Adsorption and Phase Behaviour in Nanochannels and Nanotubes [1 ed.] 9048124808, 9789048124800

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Adsorption and Phase Behaviour in Nanochannels and Nanotubes

Lawrence J. Dunne · George Manos

Editors

Adsorption and Phase Behaviour in Nanochannels and Nanotubes

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Editors Prof. Lawrence J. Dunne London South Bank University Department of Engineering Systems 103 Borough Rd. London United Kingdom SE1 0AA [email protected]

Dr. George Manos University College London Dept. Chemical Engineering Torrington Place London United Kingdom WC1E 7JE [email protected]

ISBN 978-90-481-2480-0 e-ISBN 978-90-481-2481-7 DOI 10.1007/978-90-481-2481-7 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009930956 © Springer Science+Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book is a collection of articles reviewing selected aspects of molecular adsorption and phase behaviour in nanoporous materials, from both experimental and theoretical perspectives. In a series of chapters written by some of the most active contributors to the field selected problems and experimental and theoretical strategies for studying adsorption in materials containing nanopores and nanochannels are described. Particular emphasis is given to the results of theoretical investigations and simulations of adsorption, including mixtures, in zeolites and carbon nanotubes, while other chapters review diffusion and polymer adsorption in such materials. Water adsorption in a nanopore as a model for biological membrane channels is also discussed. The book is developed out of our interest in the statistical mechanics of adsorption and our diverse interests in zeolites and carbon networks. From the outset we have been quite prepared to have overlapping contributions with different perspectives. The themes which may overlap in various chapters include: I. Nanotubes (i) Adsorption isotherms on nanotubes (ii) Theory/simulation of adsorption on nanotubes (iii) Gas adsorption in nanotubes (iv) Water structure in nanotubes (v) Polymers in nanotubes II. Zeolites/Zeotypes (i) Experimental techniques to study diffusion in zeolites (ii) Adsorption of mixtures in zeolites (iii) Structural studies of liquid adsorption in zeolites (iv) Statistical mechanical treatment of adsorption in microporous materials v

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(v) Molecular simulation of adsorption on zeolites (vi) Geometric and molecular shape effects in nanopore adsorption III. Phase Behaviour in Nanochannels and Confined Spaces (i) Freezing/melting in nanopores (ii) Vapour–liquid equilibrium in nanopores The book has been compiled with the specialist reader in mind but general introductions are given and the principles and strategies underpinning the various methodologies are outlined. Certainly, senior and graduate students may find this text useful. The approaches discussed have been developed and used by numerous people (whose work is reviewed and cited within these covers) over several decades and in part by the chapter contributors themselves. Each chapter is a review with extensive reference lists. We hope that this book will interest a wide audience including academic and industrial, theoretical and experimental physicists, chemists and biochemists, chemical and petroleum engineers and material scientists. In each chapter a local set of variables is used and hence we have not included a comprehensive key to symbols. To avoid duplication we have deliberately not included a list of references in Chapter 1 as complete details are given in the remaining text. We owe a debt of gratitude to many people. First, to the contributors themselves for all their efforts and patience. We also wish to acknowledge help and support and encouragement from Dr. Sonia Ojo and Ms. Claudia Thieroff at Springer Publishers. Particular thanks is also due to Lucy Dunne for editorial assistance. The encouragement and support of Professors Rao Bhamidimarri and Marouan Nazha is also gratefully acknowledged. Professor Sir Harry Kroto introduced us to non-planar carbon networks and nanotubes many years ago and encouraged us in this endeavour from the outset. We also wish to thank Professor Martin Chaplin for permission to reproduce Fig. 11.3(b) on the book covers. Finally, without the love and support of our wives Jackie and Eleni and daughters Katherine and Lucy, Vasiliki and Sotiria, none of this would have happened. London, UK April 2009

Lawrence J. Dunne George Manos

Contents

1 Perspective and Introduction to Adsorption and Phase Behaviour in Nanochannels and Nanotubes . . . . . . . . . . . . . Lawrence J. Dunne and George Manos

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2 Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . F.J. Keil

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3 Molecular Simulation of Adsorption of Gases on Nanotubes . . . . Erich A. Müller

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4 Molecular Computations of Adsorption in Nanoporous Materials . Ravichandar Babarao and Jianwen Jiang

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5 Polymers in Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . Guiduk Yu, Woojeong Cho, and Kyusoon Shin

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6 Statistical Mechanical Lattice Model Studies of Adsorption in Nanochannels Treated by Exact Matrix Methods . . . . . . . . . George Manos, Zhimei Du, and Lawrence J. Dunne

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7 Monte Carlo Simulation and Lattice Model Studies of Adsorption of Methane, Ethane, Carbon Dioxide and Their Binary and Ternary Mixtures in the Silicalite Zeolite . . . . . . . . George Manos, Lawrence J. Dunne, Akrem Furgani, and Sayed Jalili

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8 Molecular Packing-Induced Selectivity Effects in Liquid Adsorption in Zeolites . . . . . . . . . . . . . . . . . . . . . . . . . Joeri F.M. Denayer and Gino V. Baron

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9 Macroscopic Measurement of Adsorption and Diffusion in Zeolites Stefano Brandani

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Vapor–Liquid Equilibrium . . . . . . . . . . . . . . . . . . . . . . Joël Puibasset

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Structuring and Behaviour of Water in Nanochannels and Confined Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Martin F. Chaplin

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Freezing and Melting in Nanopores . . . . . . . . . . . . . . . . . . Kyunghee Lee, Guiduk Yu, Euntaek Woo, Soohwan Hwang, and Kyusoon Shin

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Elasticity Theory for Graphene Membranes . . . . . . . . . . . . . Juan Atalaya, Andreas Isacsson, Jari M. Kinaret, and Ener Salinas

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors

Juan Atalaya Department of Applied Physics, Chalmers University of Technology, Gothenburg, Sweden, [email protected] Ravichandar Babarao Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576, Singapore, [email protected] Gino V. Baron Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium, [email protected] Stefano Brandani Institute for Materials and Processes, School of Engineering, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JL, UK, [email protected] Martin F. Chaplin London South Bank University, Borough Road, London SE1 0AA, UK, [email protected] Woojeong Cho School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea, [email protected] Joeri F.M. Denayer Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium, [email protected] Zhimei Du Department of Chemistry, University College London, London WC1E 0AJ, UK, [email protected] Lawrence J. Dunne Department of Engineering Systems, London South Bank University, London SE1 0AA, UK, [email protected] Akrem Furgani Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK, [email protected] Soohwan Hwang School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea, [email protected] Andreas Isacsson Department of Applied Physics, Chalmers University of Technology, Gothenburg, Sweden, [email protected]

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Sayed Jalili Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK, [email protected] Jianwen Jiang Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576, Singapore, [email protected] F.J. Keil Hamburg University of Technology, Chemical Reaction Engineering, 21073 Hamburg, Germany, [email protected] Jari M. Kinaret Department of Applied Physics, Chalmers University of Technology, Gothenburg, Sweden, [email protected] Kyunghee Lee School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea, [email protected] George Manos Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK, [email protected] Erich A. Müller Department of Chemical Engineering, Imperial College London, SW7 2AZ, UK, [email protected] Joël Puibasset Centre de Recherche sur la Matière Divisée, CNRS-Université d’Orléans, 1b rue de la Férollerie, 45071 Orléans, Cedex 02, France, [email protected] Ener Salinas London South Bank University, Borough Road, London SE1 0AA, UK, [email protected] Kyusoon Shin School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea, [email protected] Euntaek Woo School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea, [email protected] Guiduk Yu School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea, [email protected]

Chapter 1

Perspective and Introduction to Adsorption and Phase Behaviour in Nanochannels and Nanotubes Lawrence J. Dunne and George Manos

Abstract This introduction reviews the themes and chapters in this book on selected topics in the adsorptive properties of nanochannels and nanotubes. Full references and diagrams are given in the remaining text of this book and are for reasons of space not duplicated here.

1.1 Background Matter adsorbed in a narrow pore or channel in equilibrium with a bulk phase can display a wide spectrum of fundamentally interesting and practically useful properties. The process of adsorption in which molecules (adsorbate) from a gas or liquid phase adhere in some way to a solid phase (adsorbent) is one of enormous importance and whose study and applications have a very long history. We can reflect, for example, on chars and carbons which have been used since antiquity. All the applications of these (for example, in medicine or wine making) involve some form of adsorptive separation of mixtures and are based on the preferential adsorption of molecules by an adsorbent. Yet, the study of adsorption carries on vigorously today benefiting from experimental, theoretical and computational advances and new industrial applications. Many applications are only poorly understood at the fundamental level because of difficult theoretical challenges and in some cases limited characterization of the adsorbent microstructure. The channels of nanotubular dimensions alluded to above exist in a variety of materials. Some are naturally occurring in rocks and clays while others are found in synthetic materials. Examples are zeolites, carbon nanotubes and the nanotubular channels of aluminophosphates, boron nitride nanotubes, metal organic frameworks (MOFs) and biological membrane channels. Disordered and defective carbon nanotubes and structures related to

L.J. Dunne (B) Department of Engineering Systems, London South Bank University, London SE1 0AA, UK e-mail: [email protected] L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_1,  C Springer Science+Business Media B.V. 2010

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the closed fullerene cage structures must exist in many chars and soots and smoke particles but we know little about the role of such disorder in adsorption. Traditional studies have provided much valuable information about adsorption isotherms and heats of adsorption and these in turn provide tests for theoretical models and fundamental insights. Heats of adsorption are measured typically from isotherms or with a calorimeter but usually only for a single component. Because of the difficulties with experiments, theoretical prediction and modelling of adsorption isotherms has long occurred and for mixtures of several components this currently seems to be the dominant tool. The key tool in obtaining high-quality adsorption isotherms has been to use the well-tested Monte Carlo simulation method which was famously developed as a straightforward way of evaluating multiple integrals. However, the theoretical prediction of adsorption isotherms is a major problem in statistical mechanics and in computational chemistry. Nevertheless, over the last 30–40 years, with advances in computational power and the development of algorithms, simulation of adsorption isotherms has become feasible at least to certain levels of approximation. This is true for heavy molecules which behave classically. Adsorbed light molecules exhibiting strong quantum behaviour, however interesting, are outside the scope of our discussions. We can usefully distinguish between chemisorption and physisorption where the latter process occurring in channels is our main focus here. In chemisorption, strong chemical bonding occurs between the adsorbate and adsorbent while not in physisorption. Adsorption is always an exothermic process where the heat of adsorption is usually an order of magnitude smaller in physisorption than chemisorption. The intermolecular forces (dispersion, polarization and electrostatic interactions) involved in physisorption are fairly well understood at least at the level of pair potentials although many body and screening effects must play some role. The interaction of adsorbed molecules with nanopore or nanochannel walls distorts the usual forces between adsorbed molecules thereby imposing an imprint of the wall structure upon the adsorbed phase itself. This in turn alters the usual properties which characterize molecules in their bulk phases. Zero point and ro-vibrational energy levels may be also altered somewhat. Thus, the effect of confinement on the nanoscale can greatly influence the phase behaviour of adsorbed molecules. The reduced dimensionality of a phase adsorbed inside a tube also has important consequences. It is expected that material adsorbed in nanochannels might exhibit unusual behaviour due to the reduced dimensionality. Actually defining the dimensionality is itself not straightforward, as behaviour which seems effectively one-dimensional at higher temperatures can cross-over to three-dimensional in an array of channels at low temperatures. The reduced dimensionality together with the competition between wall–molecule and molecule–molecule interactions introduces new chemical and physical features into the adsorbate/adsorbent system. Long-range interactions further complicate the issues here. Both the confinement itself and the structure of the pore wall and effective dimensionality may all be significant. In his contribution Puibasset discusses vapour–liquid phase behaviour of fluids confined on a nanometre scale, where the predominance of the fluid/substrate

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interactions strongly distorts properties. For instance, it is observed that the amount of fluid adsorbed in a nanoporous substrate is not a single-valued function of the chemical potential and may present a hysteresis. Understanding this phenomenon is a fundamental issue since it appears in the most frequently used method to characterize porous materials. The aim of Puibasset’s chapter is to focus on the analog of phase coexistence and the corresponding phase diagram for confined fluids. Lying between theory and experiments, molecular simulation allows accurate calculations of confined fluid properties in very realistic porous models. He focuses on heterogeneous tubular pores, which constitute a good model for MCM41, one of the most widely used molecular sieves. A new simulation framework is discussed to perform calculations of important thermodynamic properties of the confined fluid, such as a thermodynamic pressure and coexistence diagram. Freezing, boiling points and critical points can all be shifted and critical exponents (if a critical point exists at all) can all be modified in an adsorbed phase but little seems to be known about the latter for matter confined in nanochannels. In one of their chapters Shin and co-workers discuss freezing in confined spaces. In two-dimensionally confined spaces such as nanopores or nanotubes, freezing and melting occur differently from the bulk phase transition. While bulk materials are crystallized mainly via crystal growth, crystallization in nanopores is dominated by nucleation and the growth of the crystal is restricted due to the imposed spatial constraint. Different crystallization mechanisms result in different crystal structures and physical properties. Under nanoscopic cylindrical confinement, the crystals are oriented to a certain favourable direction with nucleation being dominant, and the crystal orientation can shift upon the variation of dominant crystallization mechanism. The melting temperatures (Tm ) are also influenced by the reduced dimension of crystal and interfacial interaction between crystallizable components and their environment in nanopores. Zeolites are crystalline inorganic materials with defined channel structures with sizes of molecular dimensions that enable them to discriminate between molecules. Due to their thermal stability and chemical nature of their adsorption sites they have long been key materials for adsorption and diffusion studies and there are pressing challenges. For example, the discovery of large natural gas reservoirs has led to research in the area of methane conversion to longer chain molecules such as ethane/ethene or liquid hydrocarbons. The separation of olefin and paraffin gas mixtures is an important essential process step in the overall conversion and it is usually performed by cryogenic distillation, which is energy intensive. The high capital and operating costs of a cryogenic distillation unit provide the motivation for research towards the development of alternative separation methods such as adsorptive separation processes. The time taken to do adsorption experiments can be daunting due to long equilibration periods and for mixtures the task of measuring the amount of the various adsorbed components can be expensive and problematic. Thus, because of the difficulty of experiments only a very limited amount of data and fundamental understanding exist concerning the behaviour of adsorbed hydrocarbon and small molecule mixtures (e.g. air/water) in zeolites. Yet, such data and understanding are required for the optimization of adsorptive separation processes.

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Carbon nanotubes now appear to be almost ubiquitous and although it is now some time since they were discovered, the adsorption characteristics of carbon nanotubes and other structures related to the closed fullerene cages are not fully explored due to a number of difficulties. In particular, for carbon nanotubes it is difficult to experimentally distinguish effects caused by adsorption inside and outside the tube so almost no experimental adsorption isotherms exist relating solely to adsorption on the interior or exterior of the tube. Even traces of impurities or defects causing a blockage may dominate the diffusion and adsorption characteristics of a narrow bore tube. In their contribution Jiang and Babarao review adsorption as it is a key feature of many industrially important applications such as purification, separation, ion exchange and catalysis. As the number of nanoporous materials synthesized to date is extremely large, rationally choosing a high-performance material from discovery to specific application is a significant challenge. Computational approaches at the molecular scale can offer microscopic insights into adsorption behaviour from the bottom-up, complement and secure correct interpretation of experimental results, and are imperative to new material design and advanced technological innovation. They review the recent computational studies of adsorption in nanoporous materials with a wide variety of building blocks and physical topologies, ranging from zeolites, carbonaceous materials to hybrid frameworks. The theoretical interpretation of the shape of adsorption isotherms has a number of challenging features and it is precisely this shape which provides the signature tune as to the microscopic molecular behaviour. There are numerous possible shapes of isotherms with causes such as capillary condensation, structural changes, commensurate “freezing” in the assembly of adsorbed molecules and also conformational and orientational changes. A rich isotherm structure can be present with several of these effects contributing simultaneously. In his chapter Keil describes the fundamentals of molecular simulations and their application and utility in adsorption theory and engineering. After an introduction into the historical development of adsorption theory, an account is given of recent results of simulations, which in most cases lead to results of experimental accuracy, adsorption of pure components and their mixtures in zeolites, metallo-organic frameworks (MOFs) and carbon nanotubes. First, computational approaches are described, in particular the configurational-bias Monte Carlo approach, which is of exceptional importance for the simulation of long-chain hydrocarbons. Phenomena such as "squeezing out" of molecules in mixtures and "commensurate freezing" are explained. Entropic effects in adsorption are of great importance and these are discussed. It becomes quite clear that simulations give us a much deeper insight into adsorption phenomena than could be obtained from pure experimentation alone. Interplay between theory and experiment can be the most fruitful. In his contribution Muller carries on with the molecular simulation theme and reviews the available molecular simulation methods for the modelling of adsorption of gases in single-walled carbon nanotubes. Particular attention is given to the adsorption of low molecular weight gases behaving classically. A review of commonly used intermolecular potentials, methodologies and recent results is given. He

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discusses the advantages these methods have to understanding adsorption on the nano-length scale, as it provides a sensitive control over the relevant parameters. While the adsorption characteristics of fluids on nanotubes have some parallels with other carbon adsorbents, their one-dimensional nature contributes to some of the unique adsorption properties. Furthermore, the effect of interstitial and exohedral adsorption is considered. Yet, despite their predictive capacity Monte Carlo calculations do not always allow a simple interpretation of the isotherm structure. Lattice and other models can assist here. Lattice models of adsorbed fluids have been widely discussed and can successfully model the unusual adsorption features of short alkanes in zeolites. In order to have confidence in results for quasi one-dimensional channels, an accurate statistical treatment of the adsorbed phase is essential, however. In one of the two chapters we present a review of our own work on mixture adsorption in zeolites. Adsorption isotherms have been computed by Monte Carlo simulation for methane, ethane, carbon dioxide and their binary and ternary mixtures adsorbed in the zeolite silicalite. These isotherms show remarkable differences with the ethane/carbon dioxide mixtures displaying strong adsorption preference reversal at high coverage. To explain the differences in the Monte Carlo methane/carbon dioxide, ethane/carbon dioxide mixture isotherms an exact matrix calculation of the statistical mechanics of a lattice model of mixture adsorption in Zeolites has been made. The lattice model reproduces the essential features of the Monte Carlo isotherms, enabling us to understand the differing adsorption behaviour of methane/carbon dioxide and ethane/carbon dioxide mixtures in zeolites. Later, we review our work using essentially exact (in the statistical mechanical sense) matrix treatments of several lattice models of adsorption in nanochannels. Particularly, we discuss benzene and small hydrocarbon adsorption in the zeolite silicalite and in the final part we review our lattice model of adsorption of xenon in carbon nanotubes and compare this with literature studies of Monte Carlo simulation of xenon adsorption in nanotubes. Lattice models can provide effective models for adsorption and hence a rationale for the shape of adsorption isotherms for molecules in nanopores and nanochannels. Adsorption from the liquid phase or a solution presents new challenges particularly where molecular shape becomes a key feature. For a theoretical description it is also important to have a good model of the liquid phase/solution. In their contribution Denayer and Baron review adsorption from the liquid phase on nanoporous solids such as zeolites where it can be associated with a high filling degree of the material nanopores. Given the limited space in such pores, the organization and packing of the molecules in the pores becomes an important or even dominant factor. Therefore, specific selectivity effects occur in such conditions, often completely different from the selectivity patterns observed in gas phase conditions. Selected examples are given of packing-induced selectivity effects in hydrocarbon adsorption in zeolites. Polymers adsorbed in and on nanotubes show behaviour considerably influenced by the pore geometry and these important systems raise new considerations and questions. Shin and co-workers discuss polymers in nanotubes and nanochannels in

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one of their contributions. Differently from small molecules, polymers show unique properties that are dependent on their molecular size governed by the degree of polymerization. Geometric constraints on the nanoscale changes bulk behaviour. Since the size of a polymer chain generally ranges from few nanometres to few tens of nanometres in bulk, the nanoscopic geometric constraint influences the static as well as dynamic behaviour such as mobility, crystallization mechanism, phase behaviour. Furthermore, a discussion is given of cylindrical nanoconfinement effects on the polymer behaviour. Before adsorption occurs we must have diffusion and the rate of this process can be a controlling feature in any application. Measurements of this process in zeolites are well developed but to our knowledge this is not so in nanotubes. Brandani takes up some of these issues in his chapter by giving a brief historical overview of the field of the measurement of diffusion and adsorption in zeolites and focuses on macroscopic measurements relevant to diffusion throughout an entire crystal. While measurements of diffusion may appear to be straightforward, he points out that there are problems in obtaining dependable results and the future challenges highlighted. Matter adsorbed in biological channels is of great significance where the probably the first challenge which has been taken up is to elucidate the structure and properties of water molecules adsorbed therein. In his contribution Chaplin reviews and discusses water in confined spaces which is obviously highly relevant to the structure of water in biological ion channels. The structure and properties of water in all its phases continue to be actively studied but relatively little is known about the properties of water in confined spaces. The structure of bulk liquid water is dominated by its ability to form networks of directed hydrogen bonds. Although this is also true for water in confined spaces, there are additional conflicting consequences of the extensive surface and the fit within the space available. A relatively large proportion of water molecules in confined spaces occupy the interface and their interactions with the cavity surface may govern their ability to form hydrogenbonded networks with each other. The physical properties and state of the contained water may vary widely from its bulk properties and show great dependence on the molecular characteristics of the cavity surface and the degree of confinement, as well as temperature and pressure. Apparently small changes in the surfaces or the confinement dimensions may bring about substantial changes in these properties. In the final chapter Salinas and co-workers discuss the elasticity of graphene sheets. The interest here is in the longer term to see in what way adsorbed gas atoms might influence vibrational properties and looking ahead, to the effects of stretching by adsorbed material of cavities made up of graphene sheets causing the structure to swell perhaps. Carbon nanotubes might also be studied in this way. Starting from an atomistic approach they have derived a hierarchy of successively more simplified continuum elasticity descriptions for modelling the mechanical properties of suspended graphene sheets. They find that already for deflections of the order of 0.5 Å a theory that correctly accounts for nonlinearities is necessary and that for many purposes a set of coupled Duffing-type equations may be used to accurately describe the dynamics of graphene membranes. The descriptions are validated by applying them to square graphene-based resonators with clamped edges and studying

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numerically their mechanical responses. Changes induced in these responses with adsorbed molecules are of fundamental interest for future work. Clearly there are many challenges here.

1.2 Future Speculations There now seems to be considerable confidence in the ability to understand and predict adsorption behaviour of small molecules physisorbed in nanochannels and nanotubes even when the details of complex pore geometry are considered. Less is known about phase equilibria but this is developing fast and we now have some understanding of the principles whereby shape and conformational transitions can influence adsorption characteristics. The study of adsorption isotherms of larger molecules adsorbed in channels will remain an active and developing area of research activity as will adsorption from the liquid phase and supercritical adsorption. Mixture adsorption presents many fascinating problems and challenges and lies at the heart of nearly all practical applications. On the experimental side progress in the measurement of multi-component adsorption isotherms is likely to remain a challenge. However, simulation studies appear to offer much. Certainly, it should be possible to deduce the principles underlying the cooperative adsorption or displacement of the components of mixtures in nanotubes and nanochannels. Treating the mixture as an ideal adsorbed solution may be insufficient for the task. The structure of adsorbed water in channels is being elucidated but faces the same questions as that of water structure itself. Little is known about adsorption of solutions of ionic mixtures as well as transport properties in nanochannels. Adsoption of polymers in tubes has many potential applications and is likely to be a fruitful area for fundamental studies. One major beneficiary of future research in some of the areas described in this book in the long term may be the petrochemical industry which would be able to predict more accurately the behaviour of hydrocarbon mixtures in nanoporous materials and hence aid their separation. We all indirectly benefit from this. Although the impact on global warming in the short term is less obvious, much of the work described above and being undertaken lays the foundation for such an impact in the future with long-term implications for the current energy crisis. Much more efficient use of resources and removal of pollutants from the environment are possibilities. Similarly, the separation of CO2 from mixtures by adsorption into materials containing nanochannels may have an important role to play in combating global warming. As indicated earlier for carbon nanotubes it is difficult to experimentally distinguish effects caused by adsorption inside and outside the tube. A similar question must be applicable to the adsorption properties of common soots and amorphous carbons, many of which contain nanopores and nanochannels, albeit defective ones. It is possible to imagine many structures related to graphenes and the fullerenes whose adsorptive properties we would like to know.

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Metals adsorbed in channels is another area of possible development with implications for nanoscale electronics. The filling of nanotubes with other electronically functional materials holds out exciting prospects, for example, lowdimensional nanowires and supeconductors perhaps. Other areas are medicine, biology and polymer science. In the longer term adsorption in nanopores and channels might be used for drug delivery systems with a lot of research carried out on this area already and in the highly selective separation of pharmaceuticals from reaction mixtures. Nature makes good use of channels in biological membranes and ion pumps so we may expect new and exciting physics and chemistry to emerge with our understanding of adsorption in such structures.

Chapter 2

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes F.J. Keil

Abstract Molecular simulations have clearly demonstrated their usefulness in adsorption theory and engineering. After a short introduction into the historical development of adsorption theory, recent results of simulations on adsorption of pure components and their mixtures in zeolites, MOFs, and carbon nanotubes are discussed. First, computational approaches are described, in particular the configurational-bias Monte Carlo approach, which turned out to be of outstanding importance for the simulation of long-chained hydrocarbons in confined media. Phenomena like “squeezing out” of molecules in mixtures and “commensurate freezing” are described. Various entropic effects are explicated. It is demonstrated that molecular simulations in most cases lead to results of experimental accuracy. In particular, simulations give far deeper insights into the observed phenomena than could be obtained from experiments only.

2.1 Introduction In the year 1881 H. Kayser introduced the term “adsorption” into the literature. Adsorption deals with the process in which molecules accumulate in an interfacial layer. The material in the adsorbed state is defined as the “adsorbate” and that in the bulk phase the “adsorptive.” Adsorption may occur at liquid/solid and gas/solid interfaces. Adsorption can result from van der Waals forces (physisorption) or from far stronger chemical interactions (chemisorption). Physical adsorption is very effective, in particular at temperatures close to the critical temperature of a given gas. Chemisorption usually occurs at much higher temperatures than the critical temperature and is a specific process which can occur only on some surfaces in contact with certain gases. This process is important in catalysis. Adsorption leads to a decrease in entropy and free energy of the adsorbed fluids compared to the bulk F.J. Keil (B) Hamburg University of Technology, Chemical Reaction Engineering, 21073 Hamburg, Germany e-mail: [email protected] L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_2,  C Springer Science+Business Media B.V. 2010

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phase. Therefore, adsorption is an exothermic process. Of outstanding importance is the adsorption inside porous media. In microporous materials all atoms of the adsorbent can interact with the adsorbates. Particularly, this is the case for zeolites and carbon nanotubes which mostly have diameters comparable to the molecular size. A very important conception in adsorption science is the “adsorption isotherm” which is the equilibrium relation between the amount of adsorbed material and the pressure or concentration in the bulk phase. Practical applications of adsorption processes are based on the selective uptake of individual components from their mixtures with other substances, a phenomenon which was discovered in the year 1903 by the Russian scientist Tswett. This separation property is exploited in technical applications like Pressure Swing Adsorption [1], purifying and drying of gases, purifying of liquids, chromatography [2], for obtaining a high vacuum, membrane separation, ion-exchange processes, and, for example, chiral separations of pharmaceuticals. Industrial applications of adsorption are described in many books (e.g., [3–10]). To understand the design and operation of adsorption processes, knowledge of the underlying physics of adsorption is required. Based on general thermodynamic considerations, Gibbs [11], more than hundred years ago, derived the socalled adsorption excess concept where the real interface is assumed to be an imaginary, geometrical plane, which is called the Gibbs Dividing Surface (GDS). Gibbs obtained an isotherm equation which can be combined with various equations of state resulting in adsorption isotherms, for example, Hill–de Boer isotherm, Harking–Jura isotherm, Langmuir or Fowler–Guggenheim isotherm. The Gibbs approach does not require any distinct model of the surface phase. Over the years 1914–1918 Langmuir, Eucken, and Polanyi started to develop a sound theoretical basis of adsorption processes. In particular, Langmuir introduced a clear concept of the monomolecular adsorption and derived his now famous Langmuir adsorption isotherm of which its constant parameters have a well-defined physical meaning. The Langmuir isotherm may be derived from kinetics, phenomenological, and statistical thermodynamics. Langmuir tried to extend his equation to energetically heterogeneous surfaces and multilayer adsorption, which was also described in the 1930s by Brunauer, Emmett, and Teller. Their original BET isotherm was extended the finite n-layer isotherm. Other well-known multilayer isotherms were developed by Frenkel, Halsey, and Hill (FHH) and Redhead. Based on the potential theory of adsorption by Eucken and Polanyi, Dubinin and Radushkevich (DR) published in the year 1947 the adsorption theory of the volume filling of micropores (TVFM) which has been used quite often for the description of adsorption on carbons. It could be shown that the constants of the DR adsorption isotherm are related to the porous structure of the sorbent. Over the last few decades the energetic heterogeneity, lateral interactions, adsorption kinetics, surface diffusion and surface phase mobility were investigated in more detail, based on fundamental theories like statistical thermodynamics. Adsorption in porous media and diffusion are closely connected, in particular, if one wants to describe the dynamics of adsorption or catalytic phenomena. A detailed presentation of the historical developments of adsorption theories and citations of the above-mentioned isotherms are given by Dabrowski

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[12]. Reviews of the theoretical developments are given in many books [13–24] and reviews (e.g., [25–27]). Experimental methods were compiled by Keller and Staudt [28]. The classical models of adsorption processes which are in use in chemical engineering, like Langmuir, BET, or Dubinin–Radushkevich, are based on several uncontrollable approximations. They do not start from a well-defined atomic or molecular level. Most approaches relying on statistical thermodynamics, like density functional theory by Evans [29] and Tarazona et al. [30], are difficult to apply. Therefore, molecular modeling is of increasing importance as many approximations introduced into applications of statistical mechanics can be avoided. Far more complicated systems can be investigated than with analytical description. Of course, there are also difficulties with molecular modeling. One needs to know the surface structures of the solid adsorbents on an atomic level, and accurate intermolecular potentials have to be available. A problem is the non-availability of accurate multicomponent measurements for fine tuning the potentials, or quite often even for pure components or solids. In the present review, molecular modeling results on adsorption in zeolites and carbon nanotubes (CNTs) will be presented. These materials have a well-defined porous structure.

2.2 Computational Approaches Adsorption isotherms are equilibrium properties which represent the number of particles adsorbed on a surface at a given temperature T, volume V, and chemical potential μ. In experiments the adsorbate is in contact with a bulk gas reservoir. In equilibrium the following conditions hold: μbulk = μads , Tbulk = Tads

(2.1)

The volume V, the chemical potential μ, and the temperature T, are fixed. In principle one could do a Molecular Dynamics simulation in the microcanonical ensemble for this system. But this would take by far too much computing time. Therefore, one employs a Metropolis Monte Carlo method in the grand canonical (μVT) ensemble. This approach was developed by Norman and Filinov [31]. One gives the chemical potential and temperature inside the pores, whereby the number of particles inside the pores fluctuates during simulation. Via an equation of state one can calculate the pressure corresponding to the given chemical potential. The gas phase, which is not explicitly used in the calculations, is considered as a reservoir. In a GCMC simulation, three types of moves are performed. The first one is a displacement of particles, where a particle is selected at random and given a new position. In case of molecules, rotations and/or intermolecular movements have also to be executed. The second type of trial move is the insertion of a particle at a randomly chosen position with a randomly chosen orientation; alternatively, a randomly selected particle is removed. For mixtures a fourth type of trial move can

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be used, namely swapping a particle which consists of changing the identities of the adsorbed positions or orientations. Each of the above-mentioned trial moves has its particular acceptance rule. The random moves are usually handled by the standard Metropolis scheme. Details of the grand canonical Monte Carlo (GCMC) algorithm are, for example, given by Frenkel and Smit [32], Landau and Binder [33], Sadus [34], and Nicholson and Parsonage [35]. Although the GCMC approach is very useful for atoms and simple molecules one runs into problems for complex molecules like long alkanes/alkenes, aromatic compounds, and molecules with side chains. For these molecules the acceptance rate to the insertion/removal steps drops considerably. To overcome this problem, bias techniques have been introduced. One of these is the configurational-bias Monte Carlo (CBMC) technique which makes the insertion of long-chain molecules in moderately dense fluids possible [32, 36–41]. The CBMC approach is based upon the Rosenbluth sampling scheme [42] combined with inserting the molecule chain, bead by bead, into the voids of the porous material that means the molecule is preferentially directed toward acceptable structures. The effects of these biases are then removed by modifying the acceptance rules. Initially, a fixed number of trial positions for the atom to be inserted is generated. The number of trial directions influences the accuracy of the results and the efficiency with which the results can be obtained [32]. Then one of these positions is selected according to the energy contributions from the external degrees of freedom of the molecule. The Rosenbluth weight of the entire chain is calculated and used in the acceptance rule for insertions. The CBMC method may be summed up by the following steps. In the first step a trial configuration is generated and its Rosenbluth weight determined. This involves selecting a chain and a starting position randomly, growing the chain and calculating the Rosenbluth weight for the trial chain. The starting point can occur anywhere on the chain. The number of attempted insertions is specified and the chain is grown according to a given algorithm [32]. Based on this algorithm, in a second step the Rosenbluth weight is calculated for the old configuration to retrace the old chain. In the third step the old and new Rosenbluth weights are used to either accept or reject the trial configuration. The CBMC has been extended to branched molecules [43–45] and cyclic molecules [46–48]. A modification of the CBMC scheme for all atoms models has been published by Chen et al. [49] and Macedonia et al. [50]. Jakobtorweihen et al. [51] have derived acceptance rules of any Monte Carlo trials (regrowth, insertion/deletion, identity change, and transfer between phases (Gibbs ensemble)) which are carried out in combination with the CBMC approach. It was shown that any of these Monte Carlo trials can be considered as pseudoreactions. The isosteric heat of adsorption is also a quantity of interest that can be computed from GCMC simulations:   NUads  − N Uads  ∂Uads = RT − (2.2) qst = RT −   ∂N V,T N 2 N2 where U ads is the potential energy of the adsorbed phase. In the equation above it is assumed that the gas phase is ideal. There is some confusion about the definition

2

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

13

of the isosteric heat of adsorption in the literature [26]. The heat of adsorption can be related to the temperature dependence of the Henry coefficient K H : ∂KH = qst ; β = 1/(kB · T) ∂β

(2.3)

At low pressure, the number of adsorbed molecules is proportional to the pressure P: ρa = KH P

(2.4)

The Henry coefficient can, therefore, be obtained from the low-pressure part of the adsorption isotherm. It is related to the excess chemical potential of the adsorbed molecules: KH = β exp (− βμex ) ; β = 1/(kB · T)

(2.5)

Henry coefficients are normally computed in the NVT ensemble. Molecular simulation of adsorption in microporous adsorbents is performed on a sample of solid, which for a zeolite are several unit cells with periodic boundary conditions. Adsorption on the external surface of the solid is usually ignored. The number of adsorbed molecules measured, Nm , is then related to the absolute adsorption, Na , where Na is the total number of molecules contained in the pores [52] N m = N a − Vp ρb ; ρb = P/(kB T) (ideal gas)

(2.6)

where Vp is the specific pore volume (cm3 /g) of the adsorbent, and ρ b is the density of the equilibrium gas phase. The precision of the simulation results is essentially determined by the interand intramolecular potentials [53]. This is obvious from statistical thermodynamics relations. In most simulations a rigid zeolite potential is introduced which uses a Lennard-Jones dispersion term for the interaction of oxygen atoms of the zeolite framework and the atoms of the adsorbing gases. In the case where the adsorbate molecules are polar, an electrostatic term is added, which acts between all atoms of the zeolite and of the adsorbate. Many zeolite structures are published in a book [54] and are also available on the Internet [55]. Potentials for flexible zeolites have been reviewed by Demontis and Suffritti [56]. For hydrocarbons quite often united atom potentials have been used [57], where the alkane groups CH3 , CH2 , and CH4 are considered as single interaction centers which reduce the computing time considerably compared to an all-atom potential. For the adsorbate–adsorbate interactions one has to distinguish inter- and intramolecular interactions. Various force fields for the intramolecular interactions are available. The precise parameterization of these force fields is possible based on spectroscopic data or accurate quantum chemical calculations. In any case the parameterization of the force fields has to be performed such that they describe the simulated properties over a wide range of temperatures and pressures. Zeolite–adsorbate interaction parameters are mostly

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obtained from experimental data. Smit [58] has demonstrated that fitting parameters to diffusion data may result in different parameters than from fitting to adsorption curves or Henry coefficients. Dubbeldam et al. [59] observed that steps and kinks in the adsorption isotherms are directly related to the structure of the zeolite and are very sensitive to the model parameters. A necessary and sufficient procedure for finding a consistent force field is to utilize isotherms that exhibit inflection points and to use these inflection points as calibration points for parameter optimization. The paper [59] highlights three common sources for discrepancies between experimental data sets: (1) a lack of low-pressure data, (2) a lack of high-pressure data, and (3) too short experimental equilibration times. The inflection point approach has been shown to be useful for alkenes too [60]. Pascual et al. [61] have developed a methodology to determine a transferable force field for adsorption in zeolites. They use Lorentz–Berthelot combining rules for the determination of the cross potential parameters and the anisotropic–united atom model (AUA) for hydrocarbon interactions. Pelleq and Nicholson also have developed a transferable potential model [62]. Force fields for alkenes have been developed by Pascual et al. [61], Jakobtorweihen et al. [60], and Liu et al. [63], among others, and Mallot et al. [64] introduced force fields for fluorocarbon-type molecules in zeolites. Replacing silicon by aluminum in zeolites leads to a negative charge in the zeolite framework. This charge has to be compensated by either protons or larger cations. The type and location of cations noticeably influences the adsorption properties. The location of the cations inside the zeolite is closely related to the distribution of Al over the T-sites. Calero et al. [65] have developed and validated a united atom force field for alkanes in the sodium form of FAU-type zeolites that explicitly distinguishes Si and Al atoms through the different type of framework oxygen atoms and that accounts for the density, mobility, and interactions with the adsorbate of the nonframework cation. The force field reproduces accurately the sodium positions in dehydrated FAU-type zeolites known from crystallography and the available experimental adsorption properties of n-alkanes in faujasites over a wide range of sodium cation densities, temperatures, and pressures. The force field also reproduces the adsorption of binary mixtures at high pressure. Further force fields for zeolite–cation interactions have been developed by Jaramillo and Auerbach [66], Macedonia et al. [67], Calero et al. [68], and Garcia-Perez et al. [69], among others. Many-body bond-order potential energy functions for carbon–carbon interactions have been developed by Brenner [70, 71]. These potentials have been used to model the energetics and dynamics of the covalently bonded carbon atoms in graphene sheets, nanotubes, and C60 molecules. In addition to these potentials, twobody potential energy functions describing bonded interactions within carbon nanotubes have also been proposed, for example, by Tuzun et al. [72]. A potential energy function by Walther et al. [73] has been used too. Potential energy functions for hydrogen in single-wall carbon (SWCNT) nanotubes have been developed, for example, by Wang et al. [74] and Silvera and Goldman [75, 76]. The effect of curvature of the nanotube on the force field can be taken into account [77]. The H2 –H2 interaction is very weak. Diep and Johnson [78] developed a very accurate H2 –H2 interaction potential by using quantum chemical calculations and a four-term spherical harmonics expansion. Parameters for a Lennard-Jones H2 –H2 potential are

2

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

15

presented in the paper [74]. The interaction of a hydrogen molecule with a carbon atom in a SWCNT can be presented by the Crowell–Brown potential [79]. Potential energy functions are also available for interactions of rare gases [80–82] and nitrogen [83] with SWCNTs, among others. The formulas for these potentials may be found in a book by Rafii-Tabar [84]. Also the classic book by Steele [22] should be accounted for. A few force fields based on quantum chemical ab initio methods have been published. For example, Tielens et al. [85] studied fully ab initio optimized geometries and ab initio interaction energies of butene isomers (1-butene, cis-2-butene, trans2-butene, and iso-butene) in NaY zeolites Bussai et al. [86] investigated the concentration dependence of the CH4 structure in Silicalite by Moller-Plesset perturbation theory (MP2). Klontzas et al. [87] explored the interaction of molecular hydrogen with IRMOF-1 by means of density functional theory (DFT) and Møller–Plesset perturbation theory (MP2). These authors found that basis set superposition error (BSSE) must always be taken into account and cannot be estimated. Counterpoise corrections can reach up to 50% of the binding energy.

2.3 Adsorption of Pure Components in Zeolites Simulation results up to the year 2000 on adsorption in zeolites have been reviewed by Fuchs and Cheetham [88] and Smit [89]. Smit and Maesen [90] have published a review on various aspects of zeolite simulation. Most simulations have been executed for rigid zeolite lattices. Preferentially linear alkanes and rare gases in silicalite were taken as examples for modeling, as for these molecules many experimental data are available. The CBMC approach in the grand canonical ensemble makes the simulation of long-chain alkanes in zeolites feasible. A high sensitivity of the adsorption data on small changes in the guest–host potential was found for the adsorption of methane in neutral AlPO4 -5 (an open framework zeolite with cylindrical parallel pores). The adsorption isotherm shows a step at low temperatures, instead of a smooth Langmuir isotherm. This step is caused by a phase transition of the confined methane. At first, Boutin et al. [91] found a Langmuir isotherm using a Kiselev potential. After a publication of Maris et al. [92], Boutin et al. [93] reduced their potential parameter (σ zeo–H ) by 5% which then also gave an isotherm with a step. The far more elaborate potential by Pellenq and Nicholson [94, 95] results also in an isotherm with a step for this case [96]. The Kiselev-type potential has been extended to describe the intrusion/extrusion cycle of a nonwetting fluid in a hydrophobic solid [97]. The condensation of water in silicalite-1 at room temperature could be reproduced. The TIP4P water model was employed. The linear alkanes show no steps in their adsorption isotherms, except in a few cases in certain zeolites. In Fig. 2.1 the isotherms of n-heptane in MFI are presented for four temperatures [59] together with experimental results. The inflection behavior of n-heptane is well established [101–103] also for iso-butane [98]. Smit and Maesen [104] explained this phenomenon in terms of commensurate freezing that means n-heptane has a size commensurate with the zigzag channel of MFI. At high

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F.J. Keil

1.2

Loading q/[mol/kg]

1 0.8

303K 323K 347K 372K Sun 303K Sun 323K Sun 343K Eder 347K Eder 372K

0.6 0.4 0.2 0

10–4 10–3 10–2 10–1 100 101 102 103 104 Fugacity of the bulk fluid phase f/[Pa]

105

Fig. 2.1 Isotherms of n-heptane in MFI at various temperatures [59], experimental data are taken from Sun et al. [98, 99] and Eder et al. [100]

pressures, the molecules shift from a random distribution to a distribution where the molecules are localized exclusively in the channels and not at the intersections, leaving the straight channels free for further filling of the pores which leads to a step in the isotherm. Inflection in the isotherm has been observed also for n-hexane and less pronounced for ethane. Various branched molecules show inflections, for example, 2-methylpropane adsorbs preferentially at the intersections. At a loading of four molecules per unit cell, the intersections are fully occupied, and additional molecules must be pushed in the channels requiring a significantly higher driving force. Di-branched alkanes also are preferentially located at the intersections. As these molecules are bulky, they cannot be pushed into the channels, which limits their loading to four molecules per unit cell in MFI [105]. For alkenes like 1-heptene inflection behavior was observed [60, 61] (see Fig. 2.2). The other molecules show a common Langmuir-type isotherm. 1-Octene reaches a maximum loading of 0.7 mol/kg, which corresponds to four molecules per unit cell, whereas the other isotherms in Fig. 2.2 reach a maximum loading which corresponds to approximately eight molecules per unit cell. 1-Octene does not fit completely into the zigzag channels, and therefore, blocks two adsorption sites. The inflection behavior of the 1-heptene isotherm occurs at a loading of four molecules per unit cell. The length of the 1-heptene molecule corresponds to the length of the zigzag channels. The term “siting” or “location” has a different meaning for different zeolites. In the case of a zeolite with only one type of channel (e.g., a TON-type zeolite), it refers to the location and conformation of the molecules with respect to the channels. In a zeolite with more than one type of a channel (e.g., silicalite-1), it means not only the location of the molecule in a channel, but also the distribution over the different types of channels [106]. The siting of alkanes and alkenes is not treated uniformly in the literature.

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

Fig. 2.2 Simulated adsorption isotherms of 1-pentene, 1-hexene, 1-heptene, and 1-octene in silicalite-1 at 300 K [60]

1.6 amount adsorbed/(mol kg–1)

2

1.4 1.2

17

1-pentene 1-hexene 1-heptene 1-octene

1.0 0.8 0.6 0.4 0.2 0.0 10–10 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 100 101 pressure/bar

Figure 2.3 shows the distribution of 1-heptene molecules over the different channels in silicalite-1 [60] around the intermediate plateau (see Fig. 2.2) of the adsorption isotherm. The meaning of the symbols in the legend in Fig. 2.3: is (intersection), sc (straight channel), and zz (zigzag channel). In Fig. 2.3 the entry “is-sc com sc” in the legend, for example, means that one terminal group is located inside an intersection while the other is located in a straight channel and that the center of mass (com) is also located in a straight channel. In addition the end-to-end distance with respect to the location is calculated (open symbols in Fig. 2.3). For the sake of clarity, only the locations with the greatest contributions to the total loading are shown. It is demonstrated that, up to a loading of four molecules per unit cell, nearly all 7.4 7.2

6

7.0 5

6.8

4

6.6

3

6.4

2

6.2

1

6.0

0 10–7

10–6

10–5 pressure/bar

10–4

end-to-end distance/Å

molecules per unit cell

7

zz-zz com zz is-sc com is is-sc com sc is-is com sc is-zz com zz isotherm zz-zz com zz is-sc com is is-sc com sc is-is com sc is-zz com zz

5.8

Fig. 2.3 Distribution of 1-heptene molecules and their end-to-end distance inside silicalite-1 channels at 300 K. The filled symbols represent the distributions of the different locations of the molecules inside the channels to the total loading. The open symbols represent the corresponding end-to-end distances of the molecules. The notation is explained in the text [60]

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molecules in the zigzag channels have one end in an intersection, and so block this adsorption site. The distance between the terminal groups of the molecules located in the zigzag channels is larger than the zigzag channel length. The straight channels are distinctly too short for the 1-heptene molecules. In order to insert more molecules into the zigzag channels, a reduction of the end-to-end distance is necessary. The molecules have to be coiled, which requires a higher pressure (or chemical potential), to compensate for the energetically unfavorable configurations. Increasing the pressure leads to a reduction of the end-to-end distance of the 1-heptene molecules to 6.61 Å, which is slightly smaller than the zigzag channel length, such that the molecules fit into these channels. As a consequence the straight channels can be filled. In conclusion, in the inflection region of the 1-heptene isotherm the structure of the molecules is changed so that they can be packed more efficiently. The phenomenon of a two-step adsorption was also observed for the zeolite MEL [107], which has intersecting straight channels. Dubbeldam et al. [59] and Krishna et al. [105] demonstrated that for zeolites without intersections (e.g., AFI, TON) linear and branched hydrocarbons show a similar adsorption behavior. C4 –C9 linear and branched alkanes in silicalite were investigated by Vlugt et al. [57]. For branched alkanes inflection behavior was observed for all carbon numbers studied. Linear alkanes with six and more carbon atoms also were found to exhibit inflection behavior. Hexane and heptane show inflection due to “incommensurate freezing.” Available experimental data from the literature confirm the accuracy of the CBMC simulations. This holds also for the temperature dependence of the isotherms. For purposes of fitting the CBMC simulated or measured isotherms the dual-site Langmuir model (see, for example, [24]) has been found to provide an excellent description. Van Well et al. [108] calculated adsorption isotherms and the isosteric heats of adsorption of propane, n-butane, n-pentane, and n-hexane in ferrierite FER. The results are in good agreement with experimental results. It was found that short alkanes up to C5 can access the entire two-dimensional FER pore structure, longer molecules adsorb only in the 10-ring channels and not in the 8-ring cage. For this example it was shown that the adsorption isotherms are very sensitive to the potential parameters. Liu et al. [109] detected that pure silica FER zeolites and H-FER zeolites at 333 K exhibit different adsorption behavior, which also depends on the location of the Al-atom in the zeolite framework. For a certain Al-position a very good agreement with experiments could be obtained by using the united atom force field of Dubbeldam et al. [59]. Skoulidas et al. [110] simulated adsorption isotherms of CH4 , CF4 , He, Ne, Ar, Xe, and SF6 in silicalite at room temperature. Their results are in good agreement with measurements. Mellot and Cheetham [64] performed simulations on the adsorption of a series of fluoro-, chlorofluoro-, and hydrofluorocarbons (CF4 , CF3 Cl, CF2 Cl2 , CFCL3 , CHF3 ) in siliceous Y and NaY zeolites. The authors obtained a very good correlation between experiment and simulation. The Henry coefficients increase monotonically with the chain length of alkanes. If the chain length fits exactly into the cage, the maximum Henry coefficient will be obtained. As Dubbeldam et al. [111, 112] and Krishna et al. [113] have demonstrated, a further lengthening of the carbon chains leads to a decrease in the Henry

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Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

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coefficients, as the chains have to curl and/or put part of the chain through the window, which is energetically unfavorable. For the tube-like zeolite OFF a monotonically increasing Henry coefficient was observed, in contrast to zeolites with cage structures (CHA, ERI, AFX, RHO, KFI) which exhibit a non-monotonic behavior. Krishna et al. [113] found that in cage-like zeolites adsorption goes in steps with plateaus corresponding to an integer number of molecules per cage. But tube-like zeolites with very corrugated walls may also show cage-like adsorption behavior. This has been demonstrated for AFI at low temperatures [92, 96]. The difficulties in predicting adsorption quantities for tight-fitting molecules were demonstrated by Clark and Snurr [114]. Slight changes in the zeolite structure can bring about large changes in the adsorption behavior. In this case the authors argued a flexible lattice model should give reliable results. But in most cases a rigid lattice should be sufficient, as was confirmed by the very many simulations mentioned above. Vlugt and Schenk [115] found out that only at very high loading an influence of the lattice flexibility on the adsorption isotherms is detectable. This is in coincidence with the result that sometimes at very high pressures small deviations between experiments and simulations were found. Ravikovitch et al. [116] have simulated nitrogen and toluene adsorption isotherms on ordered MCM-41 and plugged hexagonal templated silica (PHTS). The authors developed a molecular model of toluene adsorption in silica nanopores that accounts for surface heterogeneity. The model parameters were fitted to reproduce the experimental isotherm on a reference MCM-41. The model was then used to predict the hysteretic adsorption isotherm in a wider PHTS material. A good agreement with experiment was found. Maurin et al. [117] coupled molecular simulations with adsorption microcalorimetry measurements in order to understand more deeply the interactions between carbon dioxide and various types of faujasite surfaces. Their newly derived interatomic potentials for describing the interactions within the whole system, provide isotherms and evolutions of the differential enthalpy of adsorption as a function of coverage for DAY, NaY, and NaLSX which are in a very accordance with experimental results. The force field was derived from ab initio calculations for representing the interactions between carbon dioxide and zeolite adsorbents. It was transferable and allowed accurate reproduction of the microcalorimetry data. A microscopic mechanism for CO2 adsorption was proposed for each of the three zeolites, which is consistent with the evolution of the differential enthalpy of adsorption as a function of the coverage. Vitillo et al. [118] have simulated the progressive filling of 12 purely siliceous models of common zeolite frameworks with hydrogen in order to estimate the maximal amount of molecular hydrogen which could be placed into the pore system of these zeolites, regardless of the mechanism of adsorption or encapsulation. The calculated capacities have been intended as limiting values which could be achieved either in extreme physical conditions (low temperature and high pressure) or in the presence of ideal strong adsorption sites (negligible volume, high binding energy). The zeolite capacities are intrinsically limited by geometrical constraints to be at best 2.86 mass %, obtained for RAU and RHO zeolites. The H2 –H2 interaction potentials were fitted to ab initio (MP4/ang-cc-pVQZ) data.

