RECENT ADVANCES IN GAS SEPARATION BY MICROPOROUS CERAMIC MEMBRANES
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Membrane Science and Technology Series, 6
RECENT ADVANCES IN GAS SEPARATION BY MICROPOROUS CERAMIC MEMBtlANES
Edited
by
N.K. Kanellopoulos NCSR "Demokritos", Membranes for Environmental Separations Laboratory, 15310 Aghia Paraskevi Attikis, Greece
2000 ELSEVIER Amsterdam
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9 2000 Elsevier Science B.V. All rights reserved.
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First edition 2000 Library o f Congress Cataloging-in-Publication Data
Recent advances in gas separation by microporous ceramic membranes / edited by N.K. Kanellopoulos.-- Ist ed. p. em. -- (Membrane science and technology series ; 6) Includes bibliographical references and index. ISBN 0-444-50272-6 (alk. paper) 1. Gas separation membranes. 2. Ceramic materials. 3. Gases--Separation. I. KaneUopoulos, N. K. (Nick K.) II. Series. TP159.M4 R43 2000 660'.2842--dc21 00-056192
ISBN: 0-444-50272-6 ISSN: 0927-5193
(~The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.
CONTENTS
List of Contributors
VII
Preface
Xl
1. Fundamentals and Sorption in Micropores 1.1
Membrane characterisation by combination of static and dynamic techniques Th. A. Steriotis, K. L. Stefanopoulos, A. Ch. Mitropoulos and N. K. Kanellopoulos
1.2 In situ X-ray diffraction studies on micropore filling T. liyama, T. Ohkubo and K. Kaneko
35
1.3 Neutron and ion beam scattering techniques J. D. F. Ramsay
67
1.4 Application of pulsed field gradient NMR to characterize the transport properties of microporous membranes W. Heink, J. Karger and S. Vasenkov 1.5 Diffusion studies using quasi-elastic neutron scatttering H. Jobic
97 109
1.6 Frequency Response methods for the characterisation of microporous solids L. V. C. Rees and L. Song
139
1.7 Measurement of diffusion in porous solids by Zero Length Column (ZLC) methods D. M. Ruth'ven and S. Brandani
187
1.8 Characterisation of microporous materials by adsorption microcalorimetry P. Llewellyn
213
2.
Modeling of Sorption and Diffusion in Microporous Membranes
2.1 Simulation of adsorption in micropores D. Nicholson and T. Stubos
231 257
2.2
Molecular simulation of transport in a single micropore D. Nicholson and K. Travis
2.3
Simulation of gas transport in a "network of micropores". The effect of pore structure on transport properties E. S. Kikkinides, M. E. Kainourgiakis and N. K. Kanellopoulos
3.
Recent Advances in Microporous Membrane Preparation
3.1
Microporous carbon membranes S. Morooka, K. Kusakabe, Y. Kusuki and N. Tanihara
323
3.2
Microporous silica membranes N. Benes, A. Nijmeijer and H. Verweij
335
297
3.3 Zeolite membranes J. D. F. Ramsay and S. Kallus
373
3.4
Chemical vapor deposition membranes M. Tsapatsis, G. R. Gavalas and G. Xomeritakis
397
3.5
Composite ceramic membranes from Langmuir-Blodgett and Self-Assembly precursors K. Beltsios, E. Soterakou and N. K. Kanellopoulos
417
3.6
Nanophase ceramic ion transport membranes for oxygen separation and gas stream enrichment C. G. Guizard and A. C. Julbe
435
4.
Gas Separation Applications
4.1
Nanoporous carbon membranes for gas separation S. Sircar and M. B. Rao
4.2
Microporous inorganic and polymeric membranes as catalytic reactors and membrane contactors E. Driofi and A. Criscuofi
473
497
vii
List of Contributors K. Beltsios MESL, Institute of Physical Chemistry, NCSR "Demokritos", 15310 Aghia Paraskevi Attikis, Greece N. Benes Laboratory of Inorganic Materials Science, Department of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands S. Brandani Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK A. Criscuoli Department of Chemical Engineering and Materials, University of Calabria, Via Pietro Bucci, Cubo 17/C, Arcavacata di Rende (CS), 87030 Italy E, Drioli Research Institute on Membranes and Modelling of Chemical Reactors, and Department of Chemical Engineering and Materials, University of Calabria, Via Pietro Bucci, Cubo 17/C, Arcavacata di Rende (CS), 87030 Italy G. R, Gavalas Division of Chemistry and Chemical Engineering, 210-41, California Institute of Technology, Pasadena, CA 91125, USA C, G, Guizard Laboratoire des Mat6riaux et Proc6d6s Membranaires, UMR CNRS 5635, Ecole Nationale Sup6rieure de Chimie, 8, rue de I'Ecole Normale, 34296 Montpellier Cedex 5, France W. Heink Fakult~t f0r Physik und Geowissenschaften, Universit~t Leipzig, Linn6stral3e 5, D-04103 Leipzig, Germany T. liyama Physical Chemistry, Material Science, Graduate School of Natural Science and Technology, Chiba University, Yayoi, Inage, Chiba, 263-8522 Japan H. Jobic Institut de Recherches sur la Catalyse, CNRS, 2 Avenue Albert Einstein, 69626 Villeurbanne, France A. C. Julbe Laboratoire des Mat6riaux et Proc~d6s Membranaires, UMR CNRS 5635, Ecole Nationale Sup6rieure de Chimie, 8, rue de I'Ecole Normale, 34296 Montpellier Cedex 5, France M. Kainourgiakis MESL, Institute of Physical Chemistry, NCSR "Demokritos", 15310 Aghia Paraskevi Attikis, Greece S. Kallus Laboratoire des Mat6riaux et des Proc6d~s Membranaires, UMR CNRS 5635, Universit6 Montpellier II, 2 pl Eugene Bataillon, 34095 Montpellier, France
viii
K. Kaneko Physical Chemistry, Material Science, Graduate School of Natural Science and Technology, Chiba University, Yayoi, Inage, Chiba, 263-8522 Japan N. K. Kanellopoulos MESL, Institute of Physical Chemistry, NCSR "Demokritos", 15310 Aghia Paraskevi Attikis, Greece J. K~irger Fakult~t for Physik und Geowissenschaften, Universit~t Leipzig, Linn6stral~e 5, D-04103 Leipzig, Germany E, S. Kikkinides Chemical Process Engineering Research Institute, P.O. Box 361, ThermiThessaloniki 57001, Greece K, Kusakabe Department of Materials Physics and Chemistry, Graduate School of Engineering, Kyushu University, Fukuoka 812-8581, Japan Y. Kusuki Polymer Laboratory, Corporate Research and Development, Ube Industries, Ichihara 290-0045, Japan P. Llewellyn Centre of Thermodynamics and Microcalorimetry - CNRS, 26 rue du 141~rne RIA, 13331 Marseille cedex 3, France A. Ch. Mitropoulos Cavala's Institute of Technology, Department of Petroleum Technology, 65404 St. Lucas, Cavala, Greece S, Morooka Department of Materials Physics and Chemistry, Graduate School of Engineering, Kyushu University, Fukuoka 812-8581, Japan D. Nicholson Department of Chemistry, Imperial College of Science, Technology and Medicine, London SW7 2AY, UK A, Nijmeijer Laboratory of Inorganic Materials Science, Department of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands T. Ohkubo Physical Chemistry, Material Science, Graduate School of Natural Science and Technology, Chiba University, Yayoi, Inage, Chiba, 263-8522 Japan J, D. F. Ramsay Laboratoire des Materiaux et des Proc~des Membranaires, UMR CNRS 5635, Universite Montpellier II, 2 pl Eug6ne Bataillon, 34095 Montpellier, France M. B. Rao Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, PA 18195-1501, USA
L. V. C. Rees
Department of Chemistry, The University of Edinburgh, West Mains Road, Edinburgh EH9 3JJ, UK
D. M. Ruthven
Department of Chemical Engineering, University of Maine, Jenness Hall, Orono, ME 04469-5737, USA S. Sircar Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, PA 18195-1501, USA L. Song
Department of Chemistry, The University of Edinburgh, West Mains Road, Edinburgh EH9 3JJ, UK
E. Soterakou MESL, Institute of Physical Chemistry, NCSR "Demokritos", 15310 Aghia Paraskevi Attikis, Greece K, L, Stefanopoulos
MESL, Institute of Physical Chemistry, NCSR "Demokritos", 15310 Aghia Paraskevi Attikis, Greece
Th. A. Steriotis
MESL, Institute of Physical Chemistry, NCSR "Demokritos", 15310 Aghia Paraskevi Attikis, Greece
T. Stubos
MESL, Institute of Physical Chemistry, NCSR "Demokritos", 15310 Aghia Paraskevi Attikis, Greece
N. Tanihara Polymer Laboratory, Corporate Research and Development, Ube Industries, Ichihara 290-0045, Japan K. Travis Department of Chemistry, Imperial College of Science, Technology and Medicine, London SW7 2AY, UK M. Tsapatsis
Department of Chemical Engineering, 159 Goessmann Laboratory, University of Massachusetts, Amherst, MA 01003-3110, USA
S. Vasenkov
Fakult~t for Physik und Geowissenschaften, Universit~t Leipzig, Linn6stral~e 5, D-04103 Leipzig, Germany
H. Verweij
Laboratory of Inorganic Materials Science, Department of Chemical Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
G. Xomeritakis
Department of Chemical Engineering, 159 Goessmann Laboratory, University of Massachusetts, Amherst, MA 01003-3110, USA
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PREFACE
This book is dedicated to the rapidly grown field of microporous ceramic membranes with separating layers of pore diameter less than 2 nm. In spite of the recent rapid growth of the research effort directed towards the development of microporous ceramic membranes the field is still considered to be at its infancy and exhibits a significant future growth potential. The driving force for these efforts is of course the very promising opportunities for cost-effective large scale gas separation applications, which can be classified into two major categories: the "high temperature molecular sieving" and the "low-temperature reverse molecular sieving" modes of separation techniques. For high temperature applications, molecular sieving, based on the size exclusion of large molecules, is very often identified as the major mode of separation by microporous membranes. For low temperatures, the "reverse molecular sieving" is considered as a very efficient mode of separation in which preferential sorption of heavy gas mixture components results in the exclusion of the light components of the gas mixtures and the permeation of the heavy components through the membrane. The idea of the reverse sieving was initially introduced during the early sixties by professor R.M. Barrer and coworkers and even though it has been demonstrated that the method can combine high permeances with high selectivities, no major application has been developed as yet. One of the reasons, of course, is that the production of large surface areas of microporous separating layers with a minimum number of defects and a minimum thickness is an extremely challenging task. Over the recent years, significant progress has been made with respect to the development of novel microporous asymmetric membranes, mainly involving modification by means of deposition of additional material within the pores of the substrates. Most state-of-the-art technologies aiming in the development of microporous ceramic membrane are presented in chapters 3.1, 3.2 and 3.3. These include several material deposition methods and techniques on macroporous or mesoporous supports and substrates from the liquid or vapour phase, namely those involving sol-gel, zeolite and chemical vapour deposition techniques. In addition to the above-mentioned methods, the classical technique of carbonizing polymeric deposits along with one of the novel techniques of plasma-treating, organically deposited Langmuir-Blodgett films, are also presented. Finally, chapter 3.6 is dedicated to nanophase mixed ionic-electron membranes for enhanced oxygen transport, which pose a strong candidacy for a number of significant commercial applications. Another significant factor that seriously hinders further development of the microporous ceramic technology is the lack of comprehensive understanding of the equilibrium and transport properties of molecules confined within nanopores. The development of a satisfactory sorption and transport equation for the microporous membrane performance requires the development of efficient characterization techniques for the elucidation of the structural characteristics of the separating layer. Combining
xii sorption techniques, scattering and differential permeability techniques, the characterization of the complex pore structure of the microporous layer, interpenetrated by a network of larger pores can be obtained. These are analytically presented in chapter 1.1. The study of the physical state of sorbed phase confined in micropores can be determined by nondestructive scattering techniques. The recent advances of in situ X-Ray Diffraction (XRD) are presented in chaptem 1.2, whereas the principles of Small Angle Scattering techniques are outlined in chapter 1.3 along with recent developments employing the contrast matching technique. Several chapters of the first section are dedicated to the study of the diffusion processes in the micropores. "Microscopic" methods allow for the direct determination of the self-diffusion coefficient under equilibrium conditions by using two complementary methods, the pulse field gradient neutron magnetic resonance (PFGNMR) and quasi-elastic neutron scattering (QENS) techniques. "Macroscopic" or non-equilibrium methods, which allow the determination of transport diffusivities measured under the application of a concentration gradient, are also explicitly presented. The recently developed QENS technique allows the simultaneous determination of both transport and self-diffusion coefficients. In combination with the methods of differential permeability, frequency response and zero-length column chromatography, presented in chapters 1.4, 1.5 and 1.6 respectively, understanding of the effect of micropore confinement on the self- and transport diffusion coefficients may be obtained. The second section is devoted to the modeling of the sorption and transport through the complex porous structure of the microporous separating layer. Chapters 2.1 and 2.2 provide an overview of the recent advances in the simulation of sorption and transport processes at the single pore level. In chapter 2.3 the theory of networks of single pores is presented. Although the network theory is fully developed, insufficient description of the transport process in a single micropore precludes explicit assessment of the effect of the pore structm'al characteristics (pore size distribution, degree of connectivity etc.) to the overall transport and selectivity performance of the membrane. Since a micropore network model is currently under development, the analysis in chapter 2.3 is limited to networks comprised of mesopores, which are necessary for the description of transport through the larger pore network interpenetrating the network of micropores in the separating layer. In chapter 4.1 some of the most promising applications for the "low temperature reverse molecular sieving" mode of separation are presented, namely the recovery of paraffins and olefins from fluid catalytic cracking off gas along with the carbon dioxide removal from natural gas. These are two major processes that merit further consideration for full commercial exploitation. In addition to the above, some applications based on the "high temperature molecular sieving" technique are presented in chapter 4.2. It should be noted that the chapters of this book bring forward a wide range of issues, namely fundamentals of complex sorption and transport processes in micropore structures, highly innovative methods of preparation of microporous membranes and examples of their possible commercial applications. It is hoped that the reader will find useful and will take advantage of the insights presented by the distinguished investigators, who have contributed significantly to the advance of research efforts in the diverse topics presented in this book.
xiii
Acknowledgments I would like to thank the staff of Elsevier Science and especially Drs. Huub MantenWerker and the members of our lab Drs. G. Papadopoulos, F. Katsaros, G. Romanos, V. Kouvelos, G. Pilatos and N. Kakizis. The contribution of our secretary Ms. S. Botta is gratefully acknowledged. Special thanks go to European Commission for the financial assistance and to all our partners in several microporous membrane research projects funded by the European Commission and especially to Steve Tennison from Mast International Ltd.
Nick Kanellopoulos
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RecentAdvancesin Gas Separationby MicroporousCeramicMembranes N.K. Kanellopoulos(Editor) 2000 ElsevierScienceB.V. All rightsreserved.
MEMBRANE CHARACTERISATION BY COMBINATION OF STATIC AND DYNAMIC TECHNIQUES
Th. A. Steriotis l, K. L. Stefanopoulos 1, A. Ch. Mitropoulos 2, and N. K. Kanellopoulos 1
1Membranes for Environmental Separations Laboratory, Institute of Physical Chemistry, NCSR "DEMOKRITOS", 15310 Aghia Paraskevi Attikis, Greece 2Cavala's Institute of Technology, Department of Petroleum Technology, 65404, St. Lucas, Cavala, Greece *(current address)
The combination of static and dynamic techniques is a powerful tool which can provide detailed information for the characterisation of membranes. We here present the type of information obtainable by the adsorption, permeability, and small angle scattering (SAS) techniques and important combinations of them. Additionally, examples of recent applications of the methods and their selected combinations on a series of characteristic membrane materials are presented. Important conclusions are drawn regarding the pore and internal surface morphology of various types of membranes.
1. INTRODUCTION Membrane technology is of great industrial interest, as in many cases it can replace successfully traditional, pollution-prone and energy consuming separation processes. The thorough assessment of structural features of the pore network and surface physicochemical properties is probably the most important part of the characterisation of a membrane system and a number of equilibrium and dynamic experimental techniques can be used for that purpose. It is generally accepted that detailed description of a membrane system can only be possible when different techniques are combined.
In the following, we briefly survey three independent methods of membrane characterisation (adsorption, permeability, small-angle scattering (SAS) of x-rays and neutrons) and two significant combinations of them, permeability in conjunction with adsorption and adsorption in conjunction with SAS, with special emphasis on the type of information obtainable in each case. Additionally, we demonstrate examples of the application of the methods to the following materials: i. y-Al203 mesoporous pellets, prepared by symmetrical compaction of Degussa aluminium oxide of type-C (particle radius ~ 100 A) in eleven sections. During compaction, special care was taken to avoid non-homogeneity effects by applying an appropriate level of compression to each section. ii. A Vycor 7930 type porous silica. According to the manufacturer (Coming) the sample has a porosity of ~ 28% and pores of ~ 40 A diameter. iii. Asymmetric gas-separating carbon membranes obtained through carbonisation of a polymer resin precursor and subsequent activation (1). These membranes have a microporous carbon skin on top of a macroporous carbon substrate. The pore diameter of the microporous material appears to be in the 12-15 A range, while microporosity is about 38%. iv. Silicalite-1 membranes prepared by in situ hydrothermal synthesis and crystallisation of the zeolite inside the pores of ct-A1203 macroporous disk-shaped supports. Syntheses were performed by dissolving pyrogenic silica (Aerosil 380, Degussa) in aqueous solutions of tetrapropylammonium hydroxide (templating agent). After ageing the mixtures were heated with the support in Teflon lined stainless steel autoclaves. Finally, the organic template was removed by calcination (2).
2. DYNAMIC AND EQUILIBRIUM METHODS
2.1. Adsorption The adsorption isotherm, i.e. the quantity of gas (vapour) adsorbed on a solid at different pressures, at constant temperature, is a function of the surface area and the pore structure of the solid and thus can provide useful information about these two factors. To this
end adsorption isotherms (especially N 2 at 77 K) is a widely used technique for the characterisation of porous materials. The presence of certain types of pores (micropores, i.e. pore width, w, less than 20 A, mesopores with 20 A < w < 500 i~ and macropores with w > 500 A) produce different shapes of isotherms. The majority of these isotherms can be grouped into in five classes (six with the stepped isotherm) after Brunauer, Deming, Deming and Teller (3) according to the pore size of the solid and, the adsorbent-adsorbate interaction. The analysis of adsorption data can in principle produce values for the surface area and total (or micro) pore volume of the solid under investigation, by means of well-established methods such as BET, Langrnuir or DR (4). The pore size distribution (psd) of mesoporous solids can be derived through isotherm analysis methods, based on the Kelvin equation (5). Clearly the above analysis can be only applied to pores accessible to the penetrating gas (open pores), while inaccessible (closed) pores can only be detected with the aid of other methods such as SAS. Furthermore, Kelvin equation is not applicable to micropores, due to their small size (few molecular diameters) and the overlapping potential fields of neighbouring walls. In this case the processing of adsorption data using the potential theory (6) is possible and can lead to psd estimates of semi-quantitative validity. Typical examples of N2 adsorption isotherms (77 K) for mesoporous alumina and Vycor and microporous carbon membranes are shown in Figure 1, while the corresponding pore size distributions are presented in Figure 2. While a number of more or less established characterization methods exist for mesopores and macropores, the assessment of microporosity is much less advanced, due to experimental difficulties and the lack of an appropriate model for the interpretation of adsorption data. N2 adsorption at 77 K is probably the most studied technique, however obtaining accurate experimental isotherms is hampered by the long equilibration times required at the low liquid nitrogen temperature. In order to overcome this limitation the micropore structure evaluation can be based on isotherms of carbon dioxide or other vapours obtained at higher temperatures, provided that suitable equilibrium models for the sorption of non-spherical molecules are available.
200
J
180
160 140
._._ Car~n
n 120 Ior) 100 E 0 v 80
_._v,
_,,iA~
cor
....
- o - Alumina
:" /
j/
60 40 20 0
i
r
i
,
0.2
0.4
0.6
0.8
1
piP0
Figure 1. N2 adsorption isotherms of Vycor, AI203 and microporous carbon membranes, at 77K. 70
"~
60
4 - Carbon
50
- . - Vycor
II
- o - Alumin
or)
% ~
>* -~
30 20 10
10
100
r (A)
Figure 2. Pore size distributions based on the Kelvin equation (Vycor, alumina) and DubininAstakov method (carbon).
The Grand Canonical Monte Carlo (GCMC) method is ideally suited to adsorption problems because the chemical potential of each adsorbed species is specified in advance (7,8). At equilibrium, this chemical potential can be related to the external pressure by making use of an equation of state. Consequently, the independent variables in the GCMC simulations are the temperature, the pressure and the micropore volume, i.e. a convenient set, since temperature and pressure are the adsorption isotherm independent variables. Therefore, the adsorption isotherm for a given pore can be obtained directly from the simulation by evaluating the ensemble average of the number of adsorbate molecules whose chemical potential equals that of the bulk gas at a given temperature and pressure. To this end, a method for the determination of the micropore size distribution based on Monte Carlo simulation has been developed (9). In this work the mean CO2 density inside a single slit shaped graphitic pore of given width, is found on the basis of G.C.M.C. simulations for a pre-defined temperature and different relative pressures. Starting from an initial PSD guess, it is then possible to produce a computed CO 2 sorption isotherm and compare it to the measured one. After a few iterations, the procedure results in a PSD which, if desired, can be further refined at the cost of additional computational effort.
2.2 Permeability The fluid flow properties of porous media, are extremely sensitive functions of the pore size distribution (psd) and additional pore structural characteristics (shape, connectivity). To this end permeability is a dynamic technique, which can provide useful data concerning the structure of membranes and evaluate their overall quality simultaneously. We may note that open pores can be either conductive or blind (dead-end). Both open pore types contribute to adsorption, while permeation occurs through conducting pores only. The measurement of the permeability, P, of a weakly adsorbed gas (e.g. helium) through a membrane can be used for the calculation of gas diffusion coefficient, Dg (in this case P=Dg.e where e is the porosity). As the pressure gradient across the membrane increases, the flow regime changes from Knudsen to viscous (Poisseuille). Additional surface or condensate flow occurs when the gas (vapour) is adsorbed or condensed in the pores of the membrane. On Knudsen flow, which occurs when the pore radius, r, is sufficiently smaller
than the mean free path, ~., of the flowing molecules (r<0.05)~), Dg (and P) is independent of the pressure gradient. Thus, the existence of membrane defects can be detected by examining
Dg is
whether
constant over a range of pressure gradients or not. A typical example is
presented by Romanos et al. (2) for two zeolite membranes on supports of different porous size (0.08 ~m and 0.15ktm) synthesised at the same hydrothemal conditions. The membranes were examined by helium differential permeability experiments at 308 K and exhibited a typical crack (macropore) free behaviour. From Figure 3a it is clear that the change from molecular to viscous flow, for both membranes, occurs at approximatelly 12 bar, where the mean free path of helium for the aforementioned temperature is 168 ,~. By making the usual assumption that Knudsen regime changes to viscous when 0.01
7
7 Wb o
i
~"
~
.
~ O
7
I
~ oO
o
~
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o
o o
support 0 . 1 5 ~
'~
az
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x
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8 ----0
--0
support O.081a m I 3 ~------~ 0 5
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15
20
, 25
30
35
0
0.5
Pressure (bar)
(a)
1
1.5
Pressure (bar)
(b)
Figure 3. Helium differential permeances at 308K of the zeolite membranes on ct-A1203 substrates with mean pore diameters (a) 0.08 ~tm and 0.15 ~tm, (b) 0.08 ~tm and 9 ~tm.
Additionally, the structure factor kg (the ratio of the experimentally measured Dg over the theoretical one for cylindrical pores), introduced by Barrer and co-workers (11,12), can easily be calculated from experimental helium permeability values. The magnitude of this factor, in combination with other available structural information (porosity, inhomogeneity) leads to some general conclusions about the pore network geometry (12,13), as in the case of mesoporous alumina and microporous carbon membranes. The measured helium permeability values were constant for both membranes over a wide range of pressure heads (10-900 mbar) ensuring that the membranes were crack-flee and that mass transport occurred in the Knudsen regime (14). The value of kg (0.51), calculated for the alumina membrane, was higher than the expected ideal value of 1/3 for an isotropic, homoporous medium. This can be attributed to deviations from the cylindrical geometry (flattened pores) and/or the assumption of random reflection of molecules (15). On the other hand the structural factor kg for the asymmetric carbon membrane was estimated (1) to be in the range of 8xl 05. Such an extremely low value for a carbon membrane having a skin with a microporosity of approximately 0.38, may signify an unusually long (tortuous) path necessary for permeability and therefore, the porous structure should be in the vicinity of its percolation threshold. As the percolation threshold should be near 0.15 for a three dimensional pore arrangement and near 0.50 for a two dimensional arrangement, a percolation threshold in the vicinity of 0.38 indicates that the pore arrangement has a partial two dimensional character. Nevertheless, an alternative explanation for the observed very low structure factor is the presence of constrictions. To this end, with the aid of information obtained from SAS and N2 adsorption, structural models for the carbon membrane have been developed (16). Furthermore, the dependence of Dg on temperature is of major importance mainly in microporous systems, revealing whether diffusion is activated or not. Activated diffusion takes place when the molecules must surmount an energy barrier, usually produced by constrictions at the pore mouths. For alumina and Vycor the Knudsen mechanism was supported by measurements of helium permeability at different temperatures (Dg oc T~), while for all the microporous membranes the temperature dependence of the integral permeance of helium was not Knudsen. The process was found to be activated and activation energies were derived assuming an Arrhenius type (PoceEa/Rx)of behaviour. The activation energies, E~, can
-16.5
o~ -16.7
support 0.15Ix
[-.. -16.9 r~ "~
-17.1
o ~
-17.3 x
~"
support 0.08Ix -17.5 -177 2.4
2.6
2.8
3
3.2
3.4
3.6
1000/T (K l )
Figure 4. Calculation of Ea for helium on zeolite composite membranes.
be easily calculated from the slope of the In Pe vs. 1/T curves (Figure 4). The activation energy for the carbon membrane was 4 kJ/mol, while in the case of zeolite membranes, although the permeability values were different, Ea was 6 kJ/mol for both of them (Figure 4). These values imply that the pore network of the carbon membrane is constricted while the zeolite porous system, developed during synthesis, is the same in the two substrates (in terms of pore size, connectivity, constrictions etc.). The latter can only happen if the membranes are defect free and transport occurs only through silicalite-1 channels (2). On the other hand, differential permeability measurements of adsorbable gases or vapours through mesoporous media, performed at relative pressures (P/P0) ranging from 0 to 1, exhibit a maximum at a certain P/Po. This fact is attributed to the occurrence of capillary condensation in the mesopores and by using appropriate models, structural characteristics of the porous media can be deduced (17). Differential permeability experiments of CO2 on microporous carbon membranes (1) at slightly supercritical conditions (T=35 ~
have shown
that at a pressure of about 35 bar a peak analogous to the mesopore case appears (Figure 5). Further experimentation with nanoporous Vycor membranes having their surface modified by various types of continuous or near continuous microporous layers, show that for materials with similar physical chemical characteristics the maximum appears to shift to smaller values with smaller pore sizes (18). Thus, the technique may eventually lead to a method appropriate
for the assessment of the pore structure of microporous membranes. However, the correlation of such peaks with the pore structure is, at present, rather qualitative, due to the limited knowledge regarding the behaviour of the fluid in micropores. The sharp maxima which are observed can be attributed to the special arrangement of the CO2 molecules inside the micropores. Simulation studies have predicted an orientational ordering transition of the molecular axis of supercritical CO2, inside individual pore models, with size of the order of few molecular diameters (19). To this end recently (20), neutron diffraction experiments in conjunction with in situ adsorption are used to monitor changes in the nature of the adsorbed gas.
350
300 25O
~g 2:~
200 150
100
0
10
20
30
40
50
60
Pm (bar)
Figure 5. Differential CO2 permeability curve at 308 K for the composite carbon membrane.
A qualitative explanation for the observed maxima can also be given following Nicholson et al. (21). The flux in the x-direction is given by the expression:
J=Jo +Jv =--~--
1] ~,Ox)
(1)
10 with Jo : diffusive flux, driven by the chemical potential, tg~t/c~, Jv : viscous flux, driven by the pressure gradient, c~/o~x, c : molecular concentration inside the pore space, Do : diffusion coefficient, k : the Boltzmann constant, T : absolute temperature, B0 : geometrical term and 1"1 :the viscosity coefficient. Non Equilibrium Molecular Dynamics (NEMD) studies revealed that for certain micropores viscous contribution is activated above a critical density (and therefore pressure) of the fluid inside the pores. At lower pressures the diffusive flux is predominant, while in higher pressures the viscosity coefficient increases rapidly. This situation will eventually lead to a maximum in the permeability, as predicted for the permeability of supercritical CH4 through a 0.95 nm width slit-shaped carbon micropore (21). On the other hand a Monte Carlo (MC) simulation (9) for carbon dioxide in supercritical conditions is currently used to estimate the density profile (and, as a result, the mean density) of the fluid inside carbon slits at various pressures (Figure 6). We expect that ultimately a combination of such NEMD and MC simulations will permit the determination of pore width from the high-pressure differential permeability data.
c
i
A o
6
O
;
i
E v 2
0 0
t
~
,.+
t
t
t
I
10
20
30
40
50
60
70
p (bar)
Figure 6. Mean density of CO2 at 308 K inside micropores vs. pressure (Monte Carlo simulation) A. pore width = 7.5 A, B. pore width = 10.5 A, C. pore width = 15.5 A
Finally, for pores smaller than approximately 10 A, the membrane pore size may also be assessed with the help of a series of organic probe molecules. In the case of zeolites the pores are part of the crystal and their dimensions determined from crystallography. To this end the measurement of the permeability of a large probe molecule (that cannot fit in the zeolite pores) provides an accurate method to check whether transport occurs through cracks or intra crystalline voids. For example such experiments were performed for silicalite-1 membranes (2) via the Wicke~Kallenbach method. Nitrogen was used as sweep gas, flowing on the substrate side, while the zeolite side was fed with a mixture of helium and SF 6 (50%). Both sides were kept at atmospheric pressure. The lowest calculable permeance value, limited by the analytical ability of the thermal conductivity (TCD) detector, at the specific set up, was 1.4xl 0 -14 (m3/m2 sec kPa). Under the above conditions, no flux of SF 6 was detected or else, its permeance was lower than the aforementioned limit. However, it should be noted that in principle the permeability of vapours of appropriate substances would require a continuous path of pores having a minimum dimension larger than the diameter of the probe molecules. A complication that often arises in practice is that the microporous structure contains paths that allow certain probe molecules j to enter the one side of the membrane but not to exit from the other. Subsequent degassing may then become rather cumbersome and in situ burning of the probe molecules is, in general, not possible as the membranes are usually sealed and held in place with the help of an organic resin.
