling gas transport through microporous materials. The most common approach is phenomenological, and relies on the continuum equations of transport of mixtures through the material, which is described only in terms of effective properties, such its permeability. The technique is relatively easy to apply, and finds widespread application in the design of membrane modules and systems. On the other hand, the approach ignores important information about the porous material’s morphology, including its PSD and the interconnectivity of the pores, and provides little insight into the preparation process of what is important in determining final membrane performance and properties. On the opposite end of the spectrum of approaches to understanding transport and separation of mixtures through nanoporous materials have been completely atomistic- based models of both the molecules and the materials (for recent reviews see, for example, Gelb et al., 1999; Sahimi and Tsotsis, 2006). Such simulations require, however, highly intensive computations (Sahimi et al., 2006). Most of the published efforts are based on atomistic simulation of flow and transport of a fluid mixture through 38 a single nanopore, such as the work by the USC group (Xu et al., 2000, 2001). In practice, however, the pore space of a membrane layer (as well as its support) consists of a network of interconnected pores, the sizes of which are distributed according to a PSD. The network is typically three- dimensional (3D), or quasi-2D, if the membrane layer is thin. As pointed out by, for example, Seaton et al. (1997), to accurately model any membrane and the phenomena that take place in it, one must take into account the effect of its interconnected pore structure. A third approach, therefore, and the one that is used in this Thesis, is based on utilizing pore network models of porous materials. Any porous medium (including membranes) can, in principle, be mapped onto an equivalent network of pore throats (bonds) and pore bodies (nodes) (Sahimi et al., 1990; Rieckmann and Keil, 1997, 1999). In the resulting networks the pore sizes are distributed according to a PSD that can be measured by a variety of techniques. Since the network’s pores are interconnected, the effect of their connectivity, which greatly influences transport and separation of molecules through the network, is also automatically taken into account. Thus, one has a realistic model of a nanoporous membrane, which can be used to study the effect of a variety of factors on the transport and separation of gaseous mixtures through it. Although pore network models have been used extensively in the past for studying transport and reaction processes in porous catalysts and several other types of porous media, there are only few studies for the modeling of transport and separation of gaseous mixtures in microporous membranes. Tzevelekos et al. (1998) studied the 39 transport of condensable vapors through a mesoporous alumina. They simulated the process, including; adsorption, mass transfer, and capillary condensation, over the entire range of relative pressures. Sotirchos and Burganos (1988) studied transport of multicomponent gaseous mixtures in pore networks, but their study was in the context of the effective-medium approximation. Petropoulos (1991) used a network model to study gas transport in a bidisperse porous adsorbent, while Burganos (1992), used a pore network model to study Knudsen diffusion of a single gas in porous materials. Knudsen diffusion in randomly oriented capillary structures was studied by Burganos (1989). An interesting approach to the transport of gaseous mixture through membranes was developed by MacElroy (1997), Seaton (1997), and Sedigh et al. (1998), based on combining atomistic simulations and the critical-path analysis (CPA). The CPA is a technique according to which, for a sufficiently broad PSD, the transport process is dominated by a critical pore of size r c , such that pores with sizes much larger or smaller than r c do not make significant contributions to the overall rate of the transport in the membrane. The original CPA was developed for hopping conduction in amorphous semiconductors, but has since been applied to a wide variety of transport processes in disordered materials (see, for example, Sahimi, 1993, 2003). In the following chapters we first describe some of our efforts in preparing microporous SiC membranes by the CVD/CVI techniques, with the goal of generating the data that are necessary to validate the network models of the membrane formation process. Subsequently, we describe our modeling efforts using the models in describing 40 the phenomena during membrane preparation and of the important membrane characteristics that determine performance. 41 Chapter 2 Fabrication of Microporous Silicon-Carbide Membranes 2.1 Introduction This chapter summarizes the experimental procedure for fabrication of microporous SiC membranes. Making a suitable membrane support is the first and probably the most important step to prepare a good quality membrane. The support must have some unique properties in order to be applicable for membrane use. They include: (i) Strong mechanical strength to resist high pressure gradients without deforming and cracking; (ii) a well-connected macroporous structure so that it does not provide resistance to flow; (iii) similar physical characteristics with the deposited membrane layers, particularly close to the same thermal expansion coefficient; (iv) a smooth surface morphology to allow the deposition of uniform thin films. 2.2 Support Preparation As noted in chapter 1, so far, mostly commercial substrates have been utilized for fabrication of SiC membranes. Vycor® glass tubes were used in the earlier efforts (e.g., Gavalas et al. 1989), which have an isotropic, uniform pore structure (pore diameter ~ 40-50 Ǻ), but the glass is expensive, has low permeance, and is fairly fragile. Most studies in the literature have used alumina materials for membrane supports, because they 42 are relatively cheap and have good mechanical properties. However, they are not ideal as supports in the preparation of microporous SiC membranes, since they have lower resistance in chemically corrosive environments and in high-temperature operations, when compared to SiC, as well as quite different thermal expansion coefficient from that of SiC, which raises concerns about the mechanical stability of the resulting composite membrane system. As previously noted, in the USC group Suwanmethanond et al. (2000) prepared mesoporous and macroporous SiC flat disk supports by pressureless sintering of SiC powders under an inert atmosphere of argon. They were then used in the preparation of microporous SiC membranes (Ciora et al., 2004; Elyassi et al., 2007; Chen et al., 2008). Flat disks are not convenient, however, to use (in terms of sealing), particularly in the membrane reactors applications. In this study we also prepared tubular supports, in addition to flat disks. For the preparation of the green samples, we used a SiC powder with a mean particle size of 0.6 µm, which was supplied by the Superior Company (HSC059). To prepare the supports, we first mix the SiC powder together with the appropriate sintering aids, using acetone as the dispersing medium. Oleic acid is then added as the pressing aid. The resulting slurry materials are then mixed thoroughly to ensure complete homogenization using an ultrasonic bath for 20 min, and are then dried in an oven or under the hood. The dried powder is then pressed, using molds of various shapes, with pressures of up to 117.15 MPa, in order to obtain disk- 3.2 cm (1.25 in) in diameter, or green cylindrical samples with an ID of ¼ -inch and OD of ½-inch. The green SiC samples are then heated at a rate of 3 °C/min in a graphite resistance furnace 43 (Thermal Technology, Inc.) until they reach the appropriate sintering temperature - 1700- 1900 °C - and are kept there for a predetermined period of time under an inert argon atmosphere. When sintering is finished, the samples are cooled down to room temperature at a cooling rate of 6 °C/min. The resulting sintered porous SiC substrates are characterized by measurements of their overall porosity using the Archimedes method. The advantages of the method are, (i) it only measures the accessible pores that are of relevance to membrane transport, and (ii) it is not a destructive technique. In our measurements acetone was used as the buoyant liquid, instead of water, because it wets the surface of the SiC better. The key assumption of the method is that acetone penetrates into all of the accessible pores of the SiC sample. The porosity ε is then calculated using the following formula: (2.1) where W air is the weight of the substrate in air, W a the weight of the substrate in acetone, V s the bulk volume of the substrate, and ρ a the density of acetone at 25 °C. A digital linear gauge (EG-133 ± 0.0001% accuracy, manufactured by Mitutoyo) is used to measure the thickness and diameter of each substrate. The error in the repeatability of the porosity measurements was shown to be better than 0.5%. For determining the average pore size we used single-gas transport measurements using a non-adsorbing gas (e.g., Ar). The flux through the membrane support is the result of the combined Knudsen diffusion and viscous flow, and is described by 44 (2.2) (2.3) (2.4) which, after integration yields, (2.5) Here, d p is the average pore diameter (m), ε the porosity, η the viscosity (Pa-s), L the support thickness (m), D eff the effective Knudsen diffusion coefficient (m 2 /s), and M i the molar mass of Argon (g/mol). Plotting J i /ΔP as a function of the average pressure should result in a straight line, see Figure 2.1. From the intercept and slope of the line, the average pore diameter as well as the porosity-tortuosity ratio of the membrane, are calculated. 45 Figure 2.1 Argon permeance vs. average pressure. The porosity of the support, as calculated using Figure 2.1, is 0.3, while the average pore diameter turns out to be 65.3 nm. The tortuosity is 1.77. Measurement of the He permeance through the membrane supports is another important characteristic. The He/Ar separation factors for the SiC supports that we prepared varies from 2.5 to 2.9 (note that for Ar and He the ideal Knudsen separation factor is 3.16), which indicates that they are mostly macroporous. 2.3 Preparation of the Membranes Two types of membranes were prepared. One was flat-disk membranes, while the second membrane was tubular ones. Both membranes were prepared in collaboration with Dr. Chen. The procedure for their fabrication was already reported by Chen et al. 46 (2008a). Thus, we describe their preparation strictly for completeness, and focus on those aspects not discussed by them. 2.3.1 Flat-disk Membranes The schematic of the (CVD/CVI) apparatus is shown in Figure 2.2. The hot-wall CVD reactor is made out of quartz, has a diameter of 3.8 cm, a length of 73 cm, and a 20 cm long hot-zone. The disks were first treated in flowing air at 450 °C to remove any excess carbon. The support disk is sealed on the top of a quartz tube (1 cm ID) using a glass powder (#7740, purchased from the Corning Co) as a sealant. The quartz tube is then coaxially inserted into the center of the CVD reactor, connected with the aid of a Cajon Utra-Torr fitting. A ceramic cylindrical heater is utilized, which is controlled by an OMEGA temperature controller (CN 9000). 47 Figure 2.2 Schematic of flat disk SiC membrane experimental setup. Tri-isopropylsilane (TPS) (purchased from the Aldrich Company) was used without any further purification. Ultra-high purity Ar (99.999%) or He (99.999%) were used as the carrier gas, and their flow rates were controlled by mass-flow controllers (Brooks 5850E). A syringe pump (HARVARD, PHD2000) was used to inject the TPS 48 precursor into the carrier gas. The feed-lines from the pump to the furnace were heat- traced and kept at 140 °C at all times, in order to ensure that the TPS is in the vapor phase. The pressure inside the reactor was ~1.3 atm, while the pressure inside the quartz tube was ~1 atm. The pressure was monitored with the aid of pressure (or vacuum) gauges. The flow rate of the TPS ranged from 1×10 −8 mol s −1 to 5×10 −8 mol s −1 , and the argon flow rate from 2×10 −5 mol s −1 to 9×10 −5 mol s −1 . The deposition time ranged from 2 to 15 h. The deposition temperature was varied between 600-850 °C. During deposition we continuously measured the argon permeation rate. If there was a significant decrease in the permeability, we switched the carrier gas to He to determine the He/Ar selectivity. In the experiments the flow rate of the TPS was set at 4.3×10 −8 mol. s −1 . The carrier gas (Ar) has a flow rate of 4.3×10 −5 mol.s −1 . Figure 2.3 presents the typical results obtained with one of the SiC disks with an initial porosity of 0.3, which, prior to deposition, had an argon permeance (measured at 760 °C) of 1.8×10 −7 mol.m −2 . s −1 .Pa −1 . The data indicate that, in the first 12,000 s of the experiment, the permeances for both helium and argon decrease. However, the helium permeance decreases more slowly than that of argon, and as a result, as shown in Figure 2.4, the selectivity (ratio of permeances) between helium and argon steadily increases. This behavior may be explained as follows: In the beginning of the experiment, since the disk is mesoporous (average pore diameter for this disk ~ 29 nm), argon and helium are both transported by a Knudsen type mechanism. As the TPS decomposes into SiC and gradually fills the membrane pores, the average pore size decreases, which results in decreasing the permeance for both helium and argon. Helium being the smallest of the 49 two molecules, it can penetrate through a larger fraction of small unplugged pores than argon through an activated diffusion mechanism, so plugging of the pore structure affects it less than it does for argon. Figure 2.3 The permeance of the inert gases vs. injection time for flat-disk membranes (after Chen et al., 2008a). 50 Figure 2.4 The He/Ar selectivity vs. time (after Chen et al., 2008a). 2.3.2 Tubular Membranes The preparation of tubular membranes was similar to the flat-disk ones, except that the geometry was different. We also built a new CVD/CVI apparatus to accommodate the tubular SiC support tubes, which is shown in Figure 2.5. The new apparatus was made with stainless steel, so that it could be operated at pressures much higher than the quartz apparatus of Figure 2.4, in order to broaden the CVD operating conditions, as well as building a transmembrane pressure drop across the membrane without needing to evacuate one side of the reactor, thus enabling us to conveniently monitor the progress of the SiC deposition online. The most important advantage of the 51 CVD apparatus of Figure 2.5 is that it allows us to conveniently seal the reactor; since in the flat disk apparatus the glass sealant often cracked when we changed the deposition temperature. Figure 2.5 Schematic drawing of the stainless steel apparatus for tubular membranes. Some of the more important details of the experiments are as follows: Figure 2.6 shows the support tube before and after it had been glazed. The glaze makes the part of the membrane nonporous and gives a smooth surface. 52 Figure 2.6 The support for the SiC membrane before (left) and after (right) glazing. In our investigations we first focused our attention on choosing the appropriate temperature for pyrolysis of the TPS. Thus, we carried out a series of pyrolysis experiments at various reaction temperatures in the flow CVD reactor of Figure 2.5, ranging from room temperature to 760 °C, while analyzing the exit TPS concentrations by the GC/MS. We found that the TPS starts to pyrolyze at temperatures between 600 to 625 °C, and as the temperature increases, the exit concentration of the TPS decreases, indicating an increased decomposition rate, as expected. Figure 2.7 shows typical results of the experiments, where the initial concentration of the TPS in Ar was 19.1 ppm (the reactor pressure and Ar flow rate were set at 5 psig and 2 ml/s). The concentration curve starts decreasing at 625 °C and reaches a value of less than 1 ppm at 760 °C. 53 Figure 2.7 Pyrolysis of the TPS. Subsequently, we focused on understanding the effect of the various operating parameters on the characteristics of the final films produced, by focusing on the temperature and pressure of pyrolysis. When the temperature is high, any molecules that reach the surface react instantly and the decomposition of the precursor is fast. The pressure of the reactor, which also determines the pressure drop across the membrane during pyrolysis, also affects the transport rates and may become the controlling factor, particularly at the higher deposition temperatures. Other important factors include the carrier flow gas velocity, and the injection rate of the precursor that determine the concentration of the precursor, as well as the mass transport characteristics to the surface of the support. The desirable outcome here is to prepare a uniform thin film with little or no gas phase reaction occurring by adjusting the aforementioned experimental parameters. 