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Over the last few years, various investigations of the effect of cations in zeolites have been published. Van Well et al. [119] simulated the adsorption properties of linear alkanes in H-ZK-5 and K-ZK-5 at room temperature. Alkane molecules up to n-heptane are able to adsorb in the α- and γ-cages of ZK-5, but adsorption of longer alkane molecules is limited to the large α-cages as a result of size exclusion from the small γ-cages. These γ-cages are the preferred adsorption sites for propane and n-butane because of the more favorable heats of adsorption. In contrast, n-hexane and n-heptane adsorb preferentially in the large α-cages because of the larger entropy in these cages. For n-pentane, the γ-cages are the preferred adsorption sites in H-ZK-5, whereas in K-ZK-5 the α-cages are the preferred adsorption sites. This difference is caused by the presence of the potassium cations in the γ-cages of K-ZK-5 which constraints the adsorption in these cages. Lachet et al. [120] investigated the adsorption and coadsorption of m-xylene and p-xylene in the faujasites NaY and KY. Results of the single component and binary mixture studies showed, in agreement with the experiments, that the exchange of Na+ ions by K+ ions had almost no effect on the amount of adsorbed p-xylene but leads to a significant decrease of the m-xylene adsorption capacity and a reversal of the adsorption selectivity. A detailed analysis of the adsorption sites has revealed extremely different behaviors of the two zeolites studied toward xylene molecules. The NaY zeolite can accommodate up to four molecules per supercage. Whatever the loading, their simulations have shown that the xylene/NaY interactions are stronger for m-xylene than for p-xylene. Beerdsen et al. [121] performed simulations for the adsorption of methane, ethane, and propane in MFI- and MOR-type zeolites with various nonframework sodium and framework aluminum densities. The position and density of the sodium and aluminum atoms in the zeolite have a large influence on the adsorption of alkanes. An increase in the nonframework sodium density increases the C1 –C4 loading in MOR-type zeolites, but slightly decreases the loading in MFI-type zeolites. Furthermore, increasing the sodium density does not affect the selectivity of MOR-type zeolites for adsorbing n-butane from n-butane/isobutane mixtures, but it markedly increases the selectivity of MFI-type zeolites. The MOR-type channels are large enough for nonframework sodium cations to provide favorable adsorption sites for the adsorbing alkanes. The MFI channels are too small to accommodate both a sodium cation and an alkane, and therefore, nonframework sodium cations only block adsorption sites, preferably at the intersections. In the absence of accessible intersections, the selectivity of MFI-type zeolites for adsorbing linear alkanes increases, but its adsorption capacity decreases. Further papers on the influence of cations on the adsorption of alkanes and alkenes were, for example, published by Macedonia et al. [122], Tielens et al. [85], and Calero et al. [123]. Tielens et al. [85] investigated, by means of ab initio methods, the adsorption of butene isomers in a FAU-type zeolites. They obtained interaction energies by quantum chemical calculations for four different butene isomers (iso-butene, 1-butene, cis-2-butene, and trans-2-butene) with different cations in the gas phase and with a zeolite cluster containing Na+ as compensating cations. The calculated interaction energies for a

2

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

21

Na+ zeolite cluster are close to the experimental ones with a magnitude of deviation of only 2 kcal/mol. However, experimentally determined Henry constants decrease in the order: iso-butene > cis-2-butene > 1-butene > trans-2-butene, while the calculations predicted the highest interaction energy for 1-butene. Calero et al. [123] have developed a force field able to accurately describe the adsorption properties of linear alkanes in the sodium form of FAU-type zeolites. The force field reproduces the sodium positions in dehydrated FAU-type zeolites known from crystallography, and it predicts how the sodium cations redistribute when n-alkanes adsorb. Fleys and Thompson [124] calculated the adsorption isotherms of water in dealuminated zeolite Y(DAY) using the Compass force field. Supercritical CO2 adsorption on NaA and NaX have been studied by Liu and Yang [125] by means of NPT Gibbs ensemble Monte Carlo simulations. The adsorption isotherms were in good agreement with the experimental data. The cation redistribution upon the adsorption of CO2 in NaY and NaX has been simulated by Plant et al. [126]. The authors employed molecular dynamics to investigate cation motions as a consequence of CO2 adsorption. A similar investigation was performed for the cation migration upon methanol adsorption in NaY and NaX faujasite zeolites [127]. Yazaydin and Thompson [128] have simulated, by means of Monte Carlo and molecular dynamics, the adsorption of methyl t-butyl ether (MTBE) in silicalite, mordenite, and zeolite beta with different Na+ cation loadings. Although the three zeolites have similar pore volumes, zeolite beta, with its pore structure which is mostly accessible to MTBE molecules, is predicted to adsorb significantly more MTBE than silicalite and mordenite. The Na+ cation loading up to four cations does not have a significant effect on the adsorption capacity of the zeolites studied. However, for silicalite and zeolite beta increasing the Na+ content increases the amount adsorbed at very low pressures. On the other hand, the location of Na+ cations in mordenite precludes an effect, since the cations are in pores which are not accessible to MTBE molecules. Sebastian et al. [129] investigated the adsorption of N2 , O2 , and Ar in Mn(II)-exchanged zeolites A and X. They have found a good agreement of GCMC simulations and experiments. Although not a subject of this review, a few remarks should be made on metalorganic frameworks (MOFs). These materials may lead to advances in adsorption separations because the properties of these materials may be tailored in a synthetically predictable manner. For example, Sarkisov et al. [130], Düren and Snurr [131], Frost et al. [132], Düren et al. [133], Keskin and Sholl [134], Babarao et al. [135], Yang and Zhong [136], and Klontzas et al. [87] have published simulated adsorption isotherms of various MOFs. Rowsell and Yaghi [137] have examined H2 adsorption isotherms of eight MOFs for correlations with their structural features. All the MOFs displayed approximately Langmuir (Type I) isotherms with no hysteresis. The isotherms were measured at 77 K up to a pressure of 1 atm. Saturation was not reached for any of these materials under these conditions. HKUST-1, MOF-74, and IRMOF-1 adsorption isotherms at 77 K are presented in Fig. 2.4. HKUST-1 shows a high uptake capacity compared to zeolites (up to 7.2 mg/g) or activated carbons (up to 21.4 mg/g).

22

F.J. Keil 28

IRMOF-1 MOF-74 HKUST-1

24

Uptake (mg/g)

20 16 12 8 4 0

0

200

400 Pressure (Torr)

600

800

Fig. 2.4 Hydrogen adsorption isotherms of HKUST-1, MOF-74, IRMOF-1 at 77 K [137]

2.4 Adsorption of Mixtures in Zeolites Although most applications of adsorption involve mixtures, there are only a small number of measurements on such systems available. In Table 2.1 some citations from the literature are presented. The reason for this situation is the tremendous expenditure of time for such measurements [28, 138]. Therefore, simulation could give useful contributions to this field of research and industrial applications, for example, in the design of separation devices or in catalysis. Vlugt et al. [45, 105] detected some remarkable features by investigating isotherms of 50/50 binary mixtures of C5 , C6 , and C7 hydrocarbons in silicalite. The loading of the branched isomer in all three binary mixtures reaches a maximum when the total mixture loading corresponds to four molecules per unit cell. Higher loadings are obtained by “squeezing out” of the branched alkane from the silicalite and replacing these with the linear alkane. This “squeezing out” effect is found to be entropic in nature; the linear alkanes have a higher packing efficiency and higher loadings are more easily achieved by replacing the branched alkanes with the linear alkanes. Three types of entropy effects can be distinguished [105]. Size entropy effects favor the component with the smaller number of carbon atoms because the smaller molecule finds it easier to fill in the gaps within the zeolite matrix at high molecular loadings. Configurational entropy effects come into play for mixtures of alkanes that differ in the degree of branching. For a mixture of linear and branched alkanes with the same number of carbon atoms configurational entropy effects favor the linear isomer because such molecules pack more efficiently within the intersecting channel structures of, for example, zeolites. Length entropy effects come into force for sorption of linear and branched alkanes inside cylindrical channels of zeolites (e.g., AIF, MOR). Here, the double-branched alkane has the shortest length and can be packed more efficiently within the channels. An example of squeezing out is given in Fig. 2.5, where the component loadings of C3 and C4 for a 50/50 bulk

2

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

23

Table 2.1 Measurements of mixture isotherms Mixtures investigated

Experimental method

Zeolites

Citation

C1 /C2 , C1 /C3 , C1 /C4 , C2 /C3 , C1 /C2 /C3 , C1 /C2 /C4 , C1 /C2 /C4 , C1 /C2 /C3 /C4

V

MFI LTA FAU

[139]

CH4 /C2 H6 , CO2 /C2 H6

CV

FAU MFI

[140]

C7 /C10 , C7 /C14 , C7 /C11 , C10 /C11 , C10 /C14 , C11 /C14

G

LTA

[141]

C10 /H2 , C11 /H2 , binary mixtures of C6 –C10

PC

FAU

[142]

CO2 /C3 H8 , H2 S/C3 H8

V

MOR

[143]

CO2 /H2 S, CO2 /C3 H8 , H2 S/C3 H8 , CO2 /H2 S/C3 H8

V

MOR

[144]

Binary mixtures C5 –C22

G

MFI

[145]

C5 /C6 , C5 /C7 , C6 /C7 , C6 /C8 , C6 /C10 , C7 /C8 , C8 /C9 , C8 /C12

G

MFI

[146]

CO2 /C3 H8 , SF6 /C2 H6 , C2 H4 /C2 H6 , CO2 /C2 H4 , CO2 /C2 H6 , CO2 /C2 H4 /C2 H6 , SF6 /CH4

CV

FAU MFI

[147]

C3 H8 /C3 H6

MC

Na-LTA

[148]

C2 H6 /CO2

V

MCM-41

[149]

CO2 /CH4 , CO2 /N2 , CO2 /N2 /CH4

Various methods

MFI MOR ISV ITE CHE DDR

[150]

CO2 /CO

LHPG

FAU

[151]

CH4 /CO2 /N2

LPSA

13X, CMS, 3 K

[152]

C6 /2-methyl C5 , C6 /2,3 dimethyl C4 , 2-methyl C5 /2,3 dimethyl C4

NIR and manometric

MFI

[153]

CO2 /CH4

G

MFI

[154]

CO2 /N2 , CO2 /CH4 , CH4 /N2

Pulse chromatography

MFI

[155]

CO2 /N2 , CO2 /CH4 , CH4 /N2

Pulse chromatography

MFI

[156]

24

F.J. Keil Table 2.1 (continued)

Mixtures investigated

Experimental method

Zeolites

Citation

C2 /C3 , C2 /C4 , C2 /CO2 , C3 /C3 H6

G

LTA

[157]

CH3 OH/C3 H6 O

Density bottle method

MFI

[158]

C13 isomers

Liquid phase batch

MFI BEA AEL

[159]

(a) 50-50 mixture isotherm

(b) Sorption selectivity

8

102

6 4

C3, CBMC nC4, CBMC IAST

2 0

nC4/c3 sorption selectivity

Component loading/[molecules per unit cell]

Abbreviations: G, gravimetric; V, volumetric; CV, calorimetric–volumetric; PC, perturbation chromatography; LHPSG, low–high pressure gravity; LPSA, layered pressure swing adsorption

CBMC IAST Henry selectivity

101 MFI; T = 300 K

MFI; T = 300 K

100 101 102 103 104 105 106 107 108 Total pressure/[Pa]

100

0

2 4 6 8 10 12 Mixture loading, Θmix /[molecules per unit cell]

Fig. 2.5 Sorption loadings of equimolar binary mixture of C3 and n-C4 ; n-C4 /C3 sorption selectivity [160]

mixing are presented [160]. The loading of C3 continuously increases with increasing pressure. On the other hand, the loading of n-C4 reaches a plateau value for pressures in the 10–100 kPa range. Increasing the total systems pressure beyond 100 kPa leads to a decline in the loading of C4 . This is a size entropy effect that favors smaller sized molecules. Similar phenomena have been observed for many other examples (see, for example, [27], and the literature cited therein). The “squeezing out” effect has also been observed for alkenes. In Fig. 2.6 simulated adsorption isotherms of a binary mixture of ethene and 1-butene for a fixed equimolar bulk phase composition at 400 K and selectivity of 1-butene with respect to ethene are presented. As can be seen from Fig. 2.6, the IAST [161] gives results which coincide with molecular simulations. Unlike in the paper [60] in Fig. 2.6 fugacities instead of

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

25

20

2.0 1.5 1.0

16

total amount adsorbed ethene 1-butene selectivity IAST

12 8

0.5 0.0 10–3

selectivity

Fig. 2.6 Simulated adsorption isotherms for a binary mixture of ethene and 1-butene for a fixed equimolar bulk phase composition at 400 K and selectivity of 1-butene with respect to ethene [60]

amount adsorbed/(mol kg–1)

2

4

10–2

10–1 100 101 total bulk pressure/bar

0 102

pressures were employed at high pressures. Interestingly, the “squeezing out” effect is also obtained by means of IAST using the pure component isotherms as input data. Again the maximum for 1-butene in the adsorption isotherm at high pressures may be due to a size entropy effect because a difference in the size of the results in a different packing efficiency. Another reason for a maximum can be found in nonideality effects [162]. Figure 2.7 shows the component fugacities for an equimolar mixture of ethene and 1-butene at 400 K. The slope of the component fugacity of 1-butene decreases significantly for pressures above 40 bars, while the slope of the component fugacity of ethene hardly changes. As has been shown by Heyden et al. [162], a difference in the component fugacities can lead to a maximum in the isotherm, because, in addition to the potential energies, exerted by the molecules, the ration of the component fugacities decides which component “wins the competition” in the pore. 50

component fugacity/bar

ethene 1-butene

40 30 20

10

0

0

20

40 60 total bulk pressure/bar

80

100

Fig. 2.7 Component fugacities for an equimolar mixture of ethene and 1-butene at 400 K

26

F.J. Keil

Pascual et al. [61] have simulated the binary isotherms of C2 , C3 , C4 , and C7 alkane/alkene mixtures using their anisotropic united atom (AUA) potential optimized from experimental isotherms of butane in silicalite. No further adjustment of the potential parameters was necessary for predicting the behavior of alkane and alkene molecules in silicalite. Fox and Bates [163] calculated the behavior of binary and ternary mixtures of hexane, 2-methylpentane, and cyclohexane in silicalite-1, AlPO4 -5, and ITQ22 at 300 and 600 K. In silicalite-1, at 300 K, hexane dominated the adsorption, followed by 2-methylpentane and finally cyclohexane. This order was altered at 600 K, with cyclohexane adsorbing in greater numbers than hexane and, finally, 2-methylpentane. The increase in adsorption of cyclohexane coupled with a decrease in 2-methylpentane and, to a lesser extent, hexane with increasing temperature was echoed in AlPO4 -5 and ITQ-22. The authors explained this phenomenon in the following way: As the temperature increases, the shape of the cyclohexane molecules does not change their structure significantly, whereas the hexane and 2-methylpentane molecules decrease in length and increase in width, because of the increased number of gauche bonds present at high temperatures. The short, bulky molecules find it harder to fit into the channels of the zeolites and, because 2-methylpentane is further hindered by the presence of the branched CH3 group, the uptake of 2-methylpentane decreases faster than that of hexane at high temperatures. A nice example for the configuration entropy effect has been published by Krishna et al. [105]. The component loadings of an equimolar mixture of n-C6 and 3-methylpentane (3MP) as a function of pressure at 362 K are presented in Fig. 2.8. A maximum in the loading of 3MP at about 100 Pa can be observed. When the pressure is raised above 100 Pa the loading of 3MP reduces virtually to zero. The n-C6 molecules fit nicely into both straight and zigzag channels, whereas the 3MP molecules are preferentially located at the intersections between the straight chan-

Component loading/molecules per unit cell

8 nC6, CBMC 3MP, CBMC

6

4

MFI; T = 362 K

2

0 10–1 100 101 102 103 104 105 106 Total system pressure/[Pa]

(b) Sorption selectivity 100 nC6/3M P sorption selectivity

(a) 50-50 mixture isotherm

CBMC

MFI; T = 362 K 10

1 10–1 100 101 102 103 104 105 106 Total system pressure/[Pa]

Fig. 2.8 Simulations of the loadings of n-C6 and 2-methylpentane (3MP) mixtures at 362 K as a function of total pressure; n-C6 /3MP sorption selectivities as a function of total pressure [105]

2

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

27

nels and the zigzag channels. Below a total loading of four molecules per unit cell, there is no real competition between n-C6 and 3MP. The n-C6 locates within the channels and 3MP at the intersections. When all the intersection sites are occupied, to further adsorb 3MP one needs to provide an extra “push”. Energetically, it is more efficient to obtain higher mixture loadings by “replacing” the 3MP with n-C6 . This configurational entropy effect is the reason behind the curious maxima in the 3MP loading in the mixture. The sorption selectivity increases from near-unity values for pressures below 100 Pa to values around 50 near-saturation loadings. Lu et al. [164] simulated the adsorption of binary mixtures of C4–C7 alkane isomers in ISV and MOR at 300 K. The results were compared with MFI. For all mixtures of alkane isomers investigated in the same zeolite, the same trends were observed. The adsorbed amounts of linear and branched alkanes are all with increasing pressure for ISV and MOR. At higher pressures the amounts of branched alkanes were larger than linear alkanes. This was quite different from MFI. Similar phenomena have been detected by Schenk et al. [165]. Chen and Sholl [166] have shown that the transition matrix Monte Carlo (TMMC) method of Shen and Errington [167] is well suited to simulate binary adsorption in porous materials. At the completion of a TMMC simulation the adsorption isotherm for all possible bulk phase compositions and pressures is available without fitting or interpolation, in contrast to the usual GCMC approach of obtaining data for a discrete set of state points. In addition to the complete binary adsorption isotherm, TMMC also allows direct evaluation of derivatives of the term such as the binary thermodynamic correction factors. Garcia-Perez et al. [150] studied the adsorption properties of CO2 , N2 , and CH4 in all-silica zeolites using molecular simulations and measurements. The adsorption selectivity was analyzed for mixtures with various bulk compositions. Although in most cases IAST could describe binary mixture isotherms surprisingly accurate, Krishna and van Baten [168] found a segregated nature of adsorption of CO2 /CH4 , CH4 /N2 , and CO2 /Ar mixtures in DDR. CO2 and N2 molecules locate both within the cages and at the windows, whereas CH4 and AR adsorb predominantly within the cages. The IAST does not adequately describe the component loadings for mixture adsorption and there are strong non-ideality effects, in particular for CO2 /CH4 mixtures. Another important consequence of segregation is that the adsorbed CO2 and N2 molecules at the window regions hinder the inter-cage transport of partner molecules such as CH4 , Ar, N2 , or Ne. Such hindrance effect is not catered for by the Stefan–Maxwell diffusion theory. Separations of gas mixtures also have been published. A few examples are given in the following papers [168–172].

2.5 Adsorption in Carbon Nanotubes (CNTs) Carbon nanotubes form an allotrope of carbon. They were discovered in 1991 by Iijima [173]. Two types of nanotubes have been distinguished so far, namely the single-walled carbon nanotube (SWCNT) and the multiwalled carbon nanotube

28

F.J. Keil

(MWCNT). A SWCNT is generated by folding back a graphene sheet on itself and forming a seamless cylinder with constant radius. Since no dangling bonds must be present, the SWCNTs are closed off at each end by hemispherical caps. These caps can be opened by experimental techniques. Carbon nanotubes have many possible applications in different fields, such as microelectronics, chemistry, polymer science, material science, and chemical engineering. Due to its hexagonal structure, it is possible to roll up graphene sheets in various ways and, therefore, various CNT structures exist. In general the structures can be divided into three main classes: “zigzag,” “armchair,” and “chiral” structures (see Fig. 2.9). The chirality (helicity) is defined by the chiral vector C = n1 aˆ 1 + n2 aˆ 2 with integers n1 ≥ n2 , and where aˆ 1 and aˆ 2 are vectors of the graphite unit cell. The radius of a carbon nanotube is defined as  lC−C 3(n21 + n1 n2 + n22 ) rCNT = 2π

where lC–C = 0.142 nm is the carbon–carbon bond length. Further details on describing CNT structures are given in [84]. Besides hydrocarbons [174–179], adsorption of other molecules and mixtures on SWCNTs have been investigated, like hydrogen [180–182], nitrogen [183–186], nitrogen/oxygen [187, 188], water [189], neon [190], helium [191], and xenon [192, 193, 194], and Krypton [194]. Applications of CNTs in pressure swing adsorption columns have been described by Heyden et al. [193]. The “squeezing out” effect has

MWCNT

SWCNT

armchair structure (n,n)

zigzag structure (n,0)

Fig. 2.9 Structures of single-wall (SWCNT) and multiwall (MWCNT) carbon nanotubes

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

29

6

2.4

5.5

2.2

5

2

4.5

1.8

4

1.6

3.5

1.4

3

1.2

2.5

1 0.8

2

1.5

total amount adsorbed ethane ethene selectivity IAST

1 0.5 0

0.01

selectivity

amount adsorbed [mol kg–1]

2

0.1

1

10

100

0.6 0.4 0.2

0 1000

total bulk pressure [bar]

Fig. 2.10 Adsorption isotherm of a binary mixture of ethane and ethene inside a (20,0) carbon nanotube for a fixed equimolar bulk phase composition at 300 K and selectivity of ethane with respect to ethene [195]

been detected also in CNTs [162, 195]. In Fig. 2.10 simulated adsorption isotherms of the binary mixture of ethane and ethene inside a (20,0) carbon nanotube at 300 K for a fixed equimolar bulk phase composition are presented. Additionally, the selectivity is shown. Like for the pure component adsorption, at low pressure more ethane than ethene is adsorbed. At high pressures the entropic effects lead to a reversal. Thus, one observes a distinct maximum in the ethane isotherm, which is also calculated by the IAST. The development of adsorption layers for 1-butene inside a (40,0) SWCNT is presented in Fig. 2.11. Radial density profiles (RD) at 300 K and two pressures (0.1, 100 bar) are given. At 0.1 bar the first adsorption layer, at a distance of 0.4 nm from the wall, is formed. At 100 bar the third layer is formed, and the center part of the tube is filled. Figure 2.12 shows calculated Henry coefficients of n-alkanes as a function of their carbon numbers at 300 K in SWCNT of various diameters. The simulations were performed for the zero-loading limit. The very strong interaction between the (10,0) CNT and the alkanes is demonstrated by rather large Henry coefficients compared to the values for the wider tubes. As expected, the Henry coefficients increase monotonically with the number of carbon atoms. Striolo et al. [189] have used GCMC simulations to study water adsorption in (6:6), (8:8), (10:10), (12:12), and (20:20) SWCNTs in the temperature range 248–548 K. The 298 K adsorption isotherms exhibit the typical features of type V adsorption isotherms in the IUPAC classification, the water uptake is negligible at low pressure. Pores suddenly and completely fill once a threshold pressure is reached; the adsorption/desorption loops are characterized by wide hysteresis. The width of the hysteresis loops decreases as pore diameter narrows. Adsorption isotherms simulated in (6:6) and (8:8) SWCNTs at 298 K do not show adsorption/desorption hysteresis loops, and the water uptake in (6:6) SWCNTs is gradual as the pressure increases. The structure of the confined water depends on the

30

F.J. Keil pressure [bar] 100

102

40 RD isotherm

particle density [nm–3]

35

10 8

30 25

6 p = 0.1 bar

20 15

4

×

10

2

5 0

0

0.2

0.4 0.6 0.8 1 1.2 distance from CNT wall [nm]

1.4

amount adsorbed [mol kg–1]

10–2

10–4

0 1.6

pressure [bar] 10–4

10–2

100

102 ×

particle density [nm–3]

35

8

30 25

6 p = 100 bar

20

4

15 10

2

5 0 0

0.2

0.4 0.6 0.8 1 1.2 distance from CNT wall [nm]

1.4

amount adsorbed[ molkg–1]

10

40

0 1.6

Fig. 2.11 Radial density profiles (RD) represent 1-butene distributions inside a (40,0) CNT at 300 K. The adsorption isotherm of 1-butene at 300 K is plotted on the lhs axis, the state point for which the RD profile is shown is marked with a cross [195]

diameter of the SWCNTs. A pronounced layered structure is observed when water is confined within (12:12), (10:10), and (8:8) SWCNTs. Water confined in (6:6) SWCNTs at 298 K forms one-dimensional hydrogen-bonded chains in which each water molecule receives one hydrogen from the preceding molecule along the chain. Hu et al. [196] have used MC simulations and Widom’s test particle insertion method to calculate the solubility coefficients and the adsorption equilibrium constants in SWCNTs (10:10). The hydrogen adsorption isotherms at room temperature were predicted by following the Langmuir adsorption isotherm using the calculated constants S (solubility coefficient) and K (adsorption equilibrium constant). The effect of alkali doping on hydrogen adsorption was studied by incorporating K+ or Na+ ions into nanotube arrays using a Monte Carlo simulation. The results on hydrogen adsorption isotherms indicate hydrogen adsorption of 3.95 wt% for

Molecular Simulation of Adsorption in Zeolites and Carbon Nanotubes

Henry coefficient [molm–3 Pa–1]

2

1024 1022 1020 1018 1016 1014 1012 1010 108 106 104 102 100 10–2

31

(10,0) CNT (20,0) CNT (30,0) CNT (40,0) CNT

2

4 6 8 10 carbon number of n-alkane

12

Fig. 2.12 Henry coefficients of n-alkanes as a function of their carbon numbers at 300 K in CNTs of various diameters [195]

K-doping and 4.21 wt% for Li-doping, in reasonable agreement with experimental results. Jiang et al. [177] have investigated the adsorption and separation of linear (C1 –n-C5 ) and branched C5 alkanes at 300 K on (10,10) SWCNT bundles. For a five-component mixture of C1 –C5 linear alkanes, the long alkane adsorption first increases and then decreases with increasing pressure, but the short alkane adsorption continues increasing and progressively replaces the long alkane at high pressures due to the already mentioned size entropy effect. All the C5 isomers adsorb into the internal annular sites on a bundle with a gap of 3.2 Å, but only n-C5 also intercalates the interstitial channels on a bundle with a gap of 4.2 Å. Mpourmpakis et al. [197] have combined density functional theory (DFT) with GCMC simulations to investigate the dependence of hydrogen storage in SWCNTs on both tube curvature and chirality. Burde and Calbi [198] have calculated the kinetics of gas uptake on different regions of carbon nanotube bundles by means of a kinetic Monte Carlo scheme. A lattice-gas description was used to model the adsorption of particles on a one-dimensional chain of sites under two types of dynamics, namely external kinetics, in which the chain is on the bundle’s external surface directly exposed to the gas, and pore-like kinetics, expected to occur inside the tubes and interstitial channels, where adsorption occurs via gas diffusion from the ends. From the time evolution of the coverage at a fixed temperature, equilibrium times are obtained as a function of chemical potential. The equilibrium time of the external phase decreases linearly as the coverage increases toward monolayer completion; the rate at which this occurs strongly depends on the ration between the binding energy and the temperature. The adsorption rate in pore-like phases is typically two orders of magnitude slower than that of external phases. Further details on adsorption on CNTs may be found in the books [84, 199].

32

F.J. Keil

2.6 Concluding Remarks Molecular simulations have clearly demonstrated their usefulness in adsorption theory and engineering. In most cases experimental accuracy can be obtained. In particular, molecular simulations have given far deeper insights into the observed phenomena than could have been obtained only from experiments. The expenditure of time for obtaining multicomponent adsorption isotherms can be reduced considerably by employing molecular simulations. This short review is not comprehensive, but highlights some of the recent results.

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187. Jiang J, Sandler SI (2004) Nitrogen and Oxygen Mixture Adsorption on Carbon Nanotube Bundles from Molecular Simulation. Langmuir 20:10910–10918 188. Arora G, Sandler SI (2005) Air Separation by Single Wall Carbon Nanotubes: Thermodynamics and Adsorptive Selectivity. J. Chem. Phys. 123:044705/1–9 189. Striolo A, Chialvo AA, Gubbins KE, Cummings PT (2005) Water in Carbon Nanotubes: Adsorption Isotherms and Thermodynamic Properties from Molecular Simulation. J. Chem. Phys. 122: 234712/1–14 190. Gordillo MC, Brualla L, Fantoni S (2004) Neon Adsorbed in Carbon Nanotube Bundles. Phys. Rev. B 70:245420/1–7 191. Marcone B, Orlandini E, Toigo F, Ancilotto F (2006) Condensation of Helium in Interstitial Sites of Carbon Nanotube Bundles. Phys. Rev. B 74:085415/1–5 192. Simonyan VV, Johnson JK, Kuznetsova A, Yates JT (2001) Molecular Simulation of Xenon Adsorption on Single-Walled Carbon Nanotubes. J. Chem. Phys. 114:4180–4185 193. Heyden A, Düren T, Kolkowski M, Keil FJ (2001) Design of a Pressure Swing Adsorption Module based on Carbon Nanotubes as Adsorbent – A Molecular Modeling Approach. Hung. J. Ind. Chem. 29:95–104 194. Jalili S, Majidi R (2007) Study of Xe and Kr Adsorption on Open Single-Walled Carbon Nanotubes Using Molecular Dynamics Simulations. Physica E 39:166–170 195. S. Jakobtorweihen, F.J. Keil Adsorption of Alkanes, Alkenes, and their Mixtures in SingleWalled Carbon Nanotubes and Bundles, Mol. Sim. 35(2009), 90–99 196. Hu N, Sun X, Hsu A (2005) Monte Carlo Simulations of Hydrogen Adsorption in AlkaliDoped Single-Wall Carbon Nanotubes. J. Chem. Phys. 123:044708/1–10 197. Mpourmpakis G, Froudakis GE, Lithoxoos GP, Samios J (2007) Effect of Curvature and Chirality for Hydrogen Storage in Single-Walled Carbon Nanotubes: A Combined ab initio and Monte Carlo Investigation. J. Chem. Phys. 126:144–704 198. Burde JT, Calbi MM (2007) Physisorption Kinetics in Carbon Nanotube Bundles. J. Phys. Chem. C 111:5057–5063 199. Bottani EJ, Tascon JMD (Eds.) (2005) Adsorption by Carbons. Elsevier, Amsterdam

Chapter 3

Molecular Simulation of Adsorption of Gases on Nanotubes Erich A. Müller

Abstract Molecular simulation poses a unique advantage as a technique to understand the physical phenomena at the nanoscale, as it provides an utmost level of control over the relevant variables. The available molecular simulation methods for the modelling of adsorption of gases in single-walled carbon nanotubes are reviewed. Special emphasis is given to the adsorption of low molecular weight gases within classical (non-quantum) scenarios and an overview of commonly employed intermolecular potentials, methodologies and recent results is provided. While the adsorption characteristics of fluids on nanotubes have some parallels with other carbon adsorbents, their one-dimensional morphology contributes to some of the unique adsorption properties described here. Additionally, the effect of interstitial and exohedral adsorption is discussed.

3.1 Introduction There is an innate interest in the scientific community to understand the effects that confinement will have on the bulk and equilibrium properties of fluids [1]. Experimentation under these conditions encounters multiple challenges, starting by the description of the solid surfaces, frequently riddled with impurities, complex pore size distributions and structural and energetic heterogeneities; last but not least, the inhomogeneous nature of the adsorbed fluids is difficult to probe and evaluate. Molecular simulation poses a unique advantage in this area, as it provides a level of control over the variables which are thought to be important which would otherwise be impossible to study with the current experimental techniques. Of the available gas–solid systems, of particular interest are those involving carbon adsorbents, as it is the selection of choice for most proposed industrial physisorption applications due to its low cost, availability and selectivity. Carbon E.A. Müller (B) Department of Chemical Engineering, Imperial College London, SW7 2AZ, UK e-mail: [email protected]

L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_3,  C Springer Science+Business Media B.V. 2010

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is in general a very complex and ill-defined adsorbent, so plain models, based on simplified geometries have been employed within both theoretical and simulation studies. In that sense, in the last 20 years or more the slit-pore model of carbon has been an obvious choice of archetypical adsorbent, as its dimensions and structure are well known, resembling a graphitic, unactivated carbon. As such, the slit-pore geometries have served as suitable building blocks for more complex and realistic representations [2]. It thus follows naturally that recent efforts have and will be devoted to the study of the one-dimensional analogue of the slit pore, namely the carbon nanotube. The idea of using a one-dimensional cylindrical pore to study adsorption is not new. However, most of the older studies have been performed on models that assume perfect cylinders with smooth walls, with potential functions dependent on the radial position of the atoms with regard to the surface. The smooth surface presents artificial results with respect to layering on the walls and diffusion parallel to the length of the tube. Probably one of the first examples of a simulation on a one-dimensional realistic carbon pore was that by Bojan et al. [3], where corrugated pores were assembled by sequential addition of carbon molecules in a cylindrical geometry and the adsorption of methane on these structures was studied. Reasonably small diameter pores were studied with effective diameters of 1.3, 2.0 and 2.6 nm. The discovery in the early 1990s [4] of the synthetic nanotube structures immediately generated in the simulation community excitement as the perspective of being able to study, model and compare with experiments a uniquely well-defined carbon adsorbent. One of the first examples of molecular simulation of adsorption of gases on nanotubes was the simulations of the adsorption of HF molecules into “molecular straws” reported by Pederson and Broughton [5]. Today, the amount of reported simulation studies on nanotubes exceeds thousands of papers. This review attempts to gather some of the most relevant techniques and results of studies of classical molecular simulations as applied to the adsorption of gases in nanotubes.

3.2 Intermolecular Potentials 3.2.1 Fluid Force Fields The heart of any molecular simulation is the force field or intermolecular potential used. The reader is referred to the general literature for a complete review [6, 7]. The intermolecular potential functions used in molecular simulations of simple systems can, in principle, be obtained directly from ab initio quantum mechanical calculations. In practice, after lengthy calculations, one must formulate a semi-empirical mathematical model to fit the computed data, usually as a function of intrinsic molecular parameters (size, charges, etc.) and inter and intramolecular interactions (bond stretching and bending, relative distances to other molecules or functional groups, etc.). The procedure is complex and offers only partial success for simple molecules (see, for example [8]). In general, a compromise must be made between the accuracy of the details to be included in the model and the computational cost

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associated with the energy calculation of a given force field. A detailed model is most likely to reproduce a larger amount of properties in a more satisfactory way, while a simpler one is generally more limited in scope and accuracy. One must bear in mind that more than three quarters of the computer time of any fluid phase simulation is consumed by energy and/or force calculations, and the computational effort typically rises as the order of O(N2 ), where N is the total number of molecules involved in the simulation. Thus, the potential models must be chosen appropriately with due consideration of the quality of the results desired and the hardware and algorithms available. In general, most cases of interest included in this discussion may be described with available intermolecular potentials developed for vapour–liquid equilibria and fluid phase thermophysical property estimations, e.g. the Drieding [9], AMBER [10], TraPPE [11], OPLS [12] force fields, to name a few of the most popular. These semi-empirical potentials typically group the hydrogen atoms with the corresponding principal atom (O, C, etc.) in a single effective entity (a united atom). Although not usually labelled as such, these models are group contribution-type methods, as molecules may be built with the corresponding united atoms and appropriate expressions for the intra-atomic bonding potentials. The methodology is quite firmly established [6, 13], although efforts are required to improve the transferability of the potentials, i.e. the ability of the parameter set fitted to a certain molecule and/or condition to be used in a different environment. Special mention must be made of the anisotropic united atom potential [14], where the centre of force of the united atom is displaced with respect to the centre of mass, in order to take into account the presence of the hydrogen atoms.

3.2.2 Effect of the Details of the Potential Models on the Adsorption Isotherms Figure 3.1 shows the comparison of the adsorption of CO2 on a (10,10) armchair nanotube using three available potentials with different degrees of coarse graining (many other options are available, for a review of CO2 potentials, see [15]). This particular example has created controversy as to whether the detail is required [16] or it is unnecessary [17, 18]. For CO2 , one might consider a fully atomistic model, consisting of three atomic sites bonded in a linear fashion, with partial charges on each site. The six-site (three repulsion–dispersion sites and three delocalized partial charges) EPM2 model [19] is an example of such a parameterization. One may recognize that one could coarse grain the electrostatics by considering a centrally placed quadrupole moment, as the fluid is not a polar molecule and the effect of the closely located charges screen each other out. Taking the coarse graining even further, the small aspect ratio of the molecule suggest that one may use a Lennard-Jones dumbbell instead of the three atomistic sites. This three-site model (two Lennard-Jones centres and a central quadrupole moment) is referred to as the 2CLJQ potential and has been parameterized initially for CO2 by Moller and Fischer [20, 21]. Even further coarse graining could be performed by taking

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Fig. 3.1 CO2 adsorption isotherms at 298 K obtained by GCMC simulations for a (10,10) SWCNT using different fluid–fluid intermolecular potentials. Solid circles are the 2CLJQ potential [20], diamonds are the EPM2 model [19] and squares are the LJ spherical model [16, 17]

a minimalistic approach and considering the CO2 molecule as a single isotropic spherical Lennard-Jones model (LJ) [16], or better yet, an angle-averaged potential [15, 22]. If one were to believe the premise that the most detailed model (in this case EPM2) is the most precise, it is seen (Fig. 3.1) that both coarse-grained models perform reasonably well, the 2CLJQ model excelling in the low-pressure region. Discrepancies are more notable at the higher pressures, where the fluid is closely packed. The computational effort decreases significantly with the coarse graining, as computations typically scale as the square of the number of sites, i.e. simulations using the EPM2 model will take an order of magnitude longer than those using a single-site model. All else being equal, it is clear that no definitive conclusion is possible, since ultimately, the quality of the result should rely on the adjustment of some parameters to experimental data, which in the case of adsorption on nanotubes is both limited and riddled with experimental uncertainties with respect to the characterization of the samples. Additionally, Fig. 3.1 may not be representative of other systems, particularly more elongated molecules, where the confinement effect can have considerable influence on the conformations of the molecules. Here, the use of more detailed potentials might reduce the uncertainties. A further example, where the adsorption of alkenes and alkanes on carbon nanotubes is compared based on coarse gained vs fully atomistic potentials is given by Cruz and Müller [23]. A pervasive problem when performing molecular modelling studies of water is the unsuitability of common intermolecular potentials to deal with the complexity of the fluid. In spite of the very large number of potentials available, there is no unique preferred choice [24–25]. Most rigid non-polarizable intermolecular potentials in common use for simulation of water, e.g. SPC [26], TIP3P and TIP4P [27], are optimized to a small sub-set of thermodynamic properties, most commonly, the density,

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the vaporization enthalpy, radial distribution functions and apparent liquid phase dipole moment. Thus, the prediction of other thermodynamic properties, such as the correct gas phase dipole moment, the vapour pressure, interfacial tension, melting point, ice structure and adsorption properties are elusive. A review of the effect of using different models for water (SPC, SPC/E, TIP3P) [28] reports similar trends in all cases studied, suggesting that the fine details may be of little importance to study adsorption and transport properties. This is contrast with earlier studies where the energetic properties (which relate to heats of adsorption) have been seen to vary widely with the choice of potential [29]. The final conclusion has to be two sided and thus the choice of level of detail of the model used depends fundamentally on the type of results that are sought.

3.2.3 Potentials for Carbon Nanotubes Simonyan et al. [30] have presented detailed analysis which concludes that for studies of adsorption and transport behaviour within SWCNT’s, the potential should optimally be an atomistic description with rigid bonding and corrugated, i.e. all atoms in the tube should be explicitly included in the calculations. This has been subsequently confirmed in other situations [29]. Additionally, Simonyan et al. [31] compared their simulations with experiments of Xe on nanotubes suggesting that the potentials developed for graphene sheets may be transferred to the case of cylindrical pores. Nanotube corrugation and geometry seems unlikely to have any effect on the equilibrium properties, although it does have significant effect on diffusivities, especially at low loadings. SWCNT’s may be described in an atomistic way, assuming carbon atoms to be a collection of discrete Lennard-Jones spheres, with parameters taken to be those customarily used to describe the carbon atoms of graphene sheets; εC = 28.0 K and σ C = 3.4 Å [31]. The bending of a planar graphene sheet to form a carbon nanotube results in the squeezing-out of the π clouds towards the nanotube exterior volume and will tend to stretch the purely sp2 hybridized σ (C–C) bond from a value of 1.4 Å to roughly 1.42 Å for the C–C sigma bond [32]. In most cases, cross fluid–fluid and solid–fluid interaction parameters (σ ij , εij ) can be estimated according to the classical Lorentz–Berthelot combining rules [33, 34] σ ij = (σ ii +σ jj )/2, εij = (εii ·εjj )1/2 . If reliable experimental data are available, especially at low loadings, these cross parameters can be revised, as there is only a very weak justification for abiding to these combination rules [35]. A rigid surface potential, such as the one suggested, implies some simplifications with regard to the lack of flexibility (breathing) of the surface. Other implicit assumptions regard the validity of the direct application of the fluid–fluid and cross parameters to the adsorption process, the absence of end effects (i.e. consideration of only open-ended nanotubes), amongst others. One must bear in mind that any of the aforementioned assumptions may have an effect on the results, so a direct comparison of the simulation results with experiments should be made with care. Furthermore, actual experimental adsorption isotherms involve both exohedral and

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endohedral adsorption (discussed later) and are hampered by issues such as poor purity of samples and polydispersity of the tube diameters. While in principle, multiwalled nanotubes, composed of concentrically nested nanotubes can be produced, the bulk of the recent synthesis efforts are concentrated on single-walled nanotubes. Fortunately, the adsorption characteristics of single and multi walled nanotubes of the same internal diameter is roughly equal, e.g. the presence of more than one carbon layer does not change significantly the adsorption phenomenology. In order to relate the results obtained from the simulations to macroscopically observed quantities, the pore volume must be taken into account. In simulations, the positions of the surfaces are well defined, as they are described by the positions of the centres of mass of the atoms that are used to build them. However, fluid particles used in most realistic simulations are impenetrable, as they are composed of a “soft” but repulsive core. Since the distance of minimum approach of a molecule to a wall will depend on external factors such as the temperature and local density, the actual “effective” volume of a pore is ill defined. It is customary to define the nanotube diameter taking into account the circumference that includes the centres of the atoms forming the cylinder D. One must define the internal tube diameter accessible to fluid molecules as Deff , and relate this quantity with D. An expression proposed by Kaneko et al. [36] for carbon slit pores can be extended to nanotubes [37] Deff = D − (2zo − σeff ), zo = 1.32σsf

(3.1)

where zo is the distance of closest approach (determined from fit to experiments) between the fluid and the wall. In the case of heteronuclear adsorbates or mixtures, it may be simpler to adopt an expression of the type Deff = D–σ C , where σ C is the Lennard-Jones diameter of a graphitic carbon atom (3.4 Å). Either of these definitions has no effect whatsoever on the molecular simulation calculations and is only relevant when attempting to match experimental data with simulations. An unambiguous way of representing the adsorption data would be to scale the results with respect to the surface area of a sample or to the mass of the sample (for example, in µmol/m2 or mmol/g). It is clear that realistic models of carbon nanotube bundles will have to take into consideration the polydispersity of the tube diameters and the results in the adsorption isotherms will be influenced by this polydispersity. Taking from the extensive experience that has been gained in the field by working with graphite slit pores, one can envisage that similar methods will be applied. For example, a way to describe a carbon with a wide pore size distribution is to consider it as a discrete ensemble of fixed width independent slit pores. For each of these “model” pores one can obtain the adsorption isotherm from either computer simulation or density functional theory. The resulting adsorption isotherm for the complete pore can be obtained by suitably integrating and interpolating these results. Several approaches of this type have been successful in determining pure fluid adsorption [38] and mixtures [39] on slit pores and the application is straightforward.

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3.2.4 Effect of Charge Anisotropy It is usually accepted that the electrostatic interactions at the graphitic surface are negligible compared to the van der Waals repulsion–dispersion terms and are usually accounted for in the two-body-averaged potentials used in simulations. In the case of polar and/or quadrupolar molecules adsorbing on to carbons this may not be the case. It is plausible to assume that the aromatic charge displacement, produced by the fact that carbon atoms in graphite have only three covalent bonds is somehow similar to those found in planar aromatic molecules such as benzene, naphthalene, anthracene and larger polynuclear aromatics. Following the suggestion of Vernov and Steele [40], if the quadrupole moment perpendicular to the molecular plane is calculated on a per atom basis, a nearly constant value of −1.3 + −0.2 B is obtained. This suggests that when considering strongly polar or quadrupolar molecules, explicit accounting of these interactions must be made. An experimental value has been reported by Whitehouse and Buckingham [41], where Q = −0.91 +/− 0.03 B. Zhao and Johnson [42] have developed an average potential, similar to the smooth Steele potential which incorporates explicitly the effect of the wall quadrupolar interactions and which could be used directly in the case of nanotubes. However, in most situations, the inclusion of quadrupoles seems to have a negligible effect [43]. In a planar graphene sheet, carbon atoms are sp2 hybridized, with an overall charge distribution of electrical quadrupoles located perpendicular to the surface. When this sheet is bended to produce a nanotube, the sp2 orbitals tend to be “squeezed-out” towards the tube’s outer space, and therefore the original quadrupole moments will be dislocated further away from the tube’s interior volume. As the nanotube radius decreases (due to the increased bending of the original graphene plane) this effect tends to be more pronounced and confined molecules will experience a reduced charge anisotropy. Most studies neglect any such quadrupole-surface interactions and this simplification has a parallel on the approximations made when studying adsorption on graphene sheets [44]. Even in scenarios where these contributions are expected to be important, as in the case of hydrogen adsorption onto single-walled nanotubes [45], the conclusion is that the presence of charges on the nanotube walls leads to an increase of only around 15% in the adsorption capacity at T = 298 K.

3.3 Simulation Methods There are several introductory textbooks available on molecular simulation, amongst these of particular interest is the seminal book by Nicholson and Parsonage [46] dedicated to the molecular modelling of adsorption. It has been followed by textbooks on general aspects of molecular simulation, which include specific chapters and worked examples on the aspects of modelling adsorption [35, 6, 47, 48].

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3.3.1 Grand Canonical Monte Carlo Of the available techniques to study adsorption, Grand Canonical Monte Carlo excels by far as the method of choice, since the underlying statistical mechanical ensemble is well suited for this purpose. The GCMC methodology in common use today may be traced back to Adams [49]. Today several excellent textbooks have explicit details on the background and the basics of the method. The reader is referred to any of them for examples of code and further details [47, 50, 35, 51, 6, 52, 48]. In GCMC the temperature T, the volume of the pore V, and the chemical potential of each species μi are kept fixed. An equilibrium is, in principle, attained between the confined fluid and an infinite ideal reservoir at the specified conditions. The number of molecules in the pore is allowed to vary, and its average is the relevant quantity of interest. Although not of widespread use, it is possible to replace the chemical potential by the more convenient variable: activity, ξi , defined by ξi =

exp (μi /kT) 3i

(3.2)

where i is the de Broglie wavelength, which includes contributions from translational and rotational degrees of freedom and k is Boltzmann’s constant. In the ideal gas limit, the activity is numerically equal to the number density, therefore allowing for a more “physical” handle on the numerical values. Alternatively, the chemical potential may be replaced by the fugacity, which in turn is numerically equal to the partial pressure in the ideal gas limit. An inconvenience in the GCMC calculations is that in order to relate the simulation results to actual experimental data, the chemical potential (or the activity or fugacity) must be expressed in terms of the bulk phase pressure. In most cases, particularly when the fluid phase is a gas, the use of an appropriate equation of state is a sensible route. It is again in this case that the definition in terms of activity is useful. Based on the assumption that a virial equation of state, truncated after the second virial coefficient, may be accurate to describe the pressure–volume–temperature relation for the bulk gas, the pressure may be related to the activity as Pi = RTξi (1 − Bi ξi )

(3.3)

where B, the second virial coefficient, is available as a function of temperature (see, for example [52]). This is usually sufficient for most practical applications where isotherms are reported to pressures below and up to 15 bar. For the low-pressure region, up to ambient pressure, the ideal gas approximation (i.e. taking Bi ξ i ran, where ran is a random number in the interval (0,1), β = 1/kT and ∆H depends on the type of move; for displacement/rotation moves ∆H is equal to the difference in configurational energy of the system, ∆U, for an insertion attempt it corresponds to β H = ln



N+1 Vξ



+ β U

(3.4)

and for deletion attempts to β H = ln



Vξ N



+ β U

(3.5)

where N is the total number of particles present in the confined space. In the case of multicomponent systems, both the activities, number of particles and pressures referred to in Eqs. (3.4) and (3.5) correspond to the values for the particular species. Care must be taken to ensure microscopic reversibility when including multiple species. Although the difference is often subtle, it is a necessary condition that the probability of a move in a given direction (for example, the creation of a particle) be exactly counterbalanced by the same probability of the inverse move (the destruction of a particle). The results of typical GCMC simulations are the ensemble average of the number of molecules adsorbed, , in the corresponding free volume. Results are usually expressed as the excess adsorption, Ŵ, taken as the difference between the adsorbed density, and that of the corresponding bulk, ρbulk ; for a nanotube, the expression corresponds to Ŵ=



0

D/2 
− ρbulk dr V

(3.6)

If one can assume the bulk density is small corresponding to the adsorbed density, the adsorption can be estimated by the radial density profile (per unit area). As

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discussed previously, the volume and area are not uniquely defined, and care must be taken when reporting results. In addition to calculating the ensemble average adsorbed number of molecules, one can also calculate the isosteric heat, qst from fluctuation formula. For the case of the adsorption of a low density (ideal) gas, the result is qst =

< NU > − < N >< U > − kT < N 2 > − < N >< N >

(3.7)

Constant volume heat capacities may be calculated using the second-order fluctuation of the above quantity [35, 53].

3.3.2 Molecular Simulation Nowadays, Molecular Dynamics (MD) is becoming a more familiar tool to researchers partially due to the availability of off-the-shelf molecular simulation programs. Most freely available and academic versions of programs focus on MD techniques, and in this area, a significant number of mature programs can be readily available, e.g. DL_POLY (http://www.dl.ac.uk/TCSC/Software/DL_POLY/main.html), AMBER (http://amber.scripps.edu/), Gromos (http://igc.ethz.ch/gromos/), Gromacs (http://www.gromacs.org/), LAMPPS (http://lammps.sandia.gov/), NAMD (http://www.ks.uiuc.edu/Research/namd/) to name a few. Contrary to the case of Molecular Dynamics methods, freely available GCMC codes are scarcer, most possibly due to the fact that the coding is simpler and many research groups tend to have their own in-house versions. Most modern MD codes are already written with parallel computers in mind, something which is particularly interesting as multicore single user personal computers can be adapted to run these programs with almost linear scaleup. The availability of off-the-shelf quad-core machines incite the use of MD as a tool for studying adsorption, in spite that it is clearly not the natural choice. In order to use MD to study adsorption, one must usually include in the same simulation box both the adsorbent and the corresponding bulk phase (cf. Fig. 3.2). There are several problems to be dealt with here: in first place, there is no control over the final equilibrium pressure of the bulk fluid, as it will depend on the number of molecules that end up in the adsorbed phase. Hybrid MC–MD methods can be devised to overcome this issue, based on the insertion/deletion steps of the GCMC method as applied to the bulk phase [54, 55]. This is lengthy and inefficient for equilibrium determinations, but can be an effective way of determining steady state transport properties (diffusivities and thermal conductivities). A second problem appearing is the shape of the cell. If a bundle is considered, a parallepiped simulation box can be devised. Otherwise an artificial separation must be made in the cell between the bulk and the nanotube. A further option is to immerse the nanotube within the bulk fluid. A note is to be made that in this

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Fig. 3.2 Snapshot from an equilibrium configuration of the adsorption of methane (blue) , CO2 (green-brown) and water (red-white) inside a carbon nanotube of 1.5 nm internal diameter in equilibrium with a gas at 25 bar and 350 K. Initial composition of the gas was an equimolar mixture of methane and CO2 with a 10% molar humidity. Note the adsorption and clustering of water molecules in the pore, despite the hydrophobic nature of the carbon pore. (Results from MD simulations of Salih and Müller [104])

final scenario, molecules which adsorb exohedrally may also interact with the endohedrally adsorbed molecules, thus inducing artificially high adsorptions. In these cases, where the cell contents are highly heterogeneous, pressure calculations based on the virial theorem, and thus associated constant pressure algorithms give spurious results.

3.3.3 Gibbs Ensemble Monte Carlo The Gibbs Ensemble Monte Carlo method has become the de facto method to study phase equilibria in fluids [56–57]. However, one of the first (and possibly less known) applications of the Gibbs ensemble was to study the adsorption of a LJ fluid on cylindrical pores [57, 58]. While the parameters were chosen to approximate Ar adsorption on solid CO2 , it is an example of a spherical molecule in a 1-D confinement, analogous to what we now consider when studying CH4 or Xe on nanotubes. GEMC can be used for two separate purposes, either to study adsorption as such, in which one of the two boxes contains the bulk fluid (or mixture) and the other the adsorbate and adsorbing phase. In that case it does not differ too much from a GCMC simulation other than the fact that the bulk fluid is also being simulated simultaneously. The real advantage is present if one is interested in studying phase transition within the confined space, such as vapour–liquid transitions, or some other surface-induced transitions such as capillary condensation, wetting. GCMC and GEMC simulations rely on an insertion (and a deletion) step to provide the appropriate statistics that relate to the chemical potential of the named species. For very dense systems and/or for larger and chain-like molecules, the probability of randomly performing a successful insertion (which thus contributes to the required average) is pathetically small, up to a point of rendering the simulation useless. A way around this problem is to incorporate “external” information in the way the molecules are inserted, and subsequently to correct for the bias in the corresponding acceptance criteria [58, 59]. This configurational bias (CBMC) technique has been employed in studies of adsorption and effectively allows the simulation of very long and/or bulky molecules, asymmetric mixtures, and/or dense systems with

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significant success. An excellent example is the parametric study [60–61] of the adsorption of the series of C1-C5 alkanes (including branched iso and neopentane) and some mixtures on nanotube bundles using CBMC techniques as described. One of the most successful enhancements of both the GCMC and GCMC schemes is the understanding that the fluctuations seen in the systems correspond to the sampling of nearby phase space, and if collected and analysed appropriately can lead to information on others parts of the phase diagram. Thus a single simulation on a given state point not only can be used to obtain a unique state point, but also its close neighbouring states. This is the underlying concept behind the histogram-reweighing techniques, based on the seminal work of Ferrenberg and Swendsen [62].

3.4 Adsorption of Fluids 3.4.1 Hydrogen The recent interest in hydrogen as an alternative fuel has sparked a flurry of research into the technological aspects of its production and distribution. Amongst the most problematic aspect of the hydrogen economy is its storage and transport. Adsorption has been seriously considered, as the classical alternatives: liquefaction and high-pressure storage are woefully unpractical. Adsorption competes strongly with chemisorption in metal hydrides as a viable storage option. Within the possible adsorbents carbon nanotubes have stood out as possible candidates due to their intrinsic high surface area [63, 64]. For comprehensive reviews, see [65–66]. The low temperatures required for an appreciable H2 adsorption also point to the need of explicitly introducing quantum effect in the simulations. The quantum corrections necessary both to the adsorbent and the adsorbate are beyond the scope of this review. A recent review of classical and quantum-based simulations of H2 in nanotubes can be found in Yang [67] and Rafii-Talbar [68]. At high temperatures, the classical (non-quantum) behaviour is retained, and the methods described herein are suitable. The simulations of Wang and Johnson [69] are an example of parametric studies which reflect the difference in the results obtained from quantum and classical simulations. For example, at 50 atm and 77 K, the density of H2 inside a (9,9) SWCNT is 17% higher if quantum effects are neglected. At 298 K the differences shrink to about 3%. The capability of nanotube bundles to serve as effective H2 storage materials at the capacities required by commercial and automotive industries is vigourously debated [70]. It seems unlikely (see the simulation results of Knippenberg et al. [71] and those of Lamari and Weinberg [72]) that pristine carbon bundles may be used to meet the high bar set by the US Department of Energy’s guidelines for hydrogen adsorption materials. However, it is believed that doping the nanotubes, for example, by adding alkali metals [73, 74], may prove to be the way to obtain significant adsorption capacities.

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3.4.2 Methane and Noble Gases Of all the light effectively spherical molecules (with the exception of hydrogen), the adsorption of methane has been the most studied [75, 76, 56, 77–78, 18] as it is both the archetypical quasi-spherical fluid and its storage as a fuel stock is of economical and technological urgency [79]. Adsorption of perfluoromethane, CF4 , has also been discussed [55, 80, 81]. These studies are not only of technological interest, but the spherical nature of the adsorbent allows for the direct application of theory, such as non-local density functional theory (DFT), to describe the adsorption (see, for example [82]). The adsorption of the noble gases have, for the same reasons, also been given attention: Ar [83, 85, 84–85], Ne [82], Xe [86, 82, 71, 90, 87–88], Kr [81, 98] have all been studied extensively.