2.3. Small Angle Scattering Small angle scattering~(SAS) method either with neutrons (SANS) or X-rays (SAXS) provides structural information about inhomogeneities of scattering objects, such as pores, in the size range from 1-200 nm. In addition, for the study of larger inhomogeneities (up to 1 gm or even more), the area is covered by Double Crystal Diffractometers, a special kind of SANS instruments. As a result, SAS measurements are ideal for the study of mesoporous and macroporous materials. On the other hand, SAS technique although improper for the study of microporous media, is not worthless. One reason being that microporous materials have sizes less than 2 nm whereas SAS is general applicable for sizes larger than 1 nm. This means that there is an overlapping range between 1-2 nm detectable by this method. In addition, porous
12 media commonly refer to a wide range of pore sizes. For instance, coal contains all classes of pores (i.e. micro-meso and macro pores). Therefore SAS method may provide critical information for these mixed class porous systems (22-26). Again, pore network effects referring to microporous necks controlling the entrances of larger pores is another example which can successfully be studied by SAS. Finally, we may note that SAS may detect both open (either conducting or not) and closed pores. The basic principles of SAS are described in Chapter 1.2 of this book.
3. COMBINATIONS OF METHODS 3.1 Permeability in conjunction with Adsorption (Relative Permeability) Permeability in conjunction with Adsorption or Relative Permeability (PR), is a technique, which combines dynamic (permeability) with static measurements (adsorption) in a systematic manner and can become a source of detailed porous structure information (13). The method was introduced by Professor R.M. Barrer and his group, who were the first to measure
PRof He and H2 through membranes formed by compaction of carbolac, blocked by sorbed SO2 and NH 3 (27,28). Very few measurements have been reported thereafter due to experimental complications (13). In order to overcome these difficulties, a new simplified relative permeability technique has been demonstrated recently by Steriotis el al. (29). A similar technique, known as permoporometry, which is developed by Eyraud (30), Katz (31) and Cuperus (32), is based on counter diffusion measurement through the partially blocked membrane. Finally, time saving multiple membrane figs, with the unique capability to perform a broad range of characterisation and evaluation measurements, can be used (33). The relative permeability technique is based on the monitoring of the permeability, P, of a weakly adsorbed gas, such as helium, through a porous membrane partially blocked by a second dense adsorbate phase. The plot of P, normalised to the permeability at zero content, versus the volume adsorbed (normalised with respect to maximum content), or the equilibrium (relative) pressure is the PR curve. Upon adsorption of a gas or vapour on a mesoporous membrane, a layer of adsorbate is building up on the walls of the pores and, when condensation occurs additionally, all the pores with radii smaller than a critical value, given
13 by the Kelvin equation are blocked. Thus, by following the mechanism of "narrowing" and consequently "blocking" larger pores, along the PR curve, the influence of different sizes of pores on the flow properties of the medium is revealed. The data, if analyzed, can offer important structural information, relevant to the evaluation of performance of membranes, provided that an appropriate pore structural model is used for the data interpretation. The most widely used model for the membrane pore structure simulation is the heteroporous model. The pore structure is represented as a bundle of tortuous capillaries with radii obeying the pore size distribution fir) and having effective length Leer, along the axis of the flow. The model is completely defined by the f(r) and the tortuosity factor, x= Lee/L, where L is the straight distance in the direction of the flow. Alternatively, the capillary network model constitute a significant improvement over the aforementioned mentioned tortuosity model, since it can provide realistic modeling, especially for systems involving membranes partially blocked by condensed vapours. In this model the degree of connectivity of the pores, nT, is replacing the less tangible tortuosity factor x. In most cases, the gas permeability diminishes at the "percolation threshold", at which a significant portion of the pores are still not blocked; however in the heteroporous tortuous capillary model gas permeability diminishes only when all the pores are blocked by sorption and condensation, unless constrictions in the capillaries are introduced. In comparison, the network model can provide a satisfactory analysis of percolation threshold problem, without, as noted earlier, increasing the number of the model parameters. Since the tortuosity model is unable to reproduce the percolation threshold, a simple network model, introduced by Nicholson and Petropoulos (34-36) can be used instead. For the gas relative permeability, an explicit approximate analytical relation between the relative permeability and the two network parameters, namely nT and the first four moments of the pore size distribution, f(r), has been developed, based on the Effective Medium Theory Approximation (EMA) (29,37). In the Effective Medium Approximation, the actual network is replaced by an "effective" alternate network of uniform radius. The "effective" radius is estimated by requiring equal flux to result from both the effective and the actual networks at the same pressure difference. Upon adsorption the pores follow a probability distribution ftmction f(x)
14 with x=r-t (r is the pore radius and t is the thickness of the adsorbed layer). When condensation occurs, all the pores with radii smaller than a critical value Xk(Xk=rk-t) are blocked (filled with condensed vapour) so that the probability distribution function is Xk
fc(x)=fa'8(x)+f(Xk<X
is the fraction
0 of the blocked pores. From EMA,
oo
f(Pm - c " x3)" fc(x)dx
o
c-x
3
+ vP m
nT
with v = ~ - I
,
= 0
(2)
nT: connectivity (2 < ~ n T
Knudsen regime). By moment expansion of the aforementioned EMA equation the relative permeability can be related explicitly to the relevant microstructural network parameters, namely the suitable moments of the pore size distribution and the network connectivity. The relative permeability, PR, is then given as :
PR = ~c3 .(1 + al); f__b_-.(_a_Ml+ aM2 ) .(fb , ~ ' ] ~c3((1 + a l ) . a M
~, I-~.
(3)
where the relevant parameters (a and ~.) are functions of nT and the moments of psd defined in oO
(29,37) fb is the fraction of conducting pores (fb= 1-fa), Xe = ~ x. f(x) dx is the average radius of Xl, fb oo
the open pores and re = ~r. f(r)dr is the average radius of the network.
0 A typical application of this model is the calculation of the pore connectivity, as in the case of the mesoporous alumina membrane. Figure 7 illustrates the helium PR versus the amount of the adsorbed water. It becomes clear that when the amount adsorbed is between 0 and 30% of the total pore volume there is a sharp decrease in PR while for adsorbed fractions
15 over 30% the drop in
PRis smoother and approaches 0 before saturation (at Vs/Vt ~0.83). At
this point (percolation threshold) the last open continuous path of the membrane system is blocked and no flow through the system occurs. Figure 7 shows also the EMA model
PR
curves for nT=2, 8 and 20. By using the best fitting simulation curve the actual pore connectivity (nT=8 for alumina) can be estimated (29). We may also note that an EMA model has recently been developed for the condensed vapour differential permeability (17). It is shown that these measurements are very sensitive to the structural characteristics of the membrane (10) and a complementary dynamic method for the determination of nT can be based thereon. On the other hand the dominant feature of packings of spheroidal particles (as in the case of alumina pellets) is the constrictions between the tetrahedral cavities formed by the microspheres. A more realistic picture, based on the random sphere packing models, is thus required. Such models are obviously more complex. Conversely, they permit a more realistic representation of the pore space among the spheroidal particles. A preliminary model has been reported for sorption (38) and relative
permeability PR (39). A more realistic model of the sorption and permeability at different porosities has been developed. The model aims at predicting the sorption and the condensed phase permeabilities, given the porosity and the spheroidal particle size distribution (40). Another application of the relative permeability technique is the monitoring of changes in the physicochemical properties of the surfaces, brought by adsorption (or reaction). We here present in detail an example of the effect of surface geometry on transport properties as monitored with the aid of PRmeasurements on the Vycor diaphragm. The measurements were carried out or in a high-pressure multi-component apparatus described elsewhere (40). After the measurement of helium permeability, the Vycor diaphragm was loaded either with water or with CH2Br2 vapor at various relative pressures (P/Po). At each equilibrium point, helium permeabilities were recorded and the two relative permeability curves were drawn.
16 A
0.9 0.8 0.7 0.6
~, 0.5
\
0.4 +
nx=2.01
i
0.3+ 0.2~0"1T O,
0
~
b
t
0.2
0.4
0.6
,.. 0.8
1
Figure 7. Relative permeability curve for the alumina membrane (lines: PR curves calculated from EMA). For a smooth cylindrical pore in classical Knudsen theory (41) the diffusion coefficient Og is given by:
2 12RT Dg =-~d ~M
(4)
where d is the pore diameter, and M is the mass of the diffusing molecule. On the other hand, when the surface is rough, the movement of diffused molecules may be affected by surface irregularities. For a pore tunnel having a fractal pore wall, with a fractal dimension Ds, Coppens and Froment (42) suggested the following expression:
Dg
16
- ~ 3cAcc
./RTA/
8
-2
)D~
V-2-~-M- 'SMAX
(5)
17 where A is the cross sectional area of the capillary, 6 is the diameter of the diffusing molecule, ~SMAXis the upper cutoff for surface fractality, and CA, Cc are respectively a proportionality constant for the area/perimeter relationship and a factor correcting the surface curvature. Figure 8 shows the PR-CUrve whereas the adjusted drawings portray the cross section of a pore tunnel in three different stages (I. dry and fractally rough, IIa. H20 loaded and smooth, IIb. CH2Br2 loaded and fractally rough). The shape of the 1-120 PR curve suggests a maximum at p/p0=0.3. The PR-maximum can not be resolved by a trivial application of Knudsen theory. As adsorption progresses an amount of adsorbed water is deposited on the pore walls. Therefore, the radius of the pores will decrease by a value t corresponding to the thickness of the adsorbed film. Consequently, if Eq. (4) was valid,
Dg (and thus
P) should
constantly decrease as P/P0 increases, as in the case of CH2Br2. This is not the case for H20 however. Helium permeabilities through a) a dry Vycor diaphragm and b) a Vycor diaphragm loaded with H20 at p/p0=0.3 are measured equal to 0.84x10 3 and 1.70x10 3 cm%ec respectively (43). The adsorption of helium at ambient temperatures and other effects, such as helium interaction with dry or wet Vycor surfaces, are practically negligible and thus not sufficient to explain this large difference in P. A possible explanation is the following: when the Vycor sample is dry (Figure 8, stage I) the pore walls are not smooth and therefore Eq. (4) is not valid. Instead, Eq. (5) is more appropriate to describe the molecular movements within the pore network. As the amount of adsorbed water increases however (stage II), the surface geometry of Vycor changes from a rough to a smooth configuration. According to previously published data (44) at p/p0=0.3 defractalization of the system takes place. The internal surface of dry Vycor is probably disturbed by a number of silicate entities which give rise to a fractal character and affect the movements of the diffusing molecules. When H20 loaded, silicate entities fall dawn rendering the surface smooth (Stage IIa). As a result a maximum appears on the H20 PR" curve. In the case of CH2Br2 there is only a partial coverage of the silicate entities at p/p0=0.3 (Stage IIb), while defractalization occurs at p/p0=0.76 (45). In Eq. (5), CA is in general quite independent from the shape of the perimeter. For a smooth cylindrical pore CA--3.5449 and for a Koch-type fractally rough perimeter CA-3.7816 (42). Thus, an average value of CA--3.6 may be accepted. Again, at early adsorption stages the
18 tortuosity factor may be considered as invariant. By assuming the following parameters: CcM (42), 8MAx--20%,8He--2.18%, d'-40%, D-2.5 (45), porosity-0.3, and obtaining volume adsorbed and t from adsorption isotherms (Vycor and non porous SiO2 at p/p0=0.3), a ratio of effective d i f f u s i v i t i e s Dg~
~ --- 0.5 is calculated for water w h i l e Dg~
~ ,-~ 2 for CH2Br2. The results
are in excellent agreement with experimental data (see Figure 8).
Figure 8.
PR c u r v e s for He/H20 and He/CH2Br2. The insets illustrate the cross section of a pore ttmnel. Stage I: dry; Stage II: p/p0=0.3 (a) H20 loaded (b) CH2Br2 loaded.
In the case of microporous materials, relative permeability technique can as a first approach be used as an easy and quick test for defect-free membranes as pointed out by Steriotis et al. (14). In Figure 9 the relative permeability curves of two carbon membranes (C1 and C2) are presented. The membranes were equilibrated to various relative pressures of benzene vapours and helium permeability was measured at each equilibrium point. Of course the adsorption isotherm could not be assessed due to the composite nature of the adsorbent
19 (part of the vapour was adsorbed by the macroporous substrate). The shape of the relative permeability curve for C1 indictes the presence of cracks while C2 reveals a crack free behaviour.
1.0 I~,
7
0,9 0,8 defects
"
"
-- I I
0,7 0.6 - -It- - C 1
"
05 0.4
0.3 0.2
0.1 0.0 0
0.2
0.4
0.6
0.8
P/ p0
Figure 9. Defect (C 1) and defect free (C2) behaviour of two carbon membranes via PR curves for He/benzene at 25 ~
Additionally, the quality of produced zeolite membrane and the defect (macropore) free growth of the zeolite layer can be certified via COJHe relative permeability experiments at elevated pressures (2). As an example, an experiment performed in a high pressure WickeKallenbach selectivity apparatus (46) at 308 K (just above the critical temperature of carbon dioxide), is presented. The membrane was fed with 10 ml/min helium on the support side, while 10 ml/min carbon dioxide were flowing past the zeolite layer. The steady state helium permeances were measured for all the successive increments of pressure (both sides) from 1 to 10 bars. No helium was detected at the zeolite layer side when the pressure reached the value of 8 bars, which corresponds to the complete filling of pores, as indicated by the carbon dioxide isotherm at 308 K (Figure 10). We may conclude that relative permeability can be used for the characterisation of microporous membranes, since one can progressively block the micropores (Starting equilibrium at p/p0=l x 105) and study the effect of pore blocking on the permeability of the
20 microporous medium. Nevertheless, a systematic use of this test may have to wait for the development of a quantitative interpretation of micropore adsorption based on NonEquilibrium Molecular Dynamics (NEMD) modelling. 1.0 A
0 ~0 O
0.8 ~ , ,
O0 ~
~ 06
~
cro~ ~
y
oo0-
, o
2
4 Pressure
6
8
(bar)
Figure 10. Defect free behaviour of zeolite membrane via PR curves for H e / C O 2 at 3 5 ~ (PR is the relative permeability, Vs/V, is the adsorbed volume normalised to the total uptake).
3.2. Adsorption in Conjunction with SAS Adsorption in conjunction with SAS is powerful technique for providing information about adsorption mechanisms (filling of micropores, growth of adsorbed multilayers, capillary condensation, formation of fractal-like networks, physicochemical properties of the surface etc.) in porous media (44-45,47-50). According to SAS theory (51), the intensity I(Q) scattered by a two phase system is related to the corresponding electron densities (or scattering length densities, sld) p~ and P2 of the phases in terms of the expression
I(Q) oc (Pl - [32)2
(6)
Apparently, when a porous medium is fully saturated with a condensed liquid adsorbate having similar electron density (sld) with the solid matrix (pl=p2), complete contrast matching will be reached. As a result the scattering is zero and the sample becomes transparent to X-rays (neutrons). In the case of SANS, an appropriate mixture of molecules of
different hydrogen isotopes (e.g. H20/D20 or C6H6/C6D6) may be used as a contrast matching adsorbate for a wide range of materials, whereas a case-specific substance having an electron density similar to that of the solid is required for SAXS. When the sample is partially saturated, the pores filled with condensate will cease to act as scatterers and only the remaining empty pores will produce a measurable intensity. Since the electron density of pores is zero (Ppore=0) and complete contrast matching is attained (Pso~ia=Padsorbate=P),then Plp2=p. Therefore, a correlation between the scattered intensity and the amount adsorbed can be deduced. As a first approach, simple mathematical expressions have been derived based on the general case of the scattering theory. For an isotropic porous medium the spherically averaged intensity, I(Q), may be represented by the integral
| sinQrQr dr I(Q) = 4np2~(1-e)V~r27(r)
(7)
0
where V is the volume of the sample, p is the electron density or the sld, ~,(r) is the correlation function at point r, and e is the fraction of the pore volume; Q is the scattering vector (Q=4nsin0/~ where 20 is the scattering angle and ~ the wavelength). A Fourier inversion of Eq. (7) gives
1
~(1-~)7(r)=2n292V
i Q2I(Q) sinQr Qr dQ
(8)
At r=0, Eq. (8) gives the Porod invariant
oo
= ~Q2I(Q)dQ = 2nzpzve(1 - e) 0
(9)
22
Eq. (9) depends only on the porosity of the specimen and not on the form of the pores. A convenient way to quantify Eq. (9) is to plot the scattering data as a function of Q2 I(Q) vs Q and then to consider the area under the curves. According to adsorption theory, two mechanisms may result to a change in that area. One is the deposition of an adsorbed film on the surface of the pore walls, and the other is the condensation in an entire class of pores. In both cases, an apparent increase in the solid volume fraction of the dry sample, (l-e), by ~x (where ~x is the volume fraction of the amount adsorbed at p/p0--x) and an equal decrease in the pore volume fraction, e, will be observed. Relating Eq. (9) for p/p0=0 and x, the following expression can easily be deduced
(10)
q)2x + ( I - 2 ~ ) ( p x + ~ ( I - ~)(oe x - I ) -- 0
where txx = q-z-~- and 0<%<e. From Eq. (10) the amount % may be calculated and normalized q0 to u~=%/e. However, a computational error may arise due to the fact that the integral in Eq. (9) is infinite, whereas the estimation of the area under the curve requires a defined Q interval. In particular, Eq. (9) can be approximated by the relation
oo
Qmax
j'Q2I(Q)dQ ~ 0
Qmin
j" Q2I(Q)dQ + Qmin
j" Q2I(Q)dQ+ 0
oo
j'Q2I(Q)dQ
(11)
Qmax
According to Patel and Schmidt (52), the second and third terms on the right hand of Eq. (11) introduce errors of about 1 and 10% respectively. If the scattering intensity
n e a r Qmax
is proportional to Q-a, I(Q) can be approximated by the empirical expression I(Q)=K,Q%K2 where K1 and I(2 are constants. Thus, the third term equals to Kz/Qm~,. If, however, I(Q) is independent of Q
n e a r Qmax the
third term accounts only for the background. The second term
can, in general, be considered negligible.
23 Another source of errors may arise from the evaluation of the first term on the right hand of Eq. (11), especially, when the effect of the absorption of x-rays by the adsorbate fluid is appreciable. As adsorption progresses this error can become significant. According to the linear absorption theory (53) of x-rays the observed scattered intensity lobs is related to the absorption coefficient, Ix, and the path thickness, t, by the expression: Iobs=I(Q)e-"t . Assuming that IXdoes not depend on Q in the small-angle region, Eq. (10) becomes
q)2x + (1- 2e)q~x + ~(1- e)[((z x)obsexp(q~,dao,lx*t) - 1]
(12)
where IX* is the mass absorption coefficient and dad s is the liquid density of the adsorbate. By plotting Ux vs P/P0 (Eq. (10) or Eq. (12)), a reconstruction of the adsorption isotherm may be obtained (26,45). For the case of incomplete contrast matching, Eq. (7) is not valid (Psolid:C-Padsorbat~)and a scattering contrast between the solid/film and film/pore interfaces is detected. A development of a theory on incomplete CM can be justified by the following three reasons: 9 CM may not be possible if heterogeneities, other than pores, are present within the solid matrix; 9 Proton exchange can result to a drift in CM; 9 At high P/P0 where capillary condensation usually commences, CM will dramatically reduce the scattered intensity, resulting in an increase of the statistical error. For a ternary system the Porod invariant qx may be written as follows,
~x = SQ2I(Q)dQ = 2n2 (Ap2)x V o
where (Ap2/ /
/ X
(13)
is the mean square fluctuation of the contrast. Assuming that the electron
densities within the individual phases are constant and the electron density of the pores pp is negligible, this fluctuation can be written as (54):
24
AP 2
) X
=(Ps-Px)
2
2 2 +Pxq~x(~Pp-qgx)+Psq~s(q)p-q)x)
r
(14)
In this expression Ps and Px are the electron densities of the solid and the adsorbate respectively, q0s is the volume fraction occupied by the solid, q0pis the fraction occupied by the pores when the sample is dry (q0s+q)p=l) and (Px is the fraction of the amount adsorbed at p/p0-x. As adsorption progresses, q)x increases due to: (a) the deposition of an adsorbed film on the solid surface, and (b) capillary condensation. At p/p0=0 and p/p0--1, the system is binary and Eq. (14)reduces to x,(Ap2/o
= los2(ps(Pp
and ,,(Ap2/I = (Ps-Px )2 (Ps(Dp respectively.
Therefore,
q____~_=x 1 - Px
2
//2
9q~ + 1 - q)____~+x Px
q'p -PTsj
q~
1-
r
(15)
1.0 0.8 .. 0.6-~ ~
0.4 0.2 0.0 0
0.2
0.4
0.6
0.8
1
P/P0 Figure 11. The adsorption isotherm of water for an alumina membrane and the reconstruction of the adsorption isotherm from SANS data. (o) adsorption and (o) desorption branches from SANS; (m) adsorption and (cl) desorption branches volumetrically. Eq. (15) allows the calculation of q0x from SAS spectra. Figure 11 illustrates the adsorption isotherm of water for a mesoporous alumina membrane and the reconstruction of the isotherm
25 from SANS data (55). Within the limits of experimental uncertainty, the agreement between the two curves is good. In the following, we will describe a few recent applications for the characterisation of porous media by combining in situ adsorption and SAS.
(a) Calculation of co-ordination number Adsorption in conjunction with SAS can provide information regarding the film areas for each relative pressure. This may be attained by calculating the surface area via the Porod formula and by assuming that the fraction of pores that are closed is negligible. This is a quite reasonable assumption for the case of an alumina porous pellet produced via symmetric compaction of particles with a narrow distribution and a final porosity comparable to that of a random close packing (rcp) or a loose close packing (lcp) of spheres (0.36-0.40 vs. 0.40). Since symmetric (i.e. rather homogenous) compression leads to porosities similar to those of rcp or lcp structures and, in addition, the alumina material can not flow (e.g. the way a polymer particle may do upon compression), it is rather safe to postulate that the pore network of the compressed alumina structure should not differ in its coarse features from that of the rcp and lcp structures. Since the latter contain no closed pores we will assume that the fraction of closed pores in the compressed alumina is not significant. By applying appropriate models, such as those postulated by Kamaukhov and Kiselev (56) or Wade (57,58) the co-ordination number of the alumina membrane may be obtained. Although our experimental technique (adsorption in conjunction with SANS) was completely different from that of Wade, for similar samples the two results are comparable and a Cn --10 was estimated for the alumina membrane (59).
(b) Defractalization process Adsorption in conjunction with SAS may provide a detailed information about the geometrical nature of internal surface of a porous material. Such information is of interest as in many cases the nature of the internal surface itself- in addition to the porosity level - is of critical importance for a porous material application (e.g. catalysis). According to the Bale and Schmidt formula (60), at large Q values, the fractal dimension, D s, of the surface is related
26 to the scattered intensity through the expression:
I(Q) oc QDs-6. When the slope is equal to -4,
a smooth surface is exhibited, and Ds=2; i.e. it is Euclidean. When 2
27
Figure 12. SANS spectra from Vycor/H20 system at p/p0-0 (upper curve) and 0.28 (lower curve) and the corresponding cross section of a pore tunnel (I(Q) is multiplied by an arbitrary
factor). On the other hand, the SAXS data suggest that CH2Br2 defractalizes the Vycor surface rather slowly in terms of the following mechanism (Figure 13). At p/p0=0.34 (stage II), a tiling of CH2Br2 molecules occurs at the surface, causing a partial coverage of the silicate entities and the monolayer formed reduces slightly the fractal dimension (Ds=2.5). At p/p0=0.76 (stage III), the heights of the molecular tiles and largest silicate entities are equal and a smooth liquid/vapor interface is observed. Recently, Hoinkis (66) studied benzene adsorption, with in situ SANS measurements, on a porous
SiO 2
glass (CPG-10-75) with pore size 7.7 nm. He observed that the internal
surface of the dry sample is fractally rough with Ds=2.3. However, adsorption of a benzene film at p/p0--0.74 revealed a smooth liquid/vapor interface. One may note that in this discussion we have used the terms 'rough' and 'fractal' in somewhat interchangeable manner. In reality there is a physical distinction between a rough and a fractal surface, but it is very difficult to distinguish between the two via a SAS experiment. A detailed analysis of this point has been given in the pertinent literature (67). Nevertheless, this point does not affect the essence of our conclusions regarding the interaction of the adsorbate molecules with large protruding silicate entities of the internal surface.
28
Figure 13. SAXS spectra from Vycor/CH2Br 2 system at P/P0 (from top to bottom): 0, 0.34, 0.76 ) and the corresponding cross section of a pore tunnel (I(Q) is multiplied by an arbitrary factor).
(c) Size distributions A full correlation between adsorption and SAS requires a close monitoring of the Kelvin equation. Since the SAS data provide information about the average pore size of empty pores and the Kelvin equation furnishes the upper limit of pores filled with condensate, the relation between the Kelvin and SAS predictions is not straightforward and the help of appropriate models is required. As an example, the pore structure of an alumina membrane was studied by adsorption in conjunction with SANS (68). Contrast matching was achieved by using a 25% H20/D20 mixture. SANS data were modeled by assuming a system of polydisperse homogeneous permeable spheres (69,70) and pore size distributions were extracted at each pressure step according to the scattering of a rectangular distribution function (Figure 14). At low relative pressures, as the adsorption process progresses (0
29 distributions. At p/p0=0.94 there are still open pores of about 120 .~ acting as scatterers, probably due to the presence of positive and negative curvature effects (71).
0.02
dry
i"
p/pO=0.45 ---t- -
p/pO=O.
57
=_. p/pO=0.70 A
0.01
O. 0
'
--X--- p/pO=0.77
I
50
100
150
r(A) Figure 14. Predicted pore size distributions according to the scattering of a rectangular distribution function.
A simple method for evaluating average particle (pore) sizes from SAS by curved and fractal surfaces, has been developed (72). It is based on the extension of Porod and Kirste equation (73) from smooth to fractal surfaces. The scattered intensity, I(Q) can be expressed as"
I(Q) = Q6A_~,+ QsA_~s(6 - Ds)(5 - D~) ~ 1+ 3 1 1 12 qr2 8 \ r , - ~ j
(16)
where D is the fractal dimension of the surface, r] and r2 are the prime radii of curvature averaged over the intemal surface, AoocF(5-Ds)sin{n(3-Ds)/2. By introducing appropriate values for Ds and r], r2 the Porod's law, the Kirste and Porod correction, or the Bale and
30 Schmidt (60) formula may be recovered. By assuming an appropriate geometrical shape an estimation of the average particle (pore) size can be retained.
(d) Formation of pore percolation network with fractal geometry Li et al. (49) observed a very-low-Q scattering component (D=1.75), during desorption of matched water mixture (64% D20) in Vycor porous glass. This is a diffraction evidence for the
~ k.
P/Po=.O. P/Po = 9 5 %
t
interpore scale "percolation network" with fractal geometry. The correlation length is larger than 1000 A for the high humidity samples, a distance equivalent to about five times the average pore
~ , ~ - - . ~ a , , ] ~ , - -~--~
separation distance. The desorption process has been demonstrated with Monte Carlo calculations (Figure 15) by using a square 100 x 100 lattice. The pore neck sizes were distributed at random according to a given size distribution with the imposed constraint that only those pores connected to the vapor phase by sufficiently large neck channels may desorb. In this case, the percolation cluster of empty pores is clearly visible at high relative pressures (p/p0=0.95). When P/P0 is lowered (p/p0=0.75), the large-scale fractal structures penetrate deep into the bulk material. Theoretical studies suggest (74-76) that D should be 1.89 for
Figure 15. Monte Carlo simulations for desorption as a percolation process. The pores are situated on the lattice sites, with the empty pores represented as disks.
a two-dimensional cluster and 2.4 for a three-dimensional (3D) system and with little dependence on connectivity if pore has more than three connects. However, SANS measurements suggest that the real system is somewhat more complicated. The real difference may be due to the fact that Vycor is produced by spinodal decomposition followed by an etching process, which means that it has a spaghetti-like structure with very low connectivity (77).