54 A number of experiments were carried out to investigate the effect of the various experimental parameters. The first set of experiments was carried out at high reactor pressures (the permeate-side pressure was kept atmospheric, and the pressure drop across the membrane was kept constant for any given experiment ranging from 10-30 psi), relatively high flow rates of the inert carrier gas (4-10 ml/s) and of the TPS injection rates (5-20µl/hr) were used, and temperatures in the range of at 700-750 °C. Table 2.1 shows the details for all the experiments carried out. Under such experimental conditions we observed accumulation in the reactor of powder indicative of gas-phase reactions, and not much deposition on or inside the macroporous substrate. In the following discussion we focus here on those aspects not reported by Chen et al. (2008a) As Figures 2.8 and 2.9 for Experiment 1 indicate (see Table 2.1), the membrane support permeances for either argon or helium did not drop significantly, and the ideal separation factor remained close to the Knudsen value. 55 Experiment 1 Experiment 2 Experiment 3 Pressure (psi) 30 2 3 Temperature (°C) 750 750 750 Ar flow rate (ml/s) 10 2 1 TPS Flow rate (µl/hr) 20 5 5 Table 2.1 The detailed experimental conditions for the CVD experiments. Figure 2.8 Reduction in the permeance of Ar and He during deposition for Experiment 1. 56 Figure 2.9 The He/Ar separation factor vs. deposition time for Experiment 1. Reducing the pressure and flow rate significantly improves the performance. Figures 2.10 and 2.11 present the Ar and He permeances and the corresponding separation factor for Experiment 2, listed in Table 2.1. There seems to be great reduction in the argon and helium permeances and, consequently, at the end of deposition the separation factor levels to ~13 (Figure 2.11). As we also observed previously with the flat disk membranes, under such deposition conditions the permeance of argon decreases faster than the helium permeance, with the net result being that the ideal separation factor increases with deposition time. This behavior indicates that substantial decomposition of TPS occurs within the pore structure of the support in order to form SiC, which 57 progressively plugs the pores of the support. When the size of the pores becomes small enough, only helium can diffuse through them, while argon is excluded and only transports through pores with higher initial diameters, which remain substantially, unfilled. Figure 2.10 Reduction in the permeance of Ar and He during the deposition in Experiment 2. 58 Figure 2.11 The He/Ar separation factor vs. deposition time for the experiment 2. The results of Experiment 3 are shown in Figures 2.12 - 2.14. The permeance of Ar decreases from 5.79×10 −8 mol.m −2 .s −1 .Pa −1 to 9.85×10 −9 mol.m −2 .s −1 .Pa −1 , and that of He decreases from 1.55 ×10 −7 mol.m −2 .s −1 .Pa −1 to 9.93×10 −8 mol.m −2 .s −1 .Pa. −1 As a result, the He/Ar separation factor increases to ~11. Figure2.14 correlates the changes in the permeation characteristics with the amounts of the TPS used during the deposition experiments, which provides insight into how much TPS needs to be consumed to prepare a membrane. 59 Figure 2.12 He and Ar permeance vs. deposition time in Experiment 3 (after Chen et al., 2008a) After the deposition was finished in Experiment 3, we measured the permeances for both Ar and He in situ in the CVD reactor at various temperatures (by lowering the temperature in a step-wise manner, and measuring the permeances after the sample temperature had stabilized to a new value). 60 Figure 2.13 The He/Ar separation factor vs. deposition time in Experiment 3 (after Chen et al., 2008a). Figure 2.14 He and Ar permeance vs. injected amount of the TPS in Experiment 3. 61 As shown in Figure 2.15, the He permeance decreases slightly as the temperature is lowered, from 9.94×10 −8 mol.m −2 .s −1 Pa −1 at 750 °C to 9.61×10 −8 mol.m −2 .s −1 Pa −1 at room temperature, indicative of activated diffusion. On the other hand, the Ar permeance, which has a larger kinetic diameter, increases as the temperature decreases from 9.85×10 −9 mol.m −2 .s −1 Pa −1 at the 750 °C to 3.11×10 −8 mol.m −2 .s −1 Pa −1 at the room temperature, indicative of Knudsen flow as shown in Figure 2.16. following a linear correlation with 1/√T . As Figure 2.17 shows, as a result of this behavior, the He/Ar selectivity decreases as the temperature decreases form ~10.1 at 750 °C to ~3.1 at room temperature. Figure 2.15 He and Ar permeance vs. temperature (after Chen et al., 2008a). 62 Figure 2.16 Indication for Knudsen flow for argon (after Chen et al., 2008a). Figure 2.17 The He/Ar membrane selectivity vs. time 63 Chapter 3 A Continuum Model of Membrane Preparation The work in this chapter is the result of a joint project with Dr. Chen. The results were published in the paper, “Experimental studies and computer simulation of preparation of nanoporous silicon–carbide membranes by chemical-vapor deposition and infiltration techniques,” in Chemical Engineering Science 63, 1460-1470 (2008), which is referred to as Chen et al. (2008a). The main purpose of describing part of the work in this chapter is to set the stage for the pore network modeling work that will be presented in chapter 4. The work described in this chapter demonstrates the need for a pore-scale model. But, in addition, it provides the same set of equations that will be utilized in part of the work described in chapter 4. 3.1 Introduction This chapter presents the development of a dynamic, continuum model of the tubular SiC membrane fabrication process using the CVI/CVD technique, which properly accounts for the reaction, fluid mechanics, and the transport processes that take place inside the reactor and membranes. As described below, the model correctly predicts key characteristics of the membranes, such as their permeance and the evolution of their porosity as functions of the deposition time and other experimental conditions. 64 3.2 Model Development As described in chapter 2, during membrane preparation, the Sic precursor, tri- isopropylsilane, is carried by the Ar carrier gas into the support's pore space, where it reacts to form SiC on the pores' surface. This gradually shrinks the pores and reduces the membrane's permeance. Our experiments, as well as those of several other groups, indicated that the gas-phase reaction is insignificant. For example, Boo et al. (2000) found a very high sticking coefficient for the SiC source growth on the SiC substrate. Park et al. (2000) and Valente et al. (2004) reported that the SiC surface has a very strong reaction potential for the SiC source. Therefore, in the model described below it was assumed that the main reaction occurs on the pores' solid surface. As noted previously, a gas is transported through a porous membrane by four main mechanisms, namely, molecular, Knudsen, and hindered or configurational diffusion, and viscous flow. In principle, surface flow might also play a role in transport through very small pores. It is very difficult, however, to quantify the effect of surface flow without introducing several empirical parameters. Moreover, in small pores where the ratio of the molecules and the pores' sizes is close to 1, it is difficult and indeed not clear how to distinguish surface flow from hindered transport that we include in the modeling. Therefore, in the model the effect of surface flow was ignored (Chen et al., 2008a). 65 Molecular (Fickian) diffusion and viscous flow occur only in large pores. Since the support that we used has an average pore size that is smaller than the mean-free paths of He and Ar, only hindered and Knudsen diffusion were included in the model (Chen et al., 2008a). Because the disk's radius is much larger than its thickness, a one-dimensional (1D) system should suffice for modeling the phenomena. Thus, the continuity equations for two species A and B are given by, (3.1) (3.2) Here, C A represents the molar concentration of the intermediate species responsible for the SiC growth, C B the molar concentration of the carrier gas (Ar), ε p (q) the local porosity with q being the amount of deposited SiC (mol Si/m 3 ), and τ the tortuosity factor. It was assumed (Chen et al., 2008a) that while the local porosity changes as a result of the SiC deposition, the tortuosity factor remains invariant. The effective diffusivity for both species A and B is given by, (3.3) where D K and D H represent, respectively, the Knudsen and hindered diffusivities. In 66 general, the expression that relates D e to D K and D H is more complex than Equation (3.3), and involves the mole fractions of the gases in the mixture (see chapter 4). For the purposes of this preliminary model, however, Equation (3.3) provides adequate estimates of D e . The Knudsen diffusion coefficient of B (Ar) is described by, (3.4) where T is the temperature, R g the gas constant, M Ar the molecular weight of Ar, and R p the pore size which is described by, (3.5) Here, R p0 and ε p0 are the initial pore size and porosity of the substrate, prior to deposition. The Knudsen diffusivity of A is given by an equation similar to (3.4), written down for the different reactant. The local porosity ε p (q) that appears in that equation varies as a function of the deposition process according to the following equation, (3.6) where M SiC and ρ are, respectively, the molecular weight and density of SiC. For the hindered diffusivity Equations (4.7)-(4.10) were used. The bulk binary diffusivities were computed using the known relations for the diffusivity of gases (Bird et al., 2007). 67 An expression for the rate of deposition of SiC must now be specified. It was assumed (Chen et al., 2008a) that the rate is described by the following equation (Chang et al, 1997; Yamaguchi et al, 2000; Beyne and Froment, 2001; Birakayala and Evans, 2002), (3.7) which is consistent with the assumption of having a high sticking coefficient for the SiC precursor on the pore surface of the SiC membrane. Here, K d is an effective deposition rate constant, σ v the local internal pore surface area per unit volume, which depends on the porosity ε p (q), and the surface reaction exponent. It was also assumed that (Beyne and Froment, 2001) (3.8) where A is a constant, and n = 2 for a pore space consisting of (parallel) cylindrical pores (n = 2/3 if the pore space consists of spherical cavities). Therefore, putting everything together, one obtains the following set of equations that govern the evolution of the system, (3.9) 68 (3.10) (3.11) where, K = AK d . The parameters K and η are used to fit the results of the numerical solution of Equations (3.9)-(3.11) to our experimental data. In a series of preliminary simulations, we also treated the molecular weight of the reactants A as a fitting parameter. However, the fitted molecular weight turned out to be about 150, very close to that of the TPS, which is 158 and, interestingly, implies that the reactant reaches and penetrates the pores nearly intact. Thus, the molecular weight of the reactants A was set to be the same as that of the TPS, and was not treated as an adjustable parameter. To solve Equations (3.9)-(3.11) the following initial conditions were used, which are consistent with the way the deposition experiments were carried out (Chen et al., 2008a). At t = 0 and for |r|≤r 0 , (3.12) and, (3.13) where C Bi (r) is computed by solving the steady-state form of Equation (3.10) prior to the deposition, i.e., C Bi (r) is the solution of the following equation: 69 (3.14) The solution of Equation (3.14) is a linear profile for C Bi (r). One must also specify the boundary conditions in order to solve Equations (3.9)-(3.11). On the tubular membrane surface at r = R 1 (see Figure 4.1) and at times t≤t TP , one has, , (3.15) where C t is the total concentration, and superscript T indicates the tube side. For times t>t TP one has (3.16) (3.17) On the surface at r = R 2 (see Figure 4.1) - the shell side denoted by the superscript S - and for times t >0, one sets, , (3.18) In practice, however, C A nearly vanishes on the surface at r = R 2 . Here, t TP is the time at which a turning point in the porosity is reached, i.e., when the local porosity at the tube side surface reaches a value so that the corresponding average pore radius is close to the molecular diameter of the TPS. 70 The finite-element method (Finlayson, 1980) was utilized in order to solve Equations (3.9)-(3.11) and their associated initial and boundary conditions. Linear basis functions were utilized, and the time-dependent terms were discretized as, (3.19) where Δt is the size of the time step, and the concentration at grid point i after n time steps. An adaptive time step as well as an adaptive computational grid were used in order to ensure accurate solutions. When the quantities of interest varied rapidly, a smaller time step was used, whereas when they varied relatively slowly, a larger Δt was utilized. Thus, in the intervals 0<t≤9.7 h and 10.5<t <22 h we used, Δt = 20 s, while in between the two intervals Δt = 5 s was utilized since, as the results presented below indicate, the most significant changes in the quantities of interest occur in the middle time interval. As for the computational grid, in the intervals 0< r <1 and 2µm < r < 2mm the grid was uniform with the distance between the grid points being, respectively, 25 and 0.15 nm. The grid in between the two intervals was adaptive in order to ensure flexibility in the computations and accuracy of the solutions. The Newton–Raphson method was used to linearize the equations, and the linearized equations were solved using the biconjugate- gradient method. To solve the tube side and shell side equations numerically, the length of the system was divided into a number of grid blocks. 10, 20, 30, and 40 grid blocks were used, but the accuracy of the solution with 10 blocks was comparable to those obtained 71 with the denser grids. The total length of the reactor was 5 cm. The initial average pore size of the membrane (support) was 65 nm, while its initial porosity was 0.3. The governing equations for the membrane in the radial direction, adjacent to the first block in the z-direction, were then solved by the method we described above. The tube side is described by the following equations for the species A and B (the TPS and Ar, respectively), , (3.20) , (3.21) The governing equations for the shell side are similar to the tube side, except that the right sides of Equations. (3.20) and (3.21) must be evaluated at r = R 2 (see Figure 4.1). , (3.22) , (3.23) 72 3.3 Results and Discussion Figure 3.1 presents the dependence of the relative porosity, ε/ε o , on the distance r from the top surface, at several deposition times. The most significant changes in the porosity happen very near the top surface (r = R 1 ) of the support that faces the precursor flow. As the pore volume decreases, the corresponding average pore size also decreases, eventually reaching the value corresponding to the kinetic diameter of the TPS, taken to be ∼ 0.5 nm in the simulations. To obtain the results of the simulations shown in Figure (3.1), this happens after about 21 h of deposition. Beyond this time, the TPS is no longer able to infiltrate into the support while Ar, with a kinetic diameter of ∼ 0.34 nm, still permeates through. Therefore, after about 21 h, very little additional deposition can happen inside the support and, hence, the Ar permeance becomes stable. 73 Figure 3.1 Dependence of the relative porosity of the tubular membrane on the distance r from the support’s top surface, at several deposition times t (after Chen. et al., 2008a). The porosity variations shown are in qualitative agreement with the electron microscopy observations of the membranes prepared by the CVD/CVI of the TPS. Shown in Figure 3.2, for example, is a scanning-electron microscopy (SEM) image of the cross-section of a membrane, indicating a relatively uniform dense membrane layer (the darker part of the image at the top), followed by a relatively intact portion, qualitatively consistent with the simulations results. 74 Figure 3.2 The SEM image of the cross section of the SiC membrane. The darker area near the top surface is the membrane layer. Figure 3.3 compares our experimental data (Experiment 3) for the Ar permeance during the deposition process with the computed permeances, obtained by the numerical simulations of the governing equations. The agreement between the data and the numerical results (using two adjustable parameters, namely K and η) is excellent, hence indicating the overall validity of the model. 75 Figure 3.3 Comparison of the experimental data for the Ar permeance with the results of the numerical simulations (after Chen et al., 2008a). To further test the model, a second set of experiments were carried out. Figure 3.4 presents the experimental data and compares them with the fit of the numerical solution of the governing equations, using the same values of the adjustable parameters K and η as calculated in Figure 3.3. The agreement is reasonable, hence indicating once more the accuracy of the model. 76 Figure 3.4 Comparison of the experimental data for the Ar permeance with the results of the numerical simulations (after Chen et al., 2008a). 