3.4.3 CO2 and Other Quadrupolar Fluids Recent interest on the reduction of greenhouse gases from air has further sparked the interest on CO2 as it is the most relevant greenhouse gas. Within the context of the efforts to reduce greenhouse gas emissions, adsorption is posed to play a key role in separation of CO2 from flue gases. Additionally, sequestration of CO2 in deep unmineable carbon seams has been considered as an option [89]. The optimal design of process requires detailed molecular information on the adsorption desorption of CO2 on carbon nanopores. However, from an academic point of view, CO2 is also the archetypical linear quadrupolar molecule. Quadrupolar interactions are unique, in the sense that although they result from a delocalized charge distribution in a molecule, they are relatively short ranged and directional (anisotropic), with an intermolecular potential energy that decays with distance (r), as r−8 (as compared to r−6 for dispersion forces and r−3 for dipolar interactions). A few molecular simulation-based adsorption studies are available for adsorption of pure carbon dioxide [90, 18, 91], and its mixtures [97, 17, 18] in nanotubes. Nitrogen is a linear quadrupolar molecule, although its quadrupolar moment is almost half of that of CO2 . Thus, nitrogen is customarily treated as an apolar molecule, neglecting the effect that the quadrupole moment may have on the structure and thermodynamics of their condensed phases. Adsorption by nitrogen is the accepted method for experimentally determining the pore size distribution. For this to be done appropriately in the case of nanotubes, the results must be then related to adsorption isotherms determined either by simulations or by theoretical approximations. Simulations of N2 on nanotubes have been performed by Maddox and Gubbins [92], Khan and Ayappa [92], Yin et al. [93], Ohba and Kaneko [38], Yoo et al. [94], Paredes et al. [95], Arora et al. [96], Jiang and Sandler [33], Skoulidas et al. [17], and Müller [102]. The emphasis of the above-mentioned studies is on the equilibrium adsorption isotherms, isosteric heat curves and/or diffusion properties with little detail being

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given to the structure of the fluid within the pore. A notable exception is the simulation study reported by Khan and Ayappa [103] where they studied angular orientations of N2 and B2 molecules within nanotubes and that by Müller [102] where he studied nitrogen, (N2 ), carbon dioxide (CO2 ) and perfluoroethane (C2 F6 ). For the latter, a unique slanted ordering is seen in (15,0) nanotubes due to the particular combination of steric effects and the unique anisotropic attraction experienced by these fluids. Another peculiar observation has been made [97] in simulations of CO2 on the external edges of nanotube bundles where a “T” configuration in the adsorbed one-dimensional fluid has been reported. This ordering is ascribed to a combination of the strong fluid–fluid quadrupolar interactions of CO2 , and the anisotropic nature of the attraction in the vicinity of the nanotube bundle grooves.

3.4.4 Water While studies of adsorption of water on carbons have always been abundant, in recent years the trend has been shifted towards the study of adsorption of water on SWCNT. The reason for this is twofold: on the one hand, there is an obvious practical interest in this system as it is a biomime, a model for ion channels and/or mass transport through biological membranes [97–98]; on the other hand, from a purely academic point of view, the physics of the adsorption of water on carbons is quite unique. There are major differences between the strongly polar nature of the water–water interactions and the usually weaker interactions between water and most organic substrates. Carbon is commonly understood thus as a hydrophobic material, in spite of that there is evidence that water can become very strongly adsorbed within carbons. This apparent ambiguity can be explained by understanding the hydrogen-bonding nature of water and the predominant role it has on its adsorption properties. Müller et al. [99] gave a clear account of the unique adsorption mechanism of water on graphite, showing how water adsorbs on slit-pore surfaces as clusters, as opposed to the more conventional Langmuir monolayer. Later simulation studies [100, 101] showed how neat graphitic carbons could have a crossover behaviour from hydrophobic to hydrophilic behaviour by the doping of carbon surfaces with association sites, such as polar and oxygenated groups. More recent studies (cf. Fig. 3.2) show that other fluids may have a synergetic effect on the adsorption of water [102]. The parallels (and differences) between the behaviour of water in 2-D and 1-D carbon geometries have been recently pointed out by Cicero et al. [103]. It is not surprising then that a relatively large number of simulation studies have been performed using nanotubes as the adsorbent, since here, the already challenging physical picture of water behaviour in confinement is complicated by the 1-D geometry of the adsorbents. For nanotubes, at least three different fluid water adsorption patterns can be observed. For very small nanotubes, a water nanowire is formed, with water molecules forming a continually hydrogen-bonded single file [105, 106]. This single-file behaviour is unique to water and has the consequence that in order to

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maintain the HB network, molecules will engage in a pulse-like concerted movement [110, 107, 108], which could even be used as a molecular level pump [109]. Larger diameter nanotubes could in principle accommodate a layer of water on the surface. There is some contradiction in the results reported in the literature with respect to the exact nature and structure of water at these conditions. Very ordered helical structures have been reported for (7,7), (8,8), (9,9) [10] and (10,10) nanotubes [111] at ambient conditions. Other authors have reported some different ordered structures in intermediate-sized nanotubes [112–113] shedding some controversy over the actual details of the structures. The ordering has been confirmed using infrared spectroscopy compared to detailed molecular simulations [126]. These ordered water structures are expected and seen at high pressures and at ice-like conditions [114], so it is likely that their appearance at room temperature may be a result of metastable states being stabilized by the confining geometry rather than by artefacts of the different simulation methods. The presence of hysteresis in the adsorption isotherms [115, 116] is a tell-tale signature of this. In fact, at low temperatures, even DNA-like double helixes have been reported [117]. A full phase diagram of confined water in nanotubes has been described recently using MD simulations [118]. As is expected, as the nanotube diameter increases, the confined water assumes a more bulk-like behaviour. For other hydrogen-bonding fluids, such as ethanol, the effect of disturbing the bulk-phase hydrogen-bonding matrix proves to be significant [119]. The unique confinement induced by the nanotubes has a toll on the expected average number of hydrogen bonds per molecule [120–121] and on the density of the confined fluid [119, 127], lowering each of them significantly.

3.4.5 Meso and Macro Molecules The small size of the most common nanotubes precludes the adsorption of larger molecules due to steric effects. Larger molecular weight adsorbates will typically be liquids, and as such the physical phenomena are called intrusion. Nothing precludes the models mentioned here to be used, although one must take into consideration the dense nature of the equilibrium bulk phase and the inherent difficulties associated with the modelling of larger molecules with a significant number of degrees of freedom. The overall discussion of intrusion into nanotubes is beyond the scope of this work; however, some highlights are given below. The lower molecular weight alkanes are gases, and as such are amenable to being adsorbed endohedrally in nanotubes. For mixtures, some components may exclude others, not necessarily only due to size. The results of Heyden et al. [89] on alkanes (C1–C3) and CF4 mixtures exemplify this idea. A recent study by Jiang et al. [62, 63] showcased the behaviour of linear (from C1–C12) and branched C5 alkanes including the effect of confinement on phase behaviour; while others [122, 123] report adsorption of C5–C9 including a branched alkane. SWCNT provides excellent separation of alkanes from their branched isomers. Curiously, however, it is not the 1-D geometry that is of importance here, but rather the reduced pore size [124].

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Fig. 3.3 Snapshots from molecular dynamics simulations at 300 K. (a) Composite image with nanotube size increasing from left to right. In narrow nanotubes, the water adopts a single-file arrangement but becomes more disordered in a fashion similar to that of bulk water in wider nanotubes. Confined within a nanotube of a “critical” diameter (fifth from left), the water spontaneously orders into a regular array (right). Cross-sectional view of water inside the critical-sized 8.6-Å-diameter nanotube showing a multicolumnar water structure. Colours: nanotube (green), wafer (gray), water oxygens (red), water hydrogens (white). Taken from [125]

Adsorption of n-hexane [125] (Fig. 3.3), n-heptane [138] and n-decane [126] have also been studied in this context. Alkane/alkene separations (ethane/ethylene) have received attention, as they are important feedstocks for petrochemical processes. However, SWCNTs do not seem to present any advantage over zeolites or other adsorbents [23]. Neither the dispersion forces nor the charge anisotropy between the molecules are distinct enough to provide significant preferential adsorption. The common spherical allotropes of carbon C60, C70 (buckyballs) may be trapped inside suitable-sized SWCNT. Simulations of these systems have shown that these combined structures (nanopeapods) can have quite unique adsorption properties and structure [127–128]. Since the innovative suggestion that nanotubes could adsorb DNA molecules [129, 130] from aqueous solutions, several authors have considered the biomedical applications of adsorption on nanotubes (for a review see [131]). Recently, Hilder and Hill [132, 133] have reported the modelling of the inclusion of anticancer drugs inside carbon nanotubes and Liu and Wang [134] have studied the adsorption of zadaxin from an aqueous media. The long-term vision of these studies is to explore the possibility of using nanotubes as drug delivery agents [135, 136].

3.5 Exohedral Adsorption Adsorption on nanotubes may naturally take place in the external parts of the nanotubes (Fig. 3.4). This adsorption, in general, is not too different from the adsorption of fluids on graphitic surfaces, and as such has received much less attention. There are, however, differences to be taken into account: for very small diameter nanotubes, the squeezed-out electron clouds may induce changes in the local electrostatics; similarly, geometry (curvature) is different, leading in some cases to enhanced attraction. A more important aspect is that nanotubes will naturally exist in bundles or ropes (as opposed to in a dispersed state). These groupings of elongated fibre-like structures allow for gaps between the outer layers of SWCNTs. Due

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Fig. 3.4 Adsorption sites on a bundle of nanotubes. Nanotube bundle (black) is seen in the direction of the cavity of the tubes. Adsorption sites are shaded; (a) endohedral, (b) external, (c) groove, and (d) interstitial

to the cylindrical geometry, two closely packed nanotubes will form a wedge-shaped groove when both of their external surfaces are seen together (Fig. 3.4). These grooves are particularly interesting associating sites, as the wall attractions will be maximized along the length of the gap, preferentially allowing for the adsorption of smaller molecules. Furthermore, the confined (wedge-shape) geometry allows for the appearance of distinct ordered phases, amongst them, an apparently stable “three stripe” phase, which has been observed with some of the lower halogens (Ar, Kr and Ne but not Xe) [82, 137]. Another consequence of nanotube bundling is that when three or more nanotubes are bundled in a parallel fashion, a triangular (or other shape) interstitial channel is formed, which allows further molecules to adsorb. The interstitial channel is rather small, so only very small diameter molecules (H2 , He, Ne) may adsorb there. A parametric study of the plausibility of adsorption of different types of molecules in different adsorption sites has been presented [138]. The secondary porosity created between the tubes plays an important role in the overall adsorption curve. As the interstitial distance amongst the tubes is not a well-characterized property, one may attempt to describe experimental results by matching to GCMC results with varying intertube distances [139]. Exohedral adsorption may be significant and should be considered when comparing to experimental results. Agnihotri et al. [141] studied the adsorption of hexane both on the insides and outsides of nanotube bundles and concluded that under an ideal scenario that all nanotubes are open ended, the exohedral adsorption could account for a twofold increase in the total adsorption. A proper comparison between experimental data and simulation isotherms requires an account of endohedral, exohedral and the impurities of the samples [140, 141]. Exohedral adsorption of water on CNT bundles has unique aspects. At low densities, adsorption is confined to the grooves, and the unique wedge-shaped geometry of the surface allows for the adsorption of water molecules in a low density state. An increase in the equilibrium partial pressure brings in the adsorption of

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a distinctively different fluid, accompanied by a first-order phase transition [134, 135, 142]. Kondratyuk et al. [138, 139] studied the adsorption of both linear and branched alkanes in the grooves and interiors of SWCNT. Not surprisingly, linear alkanes tend to adsorb strongly one-dimensionally along the groove (as if trying to fill it) as opposed to the branched alkanes that do not fit properly due to steric hindrance. As a consequence, the space occupied by the linear alkanes is less than that of a branched isomer. This does not happen inside nanotubes, where for the larger nanotubes, the linear alkanes can presumably adopt other conformations. The strength of the exohedral adsorption is not as high as that experienced inside the nanotube; therefore, exohedral adsorption was found to occur at lower temperatures than endohedral adsorption (Fig. 3.5). A recent paper by Agnihotri et al. [160] has shown that mixtures of water and alkanes adsorbing on nanotube bundles may present peculiarities. For such systems, Agnihotri et al. [160] described how the adsorption of water will preferentially be located in the confined spaces (interstitial an endohedral adsorption) while the non-polar fluid (n-hexane) will have a preference for the exohedral adsorption. The results seem to be an equilibrium result, as the result is indifferent to the adsorption protocol, if either the water or the organic phase has been pre-adsorbed. This unique

Fig. 3.5 Schematic representation of sequential adsorption of water and organic vapours (solid lines) and the reversed sequence (dashed lines). Cyan areas represent water adsorption, whereas red zones represent adsorption of organic vapour. When the SWNT sample is presaturated with water vapour, organic molecules are prevented from adsorbing into the internal pore volume of the bundles. When the adsorption sequence is reversed, water displaces the organic molecules preadsorbed in the internal pore volume of the bundles. Taken from [143]

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site selectivity, predicted by GCMC simulations and confirmed by experiments in essence showcases a powerful mechanism for enhancing adsorption selectivity.

3.6 Nanohorns Nanotubes are only one of the many of the high-order regular structures seen as candidates for adsorption studies. A fascinating novel structure is that of nanohorns, which resemble nanotubes which are closed in one end and tapered in the other, much like an ice cream cone. Nanohorns are structures loosely related to both buckyballs and nanotubes; they are open-ended conical nanoparticles, of a few nanometres long (see Fig. 3.6(b)) which can aggregate to form dahlia-shaped structures 80–100 nm in diameter (see Fig. 3.6(a)). Due to their geometry (having a closed cone-shaped end) the solid–fluid interaction is apparently much stronger than conventional nanotubes, making it a plausible candidate for the non-cryogenic storage of hydrogen and for substrates for catalysts.

Fig. 3.6 (a) Micrograph of a dahlia-shaped assembly of carbon nanohorns. (b) Each spherical assembly is composed of a number of individual single-walled nanohorns (SWNH). (c) Adsorption of N2 on nanohorns from Ohba et al. [144]

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Recent simulations of these materials have sparked a significant interest, as the tube structure is much shorter than that of nanotubes and the geometry is better defined and easier to characterize [161, 145]. Additionally, high-purity samples are available for experimental adsorption measurements. All methods described herein can be translated and applied directly to this case.

3.7 Summary and Perspectives The molecular simulation of the adsorption of gases and dense fluids on nanotubes may be performed with relative ease. The body of knowledge accumulated over the last years about molecular simulation techniques has added to the numerous previous studies using slit-pore carbon geometries allowing us to have significant confidence in the results obtained. Still lacking is the clean comparison of simulations to experimental data, largely caused by the intrinsic difficulties in obtaining pure and well-characterized nanotube samples. While the 1-D geometry provides some noticeable differences with respect to the classical slit-pore geometries and the most likely random porous networks, its effect on the general adsorption and selectivity does not seem to be significant. However, the unique geometry of SWCNTs allows them to be incorporated into membranes. Here, not only the aspects of equilibrium adsorption are important, but new physical phenomena come into play, regarding the diffusivities and the permeation capabilities of the different molecules. The smooth linear nature of the carbon nanotubes provides a significant difference from common natural porous materials. Thus, complementary simulations must be performed to find the transport properties involved, such as diffusivities and permeabilities. These aspects are well beyond this review; however, the reader is referred to a paper by Chen and Sholl [146] where they present an excellent example of the level and type of computational studies required to evaluate the transport and equilibrium properties of a mixture (CH4 /H2 ). Similarly, simulations of nanotubes as biomimetic analogues of channels in cell membranes will most likely also require the study of transport properties on functionally substituted nanotubes, an interesting topic which is beyond this review. Acknowledgments I gratefully thank Profs. K. J. Johnson, J. P. B. Mota, S. Agnihotri, K. Kaneko and T. Ohba for contributing comments, data and figures to sections of this review. Partial financial support for this work has been given by the UK. Engineering and Physical Sciences Research Council (EPSRC), grants EP/D035171 and EP/E016340.

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79. Yang, R. T. (2000). Hydrogen storage by alkali-doped carbon nanotubes-revisited. Carbon 38(4): 623–626. 80. Froudakis, G. E. (2001). Why alkali-metal-doped carbon nanotubes possess high hydrogen uptake. Nano Letters 1(10): 531–533. 81. Muris, M., N. Dufau et al. (2000). Methane and krypton adsorption on single-walled carbon nanotubes. Langmuir 16(17): 7019–7022. 82. Calbi, M. M., S. M. Gatica et al. (2001). Phases of neon, xenon, and methane adsorbed on nanotube bundles. Journal of Chemical Physics 115(21): 9975–9981. 83. Zhang, X. R. and W. C. Wang (2002). Methane adsorption in single-walled carbon nanotubes arrays by molecular simulation and density functional theory. Fluid Phase Equilibria 194: 289–295. 84. Kim, B. H., G. H. Kum et al. (2003). Adsorption of methane and ethane into single-walled carbon nanotubes and slit-shaped carbonaceous pores. Korean Journal of Chemical Engineering 20(1): 104–109. 85. Bienfait, M., P. Zeppenfeld et al. (2004). Thermodynamics and structure of hydrogen, methane, argon, oxygen, and carbon dioxide adsorbed on single-wall carbon nanotube bundles. Physical Review B 70(3): 035410. 86. Ortiz, V., Y. M. Lopez-Alvarez et al. (2005). Phase diagrams and capillarity condensation of methane confined in single- and multi-layer nanotubes. Molecular Physics 103(19): 2587–2592. 87. Kowalczyk, P., L. Solarz et al. (2006). Nanoscale tubular vessels for storage of methane at ambient temperatures. Langmuir 22(21): 9035–9040. 88. Duren, T., L. Sarkisov et al. (2004). Design of new materials for methane storage. Langmuir 20(7): 2683–2689. 89. Heyden, A., T. Düren et al. (2002). Study of molecular shape and non-ideality effects on mixture adsorption isotherms of small molecules in carbon nanotubes: A grand canonical Monte Carlo simulation study. Chemical Engineering Science 57(13): 2439–2448. 90. Byl, O., P. Kondratyuk et al. (2003). Adsorption of CF4 on the internal and external surfaces of opened single-walled carbon nanotubes: A vibrational spectroscopy study. Journal of the American Chemical Society 125(19): 5889–5896. 91. Tanaka, H., M. El-Merraoui et al. (2002). Methane adsorption on single-walled carbon nanotube: a density functional theory model. Chemical Physics Letters 352(5–6): 334–341. 92. Maddox, M. W. and K. E. Gubbins (1995). Molecular simulation of fluid adsorption in buckytubes. Langmuir 11(10): 3988–3996. 93. Jakubek, Z. J. and B. Simard (2004). Two confined phases of argon adsorbed inside open single walled carbon nanotubes. Langmuir 20(14): 5940–5945. 94. Kosmider, M., Z. Dendzik et al. (2004). Computer simulation of argon cluster inside a singlewalled carbon nanotube. Journal of Molecular Structure 704(1–3): 197–201. 95. Rols, S., M. R. Johnson et al. (2005). Argon adsorption in open-ended single-wall carbon nanotubes. Physical Review B 71(15): 155411. 96. Kuznetsova, A., J. T. Yates et al. (2000). Physical adsorption of xenon in open single walled carbon nanotubes: Observation of a quasi-one-dimensional confined Xe phase. Journal of Chemical Physics 112(21): 9590–9598. 97. Matranga, C., L. Chen et al. (2004). Displacement of CO2 by Xe in single-walled carbon nanotube bundles. Physical Review B 70(16): 205427. 98. Jalili, S. and R. Majidi (2007). Study of Xe and Kr adsorption on open single-walled carbon nanotubes using molecular dynamics simulations. Physica E-Low-Dimensional Systems & Nanostructures 39(1): 166–170. 99. Jalili, S. and A. Vaez (2007). Xenon adsorption on defected single-walled carbon nanotubes. Chemical Physics Letters 437(4–6): 233–237. 100. White, C. M., D. H. Smith et al. (2005). Sequestration of carbon dioxide in coal with enhanced coalbed methane recovery – A review. Energy and Fuels 19(3): 659–724. 101. Matranga, C., L. Chen et al. (2003). Trapped CO2 in carbon nanotube bundles. Journal of Physical Chemistry B 107(47): 12930–12941.

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102. Müller, E. A. (2008). Staggered alignment of quadrupolar molecules inside carbon nanotubes. Journal of Physical Chemistry B 112(30): 8999–9005. 103. Khan, I. A. and K. G. Ayappa (1998). Density distributions of diatoms in carbon nanotubes: A grand canonical Monte Carlo study. Journal of Chemical Physics 109(11): 4576–4586. 104. Yin, Y. F., T. Mays et al. (1999). Adsorption of nitrogen in carbon nanotube arrays. Langmuir 15(25): 8714–8718. 105. Yoo, D. H., G. H. Rue et al. (2002). Study of nitrogen adsorbed on single-walled carbon nanotube bundles. Journal of Physical Chemistry B 106(13): 3371–3374. 106. Paredes, J. I., F. Suarez-García et al. (2003). N2 physisorption on carbon nanotubes: Computer simulation and experimental results. Journal of Physical Chemistry B 107(34): 8905–8916. 107. Arora, G., N. J. Wagner et al. (2004). Adsorption and diffusion of molecular nitrogen in single wall carbon nanotubes. Langmuir 20(15): 6268–6277 108. LyndenBell, R. M. and J. C. Rasaiah (1996). Mobility and solvation of ions in channels. Journal of Chemical Physics 105(20): 9266–9280. 109. Beckstein, O., P. C. Biggin et al. (2001). A hydrophobic gating mechanism for nanopores. Journal of Physical Chemistry B 105(51): 12902–12905. 110. Hummer, G., J. C. Rasaiah et al. (2001). Water conduction through the hydrophobic channel of a carbon nanotube. Nature 414(6860): 188–190. 111. Zhu, F. Q. and K. Schulten (2003). Water and proton conduction through carbon nanotubes as models for biological channels. Biophysical Journal 85(1): 236–244. 112. Müller, E. A., L. F. Rull et al. (1996). Adsorption of water on activated carbons: A molecular simulation study. Journal of Physical Chemistry 100(4): 1189–1196. 113. Müller, E. A. and K. E. Gubbins (1998). Molecular simulation study of hydrophilic and hydrophobic behaviour of activated carbon surfaces. Carbon 36(10): 1433–1438. 114. McCallum, C. L., T. J. Bandosz et al. (1999). A molecular model for adsorption of water on activated carbon: Comparison of simulation and experiment. Langmuir 15(2): 533–544. 115. Rivera, J. L., C. McCabe et al. (2002). Layering behavior and axial phase equilibria of pure water and water + carbon dioxide inside single wall carbon nanotubes. Nano Letters 2(12): 1427–1431. 116. Cicero, G., J. C. Grossman et al. (2008). Water confined in nanotubes and between graphene sheets: A first principle study. Journal of the American Chemical Society 130(6): 1871–1878. 117. Salih, A. and E. A. Müller (2008). http://erich.a.muller.googlepages.com/movies unpublished results 118. Hanasaki, I., A. Nakamura et al. (2008). Structure and stability of water chain in a carbon nanotube. Journal of Physics-Condensed Matter 20(1): 015213. 119. Hanasaki, I. and A. Nakatani (2006). Hydrogen bond dynamics and microscopic structure of confined water inside carbon nanotubes. Journal of Chemical Physics 124(17), p. 174714. 120. Waghe, A., J. C. Rasaiah et al. (2002). Filling and emptying kinetics of carbon nanotubes in water. Journal of Chemical Physics 117(23): 10789–10795. 121. Striolo, A. (2007). Water self-diffusion through narrow oxygenated carbon nanotubes. Nanotechnology 18(47): 475704. 122. Longhurst, M. J. and N. Quirke (2007). Temperature-driven pumping of fluid through singlewalled carbon nanotubes. Nano Letters 7(11): 3324–3328. 123. Noon, W. H., K. D. Ausman et al. (2002). Helical ice-sheets inside carbon nanotubes in the physiological condition. Chemical Physics Letters 355(5–6): 445–448. 124. Liu, Y. C., Q. Wang et al. (2005). Fluid structure and transport properties of water inside carbon nanotubes. Journal of Chemical Physics 123(23), p. 234701. 125. Mashl, R. J., S. Joseph et al. (2003). Anomalously immobilized water: A new water phase induced by confinement in nanotubes. Nano Letters 3(5): 589–592. 126. Byl, O., J. C. Liu et al. (2006). Unusual hydrogen bonding in water-filled carbon nanotubes. Journal of the American Chemical Society 128(37): 12090–12097.

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127. Alexiadis, A. and S. Kassinos (2008). The density of water in carbon nanotubes. Chemical Engineering Science 63(8): 2047–2056. 128. Koga, K., G. T. Gao et al. (2001). Formation of ordered ice nanotubes inside carbon nanotubes. Nature 412(6849): 802–805. 129. Striolo, A., K. E. Gubbins et al. (2004). Simulated water adsorption isotherms in carbon nanopores. Molecular Physics 102(3): 243–251. 130. Striolo, A., A. A. Chialvo et al. (2005). Water in carbon nanotubes: Adsorption isotherms and thermodynamic properties from molecular simulation. Journal of Chemical Physics 122(23), p. 234712. 131. Bai, J. E., J. Wang et al. (2006). Multiwalled ice helixes and ice nanotubes. Proceedings of the National Academy of Sciences of the United States of America 103(52): 19664–19667. 132. Takaiwa, D., I. Hatano et al. (2008). Phase diagram of water in carbon nanotubes. Proceedings of the National Academy of Sciences of the United States of America 105(1): 39–43. 133. Shao, Q., L. L. Huang et al. (2007). Molecular dynamics study on diameter effect in structure of ethanol molecules confined in single-walled carbon nanotubes. Journal of Physical Chemistry C 111(43): 15677–15685. 134. Gordillo, M. C. and J. Marti (2000). Hydrogen bond structure of liquid water confined in nanotubes. Chemical Physics Letters 329(5–6): 341–345. 135. Gordillo, M. C. and J. Marti (2003). Water on the outside of carbon nanotube bundles. Physical Review B 67(20): 205425. 136. Walther, J. H., R. Jaffe et al. (2001). Carbon nanotubes in water: Structural characteristics and energetics. Journal of Physical Chemistry B 105(41): 9980–9987. 137. Rana, M. and A. Chandra (2007). Filled and empty states of carbon nanotubes in water: Dependence on nanotube diameter, wall thickness and dispersion interactions. Journal of Chemical Sciences 119: 367–376. 138. Kondratyuk, P., Y. Wang et al. (2007). Inter- and intratube self-diffusion in n-heptane adsorbed on carbon nanotubes. Journal of Physical Chemistry C 111(12): 4578–4584. 139. Kondratyuk, P., Y. Wang et al. (2005). Observation of a one-dimensional adsorption site on carbon nanotubes: Adsorption of alkanes of different molecular lengths. Journal of Physical Chemistry B 109(44): 20999–21005. 140. Jiang, J. W. (2006). Pore size or geometry: Which determines the shape and inverseshape selective adsorption of alkane isomers? Journal of Physical Chemistry B 110(17): 8670–8673. 141. Agnihotri, S., J. P. B. Mota et al. (2006). Theoretical and experimental investigation of morphology and temperature effects on adsorption of organic vapors in single-walled carbon nanotubes. Journal of Physical Chemistry B 110(15): 7640–7647. 142. Supple, S. and N. Quirke (2004). Molecular dynamics of transient oil flows in nanopores I: Imbibition speeds for single wall carbon nanotubes. Journal of Chemical Physics 121(17): 8571–8579. 143. Hodak, M. and L. A. Girifalco (2003). Systems of C60 molecules inside (10,10) and (15,15) nanotube: A Monte Carlo study. Physical Review B 68(8): 085405. 144. Khlobystov, A. N., D. A. Britz et al. (2004). Observation of ordered phases of fullerenes in carbon nanotubes. Physical Review Letters 92(24): 245507 145. Baowan, D., N. Thamwattana et al. (2007). Zigzag and spiral configurations for fullerenes in carbon nanotubes. Journal of Physics a-Mathematical and Theoretical 40(27): 7543–7556. 146. Gao, H. J., Y. Kong et al. (2003). Spontaneous insertion of DNA oligonucleotides into carbon nanotubes. Nano Letters 3(4): 471–473 147. Zhao, X. C. and J. K. Johnson (2005). An effective potential for adsorption of polar molecules on graphite. Molecular Simulation 31(1): 1–10. 148. Lin, Y., S. Taylor et al. (2004). Advances toward bioapplications of carbon nanotubes. Journal of Materials Chemistry 14: 527–541. 149. Hilder, T. A. and J. M. Hill (2007). Modelling the encapsulation of the anticancer drug cisplatin into carbon nanotubes. Nanotechnology 18(27): 275704. 150. Hilder, T. A. and J. M. Hill (2008). Probability of encapsulation of paclitaxel and doxorubicin into carbon nanotubes. Micro and Nano Letters 3(2): 41–49.

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Chapter 4

Molecular Computations of Adsorption in Nanoporous Materials Ravichandar Babarao and Jianwen Jiang

Abstract Adsorption lies at the heart of many industrially important applications such as purification, separation, ion exchange, and catalysis. As the number of nanoporous materials synthesized to date is extremely large, rationally choosing a high-performance material from discovery to specific application is a substantial challenge. Computational approaches at the molecular scale can provide microscopic insight into adsorption behavior from the bottom-up, complement and secure correct interpretation of experimental results, and are imperative to new material design and advanced technological innovation. We review the recent computational studies of adsorption in nanoporous materials with a wide variety of building blocks and physical topologies, ranging from zeolites, carbonaceous materials to hybrid frameworks.

4.1 Introduction Nanoporous materials play a pivotal role in many engineering processes and industrial applications, for instance, the separation of impurities from natural gas to improve energy efficiency, the catalytic cracking of hydrocarbons, and the removal of toxic heavy metals from wastewater [1]. An overwhelmingly large number of porous materials have been and are being synthesized and characterized in the laboratory; consequently, theoretical guidelines are highly desired to properly choose a material for specific use. Due to the spatial confinement and surface interactions, fluids in nanodomain behave significantly different from bulk phases. A clear mechanistic understanding of fluid behavior in nanoporous materials is not only of fundamental interest but also of central importance for practical use. In the past, analytical approaches like density-functional theory (DFT) based on the advances in J. Jiang (B) Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576, Singapore e-mail: [email protected] L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_4,  C Springer Science+Business Media B.V. 2010

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statistical mechanics have been developed to examine the equilibrium properties of fluids in simple slit-like and cylindrical pores [2–4]. With an appropriate weighting function, DFT can accurately predict the density profiles, isotherms, configurations, layerings, etc. [5–7]. However, the extension of DFT to three-dimensional complex geometries is formidable. With ever-growing computational power, the state-of-theart molecular simulation has increasingly become a versatile tool to investigate the underlying properties of fluids confined in porous materials [8]. Simulation provides microscopic insight from the bottom-up that is otherwise experimentally inaccessible or difficult, if not impossible. In addition, simulation can be used to complement and secure fundamental interpretation of experimental results and establish the structure–function relations based on molecular description. Therefore, molecular simulation is indispensable in the rational design of novel materials of increasing complexity for new applications. In this chapter, we review the recent simulation studies of adsorption in zeolites, carbonaceous structures, and hybrid organic–inorganic frameworks. They represent three classes of fascinating nanoporous materials that are being widely used in chemical industries and/or exhibit unique properties for emerging applications. Experimental studies of adsorption in these materials have been comprehensively addressed; however, simulation studies are relatively less. Due to page limitation, the technical details of simulation principles are not described here, which are well documented elsewhere [9, 10]. Nevertheless, the commonly used simulation methods for adsorption in porous media are briefly introduced below. Stochastic Monte Carlo (MC) instead of deterministic molecular dynamics (MD) method is most frequently used for adsorption studies [11]. The reason is that MD is not straightforward to simulate adsorption isotherm as a function of bulk pressure, which is considered as one of the most important adsorption properties because isotherm can be readily determined from experiment. MD motion mimics the natural behavior of molecules through Newton’s equation of motion, that is, molecules diffuse into a porous material before adsorption equilibrium is reached. This is a slow process particularly for large molecules, such as alkanes and aromatics. In contrast, MC does not need to follow the natural pathway and thus allows random trial move so that a successful move may correspond to a large jump in phase space, thereby reaching equilibrium rapidly. Most MC simulation studies of adsorption are carried out in canonical or grand-canonical ensemble. In canonical ensemble, temperature, volume, and number of adsorbate molecules are fixed. MC simulation in canonical ensemble is usually used to simulate the distribution and energetics of adsorption sites, isosteric heat, and Henry’s constant. In grand-canonical ensemble, temperature, volume, and chemical potential of adsorbate are fixed. Grand-canonical Monte Carlo (GCMC) simulation permits adsorbate molecules to exchange between adsorbed phase and bulk fluid reservoir, and can provide adsorption isotherm easily. To compare GCMC results with experimental data, the chemical potential has to be converted into pressure using an equation of state or additional simulation. Widely used to simulate phase equilibria, Gibbs ensemble Monte Carlo (GEMC) method [12] has recently been used to simulate adsorption at a given pressure. GEMC is performed in two microscopic cells, one

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cell with the adsorbent and the other with the bulk fluid. At a given pressure, the volume of the bulk fluid is allowed to change. The total number of molecules is fixed, but molecules can be transferred from one cell to the other. In the studies reviewed here, these simulation techniques have been used to elucidate adsorption phenomena at the molecular level in zeolites, carbonaceous materials, and hybrid frameworks.

4.2 Zeolites Zeolites are three-dimensional crystalline aluminosilicate materials [13, 14]. The primary building blocks in zeolites are tetrahedral oxides SiO4 and AlO4 that connect together via corner-sharing oxygen atoms to form well-defined open pores. The pores are precisely uniform, which distinguishes zeolites from many other porous materials. Figure 4.1 shows three typical structures: MFI, FAU, and LTA. Some zeolites are not stable upon removing nonframework species, and the stability varies with topology and composition. Zeolites have been extensively used in industrial applications such as molecular sieving, catalysis, and ion exchange. Progress till 2001 in the computation of thermodynamic properties of guest molecules in zeolites was reviewed [15]. (a)

(b)

(c)

Fig. 4.1 Zeolites: (a) MFI, (b) FAU, and (c) LTA. Color code: red, O; yellow, Si; pink, Al

4.2.1 Light Gases N2 , O2 , and Ar in zeolites LTA, X-, and Y-FAU were studied using GCMC simulation; the predicted N2 /O2 selectivity in LTA was in accordance with experiment [16]. With the appropriate force field parameters, experimentally determined isosteric heats of NH3 adsorption in MOR, MFI, and Y-FAU were reproduced by simulation [17]. Adsorption of CH4 and CF4 in silicalite was measured and simulated as a function of phase composition, total pressure, and temperature; the simulated isotherms and isosteric heats were in good agreement with experiments [18]. A hierarchical modeling approach combining atomistic and coarse-grained simulations

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was proposed to predict adsorption thermodynamics of single components and binary mixtures in silicalite. The results agreed well with full atomistic simulation, but the hierarchical approach saved an order of magnitude of computational effort [19]. Long-range corrections were evaluated for simulation of CH4 and SF6 in silicalite at 298 K. A consistent use of cutoff radius was shown to be more important than the inclusion or the neglect of long-range corrections to potential energy [20]. Nonpolar (Ar and CH4 ) and quadrupolar gases (N2 and CO2 ) in pure siliceous FAU were investigated both experimentally and theoretically at ambient temperature and high pressures [21]. A systematic simulation study was reported for H2 in 12 purely siliceous zeolites, in which the cell parameters and framework flexibility were allowed to vary upon progressive filling of gas molecules, and then the maximum H2 adsorption capacity in each framework was examined [22]. CO2 and N2 both as single component and as binary mixture were studied in MFI, ITQ-3, and ITQ-7. The CO2 /N2 selectivity varies strongly with the crystal structure. Simulated binary adsorption agrees well with prediction by the ideal adsorbed solution theory (IAST) based on single-component adsorption [23]. Thermodynamics and siting of CO2 , CH4 , and their mixtures were studied in ITQ-1 which consists of two independent pores of different geometry. At the three temperatures considered, a preferential adsorption of CO2 over CH4 was found. The equilibrium selectivity was distinctly higher in its sinusoidal channel pore system than in the large cavity system over a wide range of pressures starting from the Henry law regime. A maximum in selectivity was observed at low temperature, high pressure, and CH4 rich gas-phase composition [24]. H2 O adsorption in X- and Y-FAU as well as in silicalite was computed. The isotherms and heats of adsorption were in good agreement with available experiments. The existence of cyclic H2 O hexamers located in the 12-ring windows of NaX, recently disclosed by neutron diffraction experiments, was observed [25]. A joint experimental and simulation study of H2 O adsorption in silicalite was reported. The simulation qualitatively reproduced experimentally observed condensation thermodynamic features. A shift and a rounding in the condensation transition were found with an increasing hydrophilicity of the local defect, but the condensation transition was observed above the saturation pressure of H2 O [26]. Simulation of H2 O in MFI by reducing the dipole moment of H2 O matched well with the experimental isotherm at 300 K. The work suggests that the effective intermolecular potential parameterized for bulk water is insufficient to describe the ultraconfined water molecules [27].

4.2.2 Alkanes and Alkenes With the development of configurational-bias Monte Carlo (CBMC) algorithm [28–30], adsorption of alkanes has been studied efficiently. Instead of inserting a molecule at a random position, a molecule in CBMC is grown atom-by-atom biasing toward the energetically favorable configurations while avoiding overlap with other atoms, and the bias is then removed by adjusting the acceptance rules [10]. Using the united-atom (UA) model for alkanes, Smit et al. observed an inflection

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in the adsorption isotherms of n-hexane and n-heptane in MFI, but not of shorter or longer alkanes. This inflection was attributed to the “commensurate freezing” effect as the size of the zigzag channel in MFI is commensurate with the size of n-hexane and n-heptane [31, 32]. Apart from MFI, alkanes were also simulated in other zeolites such as FER, MTW, TON, and DON using the same force field as for MFI and good agreement with experiments was found [33]. Linear alkanes ranging in length from C2 to C7 in AlPO4 -11 were examined and the isosteric heats at infinite dilution matched well with experimental data [34]. The UA model was also used to study adsorption of alkenes and their mixtures in silicalite-1, theta-1, and deca-dodecasil. For C3 H6 /C3 H8 mixture in Na-LTA, adsorption isotherm from simulation agrees well with experimentally measured results in cation-free LTA, but not in Na+ -exchanged LTA [35, 36]. The anisotropic united-atom (AUA) potential for hydrocarbons was adopted to calculate the adsorption isotherms of linear and branched alkanes in MFI; good agreement was observed with experiments and other simulation results [37]. Also, with the AUA potential, adsorption isotherms of pure alkenes and alkane/alkene mixtures in MFI were found to be closely consistent with experimental results [38]. Adsorption of n-pentane and n-hexane in FER showed that the prediction of UA potential is inconsistent with experiment, whereas the AUA potential can reproduce the subtle change in adsorption site occupancies of n-alkanes (n = 3−7) [39]. Simulation isotherms of cycloalkanes like cyclopentane, -hexane, and -heptane in MFI reproduced most features in experiments and revealed an inflection for cyclopentane but not for cyclohexane [40]. Comparing measured and simulated adsorption properties, a decrease in the Gibbs free energy of formation and adsorption was observed. Based on this observation, the selective production and adsorption of the most compact, branched paraffins in n-hexadecane hydroconversion in molecular sieves with pore diameters of ~0. 75 nm was satisfactorily elucidated [41]. Krishna et al. screened different zeolites for the separation of CH4 /CO2 mixture and of hexane isomers. The linear n-hexane has a longer “footprint” and occupies a longer segment in the zeolite channel; 2,2-dimethylbutane is the most compact isomer and has the smallest “footprint.” Consequently, a greater number of 2,2-dimethylbutane molecules were observed to be located within the MOR channels, in which the length entropy effect dictates [42, 43].

4.2.3 Aromatics Adsorption of aromatic molecules that fit tightly into zeolite frameworks has been investigated by simulations. Different guest–host force fields including all-atom, UA, and AUA models were used to simulate the adsorption of benzene, xylene, and cyclohexane. The force field considerably affects the locations of benzene and cyclohexane. This suggests that care should be taken in choosing the force field, particularly when the guest–host ratio is near the value defined for the levitation effect [44]. Benzene adsorption in different ORTHO silicalite structures was analyzed to determine the origin of a surprising factor of 3.1 difference in Henry’s constants. The results indicate that a slight change in the lattice oxygen atom positions can

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reposition the sorbates enough to sizably affect the electrostatics. The small shifts are magnified by the partial cancellation of large electrostatic terms on the hydrogen and carbon atoms [45]. Binary adsorption was simulated in silicate for benzene/CH4 and benzene/cyclohexane mixtures. Benzene and cyclohexane display preferred siting in the channel intersections, while CH4 adsorbs preferentially in the channels. In benzene/cyclohexane mixture, cyclohexane adsorbs in the intersections and benzene is pushed to the zigzag channels [46]. Adsorption in silicalite from binary mixtures of p-xylene, m-xylene, and toluene was investigated using simulation. Experimental isotherms can be well reproduced if MFI is modeled in the para form, but not in the native ortho form. This reveals that zeolite structure undergoes a transition from ortho to para in the presence of aromatic molecules [47]. Using a nine-site model for benzene, adsorption isotherms of benzene, propene, and their mixture in zeolites MOR, FAU, BEA, MFI, and MCM-22 were computed and in excellent accordance with measured results [48]. Benzene and benzene/thiophene mixtures were studied in FAU, MFI, MOR, and Na-FAU using energy-biased GCMC, and the calculated isotherms and heats in Na-FAU were consistent with experiments [49, 50]. Adsorption properties of xylenes in X- and Y-FAU were reproduced by simulation using a potential derived from Pellenq and Nicholson scheme. m-Xylene was found to be adsorbed preferentially and the p-xylene/m-xylene selectivity predicted from simulation was in good agreement with experiment [51–53]. Preferential adsorption of o-xylene over p-xylene in AlPO4 -11 was predicted by simulations, agreeing with experimental data. The selective adsorption comes from the smaller length of o-xylene along the crystallographic c-axis in AlPO4 -11 compared to p-xylene, different from the case in AlPO4 -5 and AlPO4 -8 [54].

4.2.4 Cation-Exchanged Zeolites A large number of simulation studies have been reported for adsorption in cationexchanged zeolites. Isotherms of H2 adsorption in Na-, Ni-, and Rh-exchanged XFAU were calculated at different temperatures and compared with measured data [55]. Adsorption of alkanes in two different Na-MOR zeolites from simulation was found to agree with experiment with the adequate inclusion of cations and realistic charges in Al atoms [56]. CH4 , C2 H6 , and C3 H8 in Na-exchanged MFI and MOR were simulated. The predicted positions of nonframework Na+ cations are in agreement with those determined by X-ray diffraction and so are the computed isotherms and Henry’s constants. Adsorption of alkanes is largely influenced by the position and the density of nonframework ions [57]. For a given type of cation, adsorption of alkanes in ZSM-5 was found to increase with decreasing density of the nonframework cations; for a given Si/Al ratio, adsorption increases with decreasing atomic weight of the cation [58]. Isotherm and enthalpy of N2 adsorption in X-FAU were simulated at ambient temperature, and qualitative agreement was observed with experiments for a series of monovalent and divalent cations [59]. Adsorption of CH4 in X- and Y-FAU was investigated using a newly derived force field in which the Lennard-Jones parameters between CH4 and FAU were obtained by fitting the

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potential energies from ab initio cluster calculations. The calculated isotherms and enthalpies were in good agreement with those experimentally obtained [60]. Redistribution of nonframework cations was observed to occur upon adsorption of water and aromatics in NaY, and in turn the selectivity of p-xylene over m-xylene was enhanced by a fact of 4 [61]. Single- and binary-component adsorption of CO2 , N2 , and H2 in dehydrated Na-4A was examined, and the strong selectivity for CO2 over both N2 and H2 was observed and also compared with that in MFI [62]. Supercritical CO2 adsorption in NaA and NaX was studied using GEMC, and the adsorption isotherms and enthalpies were discussed in detail and compared with available measured data [63]. Combining experimental and simulation techniques, adsorption of N2 , O2 , and Ar in Mn-exchanged zeolite-A and -X was studied. It was observed that the selectivity of O2 over Ar was higher in both zeolite-A and -X [64]. Ar, N2 , and O2 in Naand Ca-exchanged LTA were studied both experimentally and theoretically. The predicted isotherms match well with measured data and the extent of adsorption increases with increasing Ca ions in LTA [65]. Selectivity of CO2 over N2 in flue gas was found to rise remarkably in Na-ZSM-5, particularly at low pressures. This is because CO2 has a large quadrupolar moment and its adsorption is enhanced due to the electric field of cations [66]. Adsorption and enthalpy of CO2 in various types of FAU including purely siliceous DAY, NaY, and NaLSX were simulated. Typical arrangements of CO2 molecules were identified from low to high pressures [67]. Figure 4.2 shows that Na+ ions interact with a few CO2 molecules at a low pressure. With increasing pressure, Na+ ions are more and more solvated by the surrounding CO2 molecules. Consequently, CO2 /NaY interaction slightly decreases due to the solvation process, whereas CO2 /CO2 interaction increases. The combination of these two contributions leads to a relatively constant value of adsorption enthalpy as pressure rises [67]. CO2 adsorption was further simulated in LiY and NaY at various temperatures with Li+ –CO2 interaction derived from ab initio calculations. The results revealed two different types of adsorption behavior in NaY and LiY at 323 and 373 K, respectively [68]. Table 4.1 summarizes the simulation studies of adsorption for a wide variety of guest molecules in different zeolites.

Fig. 4.2 Typical arrangements of CO2 molecules in NaY at (a) 0.1 bar, (b) 1 bar, and (c) 25 bar. Na+ cations are represented in green, and the typical distances of Na–Oz and O–Oz are in Å. Reproduced with permission from [67]. Copyright (2005) American Chemical Society

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R. Babarao and J. Jiang Table 4.1 Simulation studies of adsorption in zeolites

Guests

MFI

X-FAU

Y-FAU

Other zeolites

Computed quantities

Rare gases N2

[69–76] [74, 90]

[16, 78–80] [16, 78]

[81–89] [16, 77]

P, I, Q, K, S P, I, Q, K, S

O2

[90]

[16, 77, 78] [16, 78, 91–93] [16, 78, 91–93] [92] [78]

CO CO2 SF6 NH3 H2 O Methane Ethane Propane n-Butane i-Butane n-pentane n-Hexane Cyclohexane n-Heptane

[94] [76] [17] [26, 95] [31, 70, 76, 90, 98–101]

n-Octane n-Nonane

[109, 110, 118]

n-Decane

[109–112, 118]

Ethane/butane [106] CF4 [76] CF3 Cl CF2 Cl2 CFCl3 CHF3 CHCl3 Isopropylamine Acetonitrile Benzene [45, 125, 126] o-Xylene m-Xylene p-Xylene

[125]

P, I, Q, K, S

[78] [17] [96] [80]

[31, 32, 105, 106] [31, 32, 105] [32, 105, 109–113] [111, 114, 115] [105, 109, 110, 116] [32, 105, 109, 110, 114, 116] [32, 105, 109–111, 114, 116] [109, 110, 118]

[16, 78]

[17] [96, 97] [76, 81, 83, 84, 86, 88, 98, 102–104] [103]

P, I, Q, K, S

[103, 107, 108] [87, 109, 110] [87, 103, 107–110]

P, I, Q, K, S P, I, Q, K, S

[109, 110]

P, I, Q, K, S P, I, Q, K, S

[103, 107–110]

[87, 109, 110] [87, 107–110, 114] [117] [87, 109, 110] [87, 107–110, 114]

P, I, Q, K, S

[109, 110, 118] [109, 110, 118] [109, 110, 118]

[109, 110, 118]

P, I, Q, K, S

[109, 110, 118]

P, I, Q, K, S

[109, 110, 118]

P, I, Q, K, S

[119] [76]

P, I, S P, I, Q, S P, Q, S P, Q, S P, Q, S P, Q, S P, Q, S Q, S P, S P, I, Q, K, S P, S P, I, Q, S

[122] [17]

[120] [120] [120, 121] [120] [120] [122, 123] [17]

[122, 127]

[127, 128]

[53]

[51, 52, 80, 128, 129] [51, 52, 80, 128, 129]

[17] [124] [54, 128]

[53]

P, I, Q, K, S P, I, Q, K, S P, I, Q, K, S Q, S P, I, Q, K, S P, I, Q, K, S

[54]

P, I, Q, K, S P, I, Q, K, S

P, I, Q, S

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Table 4.1 (continued) Guests Other alkylbenzenes m-Dinitrobenzene Phenol Pyridine

MFI

X-FAU

Y-FAU

Other zeolites

Computed quantities

[17]

P, I, Q, K, S P, S P, I, Q, K, S Q, S

[130] [126] [17]

[17]

P : interaction energy, I: adsorption isotherm, Q: heat of adsorption, K: Henry’s constant, S: structure of adsorbed phase.

4.3 Carbonaceous Materials Carbon atoms can possess different extents of aromatic sp2 or aliphatic sp3 hybridization and have different bonding and ring structures. As a consequence, there are several stable carbon forms ranging from naturally occurring bulk structures such as graphite, diamond, activated carbons [131] to discrete structures such as fullerenes and nanotubes [132]. Among these, activated carbons and nanotubes have been extensively studied as adsorbents for gas purification and separation.

4.3.1 Activated Carbons Activated carbon is a synthetic carbonaceous material, constructed by thermal decomposition and activation at elevated temperature. Activated carbon consists of elementary graphitic crystallites and amorphous structures. In many theoretical studies, the simplified carbon slit-like pores have been used to mimic activated carbons. Simulation of pure and mixed CH4 and CO2 adsorption in slit pores gave results close to the measured data in A35/4 activated carbon [133]. CO2 /CH4 /N2 mixtures in slit pores showed that CO2 is preferentially adsorbed and the simulation results are consistent with the predictions from DFT at various temperatures and pressures [134]. Adsorption of alkanes in slit pores was studied over a wide range of temperature, pressure, alkane chain length, and slit width to evaluate their effects. The behavior of long alkanes at high temperatures was found to be similar to short alkanes at lower temperatures [135]. The performance of 1- and 5-site models of CH4 adsorption in slit pores was evaluated. Although the two models yield comparable pore densities, the number of particles predicted by the 1-site model is always greater, regardless of whether temperature is subcritical or supercritical [136]. Adsorption and desorption of H2 O in a platelet model for activated carbon were simulated. The model included the effects of structural disorder which are missing in the slit pore model. Hysteresis observed was in good agreement with experimental results [137]. Simulation of pentane isomers and their ternary mixtures in a series of carbon nanoslits demonstrated competitive adsorption. With decreasing slit width, first shape selective adsorption occurs due to the configurational

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entropy effect, followed by inverse-shape selective adsorption that occurs due to the area entropy effect, and finally no adsorption occurs [138]. Using hypothetical C168 schwarzite to represent an amorphous nanoporous carbon, N2 , O2 , and their mixtures were investigated by GCMC and GEMC simulations [139, 140]. In addition to six-member rings, C168 schwarzite has some seven-member rings rather than five-member rings as in buckyball C60 . The curvatures of C168 and amorphous nanoporous carbon are similar; therefore, C168 may provide a similar environment for adsorbates as that found in real samples. The predictions of mixture adsorption using IAST based solely on the adsorption of pure gases agree well with simulation results. The energetic effect, by itself, cannot explain the large difference in the permeation rates of O2 and N2 observed experimentally. The entropic effect, which arises due to the size difference, is the dominant factor for the large selectivity favoring O2 over N2 [141].

4.3.2 Carbon Nanotubes A carbon nanotube (CNT) can be envisaged as a cylinder rolled from a graphene sheet and exists in three types, namely armchair, zigzag, and chiral. In 1991, Iijima discovered the first CNT [142] and has ever since triggered extensive interest in CNTs for a variety of applications. A number of earlier simulation studies were carried out in CNTs for the adsorption of N2 , O2 , CO2 , Ar, Kr, Xe, CH4 , etc. The adsorption isotherms of N2 [143], Xe [144], CH4 , and C2 H6 [145] were found to be of type I regardless of temperature, inconsistent with experimentally observed type II at subcritical temperatures. This is because the infinite periodic CNT bundles were used in these simulations; however, real CNT samples have finite diameters. As a result, the external surface of the finite CNT bundle was not included, which is available for gas adsorption as evidenced experimentally by gas adsorption on closed-ended CNTs. A handful of simulations were carried out to examine the role of external surface of CNT bundle on adsorption. Gas molecules were observed to form a quasi-onedimensional phase in the grooves at low pressures, whereas a “three-stripe” phase parallel to the grooves at high pressures, followed by monolayer and bilayer phases. The adsorption isotherms were predicted to be type II, consistent with experiment [146–148]. This suggests that the external surface must be accounted for to correctly predict adsorption in CNTs. The role of external surface was thoroughly exploded for N2 adsorption in two types of single-walled CNT bundles, as shown in Fig. 4.3 [149]. In the infinite periodic bundle, adsorption follows type I at both sub- and supercritical temperatures and occurs inside CNTs, first at the annuli and then at the centers. In the finite isolated bundle, adsorption follows type II at subcritical temperatures, as observed experimentally. Adsorption occurs first at the annuli and then at the grooves. At high pressures, adsorption also occurs at the ridges surrounding CNTs and at the centers, and on the external surface at still higher pressures. As temperature increases from sub- to supercritical, adsorption in the finite isolated bundle changes from type II to type I [149]. Adsorption of N2 and O2 mixture was

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Fig. 4.3 Energy contours of a nitrogen atom along the xy plane in (a) periodic infinite bundle and (b) isolated finite bundle. The vdW gap between CNTs is 3.2 Å. Adapted with permission from [149]. Copyright (2003) American Physical Society

further studied in the two types of CNT bundles mentioned above. The selectivity depends strongly on temperature but only weakly on the type of the bundle, and air might be separated by competitive adsorption in CNT bundles [150]. Simulation of CO2 trapped in CNT bundles showed a sequential filling of adsorption sites in CNT with the interstitial sites preceding endohedral sites. The drastic changes predicted from simulation in the binding energy distributions and density profiles qualitatively explained the frequency shifting, broadening, and integrated intensity changes in experimental infrared spectra [151]. Adsorption of an equimolar CO2 /CH4 mixture was determined in five CNTs with various diameters to investigate the effects of temperature, pressure, and pore size. The results revealed that pressure and temperature have little effect, in contrast to the pore size [152]. CH4 adsorption was examined in triangular bundles of armchair CNTs with the vdW gap varying from 0.335 to 1.0 nm. The results demonstrated that (15,15) CNT bundle with the vdW gap of 0.8 nm is optimal for CH4 storage at 300 K [153]. Adsorption and separation of linear and branched alkanes in CNTs were studied at room temperature. The results suggested the possible separation of alkane mixtures based on difference in either size or configuration, as a consequence of competitive adsorption [154]. Shape and inverse-shape selectivity were observed for C5 isomers in CNTs and the tube size was identified to be crucial as to which type of selective adsorption might occur. In (7,7) CNT, inverse-shape selective adsorption occurs due to the length entropy effect, whereas in large-size CNT, shape selective adsorption occurs due to the configurational entropy effect [155]. Real CNT samples consist of heterogeneous rather than homogeneous bundle. Shi and Johnson proposed an optimization method to construct a heterogeneous CNT bundle with different diameters, as illustrated in Fig. 4.4. At low coverages, simulated isosteric heats in the heterogeneous bundle were consistent with experimental data. In contrast, the isosteric heats in the homogeneous bundle were about 25% lower. Therefore, an accurate description of adsorption in CNT bundle must

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(a)

R. Babarao and J. Jiang

(b)

Fig. 4.4 (a) A heterogeneous CNT bundle. (b) Isosteric heat for Ar. Circles and triangles are experiments, diamonds (squares) are simulations in the heterogeneous (homogeneous) bundles. Reproduced with permission from [156]. Copyright (2003) American Physical Society

account for the heterogeneity [156]. Displacement of CO2 by Xe was simulated in the heterogeneous bundle also. The spectral information constructed from simulation qualitatively reproduced experimental intensity. The good agreement between simulation and experiment suggested that the adsorption sites associated with the intensity peaks and changes upon Xe exposure are the consequence of CO2 being displaced from the sites [157].