4. CONCLUSIONS Three independent methods of membrane characterisation (adsorption, permeability, small-angle scattering) have been briefly surveyed with emphasis given to two combinations
31 of them, namely relative permeability and adsorption in conjunction with SAS. Additionally, examples of recent application of the methods to membrane materials were reported. Relative permeability provides a means of "on line" measurement of the fluid flow properties of a membrane during adsorption. Since these properties are closely connected with microstructural characteristics, such as pore shape, connectivity and size distribution, the interpretation of the results can lead to enhanced information concerning the porous network of the membrane under investigation. Furthermore, surface changes during adsorption can be easily monitored. On the other hand the technique is extremely useful for testing the existence of defects in microporous membranes. For quantitative information on the pore network structure of microporous materials the PR data must be coupled with sophisticated micropore adsorption models, such as Grand Canonical Monte Carlo and Non Equilibrium Molecular Dynamics. Small Angle Scattering is nowadays a widely used technique for the structural characterisation of porous media. Analysis of the SAS spectra provides information about the pore size and shape, as well as the surface properties. Consequently, by combining adsorption with SAS one can obtain the whole range of the aforementioned information at every equilibrium point. To this end, adsorption in conjunction with SAS becomes a powerful tool for the investigation of the adsorption mechanism in porous media. The amount adsorbed can be correlated in terms of the scattering theory and reconstruction of the adsorption isotherm can be attained. Furthermore, the technique can detect changes in the porous structure or in physicochemical properties of the surface during adsorption. We may conclude that the combination of equilibrium and dynamic characterisation methods leads to an enhanced spectrum of information for membrane materials, in terms of pore geometry and surface properties. Thus, by combining techniques it is possible to draw self-consistent conclusions, which are broader than those obtained from a single method.
5. ACKNOWLEDGMENTS We gratefully acknowledge the TMR program of the European Commission for providing financial support and access to Berlin Neutron Scattering Centre (Hahn Meitner Institut) and ISIS pulsed neutron source, Rutherford Appleton Laboratory, UK.
32 6. REFERENCES 1. F.K. Katsaros, T.A. Steriotis, A.K. Stubos, A.Ch. Mitropoulos, N.K. Kanellopoulos, and S. Tennison, Microporous Materials, 8 (1997) 171. 2. G.E. Romanos, Th.A. Steriotis, E.S. Kikkinides, N.K. Kanellopoulos, V. Kasselouri, J.D.F. Ramsay, P. Langlois, and S. Kallus, submitted to J. European Ceramic Society. 3. S. Brunauer, L.S. Deming, W.S. Deming, and E. Teller, J. Amer. Chem. Soc., 62 (1940) 1723. 4. S.J. Gregg, and K.S.W. Sing, Adsorption, Surface Area and Porosity, 2nd ed., Academic Press, London, 1982. 5. B.F. Roberts, J. Colloid Interface Sci., 23 (1967) 266. 6. M.M Dubinin, Russ. J. Phys. Chem., 39 (1965) 697. 7. D. Nicholson, and N.G. Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press, 1982. 8. R.F. Cracknell, D. Nicholson, and N. Quirke, Mol. Phys., 80 (1993) 885. 9. S. Samios, A.K. Stubos, N.K. Kanellopoulos, R.F. Cracknell, G.K. Papadopoulos, and D. Nicholson, Langmuir, 13 (1997) 2795. 10. M.E. Kainourgiakis, A.K. Stubos, N.D. Konstantinou, N.K. Kanellopoulos, and V. Milisic, J. Membrane Sci., 114 (1996) 215. 11. R. Ash, R.W. Baker, and R.M. Barrer, Proc. R. Soc. London Ser. A, 299 (1967) 434. 12. R. Ash, R.M. Barrer and R.J. Sharma, J. Membrane Sci., 1 (1976) 17. 13. N.K. Kanellopoulos, and J.H. Petropoulos J. Chem. Soc. Faraday Trans., 1723 1, 79 (1983)517. 14. T.A Steriotis, F.K. Katsaros, A.K. Stubos, A.Ch. Mitropoulos, and N.K. Kanellopoulos, Measurements Sci. Technol., 8 (1997) 168. 15. R. M. Barrer, The solid-gas interface, vol. 2, E.A. Flood ed., Marcel Dekker Inc, NY, 1967. 16. Th. Steriotis, K. Beltsios, A.Ch. Mitropoulos, N. Kanellopoulos, S. Tennison, A. Wiedenman, and U. Keiderling, J. Appl. Polym. Sci., 64 (1997) 2323. 17. E.S Kikkinides, K.P. Tzevelekos, A.K. Stubos, M.E. Kainourgiakis, and N.K. Kanellopoulos, Chem. Eng. Sci., 52 (1997) 2837. 18. E Magoulianiti, K. Beltsios, D. Davazoglou, G. Romanos, and N.K. Kanellopoulos, Proceedings of the International Symposium on International Chemical Vapor Deposition: CVD-XIV and EUROCVD 11 Symposium, Paris, Proceedings Volume 97-25 (1997) 576. 19. G.K. Papadopoulos, and N.K. Kanellopoulos, unpublished work. 20. F.K. Katsaros, Th.A. Steriotis, K.L. Stefanopoulos, N.K. Kanellopoulos, A.Ch. Mitropoulos, M. Meissner, and A. Hoser, Physica B, in press. 21. D. Nicholson, and R. Cracknell, Langmuir, 12 (1996) 4050. 22. M. Kalliat, C.W. Kwak, and P.W. Schmidt, ACS Symp. Ser., 169 (1983) 3. 23. O.P. Mahajan, Powder Technol., 40 (1984) 1. 24. P.L. Walker, Philos. Trans. R. Soc. London Ser. A, 300 (1981) 65. 25. A.Ch. Mitropoulos, J.M. Haynes, R.M. Richardson, T.A. Steriotis, A.K. Stubos, and N.K. Kanellopoulos, Carbon, 34 (1996) 775.
33 26. A.Ch. Mitropoulos, K.L. Stefanopoulos, and N.K. Kanellopoulos, Microporous and Mesoporous Mat., 24 (1998) 29. 27. R. Ash, R.M. Barrer, and C.G. Pope, Proc. R. Soc. London Ser. A, 271 (1963) 19. 28. R. Ash, R.M. Barrer, and R.T. Lowson, J. Chem. Soc. Faraday Trans., 1 (1973) 2166. 29. Th. Steriotis, F.K. Katsaros, A. Mitropoulos, A.K. Stubos, and N.K. Kanellopoulos, J. Porous Mat., 2 (1995) 73. 30. C. Eyraud, Application of gas-liquid permoporometry to characterisation of inorganic ultrafilters, E. Dridi, and M. Nakagaki ed., Proc. Eur.-JPN Cong. Memb. Processes, 1984, 629. 31. M.G. Katz and G. Baruch, Desalination, 58 (1986) 199. 32. F.P. Cuperus, D. Bargeman, and C.A. Smolders, J. Membrane. Sci., 71 (1992) 57. 33. T.A. Steriotis, F. Katsaros, A.K. Stubos, A.Ch. Mitropoulos, P. Galiatsatou, and N.K. Kanellopoulos, Rev. Scient. Instrum., 67 (1996) 2545. 34. D. Nicholson, J.K Petrou, and J.H. Petropoulos, Chem. Eng. Sci., 43 (1988) 1385. 35. D. Nicholson, and J.H. Petropoulos, J. Chem. Soc. Faraday Trans., 80 (1988) 1069. 36. N.K. Kanellopoulos, and J.K. Petrou, J. Membrane Sci., 37 (1988) 1. 37. J.H. Petropoulos, J.K. Petrou, and N.K. Kanellopoulos, Chem Eng. Sci., 44 (1989) 2967. 38. N.K. Kanellopoulos, J.K. Petrou, and J.H. Petropoulos, J. Colloid Interface Sci., 96 (1983) 90. 39. N.K. Kanellopoulos, J.K. Petrou, and J.H. Petropoulos, J. Colloid Interface Sci., 96 (1983) 101. 40. S.A. Sasloglou, J.K. Petrou, N.K. Kanellopoulos, and G.P. Androutsopoulos, J. Microporous Mesoporous Mat., in press. 41. S. Dushman, Scientific Foundation of Vacuum Technique, John Wiley, NY, 1949. 42. Coppens, and G.F. Froment, Chem. Eng. Sci., 50 (1995) 1027. 43. P.K. Makri, G. Romanos, T. Steriotis, N.K. Kanellopoulos, and A. Ch. Mitropoulos, J. Colloid Interface Sci., 206 (1998) 605. 44. A.Ch. Mitropoulos, P.K. Makri, N.K. Kanellopoulos, U. Keiderling, and A. Wiedenmann, J. Colloid Interface Sci., 193 (1997) 137. 45. A.Ch. Mitropoulos, J.M. Haynes, R.M. Richardson, and N.K. Kanellopoulos, Phys. Rev. B, 52 (1995) 10035. 46. F.K. Katsaros, Th.A. Steriotis, A.K. Stubos, and N.K. Kanellopoulos, in preparation. 47. E. Hoinkis, and A.J. Allen, J. Colloid Interface Sci., 141 (1991) 540. 48. J.D.F. Ramsay, and G. Wing, J. Colloid Interface Sci., 141 (1991) 475. 49. J.C. Li, D.K. Ross, L.D. Howe, K.L. Stefanopoulos, J.P.A. Fairclough, R. Heenan, and K. Ibel, Phys. Rev. B, 49 (1994) 5911. 50. J.C. Li, D.K. Ross, and M.J. Benham, J. Appl. Cryst., 24 (1991) 794. 51. A. Guinier, and G.Foumet, Small-angle scattering of x-rays, Wiley, NY, 1955. 52. I.S. Patel, and P.W. Schmidt, J. Appl. Cryst., 4 (1970) 50. 53. B.D. Culity, Elements of X-ray Diffraction, Addison-Wesley, Reading, MA, 1967. 54. L.A. Feigin, and D.I. Svergun, Structure Analysis by Small-Angle X-Ray and Neutron Scattering, Plenum Press, NY, 1987. 55. A.Ch. Mitropoulos, T.A. Steriotis, F.K. Katsaros, K.P. Tzevelekos, N.K. Kanellopoulos, U. Keiderling, A. Sturm, and A. Wiedenmann, J. Membrane Sci., 129 (1997) 289. 56. A.P. Kamaukhov, and A.V. Kiselev, Russ. J. Phys. Chem., 34 (1960) 1019.
34 57. W.H. Wade, J. Phys. Chem., 68 (1964) 1029. 58. W.H. Wade, J. Phys. Chem., 69 (1965) 1395. 59. Th. Steriotis, A. Mitropoulos, N. Kanellopoulos, U. Keiderling, and A. Wiedenmann, Physica B, 234-236 (1997) 1016. 60. H.D. Bale, and P.W. Schmidt, Phys. Rev. Lett., 53 (1984) 596. 61. E. Cheng, M.W. Cole, and P. Pfeifer, Phys. Rev. B, 39 (1989) 12962. 62. F. Katsaros, P. Makri, A. Mitropoulos, N. Kanellopoulos, U. Keiderling, and A. Wiedenmann, Physica B, 234-236 (1997) 402. 63. P. Levitz, G. Ehret, S.K. Sinha, and J.M. Drake, J. Chem. Phys., 95 (1991) 6151. 64. M.J. Benham, J.C. Cook, J.C. Li, D.K. Ross, P.L. Hall, and B. Sarkissian, Phys. Rev. B, 39 (1989) 633. 65. D.W. Schaefer, B.C. Bunker, and J.P. Wilcoxon, Phys. Rev. Lett., 58 (1987) 284. 66. E. Hoinkis, Adv. Colloid Interface Sci., 76-77 (1998) 39. 67. P. Wong, Phys. Rev. B, 32 (1985) 7417. 68. K.L. Stefanopoulos, G.E. Romanos, A.Ch. Mitropoulos, N.K. Kanellopoulos, and R.K. Heenan, J. Membrane Sci., 153 (1999) 1. 69. M. Kotlarchyk, and S.H. Chen, J. Chem. Phys., 79 (1983) 2461. 70. L. Blum, and G. Stell, J. Chem. Phys., 71 (1979) 42. 71. D.H. Everett, Colston Res. Soc., 10 (1958) 95. 72. A.Ch. Mitropoulos, N.K. Kanellopoulos, K.L. Stefanopoulos, and R.K. Heenan, J. Colloid Interface Sci., 203 (1998) 229. 73. R. Kirste, and G. Porod, Kolloid-Z. Z. Polym., 184 (1962) 1. 74. V.K.S. Shante, and S. Kirkpatrick, Adv. Phys., 20 (1971) 325. 75. M. Parlar, and Y.C. Yortsos, J. Colloid Interface Sci., 124 (1988) 162. 76. A.U. Netmaann, and S. Havlin, J. Stat. Phys., 52 (1988) 203. 77. G. Mason, Proc. R. Soc. London Ser. A, 415 (1988) 453.
Recent Advancesin Gas Separationby MicroporousCeramicMembranes N.K. Kanellopoulos(Editor) o 2000 ElsevierScienceB.V. All rightsreserved.
35
In situ X-ray diffraction studies on micropore filling T. Iiyama*, T. Ohkubo, and K. Kaneko Physical Chemistry, Material Science, Graduate School of Natural Science and Technology, Chiba University, Yayoi, Inage, Chiba, 263-8522 Japan The molecular level picture of micropore filling mechanism in carbon micropores was summarized and the necessity of the structural study on molecules confined in carbon micropores with in situ X-ray diffraction is described. The detailed explanation of in situ X-ray diffraction technique for adsorption studies is given. The effectiveness of the in situ X-ray diffraction for elucidation of structures of molecules filled in micropore is shown for CCI4, water, and alcohol molecules. In situ X-ray diffraction studies shows the presence of organized structure inherent to an adsorbate molecule, although only adsorption measurement cannot distinguish the molecular state different from the bulk liquid.
1. INTRODUCTON Human welfare has been giving a great perturbation to earth. Energy saving technology has been requested in every industrial field in order to preserve the global environments.. Porous materials, in particular, microporous solids have a hopeful potential to develop new energy saving technology. Microporous solids have been helpful for efficient separation, concentration, purification, and removal through an intensive adsorption of gases in micropores. Consequently, more understanding of the adsorption of gases in micropores has been strongly required. Pores whose pore width w is less than 2 nm are called micropores according to the IUPAC classification [1]. Representative microporous solids are activated carbon and zeolite. Zeolite is a crystalline solid having well-described micropores. Activated carbon is basically ill-crystalline, because activated carbon has a random structure of nano-order graphitic units having two to four stacking layers. The void spaces between thin graphitic units provide slit-shaped micropores. However, activated carbon fiber ACF and KOH-activated carbon have considerably uniform micropores and great pore volume. Granulated activated carbon which has a wide pore size *Present address: Department of Chemistry, Faculty of Science, Shinshu University, Asahi, Matsumoto, Nagano, Japan
36 distribution from micropores to macropores, has been widely used in a variety of industries. A better understanding of micropore structures and adsorption mechanism should contribute to realize the energy saving technologies. The adsorption of vapor in micropores is called micropore filling. Originally Dubinin et al introduced the concept of micropore filling for adsorption on activated carbon [2]. They considered that mechanism of pore filling can describe the adsorption on activated carbon in a better way compared with the mechanism of successive layer formation so called BET mechanism. They assumed that the filling ratio r at an equilibrium pressure P, which is the fraction of the micropore volume occupied by adsorbed molecules, is a function of the Polanyi adsorption potential A and the parameter n associating with the distribution of the adsorption potential. A = RT In (Po/P)
(1)
Here Po is the saturated vapor pressure and A is reduced by the characteristic energy E. They chose the following Weibull distribution function to get the so-called DubininAstakhov equation. r = exp [- (A/E) n ]
(2)
r
(3)
W /Wo
where W and W o are the volume occupied by molecules at P and the micropore volume, respectively, n - 2 gives Dubinin-Radushkevich (DR) equation. Stoeckli et al took into account the pore size distribution and extended the above approach to describe adsorption in micropores even for polar molecules [3]. Sing et al [4] introduced ct s-plot to analyze the adsorption isotherm, proposing the two-stage mechanism of primary and cooperative (or secondary) fillings. However, they did not analyze sufficiently the adsorption isotherm below cts = 0.5, which is essentially important in micropore filling. Then, no clear physical model for the twostage mechanism was given. Kaneko et al [5,6] introduced the high resolution ct splot over the wide ct s ranges from 0.01 to 2.4 using the adsorption isotherm from P/Po = 1 0 -5 -- 1 0 6 . They showed the presence of two upward deviations from the linear ct splot below as = 1.0. The upward deviations from the linearity below and above ct s = 0.5 are called filling and cooperative swings, respectively. The filling swing is ascribed to an enhanced monolayer adsorption on micropore walls and cooperative swing comes from filling in the residual space between the monolayer on both micropore walls. Also Kaneko et al showed accelerated bilayer adsorption in micropore filling of He at 4.2 K [7,8], pore entrance-enriched micropore filling of n-nonane [9], dipole interaction mediated micropore filling of SO2 [11], and cluster mediated filling of water [11] on the
37 basis of the detailed analysis of their adsorption isotherms. These mechanisms of micropore filling have no physical evidences except for adsorption isotherms, although molecular simulation can be referred in some cases. We need direct physical information on the mechanism of micropore filling. As activated carbon can absorb efficiently light from ultraviolet to infra-red region, ordinary molecular spectroscopies cannot be applied to the carbon system. However, X-ray can penetrate carbon walls and thereby X-ray technique such as X-ray diffraction and small angle X-ray scattering SAXS can be applied to study micropore filling in activated carbon. This review describes assembly structures of molecules adsorbed in micropores of activated carbon, mainly ACF, which is obtained from in situ X-ray diffraction examinations and gives structural pictures on micropore filling. Although in situ Xray diffraction is powerful to elucidate the structure of the molecular assembly in micropores, it is not necessarily effective to understand the mechanism of micropore filling. The integrated information with in situ SAXS can provide a more perfect description on the micropore filling mechanism in future.
2. MICROPORE FILLING MECHANISM AND MOLECULAR PROPERTIES We will discuss the interaction of a molecule with the graphitic slit pore of the micropore model of activated carbon. The interaction between a molecule and a surface atom as a function O(r) of the distance r between them can be expressed by the Lennard-Jones potential, 9 (r) = 4 e sf[( Crsf/r)'2-( Crsf/r)6 ]
(4)
where crse and Crsfare the well depth and effective diameter for the molecule-graphitic carbon atom. These cross parameters are calculated according to the Lorentz-Berthelot _ )1/2 rules, e ~f ( t~ ss t~ ff ; O'sf = ( O'ss "[" O'ff)/2. Here, (O'ss, e ss) and (O'er, eff ) are the Lennard-Jones parameters for a surface atom and a molecule, respectively. The interaction potential O(z) for a molecule and a single graphite slab is given by the Steele 10-4-3 potential [12], (I) (Z) = 2 7~" tO C ~ sf O's? A {(2/5)( O"sJz)l~
4- 0"s4/[3 A (0.61 A +z)3]}
(5)
where z is the vertical distance of the molecule above the surface, A is the separation between graphite layers (=0.335 rim), p c is the number density of carbon atom in a graphite layer (=114 /nm3). As the micropores of activated carbon can be approximated by the slit spaces between the predominant basal planes of nanographitic units, the whole interaction potential 9 (Z)p of a molecule with the micropore of an inter-graphite surface distance H can be given by eq. (6).
38 (1)(Z)p ----- (1)(Z) + (1)(H-z)
(6)
Consequently, we can evaluate the potential profile of the molecule adsorbed in the graphitic micropore. Here H is not the effective pore width w determined by the adsorption experiment. The difference between H and w is a function of o'sf and cr~ [13]. H-w = 1.60
O'sf--
(7)
O'ff
Figurel shows potential profiles of N 2 in a slit-shaped graphite pore as a function of w using the one-center approximation. Here, the molecular position in Figure 1 is expressed by a vertical distance z from the central plane between two surfaces. The potential profile of an N2 molecule in the graphitic slit pore of w > 0.6 nm has double minima, indicating the presence of monolayer adsorption on each microporewall. As the barrier height between the double minima of the micropore of w > 0.7 nm is greater than 77 K, a distinct monolayer adsorption on each pore-wall occurs in micropores of w > 0.7 nm. The predominant monolayer adsorption occurs at the low P/Po region, compared with the monolayer adsorption on the fiat surface, because the N2-graphitic surface interaction in micropores is greater than that on the flat surface. In the case of micropores having w > 1 nm, N2 molecules should be adsorbed in the space between monolayers on both micropore-walls after monolayer adsorption. Then, we can presume that in micropore filling of N 2 by micropores of w > 1 nm, there are two elementary processes - monolayer adsorption on each micropore-wall and filling in the residual space between the opposite monolayers on micropore-walls. The separation
1000
\ -1000
= 1.5nm
t,q 0.8 nm
-2000 \
07nm
\
0.5 nm
-3000
I -0.8
o'4
o
o14
z/nm
Figure 1. Potential profile of
a n N 2 molecule
with a graphitic slit space.
39 of these elementary processes depends on the height of the potential barrier between double minima. In the micropores of w < 0.7 nm, which are called ultramicropores, a single layer or double layers can be formed between two graphitic walls. The multistage micropore filling process on the real activated carbon having pore size distribution was observed. The Grand Canonical Monte-Carlo (GCMC) simulation studies support this two-stage model of micropore filling of N 2 [14,15]. Very recently Ohba et al [16] showed clearly the role of preformed monolayer in the further adsorption by use of GCMC simulation. Figure 2(A) shows the potential profiles of a n N 2 molecule at the second layer in the monolayer-coated graphite slit pore of w = 1.24 nm. The potential depth of the second layer molecule corresponds to 60 % of that of the monolayer molecule. As the interaction potential of the second layer molecule with the monolayer is more than 70 % of the total molecular potential, adsorption of molecules at the second layer position should stem mainly from the adsorbate-adsorbate interaction. Figure 2(B) shows the simulated N 2 adsorption isotherms for monolayer and second layer formation at 77 K. Adsorption at the second layer positions occurs steeply at a narrow relative pressure region. This filling is not the capillary condensation, but this should be designated cooperative filling, as suggested by Sing et al [4,17]. The above simulation results coincide well with experimental data from the high resolution ct ~-plot by Kaneko et al [17,18]. The basic mechanism of micropore filling of vapor molecules which interact strongly with the graphitic walls is almost understood, as described above. However, still microscopic molecular states and structures are not elucidated. We need to introduce a structural technique to adsorption in micropores. As the low temperature limit of the in situ X-ray diffraction experiment is 140 K, so far we cannot apply this method to N2 adsorption at 77 K. In stead of nitrogen molecules, we must choose an appropriate vapor molecule which satisfies the following requirements: The probe molecule is a nonpolar molecule of a spherical shape and it has a great X-ray scattering ability in 1ooo m
"7 ' i A ) !!!
500
~-
I I !
I
0
! | | I
-500
9
9 9
9
II 0fO
- 1000
-1500
I!
i
k -0.4
!
.~ 500 (B) ,,.., o E '~
30o
l
1
Z
t e
t o
o
t
- - -~i! - ''~176176
_t 06
0
t
200 100
l
,, ,, . . . . . . . . . . . . .
400
0.4
-5
-4
-3
-2
-1
0
l o g ( P / P 0)
Figurer 2. (A) Potential profile of a n N 2 molecule in the monolayer-coated micropore of w =1.24 nm. The solid curve is the potential due to monolayer molecule. (B) Simulated N 2 adsorption isotherms for monolayer and second layer adsorption in micropore of w=1.24 nm at 77 K. Broken curve: monolayer, solid curve: second layer.
40
order to get a good contrast against graphitic walls. A CC14 molecule has a high symmetry of T~ which can be approximated by the spherical shape. Also the atomic number of a chlorine atom is much greater than that of the carbon atom, giving rise to a distinct electron density difference. Accordingly a CCI 4 molecule was chosen as the representative of the nonpolar vapor molecule. The basic picture on the micropore filling of nonpolar vapor in carbon micropores should have a good similarity for other vapor molecules of spherical shape. Then, X-ray diffraction study on micropore filling of CC14 will be shown in this review. The mechanism of micropore filling can be estimated from the molecular simulation studies on N2 adsorption in graphite slit pore. The basic feature of the micropore filling of N2 can be applied to that of CCI 4. Recently Suzuki et al [19] showed that CC14 molecules are adsorbed in micropores at 303 K in a similar way to N 2 at 77 K. However, the CC14 molecule is greater than the N2 molecule, inducing a specific structure formation in smaller micropores. The adsorption of water on activated carbon is not clearly understood. We have a question why water molecules can be adsorbed in hydrophobic carbon micropores. A water molecule cannot interact strongly with the graphite surface, because the induced images in the graphite surface from the permanent dipole of a water molecule is not stable under the random orientation of physisorbed molecules. Hence, the electrostatic interaction is not great and also the dispersion interaction of a water molecule with the graphite surface is not predominant, leading to very small interaction energy. Therefore, water molecules are not adsorbed on the graphite surface and even in graphitic micropores at the low P/P0 region. It is well-known that water molecules form the dimer or clusters even in the gas phase. In carbon micropores, highly concentrated Table 1 Molecular properties of adsorbed molecules N2 O'ff / A 3.798 e ff/ka / K 71.4 O'se / A 36.0 e sJka / K 45.0 3-D size / A3 2.6x2.6x3.8 D-pole / D none Q-pole -4.9 / 1040 Cm -2
Polarizability / 10 .24 cm 3
1.7403
CC14 5.947 322.7 4.67 95.6 5.43 none none
10.5-11.2
H20 CH3OH C2HsOH 2.614 3.626 4.530 809.1 489.1 362.6 3.01 3.51 3.97 151.3 116.8 101.3 2.6x2.8x3.6 3.6x3.9x4.4 3.9x4.2x6.0 1.854 1.70 1.69 8.7(xx) ..... -8.3(yy) -0.4(zz) 1.45 3.29-3.32 5.11-5.51
Here O'sf and e se are obtained from Lorentz and Berthelot rules and [s] denotes an carbon atom. 3-D size, D-pole, and Q-pole are 3-dimensional size, dipole moment, and quadruple moment, respectively.
41
O
H20 O o
(H20)5
O
~](H,O)n Molecular position
Figure 3. Hypothetical potential profile of a water molecule in graphitic micropores.
e~0
A AAAA 9
800
o
600 o ,.0 o r~
400 A
9
(3 .~i~3 O AC) 9 9
o
200 -
o 9 o~
o
<
o ~a___e~~-" 0
0.2
~
"
0.4 P/P
,
,
0.6
0.8
1
0
Figure 4. Water adsorption isotherms of P5 and P20 at 303 K. O: P5-adsorption, O: P5-desorption, A- P20-adsorption, A: P20-desorption water clusters such as the pentamer ( H 2 0 ) 5 of which size is about 0.5 nm must be formed due to the compressed conditions [20]. Water molecules are bonded with each other using hydrogen bonds in the cluster and thereby the electrostatic charge of the water molecule should be more compensated, inducing a strong Lennard-Jones type interaction with the graphitic rnicropore-walls. Figure 3 shows hypothetical interaction potential profiles of a water molecule in a graphite slit pore. For a single water molecule, the interaction is very weak, but the cluster in which point charges are used for the hydrogen bonding formation can have a deeper potential well through the
42 dispersion interaction. The cluster fitting for the micropore geometry has a deeper potential well. Consequently, water adsorption begins suddenly above the medium P/Po where the clusters grow. In case of a small alcohol molecule, more or less similar situation can be applied to adsorption in graphitic micropores. Then, we can check phenomenologically this aspect using adsorption of alcohol. As the relationship between the micropore filling mechanism and molecular interaction is not sufficiently established yet, the factors of molecules in Table 1 must be taken into account. Here the Lennard - Jones parameters ,~ ff and e sf indicate the intermolecular interaction and the molecule - graphite wall interaction, respectively. The e n of a water molecule is quite great due to the hydrogen bonding, giving a large O'sf through the Lorentz Berthelot rule. However, the O'sf for a water molecule is not a good measure. This situation holds for a methanol molecule having a strong hydrogen bonding ability. Figure 4 shows adsorption isotherms of water on two kinds of ACFs at 303 K. Here P20 and P5 have the average pore widths of 1.1 and 0.75 nm, respectively. The adsorption isotherms are of Type V. Almost no water is adsorbed below P/P0 = 0.5, while a predominant adsorption begins above P/P0 = 0.5. The rising relative pressure of P20 is higher than that of P5. The rising relative pressure of P20 treated at 1273 K in H2 which has almost no surface functional groups according to the X-ray photoelectron spectroscopy, shifts to a higher value [21]. The adsorption isotherm of P20 has a noticeable hysteresis, while that of P5 has almost no hysteresis. The adsorption hysteresis of the water adsorption isotherm is not caused by capillary condensation [22]. On the course of adsorption, water molecules are adsorbed through the above-mentioned cluster mediated filling mechanism to form a uniform adsorbed layer in micropores. On the other hand, molecules and/or small clusters are evolved from the interface between the adsorbed layer and gas phase. This different processes should provide the adsorption hysteresis. Iiyama et al is now trying to elucidate the mechanism of the cluster-mediated filling of water and the adsorption hysteresis with in situ SAXS [23].