3.4 Summary and Conclusions The results of the computer simulation of the preparation of SiC membranes by chemical-vapor deposition and infiltration techniques were presented. The study indicates that the CVI/CVD process of the TPS on the SiC support continues only so long as the pore sizes are larger than the molecular radius R TPS of the TPS. Once the pores shrink to a 77 size smaller than (or equal to) R TPS , the permeance of argon no longer changes, even if one continues the deposition. Moreover, significant porosity changes occur mostly in the region very close to the top surface. The dynamic model was developed for describing the CVI/CVD processes. Using only two adjustable parameters, the models provided accurate predictions for the membrane’s permeance. These parameters were then used in our model which is described in chapter 4. This chapter demonstrates the strengths and shortcomings of a continuum model. It indicates that, with the use of a few adjustable parameters, one may obtain reasonably accurate predictions for the experimental data. But, the model cannot provide much information for the evolution of the pore space morphology of the membrane. Moreover, the fact that the adjusted parameters provide reasonable predictions for some quantities of interest does not guarantee that the model can be predictive, if one tries to utilize it for computing any other important quantities of the process. Thus, one must develop a pore network model that can address such concerns. Such a model is described in the next chapter. 78 Chapter 4 A Network Model of Membrane Preparation 4.1 Introduction In this chapter a pore network model is developed to study the evolution of the pore space of a microporous SiC membrane during its fabrication by a chemical-vapor deposition/chemical-vapor infiltration technique. The pore space of the support is represented by a three-dimensional network of interconnected pores, in which the pores’ effective size is distributed according to a pore size distribution that closely mimics the experimental pore size distribution (PSD). The chemical reaction that generates the SiC, the various transport mechanisms in the pores, and the evolution of the pore structure during the SiC deposition on the pores’ surface are included in the model. The Maxwell- Stefan equations are used for describing the pore-level transport processes, which include the Knudsen and hindered diffusion, as well as viscous flow. The effect of pore shrinkage and blockage as a result of the deposition of the SiC on the pores’ internal surface is taken into account. The simulator monitors the PSD as the membrane’s structure develops by the CVD process and evolves. Also computed are the permeances of argon, the carrier gas, and of helium during the CVD, as well as the membrane’s selectivity. Good agreement is found between the simulation results and the experimental data. Thus, the model may be used for determining the optimal conditions under which a membrane may achieve a given value of permselectivity. 79 4.2 Background As noted in the previous chapters, SiC has many unique and desirable properties, which have made it a prime candidate for fabrication of microporous membranes (Ciora et al., 2004; Elyassi et al., 2007, 2008; Suda et al., 2006). Being permselective at high temperatures, particularly in the presence of steam, has made the SiC membranes an excellent candidate for applications in membrane reactors. Indeed, in the water-gas shift and methane steam reforming reactions, where the membrane must function in the presence of high-temperature steam, the performance of the SiC membranes has been shown to be promising. As previously described in moe detail, there are several ways to fabricate such membranes, including by the sol-gel technique, the pyrolysis of pre- ceramic precursors (Elyassi et al., 2007, 2008), and by the CVI/CVD (Chen et al., 2008, see chapter 3). It is the last technique that is of interest in this Thesis. Since the CVD method is widely used in the chemical and semiconductor industries, the evolution of the pores’ size, the transport of the reactants through the pore space, and determination of the kinetics of the CVD reaction are of much current research interest (Lin et al., 1989; Lin and and Burggraaf, 1991 and 1992; Tsapatsis and Gavalas, 1992, 1997). To develop a deep understanding of the CVD process, one must have an accurate model of the pore space in which the CVD process takes place, as well as the correct mechanism(s) by which molecules and their mixtures are transported through the pore space. If such a model is developed, it may be used for optimizing the performance of the membranes. 80 The CVD process involves the transport of the reactants in a pore space, and various approaches have been used, so far, to model such transport processes in porous materials. One approach is based on the classical equations of transport (Lin et al., 1989, 1991,1992; Gavalas et al., 1989; Langhoff et al., 2008). For example, Langhoff and Schnak (2008) modeled the CVI of pyrolytic carbon as a moving boundary problem, in order to determine the evolution of the structure of the carbon layer, using a one- dimensional continuum model. Such an approach can neither take into account the effect of the morphology of the pore space, i.e., its PSD and pore interconnectivity, on the transport of the gases, nor can it be predictive quantitatively, unless some adjustable parameters are introduced into the model. A more realistic approach is one that represents the porous material by a network of interconnected pores (or bonds), joined together at the network’s nodes (Sahimi et al., 1990; Rieckmann and Keil, 1997, 1999). Since most porous materials contain interconnected pores, the connectivity strongly affects the transport of gases through their pore space. In addition, a fraction of the pores may be plugged as a result of the deposition of the reaction products on their surface and become inaccessible to the reactants during the deposition part of the CVD or CVI. If the fraction of the inaccessible pores reaches the percolation threshold of the pore space (Sahimi, 1993, 1994, 1995, 2003), the pore space will lose its macroscopic connectivity and the transport of the gases ceases. The percolation threshold is the minimum fraction of the unplugged pores for the formation of a sample-spanning cluster of open pores. 81 The shrinkage of the pores and their blockage also occur in a variety of other important phenomena and have, therefore, been studied using pore-network models. Sahimi and Tsotsis (1985) and Arbabi and Sahimi (1991) studied the pore blockage phenomenon in the deactivation of porous catalysts. Nukunya et al. (2005) studied the clogging of the pore space due to the biomass growth from the biodegradation of organic compounds. Wood et al. (2002) developed a pore-network model that incorporated a model of diffusion and reaction in the pores and an analysis of the pore filling sequence by capillary condensation. Their model was general, allowing for any type of kinetic expression for the reactions. Beigi et al. (2009) utilized the pore-network model to study catalytic dehydration of methanol. We develop in this chapter a pore-network model for the evolution of the pore space of microporous SiC membranes during the CVI/CVD processes that we have studied experimentally (chapter 2), and investigate the transport and reaction of a binary mixture of a polymeric precursor and the carrier gas (Ar) in the pore space. The results of the computer simulations are then compared with our own experimental data. To our knowledge this is the first time that such a model has been developed for the evolution of a membrane’s pore space during the CVD and/or CVI processes. 82 4.3 Formulation of the Model Figure 4.1 shows the schematic of the model for the CVI/CVD process that is used for the fabrication of the SiC membranes. In the model we assume that the pressure drop along the axial direction is negligible, and that the pressure is uniform in both the shell and tube sides. To describe the formulation of the model, we divide the rest of this section into three parts. First, we describe the formulation of the model inside the membrane. Next, we describe the model for the tube-side, and, finally for the shell side. Figure 4.1 Schematic of the model of the tubular membrane. 83 4.3.1 Inside the Membrane In the CVI/CVD process the reaction takes place inside the pore space of the membrane. The reactant diffuses into the pore space, reaches the pores’ surface and reacts there, and forms the solid product that deposits on the pores’ internal surface. As the deposition proceeds, the pores shrink. We represent the initial pore space of the membrane’s support by a simple-cubic network, in which each pore is initially connected to 10 other pores (5 pores are connected to each end of any pore), and each node of the network connects 6 pores. The schematic of the pore network is shown in Figure 4.1. The pores - the bonds of the network - are assumed to be cylindrical with an effective radii r p , selected at random from a PSD f(r p ). We selected an analytical expression for f(r p ) such that the shape of the resulting PSD mimicked the experimental PSD of the membrane (support). It is given by (4.1) where r a and r min are the average and minimum pore radii, respectively. r min was set equal to the molecular diameter of He, which is 0.26 nm. The average pore size r a at the beginning of the deposition was taken to be the same as the average pore size of one of the SiC tubular supports, prepared in our laboratory, which is 65 nm. Figure 4.2 compares the analytical f(rp) with the experimental PSD, which was measured using the flow permporometry (FP) technique and isopropanol as the wetting liquid. It is clear that Equation (4.1) qualitatively mimics the experimental PSD. 84 We should point out that extracting the PSD from a FP experiment is, on its own, an interesting and important problem. Since the FP technique is based on controlled blocking of the pores by capillary condensation and simultaneous measurement of the flux and permeance of a gas passing through the remaining open pores in the porous sample, one measures the size distribution of the accessible or active pores, since the dead-end pores do not contribute to the measured flux. In principle, one must use a pore network model together with the data obtained by the FP experiments, in order to determine the PSD of the pore space, if the connectivity of the porous sample is known. But, because each connected path for the transport of a gas through the sample is controlled by the size of the smallest pore, the FP experiments yield the size distribution of such pores that, for now, suffices for our purpose (in chapter 5 we develop a more elaborate network model of the FP technique). Note that the experimental PSD, if determined by any other technique other than FP, can also be used directly in the computer simulations. In addition, the simple-cubic network was used for convenience. One may utilize other types of networks, including those that have a disordered topology, such as the Voronoi network that was utilized in the recent molecular simulations by our group of the transport of gaseous mixtures through the SiC membranes (Rajabbeigi et al., 2009). 85 Figure 4.2 Comparison of the experimental data for the pore size distribution with the pore size distribution generated by Equation (4.1). The transport of gaseous mixtures in the pore network is modeled based on the dusty-gas (DG) model (Mason and Malinauskas, 1983; Krishna and Wesselingh, 1997). The DG model itself is based on the Maxwell-Stefan (MS) equations (Taylor and Krishna, 1993), which are approximations of the Boltzmann’s equations for dilute gas mixtures. Since the pores’ size varies during the deposition, and the transport mechanisms depend on the pores’ size, all the relevant transport mechanisms must be included in the model. They include, (i) viscous flow, the contribution of which is significant only in the large pores, particularly at the beginning of the deposition process 86 when the support’s pores are large; (ii) hindered or configurational diffusion, which is significant when the pore size is small enough so that λ = R k /r p is not small, where λ is the ratio of the molecular size R k of component k and the pore size r p , and (iii) Knudsen diffusion, which is dominant when the collisions between the molecules and the pores’ walls are important (which is the case when the molecules’ mean-free paths are larger than a pores’ radius). Therefore, the total flux J in any pore is given by, (4.2) where J V and J D are, respectively, the viscous and diffusive fluxes. If z denotes the axial direction in each pore, J V is given by: (4.3) Here, C = [C A , C B ] T , and , with Z being the compressibility factor, R the gas constant, T the temperature, r p the pore’s radius, µ the viscosity, and C=C A +C B . The diffusive flux J D , according to the DG-MS equations, is given by: (4.4) 87 where the entries of the matrix D are given by: (4.5) with being the Knudsen diffusivity of component i, x k the mole fraction of component k, and the hindered or configurational diffusivity of component k. is given by, (4.6) where M i is the molecular weight of component i . The diffusivity D k is given by, (4.7) where the bulk diffusivity, and a function that has been determined both theoretically and semi-empirically (Deen, 1987). We estimated using the well-known expressions for the diffusion coefficients of gases (Bird et al., 2007). Hindered diffusion of a single gas in pore-network models of porous materials was studied previously (Sahimi and Jue, 1989; Sahimi, 1992; Zhang and Seaton, 1992). For Brenner and Gaydos (1977) derived the following expression, 88 (4.8) while for Mavrovouniotis and Brenner (1988) obtained (4.9) Pappenheimer et al. (1951) and Renkin (1954) proposed a semi-emprirical correlation for given by, (4.10) We used all the above expressions in the computations. The material balance for the reactant within a single pore under isothermal conditions, taking into account the chemical reactions that occur in the pore, is given by, (4.11) where R x is the consumption rate of the reactant (A corresponds to TPS) in mol/m 3 .s, and for Argon (component B) one has, (4.12) 89 We assume that the CVD/CVI process within the pores is at quasi-steady state, due to the fact that, relative to the rates of diffusion and reaction, the changes in the pore due to deposition are slow. Moreover, within each pore the transport is significant only along its axial direction. Thus, the concentration at any point within a pore represents an average over the pore’s cross-sectional area. Therefore, within each pore one has, (4.13) (4.14) where, and, , which are given by given by Equations (4.3) and (4.4). We assumed that the deposition of the solid products of the reaction on the pores’ surface forms a uniform layer. The formation rate for the solid product is given by (4.15) where n is the reaction order. Our experiments have yielded n ≅1.4 (see further discussion in chapter 3), and K ≅7.1×10 −4 m 3n−2 /s.mol n−1 , which are used in the simulations. R s is defined on the basis of the unit pore surface area, where as the consumption rate R x for 90 the reactants is based on the unit pore volume. The two rates are, therefore, linked together through, , (4.16) where n A and n s are, respectively, the stoichiometric coefficients for the reactant A and the solid product. We assumed that, n A = n s . For uniform deposition of the solid products on the pores’ surface, (4.17) which enables us to compute the change in the pore size due to the deposition of SiC on its surface. Here, M SiC is the molecular weight of the SiC, and ρ its density. The pores are connected at the network’s nodes. Assuming no adsorption or chemical reaction at the nodes, the total fluxes of the species entering each node must be equal to those of species leaving it, (4.18) 91 where is the total flux of component i in pore jl that reaches node j, S jl the cross- sectional area of pore jl, Z j is the coordination number of node j (number of open pores connected to node j), and the sum is over all the pores jl that are connected to node j. Equations (4.13), (4.14), and (4.18) govern the transport of a gaseous mixture throughout a pore network. 4.3.2 The tube side The tube side (see Figure 4.1) is described by the following equations for the species A and B (the TPS and Ar, respectively) that were also used in the continuum model described in chapter 3, , (4.19) , (4.20) The boundary condition at z = 0 were, (4.21) (4.22) where and are, respectively, the molar flow rates of the TPS and Argon, and the inlet molar flow rates, the total molar flow rate, and and the inlet mole fractions. 