4.3.3 H2 Storage High-capacity H2 storage is an essential prerequisite for the widespread deployment of next-generation fuel cells, particularly in portable devices and future automobiles. Considerable research has been undertaken over the past two decades to determine H2 capacity in different carbon structures like activated carbons, carbon nanofibers, and CNTs. Experimental studies have been reviewed recently by a few researchers [158–161]. H2 storage in primitive, gyroid, diamond, CNT, and quasi-periodic icosahedral nanoporous carbons was examined by simulation. None of these satisfies storage target, except the quasi-periodic icosahedral nanoporous carbon, which could accommodate 6 wt% H2 at 3.8 MPa and 77 K but the volumetric density does not exceed 24 kg/m3 [162]. It is clear from this work that the geometry of carbon surface can enhance capacity only to a limited extent, and the combination of most effective structure with appropriate additives should be incorporated to improve capacity. H2 adsorption in CNTs was simulated at 293 K and 77 K with an effective quantum potential derived from the Feynman–Hibbs perturbative approach [163, 164]. The capacity at 298 K predicted by quantum simulation is several percent smaller than that by classical simulation, and the quantum effect contributes 15–20% of adsorption amount at 77 K. Carbon nanohorn (CNH) is a

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Molecular Computations of Adsorption in Nanoporous Materials

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tube of typically 2–6 nm in diameter and 40–50 nm long and has a conical cap at one end. H2 isotopes in CNHs were simulated at 77 K, and the adsorption, incorporating quantum effect through the Feynman–Hibbs potential, was predicted to be in good agreement with experimental results [165]. This indicates that the quantum effect at 77 K is accurately represented by the Feynman–Hibbs potential. The effects of size, vdW gap, and CNT arrangement were investigated on H2 adsorption. Interestingly, it was observed that the size of CNT has significant influence on the amount adsorbed inside CNT and distribution in the vicinity of CNT wall, regardless of the type of CNT and vdW gap [166]. To enhance H2 storage capacity in CNTs, several approaches such as element doping [167], incorporating defects, spilling over [168], changing the size and gap of CNTs [166], introducing heteroatoms [169] as well as using silicon nanotubes (SiNTs) [170] were proposed. Both N and B doping in single-walled CNTs decrease H2 adsorption energy. The effect of alkali doping on H2 adsorption was simulated by incorporating K or Li ions into CNT arrays. H2 capacity was found to be 3.95 wt% with K doping and 4.21 wt% with Li doping, in fairly good agreement with the experimental data determined at room temperature and 100 atm [171]. There is a considerable increase in H2 binding energy by a factor of 50% in the presence of structural defects in CNTs, which in turn enhances capacity [172]. H2 adsorption in SiNTs was studied at 298 K over pressure range from 1 to 10 MPa. SiNTs exhibit a stronger attraction to H2 both inside and outside the tube compared to the isodiameter CNTs; consequently, the capacity in SiNTs is enhanced [170]. Carbon nanoscroll (CNS) structure is formed by jelly roll-like wrapping of a graphene sheet. H2 adsorption in CNSs was found to reach 5.5 wt% at 150 K and 1 MPa. Doping alkali in CNSs leads to H2 capacity of 3 wt% at ambient temperature and pressure [173, 174]. Recent analysis of thermodynamic constraints for H2 adsorption with a focus on porous carbon suggested that an optimal adsorption enthalpy of 15 kJ/mol is required to meet H2 storage target at room temperature. The enthalpy in most carbon structures is about 4–6 kJ/mol, too weak to satisfy the desired goal for H2 storage at ambient temperature. Structural modification of carbon revealed the complex relationship between adsorption enthalpy, pore volume, and the amount of H2 stored. It was observed that increasing adsorption enthalpy might reduce storage as well as delivery [175]. Numerous studies investigated the interaction of H2 with CNTs using quantum mechanics methods. Periodic DFT calculations of H2 chemisorption in CNTs showed two chemisorption sites, one inside and the other outside. H2 capacity in an empty space increases linearly with CNT diameter, and the maximum capacity is limited by the repulsive energies between H2 molecules inside CNT and those between H2 molecules and CNT wall [176]. Ab initio calculations in a cleaved cluster model revealed that boron nitride nanotubes are preferable to CNTs for H2 storage due to their heteropolar binding nature of their atoms. In addition, more efficient binding energy can be achieved by increasing nanotube diameter or equivalently decreasing the curvature [177]. H2 physisorption inside and outside achiral CNTs was examined with the CNTs modeled as curved coronene-like (C24 H12 ) graphene sheets. It was found that physisorption depends mainly on CNT diameter being

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virtually independent of chirality [178]. Another commonly used method is the two-level quantum mechanics/molecular mechanics (QM/MM) approach, in which a system is divided into two different sections; the inner section is treated by a high-level QM method and the outer section by MM method. With this method, H2 binding to the side of a (10,0) CNT was studied, in which B3LYP hybrid functional was used for the accurate calculation of reaction sites, while the universal force field (UFF) was used to describe the neighboring atoms [179]. This approach was also applied to a (4,4) CNT with 200 atoms by treating 64 carbon atoms and 32 hydrogen atoms in the inner section with a high-level theory and the outer section with a low-level theory. This study was focused on H2 coverage in CNT and the intercalation of H2 inside CNT [180]. Recently, Froudakis reviewed the existing theoretical literature on H2 adsorption in CNTs, in which the importance of simulations for understanding H2 adsorption mechanism and for improving storage capacity was underlined, and the advantages and disadvantages of both statistical and quantum mechanics modeling were discussed [181].

4.4 Hybrid Frameworks Novel hybrid inorganic–organic porous materials have been recently developed, most notably, the metal–organic frameworks (MOFs) synthesized by Yaghi and coworkers [182]. The primary building blocks in MOFs (also known as coordination networks or coordination polymers) are metal-oxide clusters and organic linkers. MOFs possess extremely high porosities (up to 90%) and large surface areas (from 500 to 6500 m2 /g). The controllable length of organic linker and the variation of metal oxide allow for tailoring the functionality, pore volume, and size over a wide atomic-scale range, as shown in Fig. 4.5 [183]. In contrast to the spherical or slitshaped pores usually observed in carbon and zeolite materials, MOFs incorporate crystallographically well-defined pores including squared, rectangular, triangular, and window-connected cages. Therefore, MOFs provide a wealth of opportunities for engineering new functional materials with tunable properties [184].

4.4.1 Light Gases In conjunction with experiment, simulation was performed to investigate the porefilling mechanism of Ar in Cu-BTC. The results agreed quantitatively with experimental isotherm up to almost complete filling of the pore network [185]. Similarly, the predicted positions and occupancies of Ar adsorption in IRMOF-1 were found to match well with experiments [186]. Binding energies of Ar, N2 , O2 , CH4 , C2 H6 , and C3 H8 were computed separately. The most preferred site is near the metaloxide cluster in the cavity where the organic linker points outward, the second preferred site is where the linker points inward, and the least preferred site is above and beneath the linker [186]. Adsorption of nonpolar gases like CH4 , N2 , and O2 was extensively studied in MOFs, particularly IRMOF-1. Different MOFs were

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Fig. 4.5 IRMOF-n (n = 1−6) structures. Zn (blue polyhedra), O (red spheres), C (black spheres), Br (green spheres in 2), amino groups (blue spheres in 3). The large yellow spheres represent the largest vdW spheres that would fit into the cavities without touching the frameworks. Reproduced with permission from [183]. Copyright (2002) American Association for the Advancement of Science

characterized for CH4 storage and compared with MCM-41, zeolites, and CNTs. The complex interplay of various factors that affect CH4 adsorption was uncovered and new, not yet synthesized, MOFs were proposed [187]. Adsorption of He, Ar, H2 , and CH4 was examined in various MOFs, including Cu-BTC, IRMOF-1, IRMOF6, IRMOF-8, IRMOF-14, MOF-2, MOF-3, and Cu-MMOM, and good agreement was observed between simulations and experiments for a number of cases and very poor agreement in other cases [188]. Simulation was performed on CH4 adsorption in a series of MOFs. The accessible surface area and the free volume were found to play a major role in adsorption at 298 K and moderate pressures [189]. From simulated adsorption isotherms of N2 , the BET surface areas of microporous MOFs were found to agree very well with the accessible surface areas estimated in a geometric fashion directly from the experimental crystal structures. In addition, the surface areas matched well with experimental reports in the literature. These results provide a strong validation that BET theory can be used to obtain surface areas of MOFs [190]. Mixture separation of CH4 and nC4 in IRMOFs was investigated to study the influence of organic linkers. The predicted selectivity was as good as or better than experimentally observed selectivity in other adsorbents, suggesting that IRMOFs are promising for hydrocarbon separation [191]. Adsorption of a mixture of pentane isomers in IRMOF-1 was simulated. Each isomer exhibits increased extent of adsorption with rising pressure, though there is less adsorption of the branched isomer due to the configurational entropy effect. Compared to CNT bundle and MFI,

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the adsorption selectivity in IRMOF-1 is smaller [192]. Gas mixtures containing CO2 , CH4 , C2 H6 in Cu-BTC were studied, and it was observed that the selectivity of CO2 over CH4 increases due to the electrostatic interaction between CO2 and the framework [193]. Mixture adsorption of CO2 and CH4 , H2 simulated in CuBTC and IRMOF-1 showed that the geometry, the pore size, and the electrostatic interaction in MOFs largely affect separation efficiency [194].

4.4.2 CO2 Storage The combustion of fossil fuels such as coal and petroleum has generated a huge amount of greenhouse gas CO2 . This has substantially led to severely adverse impacts on environment like air pollution and global warming. One of the technically feasible approaches for CO2 sequestration is to use porous media, and CO2 storage in MOFs has been explored using molecular simulations. Adsorption in three different types of materials – silicalite, C168 schwarzite, and IRMOF-1 – revealed that IRMOF-1 has the largest adsorption capacity for CO2 [195]. A different class of MOFs, based on the assembly of presynthesized molecular building units, were produced and named as MIL (Material Institut Lavoisier) series. Simulation of CO2 in MIL-53 and MIL-47 confirmed that there is a structural interchange between large and narrow pore forms of MIL-53, but not in MIL-47 [196]. CO2 adsorption was simulated in IRMOF-1 at five different temperatures and in IRMOF-3 and MOF-177 at 298 K. The results matched well with experimental data and suggested that the attractive electrostatic interactions between CO2 molecules are responsible for inflections and steps observed in the adsorption isotherms [197]. A computational study of CO2 storage was reported in MOFs with various linkers, pore sizes, and topologies. The capacity was a complex interplay of these structural properties, and the suitable pore size was found to be between 1.0 and 2.0 nm [198]. CO2 storage in a series of MOFs was simulated systematically and compared with those in CNT and Na-exchanged ZSM-5 [199]. Figure 4.6 (left) shows the snapshot for CO2 adsorption in IRMOF-1 and Fig. 4.6 (right) shows the adsorption isotherms in different MOFs. The organic linkers play a critical role in tuning the free volume and accessible surface area, which largely determine the CO2 adsorption at high pressures. As a combination of the high capacity and low framework density, IRMOF-10, IRMOF-14, and UMCM-1 were identified to be the best among those studied and even surpass the experimentally reported highest capacity in MOF-177. The simulation showed that the larger the surface area and pore volume, the higher the storage capacity achieved. Various factors like surface area, free volume, porosity, and framework density were found to correlate well with CO2 capacity near saturation [199].

4.4.3 H2 Storage H2 adsorption in MOFs has been extensively simulated toward the development of new storage media. A study of H2 in IRMOF-1, IRMOF-8, and IRMOF-18 showed

4

Molecular Computations of Adsorption in Nanoporous Materials IRMOF-1 Mg-IRMOF1 Be-IRMOF1 IRMOF1-(NH2)4 IRMOF10 IRMOF13 IRMOF14 UMCM-1

40 Nex(mmol/g)

85

30 20 10 0

0

1000

2000 3000 P(kPa)

4000

5000

Fig. 4.6 (left) Snapshot of CO2 adsorbed in IRMOF-1 at 300 K and 2000 kPa. (right) Excess adsorption isotherms of CO2 in different MOFs versus bulk pressure. Reproduced with permission from [199]. Copyright (2008) American Chemical Society

that metal-oxide clusters are the preferential adsorption sites for H2 and the effect of organic linkers becomes evident with increasing pressure [200]. Simulation of H2 in 10 noninterpenetrating MOFs revealed that H2 uptake correlates well with isosteric heat at low pressures and with the surface area and free volume at intermediate and high pressures [201]. In interpenetrating MOFs, the small pores generated by catenation play a primary role in densely confining H2 molecules so that the capacity is higher in interpenetrating frameworks than in noninterpenetrating frameworks [202]. The interactions between H2 and MOFs were artificially increased in simulation to learn the degree to which the isosteric heat must be increased to meet the current target for H2 storage. H2 density within the free volume of materials provided a useful insight and yielded a graph for the required isosteric heat as a function of the free volume to meet the storage target at room temperature and 120 bar [203]. A number of theoretical studies on the interactions between H2 and MOFs based on quantum mechanics have been reported. Generally, metal-oxide sites bind H2 more strongly (7–8 kJ/mol) than do organic linkers (4–5 kJ/mol). But as the available volume of the metal-oxide sites is small, they tend to be saturated quickly, while organic linkers play a more crucial role at higher pressures [204]. The stronger H2 binding observed at the metal-oxide sites can be attributed to the electrostatic and, possibly, orbital donation interactions, while the vdW interactions dominate at the organic linkers [205]. Several polar aromatic linkers of MOFs were examined to predict H2 binding using DFT theory, and the computed binding energies were in good agreement with experiments [206]. With the approximate resolution of identity MP2 (RI-MP2) and triple-zeta valence basis set, H2 binding energies were calculated with various substituted benzenes like C6 H6 , C6 H5 F, C6 H5 OH, C6 H5 NH2 , C6 H5 CH3 , and C6 H5 CN, in which the substituted benzenes were treated as simplified subunits of the organic linkers in MOFs. The interaction energy with C6 H5 NH2 was

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found to be the strongest (4.5 kJ/mol) [207]. Similarly, the binding energies were estimated by MP2 with the metal oxide and organic linker in IRMOF-1. GCMC simulation identified a high-energy binding site at the corner that quickly saturated with 1.27 H2 molecules at 78 K, whereas a broad range of binding sites were observed at 300 K [208]. Various IRMOFs such as IRMOF-1, IRMOF-3, IRMOF4NH2 , IRMOF-6, IRMOF-8, IRMOF-12, IRMOF-14, IRMOF-18, and IRMOF-993 were studied using RI-MP2. The highest binding energy was found in IRMOF4NH2 , even higher than in polybenzoid structures such as IRMOF-14 and IRMOF993 [209]. A significant enhancement of H2 uptake in Li-decorated IRMOF-1 was revealed with 2.9 wt% at 200 K and 2.0 wt% at 300 K. Two Li atoms strongly adsorb on the surfaces of six-carbon rings, one on each side. Each Li atom can cluster three H2 molecules with a binding energy of 12 kJ/mol [210]. Using both DFT theory and MP2 calculations, the interaction energies of H2 adsorption with benzenoid molecular linkers in MOFs were computed. Both the local-density and the generalizedgradient approximated DFT methods were inaccurate in predicting binding energy but gave a qualitatively correct prediction [211]. A strong correlation was demonstrated between H2 surface density and coordinatively unsaturated metal centers in MOFs. Quantum mechanical calculations were performed to estimate the shortest distance achievable between H2 molecules, thereby defining the surface area requirements for MOFs that can reach the target for H2 storage [212]. Table 4.2 summarizes the simulation studies of adsorption for a wide variety of guest molecules in different MOFs.

4.4.4 New Hybrid Frameworks A recent breakthrough in the development of hybrid frameworks is the crystalline, porous, covalent organic frameworks (COFs) [220–223]. Solely synthesized from light elements like B, C, O, and H, COFs consist of the organic linkers covalently bonded with boron-oxide clusters and have high thermal stability, large surface area, and porosity. These boron-oxide clusters can be regarded as analogous to the metaloxide clusters in MOFs. Composed of light elements, however, COFs have even lower density than MOFs. Currently, very few simulation studies have been reported for COFs. Adsorption isotherms of Ar, CH4 , and H2 in COF-102, -103, -105, and -108 were simulated, in which COF-102 and COF-103 showed greater affinity for CH4 due to the compact atomic structures [223]. CO2 storage in various 3D, 2D, and 1D COFs structures was simulated, and exceptionally high capacity in COF-105 and -108 was observed. The gravimetric and volumetric capacities at 30 bar correlated well with various factors such as framework density, free volume, porosity, and surface area [224]. Another breakthrough is the invention of zeolitic imidazolate frameworks (ZIFs) [225]. ZIFs have high thermal and chemical stability like zeolites and also high porosity and pore functionality like MOFs. Similar to MOFs, ZIFs showed promising storage and separation capacity [226, 227]. At the time of preparing this chapter, we were not aware of any reported simulation work in ZIFs.

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Table 4.2 Simulation studies of adsorption in MOFs Computed quantitiesb

References

M1, Cu-BTC, MI M1, Cu-BTC Cu-BTC M1-10, MOF-177, Cu-BTC, UMCM-1,MI, ML1, ML2, ML3, ML4, F-MOF1 M1, M3, M5, M6, M9, M10, ML1, Cu-BTC, CPL-2, CPL-5, MI, CU1, CU2 M1 M1, MM1 M1, M5, M6, M7, M8, MM1 M1 M1 M1 M1 M1 M1, M4, M5, M8, M9 M1, Cu-BTC M1, Cu-BTC Cu-BTC M1 M1 M1, M4

P, I, S P, I, Q, S P, I, S P, I, Q, K, S, A, F

[186, 188, 213] [186, 214] [214] [193–197, 199, 214, 215]

P, I, Q, K, S

[187, 191–193, 195, 216]

P, I, Q, K, S P, I, Q, K, S P, I, Q, K, S

[192] [192, 217] [191, 192, 217]

P, I, Q, K, S P, I P, I P, I P, I P, I, S P, I, S P, I, S P, I, Q, S P, I, S P, I, S P, I, S

[192] [218] [218] [218] [218] [191] [194] [193, 195] [214] [192] [192] [219]

M1, M4

P, I, S

[219]

M1, M4

P, I, S

[219]

Guests

Hostsa

Ar N2 O2 CO2

Methane

Ethane Propane n-Butane i-Butane n-Pentane n-Hexane Cyclohexane n-heptane Methane/butane CO2 /CH4 /H2 CO2 /CH4 CO2 /N2 /O2 C1 –C5 alkanes C5 isomers n-Butane/2methylpropane n-Pentane/2methylbutane n-Hexane/2methylpentane

a M1: IRMOF-1, M2: IRMOF-3, M3: IRMOF-4, M4: IRMOF-6, M5: IRMOF-8, M6: IRMOF10, M7: IRMOF-11, M8: IRMOF-13, M9: IRMOF-14, M10: IRMOF-16, MI: manganese formate, ML1: MIL-53, ML2: MIL-47, ML3: MIL-100, ML4: MIL-101, MM1: Cu (hfipbb) (H2 hfipbb), CU1: Cu(SiF6 )(bpy)2 , CU2: Cu(GeF6 )(bpy)2 . b I: adsorption isotherm, Q: heat of adsorption, K: Henry’s constant, A: accessible surface area, F: free volume, P: interaction energy, S: structure in adsorbed phase.

4.5 Outlook We have reviewed the recent simulation studies of adsorption for a wide variety of guest molecules in three important classes of nanoporous materials – zeolites, carbons, and hybrid frameworks. The contents reviewed are representative rather than exhaustive. From simulations, useful adsorption properties, such as binding

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sites, interaction energies, isosteric heats, isotherms, and separation factors, can be obtained and compared with available experimental data. It is important to note that, given a material, the accuracy of simulation is primarily determined by the guest– framework interactions [228]. In a large number of simulation studies, the empirical force fields have been used. For example, classical Kiselev [78] and Steele [229] potentials are commonly used for zeolites and carbon materials, respectively. For MOFs, the universal [230] and Dreiding [231] force fields are often used. These force fields were constructed by fitting to some experimental data over a limited range of conditions with certain empirical rules. Their semiempirical nature may lead to inaccurate predictions. A more rational way is to calculate guest–framework interactions from the first-principles methods. DFT has been widely used in solidstate materials; however, it fails to describe the weak physisorption interactions because the dispersion forces are not properly accounted for in the DFT theory. A major obstacle to calculate the guest–framework interactions is that high-level methods are usually required, but they are computationally very expensive, particularly for large structures. Consequently, cost-effective hierarchical approaches are adopted as a compromise of accuracy and speed. Most simulation studies on adsorption have used rigid frameworks. This allows the use of grid-interpolation techniques to compute the interactions between guest molecules and framework very efficiently. Such a simplified treatment is usually acceptable but cannot reproduce structural changes that might occur upon adsorption. The effect of framework flexibility on hydrocarbon adsorption in silicalite was examined. At low loadings, the effect is small but seems to be increasingly larger at high loadings [232]. In contrast to the relatively rigid zeolitic and carbon structures, adsorption in MOFs could easily cause structural transformation due to the existence of organic linkers. For example, a hysteresis was observed in a pillared-layer MOF, which undergoes expansion and contraction (27.9% reduction in cell volume) during adsorption and desorption [233]. However, most simulation studies scarcely include the flexibility of MOFs. H2 O in IRMOF-1 examined by molecular dynamics simulation with a flexible force field revealed that IRMOF-1 is stable at very low H2 O content but unstable upon exposed to ≥4% H2 O [234]. Exceptionally negative thermal expansion in IRMOFs was explored and two competing effects were identified. One is a local effect where all bond lengths increase with temperature and a long-range effect where the thermal movement of the linker molecules leads to a shorter average distance between corners upon heating [235]. The review is focused mainly on adsorption. To further elucidate fluid behavior in porous media, however, dynamic diffusion should be synergistically incorporated [236]. Similar to adsorption, a wealth of simulation studies have been reported for diffusion in various porous materials. Diffusion of CH4 , CO2 , and N2 in silicalite was simulated over a wide range of occupancies and compared with experimental data [237, 238]. Comparison of the self-diffusivities of CH4 /CF4 mixture in silicalite was conducted between simulation and NMR experiment, and good agreement was found [239]. A dynamically corrected transition-state theory was used in simulation to quantitatively compute the self-diffusivity of adsorbed molecules in confined systems at nonzero loading [240]. Diffusion of various gases (He, Ne, Ar, Kr, H2 ,

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N2 , CO2 , and CH4 ) was investigated in six all-silica zeolites MFI, AFI, FAU, CHA, DDR, and LTA [241]. The self- and transport diffusivities of light gases like H2 , CH4 , Ar, and Ne were simulated in CNTs and in zeolites with comparable pore sizes. It was found that diffusion in CNTs is 1–3 orders of magnitude faster [242, 243]. Mass transport of O2 , N2 , and their mixture in a CNT demonstrated that a good kinetic selectivity could be achieved for air separation by carefully adjusting the upstream and downstream pressures [244]. Self- and transport diffusions of light gases in MOF-2, MOF-3, MOF-5, and Cu-BTC as a function of pore loading were found to be in the same order of magnitude as in silicalite [245]. Simulation of benzene in IRMOF-1 revealed that the diffusion and the activation energy of benzene are considerably smaller in a flexible framework compared to a rigid one [246]. Diffusion and separation of CO2 and CH4 in silicalite, C168 schwarzite, and IRMOF1 were examined. The predictions of self-, corrected-, and transport diffusivities from the Maxwell–Stefan formulation match well with the simulation results [247]. A critical appraisal of current estimation methods was presented to predict binary mixture diffusions in a large number of porous materials, including zeolites (MFI, AFI, TON, FAU, CHA, DDR, MOR, and LTA), CNTs, titanosilicates (ETS-4), and MOFs (IRMOF-1 and Cu-BTC) [248]. In summary, molecular simulations have substantially enhanced the current state of knowledge for guest molecules in nanoporous materials. Fundamental insight has been provided to bridge the large gap between microscopic and macroscopic properties in the nanodomain. On the basis of molecular description, structure–property and function relations can be developed toward high-efficacy material screening. Molecular simulations thus facilitate and accelerate the intelligent design of new porous materials with novel topological and compositional characteristics for industrially important applications. Acknowledgments The authors are grateful for the support from National University of Singapore (Grants R-279-000-198-112/133 and R-279-000-243-123).

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Chapter 5

Polymers in Nanotubes Guiduk Yu, Woojeong Cho, and Kyusoon Shin

Abstract Differently from small molecules like gas or liquid molecules, polymers show unique properties that are dependent on the size (degree of polymerization) of the molecules. As geometric constraints on the nanoscale are imposed upon polymers, their physical behavior also tends to deviate from that in bulk. Especially in cylindrical nanopores, the imposed curvature and two-dimensional nanoconfinement are mostly considered to cause the deviation in the physical behavior. Since the size of a polymer chain generally ranges from few nanometers to few tens of nanometers in bulk, the nanoscopic geometric constraint influences the static as well as dynamic behavior in the molecular scale of polymers such as mobility, crystallization mechanism, phase behavior. In this chapter, it is generally discussed how cylindrical nanoconfinement affects the physical behavior of polymers.

5.1 Introduction A polymeric molecule consists of monomers linked by covalent bonds. As the molecular size of polymers can change with the number of monomers in the chain, some physical properties depend on the degree of polymerization: viscosity, glass transition, mechanical strength, surface tension, etc. Besides the chemical environment (the degree of polymerization), a physical environment has influence on the physical behavior of polymers. Among the physical environmental factors, geometric constraint has been especially reported to render polymeric materials unique physical properties deviating from those in bulk. So far, a lot of effort has been made to elucidate intriguing phenomena of polymers under one-dimensional (1D) confinement (e.g., the geometry of thin film). Recently, two-dimensionally (2D) confined polymers (polymeric nanotubes, nanorods, and K. Shin (B) School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea e-mail: [email protected]

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nanowires) have attracted attention not only due to the fundamental importance but also due to their potential applicability in extensive fields like photonics, sensorics, electronics, and medicine. Especially for polymers, their molecular dimensions range from few nanometers to few tens of nanometers. The two-dimensional geometric constraint, possibly controlled in the range, has been reported to bring about interesting physical behavior: enhanced diffusion mobility, change in crystallization mechanism, and confinement-induced morphology. This chapter introduces the methods to obtain 1D structured polymers (Section 5.2) and mainly discusses the physical behavior of polymers under 2D geometric constraint (Sections 5.3 and 5.4).

5.2 Infiltration of Polymers in Nanotubes In order to obtain polymeric nanotubes, nanorods, or nanowires, organic or inorganic templates with cylindrical nanopores are usually employed due to ease of dimension control (uniform diameter and length of nanopores) [1–10]. Taking advantage of template infiltration method, the curved interface and interfacial tension in addition to nanoscopic confinement exert influence over the physical behavior of polymers in cylindrical nanopores. Therefore, interesting behavior has been observed in amorphous polymers, crystalline polymers, and block copolymers whose chain size or domain size is comparable to the extent of the geometric constraint. This section covers the preparation methods to obtain polymers in the geometric structures of nanotubes, nanorods, or nanowires: polymerization of monomers directly in nanopores and introduction of presynthesized polymer in nanopores. First, polymeric nanotubes or nanowires can be prepared by electrochemical or chemical polymerization of the monomers and the polymerization initiator in nanopores [3–6]. The monomers and the initiator diffuse into nanopores and react to yield polymer molecules, and then the polymer chains favorably nucleate on the pore wall, forming a tubular structure in the beginning of the polymerization. By increasing the polymerization time, the polymeric nanotubes further turn into polymeric nanowires (Fig. 5.1(a)). Conductive polymers such as polypyrrole, polyaniline, and poly(3-methylthiophene) as well as electronically insulating polymer like polyacrylonitrile have been synthesized and obtained as nanotubes or nanowires by the direct polymerization method in nanopores. Another preparation method is to infiltrate presynthesized polymers into the nanopores [7–10]. Presynthesized polymers can be introduced in the form of either polymer solution or polymer melt. For both polymer solution and polymer melt, capillary force is the driving force to fill nanopores. Since the diameter of nanopores is in nanoscopic scale (much smaller than the capillary length), the gravity force is negligible with comparison to capillary force. On the other hand, viscosity acting as a cohesive force among polymer molecules is still influential in nanopores, which makes adhesive energy and cohesive energy always competitive. The competition

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Fig. 5.1 Schematic diagrams of preparation methods of polymeric nanotubes or nanowires. (a) Polymerization of monomers and initiator in nanopores. Depending on polymerization time, either nanotubes or nanowires are obtained [6]. Electrochemical polymerization accompanies the predeposition process of Au electrode, whereas chemical polymerization does not accompany the deposition of Au film [6] Reprinted with permission of Journal of the American Chemical Society 112, 9666 (1990). Copyright 1990 American Chemical Society (b) Wetting by polymer solution. Nanotubes are obtainable as wetting on the pore wall is preferred to filling the nanopores [8]. Reprinted with permission of Science 296, 1997 (2002). Copyright 2002 The American Association for the Advancement of Science (c) Wetting by polymer melt. Nanotubes, nanorods, or nanowires are obtained [8,10]. The SEM images, in the right, exhibit the features of polymer in cylindrical nanopores by each preparation method. [8] Reprinted with permission of Science 296, 1997 (2002). Copyright 2002 The American Association for the Advancement of Science [10] Reprinted with permission of Macromolecules 36, 4253 (2003). Copyright 2003 American Chemical Society (d) Adsorption and desorption of liquid molecules in cylindrical nanopores [11]. Liquid film with uniform thickness is observed during the adsorption on the pore walls, and partial filling with meniscus is observed during the desorption. The graph in the right shows the volume fraction in the nanopores on the variation of temperature offset ( T). Reprinted with permission of Physical Review Letters 97, 175503 (2006). Copyright 2006 American Physical Society

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between the two energies does affect the structural formation of polymers in nanopores [9]. In the infiltration of a polymeric solution, the nanopores of the template are filled with the solution within seconds via capillary force. As solvent in the nanopores evaporates, polymeric material is preferentially deposited on the walls of nanopores, resulting in the formation of nanotubes (Fig. 5.1(b)) [7, 9]. Polymer melts, another way to employ presynthesized polymer, are usually thermally annealed at the temperature well above glass transition temperature (Tg ) to infiltrate the material with enhanced fluidity (Fig. 5.1(c)). Unlike polymer solution method, nanotubes, nanorods, and nanowires are possibly obtained depending on annealing temperature, molecular weight (M), and the diameter (D) of the pores. Generally, high annealing temperature, low molecular weight, and large pore diameter are regarded to result in the formation of nanotubes [7–9]. However, complete filling is considered as the equilibrium state in cylindrical nanopores, no matter what kind of kinetic pathways (nanotubes or nanowires) the material goes through [9]. By the physical wetting methods, both polar and nonpolar polymers can be prepared in the form of nanotubes, nanorods, and nanowires. Besides high molar mass molecules such as polymers, the analogous morphologies also appear for low molar mass molecules like organic solvent to fill cylindrical nanopores [11]. Interestingly, the liquid forms nanotubes with uniform thickness during adsorption and forms partially filled morphology during desorption (Fig. 5.1(d)). By controlling the temperature difference ( T) between the nanoporous template and the liquid reservoir, the chemical potential offset ( μ) was regulated according to Kelvin equation, and volume fraction of liquid in the nanopores varies accordingly. The graph in Fig. 5.1(d) represents changes in the volume fraction of perfluoromethylcyclohexane (PMFC) in the nanopores (nanoporous alumina) on the variation of temperature offset ( T) [11]. Taking advantage of Derjaguin approximation, the total free energy (G(t)) of cylindrically symmetric liquid film on the nanopore walls is given as G(t) = 2π γ (R − t) + n μ(2Rt − t2 ) + U(t)

(5.1)

where γ is the surface tension of PMFC, t is the thickness of the liquid film, R is the radius of the nanopores, and U(t) is the van der Waals potential between the liquid and the pore walls. The second derivative of the total free energy (∂ 2 G/∂t2 ) can provide the stability of the symmetric liquid film. Capillary transition at ∂ 2 G/∂t2 =0, therefore, occurs when d reaches approximately 3.3 nm, corresponding to the critical volume fraction Vc =0.44. From that point, the conformal liquid film becomes instable until complete filling. Upon desorption cycle from complete filling, hysteresis in morphology as well as volume fraction occurs as liquid with partially filled morphology is considered to be metastable by the prediction of Cole and Saam [12, 13]. Experimentally, the metastable volume fraction was estimated to be Vm =0.23 below which the desorption curve approximately coincides with the adsorption curve. Therefore, Kelvin equation is regarded reasonable to discuss the filling transition of liquid in the cylindrical nanopores [11]. Unfortunately, the capillary transition of polymer nanotubes to complete filling has not been understood as much as that of liquids. Various experimental parameters

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are to be considered on the alteration of the morphologies of polymer in cylindrical nanopores as discussed above. However, the static behavior of polymer, predeposited in nanopores, has been widely studied: chain conformation, crystallization behavior, phase separation, etc. Furthermore, filling dynamics regarding nanowires (not nanotubes) was investigated with respect to the molecular size of polymers. The rest of this chapter accounts for the physical behavior of polymers under two-dimensional confinement that deviates from the bulk or other geometric constraints.

5.3 Structural and Dynamical Deviation of Confined Amorphous Polymers 5.3.1 Random Chain Conformation Along Rod Axes In the case of a polymer with high molecular weight, the size of polymer chain in bulk (2Rg ) can be comparable to the size of the diameter (D) of nanopores (e.g., chain size (2Rg ) of polystyrene with Mw 760 K ~ 50 nm). Under the geometric conditions of 2Rg > D, polymer chains are still capable of penetrating into nanopores via capillary force, but the chains are supposed to undergo deformation in their chain conformation in the direction of the pore diameter. However, it is investigated that a single polymer molecule maintains the unperturbed conformation in the direction of the pore axis which is apart from geometric constraint; so the chain size along the pore axis is the same as in the unperturbed bulk state [14]. The unique behavior under cylindrical confinement can be compared to that in ultrathin polymer film. In the geometry of ultrathin polymer film (1D confinement), the unperturbed Gaussian conformation is sustained in the direction parallel to the surface although the conformation is restricted in the direction of the film thickness [15]. Figure 5.2(c) schematically describes that the conformation of a single polymer chain is deformed in the direction of the pore diameter but sustains its size in the direction of the pore axis [14]. The conformation of a polymer chain in cylindrical nanopores is experimentally examined by small angle neutron scattering (SANS) by filling the mixture of hydrogenous and deuterated polymers into the nanopores. As the scattering length alters by deuterium labeling (–0.376 × 10–12 cm for 1 H and 0.667 × 10–12 cm for 2 D) [16], the mixture of deuterated polystyrene and hydrogenous polystyrene renders higher contrast between the two components (deuterated and hydrogenous polystyrenes) and improves signal-to-noise ratio of the scattering from a single chain. Therefore, a single molecule can stand out among the same molecules, which enables researchers to experimentally measure the size of a single molecule. Figure 5.2(a) is the schematic diagram of the SANS setup and the expected 2D scattering pattern, and Fig. 5.2(b) shows the typical 2D scattering pattern obtained from SANS analysis of polystyrene nanowires infiltrated in inorganic cylindrical nanopores [14]. Figure 5.3(a) exhibits that the scattering profile along qy in Fig. 5.2(b) is in good agreement with the calculated scattering profile of the unperturbed chain [14]. The match between the two curves suggests that polymer

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Fig. 5.2 SANS experiment to measure overall conformation of chains along the pore axes. (a) Schematic diagram of SANS setup. The polymer–membrane assembly is tilted at an incidence angle of 30◦ [14]. (b) 2D scattering pattern of deuterated/hydrogenous polystyrene (61% of deuterated polystyrene, M of deuterated polystyrene ~ 528 kg mol–1 and M of hydrogenous polystyrene ~ 591 kg mol–1 ) in nanopores (D ~ 30 nm). Dashed lines: the angular span averaged to analyze the overall chain dimension [14]. Reprinted with permission of Nature Materials 6, 961 (2007). Copyright 2007 Nature Publishing Group (c) Schematic illustration of a polymer chain confined in nanopores. The picture illustrates that a chain molecule maintains its overall conformation in the axial direction of nanopores even though it is deformed in radial direction

chains in cylindrical nanopores are unperturbed in the axial direction. Even by varying the degree of confinement and molecular weight, the chain size along the pore axis is found to be the same as in the unperturbed state (Fig. 5.3(b)). The unique behavior is attributed to the change in interpenetration among polymer chains. Assuming incompressibility of a polymer chain under confinement, the interpenetration along the pore axis is anticipated to decrease since polymer chains are geometrically restricted in the radial direction. It implicates that interpenetration among chains should decrease with the increase in molecular weight. Therefore, polymer chains are allowed to stay in the unperturbed conformation in the axial direction. The molecular weight dependence on interpenetration is further considered to affect the mobility of chains in the cylindrical nanopores, which is discussed in the following section [14].

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Fig. 5.3 Overall chain conformation of various molecular weights of polystyrene under cylindrical nanoconfinement. (a) Scattered intensity from SANS measurement from Fig. 5.2. Open square: the experimental data of bulk deuterated/hydrogenous polystyrene; filled square: the experimental data of deuterated/hydrogenous polystyrene in nanopores; and dots and solid line: calculated profile of an unperturbed chain [14]. (b) Chain conformation along the pore axis (Rg ) in comparison to the radius of gyration of a single chain in bulk (Rg ). Chain size along the pore axis is not affected by the degree of confinement [14]. Reprinted with permission of Nature Materials 6, 961 (2007). Copyright 2007 Nature Publishing Group

5.3.2 Weaker Dependence on Molecular Weight in Dynamics Liquid flow in microscopically/nanoscopically small capillaries is driven mostly by capillary pressure. The flow dynamics in capillaries is explained by the Lucas– Washburn equation, showing that the length that liquid flows through is proportional to t1/2 . H(t) =



γ R cos θ 2η

1/2

√ t

(5.2)

where H(t) is the length that liquid has gone through in cylindrical nanopores with time t, γ is the surface tension of liquid, R is the radius of the nanopores, θ is the contact angle between the meniscus and the pore wall, and η is the viscosity of the liquid. The equation implies that the capillary force, 2γ cos θ /R, is in balance with the viscous resistance, 4ηd[H(t)/R]2 /dt, at the length of H(t), and the rise of the meniscus of liquid is in linear proportion with t1/2 . Recent molecular dynamics (MD) simulation studies verify H(t) ∝ t1/2 relation with a Lennard-Jones (LJ) fluid in cylindrical nanopores [17–19]. By varying the wall–liquid attraction strength over a wide range, the fluid dynamics in nanopores is simulated to satisfy t1/2 law [18]. In addition to the flow of small molecules (e.g., gas, liquid), a polymer melt is also estimated to follow the t1/2 law [18]. Experimental studies in cylindrical nanopores also support the validity of the t1/2 law for both small molecules and polymer melts (Since the diameter of the

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Fig. 5.4 SAXS experiment to measure flux of polymer flow through cylindrical nanopores. (a) Schematic diagram of SAXS experiment [14]. (b) Linear decrease of scattering invariant (left) and height of polystyrene filling the nanopores (right) with square root of heating time. Circles: M ~ 114 kg mol–1 , triangles: 359 kg mol–1 , diamonds: 750 kg mol–1 , down triangles: 1032 kg mol–1 , and squares: 2880 kg mol–1 [14]. (c) The flux of polymers through cylindrical nanopores as a function of molecular weight (D~ 15 nm, L ~ 120 µm) when l ~ L/2 (filled squares), and the flux through macroscopic pores calculated on the basis of bulk viscosity (open squares). The overall mobility of polymer under nanoconfinement shows weaker dependence on molecular weight [14]. Reprinted with permission of Nature Materials 6, 961 (2007). Copyright 2007 Nature Publishing Group

nanopores is about few tens of nanometers in this study, the investigation in dynamics regards the structure of nanowires only, not nanotubes.) [14, 20]. The schematic diagram in Fig. 5.4(a) shows the setup of small angle X-ray scattering (SAXS) experiment to obtain the flux of a polymer melt into cylindrical nanopores [14]. As the polymer melts (polystyrene) fill the nanopores (nanoporous alumina), the volume fraction of polymer and template changes; thus, the scattering invariant varies with time. Equation (5.3) shows the scattering invariant (Q), where ϕ U and ϕ F are the volume fractions of the unfilled and filled spaces in the pores, ϕ P and ϕ AO are the volume fractions of the pores and alumina matrix in the membrane, and ρ PS and ρ AO are the electron densities of the polymer (polystyrene) and the membrane (alumina), respectively 2 + ϕF (ρAO − ρPS )2 ]ϕP ϕAO Q = [ϕU ρAO

(5.3)

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Herein, Q is obtained from SAXS experiment, ϕ U is substituted by 1–ϕ F ; thus the height of capillary rise changing with time is possibly estimated. The total volumetric flow rate is, consequently, obtained by calculating the difference of the heights or the difference of the scattering invariants measured at each timescale. The volumetric rate of polymer flowing into the nanopores is expressed as J = π r2

(Q/Q0 )

l = −K

t

t

(5.4)

where r is the radius of the capillary rise; l is the height of the capillary rise; K is π r2 L(ρ AO )2 /(2ρ AO –ρ PS )ρ PS . Since the length of the nanopores L in the expression of K is constant, K becomes constant. Figure 5.4(b), then, shows that the normalized difference in scattering invariants (left axis) and the height of the capillary rise (right axis) are linearly proportional to t1/2 [14]. When the overall flux is plotted as a function of molecular weight (M), it is interesting that the overall mobility of the polymer melt of a higher molecular weight in nanopores is more enhanced even though polymer chains are severely confined in nanopores (Fig. 5.4(c)) [14]. Conventionally, viscosity (η) in bulk is generally proportional to M3.4 above the entanglement molecular weight (Me ). The slope of the graph in log–log plot of flux vs. molecular weight is, therefore, estimated to be –3.4 for a polymer above Me in bulk since the flux of a polymer is inversely proportional to viscosity (η) [21]. However, the exponent for a polymer under 2D confinement is estimated to be 1.5 from Fig. 5.4(c), which shows the relation η ~ M1.5 instead of η ~ M3.4 [14]. The diffusion of a polymer melt into nanopores, in other words, shows weaker dependence on molecular weight (M). As polymer chains are confined in the space that is comparable or even smaller than the chain size (in the direction of pore diameter), the degree of interchain penetration decreases in terms of incompressibility. The polymer chains in cylindrical nanopores, therefore, exhibit enhanced dynamical behavior due to the geometric confinement on nanoscale.

5.3.3 Instability in Nanotubular Structure Polymer solution is advantageous to obtain polymeric nanotubes by template wetting method (discussed in Section 5.2). However, the nanotubes undergo morphological transformation upon heating above the glass transition temperature since the nanotubes obtained from the template wetting method are not thermodynamically stable [22]. The energetic instability results in the undulation of the surface of the polymeric nanotubes in terms of Rayleigh instability as the polymers obtain enough energy to restructure. As the surface of the nanotubes undulates with the wavelength (λ) that is greater than the circumference, the surface area of the nanotubes decreases. Hence, the structural transformation through Rayleigh instability minimizes the thermodynamic free energy by decreasing the surface area for the same volume. Recently, metallic nanowires have been reported to undergo the

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morphological transformation, which hinders the nanowires from preserving their original shape [23, 24]. Figure 5.5(a) schematically illustrates that the structural transformation of polymeric nanotubes [poly(methyl methacrylate) (PMMA)] in cylindrical nanopores upon heating at the temperature above the Tg . [22]. Thermal energy being provided with time, the undulation with the characteristic wavelength (λ) develops in its amplitude, and the ridges of the surface eventually merge, forming capillary bridges with the regular spacing in the nanotubes. Figure 5.5(b) represents the images of PMMA nanotubes changing the morphology with time [22]. Concerning the wavelength, Eq. (5.5), generally valid in thin-film geometry, shows the relation between the wavelength (λ) and the radius of capillary (b) [25] √ λ = 2π 2b

(5.5)

In other words, λ/b should be constant, irrespective of the radius of capillary. However, λ/b for nanotubes is experimentally observed to change as a function of b (Fig. 5.5(c)) [22]. The discrepancy in the relation between λ/b and b is ascribed

Fig. 5.5 Structural instability of PMMA nanotubes. (a) Schematic illustration of structural transformation of PMMA nanotubes in cylindrical nanopores [22]. (b) The TEM images of PMMA ◦ nanotubes after annealing at 200 C. A and B show PMMA nanotubes after annealing for 10 min (B: the magnified image of the square box in A); C and D show PMMA nanotubes after annealing for 20 min (D: the magnified image of the square box in C) [22]. (c) The ratio of wavelength to capillary radius (λ/b) as a function of capillary radius (b) in cylindrical nanopores where dotted line represents the relation in the geometry of thin film. λ/b in nanocylindrical geometry varies with b, whereas λ/b in thin-film geometry stays constant [22]. Reprinted with permission of Nano Letters 7, 183 (2007). Copyright 2007 American Chemical Society

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to the thickness of the polymeric nanotubes. The radius of capillary is so small in the cylindrical nanoporous structure that the original film thickness (the wall thickness of the nanotubes) should also be taken into account while that in thin film is usually ignored. In addition, the wall thickness of the polymeric nanotubes is less than 100 nm where disjoining pressure is possible to affect the surface undulation. Consequently, the radius of the capillary and the thickness of the tube wall under cylindrical nanoconfinement are considered to have influence on the structural transformation that is distinguished from the transformation in thin film [22].

5.4 Size-Dependent Crystallization and Preferential Orientation 5.4.1 Size Dependence on Crystallization Behavior For crystalline polymers, the orientation of crystallite, crystallinity, and crystallization mechanisms under cylindrical confinement deviates from that in bulk [26–29]. Since the spatial restriction affects the growth of crystals, the crystallization process is reported to be dominated by nucleation over crystal growth. Especially in small nanopores (D < 50 nm), surface-induced heterogeneous nucleation rather than homogeneous nucleation predominantly takes place. Nucleation from the wall of the nanopores is the more preferred crystallization mechanism in smaller nanopores, while nucleation from density fluctuation in melt is the main crystallization mechanism in larger nanopores [28]. Generally in cylindrical nanopores, it is studied that the crystallinity decreases, the degree of supercooling (the difference between the equilibrium melting temperature and crystallization temperature) becomes larger, and the melting temperature decreases with the decrease in the pore diameter [28, 29]. By infiltrating linear polyethylene (PE) into an inorganic nanoporous template (pore diameter approximately few tens of nanometers), the crystallization behavior of PE nanowires within the nanopores was investigated by X-ray diffraction [29]. Figure 5.6(a) illustrates the setup of X-ray diffraction experiment with the variation of angle Ψ . Ψ -dependent XRD measurement provides the information of the orientation of crystallites with respect to the geometry of the cylindrical nanopores. In Fig. 5.6(c), the reflections of (110) and (200) are clearly observed at Ψ = 0◦ , whereas only (110) at Ψ = 89.85◦ . So, a- and c-axes of orthorhombic PE crystals are identified to orient perpendicular to the long axis of nanopores (Fig. 5.6(b)) [29]. Moreover, the intensities at all reflections decrease as the nanoconfinement gets more severe (decrease in the pore diameter) [29]. The decreased intensity with the decrease in the pore size is interpreted as the decrease in crystallinity by the geometric confinement. In addition to the crystal orientation and crystallinity, crystallization mechanism is also affected by the geometric constraint [28]. Usually the relation between crystallization rate and degree of supercooling ( T) or crystallization temperature (Tc )

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Fig. 5.6 Crystallization behavior of linear PE in cylindrical nanopores as a function of the pore diameter. (a) Schematic illustration of X-ray diffraction geometry [29]. (b) Schematic illustration of crystallization mechanism in large and small nanopores. Homogeneous nucleation is dominant in larger nanopores initiated by density fluctuation. Surface-induced heterogeneous nucleation is dominant in smaller nanopores initiated from the pore walls. The z-axis corresponds to the axis of the nanopores. a- and c-axes of orthorhombic PE align perpendicular to the pore walls. (c) ◦ ◦ ◦ X-ray diffraction patterns at the tilt angles of Ψ = 0 , 45 , and 89.85 for linear PE in cylindrical nanopores with different pore diameters. Crystallinity decreases as crystallites are more severely confined [29]. Reprinted with permission of Macromolecules 40, 6617 (2007). Copyright 2007 American Chemical Society

is studied to discuss the crystallization mechanism. In the larger pores (D > 50 nm), crystallization occurs at small T and exhibits strong dependence on Tc . In smaller pores (D< 50 nm), the crystallization occurs at broader T range and shows weaker dependence on Tc , on the contrary [28]. Since the strong temperature dependence in the larger pores can be fairly interpreted by the classical homogeneous nucleation theory, the crystallization kinetics is believed to originate from homogeneous nucleation [30]. So, homogeneous nucleation prevails in larger nanopores and heterogeneous nucleation in smaller nanopores [28]. As interfacial area increases with the decrease in the pore diameter, the pore walls are considered to influence the nucleation process as described in Fig. 5.6(b). Therefore, crystallization kinetics and crystallinity are inflicted by the degree of nanoconfinement. In this section, we discuss only the size-dependent crystallization behavior of polymeric materials. The more detailed description of crystallization behavior under nanoconfinement will be given

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in Section 3.1 where the freezing and the melting behavior of various materials are discussed.

5.4.2 Preferential Orientation of Conductive Polymers Conductive nanowires in cylindrical nanopores exhibit electronic conductivities that are substantially greater than the electronic conductivities of bulk films [6, 31–33]. With decrease in the diameter of nanowires of less than 100 nm, the electronic conductivity increases exponentially. The electronic conductivity is estimated by measuring the resistance across a nanowire–template composite [31]. Considering the polymer deposited on the surfaces of the template, the resistance of a single nanowire is calculated from the resistance directly measured from the composite. Equation (5.6) represents that the resistance directly measured comprises three different resistances on the basis of its structure RM = RS1 + RPF + RS2

(5.6)

RS1 and RS2 are the resistances of the two surface layers on the composite; RPF is the total sum of the resistances of the nanowires within the nanopores; and RM is the resistance directly measured (Fig. 5.7(a)). Figure 5.7(b) exhibits the graph of the conductivity of a single nanowire on the variation of the diameter of the nanowire [polypyrrole and poly (2-methylthiophene)] [31]. It is striking that the conductivity increases with the decrease in the nanowire diameter. The electronic conductivity of a 30-nm-diameter nanowire is almost an order of magnitude higher than that of bulk material; the increment in electronic conductivity is considered to result from the preferential orientation of the polymer chains parallel to nanowire axis [32, 33]. The experiment by polarized infrared absorption (PIRA) shows that polymer chains (polyacetylene) are aligned parallel to the pore axis (Fig. 5.7(c)) [6]. The absorbance peaks are detected in the C–H stretching region by the radiation perpendicular (A⊥ ) and parallel (A ) to the axis of the nanopores. The PIRA analysis suggests that nanowires preferentially absorb the light polarized parallel to the nanowire axis, showing that the value of P (P = A /A⊥ , ratio between the integrated absorption intensities for light polarized in parallel and perpendicular to the nanowire axis, respectively) is calculated to be 1.46 in C–H stretching region [6]. Therefore, the chains of conductive nanowires under 2D nanoconfinement are preferentially oriented along the pore axis, and the preferential orientation is believed to bring about the enhancement in the electronic conductivity.

5.5 Confinement-Induced Phase Behavior of Block Copolymers Microphase separation of block copolymers (BCPs) is also perturbed by geometric constraint on nanoscale. The microphase-separated morphologies especially under cylindrical nanoconfinement are investigated to deviate from those in the bulk and

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Fig. 5.7 Change in electronic conductivity as a function of the diameter of a single nanowire.(a) Schematic illustration of conductive polymer infiltrated into cylindrical nanopores. (b) The electronic conductivity of a single nanowire as a function of the diameter. Solid line (filled circles): polypyrrole nanowire and dotted line (open circles): poly(2-methylthiophene) [31]. Reprinted with permission of Journal of the American Chemical Society 111, 4138 (1989). Copyright 1989 American Chemical Society (c) PIRA spectra in C–H stretching region for polyacetylene nanowire–membrane structure. A and A⊥ are the absorbances measured parallel and normal to the nanowire axes, respectively. Conductive polymers in nanowires exhibit exponential increase in electronic conductivity due to the chain orientation in the axial direction [6]. Reprinted with permission of Journal of the American Chemical Society 112, 9666 (1990). Copyright 1990 American Chemical Society

thin-film geometry [34–42]. The unique morphologies in cylindrical nanopores are associated with incommensurability between the repeat period of BCPs and pore diameter; interfacial interaction between the BCPs and the pore wall; and curvature of the nanopores. When the geometric dimension (e.g., film thickness, diameter of a nanowire) is not commensurate with the repeat period of BCPs, the symmetry breaking in their microphase separation appears. In thin-film geometry (oneside-supported thin film), the frustration usually leads to the deformation of the BCP structures as free surface exists: a hole and an island [34]. BCPs in cylindrical nanopores are, however, confined as well as supported by the pore walls in the radial direction; consequently, they are not allowed to change the confined dimension, but to change themselves during microphase separation. In addition to incommensurability, interfacial interaction plays an important role in microphase separation of BCPs. As for BCPs being in contact with another material, interfacial tension induces preferential segregation at the interface. For this reason

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Fig. 5.8 Microphase separation of block copolymers under cylindrical confinement. (a) TEM images of microphase separation of PS-b-PBD forming lamellar structure in bulk. The ratios of pore diameter to lamellar repeat period, D/L0 , are A: D/L0 ~ 8.1, B: D/L0 ~ 1.9, and C: D/L0 ~ 2.6. D represents the cross-sectional image of C. [35] Reprinted with permission of Science 306, 76 (2004). Copyright 2004 The American Association for the Advancement of Science (b) TEM images of microphase separation of PS-b-PBD forming cylindrical structure in bulk [35]. A:D/L0 > 4, B: the cross-sectional image of A, C:D/L0 = 1.9–2.3, D: cross-sectional image of C, E: D/L0 = 1.1–1.5, and F: cross-sectional image of D. [38] Reprinted with permission of Journal of Polymer Science Part B: Polymer Physics 43, 3377 (2005). Copyright 2005 Wiley Periodicals, Inc. (c) TEM images of inversed mesostructure of silver prepared by backfilling the silica mesostructure under different degrees of confinement [38]. A: three-layer stacked doughnuts, B: core–shell triple helix (shell: D-helix and core: S-helix), C: D-helix, D: S-helix, and E: D-helix [42]. Scale bars: 50 nm. Commensurability, interfacial interaction, and imposed curvature in nanopores influence the microphase separation of block copolymers. [42] Reprinted with permission of Nature Materials 3, 816 (2004). Copyright 2004 Nature Publishing Group

the outermost ring, segregated domain at the interface, mostly appears under 2D nanoconfinement irrespective of the degree of confinement (Fig. 5.8(a) and (b)) [35–39]. Based on the composition of BCPs, the microphase-separated morphology varies: sphere-forming, cylinder-forming, and lamella-forming structures. Figure 5.8 shows the morphologies of diblock copolymers phase separated under 2D confinement. Most of the studies exploited the diblock copolymer polystyreneblock-polybutadiene (PS-b-PBD) since PS-b-PBD strongly phase segregates due to the large difference in the chemistry of the two components [35]. Even so, polystyrene-block-poly(methyl methacrylate) (PS-b-PMMA) in weak segregation was also reported to exhibit the very similar phase separation behavior in cylindrical nanopores [39].