3. A P P L I C A T I O N O F E L E C T R O N M E T H O D TO C O N F I N E D FLUIDS
RADIAL D I S T R I B U T I O N
FUNCTION
The dimension of molecular assemblies confined in carbon micropores is limited to less than 2 nm in the width direction at least. Then, the observed X-ray diffraction of confined fluids is very broad. We need an effective analysis for the broad X-ray diffraction pattern in order to get an important information on the structure of adsorbed molecules. The electron radial distribution function (ERDF) method [24] is applied to the analysis of confined fluids. This method can extract the information on the intermolecular structure from broad X-ray diffraction patterns. The electron radial distribution function shows the average molecular number or molecular density p at a
43 distance r from a central molecule. This method was developed by distinguished pioneers such as Warren in 1930's [25]. Narten had studied liquid structures systematically with ERDF method [26]. Nishikawa et al improved the electron radial distribution function method for liquid systems [24]. We applied this improved method for the confined systems. However, we need careful corrections for accurate ERDF conclusions. 3.1. X-ray intensity correction The observed X-ray scattering intensity at s, lobs(S), of the carbon sample having adsorbed molecules is given by [27]
Io,,s(S) = I,o,(S) "P(O) "G(O) "a(O, p ) +
"A(O, p )
Here I,o,(S) is the X-ray scattering intensity of the sample at s. parameter s can be expressed, 4;rsinO s- ~ A
(8) The scattering
(9)
Here 2 tY is the scattering angle, A is the wavelength of X-ray in experiments. All of correction factors, P(tY), G(t7), A(6~,p), and Ib~ck(S) depend on the experimental arrangements. The experimental arrangement of the X-ray diffraction in this paper is shown in Figure 5. The polarization factor, P(6~), is given by [27],
e(o) =
1+
2a" cos 2 20 1 + cos2 2a
cos 2
(10)
Here 2 a' is the scattering angle on the monochrometer. The geometrical factor, G(tY), for a normalization of X-ray irradiating volume is simply given by
G(O) -
1 cos0
(11)
The Ib, ck(S) is the parasitic scattering that is mainly attributed to the scattering by windows of the sample cell. It can be determined experimentally by the measurement of the sample cell without sample in vacuo. The X-ray absorption factor, A(tY,p), depends on both of the scattering angle 2 6? and extent of molecular adsorption. It is given by,
44
A(O, p) - exp.
_ tt~ (p)'l.____.._~ cosO
- ~r (p)'lg
} (12)
where /~s(P) and /'['8(P) are the linear X-ray absorption coefficients of the sample and the gas phase which are the function of the pressure p and X-ray path length. The notation ls and Ig are the X-ray path lengths in the sample and gas phase, respectively. The first term in the exponent of eq. (12) expresses the X-ray absorption by the carbon sample and adsorbed molecules, and the second term stems from absorption by gaseous molecules in the sample chamber. The first term is much greater than the second one under these experimental conditions. Further, /z~(p) l~ can be expressed by,
Ia
t~g (p)" ts - log--~o - t,~g (p) "lg
(13)
Here Ia is the transmitting X-ray intensity with the sample and Io is that without the sample at 6? = 0 deg. Both of la and Io were measured at 20kV and 10mA using an aluminum plate of 2mm in thickness as the X-ray attenuator. The /_zg is calculated with the following equation.
,14, g
Here, (/z//O)g is the mass X-ray absorption coefficient which is calculated from the literature values [28], and M r is the molecular weight of adsorbed gas, R and T are the gas constant and the measuring temperature.
3.2. Effect of small-angle X-ray scattering Figure 5 shows the X-ray diffraction pattern of pitch-based activated carbon fiber (P20). The strong X-ray scattering is observed in a small angle region of s < 2A -~, which is a common feature of porous carbons. The X-ray scattering near s =0 ( tY= 0 ~ ) is known as small-angle X-ray scattering (SAXS). A non-uniform structure of different electron densities leads to the SAXS. The SAXS contains valuable information on mesoscopic structures of the system, elucidating latent pore structures [29,30]. However, the SAXS has a strong influence on the transformation of the X-ray diffraction data to the electron radial distribution function especially in porous carbon. The false sign curve whose wavelength is 1.5 - 2.0nm superimposes on the electron radial distribution function. Then we must remove the effect of small-angle X-ray
45
4000
3000
:~
.,. r~
2000
1000
I
I
I
I
2
4
6
8
s/A
10
1
Figure 5. X-ray diffraction pattern of pitch-based ACF (P20) in vacuo.
scattering. We assume that the total X-ray intensity, I,o,(S), can be expressed by the additive form, !
I,o , (s) = Ito t (s) + I,~ (s)
(15)
Here I,~x(S) is the small-angle scattering and Ito,' is the corrected total intensity. The porous carbons and confined fluid molecules do not have a long-range periodic structure. Therefore, corrected total intensity, I,o,' has no peak at less than 1.5z~-1. The X-ray scattering intensity at s < 1.5z~-1 can be attributed to the small-angle X-ray scattering. A log-log plot of X-ray diffraction intensity versus scattering parameter of porous carbon and confined fluid system give a straight line. Therefore, the smallangle X-ray scattering is expressed as,
ln(/,~ (s))-- a ' l n s + b
(16)
The constants a and b are determined by the least-squares method. The corrected total intensity, I,o,'(s), is derived by subtraction of the I, ax(S). The influence of the correction method of the SAXS for the ERDFs of adsorbed CC14in micropores was examined. The best method was used for further analysis.
3.3. Determination of scattering intensity of confined fluids The scattering intensity of only adsorbed molecular assembly must be extracted from
46
the observed X-ray scattering intensity Itot(S). We assume the additive form of scatterings by carbon walls (ItotC(s)) and by adsorbed molecules (Ito~a(s)), and by interference term due to adsorbed molecules and carbon walls (Imt(S)) , a s described by
I,o , (s) = I,o, c (s) + I,o," (s) + Imt (S)
(17)
Although nanographitic structures of cellulose-based ACF are effected by water adsorption [31], it is expected that adsorption of non-polar molecules does not change the structure [31,32]. Although the adsorption of polar molecules can affect the interlayer spacing of the micrographites, we assume no change of the micrographitic structures in this analysis. As the contribution of adsorbed molecules to the total diffraction is predominant, this assumption should be reasonable. Hence, the first term, I,o,C(s), can be obtained experimentally from the diffraction data of the carbon only.
lobs c (S) - I,o, c (s) " P(O) "G(O) "A(O, p = O) + Ib,,,:k(s) " A(O, p = O)
(18)
Here IobsC(s) is the observed X-ray diffraction intensity of the carbon without adsorbed fluids. These correcting factors, P(6~), G ( 8 ) , A( 8 , p), and /back(S), are already described above. The interference term is given by j,k
sin srjk
I,,,t (s) - Z Z ]'c L ~ l,k srlk
(19)
Here, fa is the molecular scattering factor of adsorbed molecules. It is calculated by atomic scattering factors of component atoms. For example, in C C I 4 case, fcc~ is approximated by, [CCI4 -'-"~/ # "J"4fc21
(20)
where fc and fcl are atomic scattering factors of carbon and chlorine atom, respectively. The rjk in eq. 19 is the distance between the j-th carbon atom and the k-th adsorbed molecule. The interference term, l,,,(s), is smaller than the second term of eq. 17 which is approximately proportional to fa" In this analysis, this interference term, Im,(s), is neglected. Therefore, I, ota(s), the scattering intensity of the adsorbed molecules can be determined as,
47
I,o, ~ (s) = I,o , (s) - I,o, c (s) .
(21)
3.4. Electron radial distribution function analysis The X-ray diffraction of the adsorbed molecular assembly, I~ota(s), consists of coherent and incoherent X-ray scatterings,
Itota(S)~le~coha(S)'l'~(s)'lmca(S)}
(22)
Here Icofl(s) and Iicoa(S) are coherent and incoherent X-ray scatterings, respectively. The coefficient 9 (s) is the ratio of the intensity observed by a detector to the incoherent scattering. This coefficient ~ (s) depends on the experimental conditions of the monochrometer. The wavelength of incoherent scattering X-ray, ~ ', increases with increase scattering angle, 2 tY. Ak, - k,'- k, = 0.024626(1- cos 20)
(23)
where 3.' and //. are wavelengths of incoherent scattering and incident X-rays, respectively, and ,4 3. is their difference. Eq. 23 can be expressed by the function of s. The monochrometer removes a part of incoherent scattering. The single crystal of graphite having a large Mosaic-Spread is used as the monochrometer in this study. However, a graphite ]nonochrometer cannot sufficiently remove the incoherent scattering in the wavelength region near the characteristic X-ray. The monochrometer removes completely the incoherent scattering when the wavelength of X-ray is far from that of the characteristic X-ray. The wavelength of X-ray which is completely removed by the monochrometer is denoted Awid. The coefficient, ~ (A//.), is given by,
[1-(IAzl/•wid)(whenlAZl ~ Zw., ) .I.(AZ) - ~ 0 (whenlAZ I > Zw., )
(24)
The wavelength Aw,d should depend on the scattering angle of the monochrometer, or, and the length of the Mosaic-Spread, Am. Am
Zw.~ = ~ tan a
(25)
48
The average length of the Mosaic-Spread on graphite single crystal is 7 " 1 0 -3 rad. Therefore, //'w,d= 0.0657 A. AS ,4 A is the function of s, eq. (26) is derived,
9 (s) -- 1-
2" 0.024626" 2,2. sZ 2
.
(26)
8,~ 'l~w,d We can use the literature value [28] for incoherent scattering, I,,fl(s), and correct numerically the incoherent scattering term, 9 (s) "l,,c(s). The coherent scattering, I, ofl (s), is fitted to the self scattering factor [28] at large s region to determine the normalization factor, le [33].
~ s
~ (27)
/ ~o~~ (Sm~x)
Here s,,,,x is a maximum of s in the experiment. In this study, we use the average of intensity over 6-12 ,~-1. Figure 6 shows the normalized, coherent, and corrected incoherent scatterings of P20.
60 50 40 ~9r
30
2 =
20
10
0
I
I
I
I
I
2
4
6
8
10
-
12
s l .~ Figure 6. The corrected X-ray diffraction intensities of P20. Fine solid line: coherent scattering intensity of carbon; dashed line: corrected incoherent scattering intensity of carbon; dotted line: sum of coherent and incoherent scattering intensities; and bold solid line: diffraction intensity of P20.
49 The electron radial distribution function (ERDF) analysis improved by Nishikawa and Iijima.[34] is applied to elucidate the structures of adsorbed molecular assembly in micropores.
The ERDF,
4 ~r2 ( ,o (r)- ,Oo), is obtained by the Fourier transformation of
the structure function, si(s). Here ,o(r) and /90 are the electron density at a distance r and the average density of adsorbed molecules, respectively. This function, 4 ~r2(,o
(r)-,Oo) ,
is a differential electron radial distribution function and
exact ERDE ERDF.
As ,0 o cannot be determined precisely,
47rr 2,0(r)
4 ~'r2(,o(r)- ioo) is
4srr2(p(r)- P~ - ~c~Z ,2 ~ si(s)'exp(-Bs2)'sin(sr)" As
where Zj is the electron number of a component atom j. distribution of the electron clouds.
Structure function,
is an
often used as
(28)
The ERDF expresses the
si(s), is given by
si(s) - s{Icoha(s) - ~ f, (s) 2}
(29)
Then ERDF can be obtained, as shown in Figure 7. Each peak in Figure 7 corresponds to the interatomic distance on graphitic structure of P20. The exponential term of exp(B sZ), is called the convergent factor that is used for reduction of Fourier noise. The Fourier noise decreased with increasing the constant B, but the position resolution of the ERDF decreased simultaneously.
B = 0.2 - 0.5 was the best for analysis. The
measuring time for getting reliable date can influence the reliability of the ERDF
10
o Q.
"~ -10
-20
' .... 0.5
' .... 1.0 r/nm
~ .... 1.5
2.0
Figure 7. Electron radial distribution function of P20.
50 because of weak X-ray beam. more than 8 h.
Necessary measuring time for the reliable analysis was
4. EXPERIMENTAL METHOD AND POROSITY OF ACTIVATED CARBON Pitch-based ACFs of P5, P10, and P20 (Osaka Gas Co.) were used as activated carbon samples. The micropore structures were determined by N2 adsorption isotherms over P/P0 = 106 to 1 at 77K. The micropore parameters of surface area, micropore volume, and average slit width w were determined from the subtracting pore effect (SPE) method using the high resolution as-plot. Table 2 shows the micropore parameters. The activated carbon samples have considerably uniform micropores. Adsorption isotherms of CC14, water, methanol, and ethanol on ACF samples were determined at 303 K after preheating at 383K and lmPa for 2h. The density of adsorbed molecular assembly in micropores was determined from the micropore volume obtained by N2 adsorption at 77 K and the amount of adsorption near the saturated vapor pressure. Table 3 collects the density of the molecular assembly in micropores of ACF at 303 K. The X-ray diffraction of molecules adsorbed in micropores of ACFs was measured by the transmission method using an angle-dispersive diffractometer (MXP3 system, MAC Science) in the scattering parameter s (=47rsin 8/A.) range of 0.7 to 12A1. The monochromatic MoKa at 50kV and 30mA was used for the diffraction measurement. The experimental arrangement of the X-ray diffraction is shown in Figure 8. The diameter of in situ X-ray diffraction chamber is 96mm. The sample holder is installed in the chamber. Mylar films are used for windows of the in situ measuring chamber and the sample holder. The sample chamber is connected to the vacuumadsorption system for controlling adsorption conditions. As the sample holder is open, gas molecules are adsorbed on carbon samples in the holder. Samples in the in situ x-ray chamber can be cooled by flowing liquid Nz down to 90K without condensation of adsorbate molecules using an indirect cooling device. The temperature of the sample is measured by a silicon diode on the sample holder. Here the distance between the silicon diode and the sample is 5mm. Table 2 Micropore structures of ACF samples Micropore volume Surface area ml.g 1 P5 P10 P20
0.336 0.622 0.971
m2.g1 900 1510 1770
External surface area m2-g-1
Pore width
5 25 44
0.75 0.82 1.13
nm
51 D
C
A
Figure 8. The schematic diagram of experimental arrangement. A: X-ray tube, B: sample, C: graphite monochrometer, D: scintillation counter, E: sample chamber, DS: dispersion slit (1.0 degree), SS: scattering slit (1.0 degree).
The in situ X-ray diffraction patterns were measured mainly at 303 K. The ground carbon samples are set in the sample holder and pre-heated at 383K and lmPa for 2h. Then X-ray diffraction of activated carbon samples in vacuo at 303K is measured which is used for correction of the diffraction contribution by carbon samples only. Then, adsorbate molecules are adsorbed and the X-ray diffraction of the molecule-adsorbed activated carbon is measured after a sufficient equilibration at P/Po = 0.9 and 303K. When the X-ray diffraction pattern of adsorbed carbon sample is measured at low temperature, the sample chamber is closed after the measurement at 303K. Only the X-ray diffraction of the water-adsorbed carbon samples is observed at 143 - 255K. The temperature of the sample is kept constant within __.0.1K for 303K and within __.5Kfor 143 - 255K during the X-ray diffraction measurement for 6h. The information on the adsorbed molecular structure is obtained as follows using the example of water: the X-ray diffraction intensity of microporous carbon with adsorbed molecules, Iais(s), expressed in terms of the additive form of the scatterings by carbon wall structures (the first term of eq.8), by adsorbed water (the second term of eq.8), and by the adsorbed water-carbon interface structures (the third term of eq.8), as given by eq.30.
X f2 + X ~ 2fw 2 sinsrk~k2 k 1 ~k9
I ~,, (s) = I
fc2 +
X J1 9 2
sr
SFklk 2
J132
+ ~.a
fc f~,
(30) sr
#
52 Here Ie is the standardization factor, and fc and fw are the atomic scattering factors of carbon and oxygen in water, respectively. The scattering caused by hydrogen atoms in water is neglected. The notation rj~ j2, rk~ k2 and rjk are the atomic distances. In this analysis, we neglect the scattering due to the adsorbed water-carbon interface. Hence, the X-ray diffraction of the adsorbed water is determined by subtraction of the first term in eq. 30 from the measured scattering. The similar procedure is applied to another adsorbate molecule. Some analytical results will be described below.
5. I M P E R F E C T PACKING STATES OF G E l 4 M O L E C U L E S The liquid structure of CC14 having Td symmetry was examined by Narten [26] and Egelstaff et al [40], suggesting a so-called bcc cluster model. Nishikawa and Murata [41] proposed the presence of a long range correlation even in a CC14 liquid state at room temperature. It is well-known that CCI 4 shows unique liquid-solid transition and it has three solid polymorphs [38]. Therefore, the molecular assembly structure of CC14 in micropores should be sensitive to the pore width. Also the study on CCI 4 confined in carbon micropores is the most reliable due to an clear contrast as mentioned before. All adsorption isotherms of CC14 were of Type I, indicating that CC14molecules are adsorbed by a representative micropore filling mechanism. As the interaction of CC14 with the graphitic micropore is stronger than that of N2, a noticeable adsorption of CC14 molecules begins at a very low pressure. The adsorption isotherms of CC14 were Table 3 The density of molecular assembly at 303 K Adsorbate
CC14
w / nm
Adsorbed density
Bulk liquid density
Bulk solid density
/ g cm -3
/ g cm -3
/ g cm -3
0.75
1.26
1.594
1.76(fcc) "1
1.1
1.43
1.87(monoclinic) .1 1.79 (Rhombohedral) .1
HzO
0.75
0.86
1.1
0.81
0.996
0.92(Ih) 0.92(Ic) .2 1.17(IceII, 123K) .2
CH3OH
0.75
0.67
0.98(113K)
C2HsOH
1.1 0.75
0.96 0.71
1.03(87K)
"J: ref.[37, 38]
1.1 ~2: ref.[39]
1.05
53
I
[[]
o 0.. i
P10 " Liquid [II] [IIIl
0
,,
0
0.5
1.0
I , , , ,
1.5
I , , ,
2.0
,I
,
2.5
,,,
3.0
r/nm Figure 9. Electron radial distribution functions of CCI 4 adsorbed on P10 and bulk liquid CC14 at 303 K. analyzed by the DR plots, leading to the adsorption capacity for CC14 in micropores. The density of adsorbed CC14 was determined using the adsorption capacity for CCla and the micropore volume determined by N2 adsorption at 77 K, as mentioned before. The density of adsorbed CCI 4 is shown in Table 3. The adsorbed density is much smaller than the bulk liquid density, indicating that the molecular size of CCI 4 is too great to fill the void space efficiently. Correspondingly, the adsorbed density of P5 (w = 0.7 nm) is smaller than that of P20 (w = 1.1 nm). The GCMC simulation study showed the same results on the density. The diffraction pattern of CCI 4 on P20 was close to that of bulk liquid, but their intensities were different from each other. The diffraction intensity of P5 was markedly weak, compared with those of P20 and the bulk liquid. The ERDFs of CC14 adsorbed in micropores of P10 and bulk liquid CC14 at 303K are shown in Figure 9. The ERDF of bulk CC14 liquid, which has main four [I], [II], [III], and [IV] peaks in the intermolecular region, is similar to the literature [40]. The [I] and [II] peaks are assigned to the nearest neighbors and second neighbors, respectively. The [I] peak of confined CC14 is the same as that of liquid CC14, while the [II] peak of the confined CCI 4 is situated at a shorter distance than that of bulk liquid. The intensities of [I] and [II] peaks are less than those of the bulk liquid. The [III] and [IV] peaks in the long-range of the confined CCI 4 are not attenuated regardless of the weak intensity compared with those of bulk liquid. Hence, the confined CCI 4 should have a more ordered longrange structure than bulk liquid regardless of the smaller adsorbed density. Figure 10 shows the ERDFs for confined CCI 4 in micropores of three kinds of ACFs. Each ACF has a different ERDF, indicating the inherent molecular arrangement. The
54
[I] peak position does not change with the pore width, while the [II] peak shifts to a smaller value with the decrease of the pore width. The [I] and [II] peak intensities decrease remarkably with the decrease in the pore width, suggesting that the serious geometrical restriction prohibits the perfect formation of the first nearest and second nearest neighbor structure around a central molecule in the narrower pore. Thus the short-range structure of the CC14 molecular assembly in the micropore is different from the bcc cluster type packing which is proposed for the bulk liquid. The confined CC14 molecules have a defective nearest neighbor structure, leading to the shorter side-shift of the second neighbors. Only the ERDF of P5 has a different peak structure in the distance range over 1.2 nm. The [II] and [III] peaks of P5 are markedly shifted from those of other ACFs in the opposite direction each other and there is an explicit new peak between [II] and [III] at 1.33 nm. This feature indicates the presence of the serious restriction of CC14 molecular packing in the narrowest pore and thereby molecules are packed in a different way from the bulk liquid. The average pore width of P5 corresponds to the 1.3 layer thickness of CCI 4. Accordingly, CC14 molecules cannot form the bcc cluster packing structure. The Grand Canonical Monte Carlo (GCMC) simulation provided the ERDF having the above features [19,42]. The CC14 molecules are packed in the narrow slit space to form an imperfect packing structure giving the observed ERDE Although the CC14 molecule can be approximated by a spherical molecule, it is noteworthy that the geometrical restriction provides the variety for the molecular packing [43,44].
, I
[II
P5 ~P10 . . . . . P20 [IIl
|
Q.. r
~
[III]
[IV]
g..
0
,
r
I: I
0
0.5
1.0
1.5
2.0
2.5
3.0
r/nm
Figure 10. Electron radial distribution functions of CC14 adsorbed on kinds of ACFs at 303 K.
three
55 4000 (A)
d
P20 " - - 303 . . . . . 223 ----143
3000 1 ~\ I~
6. d
2000
at 303 K K K K
(B)
0
1000
00
1
2
3
4
5
6
s/A;1 Figure 11. X-ray diffraction intensities of P20 at 303 K and water-adsorbed P20 as a function of temperature.
6. OREDERED STRUCTURES OF WATER IN HYDROPHOBIC MICROPORE The surface tension of water depends strongly on the curvature of the gas-liquid interface; the depression of the freezing point of water in a small space, which has been believed to be associated with biological and geological processes, is proportional to the reciprocal of the pore width [45,46]. The macroscopic properties of water confined in the small space are not sufficiently understood from the standpoint of molecular science yet. The enhanced intermolecular interaction in micropores should lead to an unusual behavior of the confined water. As the graphitic micropores of ACF can offer an excellent hydrophobic nanospaces, water confined in micropores of ACF was studied. Although adsorption of water by activated carbon has been studied for many years, the adsorption mechanism is not sufficiently established yet [47-52]. In adsorption of water on activated carbon, water molecules are presumed to form clusters prior to filling of micropores without any direct evidence. Then, X-ray diffraction study on adsorbed water has been strongly requested. The water adsorption isotherms at 303K are of Type V, as shown in Fig. 4. The saturated water adsorption amount of P5 and P20 are 290 and 790mgg 1, respectively. The fractional filling of water molecules on P5 and P-20 at 303K are 0.86 and 0.92 which are obtained using the density of the bulk liquid water at 303K and the bulk liquid N2 at 77K. The density values of water adsorbed on P5 and P20 are 0.86 and 0.81, respectively, which is smaller than that of bulk liquid and solid water. The water adsorption isotherm of P5 has almost no adsorption hysteresis, while P20 has a remarkable adsorption hysteresis.
56
(F)
........ 303 K - - 258 K ........ 223 K 143 K " . . . . Liquid
2 Q. ,
9
1
:
-1 -2 -3
0
0.5
1.0
1.5
2.0
r/nm Figure 12. Electron radial distribution functions of water adsorbed on P20 as a function of temperature and bulk liquid water at 303 K. X-ray diffraction patterns of water adsorbed P20 (w = 1.13nm) as a function of the temperature were measured, as shown in Figure 11. The X-ray diffraction patterns for all temperatures have broad peaks. In the bulk water case, the X-ray diffraction pattern changes drastically at melting point, giving sharp peaks below the melting point. X-ray diffraction patterns of bulk water at 303K and 233K are completely different from those predicted. In case of the adsorbed water there are no sharp peaks due to the bulk ice formation even at 143K. This result indicates that the water molecules in the carbon pore have only short-range order even at 143K because of space restrictions by pore walls. Both of the first peak (A) at s = 1.8 ,~-1and second peak (B) at s = 3.1/k 1 change gradually with temperature even across the freezing point of bulk water. However, Figure 13 cannot provide the detailed structure of adsorbed water itself. The ERDF analysis was also applied to this case. Figure 12 shows the ERDFs of water adsorbed in the 1.13nm-slit space at different temperatures.
The ERDF of bulk liquid water at 303K is also shown in Figure 14
for
comparison. Water adsorbed at 303K has the highest peak (F) and shoulder one (E) at 0.42 and 0.35nm, respectively. The feature at this region sensitively varies as lowering the temperature. The peak or shoulder (E) at 0.35nm is assigned to the nearest-neighbor water molecules, whereas peaks (F) and (G) are attributed to the second and third nearest-neighbor molecules, respectively.
Hence these three peaks provide the detailed
information on the short range structure of the water molecular assembly in the micropore. As bulk liquid water has an interstitial molecule at the nearest-neighbor position, and thereby the peak (E) is higher than the peak (F) in case of bulk liquid water. It is shown that the adsorbed water even at 303K has a more ordered structure
57 than bulk liquid. The ERDFs show the strong dependence with temperature. The shoulder (E) at 303K becomes a peak as lowering the temperature and at the same time the intensity decreases markedly. On the contrary, the intensity of the peak (F) increases with the decrease of the temperature. The position of the peak (E) shifts to a smaller value and the positions of the peaks (F) and (G) shift to a greater value with the decrease of temperature. The positions of peaks (E) and (F) at 143K are 0.282 and 0.456nm, respectively. These values are close to those of the bulk ice of Ih form, because the distances to the first and second nearest neighbor molecules in ice are 0.275nm and 0.449nm, respectively. The brief agreement indicates that the water molecular assembly in the 1.1nm-carbonaceous slit space has a similar structure to the bulk ice. The preliminary X-ray diffraction simulation suggests that the simulated diffraction patterns of nanocrystals having Ih and Ic structures do not coincide with the observed ones of water adsorbed in carbon micropores [53]. Also it should be stressed the fact that water molecules in micropores even at 143K have higher mobility than bulk ice, which is shown by broad peaks. The ERDF of water adsorbed at 143K has an ordered structure up to 1.3nm at best. This ordered structure range of the adsorbed water is plausible for the horizontal direction of the slit space. Although the structure of the water adsorbed was examined over the wide temperature range of 143K to 273K, we could not observe a clear phase transition behavior of water adsorbed in the micropore from X-ray diffraction. Figure 13 shows ERDFs of water adsorbed in micropores of 0.75nm in pore width as a function of the temperature and bulk liquid water at 303K. The peak intensities of (J) and (K) which are assigned to the second and third nearest-neighbor molecules increase slightly with the decrease of measuring temperature. The intensity difference between 143K and 303K is much smaller than that of the wide pore system of P20. Water molecules in P5 form a more ordered structure than in P20 and they cannot move easily even at 303K. The position of peak (J) shifts to a greater value as lowering temperature, while the position of peak (K) does not change. The first nearest neighbor molecule shows only the weak shoulder (I) near 0.3nm. Thus the structure of water adsorbed in P5 has a completely different temperature dependence from that in P20. This X-ray diffraction study clearly shows that water molecules confined in the hydrophobic nanospace have a solid like structure of the different molecular mobility. The ordering of the water molecular assembly structure depends sensitively on the pore width. We need the X-ray diffraction simulation in order to understand the water molecular assembly structure more precisely. This structural difference was ascertained by differential scanning calorimetry (DSC) [54]. The freezing DSC curve of water confined in micropores of P20 gave an exothermic peak at 220 K, whereas that of P5 had no peak. As the bulk water gave a single peak at 250 K, the DSC date were obtained under the supercooled conditions. Then, the marked depression of the freezing temperature occurs in water confined in micropores of
58
(J) (K) 2
0)
.
/~
!
......... 303 K ..... 223 K --143 K Liquid "
-
-
9 r
i
0 -1 -2 -3 L_______ 0 0.5
1.0 r/am
1.5
2.0
Figure 13. Electron radial distribution functions of water adsorbed on P5 as a function of temperature and bulk liquid water at 303 K.
P20, as observed in mesopores. Recently, GCMC simulation predicted that the freezing temperature of confined fluid in micropores depends on the pore-wall fluid interaction[55]. The freezing temperature of CC14 and benzene is noticeably elevated in micropores of ACF [56, 57]. Hence, we can conclude that even fluid confined in micropores can exhibit phase behavior. In case of water, water confined in the micropores of P20 exhibits the phase transition, whereas that of P5 does not, agreeing with the results of X-ray diffraction. Also the observed enthalpy change of water of P20 upon the phase transition was only 14 % of the bulk value. Hence, the thermodynamic state of the confined water in the carbon micropores of P20 is different from that of the bulk liquid. The confined liquid water is more stabilized due to a partial ordering and the confined solid is unstable owing to the serious restriction for formation of the perfect crystalline structure. Thus, water confined in the hydrophobic micropores of ACF has a special ordered structure[59-61].
7. ORIENTATIONAL FILLING OF A L C O H O L MOLECULES. An alcohol molecule has both of polar and nonpolar sites. The relationship between the amphilicity and the assembly structure in micropores is quite important to understand micropore filling mechanism of hydrophilic molecules. The balance between hydrophilicity and hydrophobicity can be controlled by the chain length of aliphatic group. Also alcohol molecules have hydroxyl group which can form
59 hydrogen bond.
Then, studies on alcohol confined in carbon micropores should be
helpful to understand the mechanism of water adsorption in carbon micropores.
Here,
the results on ethanol and methanol are described [61,62]. The adsorption isotherms of ethanol on ACF at 303 K are shown in Figure 14.
All
isotherms are of Type I , being analogous to the N2 adsorption isotherms. Hence ethanol molecules are adsorbed by micropore filling mechanism. The adsorption isotherm of methanol of P5 at 303 K is shown in Figure 15.
The adsorption isotherm
of ethanol on P5 at 303 K is also shown for comparison.
Although the adsorption
isotherm of ethanol has a remarkable uptake in a very low P/Po range, the adsorption isotherm of methanol has a gradual increase of adsorption below P/Po = 0.3. Therefore, the interaction of an ethanol molecule with the micropore is much greater than that of a methanol molecule. However, the methanol molecule can interact more strongly with the micropore-walls than a water molecule, because almost no water molecule is adsorbed below P/P0 = 0.3 [63]. Hence, the methanol molecule can play a role of the intermediate between water and ethanol molecules for carbon micropores. We can expect that the
-~
1000
""6
800
{
600
,~
400
"~ E <
2 0 0 ~ L~-~ ~ o
L] I} [-{-1
o
o
oo
,
,
,
0.6
0.8
P/P0
Figure
14. Adsorption
isotherms of
ethanol on ACFs at 303 K.