92 4.3.3 The Shell side The governing equations and the boundary conditions at z = 0, for and , are similar to those for the tube side, except that the right sides of Equations (4.19) and (4.20) must be evaluated at r = R 2 (see Figure 4.1). The permeate molar flow rate in this case is . , (4.23) , (4.24) The boundary condition at z = 0 were, (4.25) (4.26) 4.4 Numerical Simulations We used networks of size L x ×L y ×L z , where the lengths L x , L y , and L z are in units of l, the pores’ length, and L z represents the network (membrane) thickness from the feed to permeate side. After some preliminary simulations, we used in all the calculations, L x =L y =L z =30, but we also investigated the effect of the thickness L z (see below). Clearly, 93 if the membrane’s thickness and the number of the pores L z in the macroscopic direction are specified, the (average) length l of the pores is also estimated readily. If Equations (4.2) and (4.10), together with (4.15) and (4.16), are substituted for and in Equations. (4.13), (4.14), and (4.18), a set of coupled nonlinear equations is obtained that govern the concentrations C A and C B of the two gases in every pore of the network. To solve the equations, we first discretized them using the finite-difference (FD) approximation. The grid points within each pore were numbered from 0 to n. Then, for a grid point k inside a pore, the FD approximation yields, , (4.27) , (4.28) whereas at the two ends of each pore we used the one-sided FD discretization, , (4.29) , (4.30) all of which are accurate to , where is the distance between two neighboring grid points, with l being the pores’ length. Similar FD approximations were used for 94 discrediting the equations that govern the gas concentrations in the nodes of the pore network. Introducing Equations. (4.27) - (4.30) into Equations. (4.13), (4.14), and (4.18) leads to a large system of nonlinear equation, (4.31) where C is the vector that contains the gases’ concentrations at all the grid points. The boundary conditions used were, (4.32) for the tube side of the membrane, where C tube is the total feed concentration, and, (4.33) for the permeate or shell side (as if a sweep gas is used there). We also used a no flux boundary condition (impermeable surface) for the other four surfaces, namely, , (4.34) The size of the set of equations depends on that of the network and the number of grid points inside each pore. If the network contains N nodes and we use M grid points 95 inside each pore, then the total number of equations to be solved is about 2N(3M+1), which requires a very efficient and accurate numerical technique. The Newton method was used to solve the nonlinear system. After linearizing the equations for the Newton iterations, the resulting set of the equations was then solved using the biconjugate- gradient method. In order to provide an accurate initial guess to begin the iteration process, we first solved the linear version of the problem (n = 1, see Equation (4.15)) by assuming a first-order reaction and solving the equations using the biconjugate-gradient method. The solution for the linear problem was then used as the initial guess for the nonlinear problem. This increased significantly the efficiency of the computations. To solve the equations for the shell- and tube-sides, we divided the length of the system into ten grid blocks (after some preliminary simulations and sensitivity analysis). The total length of the reactor is 5 cm. The governing equations for the membrane in the radial direction, adjacent to the first block in the z-direction were then solved by the method described above. The TPS consumption rate, namely, and the flux of argon, both evaluated at r = R 1 , were then computed for the first block. Equations (4.19) and (4.20) for the tube side, together with their boundary conditions, were then solved numerically for the first grid block in the z-direction, using the advancing-front method. The governing equations for the shell side were similarly solved. The fluxes of A and B in the first block were then used as the boundary conditions for the second block. The governing equations for the membrane in the r- direction were then solved, the flux of Ar and the consumption rate of the TPS were 96 computed, and the governing equations for the tube- and shell side were solved. The procedure was repeated until the solution for the entire system was obtained. Pore-network simulations were carried out for the deposition of SiC on the pores’ surface during the CVD/CVI processes. We assumed the network’s thickness to be, L z =1.6 µm (estimated experimentally). To begin the computer simulations, the sizes r p of the network’s pores were distributed according to equation (4.1), with an average pore size equal to the experimentally determined value for the membrane’s support, namely, 65 nm. Deposition of the solid product of the reaction on the pores’ surface shrinks the pore sizes. If the pores become smaller than the molecule size of the TPS, R TPS ≈5 Å, no more reaction will occur in them, and only Ar can diffuse through them. Each time the solution of the governing equations for the network is obtained, the change in the size of all the pores is computed through Equation (4.17), and the pores’ sizes are updated. Then, the solution of the governing equations in the new network configuration (with shrunk pores) is obtained, the pores’ sizes are updated, and so on. In the present study we study the evolution of the membrane’s pore space during the deposition of the SiC on the pores’ surfaces, which has not been studied before using a detailed pore-level model of the type developed here. 97 4.5 Results and Discussion As one might expect, decreasing the average pore size will make it more difficult for the TPS to percolate through the pore space. At some point all the pores will either be smaller than R TPS , the molecular size of the TPS, or become inaccessible to it by being surrounded by pores with r p ≤ r TPS . In other words, some pores with r p > r TPS might be trapped between the smaller pores. At that point the He and Ar permeances begin to level off, as shown in Figure 4.3. Figure 4.3 Dependence of the gases’ permeance on the deposition time. 98 Figure 4.4 presents the experimental data for the Ar permeance along with the simulation results for several orders of reaction n (see Equation (4.15)). The experimental data for the Ar permeance are in very good agreement with the simulation results, when we used the values of n and K reported previously (Chen et al, 2008). Figure 4.4 also indicates that during the first few h of the deposition, the Ar permeance decreases relatively slowly, but as the deposition proceeds it sharply declines and eventually levels off when no pores are open and accessible to the TPS. Figure 4.4 Comparison of the experimental data for the Ar permeance in the membrane with the results of the numerical simulations for several values n, the order of the reaction that produces the SiC. 99 As already pointed out, we assumed that the deposition of the SiC on the pores’ surface is uniform. If the pores are large enough, then a uniform deposition implies that the shrinking of the pores’ sizes is a kinetically-controlled, rather than a diffusion- controlled process. To check this, we computed the Thiele modulus in the pores. As the pores’ sizes follow a PSD, the Thiele modulus was computed for a pore with a size equal to the average pore size of the PSD. Moreover, since the reaction kinetics given by Equation (4.15) is nonlinear with the order of the effective order n of the reaction being 1 <n< 2, we computed the Thiele modulus for both a first- and second-order reaction. The results are presented in Figure 4.5. They indicate that in the initial stages of the process, the reaction-deposition is indeed kinetic-controlled. But, as the size of the pores shrinks, the Thiele modulus increases, and the process moves toward a diffusion-controlled phenomenon. Note that the results indicate a transition from a kinetically-controlled process to a diffusion-limited one. The transition is particularly pronounced for the second-order reaction. Thus, we may expect the same for the reaction kinetics that we used in our simulations. 100 Figure 4.5 Time-dependence of the Thiele modulus in a pore with a size equal to the average of the PSD, for first- and second-order kinetics. Figure 4.3 compares the computed Ar and He permeances with the experimental data. Figure 4.4 shows the effect of the kinetics parameter n, indicating that the value of n=1.4 provides the best results. The agreement between the computed and measured permeances for Ar is excellent, and good for He. Figure 4.6 compares the computed ideal selectivity of the membrane S is (defined by S = K He /K Ar ) as the deposition proceeds. The maximum difference between the 101 computed and measured selectivities is about 16%. In the light of the fact that the initial PSD used in the simulations does have some differences with the experimental value (see Figure 4.2), the network model actually predicts the selectively - one of the most important properties of a membrane - very accurately. As one might expect, the selectivity improves as the deposition proceeds, and the membranes moves toward being a microporous structure. Figure 4.6 Comparison of the computed and measured selectivity of the membrane. 102 The decrease in the permeance of the two gases with the deposition is linked with the corresponding changes in the pore structure of the support, as it evolves toward a microporous membrane. Figure 4.7 presents the decrease in the average pore size of the pore network. Over the first two hours the average pore size decreases sharply, but over the same time period the Ar and He permeances (see Figure 4.3) do not decrease very much, since in the same period most of the pores are still open and accessible to both gases. As the deposition proceeds for longer periods of time, even a small change in the pore sizes may cause a significant reduction in the Ar and He permeances, because in the pores with R He =r min <r p <R Ar only He can be transported, whereas in the pores with R Ar <r p <R TPS only Ar and He can flow and diffuse. Since the TPS can no longer access such pores, no reaction occurs there and, therefore, pore shrinkage stops. On the other hand, in the pores with r p > R TPS all the gases diffuse and flow and, thus, the reaction does occur in them. Therefore, so far as the TPS is concerned, there is a percolation effect at work, namely, a fraction q of the pores given by, , (4.35) cannot be accessed by it. Clearly, if q > p c , where p c is the percolation threshold of the network, no macroscopic transport of the TPS can occur in the pore network, implying that there would be no significant reaction and clogging of the pores (except possibly near the external surface). For a 3D network of average coordination number (connectivity) Z p one has (Sahimi, 1994, 1995, 2003), p c ≅1.5/Z p . Thus, for the simple- cubic network (Z p = 6) that we used in the simulations, one has, p c ≅0.25. 103 This phenomenon can also be understood by inspecting the evolution of the network's PSD. Figure 4.8 (a,b) presents the PSD at four different times during the deposition. It indicates that as the pore space evolves, the PSD's tail becomes shorter, because larger pores shrink and are eventually closed or become inaccessible to the TPS. Thus, the TPS finds it increasingly more difficult to find a percolating path of the accessible pores throughout the pore space. This is clearly demonstrated in Figure 4.9 that presents the fraction of the pores that become closed to the TPS as the reaction proceeds. Initially, the fraction of such pores is small. But, after sometime, the fraction of the pores that are closed to the TPS sharply increases, making it more difficult for the TPS to penetrate the pore space and react. Figure 4.7 Dependence of the average pore size on the deposition time. 104 Figure 4.8 a,b Evolution of the pore size distribution with the time as the deposition and, hence, pore shrinkage and plugging, proceed. 105 Figure 4.9 Time-dependence of the fraction of the pores that are closed to the TPS. We also studied the effect of the initial connectivity Z p of the network (support) on the Ar permeance during the deposition. To do so, we used networks with an average connectivity (coordination number) Z p < 6, which we construct by randomizing the simple-cubic network. That is, we selected at random a fraction p b of the pores given by, , (4.36) and removed them from the network, i.e., set their radii equal to zero. Figure (4.10) presents the effect of Z p on the Ar permeance. The results indicate that the trends in the reduction of the Ar permeance (and, similarly, the He permeance) are not very sensitive 106 to the initial connectivity of the network, since the largest pores quickly shrink and eventually become closed or inaccessible to the TPS. This is understandable, because the deposition phenomenon is not a percolation process. As soon as the TPS reacts in any pore, the reaction product deposits on the pore's surface. This is in contrast with the transport of Ar or He through the membrane, which is a percolation process because for the gas transport to occur, there has to be a sample-spanning path of open and connected pores. Figure 4.10 Effect of the membrane connectivity on the Ar permeance during the deposition. The effect of the membrane’s thickness on its permeance and permselectivity is important and, thus, we studied it through the pore-network simulations. Figure 4.11 107 presents the dependence of the Ar permeance on the thickness L z of the network. If we keep the cross-sectional area of the network constant but increase its thickness, the permeance decreases. The reason for the decrease in the permeances is once again due to the percolation effect. Increasing the thickness, while keeping the cross-sectional area constant, implies that the pore network becomes elongated. As the thickness increases, the network resembles increasingly a two-dimensional system with lower connectivity, and it is well-known (Sahimi, 1994, 1995, 2003) that reducing the dimensionality of a network results in reduced permeances, since the probability of finding a percolating path for macroscopic transport decreases. Figure 4.11 Dependence of the Ar permeance on the membrane’s thickness. The connectivity of the pore network is Z p = 6. 108 4.6 Extension to Binary Gaseous Mixtures The three-dimensional pore network model described here has been extended in collaboration with Dr. Chen (Chen et al., 2008b) for studying the transport and separation of a binary gas mixture through a nanoporous membrane. The sizes of the pores were distributed according to a PSD. The connectivity of the network and its thickness were also varied in order to study their effect on the transport of the gases. Knudsen and hindered diffusion, as well as viscous flow, were included in the model as the mechanisms of mass transport, although our simulations indicated that the contribution of viscous flow, over the range of the pore sizes that we considered, is negligible. However, with a broader PSD and larger pores, the effect of viscous flow may become important, in which case the model can take its effect into account. It was demonstrated that, provided that the network is large enough, adjusting the thickness of the network and its average pore size – a two-parameter fit of the experiments – provide estimates for the properties of the network that are in excellent agreement with the experimental data. Moreover, the model demonstrates the fundamental importance of two factors to the permselectivity of a membrane, namely, the tail of the PSD, and the percolation effect manifested through the connectivity of the pores that are accessible to the gas molecules. In addition, the results indicate that Knudsen diffusion is the dominant mechanism of gas transport in pores as small as 0.7 nm. The details of the model and extensive results were presented by Chen et al. (2008b) and, therefore, will not be repeated here. 109 4.7 Summary A pore-network model was developed to study the evolution of the pore structure of a SiC membrane during the deposition of SiC using the CVD/CVI technique. The multicomponent diffusion and reaction phenomena were modeled based on the dusty gas model, while the pore shrinkage and blockage were described by the reaction that generates solid SiC. The results of the computer simulations are in very good agreement with the experimental data. The effect of pores interconnectivity and the membrane’s thickness on its permeance was also studied. Because the computed pore structure and the Ar and He permeances, as well as the membrane’s selectivity, agree quantitatively with the experimental data, the model may be used for determining the optimal conditions under which the membrane’s permselectivity achieves a pre-set value. Work in this direction is in progress. 