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Generally the phase-segregated morphologies under 2D confinement are frustrated as the ratio of the diameter to the repeat period (D/L0 ) is less than 3–4 and incommensurate. If D/L0 is greater than 3–4 and even incommensurate, the morphologies tend to persist the symmetric structure as observed in bulk; the frustration occurred by the geometric constraint is distributed throughout the multiple microdomains. The following subsections explain the microphase separation on the basis of the composition of BCPs under 2D confinement.

5.5.1 Lamella-Forming BCPs Figure 5.8(a) shows the microphase-separated morphologies of a lamella-forming BCP (PS-b-PBD) with the dimensional conditions of D/L0 ~ 8.1, D/L0 ~ 2.6, and D/L0 ~ 1.9, respectively [35]. When D/L0 ~ 8.1, concentric cylindrical structures appear, having the PBD microdomain with lower surface energy (darkly stained domain) for the outermost layer. Along the pore axis, each microdomain appears to align parallel to the pore axis. When D/L0 ~ 2.6; not only is the pore diameter incommensurate with the repeat period but also curvature severely affects the selfassembly. So, BCPs under the severe confinement with incommensurability exhibit fundamentally new morphology. A unique structure of stacked discs with the central spine and the outermost ring of PBD emerges parallel to the pore axis. As D/L0 further decreases to 1.9, another morphological transition comes up. Similar to the morphology in a larger pore (D/L0 > 3.2), concentric rings and parallel lamellar structure are observed normal and parallel to the pore axis, respectively [35, 36, 38]. Differently from the condition of D/L0 > 2.6, only two rings appear. Usually for lamella-forming BCPS under 2D confinement, n + 1 rings tend to appear with the condition of D/L0 ~ n, where n is an integer. Therefore, the severe geometric constraint is considered to result in the deviation of the general expectation (n + 1 rings) [38].

5.5.2 Cylinder-Forming BCPs Figure 5.8(b) exhibits cylinder-forming BCP nanostructures in cylindrical nanopores [38]. When D/L0 > 4, cylinder-forming domains (PBD) align parallel to the pore axis in PS matrix. Still, low-interfacial-energy material, PBD, is located at the interface. As D/L0 is close to 2, a single cylindrical PBD domain appears in the nanopores. When D/L0 becomes smaller (D/L0 ~ 1.1–1.5), an unusual morphology appears. PBD domain is the outermost layer in spite of the severe geometric constraint, and PS domain is, interestingly, located in the central region. Vertically, helical structure is formed with the constant torsional angle. Therefore, cylinderforming BCPs exhibit the morphological transition from simple cylinder to helical structure along the pore axis due to the commensurability and curvature [36–38].

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Nanostructure-forming BCP mixture is also reported to exhibit sophisticated mesostructures under 2D confinement. By introducing silica and triblock copolymer [poly(ethylene oxide)–poly(propylene oxide)–poly(ethylene oxide)], silica specifically interacts one of the block and is selectively located in one of the phases in the phase-separated state [42]. As the mixture of silica and the triblock copolymer is infiltrated into cylindrical nanopores and calcined, an inorganic sophisticated mesostructure, reflecting the phase-separated mesostructure of the BCP mixture under 2D confinement, is achieved. The silica mesostructures are affected by the diameter of the nanopores as the phase behavior of BCPs is largely influenced by the degree of confinement. Employing silica mesostructures, inversed mesostructure is subsequently acquired by introducing silver into the silica mesostructure. Figure 5.8(c) describes the morphologies of the inversed silver nanowires with various pore diameters [42]. Triblock copolymer mixtures show mostly complicated helical structures under 2D confinement so that the backfilled silver mesostructure also reflects the sophisticated structures. Depending on the degree of nanoconfinement in the direction of the pore diameter, various helical structures such as S-helix, core-shell triple helix, and D-helix are observed.

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14. Shin K, Obukhov S, Chen J-T, Huh J, Hwang Y, Mok S, Dobriyal P, Thiyagarajan P, Ressell, TP (2007) Enhanced mobility of confined polymers. Nat Mater 6: 961–965 15. Jones RL, Kumar SK, Ho DL, Briber RM, Russell TP (1999) Chain conformation in ultrathin polymer films. Nature 400: 146–149 16. Roe R-J (2000) Methods of x-ray and neutron scattering in polymer science. Oxford University Press, New York 17. Washburn, EW (1921) The dynamics of capillary flow. Phys Rev 17: 273–283 18. Dimitrov DI, Milchev A, Binder K (2007) Capillary rise in nanopores molecular dynamics evidence for the Lucas-Washburn equation. Phys Rev Lett 99: 054501 19. Dimitrov DI, Milchev A, Binder K (2008) Molecular dynamics simulations of capillary rise experiments in nanotubes coated with polymer brushes. Langmuir 24: 1232–1239 20. Huber P, Grüner S, Schäfer C, Knorr K, Kityk AV (2007) Rheology of liquids in nanopores – A study on the capillary rise of water, n-hexadecane and n-Tetracosane in mesoporous silica. Eur Phys J Spec Top 141: 101–105 21. Rubinstein M, Colby RH (2003) Polymer physics. Oxford University Press, New York 22. Chen J-T, Zhang M, Russell TP (2007) Instabilities in nanoporous media. Nano Lett 7: 183–187 23. Toimil-Molares ME, Balogh AG, Cornelius TW, Neumann R, Trautmann C (2004) Fragmentation of nanowires driven by Rayleigh instability. Appl Phys Lett 85: 5337–5339 24. Karim S, Toimil-Molares ME, Balogh AG, Ensinger W, Cornelius TW, Khan EU, Neumann R (2006) Morphological evolution of Au nanowires controlled by Rayleigh instability. Nanotechnology 17: 5954–5959 25. de Gennes PG, Brochart-Wyart F, Quere D (2004) Capillarity and wetting phenomena. Springer, New York 26. Steinhart M, Senz S, Wehrspohn RB, Gösele U, Wendorff JH (2003) Curvature-directed crystallization of poly(vinylidene difluoride) in nanotube walls. Macromolecules 36: 3646–3651 27. Steinhart M, Göring P, Dernaika H, Prabhukaran M, Gösele U, Hempel E, Thurn-Albrecht T (2006) Coherent kinetic control over crystal orientation in macroscopic ensembles of polymer nanorods and nanotubes. Phys Rev Lett 97: 027801 28. Woo E, Huh J, Jeong YG, Shin K (2007) From homogeneous to heterogeneous nucleation of chain molecules under nanoscopic cylindrical confinement. Phys Rev Lett 98: 136103 29. Shin K, Woo E, Jeong YG, Kim C, Huh J, Kim K-W (2007) Crystalline structures, melting, and crystallization of linear polyethylene in cylindrical nanopores. Macromolecules 40: 6617– 6623 30. Avrami M (1939) Kinetics of phase change I∗ general theory. J Chem Phys 7: 1103–1112 31. Cai Z, Martin CR (1989) Electronically conductive polymer fibers with mesoscopic diameters show enhanced electronic conductivities. J Am Chem Soc 111: 4138–4139 32. Cai Z, Lei J, Liang W, Menon V, Martin CR (1991) Molecular and supermolecular origins of enhanced electronic conductivity in template-synthesized polyheterocyclic fibrils. 1 Supermolecular effects. Chem Mater 3: 960–967 33. Parthasarathy RV, Martin CR (1994) Template-synthesized polyaniline microtubules. Chem Mater 6: 1627–1632 34. Fasolka M, Mayes AM (2001) Block copolymer thin films physics and applications Annu Rev Mater Res 31: 323–355 35. Shin K, Xiang H, Moon S, Kim T, McCarthy TJ, Russell TP (2004) Curving and frustrating flatland. Science 306: 76 36. Xiang H, Shin K, Kim T, Moon S, McCarthy TJ, Russell TP (2004) Block copolymers under cylindrical confinement. Macromolecules 37: 5660–5664 37. Xiang H, Shin K, Kim T, Moon S, McCarthy TJ, Russell TP (2005) From cylinders to helices upon confinement Macromolecules 38: 1055–1056 38. Xiang H, Shin K, Kim T, Moon S, McCarthy TJ, Russell TP (2005) The influence of confinement and curvature on the morphology of block copolymers. J Polym Sci Part B Polym Phys 43: 3377–3383

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39. Sun Y, Steinhart M, Zschech D, Adhikari R, Michler GH, Gösele U (2005) Diameterdependence of the morphology of PS-b-PMMA nanorods confined within ordered porous alumina templates. Macromol Rapid Commun 26: 369–375 40. Yu B, Sun P, Chen T, Jin Q, Ding D, Li B (2006) Confinement-induced novel morphologies of block copolymer. Phys Rev Lett 96: 138306 41. Sevink GJA, Zvelindovsky AV, Fraaije JGEM, Huinink HP (2001) Morphology of symmetric block copolymer in a cylindrical pore. J Chem Phys 115: 8226–8230 42. Wu Y, Cheng G, Katsov K, Sides SW, Wang J, Tang J, Fredrickson GH, Moskovits M, Stucky GD (2004) Composite mesostructures by nano-confinement. Nat Mater 3: 816–822

Chapter 6

Statistical Mechanical Lattice Model Studies of Adsorption in Nanochannels Treated by Exact Matrix Methods George Manos, Zhimei Du, and Lawrence J. Dunne

Abstract In this chapter a review is presented of our work using essentially exact (in the statistical mechanical sense) matrix treatments of several lattice models of adsorption in nanochannels. Particularly, we review our work on benzene and small hydrocarbon adsorption in the zeolite silicalite and in the final part we review our lattice model of adsorption of xenon in carbon nanotubes and compare this with literature studies of Monte Carlo simulation of xenon adsorption in nanotubes. Lattice models can provide effective models for adsorption and hence a rationale for the shape of adsorption isotherms for molecules in nanopores and nanochannels.

6.1 Introduction The rationalisation of the shape of isotherms for molecular adsorption in nanochannels such as those found in zeolites and carbon nanotubes has a number of challenging features. The lattice fluid model was introduced into condensed matter physics by Chen Ning Yang and T. D. Lee in 1952 [1] and has had a huge impact in many areas of physics and chemistry. The matrix methods discussed here calculate the so-called constant pressure and grand partition functions by extracting the dominant eigenvalue of a transfer-type matrix related to those used in the solution of the Ising chain by Kramers and Wannier [2].Typically, we will focus on one-dimensional models which describe either a zeolite lattice or a nanotube where each lattice point is mapped onto a cell of a volume v0 , which may be vacant or contain part of a molecule. Usually, the notation will be adopted such that if there are M molecules adsorbed on the lattice, then the number density in our models is denoted by ρ = M/N, where N is the number of lattice sites. G. Manos (B) Department of Chemical Engineering, University College London, Torrington Place, London, WC1E 7JE, UK e-mail: [email protected] This chapter has relied significantly on previous publications referred to in the text (Copyright Elsevier), which we reproduce without permission under the rights granted and retained by authors (http://www.Elsevier.com, 2008) L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_6,  C Springer Science+Business Media B.V. 2010

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Fig. 6.1 Pore structure of silicalite (taken from Ref. [3], copyright American Chemical Society)

In order to have confidence in the calculated results of these one-dimensional models, an accurate statistical treatment of the adsorbed phase is essential. In one-dimensional problems, approximate statistical mechanical treatments such as mean-field theories may be unreliable, which will be manifest as spurious jumps in adsorption isotherms. The pore structure of the zeolite has been determined by Olson et al. [3] and is shown in Fig. 6.1. There is a nanochannel structure made up of zigzag and straight tunnels and the intersections which connect these two types of channel. The channels in the three-dimensional structure are made up of well-separated layers. It is widely believed that well-defined adsorption sites are located in these channels and thus in our model the structure of zeolite is treated as quasi one-dimensional pores with the possibility of two types of pores. In the final part we review adsorption of xenon in carbon nanotubes. Adsorption of xenon in carbon nanotubes [4] (a (10,10) carbon nanotube is shown in Fig. 6.2) has been investigated experimentally by Kuznetsova, Yates Jr., Liu and Smalley [5] using computer simulation by Simonyan, Johnson, Kuznetsova and Yates Jr. [6] where endohedral adsorption isotherms show a step-like structure. A matrix method is used for calculation of the statistical mechanics of a lattice model of

Fig. 6.2 Structure of a (10,10) carbon nanotube (copyright www.jcrystal.com/steffenweber)

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xenon endohedral adsorption which reproduces the isotherm structure, while exohedral adsorption which does not display phase transitions is treated by mean-field theory. In our work we always regard the vapour phase as an ideal gas in equilibrium with the adsorbed phase in one or two types of pore, where the chemical potential of a component in such an ideal gas is given as [7] μ = μ0 + kT ln(P)

(6.1)

where the first term on the right-hand side is the standard chemical potential given as

0

μ = −kT ln



2π mkT h2

3/2

kT



(6.2)

where P is the pressure. In all our calculations this is equated to the chemical potential of the adsorbed phase calculated by our lattice models. The causes of the jumps in adsorption isotherms are numerous and include such effects as capillary condensation, changes in the zeolite structure caused by the adsorbed molecules, commensurate “freezing” in the assembly of adsorbed molecules [8] and many others. Of great interest to us is the observation made by Smit et al. [9] that at increased loading, molecules can seek to occupy sites which are energetically more costly, and this can result in jumps in adsorption isotherms. In our work which we describe below, we will take up the above theme and demonstrate that the jumps in adsorption isotherms can also occur due to reorientation of molecules or simple packing in the adsorbed phase where there is competition between open and close-packed structures on energetic grounds. Phase transitions in adsorbed phases of various types have been widely discussed [10] and reviewed by Gubbins et al. [11]. The nature of the critical phenomenon in an adsorbed phase in a tube is of considerable interest. Some texts that we have found consistently useful are Refs. [12–15].

6.2 Benzene Adsorption Many investigations of benzene adsorption in zeolites have been carried out [14–19]. Some unusual phenomena have been found. For instance, the isotherms can change from Brunauer’s type I to type IV [20] with decreasing temperatures .The heat of adsorption of benzene rises abruptly at some higher coverage as indicated by Pope [16] and Thamm [17], and where jumps in the adsorption isotherms have been observed. The diameter of a benzene molecule is close to the size of the pores in the silicalite zeolite so that a snug fit is possible. For benzene adsorption, we will introduce a lattice model which has dimer states to represent molecules lying down flat against the zeolite wall; monomer states to

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depict molecules pushed into higher energy smaller volume occupancies and vacant sites or holes to allow for only partial filling of the sites in the zeolite lattice. For the benzene adsorption problem we develop a matrix method which calculates the constant pressure partition function and which is essentially an exact statistical treatment of the lattice model. The constant pressure partition function is the discrete Laplace transform of the canonical partition function, firstly used in adsorption studies by Bell and Dunne [21] where lipid monolayers were modelled. The model is able to reproduce the characteristic isotherms of benzene adsorption and gives some insight into the changes in the heat of adsorption with coverage.

6.2.1 One-Dimensional Lattice Model of Benzene in Silicalite It is believed that for benzene adsorption in silicalite there are three types of adsorption sites: I, S and Z. X-ray diffraction studies by Mentzen et al. [22] show that the behaviour of benzene in silicalite is extremely complex. At low loading, the experiments show that I sites are predominantly occupied with at least two distinct orientations for the benzene molecule. For higher loading, six benzene molecules per unit cell are observed in I and Z channels. When the loading reaches to eight molecules per unit cell, I and S channels are mainly filled. To obtain a mathematically soluble problem or at least one which is amenable to treatment using mathematical software, a twin-channel, one-dimensional lattice model has been proposed. This model contains the closely related situation of two types of channels containing the sites -IZ-IZ-IZ- and -IS-IS-IS-. Generally, obtaining an accurate statistical mechanical lattice model of such a situation is extremely difficult. It is assumed that each lattice point is a cubic cell with volume v0 which may be vacant or contain part of a dimer or a whole monomer. To incorporate reorientational effects, the benzene molecule is assumed to be one of two states. One is the low-energy conformation in which the benzene lies down occupying two lattice sites and is termed a dimer state (d) signified by the symbol —, while the other is a higher energy state where the benzene molecule stands up perpendicular to the wall of the zeolite and takes up a single site and where this species is termed a monomer (m) denoted by the symbol |. There are M benzene molecules over N lattice sites; therefore, the number density is defined by ρ = M/N. This model is effectively that of two types of one-dimensional chains of sites, zigzag and straight, which may be occupied by benzene molecules in the monomer or the extended dimer states. This is not an exact representation of the zeolite structure but it may realistically mimic the silicalite in this generic lattice model. It is assumed that the potential energy of a single benzene molecule in an extended dimer state is denoted by U0 . The energy cost for converting from the dimer state to the monomer state is denoted by U11. These parameters may differ somewhat for the zigzag and straight pores. This parameter can also include the energy cost to cause a local change in crystal geometry to enable the molecule to be fitted into the crystal and may relate to the model proposed by Snurr et al. [23, 24]

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The arrangement of benzene molecules in silicalite has three possible lowest enthalpy states depending on the density, which are with all dimers: (i) — — — — — etc., with a maximum of four molecules per unit cell; (ii) compressed into monomer states with |||||, etc., with eight molecules per unit cell; and (iii) an alternate arrangement of these — | — | — | — |—, etc., with a maximum occupation of 5.33 molecules per unit cell. At absolute zero the stable state of an assembly with an internal pressure π in the silicalite pore should be the one giving the lowest configurational enthalpy Hc , where Hc = Ec = Nπ v0. Ec is the configurational energy, v0 is the volume associated with each lattice site in the pore and N is the number of sites occupied by the assembly at 0 K. For the arrangement (i) above, when N = 2M Hc = M(U0 + 2π v0 + Jdd )

(6.3)

where Jdd is the dimer–dimer interaction energy. For the arrangement (ii) above, when N = M Hc = M(U0 − U1 + π v0 + Jmm )

(6.4)

where Jmm is the monomer–monomer interaction energy. For the arrangement (iii) above, when N = 3/2M Hc = M(U0 − U1 2 + 3 2π v0 + Jmd )

(6.5)

where Jmd is the monomer–dimer interaction energy. The jumps in the isotherms to be shown below are attributed to transitions between ground states (i) and (ii) with the lowest enthalpy and a combination of these from two channels can give a rationalisation for the shoulder on the isotherms.

6.2.2 Exact Matrix Method for Constant Pressure Partition Function The theory presented here is the same for zigzag and straight channels. The ideal benzene vapour phase is in equilibrium with adsorbed benzene phase in the silicalite channels. Equilibrium is attained when the chemical potential of all the phases is the same, hence, μ = μad , where μ is the chemical potential of an ideal gas and μad is that of the adsorbed phase. Following the matrix method for a one-dimensional lattice fluid problem using the constant pressure partition function discussed previously by Bell and Dunne [21], a ring of N lattice points is considered. Each site is either vacant or occupied by part of or the entire benzene molecule. The vibrational degrees of freedom contribute a factor qint to the partition function which is assumed environment independent. Hence the canonical partition function for an assembly of M molecules can be written as

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Qad (M,V,T) =

qM int exp (−βEc )

(6.6)

configurations

Here β=1/kB T. kB is Boltzmann constant. The constant pressure partition function (M, T, π ) is the discrete Laplace transform of Qad (M,V,T) [21] and it is written as

qM (6.7)  (M,T,π ) = int exp (−β (Ec + π V)) V configurations

The series for –kB T ln (M, T, π ) is dominated by the largest term and for all practical purposes is equal to the Gibbs free energy G(M, T, π ) of the adsorbed phase. For an assembly of species on a ring, a configuration of the two types of species α, β, γ , δ,. . .,ω, α gives a contribution to the constant pressure partition function, which can be expressed as a product of M factors as Aαβ Aβγ Aχ δ . . . Aωα

(6.8)

Aμν = φμ1/2 φν1/2 φμν

(6.9)

where

The factor φ μν in the constant pressure partition function terms must be included for all separations on the lattice for each nearest neighbour pair of molecules μ–ν. If the interaction energy between such a pair is Jμν, then the corresponding factor in the constant pressure partition function is exp(–βJμν ), and for r vacant sites or holes between the pair, the corresponding factor is exp(–rβπ ν 0 ). Hence   φμν = exp −βJμν + exp (−βπ ν0 ) + exp (−2βπ ν0 ) + exp (−3βπ ν0 ) + . . . ,

is introduced into (M, T, π ). Summation of the geometric progression gives   φμν = exp −βJμν + exp (−βπ ν0 ) (1 − exp (−βπ ν0 )) .

(6.10)

Now since

 (M,T,π ) =

Aαβ Aβγ Aγ δ . . . Aωα

α=m,d;β=m,d;γ =m,d;ect

and using the inner-productrule for multiplying two square matrices B, C such that D=BC, where Di,j = k Bi,k Ck,j for large M, the constant pressure partition function can be rewritten as  (M,T,π ) = TraceAM = λ1M + λ2M ≈ λmaxM

(6.11)

where λ1 and λ2 are the eigenvalues of the matrix A and λmax is the largest of these. The 2×2 symmetrical matrix A for the benzene assembly in the silicalite

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pore has the diagonal elements Amm = φm φmm , Add = φd φdd and an off-diagonal 1/2 1/2 term Amd = Adm = φm φd φmd . φ m and φ d are given as φm = qint exp (−β (π ν0 + U0 − U1 ))

(6.12)

φd = qint exp (−β (2π ν0 + U0 ))

(6.13)

The maximum eigenvalue λmax of the matrix A can be given as

λmax =

Amm + Add +



(Amm − Add )2 + 4Amd2 2

(6.14)

The expression of Gibbs free energy of the adsorbed phase is G(M,T,π ) = −kB T ln  (M,T,π ) = −MkB T ln λmax

(6.15)

According to standard statistical thermodynamics, the chemical potential of the adsorbed phase can be obtained by differentiation of the Gibbs free energy 

μad =

∂G ∂M



T,P

= −kB T ln λmax

(6.16)

The equilibrium density of the adsorbed phase at a fixed internal pressure π is obtained straightforwardly using the relation V=(∂G/∂π ) and G as defined above. V=

∂ ln λmax ∂G = −MkB T ∂π ∂π V = Nv0

(6.17) (6.18)

where v0 is the volume of each site. From the above we obtain kB T ∂λmax N =− M v0 λmax ∂π

(6.19)

Therefore, the density of adsorbed phase is ρad =

1 M  = T ∂λmax N − v0kλBmax ∂π

(6.20)

Thus from a given internal pressure π , an adsorbed phase density and chemical potential can be calculated. Since the chemical potential in the adsorbed phase is set equal to that in the ideal gas phase, chemical and hence isotherms can be obtained by calculation of the mean occupation numbers of the two types of pores.

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6.2.3 Calculation Procedure • Establish the partition function of adsorbed phase. • Following the standard statistical mechanics, obtain the chemical potential for the adsorbed phase. • Set the temperature T and internal pressure π . • Calculate φ m, φ d, φ md , φ mm and φ dd . • Obtain Amm, Add and Amd and furthermore get λmax . • The adsorbed-phase chemical potential can then be found. • Since the chemical potential of adsorbed phase and gas phase is equal when equilibrium is achieved, the pressure of the gas phase can be worked from the idealgas chemical potential. • Finally, calculate the number of adsorbed molecules.

6.2.4 Numerical Results and Discussion Adsorption isotherms are calculated for a wide range of interaction parameters. If the enthalpy for dimer state molecules is lower than that for compressed arrangements, which is decided by Eqs. (3–5), then the isotherm for a single pore always shows a step, see Fig. 6.3 (taken from Ref. [25]). The cause of this is that dimer state molecules do not compress into a close-packed arrangement until the chemical potential is sufficiently high. The step in the theoretical isotherms is in no case due to a phase transition in the adsorbed system but arises from an energetically costly 9.0 8.0

molecules per unit cell

7.0 6.0 5.0 4.0 3.0 2.0

T = 273K

1.0 0.0 0

T = 283K 1

2 log pressure (pa)

3

4

Fig. 6.3 Theoretical adsorption isotherms of benzene in silicalite for the single-pore model (taken from Ref. [25], copyright Elsevier)

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10

molecules per unit cell

8

6

4

2

T = 273K T = 283K

0

0

1

2

3

4

log pressure (pa)

Fig. 6.4 Theoretical adsorption isotherms of benzene in silicalite for the two-pore model. The interaction parameters for both pores used are Jdd = −5.90×10−21 J, Jmm = 0.0 J, Jmd =1.0×10−18 J, U0 = −5.95×10−20 J, U1 = −3.0×10−21 at 273 K, but with U1 = −3.0×10−21 and −5.0×10−21 J for the two pores at 283 K, respectively (taken from Ref. [25], copyright Elsevier)

co-operative compression from dimer states to monomer states with a lower volume residence. For a combination of two types of pores with slight differences in U1 values, −3.0×10−21 J and −5.0×10−21 J, the additional shoulder is produced in the isotherms and is shown in Fig. 6.4. Here it is considered that the shoulder is due to molecules in one pore compressed before the other and for which the transition is preferred in the pore with smaller absolute U1 value. The heat of adsorption at low coverage obtained from the theoretical isotherms in standard way is close to 60 kJ/mol in satisfactory agreement with experimental values reported by Thamm [17] and Chiang et al. [26], which is shown in Fig. 6.5. There is a dip in the heat of adsorption at a coverage of approximately 4–5 molecules per unit cell due to interconversion between dimer states and compressed states also seen experimentally and a rise at higher coverage which follows the experimental trends reported by Pope [16] and Thamm [17]. The magnitude of the parameters used has a straightforward interpretation. The interaction parameter Jdd =−5.90×10−21 . J has the expected order for the dispersion interaction between two benzene rings separated by about 5.0 Å. The LennardJones 12-6 potential well depth for the spherically averaged benzene–benzene interaction is −6.07×10−21 J [7]. These parameters for the benzene–benzene interaction, Jdd and U0 = −5.95×10−20 J, are also close to those used by Snurr et al., see Fig. 14 of Snurr’s work [23]. The repulsive character of Jmm = 0.0 J, Jmd =1.0×10−18 J is consistent with that expected for molecules being compressed together. If the

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differential molar heat of adsorption (kJ/mol)

Pope et al. (1986) Thamm et al. (1987)

100

Thamm et al. (1987)

90

80

70

60

50

40 0

2

4 6 molecules per unit cell

8

10

Fig. 6.5 Differential molar heat of adsorption calculated for benzene adsorbed in the two-pore model of silicalite (taken from Ref. [25], copyright Elsevier)

parameter U1 is interpreted as representing the cost per molecule to cause a loadinginduced structural transition in the silicalite and which has an average value of −4×10−21 J, it is of interest to note that it is close to the corresponding parameter Azeo used by Snurr et al. [23, 24]. Thus in summary, for a wide range of interaction parameter, the model gives jumps in adsorption isotherms similar to those observed experimentally for benzene adsorption in silicalite. The exact calculation attributes the experimentally observed jumps in the level of adsorption with rising pressure to reorientational transitions amongst molecules in the adsorbed phase. The model also satisfactorily reproduces the essential features of the loading dependence of the heat of adsorption. Although the model is used to give a plausible interpretation of benzene adsorption in the quasi one-dimensional pores of zeolites with two types of pores, the model also has relevance to the alkane adsorption studied by Smit et al. [9, 23–24] and more generally to the adsorption of gases on surfaces where the moleculs can lie flat or perpendicular to a surface. The model described in this chapter can be extended to larger molecule adsorption in zeolites. Next, we will show that the similar model and treatment can be extended to explain the inflexion points in adsorption isotherms of simple alkanes.

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6.3 Exact Statistical Mechanical Lattice Model of Commensurate Transitions of Alkanes Adsorbed in Silicalite Fluids adsorbed in the narrow pores of zeolites can undergo “freezing” transitions whose theoretical interpretation presents a number of challenges as discussed by Smit and Maesen [9] and Du et al. [29]. Most studies of molecules in pores have focused on simple fluids, which can be modelled as almost spherical molecules. However, molecular shape and pore structure can also exert an influence on the adsorption behaviour particularly for large molecules. Thus, the adsorption isotherms of alkanes in the silicalite zeolite provide an example of this where the short-chain (C3 –C5 ) and long-chain (C10 ) alkanes have simple isotherms, whereas isotherms for ethane (C2 ), hexane (C6 ) and heptane (C7 ) have inflection behaviour, suggesting that some kind of ordering transition takes place. Computer simulations of the adsorption of straight-chain hydrocarbons in silicalite suggest that these inflection points arise as a consequence of the interplay between the length of the zigzag pores and the length of the alkane molecules. In particular, it was originally suggested from the analysis of Monte Carlo density distributions by Smit and Maesen [9] for hexane and heptane and subsequently restated by Du et al. [29] for ethane isotherms that when the length of the pore is an integer number of molecular lengths, the molecules can “freeze” in a configuration that is commensurate with the pore structure, giving an unusual shape to the isotherms. In this section a study of commensurate transitions in a lattice model of the ethane, hexane and heptane in silicalite, where there is energetic competition between open and close-packed structures, is presented. The aim is to investigate whether the unusual shapes of the isotherms can be reproduced by an accurate treatment of a model which builds in the above picture.

6.3.1 One-Dimensional Lattice Model of Linear Alkanes in Silicalite As already mentioned above, there is a well-known channel structure in silicalite that consists of zigzag and straight channels and the intersection, which joins them. The Monte Carlo simulations of ethane, hexane and heptane show that at low chemical potential, ethane, hexane and heptane move freely in the silicalite pore system, but at high pressure, two ethane molecules or a single hexane or heptane molecule is forced at some energetic cost into the zigzag pore. The geometry of the lattice restricts this ordering only to these alkanes in silicalite since all other straight-chain alkanes are too long. To obtain an exactly soluble statistical mechanical lattice model, which mimics the situation described above, a one-dimensional lattice model is considered here. To model commensurate transitions, an alkane molecule is assumed to adopt one of three possible states. The low energy extended residence (type 1) occupying δ lattice

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sites or compressed into one of two higher energy state occupying σ sites with differing orientations or conformational states (type 2 and type 3). The formalism is the same for both these possibilities. The Helmholtz free energy of adsorption of the molecule in the zeolite is denoted by U0 for extended states and U0 –U1 for compressed states. The parameter U1 can include the energy cost of any conformational transitions and should be a small fraction of U0 . The Monte Carlo density distributions observed by Du et al. [29] suggest that at low pressure, ethane molecules can move freely over the zeolite lattice, but at high pressures, they localise onto single sites. The essential feature of the model which embodies this is shown below for close-packed configurations of ethane molecules where movement from left to right represents increasing pressure. increasing pressure × ×

× ×

×

×

→

Type 1

×

×

or

×

Type 2

×

×

×

Type 3

The crosses represent lattice sites, while the lines represent molecules. The molecules on the left are type 1 occupying two sites (δ = 2), while those on the right are types 2 and 3 with σ =1. On the left side above, molecules of the appropriate length commensurate with the lattice spacing and can reside in an open energetically favourable position, occupying more lattice sites/molecules than do the higher energy compressed arrangement on the right. The model allows two orientations or conformational states of the compressed molecules as shown on the right above. Hexane and heptane are effectively twice the length of an ethane molecule; so guided by the Monte Carlo density distributions given in Fig. 2 from the work of Smit and Maesen [9], the equivalent picture of the compression for these molecules is ⎯⎯⎯⎯

⎯⎯⎯

× × × × × × Type 1

increasing pressure →

× × × × Type 2

or

× × × × Type 3

Heptane molecules are about 1.3 Å longer than hexane molecules so that the fit of heptane into the pore is tighter and energetically somewhat more costly than for hexane. Hence U1 for heptane should be larger than that for hexane. The above model is not exactly a representation of silicalite but is the simplest that embodies the essential physics and can be accurately treated. Alkane molecules thus arranged on the lattice may have three possible ground states. They are as follows: (i) all molecules in extended states (11111. . . .11), (ii) all in compressed states (22222. . . .22 or 33333. . . .33) and (iii) an alternate arrangement of extended and compressed states (121212. . .12 or 131313. . .13). At absolute zero, the stable state of an assembly with an internal pressure π in the silicalite pore will be the one giving the lowest configurational enthalpy Hc , where Hc = Ec + Nπ v0. Ec is the configurational energy. For the arrangement (i) above, when N=δM

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Hc = M (U0 + δπ v0 + J11 )

133

(6.21)

where J11 is the interaction energy between two molecules in extended states. For the arrangement (ii) above, when N=Mσ Hc = M (U0 − U1 + δπ v0 + J22 )

(6.22)

where J22 is the interaction energy between type 2 molecules in compressed states. A similar expression exits for type 3 orientations. For the arrangement (iii) above, when N=(δ+σ )/2 Hc = M (U0 − U1 + (δ + σ )π v0 /2 + J12 )

(6.23)

where J21 is the interaction energy between molecules in extended and compressed states. The jumps in the isotherms to be shown below are attributed to energetic competition between ground states (i), (ii) and (iii). It should be remarked that the interaction parameters referred to above are not those expected from a minimum of a potential energy curve for a pair of molecules in the gas phase but rather those appropriate for a specified lattice site separation, and since the model is one dimensional, the parameters required to reproduce experimental isotherms may be somewhat on the higher side.

6.3.2 Constant Pressure Partition Function The following section is an extension of that given above for benzene adsorption to allow for the three species 1, 2 and 3 on the lattice sites. For an assembly of three types of molecules on a ring (which avoids complications due to boundary effects), a configuration of the 1, 2 and 3 type species denoted by α, β, γ , δ,. . .,ω, α contributes a term to the constant pressure partition function, which can be written as a product of M factors given as Aαβ Aβγ Aχ δ · · · Aωα , where Aμv = φμ1/2 φv1/2 φμv

φ1 = qint exp [−β (δπ v0 + U0 )] φ2 = qint exp [−β (σ π v0 + U0 − U1 )] φ3 = qint exp [−β (σ π v0 + U0 − U1 )]

(6.24)

Following similar arguments to those given above   φμν = exp −βJμν + exp (−βπ ν0 ) (1 − exp (−βπ ν0 ))

(6.25)

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and for large M M M M  (M,T,π ) = λM 1 + λ2 + λ3 ≈ λmax

(6.26)

where λ1 , λ2, λ3 are the eigenvalues of the matrix ⎞ √ √ φ1 φ2 φ21 √φ3 φ1 φ31 √ φ1 φ11 ⎝ φ1 φ2 φ12 φ2 φ22 φ3 φ2 φ32 ⎠ √ √ φ1 φ3 φ13 φ2 φ3 φ23 φ3 φ33 ⎛

and λ max is the greatest of these. The Gibbs free energy of the adsorbed phase is given as G (M,T,π ) = −kB T ln  (M,T,π ) = −MkB T ln λmax

(6.27)

and hence the chemical potential obtained by differentiation of the Gibbs free energy is μads = −kB T ln λmax

(6.28)

The equilibrium density of the adsorbed phase at a fixed internal pressure π is obtained numerically using the relation V=(∂G/∂π ) and G as defined above. Thus from a given internal pressure π , an adsorbed phase density and chemical potential can be calculated, which is set equal to the ideal gas-phase chemical potential isotherms constructed by solving the above system of equations numerically.

6.3.3 Numerical Results and Discussion Adsorption isotherms for ethane, hexane and heptane were calculated for a wide range of interaction parameters and are shown in Figs. 6.6, 6.7 and 6.8(see the captions for the parameter values) with the aim to reproduce the isotherm shape. At low temperatures the U1 parameters have values close to 1.5 × 10−21 J, 11.8 × 10−21 J and 12.6 × 10−21 J for ethane, hexane and heptanes, respectively. As expected, the larger molecules are more difficult to compress into the pore. The parameters J11 given in the figure captions reflect the tail-to-tail interaction between the molecules in extended states and should be close in value to ethane and hexane (−0.132 × 10−20 J), where in both molecules the methyl group tail-to-tail distance is similar (about 4.5 Å). In the heptane extended states, the tails are closer (about 4.0 Å), thereby allowing a stronger interaction. Because of the uncertainty of the position of the points on the heptane isotherm, the J11 value lies in the range from −0.3 × 10−20 to −0.7 × 10−20 J, with the strongest inflection produced by the latter value. All these parameters are of a magnitude expected from a Lennard-Jones 12-6 potential with the well depths and hard core radii given in reference [25]. Similarly, the

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Fig. 6.6 Comparison of the adsorption isotherms calculated using the lattice model (solid curve) with that of Monte Carlo simulations and experiments for ethane in silicalite (details in Chapter 3). The Monte Carlo and experimental adsorption isotherm points are almost coincident and are shown as open circles. U0 = −6.66× 10−20 J and U1 =0.05U0 at 275 K. U0 = −6.98× 10−20 J and U1 =0.05U0 at 300 K. U0 = −7.30 × 10−20 J and U1 =0.05U0 at 325 K. U0 = −7.62 × 10−20 J and U1 =0.05U0 at 343 K. At all temperatures J11 = −0.132 × 10−20 J, J22 = −0.033× 10−20 J, J12 = −0.33 × 10−20 J, J21 = −0.66 × 10−20 J, J23 = −0.132 × 10−20 J, J32 = 0.132 × 10−20 J. By symmetry, J33 = J22 , J31 = J12 , J13 = J21 (taken from Ref. [30], copyright Elsevier)

temperature-dependent U0 values correspond to a Helmholtz free energy of adsorption and are of a magnitude expected from the 12-6 potential referred to above. The shape of the adsorption isotherms for ethane is in good agreement with the experimental and GCMC simulation results. In particular, they reproduce the inflection point of the simulation results observed at low temperatures, 275/300 K. The cause of this is that extended state molecules do not compress into a close-packed arrangement until the chemical potential is sufficiently high.

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Fig. 6.7 Comparison of the adsorption isotherms calculated using the lattice model (solid curve) with that of Monte Carlo simulations and experiments for hexane in silicalite. The solid and open triangles are Monte Carlo and experimental data, respectively (taken from the work of Smit and Maesen [9]). The open circles are the experimental results of Sun et al. (1996). U0 = −11.84 × 10−20 J and U1 =0.10U0 at 298 K. U0 = −10.75× 10−20 J and U1 =0.005U0 at 303 K. U0 = −11.22× 10−20 J and U1 =0.005U0 at 323 K. U0 = −11.30 × 10−20 J and U1 =0.020U0 at 343 K. At all temperatures J11 = −0.132 × 10−20 J, J22 = −0.033× 10−20 J, J12 = −0.5 × 10−20 J, J21 = −0.66 × 10−20 J, J23 = −0.132 × 10−20 J, J32 = 0.132 × 10−20 J. By symmetry, J33 = J22 , J31 = J12 , J13 = J21 (taken from Ref. [30], copyright Elsevier)

Rather similar remarks can be made about hexane (see Fig. 6.7) and heptane (see Fig. 6.8), where the inflection structure shown in both experiment and Monte Carlo simulations for heptane in particular is reproduced by the model presented here. Thus, in summary a study of commensurate transitions in a lattice model of adsorbed straight-chain alkanes (ethane, hexane and heptane) in the silicalite zeolite, where there is energetic competition between open and close-packed structures, is presented. A comparison is made of the calculated isotherms with those from experiments and Monte Carlo simulations. The exactly soluble one-dimensional lattice

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Fig. 6.8 Comparison of the adsorption isotherms calculated (solid curve) using the lattice model with that of Monte Carlo simulations and experiment for heptane in silicalite. The solid and open triangles are the Monte Carlo and experimental results, respectively (taken from Smit and Maesen [9]). The open circles are the experimental results of Sun et al. (1996). U0 = −12.63 × 10−20 J and U1 =0.10U0 at 298 K. U0 = −12.5 × 10−20 J and U1 =0.075U0 at 303 K. U0 = −12.7 × 10−20 J and U1 =0.10U0 at 323 K. U0 = −13.1 × 10−20 J and U1 =0.10U0 at 343 K. At all temperatures J11 = −0.33 × 10−20 J, J22 = −0.0165× 10−20 J, J12 = −0.0 × 10−20 J, J21 = −0.0 × 10−20 J, J23 = −0.0165× 10−20 J, J32 = −0.0165 × 10−20 J. By symmetry, J33 = J22 , J31 = J12 , J13 = J21 (taken from Ref. [30], copyright Elsevier)

model shows that when molecules can exist in states which occupy integer numbers of lattice sites, the molecules can make ordering transitions between configurations that are commensurate with the pore structure, giving unusual shape to the isotherms. The unusual isotherm shapes predicted by the model are in broad agreement with those from Monte Carlo simulations and experiments, thereby supporting the interpretation of such inflections in isotherms first put forward by Smit and Maesen [9].

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6.4 Statistical Mechanical Lattice Models of Endohedral and Exohedral Xenon Adsorption in Carbon Nanotubes and Comparison with Monte Carlo Simulation The discovery of carbon nanotubes [4, 32–33] by Ijima occurred almost two decades ago; yet despite much effort, unresolved questions remain about the adsorption properties of these materials and nanotubular structures in other forms [34] of carbon (for a review of nanotubular adsorption, see [35]) related to the fullerenes. Such materials exhibit potential for numerous applications due to their microscopic structure. It is expected that matter adsorbed in such nanochannels might exhibit unusual behaviour due to the reduced dimensionality. Carbon nanotubes may have unusual electrical properties sensitive to adsorption of gases. Inert gases are of particular interest and adsorption of xenon in carbon nanotubes has been investigated both experimentally by Kuznetsova et al. [5] and theoretically by Simonyan et al. [6]. Thus, adsorption isotherms obtained by Monte Carlo simulation for endohedral xenon adsorption in isolated carbon nanotubes show a step-like structure (see Fig. 6.9), while exohedral adsorption isotherms are more conventional. Here, we present the results of matrix-based statistical mechanical calculation of a lattice model of endohedral xenon adsorption in a narrow nanotube considered as sites on the walls of a cylinder using a matrix method, while exohedral adsorption is treated by mean-field theory. For a range of interaction parameters, the model reproduces the unusual features in the adsorption isotherms. We find a step in the xenon adsorption isotherm caused by intermolecular interactions and the energetic cost of packing atoms into the same neighbourhood.

Fig. 6.9 Endohedral and exohedral adsorption of xenon at 95 K (filled and open spheres, respectively) in an isolated (10,10) carbon nanotube (adapted from Simonyan, Johnson, Kuznetsova and Yates, Ref. [6])

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This work parallels similar work by Trasca et al. [36] and Trasca et al. [37] and that by Gubbins and coworkers [11]. The agreement with the Monte Carlo simulations of Simonyan et al. [6] gives us confidence in our model and method, thereby allowing us to very rapidly calculate a series of isotherms at a range of temperatures and at much less computer time than the simulations. In this work, we focus on narrow-bore (10,10) nanotubes (diameter 13.56 Å) shown in Fig. 6.2, which may be expected to differ significantly in properties from those with larger diameters. Recent work by Jiang et al. [38] has presented simulated isotherms for alkane adsorption in wide-bore (40.68 Å diameter) carbon nanotubes with smooth walls. These isotherms do show evidence for condensation phenomena probably due to the large diameter and intermolecular interactions. Of considerable interest in wide-bore nanotubes is the specific nature of any critical point (if it exists) in the adsorbed phase. We consider an ideal xenon vapour phase at ambient pressure and temperature P, T in equilibrium with an adsorbed phase of atoms in the nanotube. In our lattice model, we separate the problem into two non-interacting systems: one exohedral and the other endohedral, both at the same chemical potential as the vapour phase. Lattice models of adsorbed fluids have been widely discussed and can successfully model unusual adsorption features displayed by porous materials. In order to have confidence in our results for the one-dimensional endohedral adsorbed phase, an accurate statistical treatment is essential. Here the grand partition function for the endohedral adsorbed component in a nanotube is evaluated using a matrix method. The exohedral phase is modelled as adsorption on a two-dimensional layer, which is a reasonable approximation for a nanotube of sufficient diameter.

6.4.1 Matrix Method for Grand Partition Function for Endohedral Adsorption In our lattice fluid model, it is assumed that there are groups of adsorption sites along the nanotube walls. At the highest density the tube would contain along the tube repeating units of four Xe atoms as shown in Ref. [6]. This is not an exact representation of the adsorbed phase but is thought to realistically mimic the adsorption sites in the pore. Each adsorption site can be occupied by an adsorbed atom or is vacant (occupied by a “hole”). We ignore any issues arising from the corrugation of the tube, treating the nanotube wall as smooth. We present the theory for our model for the general case of N rings each with n=4 adsorption sites along the nanotube so that there are nN adsorption sites on the inside of the nanotube. All four sites on a ring are assumed to be energetically accessible to xenon atoms. We assume cyclic boundary conditions and consider the possible configurations of species with 0, 1, 2,. . .,4 identified by the index i on a chain of N rings. Each ring with n=4 sites may be occupied by i atoms in a variety of configurations each of which represents one of gi = n!/(n–i)!i! types of clusters of class i. We assume following Simonyan et al. [29] that the potential energy of

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a single adsorbed xenon atom is U (−2800k, where k is Boltzmann’s constant). To simplify the analysis we will assume that all clusters of atoms of type i in the same ring have an energy Bi , which arises from interactions between species on the same ring, and we treat all such clusters in the same class as a single species with degeneracy gi . The Bi interaction parameters (i=0, 1..4) are critical for controlling the isotherm steps and implicitly take into account the loss of free energy when a pair of non-spherical atoms is pushed together at short range. For two atoms per cluster, B2 = ε; for three atoms per cluster, B3 = 3ε and for four atoms per cluster, B4 =6ε. The clusters on nearest neighbour rings also interact. The parameter Jμv is the interaction energy between pairs of clusters μ,v on nearest neighbour rings and is calculated using the estimate Jμv =µνε, where ε (−221k) is the well depth in a Lennard-Jones potential for pairs of xenon atoms. The product µν is a measure of the number of pair interactions between clusters. The vibrational and translational degrees of freedom of a heavy adsorbed xenon atom confined inside the nanotube contribute a factor to the partition function of the endohedral phase which is assumed to be small and negligible. The grand partition function  for the endohedral adsorbed phase is given as

=

∞

M=0

 μM Q (M,nN,T) exp kT 

(6.29)

where μ is the chemical potential, Q(M,nN,T) is the canonical partition function for M interacting atoms absorbed over nN sites in a nanotube. We assume that occupation of sites by pairs of atoms is energetically forbidden so that there are now at most nN atoms on the lattice, which restricts the above summation to nN atoms. By inspection it can be seen that  may be written as the sum of the products of N factors given as =



(α=0,1,2,3,4) (β=0,1,2,3,4)

··········

(ω=0,1,2,3,4)

Aαβ Aβγ Aγ δ · · · · · · · · · ·Aωα (6.30)

The factors above are given as  1 Aμν = φμ φν 2 φμν φ0 = 1 

i] φi = gi exp [i·(μ−U)−B  kT  −J φμν = exp kTμν

Using the inner-product rule Dij =

 k



(6.31–6.34)

Bik Ckj for matrix multiplication of a pair

of matrices B and C, the grand partition function can be expressed as

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 N   A αα = Trace AN = (λ1 )N + (λ2 )N + (λ3 )N + (λ4 )N + (λ5 )N

(6.35) where λ1 , λ2 , . . .., λ5 are the eigenvalues of the matrix A whose elements are defined above. For macroscopically large N we then obtain  = (λmax )N

(6.36)

where λmax is the largest eigenvalue of the above matrix, which can be obtained numerically. The mean number of adsorbed atoms ρ given by the relation ρ=

kT ∂ · ln (λmax ) n ∂μ

(6.37)

Thus for a given gas-phase composition, the endohedral contribution to the adsorption isotherms at various temperatures for this one-dimensional model can be constructed.

6.4.2 Lattice Model of Exohedral Xenon Adsorption The method used above for endohedral adsorption becomes difficult to implement for exohedral monolayer adsorption on the surface of a nanotube due to the many possible configurations. Hence, we utilise mean-field theory, which should give a reasonable description of the exohedral contribution to the adsorption isotherm since computer simulations do not show evidence of exohedral phase transitions. We consider a two-dimensional arrangement of Nex lattice points and associated with each point is a cubic cell of a volume v0 =64 Å3 , which may be vacant or contain one of Mex xenon atoms. The exohedral number density is defined by Mex /Nex . We suppose that the adsorption potential Uads = −2295k, close to that calculated in Ref. [5], confines each atom to a cell. Such an atom is allowed translational degrees of freedom such that the partition function of a xenon atom in the cubic cell is [39] qint = 3 vo + exp



−Uads kT



(6.38)

where =



2π mkT h2



(6.39)

Each cell has a coordination number of Z=4 as suggested in Ref. [29] at least at low coverage. At higher coverage, an increased coordination number may be appropriate. The canonical partition function for the assembly is given as

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  Vint Nex ! Mex Q= (qint ) exp − Mex ! (Nex − Mex )! kT

(6.40)

Vint is the mean interaction energy given as ZJMex2 2Nex . J is the thermal average intermolecular interaction between pairs of molecules separated by the distance between two cell centres, which we take to be 0.4ε. The equilibrium separation is that between two wells and not the usual re in the Lennard-Jones potential. The Helmholtz free energy is then given as F= –kT ln(Q) and by standard statistical thermodynamics, the chemical potential is given as 

ρ μ = JZρ + kT ln 1−ρ



  − kT ln 3 v0

(6.41)

By equating this to the gas-phase chemical potential, the density of the exohedral adsorbed layer can be calculated. Thus for a given gas-phase composition, adsorption isotherms can be constructed.

6.4.3 Theoretical Isotherms for Xenon Adsorption in Carbon Nanotubes Isotherms for endohedral adsorption are shown in Fig. 6.10, where the correspondence with the computer simulations of Simonyan et al. [6] can be observed. The isotherms show a sharp step but are never truly discontinuous in the mathematical sense. The sharpness is governed by intermolecular interactions between xenon atoms and disappears in a non-interacting system. Thus, we have carefully established that despite this sharpness, there is no critical point in contrast to n-alkanes adsorbed in wide-bore carbon nanotubes referred to above [38]. However, it seems

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likely that in a three-dimensional bundle of nanotubes, longer range interactions would make the transition discontinuous. To obtain the total adsorption, the endohedral component must be added to the exohedral component. These are shown in Fig. 6.11, where the adsorption due to endohedral adsorption has been included to give the total adsorption and where the values of the interaction parameters are given in the captions. The shape of the isotherms quite remarkably reproduces all of the essential features of the Monte Carlo isotherms. The endohedral isotherms are extremely sensitive to temperature (see Fig. 6.10). This component to the adsorption shows a marked temperature dependence. A heat of adsorption at low coverage may be estimated from the gradient of ln P vs. 1/T in the standard way using the data given in Fig. 6.10. We estimate a value of approximately 31 kJ/mol, which is close to that expected from the total interactions experienced by the Xe atoms inside the tube as described above. This may be compared with the experimentally determined [5] heat of desorption of about 27 kJ/mol measured for the Xe phase confined in a single-walled carbon nanotube which is itself close to theoretical estimates [41].