O" P5 and
[-l" P20
{
te eoooe o o~~ o
%
eo
D 0
"~ 100 D O
po o
was determined, as collected in Table 3. The adsorbed density of P20 is greater than that of P5 for both of ethanol and
50
methanol. The adsorbed density of P20 is almost equal to the bulk solid
0
density, but that of P5 is slightly
o
0.4
DR plots.
Then, the adsorbed density
o
L
_~ 150 8
methanol
o
0.2
different from those of water and ethanol. The saturated amounts of adsorption in micropores for ethanol and methanol were obtained from the
confined
o
0
200
of
-
O!/
is
structure
,j I'1 t !'I' II
0
I
i
I
I
0.2
0.4
0.6
0.8
1
P/P0
smaller than the bulk liquid density for
Figure 15. Adsorption isotherms of
both cases.
These adsorption data
methanol and ethanol on P5 at 303 K.
should
associated
O" methanol and 0 " ethanol
be
with
the
60
~0.27
,,
2 0 -2
0.5
1.0
1.5
2.0
r/nm Figure 16. ERDFs of ethanol adsorbed on P20 as a function of the fractional filling ~ at 303 K.
structures of alcohol molecules in micropores. The diffraction patterns of the ethanol-adsorbed ACF had a broad peak due to the adsorbed ethanol at s = 1.5 ,~-1. The difference of diffraction intensities between the ethanol-adsorbed ACF and ACF itself was assigned to the adsorbed ethanol molecular assemblies themselves. Figure 16 shows the ERDFs of ethanol in micropores of P20 at different fractional fillings. The ERDFs of bulk liquid ethanol is also shown for comparison. The intensity of the peak of adsorbed ethanol in the short range is higher than that of bulk liquid. The intensity of the peak in the short range for t~= 0.27 is the greatest. The adsorbed ethanol has greater amplitudes than bulk liquid in the range of 0.3 to 1.2 nm. The first and second nearest peaks at 0.5 and 0.9 nm shift to a shorter distance compared with those of the bulk liquid. The periodical structure above 1.5 nm is observed for the adsorbed ethanol. This long-range ordering suggests that adsorbed ethanol molecules form highly ordered structure. However the incomplete removal of the scattering between adsorbed molecules and pore-walls may contribute to the long periodical structure to some extent. The higher peak and the peak shift suggest that the immobile state of the adsorbed molecule and thereby that ethanol molecules adsorbed in micropores of P20 should have an ordered structure of less mobility. Figure 17 shows ERDFs of ethanol adsorbed on P5 and P20 at P/P0 = 1 and 303 K. The amplitudes of ERDFs of confined ethanol are greater than those of the bulk liquid ethanol. The peak positions of the confined ethanol shift from those of the bulk liquid. The ERDFs of confined ethanol depend on the pore width. We calculated the radial
61
"---P5
ZPTu d
1t I
%
-2
0
0.5
1.0
1.5
r/nm
Figure 17. ERDFs of ethanol adsorbed on ACFs at saturated pressure. Solid bars show each positions and intensities for crystalline ethanol.
distribution of the crystalline ethanol using the crystallographic data [64] and atomic scattering factors of carbon and oxygen atoms. As the crystalline structure gives a discontinuous distribution of the neighbor atoms, it is shown by vertical bars of which height is proportional to the coordination number. However, this calculated distribution is different from the average density. Hence, we will use this calculated distribution for a qualitative structure determination of the confined ethanol. Especially, the peak position of adsorbed ethanol at r = 0.49 nm for P20 is close to the peak position of the crystalline state at r = 0.47 nm. These results indicate that the solid-like structure is formed in the short distance. On the other hand, the peak of adsorbed ethanol at r = 0.44 nm for P5 suggests the strong confinement of ethanol molecules in narrower space and not to form the ordinary solid structure because of the serious confinement. The peak position increases with the pore width, approaching to that of bulk liquid. On the other hand, the peak intensity decreases with the pore width. These tendencies hold for both of the peaks of the first and second nearest neighbors. Both changes indicate that ethanol molecules should be confined strongly in narrower micropores. The adsorbed density of ethanol molecules in micropores of P20 is not close to that of bulk liquid, but that of bulk solid. Also the ERDF features strongly support the ordered structure formation. The ethanol molecule is a linear molecule and both groups of C2H5 and OH have completely different hydrophobicity and hydrophilicity,
62 respectively. The ethanol molecules should be associated with each other using hydrophobic and hydrophilic pairs. J6nsson examined the crystal structure of ethnaol and showed that ethanol molecules are joined by two kinds of hydrogen bonds to form infinite zigzag chains [64]. Morishige et al. showed that ethanol molecules adsorbed on the graphite surface have a hydrogen-bonded specific structure at low temperature [65]. Thus, the hydrogen bonding plays an important role in the structure of ethanol molecular assemblies. The stabilization by the hydrogen bonding is more than 20 kJ/mol [66]. In case of the ordered structure of ethanol molecules in micropores, the hydrogen bonds should play an essential role. We must take into account the adsorbed density, the crystal structure, and the possible hydrogen bonds for the ordered structure of ethanol molecules in micropores. Figure 18 shows the models of adsorbed ethanol in P5 and P20. The model structure for P20 was constructed from the crystal structure of the bulk solid ethanol. This model is consistent with the observed ERDF and the density of ethanol adsorbed in micropores of P20. On the contrary, micropores of P5 are too small to form the same structure. The molecular potential calculation showed that P5 has no double minima, but a single deep minimum on the central plane between both graphitic micropore-walls [67]. Then ethanol molecules should form a single sheet structure in which ethanol molecules are associated with hydrogen bonds. This single sheet structure gives the observed low density. Figure 19 shows ERDFs of methanol adsorbed on ACFs and bulk liquid methanol at 303 K. Briefly speaking, the ERDF of adsorbed methanol has similar peaks to that of liquid methanol; the peak positions of adsorbed methanol are close to those of liquid methanol. However, the nearest neighbor peak of the ERDF for liquid at 0.46 nm is slightly shifted to a smaller distance for P5 (at 0.44 nm), although the peak of P20 is the same as that of liquid. Also the nearest peak of P5 has a weak peak near 0.38 nm. Tauer and Lipscomb reported that the intermolecular distances between neighbor molecules in bulk solid methanol at 113 K are 0.364 nm for CH3-CH3, 0.41 and 0.42 nm for CH3-O , and 0.40 nm for O-O [68]. Accordingly, the peak of the ERDF for highly
Figure 18. Model of adsorbed ethanol in P5 (a) and P20 (b) at 303 K.
63
P5 P?O.
4 A I
2
Q-
0
I~
-2
-4 !
0
0.5
1.0 r/nm
1.5
2.O
Figure 19. ERDFs of methanol on ACFs and bulk liquid at 303 K.
ordered structure of confined methanol can have the fine structure in the nearest neighbor peak. Methanol confined in micropores of P5 should be more ordered than liquid methanol. Also, the amplitudes of ERDF of confined methanol are much greater than that of liquid methanol, indicating that methanol molecules in micropores are immobile compared with liquid. As P5 has greater peak intensities than P20, methanol molecules on P5 should be more immobile than those on P20 regardless of the small adsorbed density. Therefore, methanol molecules should form an ordered structure of the less packing density for the narrow micropores of P5 as well as ethanol adsorption. Consequently, even methanol molecules are too great to construct a quasi three-dimensional solid structure in the micropore of P5. On the contrary, only ERDF of P20 has an explicit peak at 1.15 n m , indicating the formation of the third nearest coordination structure which is not observed in bulk liquid. Thus, methanol confined in micropores of P20 should have a considerably long range order, agreeing with the adsorbed density value. Methanol molecules do not form the orientationally ordered structure similar to ethanol, but the position-ordered structure which is combined with weaker hydrogen bonds than confined ethanol. Methanol molecules confined in micropores should have more freedom of rotational motions than confined ethanol. Also methanol confined in narrow micropores of P5 must have the small adsorbed density and the highly ordered structure. The effectiveness of X-ray diffraction for study on micropore filling is described in this article. However, still we need the integrated analysis using several methods in addition to X-ray diffraction. Anyway we need the high resolution adsorption experiment in order to elucidate the micropore filling. In addition to the adsorption experiment, we must provide other physical measurements such as low temperature
64 magnetic susceptibility, direct calorimetry, NMR, and molecular spectroscopy in order to filling process of molecules [69-76]. Molecular simulation is also quite powerful to understand micropore filling phenomena [77-80]. Such studies are described in other chapters. In future we need to understand the micropore filling for supercritical gases, because storage of H2 and CH4 is quite important subject [81-83].
Acknowledgment This work was funded by the Grant in-Aid for Scientific Research on Priority Areas No. 288 "Carbon Alloys" from Japanese Government. We thank to Professor Keiko Nishikawa for her helpful discussion on ERDF analysis and Mr. Tomonori Ohba and Miss. Takako Kimura for their preparation of this manuscript.
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RecentAdvancesin Gas Separationby MicroporousCeramicMembranes N.K. Kanellopoulos(Editor) 2000 ElsevierScienceB.V. All rightsreserved.
NEUTRON AND ION BEAM SCATTERING
67
TECHNIQUES
J.D.F. Ramsay Laboratoire des Mat6riaux et des Proc6d6s Membranaires, UMR CNRS 5635, Universit6 Montpellier II, 2 pl Eug6ne Bataillon, 34095 Montpellier, France
ABSTRACT Neutron coherent and incoherent scattering techniques can provide details of the microstructure of porous materials and the properties of sorbate molecules within micropores. Here the application of coherent scattering which includes, small angle neutron scattering (SANS), and diffraction are described. Two recent developments of SANS are illustrated. The first concerns the analysis of anisotropic porous materials (microporous carbon fibres, anodic alumina membranes). The second, concerns contrast variation methods, for the investigation of adsorption mechanisms (capillary condensation, micropore filling) in porous networks. Neutron diffraction can also provide information on the properties of sorbed phases conf'med in micropores, as illustrated by sorbed water in sol-gel oxides. The application of resonant ion (proton, a-particle) beam backscattering for the determination of pore size is illustrated by results obtained on meso and microporous oxide films; these films were deposited on substrates using sol-gel processes. Because a microfocussed beam is used with this technique, it is possible to obtain microstructural details, both as a function of depth (-~ 3 ~tm to - 100 ~tm) and across the sample (> 25 ~tm), with a high resolution.
1. INTRODUCTION Over the past decade there has been considerable progress in the development of microporous membranes (1,2). These membranes have been synthesised from several different materials which include microporous carbon (3), glass (4) and different oxides, such as silica (5-7) for example. Compared to organic polymers, such inorganic materials have superior properties (e.g. chemical, thermal stability) which makes them attractive for novel membrane applications (e.g. gas separation and catalytic membrane reactors (8)). The porous microstructure, or texture of these membranes may however be complex, and will depend on the type and conditions of the synthesis process employed, as has been discussed (9,10). Furthermore the diffusion and transport processes of gases in such microporous membranes will be controlled by the microstructure. A detailed understanding of membrane microstructure (e.g. pore geometry, size, shape, connectivity and surface properties) is thus required in order to optimise separation performance. A wide range of techniques has been used for the characterisation of porous solids; the basis and recommendations on the suitability of these techniques has recently been given by IUPAC (11). Thus the choice of a
68 particular technique is dictated by the sample characteristics, such as the nature of the material, whether it is supported of not, its size, shape, isotropy and mechanical resistance, as well as the range of pore size. The destructive nature of the technique may also require attention. Thus the sample may require pretreatment (drying, outgassing) to eliminate adsorbed species, such as water, especially in the case of microporous materials. The characteristics of membranes is more demanding than most other porous materials. Firstly, the membrane separation layer is generally thin and supported, which requires a sensitive technique capable of analysing a sample in such a form. The membrane material in powdered form may indeed have a different texture. Secondly, the structure is frequently anisotropic and may moreover be microporous. Assessment of microporosity is much less advanced compared to meso- and macroporosity, despite the widespread emphasis given to this topic recently. The two traditional techniques of mercury intrusion and gas adsorption have important limitations. Both of these are destructive methods, which require outgassing of the sample. Furthermore in the case of mercury intrusion, the sample is subjected to very high pressures to access micropores, which frequently leads to destruction of the microstructure. The application of gas adsorption isotherm measurements to analyse microporosity has recently advanced considerably, although there are reservations which still exist concerning the general application of theories to describe adsorption in such small pores in ill defined structures (12). This situation has lead to a growing interest in other complementary and specialised techniques which are suitable for the characterisation of microporosity. These include methods involving NMR and radiation scattering with X-rays and neutrons for example. In the present chapter the techniques of small angle neutron scattering (SANS) and resonant ion backscattering will be described. Although the SANS technique is not new, there have recently been important technical advances, and developments in the characterisation of porous materials. Two of these developments will be described in detail. The first concerns the analysis of anisotropic pore structures, e.g. in microporous carbon fibres and anodic alumina membranes. In the second, contrast matching methods have been applied, to investigate, in situ, the mechanisms of gas adsorption in porous materials. Other related information, on the physical state of condensed gas phases confined in microporous media, obtained from neutron diffraction measurements, will also be illustrated. Finally, the techniques of resonant back scattering, with protons and a-particles, will be described. Using such microfocussed ion beam methods, unique information on the structure of thin porous layers can be obtained, as will be illustrated with microporous oxide rims, prepared by sol gel processes.
2. CHARACTERISATION OF POROUS MATERIALS BY NEUTRON SCATTERING
2.1 General aspects of neutron scattering techniques The energy of neutrons used in scattering studies normally falls within the range - 300 to 0.4 meV, and corresponds to a wavelength, ~, between-~ 0.5 to -~ 15 A. Traditional sources of neutrons have been nuclear reactors, some of which have been designed to produce a flux within this energy range, typical of these is the high flux reactor (HFR) at ILL, Grenoble (13). These reactors have more recently been complemented by pulsed sources which employ particle accelerators to generate neutrons by the bombardment of a target, as for example in
69 the Spallation Source, ISIS, at the Rutherford Appleton Laboratory in Oxfordshire (14). Neutron scattering by matter arises either through an interaction with the atomic nucleus or magnetically, if atoms have unpaired electron spins. Since detailed treatments of these different scattering processes have been given in numerous reviews and books (15-17) the following outline will only indicate the basic principles and information which can be derived from the technique when applied in the present context. In particular we will confine our attention to the more important phenomenon of nuclear scattering. We can describe the scattering from a single rigidly fixed atom in terms of its cross section, or, where cr = 4rc b 2
(1)
here b is defined as the scattering length of the bound atom. However, when scattering occurs from matter which is composed of an assembly of non-rigidly bound atoms, there will be two distinct contributions to the total cross section, which arise from coherent and incoherent effects. The first of these, Or results in interference between the neutron waves scattered by the nuclei, and is associated with a coherent cross section given by
O'~oh = 4zCb:o h
(2)
The second, (Yinc, is due to interactions between the spin states of the neutron and the nucleus, and gives rise to isotropic scattering ; it does not exist for nuclei having zero spin, e.g. 12C and 160. Values of bcoh, O'ine, and the neutron absorption cross section Oa, for different nuclei are given in Table 1. Table 1 9Coherent scattering length, bcoh, and incoherent and adsorption cross sections, ~inc, and Oa for different elements. 1012beoh/Cm2 1024ainc/cm2 ..... 10240Jcm2 H -0.374 79.7 0.33 D 0.667 2.0 0.0005 C 0.665 0.0 0.0035 N 0.94 0.3 1.9 O 0.58 0.0 0.00019 A1 0.35 0.0 0.23 Si 0.42 0.0 0.17 Ti -0.34 3.0 6.1 Fe 0.95 0.4 2.6 Zr 0.72 0.3 0.18 Ce 0.48 0.0 0.63 Th 0.98 0.00 7.4 U 0.842 0.00 7.5 Data are for natural isotopic mixture and neutron wavelength of 1 A (as compiled by S.W. Lovesey ref. 16).
70 This shows that beohvaries erratically from element to element, and even for different isotopes - a feature which can be exploited in contrast variation studies (18), as will be described. Another important feature is the large value of 6i,r for the proton, which dominates that of other nuclei ; this makes incoherent scattering measurements particularly suitable for the study of hydrogenous materials (e.g. water, hydrocarbons, polymers etc.), in situations where other spectroscopic techniques (infrared and Raman spectroscopy, NMR) are unsuited because of absorption problems. This feature which has been exploited extensively in investigations of the diffusion and dynamics of water and hydrocarbons sorbed in oxide gels and zeolites for example, will be discussed in detail by Jobic in Chapter 1.4 of this volume. Neutron coherent scattering has its counterpart in small angle X-ray scattering (SAXS) and diffraction, for which the theory is very similar (19), although the possibilities afforded by contrast variation are unique to neutron scattering (18). Small angle scattering, SAS, arises from variations of scattering length density (see below) which occur over a distances dsAs (where dsAs ~ 2/20 corresponding to scattering angle 2 0 for radiation with a wavelength ~,) exceeding the normal interatomic spacings in solids and liquids. Such an effect thus occurs with : (i) assemblies of small particles in air or vacuum comprising a porous material (20-22) ; (ii) solid materials containing voids or pores (23) ; (iii) solid solutions such as alloys (23) ; and colloidal dispersions of particles and polymers in liquids (24-26). With many of these systems there are frequently inherent practical advantages in using neutron radiation became of its lower absorption in most materials compared to that of Xrays ; the latter usually require very thin specimens and considerable restrictions of sample environment. In the subsequent section the application of small angle neutron scattering in studies of porous materials will be described in more detail. 2.2 Small Angle Neutron Scattering (SANS)
2.2.1 Theory of SANS Small angle neutron scattering measurements of both X-rays and neutrons can provide structural details of porous materials on a scale covering a range from 1 nm to > 100 nm. SANS arises from variations of scattering length density, Pb, which occur over distances exceeding the normal interatomic spacings and occurs when solids contain pores. Details of the porosity and surface area can be obtained from measurements of the angular distribution of the scattered intensity (see Figure 1). The appropriate range (20) where this information is contained is defined by the momentum transfer, Q, and the size of the pore, d, where 4re sin O
Q =----T--
(3)
An analysis of the scattering in the range 0. I < Qd < 1 provides details of the size and form of scattering object (pores); information of the surface properties may be obtained at larger angles (Qd > > 1) as depicted schematically in Figure 2.
71
Since the theory of SAS for both X-rays and neutrons has been covered extensively in several comprehensive reviews and books (19, 27), the following outline will only indicate the basic principles briefly. The scattering associated with the coherent cross section of nuclei in a material has a spatial distribution, which is a function of the distribution of these nuclei. This scattering can be expressed as a partial coherent cross section (4)
do"cob _ I s ( O ) dO
IoN
Where Is is the scattered intensity (neutronsl,s l ) in solid angle .(2, I(Q) is the incident flux (usually expressed as neutrons,sl,cm 2) and N the number of scattering nuclei exposed to the beam. The coherent scattering cross-section per atom at small angles 2 0 is given in the static approximation as
d O = -N
R exp(iQR)
1
(5)
where bR is the coherent scattering length of the chemical species occupying a site with the position vector R in the material. By replacing ba by a locally averaged scattering length density pb(x), where r is a variable position vector, we can write
do" _
f(pb(r)exp(iQr)d3r )
d.O-
v
12
(6)
where the integration extends over the sample volume, K Debye and Bueche (28) showed that Eq. (6) can be simplified for small vectors r to give the final expression do" _ 1 _ 2 ~ , sin(Qr) d.Q - -N ~7 V .11 ( r ) 4 zcr2dr o
Qr
(7)
m2
where ~1 is the mean fluctuation in scattering density, viz 2
= (p~(r)
_
~
)2
(8)
and y ( r ) is a correlation function in scattering density defined as
(r/,r r(r) =
m2
7?
(9)
72
2O
NEUTRON BEAM
!
I
|--
I I J
- -
I
I
l
SAMPLE
I
2 D-DETECTOR
Figure 1. Schematic diagram of small angle neutron scattering system.
Guini er R a n g e
i
10-1 -
@
10-2 -
I-'-4
Pofod
10-3 -
P.ange
10-4 -
I
10 -1
i
1
I
10
Qd
Figure 2. Schematic illustration of SANS curve for objects, such as pores, with a dimension of d. In the Guinier range the scattering depends on the size and form of the object. In the Porod range information on the surface are obtained.
73 The function 7'(r ) is complex and contains all information from the effects of the form, (size, shape) of the inhomogeneities (e.g. pores) and their mutual arrangement. Although separation of this information is difficult, precise interpretation can be obtained from y(r_), in particular for two phase systems composed of discrete particles as considered here. Porod (29) considered a particular case of an arbitrary two phase system with sharp boundaries in which the scattering density of one phase, pl, was constant and that of the other zero. We can then write --2
rl = p~qk,(1-~k, )
(10)
where ~bl is the volume fraction of phase 1. This theory leads to the general expression for do~dO in the limit of high Q do')
2nS (11)
where S is the total surface between the phases. This equation predicts that in the high angle tail of the scattering curve the intensity decreases asymptotically as Q-4. Furthermore the absolute intensity of scattering in this region is dependent on only two parameters of the system : the difference in scattering length density between the two phases, and S, the total area of the interface between the two phases. Where the interfacial boundary is not sharp, deviations from Porod behaviour may occur (3032). More recently this has been demonstrated for a number of porous systems (e.g. microporous carbons and silicas) which have a surface roughness on a scale in the range of the inverse of Q covered in the SAS measurements. Such materials, which can be described as having surface fractal properties, show a decay of
I(Q).-. SQ -(6-~
(12)
where D is the fractal dimension. For smooth surfaces, D = 2, however in extreme cases where the surface is irregular, or has a curvature on a scale smaller than reciprocal Q space, D may become greater than 2. This can lead to the power law exponents which are smaller than -4. Typically D falls in a range between 2 and 3 for such systems. Two other linear parameters can also be derived from scattering measurements. The first, defined as the 'range of homogeneity' by Porod, is given by
l-~ - 4Vqk, S
(13)
and represents an average diameter of the heterogeneities comprising the phase having volume fraction ~1 (see Figure 3). Thus for a collection of spheres radius R, li = (4/3)R. The length, l/, like S/V, is a differential property of ~ r_) evaluated at r = 0 and consequently is unaffected by any effects of interference due to long range correlation between heterogeneities.
74 The integral breadth of the correlation function, gives another linear parameter, the 'distance of heterogeneity', lc oo
lc = 2 i),(r)dr
(14)
0
The scattering for many porous systems can be expressed more simply than the generalised expression (Eq. (7)) by using the classical theory developed to describe the scattering from assemblies of particles. The scattering cross section for such systems can be expressed as 2 I(Q) = V2 np(pp - Ps ) 2 P(Q)S(Q)
(lS)
where Vp is the volume of pores np their number density, pp and/as are respectively the scattering length densities of the pores and continuous solid phase and P(Q) is the single pore form factor. S(Q) is the structure factor, which is determined by the spatial ordering of the pores, and describes the effects of interference in the scattering from pores which are in close separation. In the limit of high Q, S(Q) tends to unity. The form factor, P(Q), has been evaluated for a variety of particle (or pore) shapes which include spheres (33), ellipsoids (34), rods (35) and flat discs (viz. slits) (36) for example. A general relationship for P(Q), which is valid for all shapes, and describes the decay of P(Q) in the region of low Q (see Figure 2) is given by Guinier.
Figure 3. Geometrical interpretation of the distance parameter ll, or 'range of homogeneity' as defmed by Porod. Here, ll, is represented by the alternate succession of chords passing through phase 1 in a two phase random medium.
75
/ 22)
P(Q) ~ exp
- Q Rg
3
(16)
where Rg is the radius of gyration of the particle or pore. The relationship is valid when QRg < 1. It will be noted that for anisotropic particles (or pores), P(Q) will depend on the particle orientation with respect to the incident and scattering vectors. The scattering from randomly distributed pores will thus differ from that where there is a preferred orientation, a feature which will be illustrated subsequently. The static structure factor,
S(Q), is given by
S(Q)=l+4nnP~I(g(r)-l)resinQrlo Qr
(17)
where g(r_) is the particle pair distribution function which describes the spatial distribution of the particles (pores) as a function of the mean separation distance. 2.2.2 Applications of SANS Two recent applications of SANS will now be described where some of the special features of the technique have been exploited. These include first the scattering from anisotropic porous structures and secondly the use of contrast variation for in situ sudies. Both of these areas are currently under development, and have importance in the context of the structure of micro and meso porous membranes and the mechanisms of gas adsorption in such structures. (a) Anisotropic pore structures The pore structure of materials may frequently be anisotropic. Some examples of such materials are given in Table 2. Information on the microstructure of these oriented pore systems is often important in the applications envisaged. Thus the mechanical strength can be influenced by pore anisotropy and orientation in substances such as carbon and ceramic fibres, thin ceramic films and construction materials. The pore orientation may also control the microscopic flow and diffusion of fluids in materials as diverse as membranes and geologic media. Table 2. Materials containing anisotropic pore structures Carbon fibres Ceramic fibres Ceramic membranes Sol-gel films Clay and zeolite minerals Templated mesoporous materials e.g. MCM-41 Crystalline solids (after topotactic decomposition) Bio-inorganic skeletal structures
76
For these materials unique microstructural information can be derived from SAS with both neutrons and X-rays. These details are not obtainable from bulk measurements, such as adsorption isotherms. This application of SANS has recently been demonstrated with ceramic alumina fibres (37), microporous carbon (38), sol-gel films (39) and alumina membranes produced by anodic oxidation (40). SAXS investigations of pore-orientation periodicity in porous polymer and carbon materials have also been recently reported by Olivier et al. (41) We will illustrate the information obtainable firstly by describing in outline the results obtained by SANS on activated carbon fibres (ACF) (38). In ACF the micropores are slitshaped and are formed by the parallel alignment of microcrystals of graphite along the axes of the carbon fibres (42). This has been established by SANS with ACF samples oriented in two different directions to the incident neutron beam (see Figure 4). Results showing SANS along the two-dimensional detector for ACF samples oriented horizontally indicate that the scattering is anisotropic (see Figure 5.a). In (i) the scattering arises from the surface of the microcrystals (> 103 m~g"1) which have 'smooth' surfaces; this gives a power law decrease in scattering of Q4. In (ii) the scattering arises from the edges of the microcrystals, which are 'rough' and have a surface fractal dimension of = 2.5. This gives rise to a Q-3.5 power law as shown. When the fibres are oriented with their axes parallel to the incident beam (see Figure 5.b), the scattering is isotropic, as can be inferred from theoretical analysis of scattering from oriented particles (43). In this situation the scattering contribution comes from both the surfaces and edges of the microcrystals, with the former dominating. The power law component is consequently close to Q-4.
(a) ACF NEt;fROM
BEAM
Detector (b)
ACF NEUTRON BEAM
Figure 4. SANS of oriented carbon fibres. Orientation of ACF (a) horizontal and (b) parallel to the incident neutron beam with respect to the two-dimensional detector. After (38).
77
(a)
10
6
(b) 9
_
104
0 0
_
O O
10 z
I ~(~4
0
0
0
o
l
~0 l 9
0 v
0
9
10
/
9
o ".
10 2
o IQ ( i i )~ ~(i) o ",
+ok"% ]
10-3 I
10 -3
I
10-2_1
I
I
10 -2
I
10 -1
Q//~-I
10 -1
Q/A Figure 5. SANS of oriented carbon fibres. (a) Fibers are oriented horizontaly with their axes perpendicular to the incident beam. (i) SANS along the vertical axis. (ii) SANS along the horizontal axis of the detector. (b) Fibers are oriented with their axes parallel to the incident beam. Scattering is isotropic and I(Q) is radially averaged data.
Another illustration where SANS measurements have been made on oriented samples having an anisotropic pore structure concems alumina membranes (40) which have been prepared by anodic oxidation of alumina in a suitable electrolyte. Such a process can result in the formation of a porous surface film which consists of a close-packed hexagonal array of cells, each containing a cylindrical pore (44). Hoare and Mort and later workers (45-47) have shown that the pore morphology of these films is remarkably regular and can be controlled by the electrolysis conditions. Subsequent developments have lead to techniques for detaching these films from the aluminium metal thus resulting in thin alumina membranes with a very uniform pore structure. Such membranes have more recently been commercialised (48) and are available with controlled narrow pore size distributions in the mesopore range. These membranes have applications in the ultrafiltration of biological samples and in gas separation by Knudsen diffusion, although at the present these are limited to a laboratory scale (8).
78 The highly uniform and oriented structure of a typical membrane (pore diameter - 200 nm) is illustrated by the field emission scanning electron microscopy results in Figure 6.a) and b). The schematic arrangement for the SANS measurements made with such a membrane (47 mm OD) sample is illustrated in Figures 7.(a) and 7.(b). Here the membrane has two different orientations to the incident collimated neutron beam: In (a) the membrane disc is perpendicular to the beam; in (b) the disc is almost parallel. (In practice, to obtain a sufficient sample area in the beam, the disc was oriented slightly ( - 3 ~ away from the parallel axis.) The corresponding orientation of the columnar pores to the neutron beam for these two sample configurations is illustrated in Figures 8.(a) and 8.(b). The SANS intensity distributions measured on a 2D detector for these two sample configurations are very different, as depicted schematically in Figures 7.(a) and 7.(b). For (a) the scattering was isotropic, and for (b) markedly anisotropic. On the detector this feature is depicted schematically (of. Figures 7.(a) and 7.(b)) by the closed lines of iso-intensity, viz.; circular and highly elliptical, respectively. A more quantitative representation of the anisotropy is shown in Figures 9.(a) and 9.(b). Here the intensity of scattering along the vertical axis, I(QO, and horizontal axis, l(Qn), on the detector, for the two different sample configurations, is displayed. Thus in Figure 9.(a), with the membrane in the perpendicular configuration the scattering is very similar along both axes, in accord with the isotropic pattern observed. In contrast the scattering in the parallel configuration (Figure 9.(b)) is highly anisotropic. Here I(Qv) is very weak compared to I(Q~ ( by a factor o f - 103).