110 Chapter 5 Determination of the True Pore Size Distribution by Flow Permporometry Experiments: An Invasion Percolation Model 5.1 Introduction Characterization of porous materials, and in particular porous membranes that are of interest to us in this paper, has been an active area of research for a long time. In particular, because the morphology of a pore space strongly affects its flow, transport, reaction, and separation properties, its accurate and realistic characterization has been of prime importance. The morphology of a pore space consists of its pore size distribution (PSD), the pores' connectivity, and the structure of the pores' internal surface - smooth versus rough - that affects the material's adsorptive characteristics. The focus of the present chapter is the accurate determination of the PSD of a porous material, such as a membrane. Depending on the range of the pore sizes that a porous material contains, one may use a variety of experimental techniques to measure its PSD. Such techniques include mercury porosimetry, nuclear magnetic resonance, and the Brunauer-Emmett-Teller (BET) adsorption method. They have been described in detail by Lowell et al. (2004) and Sahimi 2010. Another popular method of measuring the PSD of a porous material has been based on flow permporometry (FPP) (Eyraud, et al. (1984), Mey-Marom and Katz 111 (1986), Katz and Baruch (1986), and Cao et al. (1993)). It is the FFP method that is examined in the present chapter. Measuring certain quantities of interest in order to utilize them for the characterization of a porous material is often experimentally challenging. The correct interpretation and utilization of the data for determining the PSD and other morphological characteristics presents additional unique challenges. The data are usually utilized by fitting them to a model that is thought to faithfully represent the structure of the porous material under consideration. Such models normally contain a few adjustable parameters that are estimated when the data are correlated using the model. However, if the model does not contain enough information on the important characteristics of a porous material, it cannot be expected to be useful and accurate for providing insights into the material's morphology. Chief among such characteristics, in addition to the PSD, is the connectivity of the pores. Flow and transport processes in porous material are strong functions of how the pores are distributed in its pore space, and are connected to one another. In fact, without a proper accounting for the effect of the connectivity, any information generated on the PSD of a porous material that is based on any kind of flow or transport experiments in the material is suspect and unreliable. To provide an example, consider the classical mercury injection and withdrawal experiments that have been used for decades to determine the PSD of a wide range of 112 porous materials. While the experiments themselves are conceptually simple, and are carried out without much difficulty, the interpretation of the results is not. The measurements are typically represented by a set of data in terms of the mercury pressure - the capillary pressure (see below) - that is required in order to push it into a porous sample, versus the Hg saturation (volume fraction in the pore space occupied by Hg). To extract the PSD from such data, for a long time it was assumed that the pores of the pore space form a bundle of capillary tubes that are either in series or in parallel, which is a gross misrepresentation of most pore space morphologies. It was only about three decades ago that Larson and Morrow (1981), demonstrated convincingly that pore connectivity plays a crucial role in interpreting mercury porosimetry data, and in extracting the correct PSD. Their work was followed by those of others (Thompson et al. (1987), Tsakiroglou and Payatakes (1990, 1998, and 2000), Rigby et al. (2000, and 2002)) who developed pore network models and used the concepts of percolation theory (Sahimi (1994 and 2001), Tsuru, et al. (2001)), which quantifies the effect of the connectivity of the pores on the macroscopic properties of a porous medium. The same issue is true about the FFP method, which essentially measures the pressure needed to blow a bubble of gas through a liquid-filled porous material, such as a membrane. If one does not utilize a proper model for interpreting the measured data, the PSD of the porous material that is obtained using the FPP data cannot possibly be accurate. We demonstrate this assertion in the present paper. The purpose of this chapter is, therefore, to develop a pore network model that can be used in conjunction with the 113 FPP data, in order to accurately determine the PSD of porous membranes and films. We show that the conventional method that is used currently in order to extract the PSD of a porous material based on the FPP experiments is often in gross error. The plan of this chapter is as follows. In the next section we first describe the typical FPP experiments. We then describe a new model based on the pore network representation of a pore space and the concepts of invasion percolation (Sahimi, 2010) for simulating the FFP process. Section 5.4 describes how the model is utilized to compute the dry and wet curves that are measured by the FPP method, in order to obtain the PSD of a porous material. In Section 5.5 we describe an iterative algorithm based on the pore network and IP models, in order to utilize the wet and dry curves of the FPP experiments to determine the PSD. The results are presented and discussed in Section 5.6, while the chapter is summarized in the last section. 5.2 Flow Permporometry Experiments Figure 5.1 shows the schematic of the FPP apparatus. In a typical FPP experiment, a non-condensable gas - typically He - is injected into the porous material under an applied pressure, which is controlled by the pressure valve that is connected to a pressure transducer, and monitored by a computer. As the applied pressure increases, the gas begins to exit the porous material through the surface opposite to the injection face. 114 The volume flow rate of the exiting gas is measured by a mass flow meter. The plot of the flow rates versus the applied pressure is referred to as the dry curve. In the next step the porous material is immersed in a container that has a liquid that wets the internal pore surface of the material. In the present study, we used isopropanol with a porous silicon-carbide (SiC) material that is used as the support for a SiC membrane that our group has been fabricating (see below). Being the wetting fluid (WF), the liquid invades the porous medium - the imbibition process (Sahimi, 2010) – and fills up all the accessible pores. Then, the non-condensable gas used in the dry test is injected into the porous material to displace the WF. Since, compared to the liquid, the gas is a fluid that does not wet the internal pore surface of the material, its injection requires applying an external pressure. As the gas pressure is increased gradually, it begins to expel the liquid from the porous material, and after sometime bubbles of the gas percolate through the medium and begin to exit it. Being the non-wetting fluid (NWF), the gas penetrates the pore space through a path of pores that requires the lowest capillary pressure, which is equivalent to saying that, at any given time, the gas invades the pore with the largest radius to which it has direct access. The volume flow rate of the gas that exits the porous material is also measured and a plot of the data versus the applied pressures is prepared. We refer to such a plot as the wet curve. The intersection of the dry and wet curves defines the terminal point of the FPP experiments. 115 Figure 5.1 Schematic of the experimental apparatus The wet and dry curves are then used to construct a PSD. To do so, the Laplace equation, (5.1) is used to obtain an estimate of the pore size r corresponding to the pressure P on the wet curve, where γ is the surface tension between the gas and the WF, and θ the contact angle of the NWF with the pores' surface. Assuming the gas to be a perfectly NWF, one has, θ 116 = 180° and, therefore, . In our experiments with the SiC membrane support, γ = 23 dyn/cm. To construct the PSD, the frequency or the fraction of the pore sizes that, for any given r, fall between r and r+dr must be determined. To do so, one divides the pressure range used in the experiment into many small subintervals. Let a subinterval be denoted by (P l ,P h ), where P l and P h = P l +dP denote, respectively, the low and high end of the pressure interval. Then, using the wet and dry curves we compute the quantity, (5.2) which represents the percentage of the pores that correspond to the subinterval (P l ,P h ), where the subscripts refer to the quantities evaluated at the low and high ends of the pressure range. Finally, given the q f values and the pore sizes determined by Equation (5.1), the PSD is constructed. We refer to the resulting PSD as the FPP-PSD. The above procedure is, however, based on two key assumptions. (1) The pores are cylindrical. In general, this is not true, but it also does not represent a serious error, because in the flow of a NWF through a porous medium only the radius of the throats - the channels or passages through which fluid flow occurs - is important. Therefore, even if the throats are not cylindrical, one can use a modified form 117 of Equation (5.1) for other throat shapes and still obtain estimates of the effective pore sizes (Sahimi, 2010). (2) The second, and most serious, assumption is that the pores are non- intersecting capillaries. For example, Tsuru et al. (2003), and Lenormand and Bories (1980), used a set of detailed FFP experiments in order to characterize ceramic membranes, and to correlate the experimental measurements with the membrane's performance in terms of its separation property. The assumption that was made in the FFP experiments was that the membrane consists of non-intersecting cylindrical pores. The second assumption is, clearly, in serious error. Most porous materials consist of interconnected pores. The resistance that a pore space of interconnected pores offers to fluid flow, particularly during the two-phase flow during the FPP experiments - is vastly different from the one offered by a bundle of non-intersecting pores that are either in parallel or in series, as is assumed in the standard procedure for extracting a PSD from the FPP experiments. In this chapter, we propose a model to correct this deficiency. We demonstrate that the standard FPP-PSD is in error very significantly, and propose a pore network model, based on invasion percolation that produces the correct PSD. The model is tested by generating synthetic FPP data, and is examined further by using the FPP data for the porous support of a SiC membrane that our group has been fabricating. 118 5.3 Invasion Percolation Model of Flow Permporometry To develop a model for interpreting the experimental data obtained by the FPP and to extract the correct PSD, we represent the porous materials by a three-dimensional (3D) network of interconnected pores. An effective radius is attributed to each pore throat, represented by the network's bonds that connect to the sites, which are selected from a PSD. The goal is to determine a PSD from the wet and dry curves of the FPP experimental by using the pore network model, which matches the true PSD of the material. To compute the dry and wet curves using the pore network model, we need an algorithm that simulates the invasion of the pore network, saturated by the WF, by a non- condensable gas that acts as the NWF. To this end, we utilize the invasion percolation (IP) model. The IP model was proposed by Lenormand and Bories (1980), Chandler et al. (1982), and Wilkinson and Willemsen (1983) to model the displacement of a WF from a porous medium by a NWF, if the porous medium is initially saturated by the WF. This is precisely the same phenomenon as the process of expelling the WF from a porous material by a non-condensable gas during the FPP experiments. In the IP model, the network is initially filled with a WF called the defender (since it “defends” itself against the invading fluid). To each bond of the network is assigned a random number distributed according to a statistical distribution (in the 119 original IP model (Lenormand and Bories (1980), Chandler et al. (1982), and Wilkinson and Willemsen (1983)) thenumbers were distributed uniformly in [0,1]). The numbers are interpreted as the radii of the pore throats. Then, the displacing fluid - the invader or the NWF - is injected into the network under an external pressure, to displace the defender or the WF. To do so, at each time step a bond (pore throat) next to the interface between the WF and NWF that requires the smallest pressure is selected. According to Equation (5.1), the pore throat that requires the smallest pressure is the one with the largest radius. In the original IP model, the numbers distributed uniformly in [0,1] were interpreted as the pressures needed for invading the pores, and at each time step the bond with the smallest random number was selected, which is equivalent to selecting the pore throat with the largest radius. Once the next pore is selected, the external pressure is adjusted, so that the pore can be invaded, and the WF in it is expelled. Then, the location of the interface between the WF and NWF is updated, the largest pore throat is identified, the external pressure is adjusted based on Equation (5.1), so that the NWF can enter the selected pore, and so on. Expelling the WF from the porous material is, therefore, ranked in terms of the largest pores that the invading fluid must travel through. The breakthrough occurs once the pore throats that have been invaded by the NWF - the non-condensable gas - form a sample-spanning cluster between the injection face of the network and the opposite face. Two versions of such a model may be considered. In one model the defender (the WF) is incompressible and, therefore, if a cluster of pores filled with it is surrounded by the invader (the NWF or the gas), it becomes trapped. In the second model, trapping of 120 the WF is ignored. Knackstedt et al. (2002) showed that they differences between the two versions of the model is significant in a 2D system. On the other hand, the effect of trapping in 3D, which is in fact of interest to us in this chapter, is negligible (Wilkinson and Barsony, 1984). Moreover, pore network simulation of the IP model that we describe here requires an efficient computational algorithm. Sheppard et al. (1999) and Knackstedt et al. (2000) developed a highly efficient computational algorithm for simulating the IP model, while Hashemi et al. (1999) extended the model to include the effect of flow of thin films on the pores's surfaces, and when more than one fluid invades the pore space. In the present paper, we ignore the compressibility effect. Ebrahimi (2010) provides a good overview of the IP models and their wide variety of applications. The model described accurately mimics the experiments during the FPP. First, the gas flow is measured through a dry network, representing the membrane, as a function of the pressure in order to construct the dry curve (see below). Then, the membrane is saturated by the WF, and the gas flow that expels the WF from the membrane, simulated by the IP algorithm, is computed (see below for the details) as a function of the applied pressure. At very low pressures the pores are filled mostly by the liquid WF and the gas flow is very low. At a certain threshold pressure - the bubble-point or percolation pressure - the first sample-spanning cluster of the largest pores invaded by the gas is formed, and a measureable volume of gas through such pores commences. Further increase in the applied pressure opens up the smaller pores according to the Laplace equation, Equation (5.1). At the highest pressure the gas flow in the dry membrane must 121 be equal to that of the wet membrane invaded by the gas. If this is not the case, it implies that either there still are some smaller pores in the membrane or porous material that are filled with liquid WF and are closed to the gas flow. This is because they need very large applied pressure to be invaded which, however, may break the porous material, or that some of the larger pores that are filled by the WF are trapped by the gas-filled pores. 5.4 Computing the Dry and Wet Curves To model the FPP experiments and determine the correct PSD, we represent the porous material or membrane by a 3D network of pore throats that are connected to one another at the network's sites. An initial PSD is assumed, in order to attribute an effective pore radius to each pore throat. The pore sizes are selected from the initial PSD. The dry curve is then computed (see below). The IP algorithm described in Section 5.3 is then utilized in order to invade the pore network by the NWF, which is saturated by the WF. The breakthrough point (BTP) - which is just a percolation threshold (Sahimi, 2010, and 1994) - is reached when a sample-spanning cluster of the pore throats is formed, from which the WF has been expelled and is filled by the NWF. The BTP represents the initial point of the wet curve. As injection of the gas into the pore network continues, more pore throats are evacuated by the WF and are filled by the NWF or the gas. During the entire procedure the applied pressure is increased according to the algorithm described above. Then, for each pressure beyond the BTP we compute the volume flow rate of the gas exiting the pore space. Since the gas is assumed not to be soluble in the liquid and non- 122 condensable (so that no thin _lm of the NWF can form), each pore throat is either occupied by the liquid or by the gas. Thus, as described shortly, the computations of the gas flow rate is carried out using only that part of the pore network that the gas resides in. We now describe the computation of the dry and wet curves in more detail. A. The dry curve The procedure for computing the flow rate of helium in the pore network at each applied pressure is similar to our recent pore network modeling of transport of gases and their mixtures in nanoporous membranes (Chen et al. (2008), and Mourhatch et al. (2010)). In principle, the gas is transported through the pore space by Knudsen and hindered diffusion, as well as viscous flow. Thus, the total flux J in any pore throat is given by, (5.3) Where J V and J D are, respectively, the viscous and diffusive fluxes. If z denotes the local axial direction in each pore throat, J z V is given by: (5.4) where, , with Z being the compressibility factor, R the gas constant, 123 T the temperature, r p the pore's radius, and µ the viscosity. The diffusive flux J z D is given by: (5.5) where DK is Knudsen diffusivity is given by, (5.6) with M He being the molecular weight of helium. In principle, one must also include the hindered diffusivity. But, in the present work the minimum pore size is still too large for the hindered diffusivity to be important. The material balance for helium within a single pore under isothermal condition is given by, (5.7) where, The pores throats are connected at the network's sites. Assuming no adsorption or chemical reaction at the nodes, the total flux of helium entering each node must be equal to that leaving it, (5.8) 124 where is the total flux in pore jl that reaches node j, S jl the cross-sectional area of pore jl, Z j the coordination number or connectivity of node j - number of pore throats connected to node j - and the sum is over all the pore throats jl that are connected to node j. Equations (5.7), and (5.8) govern the transport of helium throughout a pore network. After some preliminary simulations to decide the network size, networks of size L x ×L y ×L z = 30 were utilized in all the computations described below, where the lengths Lx, Ly, and Lz are in units of l the pores' length, and Lz represents the network thickness form the injection to withdrawal faces. When Equations (5.4) - (5.6) are substituted for J z V and J z D in Equations (5.7) and (5.8), a set of coupled equations is obtained that govern the concentration C of helium in every pore throat of the network and its nodes. To solve the equations, we first discretized them using the finite-difference (FD) approximation. The grid points within each pore were numbered from 0 to n. Then, for a grid point k inside a pore, the FD approximation yields, , (5.9) , (5.10) whereas at the two ends of each pore we used the one-sided FD discretization, 125 , (5.11) , (5.12) all of which are accurate to , where is the distance between two neighboring grid points. Similar FD approximations were used for discretizing the equations that govern the gas concentrations in the nodes of the pore network. Introducing Equations (5.9)-(5.12) into Equations. (5.7) and (5.8), leads to a large system of equation, (5.13) where C is the vector that contains the helium concentrations at all the grid points throughout the pore network. The pressure at the injection face is specified, and is converted to equivalent concentration using the ideal gas law (since He is used). The resulting set of the equations was then solved using the biconjugate-gradient method. Once the concentration profile is computed for the given applied pressure, the volume flow rate of the gas at the exit plane is computed by summing the flux of He in all the pore throats that are connected directly to the withdrawal face. A new injection pressure is set and the computations are repeated, until the dry curve is constructed. 