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3. Olson DH, Kokotailo GT, Lawton SL, Meier WM (1981) Crystal-Structure and StructureRelated Properties of ZSM-5. J Phys Chem 85: 2238–2243 4. Iijima S (1991) Helical Microtubules of Graphitic Carbon. Nature 354: 56–58 5. Kuznetsova A, Yates JT, Jr, Liu J, Smalley RE (2000) Physical Adsorption of Xenon in Open Single Walled Carbon Nanotubes: Observation of a Quasi-One-Dimensional Confined Xe Phase. J Chem Phys 112: 9590–9598 6. Simonyan V, Johnson JK, Kuznetsova A, Yates JT, Jr, (2001) Molecular Simulation of Xenon Adsorption on Single-Walled Carbon Nanotubes. J Chem Phys 114: 4180–4185 7. Atkins PW, de Paula J (2006) Physical Chemistry, eighth edition, Oxford University Press 8. Smit B (2008) Molecular Simulations of Zeolites: Adsorption, Diffusion, and Shape Selectivity. Chem Rev 108: 4125–4184 9. Smit B, Maesen TLM (1995) Commensurate ‘Freezing’ of Alkanes in the Channels of a Zeolite. Nature 374: 42-44 10. Bell GM, Combs LL, Dunne, LJ ( 1981) Theory of Cooperative Phenomena in Lipid Systems. Chem Rev 81: 15–48 11. Gelb LD, Gubbins KE, Radhakrishnan R, Sliwinska-Bartkowiak M (1999) Phase Separation in Confined Systems. Rep Prog Phys 62: 1573–1659 12. Frenkel D, Smit B (1996) Understanding Molecular Simulation from Algorithms to Applications. Academic Press, London 13. Nicholson D, Parsonage NG (1982) Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press, London 14. Catlow CRA (1992) Modelling of Structure and Reactivity in Zeolites. Academic Press, London 15. Ruthven DM ( 1984) Principles of Adsorption and Adsorption Processes. John Wiley and Sons, New york 16. Pope CG ( 1986) Sorption of Benzene, Toluene, and p-Xylene on Silicalite and H-ZSM-5. J Phys Chem 90: 835–837 17. Thamm H (1987) Calorimetric Study on the State of Aromatic Molecules Sorbed on Silicalite. J Phys Chem 91: 8–11 18. Guo CJ, Talu O, Hayhurst, DT (1989) Phase Transition and Structural Heterogeneity: Benzene Adsorption on Silicalite. AIChE J 35: 573–578 19. Talu O, Guo CJ, Hayhurst DT (1989) Heterogeneous Adsorption Equilibria with Comparable Molecules and Pore Sizes. J Phys Chem 93: 7294–7298 20. Brunauer S (1944) The Adsorption of Gases and Vapours. Oxford University Press, London 21. Bell GM, Dunne LJ (1978) Theory of Anomalous Temperature Dependence in Spin-Labelled Lipid Monolayers. J Chem Soc Faraday Trans II, 74: 149–159 22. Mentzen BF, Lefebvre FL (1997) Flexibility of the MFI Silicalite Framework upon Benzene Adsorption at Higher Pore-Fillings: A Study by X-Ray Powder Diffraction, NMR and Molecular Mechanics. Materials Research Bulletin 32: 813–821 23. Snurr RQ, Bell AT, Theodorou DN (1993) Prediction of Adsorption of Aromatic Hydrocarbon in Silicalite from Grand Canonical Monte Carlo Simulations with Biased Insertions. J Phys Chem 97: 13742–13752 24. Snurr RQ, Bell AT, Theodorou DN (1994) A Hierarchical Atomistic/Lattice Simulation Approach for the Prediction of Adsorption Thermodynamics of Benzene in Silicalite. J Phys Chem 98: 5111–5119 25. Manos G, Dunne LJ, Jalili SE (2005) Matrix Computation of the Statistical Mechanics of a Lattice Model of Multistep Benzene Adsorption Isotherms in Silicalite. Chem Phys Lett 401: 430–434 26. Chiang AST, Lee CK, Rudzinski W, Narkiewicz-Michalek J, Szabelski P (1997) Energy and Structure Heterogeneities for the Adsorption in Zeolites in Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces Studies in Surface Science and Catalysis. 104: 519–572, Elsevier Science BV 27. Vlugt TJH, Krishna R, Smit B (1999) Molecular Simulations of Adsorption Isotherms for Linear and Branched Alkanes and Their Mixtures in Silicalite. J Phys Chem B 103: 1102–1118

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28. Maris T, Vlugt TJH, Smit B (1998) Simulation of Alkane Adsorption in the Aluminophosphate Molecular Sieve AlPO4 -5. J Phys Chem B 102: 7183–7189 29. Du Z, Manos G, Vlugt TJH, Smit B (1998) Molecular Simulation of Short Linear Alkanes and their Mixtures in Silicalite. AIChE J 44: 1756–1764 30. Manos G, Dunne LJ, Chaplin MF, Du Z (2001) Comparative Study of Monte Carlo Simulations and Exact Statistical Mechanical Lattice Model of Commensurate Transitions of Alkanes Adsorbed in Zeolites. Chem Phys Lett 335: 77–84 31. Rigby M, Smith EB, Wakeham WA, Maitland GC (1986) The Forces Between Molecules. Clarendon Press, Oxford 32. Iijima S, Ichihasji T, Ando Y (1992) Pentagons, Heptagons and Negative Curvature in Graphite Microtubule Growth. Nature 356: 776–778 33. Iijima S, Ichihashi T (1993) Single-Shell Carbon Nanotubes of 1-nm Diameter. Nature 363: 603–605 34. Dunne LJ, Sarkar AK, Kroto HW, Munn J, Kathirgamanathan P, Heinen U, Fernandez J, Hare J, Reid DG, Clark AD (1996) Electrical, Magnetic and Structural Characterization of Fullerene Soots. J Phys: Condens Matter 8: 2127–2141 35. Kondratyuk P, Yates JT Jr (2007) Molecular Views of Physical Adsorption Inside and Outside of Single-Wall Carbon Nanotubes. Acc Chem Res 40: 995–1004 36. Trasca RA, Calbi MM, Cole MW (2002) Lattice Model of Gas Condensation within Nanopores. Phys Rev E 65: 061607 37. Trasca RA, Calbi MM, Cole MW, Riccardo JL (2004) Lattice-gas Monte Carlo Study of Adsorption in Pores. Phys Rev E 69: 011605 38. Jiang JW, Sandler SI, Smit B (2004) Capillary Phase Transitions of n-alkanes in a Carbon Nanotube. NanoLett 4: 241–244 39. Hill TL (1960) An Introduction to Statistical Thermodynamics. Addison Wesley, Reading, Massachusetts 40. Dunne LJ, Manos G, Rekabi M (2009) Statistical Mechanical Lattice Models of Endohedral and Exohedral Xenon Adsorption in Carbon Nanotubes and Comparison with Monte-Carlo Simulations. Chem Phys 355: 99–102 41. Stan G, Cole MW (1998) Low Coverage Adsorption in Cylindrical Pores. SurfSci 395: 280–291

Chapter 7

Monte Carlo Simulation and Lattice Model Studies of Adsorption of Methane, Ethane, Carbon Dioxide and Their Binary and Ternary Mixtures in the Silicalite Zeolite George Manos, Lawrence J. Dunne, Akrem Furgani, and Sayed Jalili

Abstract Adsorption isotherms have been computed by Monte Carlo simulation for methane, ethane, carbon dioxide and their binary and ternary mixtures adsorbed in the zeolite silicalite. These isotherms show remarkable differences with the ethane–carbon dioxide mixtures displaying strong adsorption preference reversal at high coverage. To explain the differences in the Monte Carlo methane–carbon dioxide, ethane–carbon dioxide mixture isotherms, an exact matrix calculation of the statistical mechanics of a lattice model of mixture adsorption in zeolites has been made. The lattice model reproduces the essential features of the Monte Carlo isotherms, enabling us to understand the differing adsorption behaviour of methane/carbon dioxide and ethane/carbon dioxide mixtures in zeolites.

7.1 Introduction Separation of gases such as light alkanes and carbon dioxide and other gases [1–7] is challenging and of major importance for both environmental and economical reasons [8]. From the environmental point of view, it is crucial to develop effective separation technologies to reduce carbon dioxide emissions, which are regarded as the primary cause of global warming. From an economical point of view, the removal of carbon dioxide from natural gas and the recovery of alkanes are of some importance. Mixtures of simple gases can be separated by adsorption in zeolites due to the special characteristics of zeolite channels which may be designed to allow adsorption to selectively occur on the basis of molecular size and shape [9]. The zeolite silicalite G. Manos (B) Department of Chemical Engineering, University College London, Torrington Place, London, WC1E 7JE, UK e-mail: [email protected] This chapter has relied significantly on previous publications referred to in the text (Copyright Elsevier), which we reproduce without permission under the rights granted and retained by authors (http://www.Elsevier.com, 2008) L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_7,  C Springer Science+Business Media B.V. 2010

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(ZSM-5) is our primary concern here and is a highly porous material and throughout its structure it has an intersecting two-dimensional pore structure. ZSM-5 has two types of pores, both formed by 10-membered oxygen rings. The first of these pores is straight and elliptical in cross section; the second pores intersect the straight pores at right angles in a zigzag pattern and are circular in cross section as shown in Fig. 7.1. In this work, adsorption isotherms of pure methane, ethane, carbon dioxide and their binary and ternary mixtures in silicalite have been studied using Monte Carlo methods and some of these by statistical mechanical lattice models. With different methods used for gas separations, adsorption processes involving the use of zeolites [10] have demonstrated an increasing importance because these materials have a high thermal, mechanical and chemical stability. Silicalite pores have dimensions that are comparable to the molecular size of many gases, which make them ideal for adsorbing such gases [11–13].The straight channels (Y direction) of ZSM-5 have pore size of 5.4 × 5.6 Å and the zigzag channels (X–Z plane) have pore size of 5.5 Å, and the unit cell size is 20.07 Å × 19.92 Å × 13.42 Å. Figures 7.2 and 7.3 show further visualisations of the molecular structure of the ZSM-5 [14, 15]. Many experimental and simulations studies regarding adsorption of gases in different adsorbents are compiled and summarised in two recent books [16, 17]. Due to the difficulty of experiments, no adsorption isotherms have been published for methane–ethane–carbon dioxide binary and ternary mixtures at fixed gas phase composition. In this situation, to gain insights which are otherwise hard to obtain, grand canonical Monte Carlo (GCMC) simulations for methane–carbon dioxide and ethane–carbon dioxide mixture and their binary and ternary mixtures in the silicalite have been undertaken. Smit [18, 19] and colleagues have previously established that these types of simulations can reproduce the details of the experimentally determined single-component adsorption characteristics. The mixture isotherms show remarkable differences, with the ethane–carbon dioxide mixtures displaying adsorption preference reversal at high coverage. The con-

Fig. 7.1 Pore structure of silicalite

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Fig. 7.2 The microporous molecular structure of ZSM-5

Fig. 7.3 Ball and stick representation for silicalite. Silicon atoms are shown in dark grey and oxygen atoms in light grey Taken from ref [15]. Copyright Elsevier

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figurational bias Monte Carlo and the grand canonical Monte Carlo simulation techniques [35, 20–25] were used to compute adsorption isotherms at various temperatures ranging between 250 K and 353 K and over a wide range of pressures up to 200,000 kPa. In this work the united atom model was used and all components were represented as spherical Lennard-Jones molecules [37]. Carbon dioxide is a quadrupolar molecule and to save computer time it was modelled as a sphere whose LennardJones (LJ) potential parameters for “spherical” carbon dioxide molecules interacting with an effective potential in the silicalite cage were determined by fitting adsorption isotherms to experimental data. For methane and ethane, the LJ potential parameters were taken from the available literature [18, 19], while for carbon dioxide, the LJ potential parameters were determined empirically. The isosteric heats of adsorption for pure components were also calculated from the adsorption isotherms at various temperatures and at different loadings. The average calculated heats of adsorption for carbon dioxide, methane and ethane were found to be 24.7, 20.6 and 31.1 kJ/mol, respectively, in good agreement [26, 27] with some previous experimental and simulation work and were within the same range. The binary mixture adsorption isotherms were calculated for bulk compositions of 50:50 CO2 /CH4 , 10:90 CO2 /CH4 , 80:20 CO2 /CH4 , 50:50 CO2 /C2 H6 and 50:50 CH4 /C2 H6 . It has been found that at low pressures, ethane is preferentially more adsorbed than the other components due to its higher heat of adsorption, while at higher pressures, carbon dioxide is the most strongly adsorbed. Also interestingly, it has been found that at moderate pressures and when ethane adsorption starts to decrease, carbon dioxide and methane start to displace ethane molecules in the pores of the zeolite, which is attributed to entropic effects. The ternary mixtures adsorption isotherms were calculated for the equimolar mixtures of methane, ethane and carbon dioxide (33.3% each in the gas phase) and for the bulk composition of 85:15:5 CH4 /C2 H6 /CO2 . It has been found for the equimolar mixtures that ethane is the preferentially adsorbed component at lower pressures, but as the pressure increases, a reversal occurs and carbon dioxide becomes the predominantly adsorbed. For the 85:15:5 CH4 /C2 H6 /CO2 mixture, which represents a typical composition of natural gas, it has been found that initially ethane is more adsorbed and at a pressure approximately of 23,000 kPa, a reversal between ethane and methane occurs where ethane adsorption decreases and methane adsorption continues to increase. Despite the low presence of carbon dioxide (only 5%), it is observed that at very high pressures (about 200,000 kPa), carbon dioxide adsorption reaches almost the same adsorbed density as methane. However, despite their predictive capacity, Monte Carlo calculations do not provide a simple interpretation of isotherm structures. Thus to explain the differences in the Monte Carlo methane–carbon dioxide and ethane–carbon dioxide mixture isotherms, an exact calculation of the statistical mechanics of a lattice model of mixture adsorption in zeolites has been performed. The mixture grand partition function for methane/carbon dioxide and ethane/carbon dioxide mixtures adsorbed in a zeolite channel is calculated exactly using a previously described matrix method

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[28]. Approximate treatments, such as mean-field theory, are well known as being incorrect in one-dimensional systems [29, 30] by erroneously predicting phase transitions which may show as spurious steps in adsorption isotherms and hence to have confidence in our results, an accurate statistical mechanical treatment of the one-dimensional adsorbed phase is required. A single-lattice fluid model can by appropriate choice of parameters represent molecules in the alkane/carbon dioxide mixture in various configurations in the silicalite nanochannel. Vacant sites or holes are also introduced to allow for incomplete filling of the lattice sites. For a wide range of interaction parameter free energies, the lattice model gives unusual features in the shape of adsorption isotherms similar to those observed in Monte Carlo simulations for ethane–carbon dioxide mixtures adsorption in silicalite, thereby allowing us to interpret the results of the Monte Carlo simulations. At higher pressures, carbon dioxide molecules displace ethane molecules and we have named this phenomenon in a previous publication [31] as “adsorption preference reversal”.

7.2 Monte Carlo Simulations of Mixtures in Silicalite Monte Carlo simulations of small molecule adsorption in zeolites has been widely discussed and undertaken [32–34]. The method for the mixture was described in a previous publication [28] and uses united atoms. We have applied the same method to Monte Carlo simulations of ethane–carbon dioxide and methane/carbon dioxide mixtures in silicalite. All pseudo-atoms interact via Lennard-Jones-type potentials. The interactions between adsorbate–adsorbate and adsorbate–adsorbent are described by the Lennard-Jones potential.

U(rij ) = 4εij U(rij ) = 0,



 σij 12 rij

rij ≥ Rc

−



 σij 6 rij



,

rij < Rc

(7.1)

Where rij is the distance between atoms i and j, εij is the energy parameter (depth of the potential well), σ ij is the size parameter (collision diameter) and Rc is the cutoff radius of the potential. In this work, the cut-off radius is 13.8 Å. The contribution of the atoms beyond the cut-off radius to the total energy is estimated using the usual tail corrections [35]. In molecular simulations, it is customary to establish a cut-off radius Rc and ignore the interactions between atoms separated by more than Rc . This results a simpler program and large savings of computer resources. The interaction parameters between different united atoms i and j are calculated using the Lorentz-Berthelot mixing rules given as

σij =

σi + σj √ εij = εi εj 2

(7.2)

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Fig. 7.4 Experimental isotherms for adsorption of CO2 in silicalite at various temperatures compared with those generated by Monte Carlo simulation with the parameters given in Table 7.1 (taken from Ref. [38], copyright Elsevier)

The silicalite cage (see Fig. 7.1) is regarded as rigid, while methane and ethane are considered as one and two pseudo-atoms as discussed previously [34]. Ideally carbon dioxide would be represented by a polarisable quadrupolar molecule. However, this would require a very significant computational overhead. Thus, to avoid use of a prohibitively large amount of computer time, carbon dioxide is modelled as an effectively spherical molecule whose Lennard-Jones parameters for interactions inside the silicalite cage were determined by fitting the experimental adsorption isotherms [36] of pure carbon dioxide to that produced by Monte Carlo simulation shown in Fig. 7.4. The well depths and collision diameters are given in Table 7.1. Table 7.1 Lennard-Jones parameters used in the Monte Carlo simulations Lennard-Jones parameters

Well depth (/k)/[K]

Collision diameter (nm)

CO2 –CO2 CH3 –CH3 CH4 –CH4 CO2 –O (silicalite) CH3 –O (silicalite) CH4 –O (silicalite)

125.3 98.1 148.0 120.0 80.0 96.5

3.04 3.77 3.73 3.60 3.60 3.60

Carbon dioxide has been modelled in this way previously [37].

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Figures 7.5 and 7.6 show the equilibrium snapshots of the adsorbed molecules of carbon dioxide, methane and ethane in silicalite at 314 K and 50 kPa, and 277 K and 10,000 kPa, respectively. The zigzag channels are from left to right and the straight channels are from top to bottom. At the lower pressure, fewer molecules are adsorbed and therefore the sites are emptier than those at the higher pressures where the channels are more packed with adsorbates.

Fig. 7.5 Snapshots of pure methane, ethane and carbon dioxide adsorption in silicalite at 314 K and 50 kPa showing their distribution inside the silicalite channels (zigzag channels are from left to right and the straight channels are from top to bottom)

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Fig. 7.6 Snapshots of pure methane, ethane and carbon dioxide adsorption in silicalite at 277 K and 10,000 kPa showing their distribution inside the silicalite channels (zigzag channels are from left to right and the straight channels are from top to bottom)

7.3 Adsorption Isotherms of Binary Mixtures Figures 7.7, 7.8, 7.9, 7.10 and 7.11 show adsorption isotherms calculated by Monte Carlo simulation for ethane–carbon dioxide and methane–carbon dioxide mixtures in silicalite for the gas phase compositions given in the caption. It can be seen that carbon dioxide displaces ethane at higher coverage, while this does not occur with methane–carbon dioxide mixtures. This phenomenon which we have termed “adsorption preference reversal” has important implications for practical separation processes. To understand this behaviour, we have used the lattice model described below. The models used to describe the adsorption isotherms for the pure components can be used to determine the adsorption isotherms for binary mixtures as well. The simulations were carried out for the binary mixtures carbon dioxide– methane, carbon dioxide–ethane and methane–ethane. The carbon dioxide–methane mixture has been further studied by varying the gas phase compositions of the components. The simulations were carried for various temperatures ranging from 250 to 353 K and over a wide range of pressures up to 200,000 kPa.

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Monte Carlo Simulation and Lattice Model Studies 4.5

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4

253 K

CO2 CH4

4

3 2.5 2 1.5

3

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0 10

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N [m mol/g]

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CO2

313 K

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334 K

CO2

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CH4

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N [m mol/g]

1000 10000 100000 1000000 P [kPa]

4.5

N [m mol/g]

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0 0.1

P [kPa]

1

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100

1000 10000 100000

P [kPa]

Fig. 7.7 Adsorption isotherms of the equimolar binary mixtures (50% each in the gas phase) of carbon dioxide–methane in silicalite at various temperatures

7.3.1 Carbon Dioxide–Methane Mixtures The adsorption isotherms for the carbon dioxide–methane mixture were calculated for the following gas phase compositions: 50% carbon dioxide–50% methane (Fig. 7.7), 10% carbon dioxide–90% methane (Fig. 7.8) and 80% carbon dioxide– 20% methane (Fig. 7.9). At all temperatures, carbon dioxide is more preferentially adsorbed than methane as expected because of the stronger interaction between carbon dioxide and the

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3 253 K

CH4

2 1.5 1.5 1

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100 1000 P [kPa]

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0 0.1

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P [kPa]

1.8

1.8

1.6

300 K

1.4

CO2

1.6

CH4

1.4 N [m mol/g]

1.2 N [m mol/g]

CO2

277 K 2

N [m mol/g]

N [m mol/g]

2.5

CO2 CH4

1 0.8 0.6

333 K

CO2 CH4

1.2 1 0.8 0.6

0.4

0.4

0.2

0.2

0 0.1

1

10

100 P [kPa]

1000

10000 100000

0 0.1

1

10

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1000

10000 100000

P [kPa]

Fig. 7.8 Adsorption isotherms of the binary mixtures of 10% carbon dioxide–90% methane in silicalite at various temperatures

surface of the zeolite. Also as the temperature increases, methane becomes less adsorbed and at the temperature of 353 K, very small amounts of methane are adsorbed. This shows that the use of adsorption on silicalite as means of separation of carbon dioxide–methane mixtures is an excellent method. For the adsorption isotherms of mixtures of 10% carbon dioxide–90% methane: At low temperature (253 K), despite the predominant presence of methane, the two components are almost quantitatively equally adsorbed up to a pressure of 300 kPa. Above this pressure, carbon dioxide starts to be more adsorbed and replaces methane in the pores of silicalite. As the temperatures (277 K, 300 K, 333 K) increase, methane is initially more adsorbed than carbon dioxide. However, a reversal occurs and carbon dioxide becomes more adsorbed and replaces methane molecules. Interestingly, as the temperature increases, this observed reversal is delayed and higher pressures are required for carbon dioxide to be more adsorbed. At 333 K, methane is more adsorbed than carbon dioxide over the simulated pressure range. We can note that methane trend is just about to level off, while carbon dioxide trend continues to increase. This means that reversal occurs beyond the studied pressure at 333 K.

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4.5 4

253 K

3.5

CO2 CH4

277 K

4

CO2 CH4

3.5

3

3

2.5

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N [m mol/g]

N [m mol/g]

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2 1.5 1

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4.5 4

300 K

3.5

CO2 CH4

3.5

350 K

3

3

N [m mol/g]

N [m mol/g]

100 1000 P [kPa]

2.5 2 1.5

2.5 2 1.5

1

1

0.5

0.5

0 0.1

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1000

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CO2 CH4

0 0.1

P [kPa]

1

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1000

10000 100000

P [kPa]

Fig. 7.9 Adsorption isotherms of the binary mixtures of 80% carbon dioxide–20% methane in silicalite at various temperatures

For the adsorption isotherms of mixtures of 80% carbon dioxide–20% methane: At all temperatures, carbon dioxide is more strongly adsorbed than methane. As the temperature increases, the amount of carbon dioxide adsorbed is relatively less in the high temperatures than lower temperatures, yet carbon dioxide is more adsorbed than methane. As for methane at high pressures, as the temperature increases, the amount adsorbed increases in very small amounts. This is surprising as the behaviour of all pure components is the opposite.

7.3.2 Carbon Dioxide–Ethane Mixtures The adsorption isotherms for carbon dioxide–ethane were calculated for their equimolar mixtures (50% each in the gas phase) (Fig. 7.10) at different temperatures. Initially, ethane is preferentially more adsorbed than carbon dioxide. However, as the pressure increases, carbon dioxide adsorbs competitively displacing ethane and more carbon dioxide is adsorbed, while ethane becomes less adsorbed and the trend expressing ethane adsorption starts to decline, while carbon dioxide trend continues to increase. Interestingly, the initial difference between the amount of ethane adsorbed and the amount of carbon dioxide adsorbed reduces as the temperature

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3 CO2 C2H6

253 K

277 K

2.5

2

N [m mol/g]

N [m mol/g]

2.5

1.5 1 0.5

CO2 C2H6

2 1.5 1 0.5

0

0 1

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100000

1

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P [kPa] 2.5

10000

100000

2.5 313 K

2

CO2 C2H6

CO2 C2H6

353 K 2

1.5

N [mmol/g]

N [m mol/g]

1000

P [kPa]

1

0.5

1.5

1

0.5

0

0 1

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1000

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100000

P [kPa]

1

10

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1000

10000

100000

P [kPa]

Fig. 7.10 Adsorption isotherms of the equimolar binary mixtures (50% each in the gas phase) of carbon dioxide–ethane in silicalite at various temperatures

increases. This will be discussed more in the selectivities of the binary mixtures section. The reversal occurs here also and it is delayed as the temperature increases and the amount of carbon dioxide adsorbed becomes more than ethane at higher pressures. The reason behind this initial preference of ethane adsorbs more strongly than carbon dioxide at lower pressures is because of ethane’s higher adsorption energy, where ethane has a higher heat of adsorption than carbon dioxide. As the pressure increases and the sites of the silicalite become filled, the dominant forces become those between adsorbate molecules and the entropy effects become more important and that is where the reversal occurs and carbon dioxide starts to replace ethane molecules.

7.3.3 Methane–Ethane Mixtures The adsorption isotherms of methane–ethane were calculated for their equimolar binary mixtures (50% each in the gas phase) (Fig. 7.11) at different temperatures. At all temperatures, ethane is strongly more adsorbed than methane. The difference between the amount of ethane adsorbed and the amount of methane

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2.5

3 250 K

2.5

C2H6 CH4

277 K 2

C2H6 CH4

N [mmol/g]

N [m mol/g]

2 1.5 1

1.5

1

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1

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2.5 308 K

350 K

C2H6 CH4

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2

N [m mol/g]

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1000

P [kPa]

1.5

1

0.5

0.5

0 1

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1000

P [kPa]

10000

100000

0 1

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1000

P [kPa]

Fig. 7.11 Adsorption isotherms of the equimolar binary mixtures (50% each in the gas phase) of methane–ethane in silicalite at various temperatures

adsorbed decreases as the temperature is increased. As the pressure increases, the trend describing ethane reaches a plateau, and particularly at 250 K, it can be seen that at the highest pressure point, the ethane trend is just about to start to decline, whereas for the other temperatures, this is not observed probably because it requires higher pressures which are not within the studied range. As for methane it continues to adsorb even at higher pressures. The phenomenon of ethane being initially strongly more adsorbed may be attributed to the higher heat of adsorption of ethane, which is about 1.5 times that of methane. At high pressures, entropy effects become more important as the entropy of the methane-filled zeolite is plausibly much higher. This is the reason why at higher pressures the small molecules displace the bigger ones, which results in an increased adsorption of methane and a slightly decreased adsorption of ethane. This same observation has been found by Du [35] and Gallo et al. [10]. Figures 7.12, 7.13 and 7.14 are snapshots showing the distribution of equimolar mixtures of carbon dioxide–methane, carbon dioxide–ethane and methane–ethane, respectively, inside the silicalite channels. The ethane molecules are the atoms

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Fig. 7.12 Snapshots of equimolar mixture of carbon dioxide–methane adsorption in silicalite at 300 K and 100 kPa (left) and 20,000 kPa (right) showing their distribution inside the silicalite channels (zigzag channels are from left to right and the straight channels are from top to bottom)

connected with red bonds. The zigzag channels are from left to right and the straight channels are from top to bottom. In all snapshots the channels are more packed with molecules at higher pressure than lower pressures. It can be seen that at higher pressures, molecules seem to preferentially adsorb more in the intersections of the zigzag and straight channels of the silicalite.

Fig. 7.13 Snapshots of equimolar mixture of carbon dioxide–ethane adsorption in silicalite at 313 K and 40 kPa (left) and 10,000 kPa (right) showing their distribution inside the silicalite channels (zigzag channels are from left to right and the straight channels are from top to bottom) The ethane molecules are the atoms connected with the red bonds

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Fig. 7.14 Snapshots of equimolar mixture of methane–ethane adsorption in silicalite at 250 K and 70 kPa (left) and 17,000 kPa (right) showing their distribution inside the silicalite channels (zigzag channels are from left to right and the straight channels are from top to bottom). The ethane molecules are the atoms connected with the red bonds

7.4 Adsorption Isotherms of Ternary Mixtures We have also investigated adsorption isotherms for ternary mixtures of methane– ethane–carbon dioxide. The study of these ternary components is of great interest particularly for the natural gas industry. The simulations were conducted for equimolar mixtures of methane–ethane–carbon dioxide and a mixture of 85% methane–10% ethane–5% carbon dioxide.

7.4.1 Methane–Ethane–Carbon Dioxide Mixtures Figure 7.15 shows the adsorption isotherms for an equimolar (33.3% each in the gas phase) ternary mixture of methane–ethane–carbon dioxide at 250 K, 298 K and 350 K. At all temperatures and at lower pressures, ethane is initially preferentially more adsorbed and as the temperature increases, the difference between the amount of ethane adsorbed and the other components becomes less. At higher pressures, the other components start to displace ethane and the reversal between ethane and carbon dioxide occurs where carbon dioxide becomes the most strongly adsorbed component. As for methane, it is the lowest adsorbed component. However, when the reversal between ethane and carbon dioxide occurs, methane also starts to displace ethane and continues to be adsorbed as the pressure increases despite the fact that it is being adsorbed in much smaller quantities compared with carbon dioxide. Interestingly, it is observed that for methane for the same high pressures the quantities adsorbed at higher temperatures are more than those at lower temperatures, whereas for the other components, the contrary happens. This might be attributed to

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3.5 250 K 3

CH4 C2H6 CO2

298 K 2.5

2

2

N [m mol/g]

N [m mol/g]

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CH4 C2H6 CO2

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1.5 1

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P [kPa] 2.5 350 K

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CH4 C2H6 CO2

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Fig. 7.15 Adsorption isotherms for ternary equimolar mixtures (33.3% each in the gas phase) of methane–ethane–carbon dioxide at 250 K, 298 K and 350 K in silicalite

the fact that the other components are less adsorbed as the temperature is increased, thus sites are emptier and become available for the adsorption of methane molecules. Figure 7.16 illustrates an adsorption isotherm for a ternary mixture of 85% methane–10% ethane–5% carbon dioxide at 298 K. This composition represents a typical composition of raw natural gas. Despite presence of ethane being only 10%, it is initially preferentially adsorbed more than other components at lower pressures. However, the reversal between ethane and methane occurs at a pressure of 23,000 kPa and the ethane molecules become displaced by the molecules of the other components. Interestingly, despite that carbon dioxide is the lowest adsorbed component because of its low presence (5 vol.%), at very high pressures (about 200,000 kPa), it is noted that the reversal between carbon dioxide and ethane occurs and the reversal between carbon dioxide and methane is just about to occur. This shows the high energetic attraction between carbon dioxide and the silicalite sur-

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1.8 CH4 298 K

1.6

C2H6 CO2

1.4

N (m mol/g)

1.2 1 0.8 0.6 0.4 0.2 0 1

10

100

1000

10000

100000

1000000

P [kPa]

Fig. 7.16 Adsorption isotherm for a ternary mixture of 85% methane–10% ethane–5% carbon dioxide at 298 K in silicalite

face. This means that the use of silicalite as an adsorbent for the adsorption of this ternary mixture as means of separation is suitable, particularly if carbon dioxide is present in high percentages.

7.5 One-Dimensional Lattice Model of Small Alkane–Carbon dioxide Binary Mixtures in Silicalite Lattice models of fluids have been widely discussed [39–41]. Single-component adsorption isotherms for small alkanes in zeolites are well modelled by lattice gas models. The matrix method used in this paper evaluates the grand canonical partition function for a mixture and allows for conformational transitions where appropriate. The vapour phase is modelled as an ideal gas mixture in equilibrium with the adsorbed phase with reference chemical potential μ0 i for component i given as μ0i

= −kT ln



2π mi kT h2

3/2

kT



(7.3)

Here we extend our earlier treatments using matrix methods to embrace the adsorption of a mixture of these components in a zeolite pore. Silicalite has a well-known crystal structure [14], with a channel structure composed of zigzag and straight pores and their joining intersection shown in Fig. 7.1. The channels in the three-dimensional structure are made up of well-separated layers. The adsorption sites reside in these channels 4.3 Å apart. Thus, the model is effectively that of one-dimensional chains of sites, which may be occupied by

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molecules in various states. This is not an exact representation of the zeolite structure but is thought to realistically mimic the silicalite lattice. As in the Monte Carlo simulations, a carbon dioxide molecule is modelled as a rapidly rotating effectively spherical species and is assumed to occupy a single lattice site. The free energy of interaction of carbon dioxide with the zeolite framework is denoted by uaos. A methane molecule can occupy a single lattice site but the model allows for two possible conformations of an ethane molecule: lying down and standing up. We denote the potential energy of an adsorbed ethane molecule occupying a single lattice site as ubos, while that in an extended state is denoted as (ubos + uo). These phenomenological parameters ubos and uo can include changes in ro-vibrational free energies on adsorption of an isolated molecule onto the zeolite lattice from the gas phase. Pairs of ethane molecules in the lying configuration on neighbouring sites are assumed to interact with a strong repulsion, which is equivalent to occupation of two sites by each lying down ethane molecule.

7.5.1 Exact Matrix Method for Mixture Grand Partition Function Matrix methods for statistical mechanical treatment of one-dimensional lattice fluid problems have been discussed by us previously [39–41]. The grand partition function  for the alkane–carbon dioxide mixture in the adsorbed phase on the zeolite lattice with N sites may be written as the sum of the products of N factors  =

n n n

α=1 β =1 γ =1

····

n

  Aαβ Aβγ Aγ δ · · · · · · · · · Aωα

(7.4)

ω=1

where n is the number of possible species (including vacancies or holes) which can occupy a lattice site. Here n = 3, 4 methane/ethane–carbon dioxide mixtures, respectively. As is usual in this matrix method, we define the terms in Eq. (7.4) as 1  Aμν = φμ φν 2 · φμν

(7.5)

The factors in Aμν are given as

 μi − Ui kT   −Jμν = exp k·T

φi = exp φμν



(7.6) (7.7)

The parameter Jμν is the interaction energy between pairs of species μ, ν on nearest neighbour sites at the separation of the lattice sites. μi and Ui are the chemical potential and potential energy of species i, respectively. The chemical potential and the energy of vacant site or “hole” are evidently zero. The separation distance of

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165

adsorbed nearest neighbour pairs of molecules may differ from the optimal distance of separation for pairs of free molecules. By making the parameter J11 strongly repulsive, pairs of ethane molecules both in the lying down configuration can be made to effectively behave as dimers occupying two lattice sites each.  Bik Ckj for matrix multiplication of a pair Using the inner product rule Dij = k

of conformable matrices B and C, the grand partition function given in Eq. (7.2) can be expressed as  =

(α=1,2,3,4,····,n)

N  N   A αα = Trace AN = (λi )N

(7.8)

i=1

where λ1 , λ2 , λ3 , λ4 , ,. . ., λn are the eigenvalues of the matrix A given below ⎛

A11 A12 · ⎜ A21 A22 · ⎜ ⎜ · · · ⎜ ⎝ · · · · · An1

For large N we then obtain

· · · · ·

⎞ A1n A2n ⎟ ⎟ · ⎟ ⎟ · ⎠ Ann

 = (λmax )N

(7.9)

where λmax is the largest eigenvalue of the matrix A. Hence we have an evaluation of the grand partition function for the ethane–carbon dioxide or methane–carbon dioxide mixture in the quasi-one-dimensional zeolite pores. The mean density of adsorbed carbon dioxide ρa and alkane molecules ρb is given as ρa = k · T ·

∂ ln (λmax ) ∂μas

(7.10)

ρb = k · T ·

∂ ln (λmax ) ∂μbs

(7.11)

and

Extraction of the largest eigenvalue and subsequent differentiations were performed straightforwardly using the Mathcad mathematical software package. Thus for a given gas phase composition, adsorption isotherms for this exact onedimensional model can be constructed. The chemical potential of a, b components in an ideal gas mixture [29] is given as μas = μ0a + kT ln Xa P

(7.12)

μbs = μ0b + kT ln Xb P

(7.13)

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where μ0 a and μ0 b are the standard chemical potentials of carbon dioxide and alkane molecules, respectively, given by Eq. (7.3) above. P is the total pressure and Xi is the mole fraction of component i in the gas phase. Since changes in ro-vibrational free energies for adsorption into the zeolite are implicitly accounted for in uaos and ubos, then μ0 a and μ0 b include only the translational component of the reference chemical potential as given above.

7.5.2 Numerical Results and Discussion

Fractional Coverage

Isotherms are constructed using the above equations for a given set of variables Xa, Xb, total pressure P and temperature T. The shape of the isotherms shown in Fig. 7.17 quite remarkably reproduces all of the essential features of the Monte Carlo isotherms (Figs. 7.7 and 7.10). The parameters required to produce this effect are given in the captions to Fig. 7.17. Methane has less attraction with the zeolite lattice compared to carbon dioxide which is preferentially adsorbed at all coverages. At intermediate coverage, some methane adsorption occurs but this drops at higher

0.6 0.4 0.2 0 1

10

100

1·103

1·104

kPa

Fractional Coverage

Fig. 7.17 Lattice model adsorption isotherms for an equimolar mixture on silicalite at 253 K, uaos = –5.6 × 10–20 J, JCO2–CO2 = –0.75 × 10–21 J. (a) Methane/carbon dioxide for the parameters ubos = –4.55 × 10–20 J, JCH4–CH4 = –2.04 × 10–21 J (b) Ethane–carbon dioxide for the parameters ubos = –3.85 × 10–20 J, uo = –2.45 × 10–20 J, all other interactions are large and repulsive (much larger than kT). Carbon dioxide is the dotted trace in each figure (taken from Ref. [38], copyright Elsevier)

0.8

(a)

0.8 0.6 0.4 0.2

1

10

100 kPa

1·103

1·104 (b)

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coverage when it is displaced by carbon dioxide. This is observed in both Monte Carlo and lattice model. The explanation for ethane–carbon dioxide mixtures (Figs. 7.10 and 7.17) is different. For ethane, at low coverage, the interaction with the zeolite exceeds that of carbon dioxide and is thus preferentially adsorbed at low pressure. Hence, since the interaction energy of ethane molecules in the extended state with the zeolite is higher than that of carbon dioxide, at low coverage, ethane is preferentially adsorbed. At higher pressures and coverage, when the sites on the zeolite are about half filled, a conformational change occurs, with ethane molecules standing up and being displaced by CO2 molecules which fill the voids. Hence, at close packing, the free energy of the carbon dioxide molecules is lower than that of close-packed ethane molecules. At high pressures, a conformational transition occurs to facilitate better packing whereby ethane molecules stand perpendicular to the zeolite wall. In this state, the standing ethane attraction with the zeolite wall is less than the carbon dioxide–zeolite interaction, so ethane is displaced by carbon dioxide. It is quite remarkable that this adsorption preference reversal seen with the ethane–carbon dioxide mixture is driven by conformational transitions of the adsorbed ethane at least at this model system which has to be fairly compared with the Monte Carlo simulations. Future work should go beyond spherical carbon dioxide molecules and consider conformational transitions in these molecules too. Overall, we have observed a new and interesting feature in the adsorption isotherms of methane–carbon dioxide and ethane–carbon dioxide mixtures in zeolites as produced by Monte Carlo simulations as an alternative to prohibitively expensive experiments and have rationalised these features by an exact statistical mechanical calculation of a lattice model of this mixture.

References 1. Nicholson D, Gubbins KE (1996) Separation of carbon dioxide–methane mixtures by adsorption: Effects of geometry and energetics on selectivity. J Chem Phys 104: 8126–8134 2. Abdul-rehman HB, Hasanain MA, Loughlin KF (1990) Quaternary, ternary, binary, and pure component sorption on zeolites. 1. Light alkanes on Linde S-115 silicalite at moderate to high pressure. Ind Eng Chem Res 29: 1525–1535 3. Gallo M, Nenoff TM, Mitchell MC (2006) Selectivities of binary mixtures of hydrogen/methane and hydrogen/carbon dioxide in Silicalite and ETS by Grand Canonical Monte Carlo Techniques. Fluid Phase Equilibria 247: 135–142 4. Savitz S, Siperstein F, Gorte RJ, Myers AL (1998) Calorimetric study of adsorption of alkanes in high-silica zeolites. J Phys Chem B 102: 6865–6872 5. Heuchel M, Snurr RQ, Buss E (1997) Adsorption of CH4-CF4 mixtures in silicalite: Simulation, experiment, and theory. Langmuir 13: 6795–6804 6. Golden TC, Sircar S (1994) Gas adsorption on Silicalite. J Colloid Interface Sci 162: 182–188 7. Rees LVC, Bruckner P, Hampson J (1991) Sorption of N2, CH4 and CO2 in silicalite-1. Gas Separation and Purification 5: 67–75 8. Fraissard PJ, Conner CW (1997) Physical Adsorption: Experiment, Theory and Applications. Springer, New York 9. Ruthven DM (1984) Principle of adsorption and adsorption processes. John Wiley and Sons, New York

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10. Ruthven DM, Farooq S, Knaebel K (1994) Pressure Swing Adsorption. VCH Publishers, New York 11. Breck DW (1974) Zeolite Molecular Sieves. Wiley, New York 12. Babarao R, Hu ZQ, Jiang JW, Chempath S, Sandler SI (2007) Storage and separation of CO2 and CH4 in silicalite, C168 schwarzite, and IRMOF-1: A comparative study from Monte Carlo simulation. Langmuir 23: 659–666 13. Yun JH, Duren T, Keil FJ, Seaton NA (2002) Adsorption of methane, ethane, and their binary mixtures on MCM-41: Experimental evaluation of methods for the prediction of adsorption equilibrium. Langmuir 18(7): 2693–2701 14. Baerlocher Ch, Meier WM, Olson DH (2001) Atlas of Zeolite Framework Types. Fifth edition, Elsevier, Amsterdam 15. Gallo M, Nenoff TM, Mitchell MC (2006) Selectivities for Binary Mixtures of hydrogen/methane and hydrogen/carbon dioxide in Silicalite and ETS-10 by Grand Canonical Monte-Carlo Techniques. Fluid Phase Equilibria 247: 135–142 16. Keller J, Staudt R (2004) Gas Adsorption Equilibria: Experimental Methods and Adsorptive Isotherms. Springer, New York 17. Catlow CRA, Van Santen RA, Smit B (2004) Computer Modelling of Microporous Materials. Elsevier Academic Press, Amsterdam 18. Smit B (1995) Simulating the adsorption isotherms of methane, ethane, and propane in the zeolite silicalite. J Phys Chem 99: 5597–5603 19. Du Z, Manos G, Vlugt TJH, Smit B (1998) Molecular simulation of adsorption of short linear alkanes and their mixtures in silicalite. AIChEJ 44: 1756–1764 20. Vlugt TJ, Martin MG, Smit B, Siepmann JI, Krishna R (1998) Improving the efficiency on configurational Bias Monte Carlo algorithm. Mol Phys 94(4): 727–733 21. Leach AR (2001) Molecular Modelling Principles and Applications. Second edition, Prentice Hall, New York 22. Du Z (2001) Computer Simulation and Theoretical Studies of Hydrocarbons Adsorption in Zeolites. PhD Thesis, London, South Bank University 23. Siepmann JI (1990) A Method for the direct calculation of chemical potentials for dense chain systems. Mol Phys 70: 1145–1158 24. Siepmann JI, Frenkel D (1992) Configurational-bias Monte Carlo – A new sampling scheme for flexible chains. Mol Phys 75: 59–70 25. Dubbeldam D, Calero S, Garcia-Sanchez A, Van Baten JM, Krishna R, Gacia-Perez E, Parra JB, Ania CO (2007) A computational study of CO2 , N2 , and CH4 adsorption in zeolites. Adsorption 13: 469–476 26. Dunne JA, Mariwala R, Rao M, Sircar S, Gorte RJ, Myers AL (1996) Calorimetric heats of adsorption and adsorption isotherms– I. O2 , N2 , Ar, CO2 , CH4 , C2 H6 and SF6 on silicalite. Langmuir 12: 5888–5895 27. Sun MS, Shah DB, Xu HH, Talu O (1998) Adsorption equilibria of C1 to C4 alkanes, CO2 and SF6 on silicalite. J Phys Chem B 102: 1466–1473 28. Dunne LJ, Manos G, Du Z (2003) Exact statistical mechanical one-dimensional lattice model of alkane binary mixture adsorption in zeolites and comparison with Monte-Carlo simulations. Chem Phys Lett 377: 551–556 29. Hill TL (1960) Introduction to Statistical Thermodynamics. Addison-Wesley Publishing, Massachusetts 30. Huang K (1987) Statistical Mechanics. Second edition, John Wiley & Sons, New York 31. Khettar A, Jalili SE, Dunne LJ, Manos G, Du Z (2002) Monte-Carlo simulation and mean-field theory interpretation of adsorption preference reversal in isotherms of alkane binary mixtures in zeolites at elevated pressures. Chem Phys Lett 362: 414–418 32. Frenkel D, Smit B (1996) Understanding Molecular Simulation from Algorithms to Applications. Academic Press, San Diego 33. Nicholson D, Parsonage NG (1982) Computer Simulation and the Statistical Mechanics of Adsorption. Academic Press, London

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34. Schenk M, Vidal SL, Vlugt TJH, Smit B, Krishna R (2001) Separation of alkane isomers by exploiting entropy effects during adsorption on silicalite-1: A configurational-bias Monte Carlo simulation study. Langmuir 17: 1558–1570 35. Allen MP, Tidesley DJ (1987) Computer Simulation of Liquids. Clarendon Press, Oxford 36. Sun MS, Shah DB, Xu HH, Talu O (1998) Adsorption of equilibria of C1–C4 alkanes, CO2 and SF6 on silicalite. J Phys Chem B 102: 1466–1473 37. Rigby M, Smith EB, Wakeham WA, Maitland GC (1986) The Forces Between Molecules. Clarendon Press, Oxford 38. Dunne LJ, Furgani A, Jalili S, Manos G (2009) Monte-Carlo simulations of methane/carbon dioxide and ethane/carbon dioxide mixture adsorption in zeolites and comparison with matrix treatment of statistical mechanical lattice model. Chem Phys 359: 27–30 39. Bell GM, Combs LL, Dunne LJ (1981) Theories of cooperative phenomena in lipid systems. Chem Rev 81: 15–48 40. Dunne LJ, Bell GM (1980) Theory of cooperative phenomena in monolayers of hydroxyhexadecanoic acid isomers. J Chem Soc Faraday Trans (II) 76: 431–440 41. Du Z, Dunne LJ, Manos G, Chaplin M (2000) Exact statistical mechanical treatment of benzene adsorption in a zeolite twin-pore one-dimensional lattice model. Chem Phys Lett 318: 319–324

Chapter 8

Molecular Packing-Induced Selectivity Effects in Liquid Adsorption in Zeolites Joeri F.M. Denayer and Gino V. Baron

Abstract Adsorption from the liquid phase on nanoporous solids such as zeolites is associated with a high filling degree of the materials’ nanopores. Given the limited space in those pores, the organization and packing of the molecules in the pores becomes an important or even dominant factor. Therefore, specific selectivity effects occur in such conditions, often completely different from the selectivity patterns observed in gas phase conditions. In this chapter, selected examples are given for packing-induced selectivity effects in hydrocarbon adsorption in zeolites.

8.1 Introduction Although most industrial zeolite processes (e.g., catalytic cracking and hydrocracking) are performed at high hydrocarbon pressures or in the liquid phase, adsorption of organic molecules is commonly studied at low partial pressures in the gas phase. Far less data are available on adsorption of polar and apolar organic molecules in liquid phase on zeolites. In liquid phase, the adsorbent is saturated with adsorbate molecules, which approaches the industrial conditions in a much more realistic way than the zero coverage limit which is commonly studied in gas phase. Nonidealities like adsorbate–adsorbate interactions and surface heterogeneity become more important in such conditions [1–3]. For the modeling of multicomponent adsorption of liquid mixtures, it is often necessary to introduce cross correlation coefficients between components and complex expressions for the activity coefficients [1, 4] to obtain a good fitting between model and experimental adsorption J.F.M. Denayer (B) Department of Chemical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium e-mail: [email protected]

L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_8,  C Springer Science+Business Media B.V. 2010

171

172

J.F.M. Denayer and G.V. Baron Table 8.1 Liquid phase adsorption studies

Adsorbent

Adsorbate

Technique

Reference

NaX

Benzene, hexane

Brandani and Ruthven [9]

13X

Glucose, sucrose, sorbitol, water Glucose, fructose

Zero length column chromatography Liquid chromatography Liquid chromatography Liquid chromatography

NaX, KX Silicalite, Alumina NaY

H-ZSM-5

Na–Y NaX, NaY ZSM-5, ZSM-11, Mordenite

Methanol, ethanol, 1-propanol, 1-butanol, acetone, ethyl acetate Acetone, n-C6, mesitylene, cyclohexanol, cyclohexanone, t-butanol n-Hexane, toluene, p-xylene, n-propylbenzene, iso-propylbenzene Alcohol, benzene, heptane 1-Heptene

MCM-22 Silicalite

n-Butanol, iso-butanol, t-butanol, neopentane, benzene, o-xylene, n-C6, n-C8, n-C9, n-C10, n-C14, n-C16, 1-methyl butylbenzene Carbohydrates Aromatics and naphthenes o-Xylene, p-xylene, m-xylene, ethylbenzene p-Xylene, o-xylene p-Xylene, m-xylene p-Xylene, m-xylene n-C7, n-C10, n-C12, and n-C14 Thiophene C6–C7 alkanes

X,Y

Benzothiophene

FAU, BEA Y K–Y ZSM-5 Y Y, Silicalite A

Muralidharan and Ching [10] Ching and Ruthven [11] Lin and Ma [12]

Liquid chromatography

Claessens and Baron [13]

Batch uptake method

Choudhary et al. [14]

Batch uptake method

Dékány et al. [15]

Liquid chromatography Batch uptake method

Hrúzik et al. [16] Dessau [17]

Batch uptake method Batch uptake method

Buttersack et al. [18] Satterfield and Cheng [19]

Head space chromatography

Hulme et al. [1]

Batch uptake method Batch uptake method Batch uptake method Batch uptake method

Kurganov et al. [20] Moya-Korchi et al. [21] Chiang et al. [4] Sundstrom and Krautz [22] Delitala et al. [23] Dubreuil et al. [24]

Batch uptake method Inverse chromatography Batch uptake method and breakthrough experiments

Sotelo et al. [25]

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173

Table 8.1 (continued) Adsorbent

Adsorbate

Cu(I)–Y

Thiophene, Batch uptake method benzothiophene, 2ab initio MO methylbenzothiophene, calculation and dibenzothiophene Benzene, n-alkanes Batch uptake method

Yu et al. [28]

Thiophene/toluene

Laborde-Boutet et al. [29]

silicalite-1 and NaX LiY, NaY, KY, RbY, CsY H-Beta, H-ZSM-5, and SAPO-11

Y Y

Tetradecane, 3-methylpentane, 2,3-dimethylpentane, and 2,2,4trimethylpentane Ethanol, acetic acid, and water Xylene isomers Xylene isomers

RUB-41

1-Butene, 2-butene

CHA

C1–C8 alkenes and alcohols

ZSM-5

Technique

Reference Ma et al. [26] Yang et al. [27]

Breakthrough experiments Batch uptake method

Uguina et al. [30]

Batch uptake method

Bowen and Vane [31]

Headspace technique Batch uptake method

Buarque et al. [32] Minceva and Rodrigues [33] Tijsebaert et al. [34]

Batch uptake method and breakthrough experiments Batch uptake method

Daems et al. [35]

data. Adsorption properties cannot be extrapolated easily from gas phase adsorption isotherms. It is, however, instrumental to correctly describe the often dominant adsorption selectivity effects in modeling the catalysis [5–8]. Data on competitive adsorption between organic molecules with different polarity and carbon number at temperatures relevant to catalysis and elevated pressures are scarce in literature. Table 8.1 gives a far from complete overview of selected liquid phase adsorption studies with zeolites and the techniques used therefore. The present chapter gives examples of effects occurring in liquid phase adsorption on nanoporous solids. The influence of adsorbate and adsorbent polarity, zeolite pore size and topology, and finally, pressure and temperature of the external fluid phase on the adsorption of organic components (n-alkanes, iso-alkanes, aromatics, and other organic components) is discussed.

8.2 Adsorption of n-Alkanes on Non-porous and Porous Solids Adsorption of linear alkanes has been studied extensively on a range of materials, including active carbons, mesoporous silicas, and many zeolites. Nearly all of these studies are related to gas phase conditions, using pure n-alkanes. In gas phase, at low

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degree of surface coverage, when interactions between adsorbed molecules can be neglected, the adsorption enthalpy (which is a measure for the energetic interaction with the atoms constituting the surface of the adsorbent) is dominated by van der Waals (VDW) dispersive and repulsive interactions [36]. Since VDW interactions are linearly additive, a linear increase of the adsorption enthalpy with the carbon number Nc is observed [37–40], but also a linear increase of the adsorption entropy [41, 42]: − H0 = αNc + β

(8.1)

S0 = γ Nc + δ

(8.2)

Given that the free energy change upon adsorption is given by

G0 = H0 − T S0

(8.3)

resulting in a Henry adsorption equilibrium constant K′ K ′ = K ′ 0 e− H0 /RT = Ae− G0 /RT

(8.4)

this translates into an exponential increase of the Henry adsorption constant K′ with Nc at fixed temperature: K ′ = CeB.Nc

(8.5)

with B and C as temperature-dependent constants. This linear relation is independent of the precise form of A in Eq. (8.4) which depends on the definition of the reference state for the adsorption entropy, still a subject of debate [43–47]. However, this simple representation cannot be transposed directly to liquid phase conditions. In liquid phase, the pores of the adsorbent are filled with nalkane molecules competing for adsorption. The competitive adsorption equilibrium depends on the energy of interaction and the loss of freedom (adsorption entropy) of all competing components. Moreover, another major difference between gas and liquid phase adsorption is that molecules in the bulk phase surrounding the adsorbent are in close contact in liquid phase conditions. Therefore, the change in interaction energy in the transition from this bulk liquid phase to the adsorbed phase is smaller as compared to gas phase conditions, where molecules in the bulk phase are not interacting with each other. Depending on pore size, different effects have been observed in liquid phase. Mockel and Dryer [48] reported a decreasing linear relationship in the retention volume of n-alkanes on a non-porous, reversed phase column (octadecyl silica, ODS), with the adsorbate carbon number in liquid phase conditions. n-Alkanes on silica or ODS columns have little retention with C5 as mobile phase, and retention times are mainly controlled by exclusion effects. Similarly, experiments with a C18reversed stationary phase adsorbent led to the conclusion that for the given chromatographic system, alkane retention is mainly entropy driven. The excess entropy

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Molecular Packing-Induced Selectivity Effects

175

is close to zero for the component eluting with the holdup volume and increases with decreasing retention. By measuring the liquid phase retention of n-alkanes from C5 to C16 with different eluents (C6, C10, C14, and C17), Kazakevich and McNair [49] observed a linear decrease of adsorbate retention volume with an increase in the number of adsorbate carbon atoms. At the same time, an increase in the number of carbon atoms in the eluent led to an increase of retention volume for the same adsorbate. This chromatographic system does not show any preferable surface interactions, but an increase of the molecular mass leads to a decrease of the adsorbate excess entropy. Thus, for non-porous systems, no important differences in the liquid phase adsorption of alkanes with different chain length have been reported. For microporous materials, such as zeolites, different effects have been reported. As a result of the restricted space available in zeolite pores, the size of the molecule becomes very important at high loading with regard to its capability to pack efficiently.