79
(a)
neutron beam
~
.| ............................................
QH '\ (b) ~v
\ neutron beam
\
QH ann
Figure 7. Schematic arrangement for SANS measurements on oriented membranes. (a) perpendicular orientation, (b) parallel orientation to the incident neutron beam. (a)
(b)
perpendicular position
parallel position
n e utron.............
beam
n e na~ '
~
~
~
beam
Figure 8. Corresponding orientations of columnar pores in membranes having two different configurations shown in Figure 6. The stronger scattering behaviour observed for both axes in Figure 9.(a) and I(Qn) in 9.(b), has an intensity which decays with a power law close to that observed in the Porod scattering region, viz. I(Q) "-' Q-4
(18)
Such behaviour indicates that the inverse size of the scattering objects (viz. pores) is much smaller than that of the corresponding range of Q covered in the present measurements (1).
80 This implies that the pore size is >> 102 A, and is in accord with the SEM results shown in Figure 6. Further analysis of these preliminary SANS results is complex and can only be outlined here. The treatment takes as its basis, a model composed of an array of parallel cylindrical objects, length, l and cross-sectional radius, a (43). When such a system is oriented, where y corresponds to the angle between Q and the cylindrical axis, two specific cases can be defined for the scattering behaviour. For the first, where the vector Q is parallel to the cylindrical axis (y = 0) the scattering will be only a function of the axial length. Thus for a single isolated cylinder
sin(Ql/2))2 I(Q, y = O)~ K poV Ql/2
(19)
Where ,oo is the scattering-length function and V is the cylindrical volume. This situation corresponds to the SANS in Figure 9.(b) corresponding to I(Qv). Secondly, if the Q vector is perpendicular to the cylindrical axis, the scattering will be a function only of the radius. In this case I(Q,y = m ' 2 ) - K ( 2 p o V
J'(Qa)) 2Qa
(20)
Jl(X) is the first order Bessel function of the first kind. This situation corresponds to the SANS in Figure 9.(a) for both I(QH) and l(Qv) and also in Figure 9.(b) for I(QH). The origin of the anisotropy in the SANS behaviour observed is thus evident. It has also been possible to obtain an insight into the ordering or spacing of the pores in the present membrane, from measurements at much smaller scattering vectors (Q/A1 << 103). Such details can be obtained from the maxima in I(Q) which results from interference in the scattering with such an ordered system of uniform cylindrical pores. This is illustrated by the anisotropic isointensity plot in Figure 10. Here the maxima observed at Q---1.6"10 -3 A 1 correspond to a pore spacing of-~ 3.9"10 3 A, which is in satisfactory accord with the SEM results.
81
a) 10
9
9I ( Q n )
o
",0 0 O', 0
9I(Qv)
@, 0 @. ~ @6 O0
0.1
-
-.4
0.01 1E-3 ~.. 1E-4 1E-5
'
'
'
'
'
'
I
'
'
'
'
'
'
'
0.01
'
I
'
0.1 Q/A"
b) 109
Q
o
1
"I(Qv)
9i(Qv)
,j ,.0 -.0
0.1 0
cy 0.01
0
~
Oo o
1E-3
'%,. 0
1E-4 1E-5
0
'
'
'
'
'
'
I
~
O, 0 0
~ '
0.01
.
,
'
'
- I
o .I
'
'
Q/A" Figure 9. Anisotropic SANS results for an oriented alumina membrane (Anodisc). In (a) the orientation is perpendicular and in (b) parallel to the incident neutron beam respectively. I(Q~ and I(Qv) correspond to scattering along the horizontal and vertical axes of the 2D detector, respectively.
82
Figure 10. Anisotropic isointensity plot obtained by the SANS analysis of the Anodisc. 2.2.3 Contrast variation studies- adsorption processes Another recent and important development of SAS in the characterisation of porous solids has been the application of the contrast variation technique. Previously we have considered a two phase system, for simplicity, where the intensity of scattering, I(Q), is proportional to the contrast, i.e. the square of the scattering length difference between the two phases :
I ( Q ) ~ K ( p b ( I ) - p b ( 2 ) ) 2 = K ( A p b )2
(21)
For an evacuated porous solid, where pb(2) = pb(solid) the situation is simple since ]96(1) = O. However more detailed information may be derived if the pores are filled, or partially filled with an adsorbed vapour. Scattering from pores filled with a condensed liquid adsorbate may be eliminated if the scattering length, pb(1) is chosen to be the same as the solid, viz (Apb): = 0. This feature may be used to distinguish open and closed porosity, the latter being inaccessible to the adsorbate, for example. Using the same principal Hoinkis et al. (49) were able to measure the selective filling of micropores by exposing graphitic carbon to C6D6 at p/p0 ~ 0.25. In further applications of the contrast variation method, the filling of micropores, the growth of adsorbed multilayers and capillary condensation processes in micro and mesoporous oxide gels, have been investigated by SANS (50-53).
83 Other important developments have been made using SAXS and SANS in particular to probe the structure of the surfaces of porous materials which have a fractal character (54, 55). The majority of these earlier studies were made with samples which were pre-exposed to a fixed adsorbate pressure before the scattering measurement. This procedure, although simple, is limited for several reasons" firstly because of uncertainties in the equilibrium relative pressure due to temperature instabilities and containment problems, secondly due to irreproducibility between different solid samples, and thirdly the impossibility of making non-equilibrium and kinetic measurements. Advances have however been made recently (56) using SAXS with an in situ gravimetric system to study the isothermal adsorption of dibromethane (CH2Br2) on Vycor glass, both of which have comparable scattering length densities. The use of CH~Br2 in these investigations highlights the important limitation of SAXS compared with SANS. With SANS there is a wide flexibility in the choice of adsorbate (e.g. water, hydrocarbons alcohols, etc.) due to the control in Pb which can be achieved by substitution of H for D in the molecule (see Table 3). SAXS measurements have however been confmed to halogenated hydrocarbons since these have an electron density, Pc, which is sufficiently large to match that of porous solids, such as graphite and silica (see Table 3). This limitation precludes SAXS in studies of a wide range of simpler molecules, where evidentally adsorption is not dominated by the interaction of the halogen atoms with the surface. Recently the advantage of the SANS has been exploited using a specially designed apparatus which allows in situ measurements on thermostated samples under closely controlled relative vapour pressure of different adsorbates (57). Such an apparatus gives scope for equilibrium studies with mesoporous solids close to saturation vapour pressure and during an adsorption/desorption cycle. Furthermore measurements at very low P/P0, which are important in studies of microporous materials, are also feasible. Non-equilibrium and kinetic measurements, which are relevant to the rate of diffusion in the pore system are also possible. Table 3. Scattering length densities, pb, and electron densities, pc, of different materials. Substance pb/10 l~ cm2 pe/10 24 cm "3 H20 -0.56 0.334 D20 6.36 Benzene-h6 1.16 0.285 Benzene-d6 5.43 Cyclohexane-h 12 -0.28 0.268 Cyclohexane-dl 2 6.70 Silica 3.47 0.661 Graphite 7.67 0.683 Zirconia 5.25 1.568 We have recently applied the contrast variation technique using this apparatus in investigations of the adsorption of benzene in a mesoporous silica gels having different pore structures (53,58). In one case the gel was prepared by a sol-gel process, and had a welldefined model pore structure as had been established previously (59). To illustrate this investigation, the effect of progressive benzene (59 % C6D6) adsorption at 310 K on the SANS of the gel is shown in Figure 11. For the outgassed sample (P/p0 = 0) the scattering (full line, 11.(a)) is characteristic of a structure formed by the packing of spherical sol particles. Thus the pronounced maximum at Q of 0.025 A arises from the interference in the scattering from a partially ordered structure, where the interparticle separation is given approximately by
84 2x/Qmax viz. 24 nm. The inflexion at Q --- 0.045 A "l results from the form factor, P(Q), of the spherical particles. Thus for monodispersed spheres of radius R, the decay of P(Q) has a primary maximum at QR ~ 5.9. Although this feature is smeared here, the position correspondends to a particle diameter of 26 nm, in reasonable agreement with electron microscopy and SANS measurements on dilute sols. Beyond this inflexion, I(Q) decays linearly with Q-4 in accord with the Porod law. Changes in the SANS after equilibration with benzene having the same scattering length density as silica are shown for selected values of P/Po. At low relative pressure (P/Po ~ O.4) the scattering is little changed, but on further uptake progressive differences are noted: the almost total suppression of the interference peak is most striking. Less evident are the suppression of the inflexion due to the particle form factor, P(Q), and the reduction in intensity in the Porod region.
10 2
(b) i
I0
9
..
~.~
.'--
\\
~-4
9
XX \ f 10
\\
ooqo% ...
-1 -
10-2
10 -1
Q/A -1 Figure 11. SANS of silica gel (pore radius-~ 2.9 nm) exposed to contrast matched benzene at increasing relative vapour pressures, P/P0: (a), 0.0 ; (b), 0.42 ; (c), 0.50 ; (d), 0.55 ; (e), 0.60 ; (f), 0.67 and (g), 0.90.
85 The increase in I(Q) at low Q, associated with changes in the interference peak, can be ascribed to the growth of an adsorbed film and annular meniscus at the points of sphere contact (60) (see Figure 12.a). Such changes have been described previously for the SANS during water uptake in this gel system (50). For P/Po > 0.67 the changes become more marked and correspond to an enhanced uptake of benzene due to the development of capillary condensate at the sphere contact zones with increases in P/Po (Figure 12.b). This feature is illustrated in Figure 13, which shows the kinetics of the benzene adsorption after a stepwise increase of P/Po from 0.67 to 0.73. Here the SANS curves correspond to sequential measurements at intervals of-~ 5 min. At P/Po > 0.73 onset of the menisci coalescence in the pore throats occurs, leading to spontaneous pore filling. This results in the dramatic drop (by ~102) in scattered intensity observed at P/Po = 0.9. Other results showing the SANS on desorption, following the hysteresis loop have been described (53). More detailed theoretical analysis of such measurements can provide details of the mechanisms of capillary condensation and percolation in other model mesoporous structures of different geometry. This technique also offers future scope for investigations of both microporous and non-porous solids. In particular, details of the confinement and packing of sorbate molecules, either in micropores or monolayers on non-porous surface may in future be derived from measurements using different sorbates.
(a)
(b)
(c)
1 /
Figure 12. Depiction of adsorption and capillary condensation in sphere packings; (a) adsorbed film and annular meniscus at points of sphere contacts, (b) development at contact zone on progressive adsorption, and (c) onset of menisci coalescence in pore throat, leading to spontaneous filling (aider (60)).
86
0
"
"
["
"
I
.
.
.
.
.
.
.
.
.
I
"'"
"
"
I
.
.
.
.
!
.
.
.
.
~
.
.
.
.
(a)
60
(Y
30
(c
d) 0.00
0.02
0.04
0.06
Q/~-I Figure 13. Kinetics of benzene adsorption on silica gel after a stepwise increase of P/P0 from 0.67 to 0.73 at 310 K. Measurement time for each SANS curve was 305 s. The SANS for the outgassed gel is shown in (a) and for equilibrium at P/P0 of 0.67, 0.73 and 0.90 in (b), (c) and (d) respectively.
2.3 Neutron Diffraction- properties of sorbed phases As previously discussed, coherent neutron scattering has its counterpart in X-ray scattering which is an elastic scattering process. However the scattering cross sections of atoms for Xrays and neutrons are different (see Table 3). This feature has been exploited to investigate the mechanisms of adsorption in porous solids using both diffraction, as well as SANS, which we have already considered above.
87 Diffraction measurements can be used to distinguish the structural organisation of adsorbed atoms or molecules in porous media. This technique although not widely exploited has also been used to investigate water behaviour in meso and microporous oxides (61), as will be briefly described here. More recently neutron diffraction has been applied successfully by Rouquerol and coworkers (62-64) to study phase transition behaviour of different gases (argon, krypton, methane, nitrogen , carbon monoxide) in MFI-type zeolites. Other investigations include water in hydrated clay systems (65), and microporous carbon (38). The diffraction of bulk liquid water (D20) and ice are markedly different (see Figure 14.a and b). For ice the diffraction is characteristic of a hexagonal structure where there are four Hbonds associated with each oxygen atom as shown in Figure 15.a. The structure of liquid water is shown schematically in Figure 15.b. Here there are statistically between 3 and 4 Hbonds per oxygen atom. Although there is no long range order in liquid water there is however a short range structure, which fluctuates dynamically. This dynamic short range intermolecular structure is influenced by temperature in the bulk (66) and may be perturbed in a porous medium. It is well established that the freezing of water in mesoporous silica and other solids occurs at temperatures below 273 K - the bulk temperature. The depression of the freezing temperature is related to the pore dimension and this feature can indeed be used to determine pore size and shape (67). However when the size approaches the micropore range, -~2 nm, a regular H-bond network is unable to form in the confined pore space and the water remains in a supercooled non-crystalline state. This is demonstrated in Figure 16, which shows the effect of temperature on the neutron diffraction of hydrated porous silica (pore size -~2 nm). It will be noted that the band at 2 0 = 2 2 ~ narrows progressively, but even at 123 K the water remains non-crystalline and the diffraction is different from the bulk ice (cfFigure 14). Another application of neutron diffraction to determine the organisation of water in conf'med pore geometry has been demonstrated with smectite clay gels, such as montmorillonite and hectorite (65). Here the porous structure is formed by the parallel alignement of the thin (~ 1 nm) sheet like particles to give slit shaped pores. The interlayer zone in these structures contains water and as the uptake increases the sheet swell apart but remain highly oriented (43). It is indeed possible to examine the ordering of water molecules with respect to the clay surface as a function of interlayer spacing. In such neutron diffraction experiments the bulk samples are oriented with respect to the neutron beam (viz. either parallel or horizontal). Such investigations have shown that water molecules remain highly oriented to the clay surface to uptakes exceedings three layers.
88
(a)
3
c-"
I
>,, I1,,,,.
I
I
I
I
100
Ik=
F, f,,,.. >,,
I
I
(b) 002
15
o ,,,,,n
I/} c"
10
(c)
m
5-
0
I 10
I I I I I 20 30 40 2O/deg., X - 1.37 A
I 50
o
Figure 14. Neutron diffraction of (a) liquid D20 at 298 K (b) and (b) ice at 263 K. The peaks indexed in (b) correspond to the structure of hexagonal ice, Ih.
89 (b) WATER O OXYGEN
{a) ICE
9 HYDROGEN
. .....
l 9
v
l t
l |
i I
~
,
!
.z"~ i ,v~.,~-,4...._
,"
P
',,~---."~--L
~[
9
4. ! I
I ,
_
I ,
9
.,,S
Figure 15. Diagram depicting the hydrogen-bonded structure in (a) hexagonal ice and (b) liquid water.
8
)
(ct)
(c]
to 6 ~
c-
~>, 4 ~ 2
~
33 12I
~
8
r-.
-,-- 4 C
2 0
i 10
I 20
I 30
I 40
I // 50
I
I
10
20
I
I
30 40
I //
50
i
I
1
10
20
30
I
I
40 50
20/deg., X= 1.3655/~ Figure 16.Effects of temperature on the neutron diffraction of hydrated porous silica, SI (27% w/w D20). Temperatures (K) are" (a) 298 ; (b) 255 ; (c) 250 ; (d) 245 ; (e) 233 ; and (f) 123, respectively.
90 3. SCATTERING ION-BEAM SCATTERING TECHNIQUES The application of ion beam scattering techniques to determine pore size and pore volume, or density of thin silica gel layers was first described by Armitage and co-workers (68,69). These techniques are non-destructive, sensitive and ideally suited for the analysis of thin porous films such as membrane layers. However, apart from a more recent report on ion-beam analysis of sol-gel films (70) using Rutherford backscattering and forward recoil spectrometry, ion beam techniques have not been developed further despite their potential for membrane characterisation. This is probably due to the limited availability of ion beam sources, such as charged particle accelerators. The work of Armitage et al was carded out using both tandem Van der Graaff and linear accelerators as sources of focused proton and a-particle beams. The method involves observing the energy distribution of ions elastically backscattered after exciting a resonace in one of the nuclei of the sample. In this work the resonances for protons (2.66 MeV) and ctparticles (3.05 MeV) scattered from 160 were employed. An illustration of the experimental arrangement with an example of a spectrum obtained from porous SiO2 sample with 3.3 MeV a-particles is shown in Figure 17, together with the spectrum from a sample of non-porous SiO2 glass. It will be noted that the resonance peak for the porous sample is broader. This peak is associated with the 3.05 MeV resoncance arising from the elastic scattering of orparticles from 160. A full description is given in (69) of the methods of calculation of pore size from such backscattering data, with various assumptions of the pore and interpore path length distribution. Here only an outline of the theory can be given.
I
•
FOCUSSED_
.
i
I~eARTICLE
f,o.
SEAm
COLLIMATOR
RESONANCE'~
I0000
~ETECTOR SAMPLE]
PEAK 4,
Eo = 3.3 MeV
5000
I
"""
"~t. ~
I
SILICON
I ,, 1 2 ENERGY OF BACKSCATTERED PARTICLES (Mel0
Figure 17. Typical energy spectrum of a-particles backscattered from porous (full line) and non-porous silica thin films (dotted line). The inset shows the experimental arrangement. (after (69))
91
3.1 Theoretical principles Although details are not appropriate here, an outline of the principles of the method can be appreciated from the following simple description. The enhanced width of the oxygen resonance arises because individual backscattered ions pass through different numbers of pores before leaving the sample. Thus the total amount of solid material traversed by the ions, and hence the loss on emergence, is variable. Firstly we consider the case of an ion beam transmitted through a porous sample where the pore length through each pore is constant and equal to a. The resulting variance in path length through a random distribution of pores will be Na 2, where N is the mean number of pores encountered. The energy spread of the beam transmitted through the porous specimen will be increased over that obtained with a non porous specimen having the same areal density. The energy spectrum of the transmitted ions may be considered Gaussian so that the contribution of the pores to the measured fwhm = 2.35 (AE2) w, where (AE 2) is the corresponding variance in the energy distribution. Thusfwhm = 2.35 N1/2a dE/dx. We now consider the more complex case of backscattering from a porous sample. This arises when a beam of ions incident on the sample has an energy significantly higher than that of the resonance. If we ignore the effects of straggling, all the ions backscattered at the energy of the resonance will have penetrated the same thickness of solid material. However, the number of pores encountered will be variable with the result that the distribution of depths penetrated will have a variance Na:. After backscattering at the energy of resonance the ions will leave the surface having traversed a total distance in solid material whose variance is Na 2. The contribution of the pores to the width of the resonance follows from the transmission case, so that
fwhm = 2.35 (2Na2) m dE/dx
(23)
where dE/dx is evaluated at the emergent energy, Ed.
The total pore path length after scattering, Na, can be equated to the difference between the total path, to and the path through solid matter, te, viz.
Na = to- te
(24)
Since the porosity, e can be written as (to-te)/to we have
Na = [g/(1-6)]te
(24)
Furthermore it can be shown that
t~
= (E
r -
E a ) / dE / dR 1
where Er is the energy after scattering and dE~dR, is evaluated at 89 energy after scattering.
(25) i.e. the average
92 From the above we obtain
a=l[fwhm]2(1-o~] 2.35 ) k. 6-
l (dElr
(26)
The procedure then is to measure the fwhm of the resonance peak obtained with an equivalent non porous specimen, and to substract this in quadrature from the peak obtained from the porous sample. This ensures that the fwhm obtained in equation (26) is then free of any contributions arising from such effects as energy straggling, detector resolution, initial energy spread of the beam and natural width of the resonance. Values of dE/dR and dE/dR, can be obtained from range energy relationships (69). More sophisticated models to take account of the problem of the distribution of total path lengths through porous materials, based on the theory developed by Clement (71), were also described by Armitage et al. Such models to define pore path length dispersion were analysed on the basis of the known pore geometries of porous gels, which were formed by the packing of spherical particles, of almost uniform size.
3.2 Practical applications Armitage et al used models, as discussed above, to describe the distribution of the paths through and between the pores corresponding to the energy distribution of ions elastically backscattered from meso and microporous oxide gel films. This statistical description of the pore structure was shown to be in good agreement with pore size data obtained by other methods, such as gas adsorption isotherms. In the work with silica gels the pore structure was formed by the packing of spherical sol particles of different size and packing density to obtain samples with a range of pore size ( -- 20 to 100 nm). Other measurements were made on microporous oxide gels (e.g. titania, alumina) with pore size <2nm. These microporous gels had a known pore structure and were also formed by the packing of sol particles. However here interesting differences were observed which depended on the particle shape. Thus in the case of titania, where the gel was derived by drying a concentrated sol of relatively monodispersed rhombohedral shaped particles, the pore size parameter, a was found to be 1.9 nm. This was in accord with the size of the pore throats in such a packed structure, which had a pore geometry similar to the silica gels. However the hydrated alumina gel had a highly anisotropic structure composed of lamella shaped particles. Here the value of a determined was --3.8 nm, which was significantly greater than the known interlameUar spaceings.This difference was not unexpected, as it was noted that the average of the pore path lengths through parallel lamellae would be greater than that of the interlamellar spacing. This method, using a microfocused ion beam, has unique advantages over other techniques which could be very useful in membrane characterisation. Thus in the above work examples of measurements made on gel layers as a function of sampling depth (from-~3 ~tm to 100 ~tm) and as a function of distance across the sample were illustrated Figure 18. It will also be noted that the technique is equally appropriate for measurements in the microporous and mesoporous ranges.
93
9 x SAMPLE
E
c ,_..,
ALPHA
DATA
PROTON
DATA
---GAS ADSORPTION VALUE
B
-
"•'T l m"
.
50
X
LU N LU
r,-
0
O CL
-SAMPLE
C
Z .< LU
}-
50
=.
0--
I
1
1 1 I ! 111
1
10 to(DEPTH SAMPLED)[pm]
!
I
1 I I I1
100
Figure 18. Values of mean pore size (~2) in highly porous silica gel specimens obtained as a function of sampling depth. (after (60))
4. CONCLUSIONS In this review two different radiation scattering techniques have been described that can provide information on the microstructure of porous solids and the mechanisms of adsorption processes at surfaces and within pores. Although neutron scattering methods are relatively sophisticated, and not more readily accessible, they can give unique information which complements that available from more classical techniques of pore structure characterisation. This feature is illustrated here by two specialised applications of small angle neutron scattering which are currently under development. The first concerns the characterisation of anisotropic pore structures, where two examples are given, viz. microporous carbon and mesoporous anodic alumina membranes. The second illustrates the use of contrast matching methods to investigate the mechanisms of adsorption in porous networks. This development which is applicable to both micro and mesoporous media, may provide insight into the separation of gas mixtures in microporous membranes where either one or more components is a condensable phase. For example highly efficient separations involving pore "blocking" processes have been reported with microporous membranes (72). Although the mechanisms are still not fully understood, they are of considerable technical interest. Neutron diffraction may also provide information on the nature of the sorbed phase conf'med in micropores, as also discussed here, with reference to water in different microporous systems.
94 The other technique described- resonant ion beam backscattering, has received little development since its first application in pore structure characterisation. The method has the same advantages as SANS, by being an in-situ and non-destructive technique, which is equally appropriate for measurements on micro and meso porous materials. It does however have a unique advantage over other techniques, which could be very useful in the characterisation of microporous membranes. This results from the use of a microfocussed beam, which allows measurements as a function both of sampling depth (from-~3 ktm to 100 lam), and as a function of distance across the sample, as is illustrated here.
5. ACKNOWLEDGEMENTS Many of the neutron scattering experiments described here were carried out at the Institut Laue Langevin, Grenoble, the LLB, Saclay and the Hahn Meitner Institut, Berlin. Access and support at these facilities is gratefully acknowledged. This work was carried out with many colleagues to whom I much indebted. In particular these include Drs. L. Auvary, S. Kallus, and E. Hoinkis. Work on the ACF material was done in collaboration with Professor K. Kaneko and Dr. A. Matsumoto. 6. REFERENCES
1. A.J. Burggraaf and L. Cot (eds.), Fundamentals of Inorganic Membrane Science and Technology, Elsevier Science B.V., Amsterdam, 1996 2. V.N. Burganos (ed.) in: MRS Bulletin "Membranes and Membrane Processes", 24 (1999) 19 3. M.B. Rao and S. Sircar, J. Membr. Sci., 85 (1993) 253 4. A.B. Shelekhin, A.G. Dixon and Y.H. Ma, J. Membr. Sci., 75 (1992) 233 5. R.S.A. de Lange, J.H.A. Hekkink, K. Keizer and A.J. Burggraaf, J. Membr. Sci., 99 (1995) 57 6. R.M. de Vos and H. Verweij, Science, 279 (1998) 1710 7. R. Vacassy, C. Guizard, V. Thoraval and L. Cot, J. Membr. Sci., 132 (1997) 109 8. A.J. Burggraaf, "Transport and Separation Properties of Membranes with Gases and Vapours", in Fundamentals of Inorganic Membrane Science and Technology; A.J. Burggraaf and L. Cot (eds.), Elsevier Science B.V., Amsterdam, 1996 9. R.R. Bhave (ed.), "Inorganic Membranes, Synthesis, Characterisatics and Applications", Van Nostrand Rheinhold, New York, 1991 10. A. Julbe and J.D.F. Ramsay, "Methods for the Characterisation of Porous Structure in Membrane Materials", in Fundamentals of Inorganic Membrane Science and Technology, A.J. Burggraaf and L. Cot 8eds.) p. 67, Elsevier Science B.V., Amsterdam, 1996 11. J. Rouquerol, D. Avnir, C.W. Fairbridge, D.H. Everett, J.H. Haynes, N. Pernicone, J.D.F. Ramsay, K.W.S. Sing and K.K. Unger, Pure Appl. Chem. 66(8) (1994) 1739 12. S.J. Gregg and K.S.W. Sing in: "Adsorption, Surface Area and Porosity", Acad. Press, 2 nd Ed. London, 1982 13. K. Ibel (Ed.), Neutron Research Facilities at the ILL, ILL, Grenoble, 1994 14. J.W. White, C.G. Windsor, Rep. Prog. Phys. 47 (1984) 707 15. G.E. Bacon, Neutron Scattering in Chemistry, Butterworth, London, 1977
95 16. S.W. Lovesey, Theory of Neutron Scattering from Condensed Matter, vols. 1 and 2, Clarendon Press, Oxford, 1984 17. B.T.M. Willis (Ed.), Chemical Applications of Thermal Neutron Scattering, Oxford University Press, London, 1973 18. B. Jacrot, Rep. Prog. Phys. 39 (1976) 911 19. A. Guinier, G. Foumet, Small Angle Scattering of X-rays, Wiley, New York, 1955 20. B.O. Booth, J.D.F. Ramsay, in: J.M. Haynes, P. Rossi-Doria (Eds.), Principles and Applications of Pore Structural Characterisation, Arrowsmith, Bristol, 1985, p. 97 21. J.D.F. Ramsay, Neutron Scattering from Porous Solids, in K.K. Unger, J. Rouquerol, K.S.W. Sing, H. Kral (Eds.), Characterisation of Porous Solids, Studies in Surface Science and Catalysis, vol. 39, Elsevier, Amsterdam, 1988, p. 35 22. P.W. Schmidt, Small-angle scattering studies of porous solids, in: K.K. Unger, J. Rouquerol, K.S.W. Sing, H. Kral (Eds.), Characterisation of Porous Solids, Studies in Surface Science and Catalysis, vol. 39, Elsevier, Amsterdam, 1988, p. 35 23. G.G. Kostorz, in: H. Hermann (Ed.), A Treatise on Materials Science and Technology, Academic Press, New York, 1979, p. 227 24. R.H. Ottewill, in: J.W. Goodwin (ed.), Colloidal Dispersions, Royal Society of Chemistry, London, 1982, p. 143 25. J.D.F. Ramsay, Chem. Soc. Rev. 15 (1986) 335 26. J.S. Higgins, K. Ma, L.K. Nicholson, J.B. Hayter, K. Dodgson, J.A. Semlyen, Polymer Vol. 24 (1983) 793 27. O. Glatter, O. Kratky (Eds.), Small Angle X-ray Scattering, Academic Press, London, 1982 28. P. Debye, A. Bueche, J. Appl. Phys. 20 (1949) 518 29. G. Porod, Kolloidn Zh 124 (1951) 83 30. H.D. Bale, P.W. Schmidt, Phys. Rev. Lett. 53 (1983) 59b 31. J.K. Kjems, P. Schofield, in R. Pynn, A. Skeltorpe (eds.), Scaling Phenomena in Disordered Systems, Plenum, 1985, p. 141 32. J. Teixeira, J. Appl. Cryst. 21 (1988) 781 33. Lord Raleigh, Proc. R. Soc. London A90 (1914) 219 34. A. Guinier, Ann. Phys. 12 (1939) 161 35. A. Guinier, G. Fournet, Small Angle Scattering of X-rays, Wiley, New York, 1955 36. O. Kratky, G. Porod, J. Colloid Sci. 4 (1949) 35 37. M.H. Stacey, in: Studies in Surface Science and Catalysis, vol. 62, Elsevier, Amsterdam, 1991, p. 165 38. A. Matsumoto, K. Kaneko, J.D.F. Ramsay, in: Studies in Surface Science and Catalysis, vol. 80, Elsevier, Amsterdam, 1993, p. 405 39. L. Auvary, A. Ayral, T. Dabadie, L. Cot, C. Guizard, J.D.F. Ramsay, Faraday Discuss. Chem. Soc. 101 (1995) 235 40. L. Auvray, S. Kallus, G. Golemme, G. Nabias and J.D.F. Ramsay, Proc. of Characterisation of Porous Solids-V, Heidelberg, 1999, to be published in: Studies in Surface Science and Catalysis. 41. B.J. Olivier, R.R. Lagasse, D.W. Schaefer, J.D. Barnes, G.G. Long, Macromolecules 29 (1996) 28615 42. K. Kaneko, K. Kakei, T. Suzuki, Langmuir 5 (1989) 879 43. J.D.F. Ramsay, S.W. Swanton, J. Brunce, J. Chem. Soc. Faraday Trans. 79 (1990) 3919 44. T.P. Hoare and N.F. Mott, J. Phys. Chem. Solids, 9 (1959) 97
96 45. J.P. O'Sullivan and G.C. Wood, Proc. Roy. Soc. Lond. A, 371 (1970) 511 46. A.W. Smith, J. Electrochem. Soc., 120 (1973) 1068 47. G.E. Thompson, R.C. Fumeaux, G.C. Wood, J.A. Richardson and J.S. Goode, Nature, 272 (1978) 433 48. R.C. Furneaux and M.C. Thornton, Brit. Ceram. Proc., No. 43 (1988) 49. E. Hoinkis, A.J. Allen, J. Colloid Interface Sci. 145 (1991) 540 50. J.D.F. Ramsay, G. Wing, J. Colloid Interface Sci. 141 (1991) 475 51. J.-C. Li, D.K. Ross, M.J. Benham, J. Appl. Cryst. 24 (1991) 794 52. M.Y. Lin, S.K. Sinha, J.S. Huang, B. Abeles, J.W. Johnson, J.M. Drake, G.J. Glinka, Mater. Res. Soc. Proc. 166 (1990) 449 53. J.D.F. Ramsay, E. Hoinkis, in Proceedings of Characterisation of Porous Solids IV, Royal Society of Chemistry, London, 1997, p. 33 54. D.W. Hua, J.V.D. Souza, P.W. Schmidt, D.M. Smith, in: J. Rouquerol, F. RodriguezReinoso, K.S.W. Sing, K.K. Unger (Eds.), Studies in Surface Science and Catalysis. Characterisation of Porous Solids III, vol. 87, Elsevier, Amsterdam, 1994, p. 255 55. C.J. Glinka, L.C. Sander, S.A. Wise, N.F. Breck, Mater. Res. Proc. 166 (1990) 415 56. A. Ch. Mitropoulos, J.M. Haynes, R.M. Richardson, N.K. Kannelopoulos, Phys. Rev. B 52 (1995) 10035 57. E. Hoinkis, Langmuir 12 (1996) 4299 58. J.D.F. Ramsay, S. Kallus and E. Hoinkis, Proc. of Characterisation of Porous Solids-V, Heidelberg, 1999, to be published in: Studies in Surface Science and Catalysis: 59. J.D.F. Ramsay, B.O. Booth, J. Chem. Soc. Faraday Trans. 79 (1983) 173 60. B.G. Aristov, A.P. Kamaukhof, A.V. Kiselev, Russ. J. Phys. Chem. 36 (1962) 1159 61. J.D.F. Ramsay, Stud. Surf. Sci. Catal. 87 (1994) 235 62. P.L. Lewellyn, J.P. Coulomb, Y. Grillet, J. Patarin, H. Lauter, H. Reichert, J. Rouquerol, Langmuir 9 (1993) 1846 63. P.L. Lewellyn, J.P. Coulomb, Y. Grillet, J. Patarin, G. Andre, J. Rouquerol, Langmuir 9 (1993) 1852 64. H. Reichert, U. Mialler, K.K. Unger, Y. Grillet, F. Rouquerol, J. Rouquerol, J.P. Coulomb, Stud. Surface Sci. Catal. 62 (1991) 535 65. J.D.F. Ramsay, unpublished work 66. D. Eisenberg, W. Kauzmann, The Structure and Properties of Water, Oxford, 1969 67. M. Brun, A. Lallemand, J.F. Quinson, C. Eyraud, Thermochim. Acta 21 (1977) 59 68. C.D. McKenzie and B.H. Armitage, Nucl. Instr. and Meth. 133 (1976) 489 69. B.H. Armitage, J.D.F. Ramsay and F.P. Brady, Nucl. Instr. and Meth. 149 (1978) 329 70. J.L. Keddie and E.P. Giannelis, J. Am. Ceram. Soc., 73 (1990) 3106 71. C.F. Clement, J. Phys. D: Appl. Phys., 5 (1972) 793 72. R.M. Barrer, "Surface and Volume Flow in Porous Media" in "The Solid Gas Interface", Vol. 2, p. 557, E.A. Flood (ed.), E. Arnold Ltd., London, 1967
Recent Advancesin Gas Separationby MicroporousCeramicMembranes N.K. Kanellopoulos(Editor) o 2000 ElsevierScienceB.V. All rightsreserved.