126 B. The wet curve Computing the wet curve is similar to that of the dry curve, except that the above procedure is used using only that part of the pore network that is occupied by the gas, He, because the gas is not soluble in the liquid and, therefore, the pore throats that contain the liquid do not have any gas in them. Thus, at the BTP there is a very small amount of gas exiting the pore network, but as the applied pressure - set by the required capillary pressure, Equation (5.1), for invading the pore throats - increases, the volume of the gas exiting the pore network increases. For each injection (capillary) pressure beyond the BTP, the computations are repeated and the volume of the gas leaving the pore network is computed, in order to construct the wet curve. Once the wet curve intersects the dry curve, the computations are terminated. 5.5. Computing the Correct Pore Size Distribution The procedure to extract the correct PSD, given the experimental wet and dry curves, is as follows. (i) A pore network is generated in which the sizes of the pore throats are selected from an initial PSD. The PSD used is parameterized, i.e., it contains a few 127 parameters that must be adjusted in such a way that the computed wet and dry curves match the experimental ones as closely as possible (see below). (ii) Given the initial pore network, the wet and dry curves are computed over the same pressure range as those in the actual FPP experiments. (iii) If the computed wet and dry curves match the experimental ones, the computations are terminated. In that case, the PSD of the network with which the matching wet and dry curves were computed represent the true PSD of the porous material. (iv) If the computed and experimental wet and dry curves do not match according to some criterion (see below), the PSD is adjusted by changing one or more of its parameters. The most sensible way of doing so is perhaps holding all but one of the parameters constant and vary the remaining one, in order to understand its effect on the computed wet and dry curves. Note that, the connectivity of the pore network is, in principle, another parameter that can be adjusted. However, there are already certain procedures that use adsorption isotherms and the concepts of the percolation theory, in order to estimate the connectivity of a porous material (Sahimi, 2010, Seaton, 1995, Liu et al. 1992, and Murray et al. 1999). Thus, we assume that the connectivity of the porous material is estimated independently, and the pore network that is generated has the same connectivity. 128 (v) Once the PSD is adjusted, steps (ii) and (iii) are repeated. If the newly computed wet and dry curves match the experimental curves, the computations are terminated. Otherwise, the PSD is adjusted again and the procedure is repeated until the computed wet and dry curves converge to the experimental ones. The final adjusted PSD represents the actual PSD of the porous material, and is referred to as the IP-PSD. The criterion for the convergence of the computations to the true PSD is based on minimizing the sum of the squares of the differences between the computed dry and wet curves and the corresponding experimental data. In the computations described below, the size of the pore throats was distributed according to the following PSD, (5.14) where r a and r min are the average and minimum pore radii, respectively. Thus, the PSD contains two parameters, r min and r a . The minimum pore size is set by the limit of validity of Equation (5.1) used both in the FPP experiments and the pore network simulations. Thus, in the computations r min was fixed at 2 nm, which is the lower limit of the validity of Equation (5.1). If much smaller pores exist, one may used a modified form of Equation (5.1), the augmented Laplace equation, which adds a term that represents the disjointed 129 pressure. The average pore size r a is, therefore, the only parameter that must be adjusted, in order to compute wet and dry curves that match the experimental ones. The reason that we use Equation (5.14) is that it contains the general features of a typical PSD of membranes: (1) It is not symmetric (Gaussian), but skewed toward the smaller pore throats. (2) It has a maximum pore size (hence, not uniform). (3) It has a tail that can be quite long (representing the meso- and macropores). Clearly, any other functional form for the PSD with a number of parameters can be used, as the pore networkand the IP models are completely general. 5.6 Results and Discussion To test the method, we first generated a pore network in which the PSD was given by Equation (5.19). We used a simple-cubic network (the connectivity of which is 6), set the average pore size ra to be 45 nm, and computed the wet and dry curves by the method described in Section 5.4. We then assumed that the computed wet and dry curves represent actual (synthetic) FPP data, and attempted to use the iterative algorithm described in Section5 to determine the PSD. Thus, the iterative method was utilized in order to compute the wet and dry curves that most accurately match the synthetic data. 130 The results are shown in Figure 5.2, where the originally-computed wet and dry curves, taken as the synthetic FPP data, are compared with the curves calculated by the iterative method of Section 5.5. The agreement is excellent, which implies that the last adjusted IP-PSD, obtained by the iterative algorithm, should also match the original PSD with which the pore network and the synthetic data for the wet and dry curves were computed. Figure 5.3 compares the last adjusted IP-PSD with the original PSD. The agreement between the two is excellent. On the other hand, if we use the synthetic wet and dry curves and the usual FPP procedure that is currently used for determining the PSD, we obtain the PSD that is also shown in Figure 5.3. It is clear that the current FPP procedure yields a PSD that is grossly in error. In particular, it yields a PSD that is considerably narrower than the actual PSD, with an average pore size of 28 nm that is much smaller than the actual value of 45 nm. Figure 5.2 Comparison of synthetic wet and dry curves with the curves calculated by IP (r a =45nm). 131 Figure 5.3 Comparison of FPP-PSD with IP PSD (r a =45nm). To test whether the accuracy of the results depends sensitively on the details of the original PSD, we generated another pore network with the PSD (5.19), but with an average pore size, r a = 100 nm. The wet and dry curves were then computed by the algorithm described in Section 5.4, and were assumed to represent the (synthetic) FPP data. Then, the iterative algorithm was used to compute the wet and dry curve by adjusting the PSD, using the synthetic FPP data as the basis for the computations. Figure 5.4 compares the two sets of wet and dry curves. Once again, the agreement is excellent. Figure 5.5 compares the last adjusted IP-PSD with the original PSD with an average pore size of 100 nm. The agreement is, once again, excellent. Also shown is the FPP-PSD that was computed based on the synthetic wet and dry curves and utilizing the currently-used FPP procedure for determining the PSD, described in Section 5.2. Once again, the PSD 132 obtained by the usual FPP procedure is grossly in error, and is much narrower than the actual PSD. Figure 5.4 Comparison of synthetic wet and dry curves with the curves calculated by IP (r a =100 nm). Figure 5.5 Comparison of FPP-PSD with IP PSD (r a =100nm). 133 We now turn our attention to actual experimental data. In this case, we use a porous SiC material that we use as the support for fabricating SiC nanoporous membranes (Chen et al. 2008, and Mourhatch et al. 2010). The measured wet and dry curves are presented in Figure 5.6. Also shown is the best match of the dry and wet curves. The agreement is excellent. To carry out the computations we assumed that the PSD is given by Equation (5.19) and used the iterative method to adjust the average pore size r a . Figure 5.7 shows the progression of the convergence of the computed wet curve toward the experimental one. The computations began by assuming that the PSD was given by Equation (5.19) with an initial average pore size of 65 nm (which is what the FPP-PSD, obtained by the usual procedure of Section 5.2, indicated). After three series of iterations, the computed wet curve is in excellent agreement with the experimental curve, and one obtains a PSD with an average pore size, r a = 80 nm. Figure 5.6 Comparison of experimental wet and dry curves with the curves calculated by IP method. 134 Figure 5.7 The progression of the convergence of the computed wet curve toward the experimental one. Figure 5.8 presents the last adjusted IP-PSD, obtained by the proposed iterative algorithm, and compares it with the FPP-PSD obtained by the usual FPP procedure described in Section 5.2. The computations indicate that the average pore size of the SiC support is about 80 nm, whereas the FPP-PSD yields an average pore size of 65 nm. Thus, both the synthetic and actual experimental data indicate that the usual FPP procedure underestimates the average pore size. The reason is that the usual FPP method yields a much narrower PSD than the true one. Given that the accuracy of the IP model was demonstrated by the synthetic data, we are confident that the computed IP-PSD shown in Figure 5.8 is accurate and representative of the true PSD of the SiC porous support. The comparison shown in Figure 5.8 also exhibits the gross error in using the 135 usual FPP procedure for determining the PSD, which assumes that the pore space consists of a bundle of non-intersecting pores. Figure 5.8 Comparison of FPP-PSD with IP PSD (experimental data). It is instructive to study how the computed PSD evolves as the gas is injected into the porous material. Figure 5.9 presents the PSD computed at various stages of the simulations, using the synthetic data of Figure 5.2. As pointed out earlier, the gas is a NWF and, therefore, invades the largest pores at the lowest applied pressures. Thus, the computed PSD at such pressures consists mostly of very large pores. But, as the applied pressure increases, an increasing percentage of the smaller accessible pores are invaded and, therefore, the PSD gradually shifts to smaller pore sizes, to the point that the final IP-PSD matches the data, but also contains a long tail of the larger pores. 136 Figure 5.9 PSD computed at various stages of the simulations, using the synthetic data of Figure 5.2. 5.7 Summary The basic principle of flow permporometry (FPP), widely used for determining the PSD of porous materials and inorganic membranes, is based on the capillary pressure and the displacement of a wetting fluid (WF) in the pore space by a non-condensable gas that represents a non-wetting fluid (NWF). The currently-used procedure for determining the PSD by the FPP experiments is, however, is based on the assumption that the pore space consists of a bundle of non-intersecting pores. Flow properties of any porous material, and in particular those for two-phase flow, are strongly influenced by the interconnectivity of its pores. 137 In this chapter we demonstrated that the bundle of pores assumption leads to erroneous PSDs that do not represent the actual geometry of the pore space. It was shown that the currently-utilized procedure for using the FPP data to determine the PSD yields much narrower PSDs with a significantly lower average pore size than the actual ones. To demonstrate this, we developed a pore network model for the FPP experiments that utilizes the invasion percolation (IP) model. In the IP model the pore space is initially saturated by a WF and then is invaded by a condensable gas - the NWF - that displaces the WF and expel it from the pore space, in a manner completely similar to what happens during typical FPP experiments. We then proposed an iterative algorithm based on the pore network and the invasion percolation models for determining the true PSD of a membrane. The accuracy of the models was first demonstrated using synthetically generated models of porous materials and the FPP curves, and then examined further by using the FPP wet and dry curves for the porous support of a silicon-carbide membrane. 138 Chapter 6 Fabrication of Silicon Carbide Nanotubes 6.1 Introduction Carbon and other types of nanotubes have generated great scientific and commercial interest because of their many applications that stem from their extraordinary materials properties. Included among these properties are high mechanical strength, high thermal conductivity, and attractive electronic properties. They find applications in sensors, actuators, as well as in storage, and separation processes. Among the limitations for carbon nanotubes is their inability to survive in high-temperature, harsh-environment applications. SiC (SiCNT) nanotubes are superior to carbon nanotubes in that regard. Various synthetic methods have been used by several groups in the past to grow SiCNT. They include the template-based synthesis (Rummeli et al., 2005) and the evaporation–condensation method (Li et al., 2008). A method that is attracting recent attention is the shape memory synthesis (SMS) method, whereby macrostructural features of a solid are retained after synthesis. For instance, SiCNT are prepared by heating a mixture of Si and SiO 2 at 1200–1350 ◦ C to generate SiO vapor (Han et al., 1997). Then, this vapor is allowed to come in contact with carbon nanotubes to produce SiCNT. The reaction between the carbon nanotubes and SiO vapor generates CO as a by-product, which needs to be pumped out of the system, in order for the equilibrium to shift towards 139 the formation of SiC. When the SMS method is employed to prepare SiCNT, then these nanotubes have the same tabular structure with the starting carbon nanotubes. In the SMS method one can produce SiCNT with different sizes by simply controlling the size of the precursor carbon nanotubes (Nhut et al., 2002). Sun et al., (2002), were able to synthesize one-dimensional silicon-carbon nanotubes and nanowires of various shapes and structures, via the reaction of silicon (produced by disproportionation reaction of SiO) with multiwalled carbon nanotubes (as templates) at different temperatures. They observed multi-walled silicon carbide nanotube (SiCNT). Li, et al., (2008), have synthesized SiC nanowires, SiC/SiO2 core– shell nanocables, and SiC nanotubes simultaneously by direct heating of Si powders and multiwall carbon nanotubes (MWCNTs). The fabricated nanotubes have a outer diameters of about 20 nm. Theoretical studies have shown that SiCNTs have better reactivity than CNTs due to their polar nature. For example, SiCNTs show better hydrogen storage performance than CNTs (Mpourmpakis et al., 2006) in that hydrogen molecules can bind to the side walls of SiCNTs with larger binding energies than those of CNTs. Unlike CNTs, SiCNTs are semiconducting regardless of chirality and thus are intrinsically suitable for gas sensor application. Wu et al., 2008, theoretically examined, the possibility of SiCNTs as gas sensors to detect CO and HCN gases. 140 The aim of the present chapter is to report on preliminary work on the preparation and characterization of silicon carbide nanotubes for use in membrane preparation as well as sensor applications. Several characterization techniques are used such as powder X-ray diffraction (XRD), surface area measurements using BET, thermogravimetric analysis (TGA), and transmission electron microscopy (TEM) to characterize these materials. 