8.2.1 Effect of Degree of Pore Filling: Gas Phase Versus Liquid Phase Figure 8.1 shows zero coverage adsorption enthalpies of n-alkanes on a series of zeolites. Zeolite Y is a large pore zeolite of the faujasite family, with a threedimensional pore network, consisting of large spherical cavities with a free diameter of ca. 12 Å, interconnected through windows with a diameter of about 7.3 Å. Beta is a large pore zeolite with a complex structure. The channel system is constituted of three interconnecting pore systems. Two 12-ring linear channels (5.7 × 7.5 Å) in different crystallographic directions intersect partially. A third, sinusoidal channel (5.6 × 6.5 Å) is formed by these intersections. The cavities have a diameter of 7.6 Å. Mordenite is another large pore zeolite with a unidimensional

130

–ΔH0(kJ/mol)

110 90 70 ZSM-22 ZSM-5

50

Mordenite Beta

Fig. 8.1 Adsorption enthalpies of n-alkanes at zero coverage (from [41] with permission)

Y

30

4

6

8 Nc

10

12

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J.F.M. Denayer and G.V. Baron

channel structure with blind side pockets. The 12-ring channel has a diameter of 7.0 × 6.5 Å, the other channel is a complex 8-ring (2.6 × 5.7 Å) [50] which is not accessible to hydrocarbons. ZSM-5 is a medium pore size zeolite with two intersecting channel systems, one sinusoidal, near circular (5.4 × 5.6 Å) and one straight, elliptical (5.1 × 5.5 Å) [51]. ZSM-22 has the TON structure type, with non-intersecting undulated 10-membered ring channels with a cross section of 4.5 × 5.5 Å. The adsorption enthalpies increase almost linearly with carbon number in the zero coverage limit in gas phase (Fig. 8.1). The three faujasite zeolites have about the same adsorption enthalpy, although they differ strongly in total Al content. Higher adsorption enthalpies are observed on Beta and Mordenite, which have smaller pores than the faujasites. ZSM-22, which is the zeolite with the smallest pore diameter studied here (4.5 × 5.5 Å), has the highest adsorption enthalpy. The strongest increase of adsorption enthalpy with Nc is also observed with ZSM22, accounting for 12–13 kJ/mol per extra carbon group. On Beta, an increment of 10–11 kJ/mol was measured. On zeolite Y, the limiting adsorption enthalpy increases with 6–7 kJ/mol per additional carbon group. Also the adsorption entropy varies in a linear way with the alkane chain length (Fig. 8.2). At low loadings, alkane molecules are adsorbed preferentially on the Brønsted acid sites or on the cations. On Beta and Mordenite, the larger adsorption enthalpies are accompanied by a stronger increase of adsorption entropy with the carbon number, corresponding with 8–9 J/mol/K. Finally, on the 10-MR zeolite ZSM-22, the entropy of adsorption is significantly higher and increases with 15 J/mol/K. Alkane molecules are held strongly in the small pores of ZSM-22, resulting in a high loss of degrees of freedom. The combination of the linear increase of adsorption entropy and enthalpy with Nc results in an exponential increase of the Henry constants with the carbon number (Fig. 8.3). The rate at which the Henry constant increases with Nc (coefficient B of

–ΔSθ0,local (J/mol/K)

200

160

ZSM-22

Mordenite

Beta

Y (Si/Al 2.7)

Y (Si/Al 15)

Y (Si/Al 30)

120

80

40

4

5

6

7

8

9

10

Nc

Fig. 8.2 Localized adsorption entropies of n-alkanes against carbon number on Y zeolites, Beta, Mordenite, and ZSM-22 (from [41] with permission)

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Molecular Packing-Induced Selectivity Effects

177

1 10–1 Beta

1 10

–2

K' (mol/kg/Pa)

Mordenite

1 10–3

Y US-Y(Si/Al 30)

1 10–4

ZSM-22

1 10–5 1 10–6 1 10–7

4

6

8

10

12

14

Nc Fig. 8.3 Henry constants of n-alkanes at 325◦ C (from [42] with permission)

Eq. 8.5) varies with the zeolite type. Typically, per additional carbon atom in the molecule, the Henry constant is multiplied by a factor of 2–5. When the partial pressure of the adsorbing components, and herewith the degree of pore filling, is increased, the differences in adsorption between short and long alkanes diminish. Figure 8.4 shows separation factors α between n-octane and npentane on a Y zeolite (Si/Al 2.7) as a function of the number of molecules adsorbed per supercage. When the number of adsorbed molecules exceeds one, the separation factor drops strongly. The presence of more than one molecule in the supercage strongly affects the freedom of the adsorbing molecules in the pores (adsorption entropy). At higher degree of pore filling, the adsorption equilibrium is affected 20

αCn+3/Cn

16 12 8 4 0

0

0.5

1 molecules/cage

1.5

2

Fig. 8.4 Separation factors between n-octane and n-pentane on Y zeolite (Si/Al 2.7) as a function of zeolite loading (233◦ C) (adapted from [41] with permission)

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J.F.M. Denayer and G.V. Baron

by interactions between the adsorbed molecules. As a result of steric constraints, adsorbed molecules are no longer capable of adopting the energetically most favorable configuration inside the pores. Obviously, the competition between alkanes of different chain length depends strongly on the partial pressure of the adsorbates and thus the degree of pore filling. The effect of temperature and pressure of the fluid surrounding the porous adsorbent on the adsorption properties is demonstrated even more clearly by liquid phase pulse chromatography experiments using a column packed with Na-USY crystals at elevated temperature and varying pressures [41, 52]. Partition coefficients (ratio of concentration in solid to concentration in bulk liquid phase, reflecting the affinity of the components for the adsorbent) are shown in Fig. 8.5 as a function of the carbon number at 230◦ C. At this temperature, the transition from gas to liquid phase occurs at about 9.5 bar. At a mean column pressure of 2.6 bar, partition coefficients increase from 0.4 for n-pentane to about 6 for n-hexadecane. When the mean column pressure is increased, differences in K become smaller. At 9.8 bar finally, the partition coefficients of all n-alkanes are equal, meaning that short and long alkanes are adsorbed in an unselective way. When the mean column pressure exceeds the transition pressure, the competitive adsorption between the injected alkanes and the n-octane mobile phase disappears. In Fig. 8.6, it can be seen that K increases with increasing temperature if the pressure is held constant. In liquid phase conditions, the zeolite cavities are completely filled with carbon chain elements. Several alkane molecules are present in the large supercages of zeolite Y and mainly undergo each others’ force field. This means that only a fraction of each alkane molecule experiences the force field exerted by the zeolite structure. As a result of the relative large size of the supercages, the average adsorption potential experienced by the adsorbed molecules is not significantly larger than the

7 2.6 bar 4.0 bar 7.9 bar 9.8 bar

6 5

K

4 3 2 1 0

4

6

8

10 Nc

12

14

16

Fig. 8.5 Pressure dependence of partition coefficients of n-alkanes on Na-USY at 230◦ C as determined by HPLC experiments using n-octane as mobile phase (from [52] with permission)

8

Molecular Packing-Induced Selectivity Effects

179 7 6 5 4 K 3 2

265

1

230 T (°C)

0

210 195 5

6

9

11

12

14

16

Nc

Fig. 8.6 Temperature and carbon number dependence of partition coefficients of n-alkanes in noctane on Na-USY at a mean column pressure of 3.5 bar (from [52] with permission)

intermolecular interactions prevailing in the surrounding bulk liquid phase. Increasing the temperature of the alkane mobile phase reduces the density and thus the adsorbate–adsorbate interactions in the external fluid phase. Further, the increased temperature results in a less strong adsorption of the molecules and eventually to a reduction of the degree of pore filling with adsorbate molecules. Consequently, components will have more free space to interact with the zeolite wall. In these conditions, longer n-alkanes have a stronger interaction and are adsorbed preferentially (Fig. 8.6), as is the case in the gas phase at low surface coverage. When the pressure is isothermally increased, the density of external fluid phase increases and the zeolite returns gradually to complete saturation. When the pressure exceeds the transition pressure at that particular temperature, all differences in retention between long and short n-alkanes disappear again. The competition between alkanes is thus mainly determined by the zeolite loading, which in turn depends on the pressure and temperature of the external phase. It should be emphasized that the reference state is completely different in the liquid phase and the diluted gas state. In liquid phase, molecules in the external bulk phase are in much closer contact with each other than in the gas phase, and have less freedom than the freely moving gas phase molecules. Consequently, adsorption enthalpies and entropies differ for gas- and liquid phase adsorption.

8.2.2 Effect of Pore Size In the previous section, it was demonstrated that the large differences in adsorption, as perceived in gas phase conditions, tend to disappear completely at even higher

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degree of pore filling, in liquid phase conditions on a large pore zeolite. It is to be expected that adsorptive competition effects depend on the zeolite pore size, since smaller pore sizes lead to larger adsorption potentials [53]. Figure 8.7 shows chromatograms, obtained by injecting liquid n-alkanes in methanol as mobile phase on columns packed with zeolites with different pore size: ZSM-22, Silicalite, Beta, Na-USY, and MCM-41. On ZSM-22 and MCM-41, all n-alkanes elute at the same time and travel through the chromatographic column at the same velocity as the mobile phase. In particular, all n-alkanes have the same retention time as 2,2,4-trimethyl-pentane on ZSM-22, a component that is too bulky to enter the pores of this 10-MR zeolite. On Na-USY, the retention time increases very weakly with carbon number. With Beta and silicalite, significant differences in adsorption behavior are observed between short and long n-alkanes. Table 8.2 summarizes the partition coefficients obtained on all tested materials. The highest separation power is obtained with silicalite. The pores of MFI-type zeolites (ZSM-5, silicalite) have an average diameter of about 5.4 Å, hence adsorbed linear alkanes (having a critical diameter of 4.3 Å) are stretched along the pore axis, in this way maximizing their interactions (attractive dispersion forces) with the pore walls. The pores of Beta are somewhat larger (7.6 Å); hence the attractive forces are already smaller as compared to silicalite, explaining the lower partition coefficients. With Na-USY, having supercages with a diameter of 12.3 Å and large mesopores, the adsorption potential is too low to induce selective enrichment in the pores. The differences in Gibbs free energy related to the transition of an alkane molecule from the mobile phase to the adsorbed state (on the surface of the stationary phase) between short and long alkanes are minor [49]. This also explains the complete absence of selectivity with MCM-41, with a pore diameter of about 40 Å. Apparently, pores smaller than 10 Å are needed to discriminate between short and long n-alkanes. In the same line of reasoning, it would be expected that the largest partition coefficients would be obtained with ZSM-22, having the smallest (unidimensional) pores, with a cross section of 4.5 × 5.5 Å, but this is obviously not the case (Fig. 8.7a). With this material, diffusional barriers prevent uptake of the n-alkane molecules in flow conditions. A competitive batch adsorption experiment with an equimolar C6/C9 mixture (dissolved in iso-octane) indeed showed a preference for the longest chain, with a selectivity (= (qC9/CC9)/(qC6/CC6)) of 2.4. So, although ZSM-22 preferentially adsorbs longer n-alkanes over short ones, slow mass transfer in its small, one-dimensional pores hampers its performance in liquid chromatography separations. Partition coefficients clearly increase with increasing Si/Al ratio (Table 8.2). With decreasing Al content, the polarity of the zeolite decreases, hence the affinity for the polar mobile phase (methanol) also diminishes, making the adsorption of the highly apolar n-alkanes even more favorable. Furthermore, it is observed that the partition coefficients of the n-alkanes on ZSM-5-type zeolites depend strongly on the mobile phase used, and increase with increasing mobile phase polarity: n-hexane < acetone < acetonitrile < propanol < ethanol < methanol (Table 8.2, Fig. 8.8). Figure 8.9 shows the binary C9 /C13 liquid phase adsorption isotherm on ZSM-5. At all mixture compositions, the pores of ZSM-5 are enriched in C13 . n-Alkane

8

Molecular Packing-Induced Selectivity Effects C16

181 Silicalite

ZSM-22

C11

S (a.u.)

S (a.u.)

C5 C7 2,2,4-trimethylpentane

0

100

C6

C5

200

300 t (s)

400

500

0

600

100

200

300

400 t (s)

500

700

Na-USY

Beta C6

C16

S (a.u.)

C12

S (a.u)

600

C7 C8

C5

0

500

1000

1500 t (s)

2000

2500

0

50

100

150

200

t (s)

Fig. 8.7 Chromatograms of n-alkanes in methanol as mobile phase on ZSM-22, Silicalite, Beta, Na-USY, and MCM-41 at room temperature (from [53] with permission)

molecules are stretched along the pore axis, thus all –CH2 groups of the molecules interact strongly with the pore walls. Hence longer molecules have a higher total interaction energy and are adsorbed preferentially. This is nicely illustrated for a series of binary mixtures with C9 (Fig. 8.10). With increasing difference in chain length between C9 and the other component (C8 , C11 , C12 , C13 ) of the binary mixture, a more pronounced selectivity is observed. The competition between C9 and these components was analyzed quantitatively by applying a multicomponent Langmuir model to the experimental data. Table 8.3

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Table 8.2 Partition coefficients of n-alkanes on various zeolites at room temperature as determined by HPLC experiments using methanol as mobile phase (from [53] with permission) Na-USY

ZSM-5

ZSM-5

Silicalite

Beta

ZSM-22

MCM-41

(Si/Al 30) (Si/Al 13) (Si/Al 137) (Si/Al ∞) (Si/Al 12.5) (Si/Al 30) (Si/Al ∞) n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tetradecane n-hexadecane

0.60 0.63 0.67 0.69 0.76 0.89 0.84 0.95 1.09 1.81

2.17 3.68 3.62 7.19 8.99 10.6 15.4 – – –

12.6 22.8 17.8 47.1 58.4 – – – – –

Fig. 8.8 Partition coefficients of n-alkanes on silicalite with different mobile phases at room temperature (from [53] with permission)

15.1 24.1 35.9 50.2 – – – – – –

–

0.14

4.58 8.61 17.4 – – – – – -

0.14 – – – 0.14 – – 0.14

0.81 0.81 0.81 0.81 0.81 0.81 – 0.81 – 0.81

60 hexane Acetone Acetonitrile Propanol Ethanol Methanol

50

K

40 30 20 10 0 5

6

7

8

9

10

Nc

5 4.5

q (molec./UC)

4 3.5 3 2.5 2 1.5

C9 C13 Total

1 0.5 0 0

0.2

0.4

0.6

0.8

1

X nC9

Fig. 8.9 Binary C9 /C13 isotherm on ZSM-5 (Si/Al 40) at 20◦ C (from [53] with permission)

12

8

Molecular Packing-Induced Selectivity Effects

183

1

0.8

yi

0.6

0.4

C13/C9 C12/C9

0.2

C11/C9 C9/C8

0 0

0.2

0.4

xi

0.6

0.8

1

Fig. 8.10 Selectivity diagrams for binary n-alkane adsorption on ZSM-5 (Si/Al 40) at 20◦ C (from [53] with permission)

gives the ratio of adsorption constants of the longest chain of the mixture (KL ) divided by that of the shortest component (KS ). The ratio of adsorption constants is almost doubled per additional –CH2 group in the competing molecule. This exponential relationship between adsorption constants and carbon number is similar to that observed in the gas phase at low coverage, but much less pronounced. The adsorption constants are related to the adsorption enthalpy in an exponential way via the van’t Hoff equation, so the linear additivity of the dispersive forces is translated into an exponential increase of the adsorption constants. The smaller increase of adsorption constants in liquid phase compared to the gas phase is first of all explained by the difference in reference state. In gas phase adsorption, molecules are adsorbed from a dilute medium, where they have no interactions with each other, into a very high-energy region inside the pores, whereas in liquid phase conditions, molecules in the external phase already have a much larger interaction with each other. Second, entropy effects (organization of the molecules in the pores of ZSM-5) are much larger at high zeolite loading in liquid phase conditions, and counteract the positive effect of the gained interaction energy upon adsorption [54]. It should be noted that this regular increase of the adsorption constants in liquid phase conditions on ZSM-5 is only valid for a certain range of n-alkanes, as will be explained in the following section.

Table 8.3 Ratio of Langmuir adsorption constants for the adsorption of binary mixtures on ZSM-5 (Si/Al 40) at 20◦ C (from [53] with permission)

Mixture

#CH2

K L /K S

C8 /C9 C9 /C11 C9 /C12 C9 /C13

1 2 3 4

4.3 7.2 11.1 22.0

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8.3 Molecular Packing Effects in Liquid Phase Adsorption Given the molecular size of zeolite pores, steric effects govern the adsorption behavior. Entropic effects are often at least as important as energetic effects in the adsorption of molecules in confined environments [55–58]. This is certainly true for adsorption in liquid phase or at high pressure, where the organization and packing of molecules in the fully saturated zeolite pores becomes critical. Molecular packing effects can lead to selectivity reversal in liquid phase conditions, i.e., selectivities which are opposite to those encountered in gas phase conditions at low degree of pore filling. Three examples are given below.

8.3.1 Chain Length-Induced Selectivity Reversal in ZSM-5 Among all zeolites, ZSM-5 is undoubtedly the most studied one, both from the adsorption and catalysis point of view. Several experimental studies reported the occurrence of a kink in the pure component adsorption isotherms of alkanes in gas phase on ZSM-5 [39, 54, 59, 60] and an inflection point in the adsorption enthalpy and entropy versus zeolite loading curves [61, 62]. Generally, the two-step behavior of these linear and branched alkanes is interpreted in terms of the ZSM-5 channel geometry. For example, it has been observed that at low partial pressures, isobutane is adsorbed preferentially in the channel intersections of ZSM-5, while at higher pressures, these branched molecules are “pushed” into the channel segments, resulting in a significant loss of entropy and a kink in the adsorption isotherm [63]. The importance of entropy effects in the adsorption of alkanes on ZSM-5 has been demonstrated by computational techniques [64–66]. Also in liquid phase, remarkable chain length-dependent adsorption effects have been observed on ZSM-5 [67–69]. Figure 8.11 shows the adsorption saturation capacity of the C5–C22 n-alkanes, expressed in number of molecules and total number of –CH x groups adsorbed per unit cell. A drop in number of molecules adsorbed per unit cell is observed between C6 and C8 . From C8 on, the number of molecules adsorbed per unit cell decreases gradually. For C5 , a unit cell contains 36 –CH x groups, for C6 and C7 44 –CHx groups are adsorbed per unit cell, while for C8 , only 32 –CH x groups are adsorbed per unit cell. From C8 on, the number of –CH x groups per unit cell increases steadily, to reach a maximum of about 51–52 –CH x groups per unit cell from C14 on. For molecules shorter than C14 , it appears that the available space in the pores of ZSM-5 is thus not completely used, and gaps must be present between the adsorbed molecules. Pentane and hexane have lengths less than the distance between intersections, and occupy the channel space between two intersections [67]. Longer chains cross the intersection and extend into adjacent channel space, leading to a less efficient pore filling. With increasing chain length, the n-alkanes become more flexible, and can bend such that they start in, e.g., a linear channel segment and end in a sinusoidal channel segment. From C13 on, the molecules are flexible enough to completely

Molecular Packing-Induced Selectivity Effects

185 60

8 7

50

6 40

5

30

4 3

20

2 molecules per unit cell

1 0

10

q (-CHx per unit cell)

Fig. 8.11 Saturation capacity of the C5–C22 n-alkanes, expressed in number of molecules and total number of carbon atoms adsorbed per unit cell (adapted from [68] with permission)

q (molecules per unit cell)

8

C-atoms per unit cell 5

10

15

20

0 25

Nc

fill up the pore system. These differences in packing mechanisms between alkanes of different chain length lead to irregular selectivity patterns. Compositions in the adsorbed phase are plotted versus the compositions of the external liquid phase for selected binary alkane mixtures in Fig. 8.12. In some cases, preferential adsorption of the longest alkane chain was observed (e.g., C9/C13). With C6/C10 and C5/C7, reversal of selectivity occurs at a specific composition where adsorbate and liquid have identical composition. Finally, for C14/C15 and C15/C16, the lightest alkane was selectively adsorbed. This remarkable adsorption selectivity is the result of the organization of the n-alkane molecules. The strong interaction with the pore walls implies a high adsorption enthalpy, but also a significant loss of entropy. At high loading, the dense packing of the molecules limits the reorganization capabilities of individ1

Mole fraction longest chain in adsorbed phase

C10/C6 C7/C5

0.8

C22/C20 C15/C14

0.6

C16/C15

0.4 0.2 0 0

0.2 0.4 0.6 0.8 1 Mole fraction longest chain in external phase

Fig. 8.12 Selectivity diagrams for binary n-alkane mixture adsorption in ZSM-5 (adapted from [69] with permission)

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ual molecules in the adsorbate phase. The complete filling of the two-dimensional pore system of ZSM-5 by long alkanes requires a very high flexibility and degree of organization. Pronounced adsorption selectivity for the short chain is observed with C14/C15 and C15/C16. The chain length of C14 corresponds to the length of two sinusoidal channel segments and an intersection, but also fits into two linear channel segments and two intersections. The length of a C15 molecule exceeds the dimensions of two sinusoidal channel segments and an intersection but still fits into two linear channel segments and two intersections. A stretched C16 molecule does not fit into neither of these combinations, and thus always blocks an additional intersection compared to C15 or C14. The favorable adsorption of C14 and C15 is ascribed to this matching with characteristic dimensions of the pore system. The adsorption of mixtures containing C5 or C6 can be explained in a similar way. These molecules fit neatly into a sinusoidal channel segment, whereas longer alkanes block intersections. Admixing C5 or C6 with a longer alkane fitting less well with characteristic pore length gives rise to azeotrope behavior (e.g., C5/C7, C6/C10). These mixtures fill the pores only partially. At low concentration of the long chain, it is preferentially adsorbed due to its higher adsorption enthalpy. From a critical concentration of these long chains on, C5 or C6 adsorption is preferred because of their better fitting with the sinusoidal channels.

8.3.2 Packing Effects in the Adsorption of Alkanes/Alkenes/Aromatics Generally, alkenes and aromatics are more strongly adsorbed compared to the apolar alkanes because of their stronger electrostatic interactions with the zeolite cations [70–72]. Further, it is accepted that an increase in cation content (decrease in Si:Al) increases the selectivity of adsorption of aromatics, alkenes and even alkanes in gas phase conditions [73–76]. In liquid phase conditions, different selectivity patterns are observed. The adsorption of alkenes with different chain length (C6–C12) from alkane solvents (C5–C14) on NaY (Si:Al 2.79) was studied using a batch experimental technique [77, 78]. Under these conditions, the zeolite micropores are close to saturation, since the solvent (alkane) will show a tendency to fill up the remaining free space. Already at low alkene concentrations, the alkenes are selectively adsorbed from their mixture with an alkane as a result of the interaction of the π-electrons of the double bond with the zeolite cations. The amount of alkene adsorbed depends on the chain length of both the alkene and the alkane solvent. Two remarkable effects are observed (Figs. 8.13 and 8.14): (1) shorter alkenes are preferentially adsorbed compared to longer alkenes and (2) with shorter alkanes, the alkene selectivity decreases. These observations are opposite to the increase in adsorption with carbon number as observed in gas phase conditions. Shorter linear hydrocarbons, having a smaller

Molecular Packing-Induced Selectivity Effects

187

0.8

0.18

0.7

0.16 0.14

0.6

0.12

0.5

0.1 0.4 0.08 0.3

0.06

0.2

q total (ml/g)

q (m mol alkene/g NaY)

8

0.04 dodecene hexene

0.1

0.02 0

0 C5

C7

C8

C10

C11

C14

alkane solvent

Fig. 8.13 Liquid phase adsorption of an equimolar hexene/dodecene mixture (2 mol%) from alkane solvents (96 mol%) on NaY (room temperature) (adapted from [77] with permission) 3.5

# alkene molecules/SC

3 2.5 2 1.5 1 dodecene

0.5 0 0

hexene 2

4 6 8 10 12 External alkene concentration (mol%)

14

Fig. 8.14 Adsorption isotherms of hexene and dodecene from their mixture with decane on NaY (room temperature) (adapted from [77] with permission)

number of C-atoms, pack more efficiently at higher loading and are, in other words, favorably adsorbed because they can easily fill gaps within the zeolite matrix. Such an effect was not expected to occur on cage-type zeolites capable of hosting multiple molecules per supercage.

8.3.3 Cage and Window Effects in Liquid Phase Adsorption Other effects are observed with zeolites containing cages connected via small windows. It can be easily visualized that a higher number of small molecules fit into, e.g., a spherical zeolite cage as compared to large molecules. In zeolite literature,

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much interest has been shown for the so called “window effect” (i.e., a periodic rise and fall of the diffusion coefficients with n-alkane chain length). This effect, first reported by Gorring in 1973 [79], remained controversial for a long time [80–85]. Only recently, computer simulations and neutron spin echo experiments with the zeolite 5A have shown abnormally low diffusion coefficients when the n-alkane chain length becomes equal to or larger than the zeolite cage dimensions [86, 87]. Most experimental work related to the window effect was performed in dilute, gas phase conditions, at low degree of pore filling. In liquid phase conditions, however, the appearance of window effects is more obvious. Figure 8.15 [88] shows the adsorption capacities of a homologous C5–C24 n-alkane series on 5A measured in liquid phase at room temperature. From C5 to C9, the number of molecules adsorbed per supercage (SC) gradually decreases with carbon number. For C9–C13 n-alkanes about 1.5 molecules are adsorbed per SC. For longer alkanes the number of adsorbed molecules per SC decreases again, to reach an adsorption capacity of one molecule per supercage for C16. Unexpectedly, a sudden capacity drop was observed for C18 and C19. Expressed in number of carbon atoms adsorbed per SC, a periodical increase and decrease is observed. Between C5 and C17 two local maxima are observed for C7 (~17 C-atoms/SC) and C12 (~19 C-atoms/SC). An absolute minimum is reached for C19 occupying only 2.4 C-atoms/SC. This carbon number dependency of the adsorption capacity is explained in terms of molecular packing in the zeolite cages [88]. Alkanes up to C7 are small enough to fit in a stretched conformation in the supercage. Octane and nonane must bend in order to fit in a supercage, leading to a decrease in packing efficiency (Fig. 8.16). From C9 to C12 about 1.5 molecules are adsorbed per SC. This corresponds to a configuration in which one alkane is adsorbed in the supercage with its head and tail CH3 group in the center of 8-MR windows diagonally facing each other, while a second one is distributed between the same cage and a neighboring cage (Fig. 8.16). When the maximum number of C-atoms that can adsorb inside a supercage (~19 C-atoms) is reached (for C18 and C19), the saturation capacity

3

20

2.5

q (-CHx /supercage)

16

Fig. 8.15 Adsorption capacities of n-alkanes on zeolite 5A at room temperature (adapted from [88] with permission)

14

2

12 1.5

10 8

1

6 4

0.5

2 0

5

7

9

11

13

15 Nc

17

19

21

23

0

q (molecules/supercage)

18

8

Molecular Packing-Induced Selectivity Effects

189

Fig. 8.16 Snapshots showing different conformations of n-alkanes adsorbed in 5A during molecular simulations. Left: two C14 molecules coiled in adjacent supercages and one molecule distributed between the two cages. Right: highly coiled C16 molecule in a supercage (from [88] with permission)

decreases drastically. These alkanes adsorb very tightly in a highly coiled configuration in the SC, hampering uptake of other molecules. The competitive adsorption and diffusion of mixtures on 5A is strongly affected by these packing effects, leading to selectivity reversal between short and long chains and a remarkable dependency of diffusion coefficients on chain length [88].

8.4 Conclusions The adsorption of hydrocarbons in nanoporous solids is governed by a combination of enthalpic and entropic effects. At low degree of pore filling, when the molecules in the pore system are not hampered by the presence of other adsorbing molecules, large energetic interactions between molecular structures (e.g., double bond of alkene of aromatic ring) and adsorption sites lead to a preferential adsorption of specific species. However, liquid phase adsorption implies a high degree of pore filling. In these conditions, entropic packing effects outweigh normal tendencies based on adsorbate properties (e.g., number of C-atoms) and structural properties (aluminum contents) observed at low coverage. Even in adsorbents or catalysts with relatively large micropores, molecular selectivity is achieved at high degree of pore occupancy as a result of the organization of molecular assemblies in such confined spaces. These selectivity effects depend in a subtle way on molecular size and shape, functional groups, pore size end geometry, presence of solvents, and so on. This concept of packing-induced selectivity offers perspectives for new or optimized catalytic and/or separation processes.

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21. Moya-Korchi V, Tavitian B, Méthivier A (1996) Thermodynamics of adsorption of C8 aromatics solutions on Y zeolites. In: Douglas LeVan (ed.), Proc. 5th international conference on fundamentals of adsorption, 651–658. Kluwer, Boston 22. Sundstrom DW, Krautz FG (1968) Equilibrium adsorption of liquid phase normal paraffins on type 5A molecular sieves. J. Chem. Eng. Data 13: 223–226 23. Delitala C, Cadoni E, Delpiano D, Meloni D, Melis, Ferino I (2008) Liquid-phase thiophene adsorption on MCM-22 zeolites. Acidity, adsorption behaviour and nature of the adsorbed products. Microporous Mesoporous Mater. 110(2–3): 197–215 24. Dubreuil AC, Jolimaitre E, Tayakout-Fayolle M, Methivier A (2008) Measurement of monobranched alkane mobility in the silicalite framework in the presence of dibranched and linear molecules. Ind. Eng. Chem. Res. 47(7): 2386–2390 25. Sotelo JL, Uguina MA, Agueda VI (2007) Fixed bed adsorption of benzothiophene over zeolites with faujasite structure. Adsorption 13(3–4): 331–339 26. Ma L, Yang RT (2007) Selective adsorption of sulfur compounds: Isotherms, heats, and relationship between adsorption from vapor and liquid solution. Ind. Eng. Chem. Res. 46: 2760–2768 27. Yang FH, Hernandez-Maldonado AJ, Yang RT (2004) Selective adsorption of organosulfur compounds from transportation fuels by pi-complexation. Sep. Purif. Techn. 9(8): 1717–1732 28. Yu M, Hunter JT, Falconer JL, Noble RD (2006) Adsorption of benzene mixtures on silicalite1 and NaX zeolites. Microporous Mesoporous Mater 96(1–3): 376–385 29. Laborde-Boutet C, Joly G, Nicolaos A, Thomas M, Magnoux P (2006) Selectivity of thiophene/toluene competitive adsorptions onto zeolites. Influence of the alkali metal cation in FAU(Y). Ind. Eng. Chem. Res. 45(24): 8111–8116 30. Uguina MA, Sotelo JL, Rodriguez A, Gomez-Civicos JI, Lazaro JJ (2006) Liquid adsorption of linear and branched paraffins onto microporous adsorbents – Influence of adsorbent structure and Si/Al molar ratio. Sep. Purif. Techn. 51(1): 72–79 31. Bowen TC, Vane LM (2006) Ethanol, acetic acid, and water adsorption from binary and ternary liquid mixtures on high-silica zeolites. Langmuir 22(8): 3721–3727 32. Buarque HLB, Chiavone O, Cavalcante CL (2005) Adsorption equilibria of C-8 aromatic liquid mixtures on Y zeolites using headspace chromatography. Sep. Purif. Techn. 40(9): 1817–1834 33. Minceva M, Rodrigues AE (2004) Adsorption of xylenes on faujasite-type zeolite – Equilibrium and kinetics in batch adsorber. Chem. Eng. Res. Des. 82: 667–681 34. Tijsebaert B, Varszegi C, Gies H, Xiao FS, Bao XH, Tatsumi T, Muller U, De Vos D (2008) Liquid phase separation of 1-butene from 2-butenes on all-silica zeolite RUB-41. Chem. Commun. 21: 2480–2482 35. Daems I, Singh R, Baron GV, Denayer JFM (2007) Length exclusion in the adsorption of chain molecules on Chabazite type zeolites. Chem. Commun.13: 1316–1318 36. Ruthven DM (1984) Principles of Adsorption and Adsorption Processes. John Wiley and Sons, New York 37. Eder F, Lercher JA (1997) On the role of the pore size and tortuosity for sorption of alkanes in molecular sieves. J. Phys. Chem. B 101: 1273–1278 38. Hufton JR, Danner RP (1993) Chromatographic study of alkanes in silicalite – equilibrium properties. AIChE J 39(6): 954–961 39. Richards RE, Rees LVC (1987) Sorption and packing of normal-alkane molecules in ZSM-5. Langmuir 3: 335–340 40. Stach H, Lohse U, Thamm H, Schirmer W (1986) Adsorption equilibria of hydrocarbons on highly dealuminated zeolites. Zeolites 6: 74–90 41. Denayer JFM, Baron GV (1998) Influence of loading on the adsorption of n-alkanes on zeolites. In: Meunier F (ed.) Fundamentals of Adsorption, pp. 99–104. Elsevier, Amsterdam 42. Denayer JFM, Baron GV, Martens JA, Jacobs PA (1998) Chromatographic study of adsorption of n-alkanes on zeolites at high temperatures. J. Phys. Chem B 102: 3077–3081 43. Dremetsika AV, Siskos PA, Katsanos NA (2007) Determination of adsorption entropies on solid surfaces by reversed-flow gas chromatography. J. Hazard. Mater. 149(3): 603–608

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44. Katsanos NA, Kapolos J, Gavril D, Bakaoukas N, Loukopoulos V, Koliadima A, Karaiskakis G (2006) Time distribution of adsorption entropy of gases on heterogeneous surfaces by reversed-flow gas chromatography. J. Chromatogr. A 1127(1–2): 221–227 45. Maesen TLM, Krishna R, van Baten JM, Smit B, Calero S, Sanchez JMC (2008) Shapeselective n-alkane hydroconversion at exterior zeolite surfaces. J. Catal. 256(1): 95–107 46. Myers AL (2004) Characterization of nanopores by standard enthalpy and entropy of adsorption of probe molecules. Colloids Surf. A: Physicochem. Eng. Aspects 241(1–3): 9–14 47. Ocakoglu AR, Denayer JFM, Marin GB, Martens JA, Baron GV (2003) Tracer chromatographic study of pore and pore mouth adsorption of linear and monobranched alkanes on ZSM-22 zeolite. J. Phys. Chem. B 207: 398–406 48. Möckel HJ, Dreyer U (1993) 2nd-order retention effects in reversed-phase liquidchromatography – 1. Influence of solute size. Chromatographia 37(3–4): 179–184 49. Kazakevich YV, McNair HM (2000) Low-energy interactions in high-performance liquid chromatography. J. Chromatogr. A 872: 49–59 50. Baerlocher Ch, McCusker LB, Olson DH (2007) Atlas of Zeolite Structure Types. Elsevier, Amsterdam 51. Kokotailo GT (1978) Structure of synthetic zeolite ZSM-5. Nature 272: 437–438 52. Denayer JFM, Bouyermaouen A, Baron GV (1998) Adsorption of alkanes and other organic molecules in liquid phase and in the dense vapor phase: Influence of polarity, zeolite topology, and external fluid density and pressure. Ind. Eng. Chem. Res. 37(9): 3691–3698 53. Denayer JFM, van der Beken S, De Meyer KMA, Martens JA, Baron GV (2004) Chromatographic liquid phase separation of n-alkane mixtures using zeolites. Stud. Surf. Sci. Catal. 154: 1944–1949 54. Smit B, Maesen TLM (1995) Commensurate freezing of alkanes in the channels of a zeolite. Nature 374: 42–44 55. Calero S, Schenk M, Smit B, Maesen TLM (2004) Alkane hydrocracking: Shape selectivity or kinetics? J. Catal. 221: 241–251 56. Denayer JFM, Daems I, Baron GV (2006) Adsorption and reaction in confined spaces. Oil Gas Sci. Technol. 61(4): 561–569 57. Krishna R, van Baten JM (2008) Segregation effects in adsorption of CO2-containing mixtures and their consequences for separation selectivities in cage-type zeolites. Sep. Purif. Technol. 61: 414–423 58. Schenk M, Calero S, Maesen TLM, Van Benthem LL, Verbeek MG, Smit B (2002) Understanding zeolite catalysis: Inverse shape selectivity revised. Angewandte Chemie – International Edition 41: 2499–2502 59. Sun MS, Talu O, Shah DB (1996) Adsorption equilibria of C-5-C-10 normal alkanes in silicalite crystals. J. Phys. Chem. 100 (43): 17276–17280 60. Zhu W, van der Graaf JM, van den Broeke LJP, Kapteijn F, Moulijn JA (1998) TEOM: A unique technique for measuring adsorption properties. Light alkanes in silicalite-1. Ind. Eng. Chem. Res. 37: 1934–1942 61. Millot B, Methivier A, Jobic H (1998) Adsorption of n-alkanes on silicalite crystals. A temperature-programmed desorption study. J. Phys. Chem. B 102(17): 3210–3215 62. Yang Y, Rees LVC (1997) Adsorption of normal hexane in silicalite-1: An isosteric approach. Microporous Mater. 12: 117–122 63. Zhu W, Kapteijn F, Moulijn JA (2000) Adsorption of light alkanes on silicalite-1: Reconciliation of experimental data and molecular simulations. PCCP 2: 1989–1995 64. Krishna R, Paschek D (2001) Molecular simulations of adsorption and siting of light alkanes in silicalite-1. PCCP 3: 453–462 65. Smit B, Siepmann JI (1994) Computer-simulations of the energetics and siting of n-alkanes in zeolites. J. Phys. Chem. 98: 8442–8452 66. Maginn EJ, Bell AT, Theodorou DN (1995) Sorption thermodynamics, siting, and conformation of long n-alkanes in silicalite as predicted by configurational-bias monte-carlo integration. J. Phys. Chem. 99: 2057–2079

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67. Chempath S, De Meyer K, Denayer JFM, Baron GV, Snurr RQ (2004) Adsorption of alkane mixtures from liquid phase in silicalite: simulations and experiment. Langmuir 20(1): 150– 156 68. De Meyer K, Chempath S, Denayer JFM, Martens JA, Snurr RQ, Baron GV (2003) Packing effects in the liquid phase adsorption of C5-C22 n-alkanes on ZSM-5. J. Phys. Chem. B 107: 10760–10766 69. Denayer JFM, De Meyer K, Martens JA, Baron GV (2003) Molecular competition effects in liquid phase adsorption of long chain n-alkane mixtures in ZSM-5 zeolite pores. Angewandte Chemie 42: 2774–2777 70. Da Silva FA, Rodrigues AE (1999) Adsorption equilibria and kinetics for propylene and propane over 13X and 4A zeolite pellets. Ind. Eng. Chem. Res. 38(5): 2051–2057 71. Harlfinger R, Hoppach D, Hofmann HP (1980) Adsorption behavior of c4-carbohydrates on synthetic zeolites of the faujasite-type – 2. Calorimetric studies of adsorption equilibrium. Z. Phys. Chemie 261(1): 33–42 72. Jentys A, Mukti RR, Tanaka H, Lercher JA (2006) Energetic and entropic contributions controlling the sorption of benzene in zeolites. Microporous Mesoporous Mater. 90: 284–292 73. Dzhigit OM, Kiselev AV, Rachmanova TA (1984) Henry’s constants, isotherms and heats of adsorption of some hydrocarbons in zeolites of faujasite type with different content of sodiumcations. Zeolites 4: 389–397 74. Herden H, Einicke WD, Messow U, Quitzscj K, Schöllner R (1982) Adsorption of paraffinolefin-mixtures on Y-zeolite and Y-zeolite from liquid solution – 2. Experimental results on NaX-zeolite and NaY-zeolite. Chem. Techn. 34(7): 364–367 75. Jänchen J, Stach H (1985) Dependence of the adsorption equilibrium of n-decane on the Si/Alratio of faujasite-zeolites. Zeolites 5: 57–59 76. Papaioannou CHR, Petroutsos G, Gunsser W (1997) Examination of the adsorption of hydrocarbons at low coverage on faujasite zeolites. Solid State Ionics 799: 101–103 77. Daems I, Leflaive Ph, Méthivier A, Denayer JFM, Baron GV (2005) A study of packing induced selectivity effects in the liquid phase adsorption of alkane/alkene mixtures on NaY. Microporous Mesoporous Mater. 82: 191–199 78. Daems I, Methivier A, Leflaive P, Fuchs AH, Baron GV, Denayer JFM (2005) Unexpected Si:Al effect on the binary mixtures liquid phase adsorption selectivities in faujasite zeolite. J. Am. Chem. Soc. 127(33): 11600–11601 79. Gorring RL (1973) Diffusion of normal paraffins in zeolite T. Occurrence of window effect. J. Catal. 31: 13–26 80. Cavalcante Jr CL, Eic M, Ruthven DM, Occelli ML (1995) Diffusion of n-paraffins in offretite-erionite type zeolites. Zeolites 15: 293–307 81. Gunadi A, Brandani S (2006) Diffusion of linear paraffins in NaCaA studied by the ZLC method. Microporous Mesoporous Mater. 90: 278–283 82. Kulprathipanja S, Johnson JA (2002) Liquid separations. In: Schüth F, Sing KSW, Weitkamp J (eds.) Handbook of Porous Solids, 2568, Wiley-VCH: Weinheim 83. Magalhaes FD, Laurence RL, Conner WC (1996) Transport of n-paraffins in zeolite T. AIChe J. 42: 68–86 84. Ruthven DM (2006) The window effect in zeolitic diffusion. Microporous Mesoporous Mater. 96(1–3): 262–269 85. Yoo K, Tsekov R, Smirniotis PG (2003) Experimental proof for resonant diffusion of normal alkanes in LTL and ZSM-12 zeolites. J. Phys. Chem. B 107: 13593–13596 86. Dubbeldam D, Smit B. (2003) Computer simulation of incommensurate diffusion in zeolites: Understanding window effects. J. Phys. Chem. 107: 12138–12152 87. Jobic H, Méthivier A, Ehlers G, Farago B, Haeussler W (2004) Accelerated diffusion of longchain alkanes between nanosized cavities. Angew. Chem. 43: 364–366 88. Daems I, Baron GV, Punnathanam S, Snurr RQ, Denayer JFM (2007) Molecular cage nestling in the liquid phase adsorption of n-alkanes in 5A zeolite. J. Phys. Chem. C 111: 2191–2197

Chapter 9

Macroscopic Measurement of Adsorption and Diffusion in Zeolites Stefano Brandani

Abstract A brief introduction to the measurement of adsorption and diffusion in zeolites is presented with a focus on macroscopic measurements, i.e. when adsorption through entire crystals is studied. Particular attention is given to the difficulties in obtaining reliable results for diffusion. Three case studies, the window effect, the volumetric/piezometric experiment and the transient analysis of products (TAP) apparatus, are discussed and ways to improve either the experiments or the way in which the results are analyzed are presented. The author’s perspective into what are the current challenges in the field concludes the chapter.

9.1 Introduction Zeolites are both naturally occurring and synthetic crystals that have very wellcharacterized nanopores [1]. This feature makes these materials extremely useful in several applications in catalysis and adsorption separation processes. These applications are controlled by both adsorption equilibrium and mass transfer kinetics and as a result many different experimental techniques have been developed to measure these quantities. The fact that the crystal structure of zeolites is apparently well characterized makes them a suitable candidate for the development of theories of diffusion in confined geometries and thus of great importance for a purely fundamental study also. Equilibrium measurements of pure component isotherms are typically carried out using either gravimetric or volumetric experiments for which commercial equipment is available [2, 3]. For strongly adsorbed components at low pressures, the main alternative is to carry out chromatographic or breakthrough experiments in dilute conditions [4]. In the gravimetric experiment a microbalance is used and the net weight of the sample is measured. In a volumetric experiment a known S. Brandani (B) Institute for Materials and Processes, School of Engineering, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JL, UK e-mail: [email protected] L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_9,  C Springer Science+Business Media B.V. 2010

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quantity of sample is introduced in a calibrated volume and the pressure is measured (for this reason, this technique is also called piezometric) and the adsorbed amount is obtained from a mass balance, therefore care should be taken to use appropriate materials with polished surfaces in order to avoid significant adsorption of molecules on the walls of the apparatus. In both experiments, for an accurate measurement of the absolute adsorption isotherm, the adsorbed amount is calculated through a correction, i.e. both techniques measure the excess adsorbed amount [5]. Particular care needs to be given when measuring adsorption isotherms of light gases such as hydrogen. In this case, careful definition of the Gibbs dividing surface is needed [6] and adsorption of helium has to be taken into account [7, 8]. In order to design catalytic and adsorption processes it is necessary to have accurate equilibrium isotherms that are thermodynamically consistent and can be used to calculate derived quantities reliably [5], such as multicomponent selectivities; enthalpies of adsorption; and partial derivatives with respect to concentration for kinetic models (macropore diffusion and Darken correction for transport diffusion in crystals). The key challenges in this field are the development of databases of measurements of several properties for the same system, especially for multicomponent systems with more than two adsorbates [9]. Measurement of diffusion in zeolites can be divided into two differing approaches: microscopic studies, which typically measure the movement of molecules and relate the root-mean-square displacement over time to the diffusion coefficient using Einstein’s equation, and macroscopic studies, which measure the mass transfer between the gas and the solid adsorbent over the entire crystal and solve Fick’s law of diffusion to obtain the diffusional time constant. In general the different techniques measure different diffusivities: either tracer/self-diffusivities or transport diffusivities. If the experiment is carried out at a constant concentration in the adsorbed phase, q, then self-diffusivities are measured. Most microscopic experiments are carried out in this manner. Macroscopic techniques need marked molecules, such as deuterium-substituted hydrocarbons, to carry out tracer exchange experiments. One main advantage of these techniques is that the system is always linear and isothermal. In transport experiments a concentration gradient is present and the diffusivity has a thermodynamic contribution due to the fact that the true driving force for mass transfer is the gradient of chemical potential, μA . For single components, assuming an ideal gas mixture, it is possible to find the relationship between the corrected diffusivity D0 and the transport diffusivity DT from DT = D0

q ∂μA ∂ ln PA = D0 RT ∂q ∂ ln q

(9.1)

where the thermodynamic factor on the RHS is often called the Darken correction. Note that this correction is the ratio of the secant and the tangent to the isotherm at a given concentration, which for a type I adsorption isotherm can be quite large and is always greater than 1. For a single adsorbate, the corrected diffusivity coincides with the Maxwell–Stefan diffusivity.

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To avoid confusion, when comparing different data sets, one should remember that while the values of the self-diffusivity, the corrected diffusivity and the transport diffusivity are all different except in the limit of infinite dilution, they should obey a simple order DT ≥ D0 ≥ DS . This order was confirmed theoretically by Paschek and Krishna [10], who showed that at a fractional coverage θ 1 1 θ = + , DS D0 D11

D0 ≥ DS

(9.2)

and determined experimentally [11] as shown in Fig. 9.1. Figure 9.2 shows a condensed historical overview of the development of the field of measurement of diffusion in zeolites. While the specific techniques are very interesting on their own merits, what is important to note is the fact that, apart from the pioneering work of Tiselius [12], who observed the adsorption front of water in a very large crystal of natural heulandite by microscopy, for a considerable period of time, uptake rate experiments, introduced originally by Barrer [13], were considered to be the means of carrying out these measurements. Following the mass of the adsorbent as it changes in time, in the presence of adsorbate molecules, seemed to be the obvious way to measure diffusivities in zeolites. This can be achieved through a dynamic measurement either of the mass or of the gas pressure in a closed system, i.e. measuring the transients using the equipment developed for equilibrium experiments. The main difference is that typically for kinetic experiments, one aims to carry out the measurement with the smallest amount of sample possible in order to minimise the intrusion of other kinetic phenomena, while for equilibrium measurements, the sample mass is large in order to maximise the accuracy of the final result. Figure 9.2 clearly shows a dramatic event at the end of the 1960s, early 1970s, with the introduction of the first NMR measurements [14, 15]. With the advent of the PFG-NMR technique, pioneered by Pfeifer and Kärger, a number of inconsistencies in the results reported in the literature came to light. This led to a re-examination of

Fig. 9.1 Diffusivities for deuterium in NaX [11]. Copyright (1999) by the American Physical Society

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Direct Visual Observat Tiselius (1934)

Chromatography Haynes, Ruthven (1973)

Effectiveness Factor. Haag, Post (1981)

Transient Uptake

IR and IR/FR Grenier, Meunier (1998)

FR Yasuda Rees (1982)

Coherent QENS Jobic (1999)

ZLC

TAP

Eic, Ruthven (1988)

Nijhuis et al., Baerns Keipert (1997)

Barrer (1938) Membrane Permeation Hayhurst, Weernick (1983)

IFM Kärger, Schemmert (1999)

IR Spectroscopy Karge (1974)

E q u i l i b r i u m

Tracer Exchange Barrer (1941)

NMR Relaxation Resing, Pfeifer, Michel (1967)

Incoherent QENS Cohen de Lara, Jobic (1983)

PFG-NMR

1940

1950

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1970

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Tracer ZLC Hufton, Brandani, Ruthven (1994)

Exchange NMR Chmelka (1998)

Pfeifer, Kärger (1971)

1930

PEP van Santen (2000)

FTIR Karge (1991)

1980

1990

2000

Year

Fig. 9.2 Development of the measurement of diffusion in zeolites [11–30]. Adapted from [31]

many fast diffusing systems for which gravimetric measurements were affected by spurious kinetic resistances. To overcome these limitations, many new techniques were proposed. In many cases, the earlier results were shown to be incorrect and by the 1980s and early 1990s, most researchers in the field came to the conclusion that only microscopically determined diffusivities were reliable and that macroscopic methods, when not in agreement with these measurements, reflected the kinetic limitations due to heat transfer, external film, bed effects and surface barriers. For a detailed discussion, one should refer to the monograph [32]. This view has changed over the last 10 years. Figure 9.3 shows results published only in the past 20 years for the system n-hexane–silicalite. These measurements were carried out by groups who knew of the earlier difficulties and ensured that their respective systems were not affected. While the data correspond to different quantities, i.e. self-diffusivities and corrected diffusivities, they should be all of the same order of magnitude for the system considered. For particular adsorbents, in this case silicalite, it is apparent that the measured diffusivity is dependent on the length scale and time of the observation. The current general consensus is that all these measurements are correct, within their respective uncertainties, and the results reflect the fact that large silicalite crystals have internal imperfections [33] that dominate the macroscopic measurements. The mass transfer constant measured

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n-Hexane in Silicalite after 1989 1.E-08

3 orders of magnitude!

D (m2/s)

1.E-09 1.E-10 1.E-11 1.E-12 1.E-13 1.E-14 1.7

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Grav-Uptk (P.Voogd)

ZLC

Grav-Uptk (SU.Kulkarni)

Grav-Uptk (J.Caro et al.)

Grav-Uptk (M.Jama & Ruthven)

Fig. 9.3 Recent diffusion in zeolites data reported in the literature [34–42]

by macroscopic techniques differs by up to three orders of magnitude from what it would be in an ideal crystal, based on the results from neutron scattering and PFG-NMR. The important conclusion that can be derived from all this is that, to date, we do not have a valid theory that will be able to predict a priori the mass transfer kinetics in non-ideal zeolite crystals. As a consequence it will be necessary to carry out macroscopic measurements for the foreseeable future. This field has relatively few groups who specialize in macroscopic diffusion measurements in nanoporous materials. The majority of the data that are being published are mainly from those who investigate mass transfer kinetics as part of a wider project, who sometimes do not fully understand the underlying difficulties in making such measurements. There are no commercially available instruments designed for this purpose, so researchers often use equipment intended for equilibrium purposes or have to develop their own systems. Of the techniques shown in Fig. 9.2, the most useful for slow systems (time constants greater than 1 min) are the gravimetric and volumetric/piezometric methods with small samples, while for fast systems (time constants greater than 1 s), the zero length column (ZLC) and the frequency response (FR) techniques are the most reliable and relatively inexpensive to use. The fact that incorrect kinetic data are reported in the literature leads to the conclusion that one of the real challenges is that of providing simple guidelines that can be used to identify when the experiments are likely to be successful. We will discuss this in detail with the aid of three case studies: the “window effect” of Gorring [43] and the importance of checking the experimental procedure; the volumetric/piezometric experiment and how to modify it to prove kinetic limitations; the transient analysis of products (TAP) experiment to show how to develop simple checks for seemingly complicated experiments.

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9.2 The Window Effect or Resonant Diffusion Gorring carried out kinetic experiments to measure the diffusivities of n-alkanes (C2 –C14 ) in zeolite T and found a remarkable dependence on chain length [43]. He observed a decreasing diffusivity from C2 to C8 , followed by a rapid increase to a maximum at C12 and a subsequent fall at C14 . The results were widely accepted since they explained catalytic selectivities that had been reported previously [44]. There was no question raised as to the validity of the measurements and theories were developed to explain the results [45–47]. Only in the early 1990s, independent of each other, two groups tried to reproduce Gorring’s experimental results [48–49], but both found a monotonic decrease in diffusivity with chain length. The data at 300◦ C are shown in Fig. 9.4. Gorring’s paper is still cited widely and more detailed theories have been shown to agree with the experimental observations [51–52]. Few people probably carefully read Gorring’s paper [43] and pay attention to the conditions under which he ran the gravimetric experiments. He used 5.38 g of zeolite placed in an annular basket and pretreated the sample at high temperature using dry helium to purge the system. The experiments were all performed at a total pressure of 1 atm, i.e. at time zero a stream of pure hydrocarbon was fed to the system to replace the helium present and flow switchover was 2s. What is remarkable is that in the paper no actual experimental curves are shown, only diffusivities are reported which were extracted using the solution to Fick’s law. To put this information into perspective, current state-of-the-art gravimetric systems for kinetic experiments use 10–20 mg of sample carefully distributed; pressure steps are typically a few kilopascals to avoid nonlinearities and limit heat effects (which may still be present); for flow through systems such as the ZLC, sample mass is limited to less than 1–2 mg.

Fig. 9.4 Diffusivities in zeolite T (Source: Ruthven (2006) [50]).  [42],  [47], [48]

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So qualitatively what happened? When the hydrocarbon is first introduced, it displaces the helium in the cell but not in the space between the crystals in the basket. The crystals are approximately 1 µm in size, so the hydrocarbon will have to diffuse through the helium in the gas phase to adsorb. The initial uptake is controlled by mass transfer through the bed of solids, most likely macropore diffusion controlled, followed by a sharp temperature increase due to adsorption. Cooling will then result in a further increase in mass, thus the final uptake is most likely affected by heat transfer. The problem is further complicated by the large pressure step used. Furthermore, Gorring never changed his sample size or arranged it differently to establish any robust experimental evidence for the validity of the results, since most of the critical analysis of the uptake experiments took place after his study. The correct interpretation of these results has been shown recently by Ruthven [50], who used literature values for the diffusivities [48–49] and a relatively simple nonisothermal model to “predict” the results of an experiment carried out as in the original paper. The comparison of the 90% uptake times is shown in Fig. 9.5, indicating that a combination of mass and heat transfer kinetics accounts for the results using a monotonically decreasing diffusivity. A conclusive experimental proof of the window effect using macroscopic measurements has not been presented. A recent study [53] claims to present gravimetric experiments that show the evidence of resonant diffusion in zeolites. Similar to Gorring no experimental uptake curves are shown in the paper and this is very remarkable since the words “experimental proof” are in the title. In this case, the authors used small samples (approximately 10 mg) and 0.5% concentration steps, so it would appear that the measurements are reliable. To understand the problem with these results, simply consider the model used to interpret the uptake. The authors used the solution to the diffusion equation in a sphere, subject to a step change in the surface concentration, which is plotted in Fig. 9.6.

Fig. 9.5 Uptake times in zeolite T (Source: Ruthven (2006) [50]). , from Gorring’s data; , non-isothermal model

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0.8

mt/m∞

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

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2

Dt/R

Fig. 9.6 Uptake according to the diffusion in a sphere model

From Fig. 9.6, it is apparent that an ideal uptake experiment is virtually complete for Dt/R2 ≈ 0.55, where D is the diffusivity in the zeolite and R is the equivalent radius of the crystals. In the paper the authors claim to have measured a diffusivity of 17 × 10–10 cm2 /s on 0.85-µm crystals (mean diameter). Based on these data, the uptake would be complete only after 0.6 s. No gravimetric system can introduce a step change in concentration in less than 1 s and produce a sufficiently stable signal to extract a meaningful diffusivity. Common sense suggests that the deviation from the simple monotonically decreasing diffusion coefficient, which is observed for the smaller chain molecules (C2 to C7 all with apparent D ≥ 5 × 10–10 cm2 /s), can be attributed to the experiment being attempted on systems that were too fast to measure.