97
Application of pulsed field gradient NMR to characterize the transport properties of microporous membranes W. Heink, J. K[irger and S.Vasenkov Fakult~it fiir Physik und Geowissenschaften, Universit[it Leipzig, Linn6stra6e 5, D-04103 Leipzig, Germany The potentials of pulsed field gradient (PFG) NMR in elucidating molecular transport in microporous membranes are presented. Particular emphasis is given to structure-correlated diffusion anisotropy and single-file diffusion. First results of PFG NMR studies with A type zeolite membranes are presented.
1. FUNDAMENTALS OF PFG NMR Diffusion measurements by the pulsed field gradient (PFG) NMR method are based on the fact that most nuclei posses a non-vanishing magnetic moment. By appropriate sequences of radio frequency (rf) pulses, under the influence of a constant magnetic field, these magnetic moments may give rise to a transient M R signal, the so-called spin echo. The intensity of this signal is reduced if during the rf pulse sequence, over two short time intervals, the constant magnetic field is superimposed by an inhomogeneous field
Baaa = gz,
(1)
the so-called field gradient pulses. The signal attenuation obeys the relation ~ ( S g , t ) = exp(-y262g2D,)
(2)
where 6 and t denote the duration and the separation of the field gradient pulses, y stands for the gyromagnetic ratio which is a characteristic quantity for the given nucleus (e.g. 2.675 x 108 T 1 s1 for protons, i.e. the nucleus of hydrogen). For simplicity, it is assumed that the duration of the field gradient pulses is sufficiently small in comparison with their separation (6/3 << t) and that molecular diffusion- with the self-diffusivity (or coefficient of self-diffusion) D - is isotropic. Einstein's relation (rZ(t)) = 6 Dt
(3)
correlates the self-diffusivity D with the mean square displacement (r2(t)) of the molecules under study during the (observation) time t. Using this relation, eq. (2) may be transferred into
= e.p(-
It))' 6).
(4)
98 Eq. (4) helps to intuitivley understand the principle of diffusion measurement by NMR: If during the two field gradient pulses the individual nuclei (and hence the atoms and molecules containing these nuclei) occupy different positions, as indicated by the mean square distance (r2(t)) they do not exactly obey one and the same resonance condition. This difference in the resonance conditions leads to a signal attenuation. In the case of diffusion anisotropy, only molecular displacements in the direction of the magnetic field gradient (which by eq. (1) was chosen to be given by the z coordinate) is relevant. In this case, eq. (4) has to be replaced by ~(~g, t ) : exp(- ?'282g2 (z2(t))/2).
(5)
In more complex systems, where the intrinsic molecular transport does not follow anymore normal diffusion as characterized by eq. (3), the PFG NMR spin echo attenuation may be described by the more general relation
~(gg, t) :
~P(z,t)exp(izSgr)dz
(6)
with P (z, t) denoting the propagator, i.e. the probability (density) that during a time inerval t an arbitrarily selected molecule is shifted over a distance z in the direction of the field gradient pulses. Under the conditions of normal (i.e. Fickian) diffusion, the propagator is a Gaussian
P(z,t) = (4rcDt) - ~ exp(- z 2/4Dt)
(7)
being exclusively controlled by the coefficient of self-diffusion. Inserting eq. (7) into eq. (6) clearly yields eq. (2). A detailed description of the fundamentals of PFG NMR may be found in the literature [1-3].
2. SELF-DIFFUSION AND MOLECULAR TRANSPORT IN MEMBRANES The utility of microporous membranes is based on their permeation properties. The flux density through a membrane is determined by Fick's first law
J = _Dr(c) dc dx'
(8)
where dc/dx denotes the concentration gradient in the direction of the diffusion flux (x-direction) within the membrane. Dr(c) stands for the transport diffusivity. The term transport has been introduced [1] for clearer distinction from the self-diffusivity D. The self-diffusivity may be defined either by eq. (3) or by a modification of Fick's law (eq. (8)):
99
j
*9
~
dx,
(9)
where one considers a flux of labelled molecules (/'*) generated by their concentration gradient (dc*/dx), under the condition that the total concentration c (i.e. the sum of the concentrations of the labelled and unlabelled molecules) is uniform over the sample. Both definitions are equivalent [1]. For non-interacting molecules, i.e. in the limit of sufficiently small concentrations, the coefficients of transport diffusion and self-diffusion have to coincide (being independent of concentration). For larger concentrations, transport and self-diffusion are generally assumed to be related to each other by a Darken-type equation [ 1, 4]
DT ( C) = D( c) . d ln p
dln-----~'
(10)
where p(c) denotes the gas pressure necessary for maintaining the adsorbate at concentration c. The limits of the applicability of eq. (10) are still a subject of discussion [5-7]. Combining eqs. (8) and (10) the flux through a membrane is found to depend on three different quantities, viz. the permeate concentration gradient (being determined by the pressure difference and the adsorption potential), the "thermodynamic factor" dlnp/dlnc which is a measure of the deviation from proportionality between pressure and adsorbate concentration, and the molecular mobility in the membrane as represented by the self-diffusivity D(c). It is this latter quantity which is investigated by the PFG NMR method.
3. PECULIARITIES OF MOLECULAR DIFFUSION IN MICROPOROUS MATERIALS The intimate contact between the permeate molecules and the intemal surface of the membrane material leads to a number of special features in molecular diffusion which do not occur in bulk fluids. These peculiarities are particularly pronounced in zeolite membranes, as a consequence of their well-defined pore structure. Zeolite membranes combine the general advantages of inorganic membranes (temperature stability, solvent resistance) with a perfect shape selectivity [8, 9]. In this section, we are going to illustrate two features of zeolitic diffusion which are of immediate relevance for molecular transport in zeolite membranes. 3.1. S t r u c t u r e - C o r r e l a t e d D i f f u s i o n A n i s o t r o p y
Most of the so far manufactured zeolite membranes are of MFI type, i.e. of type silicalite or ZSM-5 [8-11]. The reasons possibly explaining this situation include the high level of accumulated knowledge in MFI synthesis, the relative ease of preparation, the suitable pore diameter of 0.55 nm which is interesting for industrial application and the high thermal and chemical stability [9]. As a special structural feature, MFI type zeolites are traversed by two types of channels, viz. straight channels in y-direction and sinusoidal ones in x-direction. There is no channel system in z-direction, and molecular transport in z-direction has to be accompolished, therefore, by interchanging periods of migration along the channel segments in x- and y-direction. As a unique feature of molecular diffusion in zeolites, there must exist, therefore, a correlation between the
100 principal elements of the diffusion tensor, i.e. between the diffusivities in x-, y- and z-direction. Under the assumption that molecular propagation from channel intersection to channel intersection via the individual channel segments may be considered as a random walk, the diffusivities may be shown to obey the relation [ 12, 13]
c2/Dz=a2/Dx+b2/Dy,
(11)
with a (~, 2 nm), b (z 2 nm) and c (~ 1.34 nm) denoting the unit cell dimension of an MFI framework in x-, y- and z-direction, respectively. It has been demonstrated by uptake measurements on macroscopic ensembles of oriented crystals [14] as well as by MD simulations [15, 16], that eq. (11) does in fact provide a reasonably good estimate of the correlation between the relevant diffusivities. Still better agreement may be attained if possible correlations between subsequent shifts along the channel seqments are - at least partially- taken into account in a "two-step" model [ 17]. The first unequivocal evidence of diffusion anisotropy in zeolite ZSM-5 has been provided by PFG NMR studies with macroscopic ensembles of oriented crystals [ 18] as well with powder samples [ 19]. The limited accuracy of these measurements, however, did not allow the separate determination of all three principal tensor elements. In the measurements with macroscopic ensembles of oriented crystals [18] the separate determination of D~ and (Dx + Dy)/2 has been possible. The obtained values were in fact found to be compatible with the correlation rule provided by eq. (11). In a powder sample of zeolite crystallites, the PFG NMR spin echo attenuation results as a superposition of exponentials of the type of eq. (2), with the respective orientation-dependent diffusivities. For isotropically distributed orientations one thus obtains 1
~(fig,t) = ~
2~r
f o
!
fexp(-g2fi2g2t[DxcoS 2 O+ DySin20COS2~ffq- Dzsin2/gsin2 g/])dq~(cosO). --1
(12) In principle, the three principal values of the diffusion tensor should be accessible by a fitting procedure of eq. (12) to the experimental data ~(Sg, t) of echo attenuation. In reality, the accuracy of the data is not high enough to allow the determination of three different parameters. Implying the validity of eq. (11), however, such a fitting procedure becomes possible. The thus obtained data were in fact found to be in satisfactory agreement with the diffusivities observed in the measurements with oriented samples [18, 19]. Diffusion anisotropy may substantially affect the permeation properties of membranes with oriented zeolite crystallites [20-22], making them totally different from those with random orientation [23, 24].
101
3.2. Single-File Diffusion Zeolites consisting of parallel channels A1PO:5, -8, -11, L, Omega, EU-1 and anisotropy: two principal elements of the PFG NMR spin echo attenuation, eq. (12),
with impenetrable walls - like ZSM-12, -22, -23, -48, VPI-5 - give rise to an extreme case of diffusion diffusion tensor degenerate to zero. In this case, the simplifies to
1 1
~(6g, t) = ~ ~exp(- y262g2Dtx2)dx
(13)
-1
with D denoting the diffusivity in channel direction. If the permeate molecules are too large to pass each other within one channel (i.e., roughly speaking, if the molecular diameters are larger than the channel radii), molecular transportation does not follow the laws of ordinary diffusion. This means in particular that eqs. (3), (7) and (9) are not valid, anymore. Due to the fact that like with a file of strung pearls, one molecule (or pearl) can only move if the adjacent one has been shifted into the same direction, molecular motion is highly correlated. Molecular motion under such conditions has been termed single-file diffusion [3, 25, 26]. It can be shown [27] that also under single-file conditions the propagator is still a Gaussian. In contrast to normal diffusion, however, the molecular mean square displacement
(r2(t))=2F4~
(14)
increases in proportion to the square root of the observation time rather than to the observation time itself. F is the mobility factor of single-file diffusion which had been introduced in analogy to the diffusivity in Einstein's law, eq. (3) [26]. The fact that under single-file conditions the molecular mean square displacement increases less than linearly with the observation time is a consequence of the correlation between subsequent displacements: Under the conditions of ordinary diffusion it is always possible to choose a time interval large enough so that the displacements during two subsequent time intervals are independent from each other. Hence, the mean square displacement over a period of time t is equal to the sum of the mean square displacements over the individual time intervals At, with t = ~ A t. Proportionality between the mean square displacement and the observation time is a consequence of this fact. Under singlefile confinement, however, displacements in one particular direction are more likely followed by displacements in the opposite direction, leading to a less than linear increase of the mean square displacement with the observation time. Being exclusively a subject of theoretical consideration over many years [25-30] it was not before the successful application of PFG NMR [31-34] that eq. (14) as the key equation of single-file diffusion could be confirmed experimentally. Experimental verification of eq. (14) is complicated by the influence of the boundary conditions on molecular displacements which is much more pronounced in the case of single-file diffusion than under the conditions of normal diffusion [35]. For practical use in catalysis or gas separation, it is the exchange with the surrounding gas phase rather than the time dependence of molecular displacement which is of immediate interest. Molecular exchange between an adsorbent and the surrounding atmosphere may be characterized
102 by the residence time distribution qffx). It is worthwhile to note that the residence time distribution is closely related to two other characteristic functions, viz. the tracer exchange curve Y(O by t
~o(r)dt
r(t)=
(15)
o
and the effectiveness factor rl(k) by oo
rl(k ) = ~e -*r ~o(r)dt ,
(I 6)
0
where k denotes the intrinsic reactivity in first order reactions [36, 37]. The mean residence or exchange time Xi,~a may be determined on the basis of the characteristic functions via the relations [37] oo
ri.tr a = f[1-7(t)]dt0
oo
frq~(~)dT.
(17)
0
For single-file systems and sufficiently large file lengths, the mean exchange time is found to obey the relation [38, 39]
"t'intra oCL3
(18)
with L denoting the file length. It is interesting to note that this dependence is in contrast with the expressions ,,.a.
"t'intr a o(2
L2
(19)
for normal diffusion [26, 40] as well as with the result of scaling arguments, "['intra oC
L4
(20)
which would be attained on the basis of eq. (19) by considering the equivalence of eqs. (3) and (14) [26]. Relation (18) may intuitively be understood by realizing that displacements, not negligibly small in comparison with the file length, are again controlled by a mechanism equivalent to ordinary diffusion with a diffusivity being proportional to L ~ [35, 38, 41 ]. Thus, by introducing a file-length-dependent diffusivity into eq. (19) one does in fact obtain the dependence as given by relation (18).
103 Membranes with zeolites of single-file type are subjected to two peculiarities. The influence of crystal orientation on the permeation properties is still more pronounced than considered in the previous section on structure-correlated diffusion anisotropy: Zeolite crystallites with channels parallel to the membrane surface would in no way contribute to the permeation flux. Moreover, as a consequence of the exclusion of mutual passages of the individual molecules, the molecules have to stick to their order within the channels, irrespective of their individual mobilities. In the literature, this phenomenon has been termed "constrained" single-file diffusion [42]. Under such conditions membranes would not be suitable for gas separation, even if the permeation rates of the individual components under the conditions of single-component adsorption were extremely different [20, 22]. Such a tendency may even be observed in systems which do not obey the single-file condition. ZSM-5 type membranes, e.g., at sufficiently high loadings, did not reveal any separation effect on mixtures of o- and p-xylene [43], while the p-xylene diffusivity at single-component adsorption is well-known to be by 3 orders of magnitude larger than that of o-xylene [44]. It is noteworthy that at sufficiently low concentrations - i.e. under conditions where mutual hindrance of the molecules is essentially excluded- both components are easily separated from each other in the same membranes [45].
4. M E M B R A N E DIFFUSION STUDIES BY PFG N M R
Diffusion measurement by PFG NMR clearly implies the existence of a measurable NMR signal. Therefore, the molecules under study must contain "NMR active" atoms, i.e. (cf. Section 1) nuclei with a non-vanishing gyromagnetic ratio, which have to occur at a sufficiently large concentration. With sample volumes of about 300 mm 2 for hydrogen, which offers the best measuring conditions with respect to both the minimum number of diffusants and minimum displacements, typical minimum concentrations are of the order of one hydrogen atom per 10 nm 3 which corresponds to about 0.1 moles per litre [3]. Hence, PFG NMR does not apply anymore if the averaged concentration of the diffusants under study within the sample volume is below this value. According to eq. (8), the permeation rate is maximum for maximum concentration gradients. This means that for a given concentration difference over the membrane, the membrane thickness should be chosen as small as possible, with values in the 10 ~tm range. Moreover, for guarenteeing mechanical stability, the membrane is generally produced as a small layer on a much bulkier support. The permeability of this support is unspecific and generally of minor interest. Since it would occupy by far the largest part of the sample volume, as a consequence of the above described limited sensitivity, PFG NMR studies have to be carried out with samples exclusively containing the microporous parts of the membranes at an utmost packing density. It is due to this reason that microporous membranes have not yet become a standard system for PFG NMR studies. In the following, a first example of PFG NMR diffusion studies with microporous materials suitable for membrane fabrication is given [46]. The particular materials used in this study were pieces of membranes that were fragmented during handling. From scanning electron micrographs, these pieces appeared to be crystallite
104
agglomerates with diameters of tens to hundreds of micrometers, where the crystallite diameters were of the order of micrometers. The membranes were prepared from homogeneous solution with a composition given by the molar ratio 5 SiO2 9A1203 " 55.12 Na20 91004.66 H20. An aluminate solution was prepared by dissolving small pieces of aluminium wire in aqueous NaOH over a 48 h period. A sodium silicate solution was prepared by dissolving NaOH pellets in distilled water then adding an aqueous silicate solution with vigorous stirring. Alter 15 min of stirring, the pre-heated aluminate solution was added, and the stiring continued. The resultant solution was transferred to a preheated container for membrane preparation. Free-standing samples of zeolite membrane were prepared by placing the above solution in a sealed Teflon container which was placed in an oven at 50 ~ for 48 h. The container was removed from the oven, drained and the zeolite film that lightly adhered to the container walls was thoroughly rinsed with distilled water. Samples of the membrane detached themselves during this procedure and were ion exchanged to the Ca-form by soaking it in 0.1 M Ca(NO3)2. Further details of membrane preparation may be found in [46]. For the preparation of the PFG NMR samples, the zeolite material was filled into glass tubes of 7.5 mm o.d. and activated by heating at a rate of 10 K h 1 under continuous pumping up to a fmal temperature of 400 ~ where it was kept for 1Oh. After cooling to room temperature the probe molecules were introduced by chilling from a calibrated gas volume with a set pressure. The diffusion measurements have been carried out by means of the home-built PFG NMR spectrometer FEGRIS 400 at a proton resonance frequency of 400 MHz with magnetic field gradient amplitudes up to 24 T m l [47]. The attenuation of the NMR signal (the 'spin echo') v e r s u s the squared gradient amplitude was essentially found to follow a monoexponential decay, as to be expected for normal diffusion in a quasi-homogeneous medium on the basis of eq. (2). The effective diffusivity was determined by comparing the signal decay with that of a standard [water, D = (2.04 + 0.08) x 10.9 m 2 sl at 20 ~ [48]. The fact that molecular transport in zeolite NaCaA is found to follow normal diffusion as described by eq. (3) deserves some further discussion. Zeolites of this type are well-known to be percolation systems [1, 3, 49], since molecular propagation from one cavity ("site") to an adjacent one is only possible if the connecting window ("bond") is permeable. This permeability may be significantly affected by the presence of cations. As a consequence of the blockage of a fraction of the windows, the mean square value of molecular displacements taken over the sample may substantially deviate from the behaviour described by eq. (3) [49]. The deviations are caused by the fact that certain fractions of the molecules experience strong confinement effects due to closed windows while other fractions are on pathways through open windows, thus being able to cover large distances. For the system under study, the blockage of the windows is known to be incomplete [50]. In PFG NMR studies, as a consequence of the limited spatial resolution, molecular displacements over distances of at least 100 nm, i.e. passages through thousands of cavities, are observed. The fmite permeability of the "closed" windows leads to the situation that over the observation time t of the PFG NMR experiment- at least in a statistic sense - all molecules experience the same history. As a consequence, over the relevant time scale, molecular transport can again be expected to obey the laws of ordinary diffusion [51 ].
105
(a)
(b)
D/m2s-1
- - u - - 173 K _ ) = - - ref. - - o - - 238 K - - x - - ref. _ & _ 294 K _ + - - ref. - - v - - 373 K - - e - - ref. 2 molecules per cavity
+T
v\
&~"& '~
1E-10
1E-10
;~
~ o
\
eL'~ u
1E-11
ref. O
5'0
~
_ = = _ 173 - - e - - 238 _ & _ 294 - - v - - 373 2 mol. per
K K K K cavity
jr
~I'~Y%--V--~_ V~V~rAr~v
e-._~_ A...a~a______a__a_Aa
m~ll-I...
'"~0%0_0
9
1(~0
1~0 " 2~0
2~0
3;0 t/ms
350 1E-11
......
~
. . . . . . .
1'o
1/2 / pm
Fig. 1 (a) Ethane diffusivity in a NaCaA membrane as a function of the observation time at different temperatures; ref." results of previous PFG NMR studies with crystalline zeolite NaCaA. (b) Ethane diffusivities in NaCaA membrane as a function of the root mean square displacement at different temperatures. [46]
Fig. 1(a) shows the dependence of the ethane diffusivity in the zeolite membrane at a loading of two molecules per cavity for different temperatures. For comparison, the diffusivity data of previous PFG NMR studies [52] with ethane in NaCaA zeolite crystallites are also included. It appears that these data are in reasonably good agreement with the results of the present study for sufficiently short observation times. The information provided by these studies becomes more obvious in the representation of the diffusivities v e r s u s the covered displacements [Fig. 1 (b)]. There is a pronounced decay for displacements around 1 rtm, which should be attributed to transport resistance, most likely situated at the interfaces between the individual crystallites. For shorter displacements, the majority of the diffusants do not interfer with these barriers. In this case one is able to trace the true intracrystalline diffusion. For larger displacements, the molecules repeatedly have to overcome these transport resistances. This leads to a slight reduction of the diffusivity. The enhancement observed in the high-temperature case for displacements > c a . 10 ~tm may be associated with the occurence of long-range diffusion, when more and more molecules are able to get out of the zeolite bulk phase, either through cracks in the material or through the outer surface of the membrane particles.
106 It should be pointed out that the range of apparently genuine intracrystalline self-diffusion is only observable at the lowest measuring temperature of 173 K (fig. 1b). Under these conditions, however, the measuring conditions are poorest and one cannot exclude that the diffusivity is further increasing. Such a behaviour could be explained by the fact that there are transport resistances within the individual crystallites with spacings of the order of or smaller than the minimum displacements. The existence of substantial intracrystalline transport resistances as a consequence of deviations from an ideal crystalline structure has been demonstrated in comparative permeation measurements with twinned and single crystals [9].
5. CONCLUSIONS The PFG NMR method has been shown to be able to monitor the time dependence of molecular propagation. In this way, in microporous materials a large variety of peculiarities of molecular transportation may become observable. Molecular mobility is in fact one of the rate determining features of membrane permeation. Application to diffusion studies with actual microporous membranes is limited by the relatively low sensitivity of PFG NMR with respect to the number of diffusants necessary for an appropriate signal generation. The potentials of such an application are illustrated by first PFG NMR measurements with A type zeolite membranes.
6. ACKNOWLEDGEMENT Financial support by the European Community (Joule JOE 3-CT95-0018; Brite-EuRam II; BE-97-4783) and the Deutsche Forschungsgemeinschaft (SFB 294) are gratefully acknowledged. We are grateful to Jiirgen Caro for giving ref. [9] to our knowledge before publication.