6.2 Preparation Technique We have used as the support for the growth of SiC nanotubes Anopore™ inorganic membrane disks (Anodisc™), which are composed of a high-purity alumina matrix that is manufactured electrochemically. For the preparation we used the hot-wall CVD technique. Tri-isopropylsilane (TPS) was used without any further purification as the polymeric precursor. Ultra-high purity Ar (99.999%) together with hydrogen were used as the carrier gases, and the flow rate was controlled by mass-flow controllers (Brooks 5850E). The hot-wall CVD reactor, has a diameter of 3.8 cm, a length of 55 cm, and has a 20 cm long hot-zone. The porous Anodisc supports, inserted into the center of the reactor. The quartz tube and the reactor are connected with the aid of a Cajon Utra-Torr fitting. A ceramic cylindrical heater is used to heat the reactor, controlled by an OMEGA temperature controller (CN 9000). A syringe pump (HARVARD, PHD2000) is used to inject the TPS precursor into the carrier gas at a pre-determined rate. The feed-lines from 141 the pump to the furnace are heat-traced, and kept at 140 °C at all times, in order to ensure that the TPS is in the vapor phase. The flow rate of the TPS was set at 8×10 −5 mol s −1 , the hydrogen flow rate at 4×10 −4 mol s −1 and the argon flow rate varied between 9×10 −5 to 2×10 −3 mol s −1 (introducing hydrogen along with Ar will help eliminate any excess carbon which maybe produced as a by product). The deposition time was set at 13 hr, while the deposition temperature was varied between 850 to 950°C. After deposition, the sample was cooled down to room temperature and was placed in hydrofluoric acid solution in order to dissolve the support. The solution was then placed in a sonicator for 48 hr. By using centrifuge we separated the solvent from the sample after assuring that all the support has disappeared (the XRD analysis on the powder will assure there is no trace of alumina left in the sample). As a next step we wash the sample using methanol in order to increase the pH to ~7. After each wash we centrifuge the sample and remove the solvent and wash it again until we reach the desired pH. The samples are then heated at a rate of 3 °C/min in a graphite resistance furnace (Thermal Technology, Inc.), until they reach the appropriate sintering temperature (1600- 1650 °C), and are kept there for a 3 hr under an inert argon atmosphere. Upon completion of the sintering process, the samples are cooled to room temperature at a cooling rate of 5 °C/min. 142 6.3 Characterization We have characterized nanotubes using TEM. Three typical TEM images of materials grown in our laboratory, which allow for direct visualization of these structures, are shown in Figures (6.1 a,b,c). a Figure 6.1 TEM pictures of SiC nanotubes. 143 Figure 6.1: Continued. b c 144 The TEM pictures show that the nanotubes have a wall thickness of 8-10 nm, diameter of 250 nm, and a length between 3-5 µm. We also used BET to determine the surface area of the nanotubes. The BET surface area is 140.08 m 2 /g with the median pore diameter of ~7Å. The relatively high surface area of these nanotubes make them a potential candidate for hydrogen storage application (if some way can be found to scale-up the preparation technique). The summary of the BET results are presented in Table 6.1 and Figure 6.2. 145 Table 6.1 Summary of BET results for SiC nanotubes. 146 Figure 6.2 BET Isotherm plot for SiC Nanotubes. XRD analysis is used in order to better understand the crystalline structure of the materials. This technique is based on observing the scattered intensity of an X-ray beam hitting a sample as a function of incident and scattered angle, polarization, and wavelength or energy. As shown in Figureure 6.3 the nanotubes consist of polycrystalline 147 β-SiC, which is reported to have a band-gap of 2.36 eV and a thermal conductivity of 3.6 W/(cm.K) (www.wikipedia.org). Figure 6.3 XRD analyses for SiC Nanotubes The above materials are currently been utilized by the USC group in the preparation of sensors and nanoporous membranes. These efforts are in a very early stage for results to be reported here, but results will be included in future applications by the USC Group. 148 References Aida, H., Nakatani, K., Gopalakrishnan, S., Sugawara, T., Ishikawa, T., Kawamura, M., Nakao, S., Nomura, M., Seshimo, M., Preparation of a catalyst composite silica membrane reactor for steam reforming reaction by using a counterdiffusion cvd method. Ind. Eng. Chem. Res., 45 (2006) 3950–3954. Ambegaokar, V., Halperin B.I., and Langer, J.S., Hopping conduction in disordered systems, Phys. Rev. B 4 (1971) 2612. Ando, K., Kusakabe, K., Morooka, S., Sea, B. K. Separation of hydrogen from steam using a SiC-based membrane formed by chemical vapor deposition of trisopropylsilane. J. Membr. Sci., 146 (1998) 73–82. Angelescu, A., Kleps. I. Correlations between properties and applications of the CVD amorphous silicon carbide films. Applied Surface Science, 184 (2001) 107–112. Arbabi, S., Sahimi, M., Computer simulation of catalyst deactivation - I. Model formulation and validation, Chem. Eng. Sci. 46 (1991) 1739. Arbabi, S., Sahimi, M., Computer simulation of catalyst deactivation - II. The effect of morphological, transport, and kinetic parameters on the performance of the catalyst, Chem. Eng. Sci. 46 (1991) 1749. Baker, R., Membrane Technology and Applications (Second Edition), John Wiley & Sons, Ltd, 2004. Beigi, H., Dadvar, M., Halladj, R., Pore network model for catalytic dehydration of methanol at particle level, A.I.Ch.E. J. 55 (2009) 442. Bernardo, P., Drioli, E., Golemme, G., Membrane Gas Separation: A Review/State of the Art, Ind. Eng. Chem. Res. 48 (2009) 4638–4663. Beyne A.O.E. and Froment, G.F. The effect of pore blockage on the diffusivity in ZSM5: a percolation approach, Chemical Engineering Journal 82 (2001) 281. Birakayala, N. and Evans, E.A. A reduced reaction model for carbon CVD/CVI processes, Carbon 40 (2002) 675. Bird, R.B., Stewart, W.E., and Lightfoot, E.N., Transport Phenomena (second revised ed), Wiley, New York (2007). 149 Boo, J.H., Lee, S.B., Yu, K.S., Sung, M.M., and Kim, Y., High vacuum chemical vapor deposition of cubic SiC thin films on Si(0 0 1) substrates using single source precursor, Surface and Coating Technology 131 (2000) 147. Brenner, H., and Gaydos, L.J., The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius, Journal of Colloids and Interface Science 58 (1977) 312. Brinkman, H.W., Cao, G.Z., Meijerink, J., de Vries, K.J., and Burggraaf, A.J., Modelling and analysis of CVD processes for ceramic membrane preparation, Solid State Ionics 63– 65 (1993) 37. Bungay, P.M., Brenner, H., The motion of a closely fitting spheres in a fluid-filled tube. Int. J. Multiphase Flow 1 (1973) 25. Burganos V.N., and Payatakes, A.C., Knudsen diffusion in random and correlated networks of constricted pores, Chem. Eng. Sci. 47 (1992) 1383. Burganos V.N., and Sotirchos, S.V., Knudsen diffusion in parallel multidimensional or randomly oriented capillary structures, Chem. Eng. Sci. 44 (1989) 2451. Burganos, V.N., and Sotirchos, S.V., Diffusion in pore networks. Effective medium theory and smooth-field approximation, AIChE J. 33 (1987) 1678. Cao, G.Z., Meijerink, J., Brinkman, H.W., Burggraaf, A.J., Permporometry study on the size distribution of active pores in porous ceramic membranes, J. Membr. Sci. 83 (1993) 221. Chandler R, Koplik J, Lerman K and Willemsen J F 1982 J. Fluid Mech. 119 249 Chang, H.-C., Morse, T.F., and Sheldon, B.W., Minimizing infiltration times during isothermal chemical vapor infiltration with methyltrichlorosilane, Journal of the American Ceramic Society 80 (1997) 1805. Chen, F., Mourhatch, R., Tsotsis, T.T., Sahimi, M., 2007. Pore network model of transport and separation of binary gas mixtures in nanoporous membranes. Journal of Membrane Science, 315 (2008) 48-57. Chen, F., Mourhatch, R.,Tsotsis, T.T., and Sahimi, M. Experimental studies and computer simulation of preparation of nanoporous silicon–carbide membranes by chemical-vapor deposition and infiltration techniques, Chem. Eng. Sci. 63 (2008) 1460- 1470. 150 Chen, P. Wu, X., Lin, J., Tan, K.L., High H2 Uptake by Alkali-Doped Carbon Nanotubes Under Ambient Pressure and Moderate Temperatures, Science 285 (1999). Ciora, R.J., Fayyaz, B., Liu, P.K.T., Suwanmethanond, V., Mallada, R., Sahimi, M., and Tsotsis, T.T., Preparation and reactive applications of nanoporous silicon carbide membranes, Chemical Engineering Science 59 (2004) 49-57. Cuperus, F.P., Bargeman, D., Smolders, C.A., Permporometry: The determination of the size distribution of active pores in UF membranes, J. Membr. Sci. 71 (1992) 57. Deen, W.M., Hindered transport of large molecules in liquid-filled pores, A.I.Ch.E. J. 33 (1987) 1409. Dollet, A., Multiscale modeling of CVD film growth—a review of recent works, Surface and Coating Technology 245 (2004) 177–178. Duren, T., Jakobtorweihen, S., Keil F.J., and Seaton, N.A. Grand canonical molecular dynamics simulations of transport diffusion in geometrically heterogeneous pores, Phys. Chem. Chem. Phys. 5 (2003) 369. Duren, T., Keil, F.J., and Seaton, N.A., Composition dependent transport diffusion coefficients of CH 4 /CF 4 mixtures in carbon nanotubes by non-equilibrium molecular dynamics simulations, Chem. Eng. Sci. 57 (2002) 1343. F. Ebrahimi, Invasion percolation: A computational algorithm for complex phenomena. Comput. Sci. Eng. 12 (No. 2) (2010) 84. Elyassi, B., Sahimi, M., and Tsotsis, T.T., Silicon carbide membranes for gas separation applications, Journal of Membrane Science 288 (2007) 290-297. Elyassi, B., Sahimi, M., Tsotsis, T.T., A novel sacrificial interlayer-based method for the preparation of silicon carbide membranes, J. Membr. Sci., 316 (2008) 73-79. Eyraud, C., Betemps, M., Quinson, J. F., Chatelut, F., Brun, M. , Rasneur, R., Determination de la repatition des rayons de pores d'un ultra_ltre, Bull. Soc. Chim. France 9-10 (1984) I-237. Finlayson, B.A. Nonlinear Analysis in Chemical Engineering, McGraw-Hill, New York (1980). Fogler, H.S., Elements of Chemical Reaction Engineering. Pearson Education International, Upper Saddle River, NJ, fourth edition, (2006). 151 Fukushima, M., Zhou, Y., Yoshizawa, Y-I, Hirao, K., Water vapor corrosion behavior of porous silicon carbide membrane support, Journal of the European Ceramic Society, 28 (2008) 1043-1048. Gavalas, G.R., Megiris, C.E., Nam, S.W., Deposition of H2 permselective SiO2-films, Chem. Eng. Sci. 44 (1989) 1829. Gelb, L.D., Gubbins, K.E., Radhakrishna, R., and Sliwinska-Bartkowiak, M., Phase separation in confined systems, Rep. Prog. Phys. 62 (1999) 1573. Ghassemzadeh, J., Xu, L., Tsotsis, T.T. and Sahimi, M., Statistical mechanics and molecular simulation of adsorption of gas mixtures in microporous materials: pillared clays and carbon molecular sieve membranes, J. Phys. Chem. B 104 (2000) 3892. Gopalakrishnan, S., João, C., da Costa D., Hydrogen gas mixture separation by CVD silica membrane, Journal of Membrane Science, 323 (2008) 144-147. Gopalakrishnan, S., Sugawara, T., Nakao, S., Nomura, M., Ono, K. Preparation of a stable silica membrane by a counter diffusion chemical vapor deposition method. Journal of Membrane Science, 251 (2005) 151–158. Gopalakrishnana, S., Sugawaraa, T., Nakaoa S., Yamazakib, S., Inadab, T., Iwamoto Y. Nomura, M., Aidaa, H. Steam stability of a silica membrane prepared by counter diffusion chemical vapor deposition. Desalination, 193 (2006) 1–7. Han, W., Fan, S., Li, Q., Liang, W., Gu, B., Yu, D., Chemical Physics Letters 265 (1997) 374–478. Hashemi, M., Dabir, B., Sahimi, M., Dynamics of two-phase flow in porous media: Simultaneous invasion of two fluids. AIChE J. 45 (1999) 1365. Hashemi, M., Sahimi, M., Dabir, B., Percolation with two invaders and two defenders: volatile clusters, oscillations, and scaling. Phys. Rev. Lett. 80 (1998) 35-48. Hashemi, M., Sahimi, M., Dabir, B., Monte Carlo simulation of two-phase flow in porous media: Invasion with two invaders and two defenders. Physica A 267 (1999) 1. Hatori, H., Yamada, Y., Shiraishi, M., Nakata, H., Yoshitomi, S. Carbon molecular sieve films from polyimide. Carbon 30 (1992) 305. Jensen, K.F., Einset, E.O., and Fotiadis, D.I., Flow phenomena in chemical vapor deposition of thin films, Annual Reviews of Fluid Mechanics 23 (1991) 197. 152 Jones, C. W., Koros, W. J. Carbon molecular sieve gas separation membranes. I. Preparation and characterization based on polyimide precursors. Carbon 32 (1994) 1419. Jorda´-Beneyto, M., Sua´rez-Garcı´a, F., Lozano-Castello´, D. Cazorla-Amoro´s , D., A Linares-Solano Hydrogen storage on chemically activated carbons and carbon nanomaterials at high pressures Carbon 45 (2007) 293–303. Katz, M.G., Baruch, G., New insights into the structure of microporous membranes obtained using a new pore size evaluation method, Desalination 58 (1986) 199. Keil, F. J. ,Diffusion and reaction in porous networks, Catalysis Today 53 (1999) 245. Knackstedt, M.A., Sahimi, M., Sheppard, A.P., Invasion percolation with long-range correlations: First-order phase transitions and nonuniversal scaling properties. Phys. Rev. E 61 (2000) 4920. Knackstedt, M.A., Sahimi, M., Sheppard, A.P., Nonuniversality of invasion percolation in two-dimensional systems. Phys. Rev. E 65 (2002) 035101. Komiyama, H., Shimogaki, Y., and Egashira, Y., Chemical reaction engineering in the design of CVD reactors, Chemical Engineering Science 54 (1999) 1941. Koresh, E., Soffer, A. Molecular sieve carbon permselective membrane. Part 1. Presentation of a new device for gas mixture separation. Sep. Sci. Technol. 18 (1983) 723. Koutsonikolas, D., Kaldis, S., Sakellaropoulos, G.P., A low-temperature CVI method for pore modification of sol–gel silica membranes, Journal of Membrane Science 342 (2009) 131-137. Krishna, R., Wesselingh, J.A., The Maxwell-Stefan approach to mass transfer, Chem. Eng. Sci. 52 (1997) 861. Kusakabe, K., Akiyama, Y., Morooka, S., Yan, S. Formation of hydrogen- permselectiveSiO 2 membrane in macropores of α-alumina support tube by thermal decomposition of TEOS. J. Membr. Sci., 101 (1995) 89–98. Kusakabe, K., Morooka, S., Kim, S.-S., Sea, B.-K., Watanabe, M., Formation of hydrogen permselective silica membrane for elevated temperature hydrogen recovery from a mixture containing steam. Gas Sep. Purif., 10 (1996) 187–195. Kusakabe, K., Morooka, S., Akiyama Y., Yan, S., Maeda, H. Hydrogen-permselective SiO 2 membrane formed in pores of alumina support tube by chemical vapor deposition with tetraethyl orthosilicate. Ind. Eng. Chem. Res., 33 (1994) 2096–2101. 153 Kusakabe, K., Yokoyama, S., Morooka, S., Hayashi, J., and Nagata, H., Development of supported thin palladium membrane and application to enhancement of propane aromatization on Ga-silicate catalyst, Chemical Engineering Science 51 (1996) 3027. Lai, H.T., Hong, L.S. Pores structure modification of alumina support by SiC−Si 3 N 4 nanoparticles prepared by the particle precipitation aided chemical vapor deposition. Ind. and Eng. Chem. Research, 38 (1999) 950-957, 1999. Langhoff, T. A., Schnack, E., Modelling chemical-vapour infiltration of pyrolytic carbon as moving boundary problem, Chem. Eng. Sci. 63 (2008) 39-48. Larson, R.G. Morrow, N.R. Effects of sample size on capillary pressure in porous media. Powder Technol. 30 (1981) 123. Lenormand R. and Bories S., C. R. Acad. Sci. B, 291 (1980) 279. Li, B.S., Wu, R.B., Pan, Y., Wu, L.L., Yang, G.Y., Chen, J.J., Zhu, Q., Journal of Alloys and Compounds 462 (2008) 446–451. Li, K., Ceramic membranes for separation and reaction, John Wiley and Sons, 2007. Lin X, Chen X, Kita H and Okamoto K, Synthesis of silicalite tubular membranes by in situ crystallization. AIChE Journal 49 (2003) 237-247. Lin X, Kita H and Okamoto K, Silicalite membrane preparation, characterization, and separation performance. Ind Eng Chem Res 40 (2001) 4069-4078. Lin X, Kita H and Okamoto K-I, A novel method for the synthesis of high performance silicalite membranes. Chem Commun : (2000) 1889-1890. Lin, Y.S., and Burggraaf, A.J. CVD of solid oxides in porous substrates for ceramic membrane modification, A.I.Ch.E. Journal 38 (1992) 445. Lin, Y.S., and Burggraaf, A.J. Modeling and analysis of CVD processes in porous media for ceramic composite preparation, Chemical Engineering Science 46 (1991) 3067. Lin, Y.S., de Vries, K.J., Burggraaf, A. J., CVD modification of ceramic membranes: simulation and preliminary results, J. Phys. Colloque 50 (1989) 861. Liu, C., Fan, Y.Y., Liu, M., Cong, H.T., Cheng, H.M., Dresselhaus, M.S. Hydrogen Storage in Single-Walled Carbon Nanotubes at Room Temperature. SCIENCE VOL 286 5 NOVEMBER 1999. 154 Liu, H., Zhang, L., Seaton, N.A., Determination of the connectivity of porous solids from nitrogen sorption measurements. II. Generalisation. Chem. Eng. Sci. 47 (1992) 4393. Liu, P. K. T., Johnston, G., Sahimi, M., Suwanmethanond, V., Goo, E, Tsotsis, T.T., Porous silicon carbide sintered substrates for hightemperature membranes. Ind. Eng. Chem. Res., 39 (2000)3264–3271. Liu, P. K. T., Suwanmethanond, V., Malladab, R., Sahimi, M., Tsotsis, T.T., Cioraa, R.J., Fayyaz, B., Preparation and reactive applications of nanoporous silicon carbide membranes. Chemical Engineering Science, 59 (2004) 4957–4965. Lowell, S., Shields, J. E., Thomas, M. A. and Thommes M., 2004 Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density (Dordrecht: Kluwer Academic) MacElroy, J.M.D., Seaton, N.A., and Friedman, S.P., Sorption rate processes in carbon molecular sieves. Equilibria and Dynamics of Gas Adsorption in Heterogeneous Solid Surfaces, vol. 104, Elsevier, Amsterdam (1997) 837. Maddocks, A.R., Harris, A.T., Synthesis of porous silicon carbide from cellulose fibre templates infiltrated with polycarbosilane, Materials Science and Technology, 26 (2010) 375-378. Marella, M.,Tomaselli, M., Synthesis of carbon nanofibers and measurements of hydrogen storage, Carbon 44 (2006) 1404–1413. Mason, E. and Malinauskas, A.P., Gas Transport in Porous Media The Dusty Gas Model, Elsevier, Amsterdam (1983). Mavrovouniotis, G.M., and Brenner, H. Hindered sedimentation, diffusion, and dispersion coefficients for Brownian spheres in cylindrical pores, J. Colloid Interf. Sci. 124 (1988) 269. Mey-Marom, A., Katz, M. G. ,Measurement of active pore size distribution of Microporous membranes. A new approach, J. Membr. Sci. 27 (1986) 119. Mourhatch, R., Tsotsis, T.T., Sahimi, M., Network model for the evolution of the pore structure of silicon-carbide membranes during their fabrication”. Journal of membrane science, 356 (2010) 138–146. Morooka, S., Sea, B.