REFERENCES
1. J. Kfirger, D.M. Ruthven: Diffusion in Zeolites and Other Microporous Solids, Wiley & Sons, New York, 1992, 605 2. R. Kimmich, NMR Tomography, Diffusometry, Relaxometry, Springer, Berlin, 1997 3. J. Kfirger, G. Fleischer, U. Roland, PFG NMR Studies of Anomalous Diffusion, in: J. Kfirger, P. Heitjans, R. Haberlandt (Eds.), Diffusion in Condensed Matter, Vieweg, Braunschweig/ Wiesbaden, 1998, pp. 144 4. M.F.M. Post, Stud. Surf. Sci. Catal. Vol. 58 (1991) 391, Elsevier Science B.V., Amsterdam 5. K~ger, D.M. Ruthven, Self-Diffusion and Diffusive Transport in Zeolite Crystals, in: H. Chon, S.-K. Ihm and Y.S. Uh, (Eds.) Progress in Zeolite and Microporous Materials Stud. in Surf. Sci. and Catal., Vol. 105 (1997) 1843, Elsevier Science B.V., Amsterdam 6. J. K~ger, Molecular Transport in Zeolites - Miracles, Insights and Practical Issues, in Proc. 121nt. Zeol. Conf. (Eds.: Treacy, M.M.J; Marcus, B.K.; Bisher, M.E.; Higgins, J.B.), Mater. Res. Soc., Warrendale, 1999, 35 7. Krishna, Chem. Eng. Sci. 48 (1993) 845 8. E.R. Geus, H. van Bekkum, W.J.W. Bakker and J.A. Moulijn, Microporous Mat. 1 (1993) 131
107 9. J. Caro, M. Noack, P. K61sch and R. Sch/afer, Microporous and Mesoporous Mat., in press 10. J.C. Jansen and G.M. Rosmalen, J. Cryst. Growth 128 (1993) 1150 11. A. Gouzini and M. Tsapatsis, Chem. Mater. 10 (1998) 2497 12. J. Kfirger, J. Phys. Chem. 95 (1991) 5558 13. J. K/irger and H. Pfeifer, Zeolites, 12 (1992) 872 14. J. Caro, M. Noack, J. Richter-Mendau, F. Marlow, D. Petersohn, M. Giepentrog and J. Komatowski, J. Phys. Chem. 97 (1993) 13685 15. E.J. Maginn, A.T. Bell and D.N. Theodorou, J. Phys. Chem. 100 (1996) 7155 16. R.Q. Snurr, A.T. Bell and D.N. Theodorou, J. Phys. Chem. 98 (1994) 11948 17. J. K/arger, P. Demontis, G.B. Suffritti and A. Tilocca, J. Chem. Phys. 110 (1999) 1163 18. U. Hong, J. K/~rger, R. Kramer, H. Pfeifer, G. Seiffert, U. Mtiller, K.K. Unger, H.B. Ltick and T. Ito, Zeolites 11 (1991) 816 19. U. Hong, J. K~irger, H. Pfeifer, U. Mtiller and K.K.Unger, Z. Phys. Chem. 173 (1991) 225 20. M. Noack, P. K61sch, D. Venzke, P. Toussaint and J. Caro, Microp. Mat. 3 (1994) 201 21. P. K61sch, D. Venzke, M. Noack, P. Toussaint and J. Caro, J. Chem. Soc., Chem. Commun. (1994) 2491 22. I. Girnus, M.-M. Pohl, J. Richter-Mendau, M. Schneider, M.Noack, and J. Caro, Adv. Mat. 7 (1995) 711 23. M.C. Lovallo and M. Tsapatsis, AIChE-J. 42 (1996) 3020 24. M.C. Lovallo, A. Gouzinis and M. Tsapatsis, J. Cryst. Growth 128 (1993) 1150 25. L. Riekert, Adv. Catal. 21 (1970) 281 26. J. Kfirger, M. Petzold, H. Pfeifer, S. Ernst and J. Weitkamp, J. Catal. 136 (1992) 283 27. J. Kfirger, Phys. Rev. E, 47 (1993) 1427 28. J.L. Lebowitz and J.K. Percus, Phys. Rev. 155 (1967) 122 29. R. Kutner, Phys. Lett. 81A (1981) 239 30. H. van Bejeren, K.W. Kehr and R. Kutner, Phys. Rev. B 28 (1983) 5711 31. V. Gupta, S.S. Nivarthi, A.V. McCormick and H.T. Davis, Chem. Phys. Lett. 247 (1995) 596 32. V. Kukla, J. Komatowski, D. Demuth, I. Girnus, H. Pfeifer, L.V.C. Rees, S. Schunk, K.K. Unger and J. K~ger, Science 272 (1996) 702 33. K. Hahn, J. K~xger and V. Kukla, Phys. Rev. Lett. 76 (1996) 2762 34. M. Ylibautala, J. Jokisaari, E. Fischer and R. Kimmich, Phys. Rev. E 57 (1998) 6844 35. K. Hahn and J. K~ger, J. Phys. Chem. B 102 (1998) 5766 36. C. R6denbeck, J. K~rger and K. Hahn, Ber. Bunsenges. Phys. Chem. 102 (1998) 929 37. C. R6denbeck, J. K~ger, H. Schmidt, T. Rother and M. R6denbeck, Phys. Rev. E 60 (1999) 2737 38. C. R6denbeck and J. K~ger, J. Chem. Phys. 110 (1999) 3970 39. P.H. Nelson and S.M. Auerbach, J. Chem. Phys. 110 (1999) 9235 40. R.M. Barrer, Zeolites and Clay Minerals as Sorbents and Molecular Sieves, Academic Press, London, 1978 41. H. Rickert, Z. Phys. Chem. N.F. 43 (1964) 129 42. D.S. Sholl and K.A. Fichthorn, J. Chem. Phys. 107 (1997) 4384 43. C.D. Baertsch, H.H. Funke, J.L. Falconer and R.D. Noble, J. Phys. Chem. 10 (1996) 7676 44. D.H. Olson, G.T. Kokotailo, S.L. Lanton and W. Meier, J. Phys. Chem. 85 (1981) 2238 45. K. Keizer, A.J. Burggraaf, Z.A.E.P. Vroon and H. Verweij, J. Membr. Sci. 147 (1998) 159 46. W. Heink, J. Kfirger, T. Naylor and U. Winkler, Chem. Commun. (1999) 57
108 47. J. K~ger, N.K. B~, W. Heink, H. Pfeifer and G. Seiffert, Z. Naturforsch. 50 a (1995) 186 48. H. Weing/~tner, Z. Phys. Chem. N.F. 132 (1982) 129 49. A. Bunde and S. Havlin, Fractals and Disordered Systems, Springer, New York, 1996 50. D.M. Ruthven, Can. J. Chem. 52 (1974) 3523 51. S. Vasenkov, J. K~ger, D. Freude, R.A. Rakoczy and J. Weitkamp, J. Mol. Catal., in press 52. W. Heink, J. K~ger, H. Pfeifer, K. P. Datema and A. K. Nowak, J.C.S. Faraday Trans., 88 (1992) 3505
Recent Advances in Gas Separation by Microporous Ceramic Membranes N.K. Kanellopoulos (Editor) 2000 Elsevier Science B.V. All rights reserved.
109
Diffusion studies using quasi-elastic neutron scattering H. Jobic Institut de Recherches sur la Catalyse, CNRS, 2 Avenue Albert Einstein, 69626 Villeurbanne, France Quasi-elastic neutron scattering allows to study molecular diffusion in condensed or adsorbed phases. Diffusion coefficients can be derived, without a model, over space scales where molecular motion follows Fick's law. Furthermore, using jump diffusion models, one can determine characteristic times and distances for the elementary jumps. For isotropic diffusion, the existing jump models are discussed and a new model, which takes into account the delocalisation of the molecule on its site, is proposed. The cases of normal one-dimensional diffusion and of single-file diffusion are examined. Applications of the technique reviewed here concern mainly molecules adsorbed in zeolites, some results obtained in microporous silica are also presented.
1. INTRODUCTION Neutron scattering is one of the many physical techniques used to characterize solid materials and adsorbed species. There are in fact several neutron-based techniques: neutron diffraction yields structural information [1], textural studies can be performed by small-angle neutron scattering [2], vibrational modes can be measured by inelastic neutron scattering [3], and translational and rotational motions of adsorbed molecules can be characterized by quasi-elastic neutron scattering (QENS). We will limit ourselves here to QENS, which is now regarded as a useful method to determine diffusivities in porous materials. This technique has also been used to study the diffusion of hydrogen in metals [4] and of ions in oxides or solid electrolytes.
2. THEORY 2.1. Interaction of neutrons with matter
Neutrons are electrically neutral particles with a spin of 1/2, a mass mn of 1.675• -24 g, and a magnetic moment/J = -1.913 nuclear magnetons. Neutrons can also be considered as plane waves, with a wave vector k of magnitude 2~X. The neutron wavelength is given by the de Broglie's relationship ~, = h/mnV, where h is Plank's constant and v the neutron velocity. For a wavelength of 1 A, this corresponds to a neutron speed of 3956 m/s. Because of its mass, the neutron has a sizable momentum p = mnV = h k. The neutron energy is given by its kinetic energy, E = 89mnV2 = h k2/2mn. Since this energy can be varied in a wide range, this means that both the structure and the dynamics of a sample can be studied.
110
Neutrons interact with the atomic nuclei on a very short range, = 10 14 m, (we leave aside magnetic scattering). It is useful to split the total scattering cross-section into two parts, a coherent and an incoherent one. Coherent scattering corresponds to an average of the potential over the nuclei and gives rise to interference effects. Incoherent scattering depends on deviations from the mean potential, due to isotopic or spin effects. Values are given in Table 1 for some elements. A cross-section, which has the dimension of a surface, is measured in barns (1 barn = 10 .28 m2). This shows that the interaction of neutrons with matter is relatively weak. Neutron adsorption is small for most elements, except for a few nuclei like boron. It appears from Table 1 that the scattering cross-sections vary irregularly from one atom to another, even from one isotope to another, unlike with X-rays. For an assembly of nuclei containing only one isotope, with zero spin, the scattering is totally coherent (e.g. 12C). For each natural element, an average over all stable isotopes must be performed. In the case of diffraction experiments, it is usually desirable to use deuterated compounds, since incoherent scattering from 1H would increase the background. For quasi-elastic or inelastic studies concerning adsorbed molecules, the best contrast will be obtained with hydrogenated species. Table 1 Coherent, incoherent, and absorption cross-sections in barns for some elements (1 barn = 10 .28 m 2, the absorption cross-section is proportional to k, here ~, = 1 A) Element
Atomic Number
H
1
Li Be B C
3 4 5 6
N O Na AI Si P Ni Pt
7 8 11 13 14 15 28 78
Mass number 1 2= D
12 13 23 27 31
~coh 1.7586 1.7599 5.597 0.454 7.63 3.54 5.554 5.563 4.81 11.01 4.235 1.66 1.495 2.163 3.307 13.3 11.65
O'inc 79.9 79.91 2.04 0.91 0.005 1.7 0.001 0. 0.034 0.49 0. 1.62 0.009 0.015 0.006 5.2 0.13
(~abs 0.185 0.185 0. 39.2 0. 426. 0. 0. 0. 1.06 0. 0.3 0.13 0.1 0.1 2.5 5.72
The unit most used in neutron scattering is the meV. Some useful conversion relationships are 91 meV = 8.065 cm 1 = 0.24 THz = 11.6 K = 23.06 cal.
]]l
2.2. Molecular dynamics
The two important quantities in a neutron scattering experiment are the energy transfer h(o and the scattering vector Q defined as h(_O - E 0 - E 1 = ( h 2 / 2 m n
)(k ~)- k 12)
(1)
Q=ko-k 1 (2) where ko and k1 are, respectively, the incident and final wave vectors (the momentum transfer is hQ). Elastic scattering corresponds to k] = k 0 (hco = 0), so that only momentum is transferred. Quasi-elastic scattering occurs at very small energy transfers, + 2 meV, around the elastic peak. Experimentally, one measures the double-differential cross-section which represents the number of neutrons scattered into the solid angle dD with energy in the range dE (or h dco). This cross-section can be split into incoherent and coherent contributions [5,6]
d2t7
dD dE
k1
N
k o 4re h
[GincSinc(Q,o))+GcohScoh(Q,(o)]
(3)
Because of the large scattering cross-section of the hydrogen atom, orinc, it is essentially the diffusion of hydrogenated molecules which has been studied by QENS. The scattering functions, S(Q, ar), are Fourier transforms of correlation functions which describe the motions of the scattering particles in space and time. The incoherent scattering function is defined as ]
Sinc(Q,(o)=-~dr dt exp{i(Q.r-(ot)} Gs(r,t) system (ksT>> ho~), Gs(r,t) corresponds to
(4)
For a classical the probability to find a particle at position r at time t, if the same particle was at the origin at time zero (the subscript s stands for self) a s (r, t) = ~-
S [r + r i ( 0 ) - ri(t)]
(5)
Therefore, one follows the motion of one individual proton or of one given molecule, and the self-diffusion coefficient can be obtained, like in PFG NMR. In principle, Gs(r,t) can be derived from the measured intensities by inverse Fourier transform. However, the range of Q and co covered by a neutron spectrometer is too limited to obtain the self-correlation function with sufficient accuracy. What is clone in practice is to use a theoretical or model expression for Gs(r,t) and compare the calculated scattering function with the measured intensities. Even if one is only interested by the long-range diffusion, the other molecular motions of rotation and vibration have also to be taken into account. The position vector of a scattering nucleus in a molecule can be decomposed into three components: r=c+b+u (6) where c describes the position of the center of mass of the molecule, b the mean distance of the nucleus from the center of mass, and u the displacement of the nucleus from its equilibrium position under the effect of vibrations. A key
112 approximation is to consider that the translational, rotational, and vibrational motions are not coupled. Indeed for most systems, these motions are observed on different time scales. The total incoherent scattering function can then be written as a convolution product in to (symbolised by | of the different scattering functions : S inc (Q,
to) = ~'inc ~ trans (Q, to)|
.~ rot (Q,to)|
.~ vib (Q, co) ~.mc
(7)
- The energy transfers involved in QENS experiments, + 2 meV, are too small to excite intramolecular vibrations so that the vibrational term in Eq. (7) consists only of a Debye-Waller factor, exp(-Q2 2), where 2 is the mean-square amplitude. The effect of this factor is to decrease the intensity of the QENS spectra, for increasing Q values (the intensity goes into the inelastic domain). For a rotational motion, the incoherent scattering function can be separated into an elastic component and a quasi-elastic contribution : sr~ Ao(Q) &(co)+ ~ Ae (Q) L(to,Fe ) (8) e An elastic component occurs because the self-correlation function has a finite value at long times (or co~0). This happens for example if an atom is rotating around a fixed position. The factor Ao(Q), which governs the intensity of the function is called the elastic incoherent structure factor (EISF). This factor can be shown to correspond to the space Fourier transform of the proton trajectory averaged over a long time ('long' depending on the time scale of the experiment). The variation of the EISF with Q gives information on the geometry of rotation. The experimental EISF can be compared with various models, e.g. rotational diffusion on a sphere, diffusion within a sphere, uniaxial rotation, and fluctuations around equilibrium positions. Characteristic times of the rotational motions can be obtained from the quasi-elastic component in Eq. 8, which usually consists of a sum of Lorentzians Le(to,Fe) whose number e and width F e depend on the model. - The most interesting molecular motion in this review is the translation, it will be described in more detail in section 4. -
The scattering from adsorbed non-hydrogenated molecules can also be studied by QENS, although it requires a high-flux spectrometer. In this case, the scattering will be partially coherent. The coherent scattering function corresponds to l
Scoh(Q,to)=-~dr dt exp{i(Q.r- tot)} G(r,t)
(9)
where G(r,t) gives the probability to find a particle at position r at time t, if any particle was at the origin at time zero
N N (t)] \/ (10) ~t~ [r +ri(O)-rj N \i=lj=l Here, collective motions are probed and the transport diffusivity, which measures the evolution of local concentration gradients, can be derived. These experiments are more difficult because the coherent and incoherent contributions must be separated [7,8]. A characteristic of coherent scattering is that the intensity depends strongly on Q, whereas the incoherent intensity has a relatively smooth Q dependence, being
G(r,t)=l[~,
113
only influenced by the Debye-Waller factor. The coherent structure factor is defined as
S(Q) = ~S ( Q , o ) dco (11) In a liquid, or for identical molecules adsorbed in a cubic zeolite, S(Q) is related through Fourier transform to a pair-distribution function, g(r), reflecting short-range order. 3. EXPERIMENTAL Neutrons are produced either from the fission of uranium or from the bombardment of a target by protons. The first possibility leads to steady-state reactors (e.g. the Institut Laue-Langevin, ILL, in Grenoble, France), the second to pulsed sources (e.g. the spallation source ISIS, Didcot, UK). For QENS experiments, the ILL is at the time being the best place in the world because there are several types of spectrometers:time-of-flight, back-scattering and spin-echo instruments, which are able to cover a wide range of energy and momentum transfers. The instrumental resolution is an essential parameter:for a resolution width Ao~, the time scale over which an average of G(r,t) will be performed is of the order of 1/Aco. By reducing A o , one can follow slower motions (the energy transfer range will accordingly be restricted). With a back-scattering spectrometer, the resolution is of = 1 peV so that l/Aco is = 1 ns. With neutron spin-echo instruments, the resolution can be as high as 1 neV giving access to characteristic times of = 1 ps. The Q resolution is less critical although one must be able to measure in between the Bragg peaks of the zeolite. The neutron cells can be cylindrical or slab-shaped, and they are usually made of aluminum. Sample quantities for QENS measurements are typically of a few grams, the exact amount depending on the scattering power of the adsorbate. Special care must be taken in order to avoid pollution of the substrate after activation, especially by water. It must be transferred into the neutron cell in a glovebox or under vacuum. 4. ISOTROPIC DIFFUSION Theories describing transport phenomena in liquids are based on the assumption that the medium is continuous. In fact, different experiments and computer simulations show that a liquid has a short-range order, leading to a diffusive and oscillatory behaviour. The interest of QENS is that continuous diffusion can be measured at small Q values and that jump diffusion can be characterised at larger Q values. 4.1. Continuous diffusion In a liquid, it is well known that a Gaussian approximation for Gs(r,t) is valid as soon as the diffusion is followed over a distance of a few tens of molecular diameters (Qd << 1, where d is the molecular diameter)
114
1
r
2
(12)
Gs(r't)= (4JrD t)3/2 exp(- 4--~t )
where D is the self-diffusion coefficient. Since the self-correlation function can be related to a particle density and since Gs(r,0)=&(r), Eq. (12) is an appropriate solution of the diffusion equation or Fick's second law
3Gs(r,t) =DV2Gs(r,t) 3t
(13)
The QENS intensities can then be calculated from Eq. (4). First, the space Fourier transform gives the intermediate scattering function
I S (Q,/) - j"exp(iQ.r) GS (r,t) dr = exp(-D Q2 t)
(14) This function can be measured with neutron spin-echo, for a given Q value. In traditional QENS, one measures energy spectra simultaneously at different Q values. The scattering function is obtained after a second Fourier transform, in time
Sinc(Q, oo)= j'exp(-i(.o t) Is(Q,t ) dt=
]
DQ 2
n" 092 + ( D Q 2
)2
(15)
The shape of the energy spectra is a Lorentzian, with a half-width at halfmaximum (HWHM) DQ 2. By plotting the width as a function of Q2, one may check that Fick's law applies if a straight line is obtained. The slope gives directly the selfdiffusion coefficient. While a linear variation of the broadening is often found at small Q values, it is usually observed that the width deviates from a straight line at larger Q values. This indicates that the continuous diffusion model is no more valid at small distances and that the details of the elementary diffusive steps have to be taken into account. Fick's phenomenological equations were demonstrated by Einstein from statistical considerations [9], showing the relationship between transport phenomena and Brownian or random motions. The concept of random walk for a particle diffusing by successive jumps on a cubic lattice can be applied to a hydrogen atom in a metal or to a molecule in a zeolitic network. In this model, the jump length g is a constant as well as the jump rate, ~:-1, where ~: is the residence time on a given site. Provided that the jumps are uncorrelated, the following relation is derived g2 O(16) 6"r The self-diffusion coefficient is independent of the number of jumps. This can be compared with the mean-square displacement calculated from Eq. (8) 4~
do
r2
-~r2Gs(r,t)dr=(4zrOt)3/2 ~r 4 exp()dr=6Ot 0 4Dt
(17)
This shows that a macroscopic diffusion coefficient can be related to jumps on an atomic scale. Provided that the interactions between the particles are negligible, the s a m e diffusivities should be obtained.
115
I
I
/
5I
4-3 2 ,,.
i
0.0
I
I
I
I
.2
.4
.6
.8
Q2
(A-2)
1.0
Fig. 1. Broadenings calculated as a function of Q2 for different models" ChudleyElliott (solid line), Singwi-Sj61ander (dotted line), Hall-Ross (dashed line) and Fick (dotted-dashed line). The HWHM are calculated for the same values of < r 2 > = 100 A 2 and "c, therefore for the same diffusion coefficient.
4.2. Jump diffusion While Fickian diffusion is obtained at small Q values (Figure 1), jump diffusion implies a deviation from the straight line at shorter distances. The interpretation of the QENS spectra at larger Q values requires a model which contains as parameters the characteristic lengths and times of the elementary steps. 4.2.1. Existing models : Chudley-Elliott, Singwi-SjSlander, Hall-Ross The first model by Chudley and Elliott (CE) was developed for diffusion in liquids, where a lattice-like structure is assumed to exist locally [10]. In fact, this model has found many applications in solid state diffusion. The CE model is based on the following hypotheses: an atom vibrates on a given site during a time .c. After this time, it jumps to another site situated at a distance d (the time taken for the jump is much shorter than the residence time), an atom having n jump possibilities. CE have shown that with these hypotheses, the incoherent scattering function is still a Lorentzian function
116
S(Q,co) =
l
Aco(Q)
(18)
/1:(0 2 + (A(o(Q)) 2
but the HWHM has a different value compared with Eq. (15)
Aa)(Q) = __1~[1- exp(iQ.d)]
(19)
nz" d
Here, all the sites are supposed to be equivalent. However, the model can be generalised to deal with crystallographically or energetically different sites [11]. The diffusion of hydrogen in metals can be well described by this model. Single crystal experiments allow to orient the vector Q along selected crystallographic directions. 4 3
v
=1A-1
-
>.,
~9 2 -
E
c-
1
-
i !
!
.
=2A-1
o
.
,|
i
i
i
4
=3A-1 o
o
!
-1 .o
-.5
0.0
,,
i
.
i
.5
1.0 Energy
Fig. 2. Simulated spectra (without convolution with instrumental resolution) for jumps between octahedral sites in a fcc lattice (the jump distance is of 3 A). The continuous line is obtained after powder averaging, the dotted line corresponds to a Lorentzian fitted on the exact profile.
117
If only polycrystalline samples are available, Eq. (18) must be averaged to take into account all possible orientations of the crystallites with respect to the scattering vector. Accurate numerical methods exist, based on the generation of optimally chosen points in the three dimensional space [12,13]. For each Q value, one obtains a sum of Lorentzians with different widths so that the total profile can be different from a Lorentzian. At small Q values, however, the calculated profile is well described by a Lorentzian function, as shown in Figure 2. This example corresponds to jumps between octahedral sites in a fcc lattice (a jump distance of 3 A has been arbitrarily selected). At larger Q values, the profile deviates from a Lorentzian function, however the exact value of the HWHM is not much different from the one obtained assuming a Lorentzian profile. For a random motion, the mean-square displacement is proportional to the square of the jump distance =Nd 2 (20)
where N is the number of jumps. In QENS, one can measure broadenings for Q values of about 0.1 A~, which corresponds to distances in real space of 2rdQ = 60 A. For jump lengths of 3 A, this means that one is able to follow the motion of a particle after 400 jumps. On this space scale, the vector d can take any direction, the isotropic approximation is justified, and the term IQ.dl 2 can be replaced by Q2d2 /3. The limit of Eq. (19), for small Q values is then Aco(Q)_ -O2d2 -_002 (21) 6"r Therefore using Einstein's relation, Eq. (16), one recovers the broadening characteristic of a macroscopic system. Chudley and Elliott have also shown that if the jumps of a fixed length d may occur in any direction, like in a liquid, the sum in Eq. (19) is replaced by an integral
Aco(Q) =
1 - -~ o~dO sinO exp(iQdcosO) 1(sin(Qd)) 1- Q-- 7 -
(22)
This function is plotted in Figure 1 for a jump length of 10 A. It has a maximum for Q = 3rd2d (Q2 = 0.2 A 2) and oscillates for larger Q values around ,(1, the jump rate. At small Q values, an expansion of the sinus function up to the third order yields again the DQ 2 law. This can be checked in Figure 1 where the lines corresponding to Fickian diffusion and to the CE model are indistinguishable below Q2 = 0.03 A 2 (Q = 0.17 A-l). The CE model was found to describe the diffusion of molecules in zeolites where specific interactions occur. For example, in the case of n-pentane in NaX [14], broadenings derived from individual fits of the spectra show a maximum at Q2 = 0.4 A 2 (Figure 3). This maximum is characteristic of a jump diffusion process with jumps of a fixed length occurring in random directions. The jump length, 7 A, is shorter than the distance between the centres of two adjacent supercages, = 11 A, which means
118
.04 (b)
.03 -
-l-
> v
E
.02 -
"1"1-
.01 0.00
0.0
!
!
!
!
.2
.4
.6
.8
Q2(A-2)
1.0
Fig. 3. Broadenings due to diffusion of n-pentane in NaX at" (a) 300 K, (b) 380 K. The points were obtained by fitting each spectrum individually, the curves correspond to simultaneous fits with all spectra using the CE model. that the long-range diffusion does not correspond to jumps from one cage to another. This picture is in agreement with recent MD simulations [15]. Another notable result obtained by Chudley and Elliott is that if the jumps have a distribution p(r), the broadening takes the form
Aro(Q)= ~1 ( 1- sin(Qr))Q...p(..7r) dr / ~p( r) dr
(23)
Since we will use only normalised distributions, Eq. (23) reduces to
Aro(Q)=~II l-~ sin(Qr) Qr p(r)dr 1
(24)
For the normal CE model, this distribution is of the form &(r-d). The other models which were proposed afterwards can be recovered starting from Eq. (24). It seems that Egelstaff [16] was the first to demonstrate that if a distribution of the form p(r)-rexp(-r/to) was used, a simplified version of the Singwi-Sj6lander model [17] could be obtained, neglecting the time taken for the jump. In order to compare the different models, we will use for the SS model a normalised distribution
0 ss (r)=
rro2 exp/r/ro
This distribution is plotted in Figure 4 as a dotted line. The mean-square jump length corresponding to this distribution is
119
=~r2p(r)dr=
1---~r3exp r2o
o
(26)
dr=6ro2
From Eq. (24), one obtains for the HWHM A(_OSs(Q)= l I 1-
=
I
1 oo Isin(Qr)exp ("~o) - r dr ] Qdo
1 ) l
1-
Q2
=
(27)
Q2ro2 22
1+ r2 1+0 ro Using Eq. (26), the broadening can also be written
AcoSS (Q) = 1
Q2 < r 2 > 6Vl+Q 2 /6
(28)
In the low Q limit, the broadening tends to < r 2 > Q2 / 6v, that is DQ 2, using a slightly different version of Einstein's relation D(29) 6~:
.12 .10 v
Q_
..."
.08 -
.1/
9
/
9
/
9
.06 -
9
J
0
/
",
/ /
9 9
\
~
/
9
.02 -
'\
/
9
.04 -
\
/
9
0.00
r\
/
/ /
/
\
\
\
\
\
\
\ 9~
N
~ ~
,
~~'.: ~ ~
!
5
I
10
15
I
20
~ ~
~
............. I
25
30
r(A) Fig. 4. Jump length distributions corresponding to the models of Chudley-Elliott (8 function at 10 A), Singwi-Sj61ander (dotted line), and Hall-Ross (dashed line). The two curves are calculated for the same value of < r 2 > = 100 A 2", this implies to= 4.082 A for SS and ro = 5.773 A for HR.
120
where the mean-square jump length replaces the square of the jump distance (cf. Eq. 16). This model was found to apply in the case of ammonia in silicalite [18]. The broadenings plotted in Figure 5 for the individual fits do not show a maximum, but converge progressively to an asymptotic value. All the spectra could be fitted simultaneously with the SS model, yielding a mean jump length of 5 A.
.06 .05 -
~"
.04 -
E
+
.03 -
d
.02
r
.01 0.00 0.0
I
!
I
I
I
!
.2
.4
.6
.8
1.0
1.2
Q2
1.4
(A-2)
Fig. 5. Broadenings measured for ammonia in silicalite (4.3 molecules per u.c., T = 360 K). (+) individual fits of the spectra, the solid line corresponds to a simultaneous fit with all spectra using the SS model, the dashed line to Fickian diffusion. The third model, proposed by Hall distribution of the form [19]
and
Ross
(HR), is based on a jump length
p H R ( r ) - r3(2~:)1/2 exp -
(30)
This distribution, which is normalised, is plotted in Figure 4 as a dashed line. Compared with the SS distribution, the maximum is situated at a larger r value, so that the width reaches its asymptotic value more rapidly (Figure 1). The mean-square jump length corresponding to the HR distribution is < r2 > = I r2p(r) dr = 2 I r4 exp 0 r03(2/~) 1/2 0
For this model, the HWHM is given by
dr = 3 r2
(31)
121
A(o HR (Q) - 4~
_l
[
~ rsin(Qr) exp Qr3(21c) 1/2 0
l - exp -
02 r2 ]
(32)
6
It is worth noting that Hall and Ross obtain the same result using a different derivation [19]. In fact, since all the models are based on the same postulates: Markovian random walk with one mean jump rate, one can always obtain the width through Eq. (24), only the jump distribution varies. The HWHMs for the different jump models, represented in Figure 1, correspond to the jump length distributions plotted in Figure 4. All the widths were calculated for the same values of < r 2 > and ~:, hence for the same diffusion coefficient D. This is manifested in Figure 1 by the fact that all the curves share the same linear variation at low Q, in the Fickian regime. It can also be noted that only the Chudley-Elliott model has an oscillatory behaviour, the two other models converge more or less rapidly towards 1/'c at large Q. 4.2.2. New Model In metals, the adsorption sites for the hydrogen atoms are well defined. In zeolites without cations, the spatial distribution of adsorbed molecules at finite temperatures can fill a channel segment or a cavity. For example, Monte Carlo simulations have shown that the centre of mass of linear alkanes is elongated in the channel segments of silicalite [20]. If one defines the distance between two sites by d o and if the delocalisation of the molecule on its site is taken into account by an additional parameter ro (physically one should have ro < d o), one can propose a new jump length distribution p(r) =
r / (r-do)2 / exp d O r0 (2~) 1/2 2 r2
(33)
This distribution is normalised, it is represented in Figure 6(a) for different values of r0 and d o. The mean-square jump length corresponding to this new model is given by ~'
< r2 > = I r 2 p ( r ) dr -
0
~ ((r-do) 2) 2 ~ r 3 exp dr = d 2 + 3 r2 d o r0 (2to) 1/2 0 2r2
(34)
The distributions plotted in Figure 6(a) were calculated for several values of ro, the value of d o being derived from Eq. (34). For this model, the following analytical expression is obtained for the HWHM of the Lorentzian l ll ~sin(Qr)expA(_o(Q) = "c Q d 0 r0 (2/01/2 0
[ sin0o, / 02ro)l
=l
l - ~ e x p -
T
Qd o
2
, dr 2r2 (35)
122
.3
/~ / \
Q.2-
,1
/
\
(a)
"
o.o
-~,.~ 0
5
f
,
" -,~-'----,--
10
I'~-
15
~
.
.
.
20
r(A)
.
:~ 3
"T" -r
25
(b)
2
0.0
!
!
i
.2
.4
.6
!
.8 Q2
(A-2)
1.0
Fig. 6. Results obtained for the new jump diffusion model" (a) jump length distributions calculated for the same value of < r 2 >= 100 A 2" the dotted line corresponds to ro = 0.1 A (d o = 9.998 A), the dashed-dotted line to to= 2.2 A (d o = 9.246 A), and the dashed line to to= 4.5 A (do= 6.265 A), (b) broadenings calculated with the same values of the parameters as in (a). y
It appears from Figure 6 that the new model is able to reproduce the variations of the CE and HR models, as well as intermediate cases. For a small delocalisation of the molecule (to= 0.1 A), the broadening is equivalent to the one obtained with Eq. (22). For a large delocalisation (r o= 4.5 A), the jump length distribution and the width are similar with the ones computed with HR model (see Figure 1). This new model is able to describe the small hump in the broadenings found for small nalkanes in the MFI structure, both experimentally [21] and theoretically [21,22]. Eq. (35) has the expe