-K., Kusakabe. K., Pore size control and gas permeation kinetics of silica membranes by pyrolysis of phenyl-substituted ethoxysilanes with cross-flow through a porous support wall. J. Membr. Sci., 130 (1997) 41–52. 155 Mpourmpakis, G., Froudakis, G. E., Lithoxoos, G. P., Samios, J., SiC Nanotubes: A Novel Material for Hydrogen Storage, Nano Lett. 2006, 6-8 (2006) 1581–1583. Murray, K.L., Seaton, N.A., Day, M.A., Use of mercury intrusion data, combined with nitrogen adsorption measurements, as a probe of network connectivity. Langmuir 15 (1999) 8155. Nam, S.W., Gavalas, G.R., Megiris, C.E., Deposition of H 2 permselective SiO 2 films. Chem. Eng. Sci., 44 (1989) 1829–1835, 1989. Nhut, J.M., Vieira, R., Pesant, L., Tessonnier, J.P., Keller, N., Ehret, G., Pham dan, C.H., Ledoux, M.J., Synthesis and catalytic uses of carbon and silicon carbide nanostructures, Catalysis Today 76 (2002) 11–32. Nomura, M, Ono, K., Gopalakrishnan, K., Sugawara, T., Nakao, S., Preparation of a stable silica membrane by a counter diffusion chemical vapor deposition method, Journal of Membrane Science, 251 (2005) 151-158. Noordman T.R., and Wesselingh, J.A., Transport of large molecules through membranes with narrow pores: The Maxwell-Stefan description combined with hydrodynamic theory. J. Memb. Sci., 210 (2002) 227. Nukunya, T., Devinny, J. S., Tsotsis, T. T., Application of a network model to a biofilter treating ethanol vapor, Chem. Eng. Sci. 60 (2005) 665. Panella, B., Hirscher, M., Roth, S., Hydrogen adsorption in different carbon nanostructures. Carbon 43 (2005) 2209–2214. Pappenheimer, J.R., Renkin, E.M., and Barrero, L.M. Filtration, diffusion, and molecular sieving through peripheral capilliary membranes pores, Am. J. Physiol. 67 (1951) 13. Pappenheimer, J.R., Renkin, E.M., Barrero, L.M., Filtration, diffusion, and molecular sieving through peripheral capilliary membranes pores, Am. J. Physiol. 67 (1951) 13. Park, S.-C., Kang, H., and Lee, S.B., Reaction intermediate in thermal decomposition of 1,3-disilabutane to silicon carbide on Si(1 1 1): comparative study of Cs + reactive ion scattering and secondary ion mass spectrometry, Surface Science 450 (2000) 117. Petropoulos, J.H., Petrou, J.K., and Liapis, A.I. Network model inverstigation of gas transport in bidisperse porous adsorbents, Ind. Eng. Chem. Res. 30 (1991) 1281. Pieson, H.O., Handbook of Chemical Vapor Deposition (CVD). Noyes Publications, Park Ridge, NJ, 1992. 156 Pinnavaia T.J., and Thorpe, M.F., Editors, Access in Nanoporous Materials, Plenum, New York (1995). Rajabbeigi, N., Elyassi, B., Tsotsis, T.T., Sahimi, M., Molecular pore-network model for nanoporous materials. I: Application to adsorption in silicon-carbide membranes, J. Membr. Sci. 335 (2009) 5. Rajabbeigi, N., Tsotsis, T.T., Sahimi, M., Molecular pore-network model for nanoporous materials. II. Application to transport and separation of gaseous mixtures in silicon- carbide membranes, J. Membr. Sci. 345 (2009) 323-330. Renkin, E.M., Filtration diffusion and molecular sieving through porous cellulose membranes, Journal of General Physiology 38 (1954) 225. Rieckmann, C., and Keil, F.J., Multicomponent diffusion and reaction in three- dimensional networks: general kinetics, Ind. Eng. Chem. Res. 36 (1997) 3275. Rieckmann, C., and Keil, F.J., Simulation and experiment of multicomponent diffusion and reaction in three-dimensional networks, Chem. Eng. Sci. 54 (1999) 3485. Rigby, S.P. , A hierarchical structural model for the interpretation of mercury porosimetry and nitrogen adsorption. J. Colloid Interf. Sci. 224 (2000) 382. Rigby, S.P., Fletcher, R.S., Riley, S.N., Determination of the multiscale percolation properties of porous media using mercury porosimetry. Ind. Eng. Chem. Res. 41 (2002) 1205. Rummeli, M.H. Palen, E.B., Gemming, T., Knupper, M., Biedermann, K., Kalenczuk, R.J., Pichler, T., Applied Physics A 80 (2005) 1653–1656. Sahimi, M., Flow and Transport in Porous Media and Fractured Rock, VCH, Weinheim, Germany (1995). Sahimi, M., Applications of Percolation Theory, Taylor and Francis, London (1994). Sahimi, M., Heterogeneous Materials I Linear Transport and Optical Properties, Springer, New York (2003). Sahimi, M., Flow phenomena in rocks: from continuum models to fractals, percolation, cellular automata, and simulated annealing, Rev. Mod. Phys. 65 (1993) 1393. Sahimi, M., 2010, Flow and Transport in Porous Media and Fractured Rock, 2nd ed. (Weinham: Wiley-VCH). 157 Sahimi, M., Gavalas, G.R., and Tsotsis, T.T., Statistical and continuum models of fluid– solid reactions in porous media, Chem. Eng. Sci. (1990) 1443. Sahimi, M., Jue, V.L., Diffusion of large molecules in porous media, Phys. Rev. Lett. 62 (1989) 629. Sahimi, M., Nonlinear transport processes in disordered media, AIChE J. 39 (1993), p. 369. Sahimi, M., Transport of macromolecules in porous media, J. Chem. Phys. 96 (1992), 4718. Sahimi, M., Tsotsis, T.T., A percolation model of catalyst deactivation by site coverage and pore blockage, J. Catal., 96 (1985) 552. Sahimi, M., Tsotsis, T.T., Molecular pore network models of nanoporous materials, Physica B 338 (2003) 291. Sahimi, M., Tsotsis, T.T., Rieth, M., and Schommers, W., Handbook of Theoretical and Computational Nanotechnology, American Scientific, New York (2006) Chapter 10. Sanchez, J., and Tsotsis, T.T., Catalytic Membranes and Membrane Reactors, Wiley- VCH, Berlin (2002). Sato, T., Itoh, N., Akiha, T., Preparation of thin palladium composite membrane tube by a CVD technique and its hydrogen permselectivity. Catalysis Today, 104 (2005) 231– 237. Sea, B.-K., Ando, K., Kusakabe, K., and Morooka, S. Separation of hydrogen from steam using a SiC-based membrane formed by chemical vapor deposition of triisopropylsilane, Journal of Membrane Science 146 (1998) 73. Seaton, N.A. Determination of the connectivity of porous solids from nitrogen sorption measurements. Chem. Eng. Sci. 46 (1991) 1895. Seaton, N.A., Friedman, S.P., MacElroy, J.M.D., and Murphy, B.J., The molecular sieving mechanism in carbon molecular sieves: a molecular dynamics and critical path analysis, Langmuir 13 (1997) 1199. Sedigh, M.G., Jahangiri, M., Liu, P.K.T., Sahimi, M., and Tsotsis, T.T., Structural characterization of polyetherimide-based carbon molecular sieve membranes, AIChE J. 46 (2000) 2245. 158 Sedigh, M.G., Onstot, W.J., Xu, L., Peng, W.L., Tsotsis, T.T., and Sahimi, M., Experiments and simulation of transport and separation of gas mixtures in carbon molecular sieve membranes, J. Phys. Chem. A 102 (1998) 8580. Sedigh, M.G., Xu, L., Tsotsis, T.T., and Sahimi, M., Transport and morphological characteristics of polyetherimide-based carbon molecular sieve membranes, Ind. Eng. Chem. Res. 38 (1999) 3367. Sheppard A.P., Knackstedt M.A., Pinczewski W.V. and Sahimi, M. Invasion percolation: new algorithms and universality classes, J. Phys. A: Math. Gen. 32 (1999) 521. Shibata, N., Kubo, Y. Takeda, Y. SiC coating on porous γ-Al2O3 using alternative supply CVI method. J. of the Ceram. Soc. of Japan, 109 (2001) 305-309. Smith, G.W., Flowers, D.L., Liu, P.K.T., Wu, J.C.S., Sabol H. Characterization of hydrogen-permselective microporous ceramic membranes. J. Membr. Sci., 96 (1994) 275–287. Sotirchos, S.V., Multicomponent diffusion and convection in capillary structures, AIChE J. 35 (1989) 1953. Suda, H., Haraya, K., Molecular sieving effect of carbonized Kapton polyimide membrane. J. Chem. Soc., Chem. Commun. (1995) 1179. Suda, H., Yamauchi, H., Uchimaru, Y., Fujiwara, I., Haraya, K., Structural evolution during conversion of polycarbosilane precursor into silicon carbide-based microporous membranes, J. Ceram. Soc. Jpn. 114 (2006) 539. Sugawara, T., Nakao, S., Gopalakrishnan, S., Nomura, M. Preparation of a multi- membrane module for high-temperature hydrogen separation. Desalination, 193 (2006) 230–235. Sun, X.H., Li, C.P., Wong, W.K., Wong, N.B., Lee, C.S., Lee, S.T., Teo, B.K., Formation of Silicon Carbide Nanotubes and Nanowires via Reaction of Silicon (from Disproportionation of Silicon Monoxide) with Carbon Nanotubes, J. Am. Chem. Soc. 124 (2002) 144-164. Sung, I. K., Park, K. H., Kim,D. P. A facile route to prepare high surface area mesoporous SiC from SiO 2 sphere templates. J. Mater. Chem., 14 (2004) 3436-3439. Suwanmethanond, V., Goo, E., Liu, P.K.T., Johnston, G., Sahimi, M., and Tsotsis, T.T., Porous silicon carbide sintered substrates for high-temperature membrane, Industrial and Engineering Chemistry Research 39 (2000) 3264. 159 Takeda, Y., Shibata, N., and Kubo, Y., SiC coating on porous γ-Al 2 O 3 using alternative- supply CVI method, Journal of the Ceramic Society of Japan 109 (2001) 305. Taylor, R., Krishna, R., Multicomponent Mass Transfer, Wiley, New York (1993). Thompson, A.H., Katz, A.J., Rashke, R.A., Mercury injection in porous media: A resistance devil's staircase with percolation geometry, Phys. Rev. Lett. 58 (1987) 29. http://www.timedomaincvd.com. Tsai, D.S., Lee, L. Silicon carbide membranes modified by chemical vapor deposition using species of low-sticking coefficients in a silane/acetylene reaction system. J. Am. Ceram. Soc., 1-81 (1998) 159–165. Tsakiroglou, C.D., Payatakes, A.C., Mercury intrusion and retraction in model porous media, Adv. Colloid Interface Sci. 75 (1998) 215. Tsakiroglou, C.D., Payatakes, A.C., A new simulator of mercury porosimetry for the char- acterization of porous materials, J. Colloid Interface Sci. 137 (1990) 315. Tsakiroglou, C.D., Payatakes, A.C., Characterization of the pore structure of reservoir rocks with the aid of serial sectioning analysis, mercury porosimetry and network simulation, Adv. Water Resour. 23 (2000) 773. Tsapatis, M., Gavalas, G.R., A kinetic model of membrane formation by CVD of SiO 2 and Al 2 O 3 , A.I.Ch.E. Journal 38 (1992) 847. Tsapatsis, M., Gavalas, G.R., Modeling of SiO2 deposition in porous vycor: Effect of pore network connectivity, A.I.Ch.E. J. 43 (1997) 1849. Tsuru,T., Hino T., Yoshioka, T., Asaeda, M., Permporometry characterization of microporous ceramic membranes, J. Membr. Sci. 186 (2001) 257–265. Tsuru, T., Takata, Y., Kondo, H., Hirano, F., Yoshioka, T., Asaeda, M., Characterizationof sol–gel derivedmembranes and zeolitemembranes by nanopermporometry, Sep. Purif. Technol. 32 (2003) 23–27. Tzevelekos, K.P., Kikkinides, E.S., Stubos, A.K., Kainourgiakis, M.E., Kanellopoulos, N.K., On the possibility of characterising mesoporous materials by permeability measurements of condensable vapours: theory and experiments, Advances in Colloid and Interface Science, 76-77 (1998) 373-388. Valente, G., Wijesundara, M.B.J., Maboudian, R., and Carraro, C., Single-source CVD of 3C-SiC films in a LPCVD reactor: reactor modeling and chemical kinetics, Journal of Electrochemical Society 151 (2004) C215. 160 van Rijn, C.J.M., Nano and Micro Engineering Membrane technology. Elsevier, first edition, 2004. Vidales, A. M., Miranda, E., Nazzarro, M., Mayagoitia, V., Rojas, F. and, Zgrablich, G. Invasion percolation in correlated porous media, Europhys. Lett., 36 (4) (1996) 259-264. Vieira-Linhares A.M., and Seaton, N.A., Pore network connectivity effects on gas separation in a microporous carbon membrane, Chem. Eng. Sci. 58 (2003) 5251. Weidenthaler, C., Krawiec, P., Kaskel, S. SiC/MCM-48 and SiC/SBA- 15 nanocomposite materials. Chem. Mater., 16 (2004) 2869–2880. Wilkinson, D., Barsony, M., Monte Carlo study of invasion percolation clusters in two and three dimensions. J. Phys. A 17 (1984) L129. Wilkinson D. and Willemsen J Invasion percolation: a new form of percolation theory J. Phys. A: Math. Gen. 16 (1983) 3365-3376. Wood, J., Gladden, L.F., Keil, F.J., Modelling diffusion and reaction accompanied by capillary condensation using three-dimensional pore networks. Part 2. Dusty gas model and general reaction kinetics, Chem. Eng. Sci. 57 (2002) 3047. Wu, R. Q., Yang, M., Lu, Y.H., Feng, Y.P., Huang, Z.G., Wu, Q.Y., Silicon Carbide Nanotubes As Potential Gas Sensors for CO and HCN Detection The Journal of Physical Chemistry C, 112-41 (2008)15985-15988. www.wikipedia.org Xu, B and Yortsos,Y. C., Salin, D., Invasion percolation with viscous forces, PHYSICAL REVIEW E ,(1)57 (1998). Xu, L., Tsotsis, T.T., and Sahimi, M., Nonequilibrium molecular dynamics simulations of transport and separation of gas mixtures in nanoporous materials, Phys. Rev. E 62 (2000) 6942. Xu, L., Tsotsis, T.T., and Sahimi, M., Statistical mechanics and molecular simulation of adsorption of ternary gas mixtures in nanoporous membranes, J. Chem. Phys. 114 (2001) 7196. Yamaguchi, T., Ying, X., Tokimasa, Y., Nair, B.N., Sugawara, T., and Nakao, S.-I., Reaction control of tetraethyl orthosilicate (TEOS) and tetramethyl orthosilicate (TMOS) counter diffusion chemical vapour deposition for preparation of molecular-sieve membranes, Physical Chemistry Chemical Physics 2 (2000) 44-65. 161 Yan, S., Kusakabe, K., Watanabe, M., Morooka, S., Kim, S.S., Separation of hydrogen from an H 2 −H 2 O−HBr system with SiO 2 membrane formed in macropores of an α- alumina support tube. Int. J. Hydrogen Energy, 21 (1996) 183–188. Zhang, L., and Seaton, N.A., Prediction of the effective diffusivity in pore networks close to a percolation threshold, AIChE J. 38 (1992) 1816.
Abstract (if available)
Abstract
Silicon carbide (SiC) is a material with very attractive properties. Its excellent mechanical strength, high chemical stability and thermal conductivity, strength and abrasion resistance at high temperatures, low thermal expansion, biocompatibility, and resistance to acidic and alkali environments, have made SiC a great candidate for use as material for the preparation of high-temperature membranes.
Linked assets
University of Southern California Dissertations and Theses
Conceptually similar
PDF
Continuum and pore netwok modeling of preparation of silicon-carbide membranes by chemical-vapor deposition and chemical-vapor infiltration
PDF
Silicon carbide ceramic membranes
PDF
Fabrication of silicon carbide sintered supports and silicon carbide membranes
PDF
Fabrication of nanoporous silicon carbide membranes for gas separation applications
PDF
Molecular modeling of silicon carbide nanoporous membranes and transport and adsorption of gaseous mixtures therein
PDF
A process-based molecular model of nano-porous silicon carbide membranes
PDF
Fabrication of nanoporous silicon oxycarbide materials via a sacrificial template technique
PDF
Exploring properties of silicon-carbide nanotubes and their composites with polymers
PDF
Thermal properties of silicon carbide and combustion mechanisms of aluminum nanoparticle
PDF
Fabrication of silicon-based membranes via vapor-phase deposition and pyrolysis of organosilicon polymers
PDF
Development of carbon molecular-sieve membranes with tunable properties: modification of the pore size and surface affinity
PDF
Biogas reforming: conventional and reactive separation processes and the preparation and characterization of related materials
PDF
Preparation of polyetherimide nanoparticles by electrospray drying, and their use in the preparation of mixed-matrix carbon molecular-sieve (CMS) membranes
PDF
The use of carbon molecule sieve and Pd membranes for conventional and reactive applications
PDF
Controlling polymer film patterning, morphologies, and chemistry using vapor deposition
PDF
In situ studies of the thermal evolution of the structure and sorption properties of Mg-Al-CO3 layered double hydroxide
PDF
Molecular-scale studies of mechanical phenomena at the interface between two solid surfaces: from high performance friction to superlubricity and flash heating
PDF
Dynamics of water in nanotubes: liquid below freezing point and ice-like near boiling point
Asset Metadata
Creator
Mourhatch, Ryan
(author)
Core Title
Experimental studies and computer simulation of the preparation of nanoporous silicon-carbide membranes by chemical-vapor infiltration/chemical-vapor deposition techniques
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Chemical Engineering
Publication Date
08/10/2010
Defense Date
04/06/2010
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
chemical vapor deposition,membrane,OAI-PMH Harvest,pore network,silicon carbide
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Sahimi, Muhammad (
committee chair
), Lu, Jia Grace (
committee member
), Tsotsis, Theodore T. (
committee member
)
Creator Email
mourhatc@usc.edu,ryan.mourhatch@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-m3339
Unique identifier
UC1463728
Identifier
etd-Mourhatch-3962 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-379063 (legacy record id),usctheses-m3339 (legacy record id)
Legacy Identifier
etd-Mourhatch-3962.pdf
Dmrecord
379063
Document Type
Dissertation
Rights
Mourhatch, Ryan
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Repository Name
Libraries, University of Southern California
Repository Location
Los Angeles, California
Repository Email
cisadmin@lib.usc.edu
Tags
chemical vapor deposition
membrane
pore network
silicon carbide