9.3 Volumetric/Piezometric Method: A Closed System To understand another common difficulty, we consider a very simple system, the volumetric or piezometric experiment, which is shown schematically in Fig. 9.7. This system was developed for equilibrium measurements. For this purpose, the dosing and uptake (where the zeolite crystals are placed) cells are connected to vacuum and the valve is then closed. The dosing volume is charged to a known pressure, i.e. a finite known amount of gas is introduced into the system. By opening the valve and measuring the final pressure, one can calculate the amount of gas

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Fig. 9.7 Volumetric/ piezometric system

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P T control

Uptake cell

Dosing cell

adsorbed from a simple mass balance. For equilibrium measurements, one needs only one pressure transducer in the dosing volume. In principle this system can be used for kinetic measurements also, if the pressure vs. time curve is recorded. Consider the following example: 1) Measurements are carried out without the zeolite present to investigate the flow characteristics of the valve. 2) Measurements are carried out with the zeolite present. If a significant difference is observed from the previous experiments, one concludes that mass transfer kinetics is being measured. An example of this is shown in Fig. 9.8. Is there something obviously wrong with proposition 1 and 2? Not at first sight. An experimentalist would comment that flow through a valve is nonlinear and therefore proposition 1 would establish the characteristics of the valve in a region 1

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that is not representative of the adsorption experiments. When adsorption is taking place, the gas flowing through the valve can be much larger than that in the non-adsorbing case if fast diffusing and strongly adsorbed components are being investigated. The second issue is that adsorption is a phase change phenomenon and heat is being generated, so one should consider the effect of heat transfer resistances also. If we limit the discussion to single-component systems, one has to be careful also with the way the solid is arranged as this will change the heat transfer characteristics. Finally, if the pressure step is relatively large, the equilibrium isotherm is not necessarily linear, introducing additional complications in the analysis of the results. We can conclude that an accurate measurement is more complicated than originally thought. The student with a passion for modelling may suggest to include these effects in the model equations and to determine additional parameters from the experiments. After all, we observe a difference between the adsorbing and the non-adsorbing case, so we should be able to measure kinetics. What we have missed, so far, by focusing too much on the experimental details is the more general question: can we use the comparison with a non-adsorbing system to confirm that mass transfer kinetics is being measured? The answer to this question is no! This appears to be an obvious conclusion to very few researchers. I will try to justify this answer by looking at Fig. 9.7 and proposing a thought experiment: replace the valve with a very small capillary tube so that the flow between the dosing and the uptake volume is slow. The pressure changes, but the change is so slow that the adsorbed phase equilibrates with the gas continuously. In this case the experiment with and without the adsorbent will be quite different, but the dynamics can be described entirely using only the equilibrium properties and the flow characteristics of the capillary. By looking at Fig. 9.7 we can therefore identify two characteristic time constants, one linked to the valve conductance and the other due to the diffusion in the zeolite. It is the ratio of these time constants that dictates in which regime we are and if we can measure diffusion. Figure 9.8 also shows the limiting case of equilibrium control where the adsorbed phase is always at equilibrium with the gas. Qualitatively the equilibrium control model has a shape that is very similar to that of the full curve, so how can we be sure that we are indeed measuring diffusion? The answer to this question can be the use of accurate models coupled to dynamic sensitivity analysis to distinguish between them, but this solution may lead to cases where one is not certain of the result. Is there an unequivocal way to prove directly from the experiments that mass transfer kinetics is being measured? Again look at Fig. 9.7. If we add a second pressure transducer and measure the pressure in the uptake cell, we now have unequivocal experimental proof of what we are measuring. This is because while the pressure in the dosing cell, whichever the controlling mechanism, will be a monotonically decreasing function in time, the pressure in the uptake volume should initially increase, go through a maximum and then decrease to the final equilibrium value if mass transfer or heat transfer limits the dynamics of this system (see Fig. 9.9). The pressure in the uptake cell will be increasing monotonically if the adsorbed phase is always at equilibrium with the gas. Therefore, it

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Fig. 9.9 Pressure transient in the uptake cell

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is possible to distinguish between these two cases from a direct observation of the shape of the pressure vs. time curve of the uptake cell. The second pressure measurement offers an additional advantage. It allows the calculation of the amount of gas present in the adsorbed phase at any time, i.e. the average adsorbed amounts are known. The knowledge of the average solid concentration and the boundary condition on the solid, given by the measured pressure in the uptake cell, allows the interpretation of the uptake data, without the need for an empirical valve model [54]. Based on this discussion we have shown that it is possible to establish experimentally if kinetics are being measured and that a modification to an equilibrium apparatus can lead to a much improved system as well as ease in the analysis of the results. For a detailed discussion of this system, one should refer to [54], which also reports an analysis of the data for the system benzene–NaX measured using this technique [55]. The measurement of both pressures was implemented by Schumacher and Karge [56], but one still finds in the recent literature data measured using a volumetric/piezometric system with very fast diffusion and only the pressure on the dosing cell being recorded [57].

9.4 Transient Analysis of Products (TAP) Method: A Flow System The TAP system is another apparatus that has been used to measure diffusivities in zeolites. In its simplest form, a TAP bed consists of a tube in which the solids are loaded. The column is then exposed to a high vacuum chamber at the outlet. Once

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Bed of adsorbent particles

Fig. 9.10 TAP system

the system is equilibrated, a very small pulse of gas is introduced at the inlet of the column. The flux coming out of the adsorbent bed is measured with time and diffusion in the solid is thus calculated. A schematic diagram of a TAP bed is shown in Fig. 9.10. The fact that the entire system is under vacuum ensures that the experiment is carried out in the region where the adsorption isotherm is linear. The system can be assumed to be isothermal because as the pulse is travelling along the column, an approximately equal amount of molecules are being adsorbed and desorbed. One should also consider that there is a very large heat capacity of the bed and metal components in comparison with the heat generated by the relatively small amount of adsorbate molecules. Finally the gas is introduced pure and there are no external mass transfer limitations due to the presence of a second carrier gas, i.e. diffusion through an external film. This seems to be an ideal system to measure the diffusivity at low concentrations. It is not surprising that with this type of system Nijhuis et al. [27] appear to be the only researchers to date who claim to have measured diffusivities macroscopically in large silicalite crystals in agreement with PFG-NMR results. Nijhuis adopted the following approach: 1) The column was loaded with silicalite that had the template molecule needed for the synthesis still present. In this case there are no empty nanopores. Experiments were carried out on these crystals, thus showing the response without adsorption. 2) The column was exposed to air at high temperature and the template was burned off, thus opening the nanopores. Experiments were repeated under these conditions and a large difference was observed, as shown qualitatively in Fig. 9.11. Nijhuis used a detailed model to interpret his results and claimed that, under the conditions in which he carried out his experiments, statistical analysis of his parameter-fitting procedure indicated that he obtained reliable results. This example gives the opportunity to explain why I believe that one of the challenges in this field is that of determining directly from the experiments if diffusion is being measured. I will explain this without the use of any mathematical model but observe the following: 1) This system, neglecting any other complicating details such as premixing and postmixing, has at least two time constants: one linked to diffusion in the zeolite crystals and the second due to Knudsen diffusion in the bed.

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2) If we are able to normalize the experimental response using only one time constant, then the system must be controlled by Knudsen transport which is always present. Nijhuis et al. [27] presented complete experimental curves at 527 K and 623 K for the system n-butane–silicalite-1. They reported the signal as measured by the mass spectrometer and the procedure that one should carry out on their data is as follows: i. Calculate the integral of the signal as a function of time and divide the signal by this amount. An advantage of this is that now the curves do not depend on the actual amount injected in the system. ii. Divide the time by a characteristic time, such as the time corresponding to the maximum of the curve, or any constant time. iii. Multiply the signal/integral by this characteristic time so that the area under the curve is 1 and the plotted curve is dimensionless. If different curves are made dimensionless in both measured quantity and time in this way and it is not possible to distinguish between them, then only one time constant can be extracted from the experiments. This has to be that related to Knudsen flow as explained above. Figure 9.12 shows the result of the procedure outlined above when applied to the data of Nijhuis et al. [27]. The procedure is extremely simple and can unambiguously prove that the claims of Nijhuis are not correct. Using a very similar apparatus, Keipert and Baerns [58] also studied the same systems but concluded that they were not able to measure the fast diffusivities. In fact the TAP can be used only in a very small window of conditions, because if the diffusivity in the solid is too low, the adsorbate will exit the column without interacting with the solid, while if the diffusivity is too fast, the solid and gas will be at equilibrium.

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Fig. 9.12 Normalized plot of experimental data

9.5 Equilibrium Control What should be apparent by now is that for closed systems there is a need to test the dynamic response of the apparatus in the absence of adsorption to establish an order of magnitude for its response time. In the case of a flow system, one needs to test the non-adsorbing case to be sure that a significant portion of molecules enters the adsorbent during the experiment. These checks are usually carried out by experimentalists. For both systems it is also true that there is a second limit that has to be considered but does not appear to be as obvious. We term this limit equilibrium control. To understand what is meant, we return to the data of Nijhuis and ask: what did he measure? Nijhuis argues that DK /τ L2 ≫D/R2 , where DK is the Knudsen diffusion in the column, L is the length of the column and τ is the tortuosity in the packed bed (typically 2–3). So how can the Knudsen flow be slow compared to the diffusion in the zeolite? The problem in this argument is that we should be comparing the flux in the gas, which is proportional to DK /τ L2 , with the flux in the zeolite, which is proportional to HD/R2 , where H is the equilibrium Henry’s constant. In zeolites this is often very large. The best way to explain this is that in the equilibrium control limit, the model of the TAP column reduces to Fick’s equation. In the TAP bed we can write the mass balance in the gas phase as DK ∂ 2 c ∂c 1 − ε ∂ q¯ + − =0 ∂t ε ∂t τ ∂z2

(9.3)

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where the three terms in the equation are the accumulation in the fluid, the accumulation in the solid and the dispersion due to diffusion. The mass balance in the solid phase is ∂q − D∇ 2 q = 0 ∂t

(9.4)

where q¯ is the average adsorbed phase concentration. These two equations are coupled with the relevant boundary conditions. If we assume that diffusion in the solid is the controlling mass transfer resistance, then the concentration at the solid surface will be at equilibrium with the gas: qS =Hc . This general model has a limit corresponding to the non-adsorbing case if one takes either D = 0 or H = 0. The second limit is that of equilibrium control, which is obtained when D = ∞. In this second limit, there is no internal gradient in the solid particle (otherwise there would be an infinite flux), therefore the average concentration is equal to the concentration at the surface Hc and ∂c 1 − ε ∂c DK ∂ 2 c + H − =0 ∂t ε ∂t τ ∂z2

(9.5)

This can be rewritten in the form of the diffusion equation ∂c − ∂t



DK ε τH 1 − ε



∂ 2c =0 ∂z2

(9.6)

Therefore Nijhuis in his experiments measured an effective Knudsen diffusivity (the term in parenthesis), which is only a function of the equilibrium constant!

9.6 Conclusions We have highlighted the importance of reliable experimental measurements of both equilibrium and kinetic properties of zeolites. While both types of experiments can be difficult to carry out, equilibrium measurements are in general easier to interpret. The simple examples considered have shown that for macroscopic measurement of diffusion in zeolites it can be challenging to detect if the results are reliable and even data reported in reputed journals can be incorrect. In general, as shown in the case studies, it is necessary to identify all the kinetic time constants of a system. It should then be possible to devise experiments or normalize the results in order to have direct evidence that the conditions are far from the equilibrium control limit. Since equilibrium can be measured through independent experiments or from the final result in closed systems, even if experimental validation cannot be achieved, one should always compare the dynamic response to that which would be predicted under equilibrium control. Only if a clear difference in the predicted equilibrium control curve and the measured dynamics can be observed, one can conclude that mass or heat transfer kinetics is quantifiable.

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There are no commercially available apparatuses that are designed specifically for the measurement of diffusion in nanoporous materials. Even if these become available, it is unlikely that a single system will be able to provide accurate results for the wide range of time constants that may be encountered in all the materials that need to be tested. This is especially true if one considers also the extension to formed materials, such as pellets and monoliths, in which diffusion in the macropores through molecular, Knudsen or viscous flow may be relevant. In order to have reliable equipments that can be used by the non-expert, the main challenges that will have to be resolved are the following: a) The need for experimental protocols that unequivocally confirm the validity of the results. b) A systematic approach to establish the experimental conditions in which physical parameters may be measured in different apparatuses. c) The development of automated systems, which are particularly important in industrial applications. d) The development of software tools that allow the simultaneous analysis of multiple experiments to extract the diffusion coefficient, combined with dynamic sensitivity analysis.

References 1. Ch. Baerlocher, W.M. Meier, D.H. Olson, Atlas of Zeolite Framework Types, 5th revised edition, Elsevier, Amsterdam, 2001. 2. F. Rouquerol, J. Rouquerol, K. Sing, Adsorption by Powders & Porous Solids, Academic Press, San Diego, 1999. 3. J. Keller, R. Staudt, Gas Adsorption Equilibria, Springer, New York, 2005. 4. D.M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, New York, 1984. 5. A.L. Myers (2002) Thermodynamics of adsorption in porous materials. AIChE J. 48: 145–160. 6. O. Talu, A.L. Myers (2001) Molecular simulation of adsorption: Gibbs dividing surface and comparison with experiment. AIChE J. 47: 1160–1168. 7. S. Gumma, O. Talu (2003) Gibbs dividing surface and helium adsorption. Adsorption 9: 17–28. 8. S. Sircar (2001) Measurement of Gibbsian Surface Excess. AIChE J. 47: 1169–1176. 9. S. Sircar (2006) Basic research needs for design of adsorptive gas separation processes. Ind. Eng. Chem. Res. 45: 5435–5448. 10. D. Paschek, R. Krishna (2001) Inter-relation between self- and jump-diffusivities in zeolites. Chem. Phys. Lett. 333: 278–284. 11. H. Jobic, J. Kärger, M. Bée (1999) Simultaneous measurement of self- and transport diffusivities in zeolites. Phys. Rev. Lett. 82: 4260–4263. 12. A.W.K. Tiselius (1934) The diffusion of water in zeolite crystals. An article on the question of absorbed molecule mobility. Z. Phys. Chem. A 169: 425–458. 13. R.M. Barrer (1938) The sorption of polar and non-polar gases by zeolites. Proc. Roy. Soc. London A 167: 392–420. 14. R.M. Barrer (1941) Migration in crystal lattices. Trans. Faraday Soc. 37: 590–599. 15. H.A. Resing, J.H. Thompson (1967) NMR relaxation in adsorbed molecules. 5. SF6 on Faujasite − dipolar coupling of flourine nuclei to ferric-ion impurities. J. Phys. Chem. 46: 2876–2880.

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16. H. Pfeifer, in NMR − Basic Principles and Progress 7, Springer, Berlin, 1972, pp. 53–153. 17. H.W. Haynes, P.N. Sarma (1974) Application of gas chromatography to measurements of diffusion in zeolites. Adv. Chem. 133: 205–217. 18. H.G. Karge, K. Klose (1974) IR-spectroscopic study of pyridine diffusion in zeolites at high temperatures. Ber. Bunsen. Ges. Phys. Chem. 78: 1263. 19. W.O. Haag, R.M. Lago, P.B. Weisz (1981) Transport and reactivity of hydrocarbon molecules in a shape-selective zeolite. Faraday Discuss. Chem. Soc. 72: 317–330. 20. Y. Yasuda (1982) Determination of vapour diffusion coefficients in zeolites by the frequencyresponse method. J. Phys. Chem. 86: 1913–1917. 21. A. Paravar, D.T. Hayhurst (1984) Direct measurement of diffusivity for butane across a single large silicalite crystal. In Proc. 6th Int. Zeolite Conf., Reno 1983, D. Olson, A. Bisio, eds., Butterworth, Guildford, pp. 217–224. 22. E. Cohen de Lara, R. Kahn, F. Mezei (1983) Determination of the intracrystalline diffusioncoefficient of methane in A-zeolites by means of neutron spin-echo experiments. J. Chem. Soc. Faraday Trans. I 79: 1911–1920. 23. M. Eic, D.M. Ruthven (1988) A new experimental technique for the measurement of intracrystalline diffusivity.Zeolites 8: 40–45. 24. H.G. Karge, W. Niessen (1991) A new method for the study of diffusion and counter-diffusion in zeolites. Catal. Today 8: 451–465. 25. S. Brandani, J.R. Hufton, D.M. Ruthven (1995) Self-diffusion of propane and propylene in 5A and 13X zeolite crystals studied by the tracer ZLC method. Zeolites 15: 624–631. 26. V. Bourdin, Ph. Grenier, F. Meunier et al. (1996) Thermal frequency response method for the study of mass-transfer kinetics in adsorbents. AIChE J. 42: 700–712. 27. T.A. Nijhuis, L.J.P. van den Broeke, J.M. van de Graaf et al. (1997) Bridging the gap between macroscopic and NMR diffusivities. Chem. Eng. Sci. 52: 3401–3404. 28. D.J. Schaefer, D.E. Favre, M. Wilhelm et al. (1997) Site-hopping dynamics of benzene adsorbed on Ca-LSX zeolite studied by solid-state exchange C-13 NMR. J. Am. Chem. Soc. 119: 9252–9267. 29. U. Schemmert, J. Kärger, J. Weitkamp (1999) Interference microscopy as a technique for directly measuring intracrystalline transport diffusion in zeolites. Microporous Mesoporous Mater. 32: 101–110. 30. R.R. Schumacher, B.G. Anderson, N.J. Noordhoek et al. (2000) Tracer-exchange experiments with positron emission profiling: diffusion in zeolites. Microporous Mesoporous Mater. 35–36: 315–326. 31. D.M. Ruthven (2006) Transport in microporous solids − Part II: Measurement of micropore diffusivities. In Fluid Transport in Nanoporous Materials, W.C. Conner, J. Fraissard, eds., Springer, Dordrecht, pp. 151–186. 32. J. Kärger, D.M. Ruthven, Diffusion in Zeolites and Other Microporous Solids, John Wiley, New York, 1992. 33. S. Vasenkov, J. Kärger (2002) Evidence for the existence of intracrystalline transport barriers in MFI-type zeolites: a model consistency check using MC simulations. Microporous Mesoporous Mater. 55: 139–145. 34. P. Voogd, H. van Bekkum, D. Shavit et al. (1991) Effect of zeolite structure and morphology on intracrystalline normal-hexane diffusion in pentasil zeolites studied by the zero-length column method. J. Chem. Soc. Faraday Trans. 87: 3575–3580. 35. W. Heink, J. Kärger, H. Pfeifer et al. (1992) High-temperature pulsed field gradient nuclearmagnetic-resonance self-diffusion measurements of n-alkanes in MFI-type zeolites. J. Chem. Soc. Faraday Trans. 88: 3505–3509. 36. J. Caro, M. Noack, J. Richtermendau et al. (1993) Selective sorption uptake kinetics of nhexane on ZSM-5 – A new method for measuring anisotropic diffusivities. J. Phys. Chem. 97: 13685–13690. 37. A. Micke, M. Bülow, M. Kocirik et al. (1994) Sorbate immobilization in molecular-sieves – Rate-limiting step for n-hexane uptake by silicalite-I. J. Phys. Chem. 98: 12337–12344

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38. M.A. Jama, M.P.F. Delmas, D.M. Ruthven (1997) Diffusion of linear and branched C-6 hydrocarbons in silicalite studied by the wall-coated capillary chromatographic method. Zeolites 18: 200–204. 39. L.J. Song, L.V.C. Rees (1997) Adsorption and transport of n-hexane in silicalite-1 by the frequency response technique. J. Chem. Soc. Faraday Trans. 93: 649–657. 40. O. Talu, M.S. Sun, D.B. Shah (1998) Diffusivities of n-alkanes in silicalite by steady-state single-crystal membrane technique. AIChE J. 44: 681–694. 41. H. Jobic (2000) Diffusion of linear and branched alkanes in ZSM-5. A quasi-elastic neutron scattering study. J. Mol. Catal. A: Chem., 158: 135–142. 42. W. Zhu, F. Kapteijn, J.A. Moulijn (2001) Diffusion of linear and branched C-6 alkanes in silicalite-1 studied by the tapered element oscillating microbalance. Microporous Mesoporous Mater. 47: 157–171. 43. R.L. Gorring (1973) Diffusion of normal paraffins in zeolite-T – Occurrence of window effect. J. Catal. 31: 13–26. 44. N.Y. Chen, S.J. Lucki, E.B. Mower (1969) Cage effect on product distribution from cracking over crystalline aluminosilicate zeolites. J. Catal. 13: 329–332. 45. E. Ruckenstein, P.S. Lee (1976) Resonant diffusion. Phys. Lett. A 56: 423–424. 46. J.M. Nitsche, J. Wei (1991) Window effects in zeolite diffusion and Brownian-motion over potential barriers. AIChE J. 37: 661–670. 47. R. Tsekov, E. Ruckenstein (1994) Resonant diffusion of molecules in solids. J. Chem. Phys. 100: 3808–3812. 48. C.L. Cavalcante, M. Eic, D.M. Ruthven et al. (1995) Diffusion of n-paraffins in offretite– erionite type zeolites. Zeolites 15: 293–307. 49. F.D. Magalhaes, R.L. Laurence, W.C. Conner (1996) Transport of n-paraffins in zeolite T. AIChE J. 42: 68–86. 50. D.M. Ruthven (2006) The window effect in zeolitic diffusion. Microporous Mesoporous Mater. 96: 262–269. 51. D. Dubbeldam, S. Calero, T.L.M. Maesen et al. (2003) Understanding the window effect in zeolite catalysis. Angew. Chem. Int. Ed. 42: 3624–3626. 52. D. Dubbeldam, B. Smit (2003) Computer simulation of incommensurate diffusion in zeolites: Understanding window effects. J. Phys. Chem. B 107: 12138–12152. 53. K. Yoo, R. Tsekov, P.G. Smirniotis (2003) Experimental proof for resonant diffusion of normal alkanes in LTL and ZSM-12 zeolites. J. Phys. Chem. B 107: 13593–13596. 54. S. Brandani (1998) Analysis of the piezometric method for the study of diffusion in microporous solids: Isothermal case. Adsorption 4: 17–24. 55. M. Bülow, W. Mietk, P. Struve et al. (1983) Intracrystalline diffusion of benzene in NaX zeolite studied by sorption kinetics.J. Chem. Soc. Faraday Trans. I 79: 2457–2466. 56. R. Schumacher, H.G. Karge (1999) Sorption kinetics study of the diethylbenzene isomers in MFI-type zeolites. Microporous Mesoporous Mater. 30: 307–314. 57. R.S. Todd, P.A. Webley, R.D. Whitley et al. (2005) Knudsen diffusion and viscous flow dusty-gas coefficients for pelletised zeolites from kinetic uptake experiments. Adsorption 11: 427–432. 58. O.P. Keipert, M. Baerns (1998) Determination of the intracrystalline diffusion coefficients of alkanes in H-ZSM-5 zeolite by a transient technique using the temporal-analysis-of-products (TAP) reactor. Chem. Eng. Sci. 53: 3623–3634.

Chapter 10

Vapor–Liquid Equilibrium Joël Puibasset

Abstract Every time a fluid is confined at a nanometer scale, the predominance of the fluid–substrate interactions strongly distorts its intrinsic properties. For instance, it is observed that the amount of fluid adsorbed in a nanoporous substrate is not a single-valued function of the chemical potential and may present a hysteresis. Understanding this phenomenon is a fundamental issue since it appears in the most frequently used method to characterize porous materials. The aim of this chapter is to focus on the analog of phase coexistence and the corresponding phase diagram for confined fluids. Lying between theory and experiments, molecular simulation allows accurate calculations of confined fluid properties in very realistic porous models. We focus on heterogeneous tubular pores, which constitute a good model for MCM-41, one of the most widely used molecular sieves. Little is known about the consequences of morphological or chemical heterogeneities in these systems. To keep advantage of the cylindrical symmetry, a new simulation framework is used to perform calculations of important thermodynamic properties of the confined fluid, such as a thermodynamic pressure and coexistence diagram.

10.1 Introduction In the wide field of physicochemical phenomena in condensed matter physics, we will focus on those arising for a fluid which is confined to a limited portion of space, for instance, inside a porous material [1]. In such circumstances, the fluid–substrate contact area is very large, which greatly influences the physicochemical properties of the confined fluid [2, 3]. Such situations are ubiquitous in nature and many industrial processes. These confinement effects are also used for the characterization of the porous materials themselves. It is then necessary to have a global understanding of fluid/substrate interactions and the effect of confinement. J. Puibasset (B) Centre de Recherche sur la Matière Divisée, CNRS-Université d’Orléans, 1b rue de la Férollerie, 45071 Orléans, Cedex 02, France e-mail: [email protected] L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_10,  C Springer Science+Business Media B.V. 2010

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The very first observation of confinement effect was seen in porous solids like charcoal which are able to take up large volumes of condensable gas (capillary condensation in mesopores). This effect was soon related to a large area of the exposed surface and the presence of pores. Zsigmondy [4] proposed the first explanation of this phenomenon based on the macroscopic Kelvin equation. Another important feature is that, for a given temperature, the amount of fluid adsorbed in the porous substrate is not a unique function of the adsorbate vapor pressure above the porous material (which determines the chemical potential) and depends on whether one performs an adsorption or a desorption experiment (hysteresis). This hysteresis is sometimes seen as an analog of the one associated with the liquid/gas transition, with associated coexistence phenomena. Various scenarios for adsorption and desorption were proposed by Cohan [5] to explain the hysteresis. The same year, Brunauer, Emmett, and Teller [6] proposed an improved model, allowing the extraction of specific surface area of adsorbent from experimental data. The film thickness or t-plot was introduced to improve the model predictions [2]. However, the nature of the hysteresis remains uncertain. It may be an intrinsic metastability, due to different curvature radius between adsorption and desorption [5], or may be interpreted as an instability limit of adsorbed film [7–10]. Recent theoretical works also confirm this view point that hysteresis is due to intrinsic metastability in a single idealized pore [11]. On the other hand, Mason [12, 13] has developed a very different approach where the hysteresis shape is related to the nature of the interconnected network or pores. Swift and coworkers [14] have also studied the combined effect of confinement and interconnectivity in Ising-like models. Very recently, Kierlik and coworkers [15, 16] introduced a coarse-grained approach which allowed the interpretation of the hysteresis loop as the signature of a very irregular energy landscape with many local minima induced by the disordered porous substrate. The effect of the temperature has also been widely studied [17–22]. Essentially, increasing the temperature reduces the width of the hysteresis loop. The improvement in material production also allows studying the effect of pore structure [23, 24]. Intensive research in this area helps to understand some experimental data (see [25] and references therein). However, due to the complexity of the real porous materials, a full understanding of adsorption phenomena is still not complete. Computational methods (molecular simulation, density functional theories, lattice models, etc.) are probably among the most powerful tools to study confinement effects. They have proven to be very efficient for simple pore geometries (slits or cylinders) and indispensable to take into account the recent improvement in porous materials description (chemical heterogeneities, morphological as well as topological disorder, etc.) [16, 25–50]. In this simulation work, we focus on a simple system to address some of the issues associated with hysteresis and coexistence. A new approach is introduced which allows the study of thermodynamic properties of fluid confined in tubular pores presenting physicochemical heterogeneities while keeping a rotational invariance around the pore axis. The corresponding generalized isobaric–isothermal ensemble is introduced for direct calculation of the thermodynamic pressure of the confined fluid. The generalized Gibbs ensemble is also presented, which allows the calculation of coexistence properties for any fluid with

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short-range interactions confined in heterogeneous tubular pores. Our results show how the confined fluid behavior rapidly changes from a simple analogy with liquid/gas transition in perfectly cylindrical pores to a description in terms of local minima in a complex free energy landscape with increasing disorder.

10.2 Molecular Model Many characterization techniques use noble gases as adsorbate because they are monoatomic and describable with simple models. For the same reasons, many experimental, theoretical, and simulation works are performed with such simple gases to study the fundamentals of adsorption and desorption phenomena, such as the hysteresis loop, its temperature dependence, and, for example, the nature of the capillary condensation transition. The fluid–fluid interactions are described by the simple (12-6) Lennard-Jones intermolecular potential with the parameters for Argon given in Table 10.1. Generally a distance cutoff is introduced such that interactions are not calculated beyond a given distance. The intermolecular potential is also shifted so that it reaches zero for that specific distance. Since the simulation systems are not very large in this preliminary study, a large cutoff is introduced (larger than 1.5 nm). It results inevitably in longer simulation runs but has the advantage of not artificially reducing the range of the interactions. Furthermore, the intermolecular potential does not need to be shifted, and long-range corrections are negligible. These corrections are hardly calculable in heterogeneous phases as in cylinders, and Panagiotopoulos [27] has shown that they have minimal influence on the calculations. The fluid–fluid parameters provide natural units to measure the physical quantities introduced in this paper (the unit for distance is σ f–f ; for energy, it is εf–f ; for temperature, it is εf–f /kB , where kB is the Boltzmann’s constant; for pressure, it is εf–f /σ f–f 3 ; etc.). The reduced quantities will be denoted by an asterisk. In the same spirit of simplicity, the Lennard-Jones potential is also used to describe the fluid–substrate intermolecular potential. This approximation neglects many aspects of fluid–wall interactions such as polarization, electrostatics, hydrogen bonding, surface reactions, chemical adsorption. Such phenomena are widely studied in the literature with various sophisticated techniques which are generally time consuming. The capillary condensation phenomenon appears in quite large systems and is largely independent of the details of the fluid–substrate interactions. The simple Lennard-Jones potential is an ideal compromise between computing capabilities and the essential features of capillary condensation physics. The

Table 10.1 Argon–solid CO2 interaction parameters Fluid–fluid (f–f) Wall–fluid (w–f)

ε/kB (K)

σ (Å)

119.8 153.0

3.405 3.725

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fluid–substrate interaction parameters are chosen more attractive than the fluid–fluid ones. Solid carbon dioxide has already been chosen as adsorbent since it fulfills this condition, which results in enhanced confinement effects in adsorption phenomena [27, 28, 51–53]. The corresponding Lennard-Jones interaction parameters are given in Table 10.1. Several pore models are considered here, which are chosen as follows. Since we are mainly interested in adsorption phenomena in mesoscopic pores (few nanometers), an atomistic description of the adsorbent is not an absolute requirement. The curvature radius of the adsorbent surface is large enough compared to molecular size to smear its atomic roughness. However, it should be mentioned that a realistic model should take into account the atomistic nature of the surface as well as its natural microtexture to correctly reproduce its adsorption properties in the lowcoverage regime. For instance, the surface roughness is known to greatly influence the initial heat of adsorption. This was shown, for instance, in the case of argon or water adsorbed in mesoporous silica [40, 44, 47, 50, 54–57]. However, the purpose here is not on giving an accurate description of the very first stages of fluid adsorption, which would have required a much more realistic description of the fluid–substrate interaction rather than the simple Lennard-Jones potential. Furthermore, the capillary condensation phenomenon is generally not greatly influenced by surface roughness at atomistic scale because it generally takes place after the first monolayer of fluid has adsorbed. The adsorption features observed in this chapter are then expected to be valid even in the presence of atomistic disorder. According to this smooth-wall approximation, the substrate molecules are replaced by a uniform distribution of interacting sites of density equal to that of the bulk adsorbent. The porous substrate is also supposed to be rigid; its structure or geometry is independent of the amount of fluid adsorbed. As a consequence, the fluid evolves in a constant potential field created by the substrate. This external field is calculated as the integral of the fluid–substrate intermolecular potential over the uniform distribution of interacting sites. It can be calculated at the beginning of the simulation and saved on a grid. Peculiar symmetries of the pore model may reduce the dimensionality of the grid and, as a consequence, the required memory. The simplest pore geometries are cylindrical or slit pores. Both have been extensively studied during last decades [11, 26–29, 31, 37, 38, 48, 53, 58–92]. The cylindrical geometry is probably the most natural to describe porous materials, generally thought as a collection of possibly interconnected tubular pores. But slit pores are also frequently encountered in carbon-made porous substrates [93]. Since the work of Evans, where he explains that a true first-order phase transition cannot occur in cylindrical pores while it is possible in slit-like geometry, most of the studies focused on slit pores. For instance, many researchers have calculated the “phase diagram” for fluid confined in heterogeneous slit pores (strip-like heterogeneities) [37, 77, 78, 86, 94–101], while the heterogeneous tubular pores have not been considered. However, infinite cylindrical or slit pores are unrealistic, the real adsorbents being generally made of a collection of interconnected elemental pieces of matter which can be modeled by portions of cylindrical or slit pores. Considering the small

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dimensions of each elemental constituent of the porous material, the intrinsic topological difference between slit and cylinder is actually irrelevant. On macroscopic scale, the porous material is made of a collection of such elemental slits or cylinders, and the nature of the transition is governed by interconnections or disorder [12, 13, 15, 16, 18, 41, 46, 102–111]. As a consequence, the true “phase diagrams” obtained in slit pores may not be a better description of experimental hysteresis than the “hysteresis diagrams” of cylindrical pores. It is believed that both geometries are relevant to describe porous materials. Since most studies concerning the effect of physicochemical heterogeneities concentrate on slit pores, it was decided to focus on tubular pores. As a consequence, the first and simplest pore to be considered is a simple cylinder. A cylindrical pore of diameter 8σ f–f (denoted as pore cyl) is drilled in a uniform density of infinite extension of interacting sites reproducing the solid carbon 3 = ∗ dioxide. The substrate site-reduced density is taken equal to ρsolid = ρsolid σf−f 0.8265, which corresponds to solid CO2 of density 1530 kg/m3 . Due to the cylindrical symmetry, the external field created by the pore has to be calculated as a function of the radial distance from the axis only. Let us denote the pore axis by z. After complete numerical integration, and taking into account the density of sites, one gets the external field ψ cyl (r, z) = ψ cyl (r) created by the solid substrate into the cylindrical pore (see Fig. 10.1a). Note that this external field is calculated for an infinitely long pore, while the numerical simulations will be performed in a finite box about 12σ f–f in the axial direction. It has been checked that this simulation box size is large enough not to introduce visible finite size effects. Some of the calculations have been performed in twice as large a system and give the same results. Any real porous material contains physicochemical heterogeneities of various origins [2, 3]. For tubular pores, the simplest heterogeneity is a sinusoidal modulation of a given wavelength along the pore axis. It is expected that a local variation of the curvature radius results in external field modulation. Similarly, for a given geometry, the external potential depends on the local chemical composition of the substrate. In the following, chemical modulation refers to such external field variation at fixed geometry. On the other hand, geometric modulation will refer to spatial deformation of the external potential without modification of its depth. Note that in a real pore, any geometric deformation of the surface is always accompanied by a modification of the well depth. However, to disentangle both geometric and chemical effects, spatial deformation of the external field with constant depth will be considered. The geometric heterogeneity is then produced by spatially deforming the potential, while in the second case, chemical heterogeneity is created by modulating the amplitude of the initial cylindrical potential. In both cases the modulation along the axial direction z introduces an explicit dependence of the potential on r and z, the radial and axial coordinates. The simplest model is a sinusoidal modulation mimicking a series of deformations or a series of more or less attractive domains. The reduced potential of the geometrically undulated pore (denoted as pore geom, see Fig. 10.1b) is given as

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Fig. 10.1 3D representation of the potential field in cylindrical pore, geometrically undulated pore, and chemically undulated pores of nominal radius R∗ = 4

⎛

r

⎞

∗ ∗ ⎝   ⎠ ψgeom (r,z) = ψcyl 1 − 0.25 cos 2πL z

(10.1)

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∗ (r) is the reduced external potential of the perfectly cylindrical pore. The where ψcyl reduced potential of the chemically heterogeneous pore (denoted as pore chem I, see Fig. 10.1c) is given as ∗ ψchem I (r,z)

   2π z ∗ ψcyl (r) = 1 + 0.25 cos L

(10.2)

where L is the length of the simulation box, equal to 12σ f–f in the direction of the z coordinate. In order to study the effect of the number of domains or heterogeneities, a second chemically heterogeneous pore is introduced (denoted as pore chem II), made of four domains with different fluid affinities    4π z ∗ ∗ ψcyl (r,z) = 1 + a(z)sin (r) (10.3) ψchem II L where a = 0.3 for z < 0 and a = 0.2 for z > 0, and L = 24σ f–f is the simulation box length (twice as long as pore chem II, see Fig. 10.1d). In all heterogeneous pores, the domains have the same spatial extension of a few molecular diameters along the axial direction. It is emphasized that the heterogeneities considered in this study are weak (< 30%). As can be seen, the external potential in the perfectly cylindrical pore is invariant along the axial coordinate, which is not the case for both undulated pores. The radius of the geometrically undulated pore varies between 3σ f–f for z = 0 and 5σ f–f for z = ± L/2. In this case, the heterogeneity consists in an enlargement of the pore size distribution up to 25% of its average value. As a consequence, the accessible volume, defined as the volume of space where the external field is finite (not divergent), is modified. Simple calculations show that the geometrically undulated 3 is the accessipore has a volume Vgeo = (1+a2 /2)Vcyl , where Vcyl = π R∗2 L∗ σf−f ble volume in the perfectly cylindrical pore and a = 0.25 in the undulation amplitude. For the chemically undulated systems, the pore radius is constant (volume 3 ), but the intensity of the fluid–wall interaction varies Vchem I = Vcyl = π R∗2 L∗ σf−f within 30% of its average value; this mimics some chemical heterogeneity with more or less attractive sites distributed along the pore. It is important to emphasize that the geometric modulation has not been applied to the solid CO2 prior to the potential calculation. This would have been certainly more realistic, but it would have induced the appearance of zones of higher potential in the regions of higher curvature. In this work we wanted to specifically decouple geometric from amplitude modulation effects.

10.3 Grand Canonical Monte Carlo Adsorption– Desorption Isotherms Most adsorption/desorption experiments follow the same experimental protocol. The measurement of the whole isotherm proceeds in several steps. Each time a given volume of gas is introduced in a vessel containing the adsorbent, the gas pressure

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is measured after the new equilibrium between the adsorbent and the surrounding gas has been reached. This way, the uptake of gas can be evaluated as a function of the pressure of the vapor above the porous material. The adsorption–desorption isotherms are generally given as a function of the pressure of the bulk gas above the adsorbent, which is sometimes misleading since this pressure actually does not refer to the adsorbed fluid but the bulk gas above. The chemical potential of the surrounding gas should be used instead, since it is a well-defined quantity which applies to the adsorbed phase as well. To avoid any confusion, the chemical potential will then be used most widely, and when pressure is used, it will be emphasized if it is the gas pressure above the substrate, or the thermodynamic pressure of the confined fluid, to be properly defined later. The confined fluid exchanges energy and particles with the surrounding gas which acts as a reservoir which imposes its temperature and chemical potential. The corresponding statistical ensemble is the grand canonical ensemble [112, 113]. The application of the Metropolis algorithm [114] to sample this ensemble by Monte Carlo methods (grand canonical Monte Carlo, GCMC) in continuous fluids was introduced by Rowley, Nicholson, and Parsonage [115]. The GCMC consists in generating a series of molecular configurations according to the grand canonical statistical ensemble, where the system can exchange particles and energy with a reservoir which imposes constant temperature and chemical potential. Therefore, it mimics a real adsorption experiment where volume and temperature are controlled and the chemical potential is fixed through the pressure of the gas reservoir. From a technical point of view, the simulation consists, on the one hand, in thermal equilibration of the simulation box through molecular displacements, with acceptance probabilities determined by the ratio of the difference in energy to the external temperature, and, on the other hand, in chemical equilibration with a particle reservoir through addition or removal of particles in the simulation box, with probabilities determined by the imposed chemical potential. For more technical details, see Refs. [116–118]. In this section, the systems studied consist of argon particles confined by the four ∗ , ψ∗ ∗ ∗ external fields ψcyl geom , ψchem I , and ψchem II (see Fig. 10.1) which represent the external potential fields in tubular pores with or without geometric or chemical undulations [70]. The size of the simulation box in the axial direction equals 12 or 24 σ f–f . It has been checked that this simulation length avoids finite size effects. The extension in the radial direction is limited by the external potential itself. The fluid– fluid interactions are calculated without distance cutoff and with minimal image convention (the system is made periodic in the axial direction). The initial configuration for each point of an adsorption isotherm is one of the equilibrated configurations of the previous point. At least 106 Monte Carlo trials per particle are performed to reach an equilibrium state. Then, twice as many Monte Carlo trials per particle provide the Markov chain used to calculate the statistical averages. In order to “thermalize” the molecular configuration between two addition or removal trials, one displacement trial is performed for each molecule. The maximal length of displacement trial is rescaled to “optimize” the acceptance ratio to around 50% [116]. Typically, the number of particles in the simulation box is several hundred, which

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results in rapid simulations compatible with high precision and complete analysis of temperature dependence of the adsorption–desorption isotherms. However, at high chemical potential, the adsorbed phase is so dense that the addition or rejection trials are much less efficient. This results in a slower convergence and less efficient sampling of the phase space for the liquid-like state and, as a consequence, much longer simulation runs. During the course of the simulation, the total number of particles in the box and the total reduced configurational energy are obtained. This allows the calculation of their average and fluctuations. Studying their correlations also provides important information on the adsorbed phase stability. See for instance the recent work of Kuchta and coworkers [48]. Here we focus on the average number of particles adsorbed in the pore, denoted by N. For each value of the reduced temperature T∗ and the reduced chemical potential μ∗ considered as parameters, we calculate the ∗ , where V ∗ reduced average density Ŵ ∗ = N/Vpore pore is the reduced accessible volume. By definition, the accessible volume is the volume of space where the potential is not infinite. This definition is identical to the one used by other authors in previous publications [27, 28, 30, 52, 53]. However, it should be noted that this definition does not take into account the van der Waals radius of the substrate species. As a consequence, the truly accessible volume is overestimated. For instance, in the case of the perfectly cylindrical or chemically undulated pores of length L∗ = 12 (pores cyl ∗ = V∗ and chem I) and reduced radius 4, the accessible volume is Vcyl chem I = 192π . For the geometrically undulated pore, it is given as Vgeo = 198π . It is slightly larger than the cylindrical volume. In a real adsorption experiment, the adsorbate is introduced progressively for adsorption into the initially outgassed porous sample. Similarly, the initial molecular configuration is the empty pore used as the starting point of the simulation at very low chemical potential. The chemical potential of the gas reservoir is then increased step by step to calculate the whole adsorption isotherm. For each step, the initial configuration is the final configuration of the preceding step. When the saturation of the sample is reached, one can start to calculate the reverse path by decreasing the pressure. The obtained so-called desorption branch is generally expected not to be identical to the adsorption branch (hysteresis). The adsorption and desorption isotherms have been calculated at five reduced temperatures: 0.6, 0.8, 0.9, 1.0, and 1.2. The results are shown as a function of the chemical potential in Fig. 10.2 for the four pore models. It has been checked elsewhere [70] that all adsorption isotherms calculated for the interacting gas converge to the ideal gas limit at low coverage, where fluid–fluid interactions are negligible. As a general remark, it can be seen that for a given chemical potential, the higher the temperature, the larger the amount of adsorbed fluid. However, for high chemical potential, the density in the porous substrate saturates independently of the temperature due to hard core repulsions in the fluid. Let us now focus on the results obtained in the perfectly cylindrical pore (Fig. 10.2a). The low-temperature curves show jumps and large hysteresis in the amount adsorbed. For instance, at T∗ = 0.6, the amount adsorbed follows the ideal gas approximation up to μ∗ = –9.6, then increases faster up to μ∗ = –9.0 and

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a) pore cyl 0.6

0.4

0.2

0.0 0.6

b) pore geom

reduced density N σAr-Ar3/V

0.4

0.2

0.0 0.6

c) pore chem I

0.4

0.2

0.0 0.6

0.4

d) pore chem II k T / εAr-Ar 0.6 0.8 0.9 1.0 1.2

0.2

0.0

–14

–12

–10

–8

reduced chemical potential μ /εAr-Ar

Fig. 10.2 Adsorption–desorption isotherms for a Lennard-Jones fluid at several temperatures in the homogeneous cylindrical pore, the geometrically undulated pore, and the chemically undulated pores. Schematic representations of the fluid state in the pores are given in circles

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Ŵ ∗ = 0.034 where it abruptly jumps to Ŵ ∗ = 0.64. The chemical potential was increased by small increments in this region of jump, which allows showing an almost vertical line. It is important to note that despite the presence of vertical lines (guides to the eyes), it was actually impossible to stabilize the reduced density to any intermediate value between Ŵ ∗ = 0.034 and 0.64 (absence of simulation points in Fig. 10.2 along the vertical lines). If the chemical potential is now decreased, one follows the T∗ = 0.6 desorption (liquid-like) branch, down to μ∗ = –10.36, that is to say, one does not follow the previous adsorption isotherm: the adsorption–desorption phenomenon is not reversible in our simulation box at such a low temperature and presents a hysteresis. As can be seen, the higher the temperature, the narrower the hysteresis loops. At T∗ = 1.0, the hysteresis is very thin, but the vertical jump is still visible. However, at T∗ = 1.2, the adsorption curve is perfectly reversible. This is in qualitative agreement with experimental observations [14, 15]. Below the so-called pseudo-critical temperature (by analogy with a first-order phase transition), the amount of adsorbed fluid is a multivalued function of μ. As can be seen, the simulation points group into branches. It has been checked that, along these branches, the adsorption–desorption is reversible. These branches, or pseudo-phases, are not true phases but deep local minima in the free energy landscape describing the system (the grand potential). The energetic barriers separating these minima are generally large compared to thermal fluctuations; the system is then trapped, which explains the reversibility of the adsorption–desorption along these branches. However, for some particular values of μ, the barriers may become small enough so that thermal fluctuations allow the system to jump into a deeper adjacent local minimum. These values define the limit of stability of the pseudophases. The vertical lines indicate the new local minimum reached by the system (see Fig. 10.2a). The introduction of a geometric undulation is a priori expected to strongly affect this set of adsorption isotherms due to confinement effects. However, inspection of Fig. 10.2b shows only small differences between the geometrically undulated pore and the perfectly cylindrical pore. The explanation is that in a realistic confinement situation, the external field is not simply deformed by the geometric undulation but is also affected in amplitude close to the spatial heterogeneity. Conversely, in our simulation study, the geometrically undulated pore is obtained by a purely geometrical deformation of the external potential, without changing the distribution of volume of space having a given value of external potential. The similarity of the adsorption isotherms shows, first, that the topologies of both pores are close enough not to introduce differences and, second, that the adsorption properties are not very sensitive to the spatial shape of the potential, but are rather related to the volume distribution of the external potential. However, there are some differences: the hysteresis loops are slightly narrower in geometrically undulated pores compared to perfectly cylindrical pore. This might be due to some destabilization of the adsorbed fluid due to geometric undulation (local curvature) which could help the nucleation of the new “phase.” On the other hand, the introduction of a chemical heterogeneity in the pore strongly affects the adsorption isotherms, as can be seen in Fig. 10.2c. First,

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quantitative changes can be noted: larger amount of adsorbed fluid for a given chemical potential, narrower hysteresis loops, and its disappearance at lower temperature (at T∗ = 1.0 the adsorption–desorption isotherm is reversible in the chemically undulated pore, while it presents a hysteresis in the perfectly cylindrical pore). Second, the isotherms are also qualitatively strongly modified, since now three branches (or “phases”) can be observed for the lowest temperatures. For instance, if one follows the isotherm at T∗ = 0.8, starting from low chemical potential, the average density jumps to an intermediate value of 0.28 for μ∗ = –10.96 before it again jumps to a liquid-like value of 0.53 for μ∗ = –10.89. If one decreases the chemical potential now, the average density will drop abruptly directly to the gas-like value for μ∗ = –11.00, without going through the “intermediate phase.” However, it is possible to jump from the “intermediate phase” to the gas-like phase by decreasing the chemical potential directly from μ∗ = –10.96 without passing by the liquid-like phase. The instability point in this case is at μ∗ = –10.97, that is to say, higher than the liquid instability point (μ∗ = –11.00), which explains why the liquid branch jumps directly to the gas branch when decreasing the chemical potential. This state of intermediate density has already been observed by other authors in strip-like geometry (slit pores) [94–97] and has been called “bridge phase.” It is essentially made of the accumulation of molecules in the attractive part of the pore (see next section). The T∗ = 0.9 isotherm does not exhibit three branches. Actually, the gas-like branch is reversible up to an average density corresponding to the intermediate “bridge phase.” However, the bridge-like to liquid-like “transition” is still visible with a small hysteresis loop for T∗ = 0.9. Let us now study the effect of introducing an extra chemical heterogeneity. The simple tubular model with four chemically different regions (pore chem II) exhibits several intermediate states in the adsorption–desorption isotherms and very interesting features. For instance, for a reduced temperature of 0.8, starting from the empty system and gradually increasing μ results in continuous filling of the system up to μ = –10.84 where the system jumps into the second local minimum. Increasing the chemical potential further makes the system jump successively into the third, fourth, and finally the fifth local minimum. Upon desorption, the system does not visit the local minima in the reverse order. After the first step, which consists in the formation of a bubble in the less attractive domain (i.e., the system jumps from the fifth into the fourth local minimum), further decrease in chemical potential makes the system jump directly into the first local minimum (gas-like branch) without visiting the third and second minima. This simple model shows that the hysteresis loop is related to a complex free energy landscape and may present a dissymmetry in adsorption and desorption. Furthermore, it is shown that the main adsorption and desorption isotherms are made of several branches with small discontinuities. As a consequence, the Gibbs equation ∂/∂μ = –N fails to give the grand potential  by simple integration of the amount adsorbed N along the isotherms. This model also gives a simple picture of adsorption–desorption subloops (scanning curves) as irreversible paths between the various local minima.

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10.4 Local Density Profiles Let us now give a molecular picture for these local minima. To do this, we study the density profiles of the confined fluid. These are measured through the averaged local density of the center of mass of the particles. Due to angular invariance, the local density is expected to be dependent on the radial and axial coordinates only: ρ(r, z). The result is averaged during the course of the simulation for each simulation box (pore cyl, geom, and chem I) and saved on a grid with 40 divisions in the radial direction (sharp variations) and 20 in the axial direction (smooth variations). The local density in the perfectly cylindrical pore (pore cyl) has been calculated for different temperatures. Figure 10.3 shows the result for a reduced temperature of 0.6 for the gas-like and liquid-like branches (only two branches, see Fig. 10.2a). As can be seen, the so-called gas-like state (Fig. 10.3, upper panel) is actually an adsorbed phase on the substrate plus a very dilute gas in the core of the pore. Since

gas state reduced density

0.06 0.05

perfectly cylindrical pore R* = 4.0 T* = 0.60

0.04 0.03 0.02 0.01 0.00 1

r*

liquid state reduced density

8.00

Fig. 10.3 Local density in the cylindrical pore as a function of the reduced radial and axial coordinates, at a reduced temperature of 0.6, for the gas-like state (upper panel) and liquid-like state (lower panel)

2

3

–4

0 –2

2

4

6

z*

perfectly cylindrical pore R* = 4.0 T* = 0.60

6.00

4.00

2.00

0.00 1

2

r*

3

–4

–2

0

2

4

z*

6

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both adsorbed and diluted components are intimately linked together, and there is no discontinuity between them (in density, for instance), they will be considered as being one gas-like state. Inspection of Fig. 10.3 shows a uniform density along the axial coordinate (within uncertainty), as expected since the potential has this translation invariance. The liquid-like state (Fig. 10.3, lower panel) also shows an adsorbed layer on the substrate and a dense liquid in the rest of the pore. Both components are again to be considered to belong to a unique “phase.” Due to the small size of the pore, layering effects are enhanced up to the fourth layer, which is actually a chain of molecules in the very center of the pore. The high intensity of the fourth layer peak located on the axis of the pore is not induced by a larger number of molecules in that region but by an extreme localization due to the commensurability between the pore diameter and the average distance between two particles at T∗ = 0.6. The molecules located in the other layers can move freely in the axial and angular direction (at constant distance from the center), whereas the central molecules can move freely only in the axial direction. Let us now focus on the density profiles in the geometrically undulated model (pore geom). As for the cylindrical pore, the local particle density is calculated and shown in Fig. 10.4 for the gas-like and the liquid-like states at a reduced temperature of 0.6. For both states, the adsorbed layer is now deformed, accommodating the substrate geometry. Despite the geometric deformation, the density peak intensity of the adsorbed layers (Fig. 10.4) is comparable to the previous cylindrical case (Fig. 10.3). The local density of fluid close to the wall is then directly related to the depth of the external potential, irrespective of the geometry of the surface, at least for large enough local curvature radius (several atomic diameters). On the other hand, the local density is not uniform any more along the undulated pore wall (or along the line of minimum potential). As a matter of fact, one can see short wavelength variations in the density. This can be explained by the lack of translation invariance along the pore axis. Due to geometric constriction, the adsorbed layer cannot move in one piece along the axis. It is anchored by geometric disorder to a minimum in energy, which induces localization points in the local density. Furthermore, the local layer structure of the liquid-like phase is much more affected due to size variations of the pore. The number of layers the fluid can accommodate in the pore varies along the z coordinate. This induces frustration in the fluid structure. If chemical heterogeneity is introduced, drastic changes are observed. As previously mentioned, three states are now possible below T∗ = 0.8. The local density profiles are shown in Fig. 10.5 for a reduced temperature of 0.75 where the three pseudo-phases (gas-like, liquid-like, and bridge-like) can coexist (see later). The gas-like state exhibits an adsorbed layer on the substrate. However, unlike the two previous cases, the adsorbed layer is not uniform. The density is modulated along the axial direction, according to the external potential, and the second layer is visible. Concerning the liquid state, one can see again the adsorbed layer and up to three subsequent layers. As for the adsorbed gas-like phase, the density is modulated along the z-direction, the modulation being larger close to the surface and disappearing in the center. The third state of intermediate density appearing for temperatures lower than 0.8 also exhibits layering effects and a strong modulation along the axial

Vapor–Liquid Equilibrium

gas state reduced density

10

0.06

227

geometrically undulated pore T = 0.60

0.05 0.04

6

0.03

4

0.02

2

0.01

0

0.00 1

z*

–2 –4

2

r*

3

liquid state reduced density

geometrically undulated pore T = 0.60

4.00 3.00 6 2.00

4 2

1.00 0 0.00 1

z*

–2 2

r*

–4 3

Fig. 10.4 Local density in the geometrically undulated pore as a function of the reduced radial and axial coordinates, at a reduced temperature of 0.6, for the gas-like state (upper panel) and liquid-like state (lower panel)

direction. In the regions of low potential, the density of the pseudo-phase is actually almost liquid-like, whereas in the regions of high potential, the density is very low (gas-like). This shows that the molecules are concentrated in a bridge anchored to the attractive sites of the substrate (bridge-like state). When several chemical heterogeneities are introduced, the number of pseudophases increases. A systematic study of density profiles for the pore chem II (four chemical heterogeneities) is not shown here due to the complexity of the graphs. However, analysis of the profiles shows that the gas-like state is made of an adsorbed layer at the wall, with density varying along the pore axis according to the external potential intensity. The second pseudo-phase (Γ around 0.2) is made of a liquid bridge anchored in the most attractive region of the pore, very similar to the previously described bridge state in pore model chem II. The third state (Γ around 0.3) is

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gas state reduced density

chemically undulated pore T = 0.75

2.00

1.00

bridge state reduced density

0.00

2

r*

6

z*

3

3.00

2.00

1.00

0.00

liquid state reduced density

1

2 0 –2 –4

4

1

2

2 0 –2 –4

r*

4

6

z*

3

3.00

2.00

1.00

0.00

Fig. 10.5 (continued)

1

2

0 –2 –4

r*

3

2

4

6

z*

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made of two liquid-like menisci in the two attractive regions of the pore. The fourth state is made of a single large meniscus bridging the two attractive domains. Finally, the fifth state is made of a liquid-like fluid completely filling the pore. Schematic representations of the local fluid density are given for all branches in Fig. 10.2, in the circles on the right hand side of the panels.

10.5 Coexistence Properties in Heterogeneous Tubular Pores: The Generalized Gibbs Ensemble 10.5.1 The Algorithm The grand canonical Monte Carlo method mimics the experimental determination of the adsorption–desorption isotherms since the temperature and the chemical potential are imposed. However, it is generally lengthy to acquire a complete set of isotherms, a requirement to calculate the complete hysteresis diagram of the confined fluid [32, 36, 40, 94, 119–121]. Nevertheless, obtaining such a diagram is essential from a theoretical point of view because it makes it possible to classify the behavior of the different systems and also from a practical point of view because the equilibrium or coexistence properties are needed in industry for phase separation purposes. Panagiotopoulos [122] proposed an entirely new simulation approach to determine directly such phase diagrams by working in the Gibbs ensemble (Gibbs ensemble Monte Carlo, GEMC), which consists in mechanically and chemically equilibrating the liquid and gas phases in a single simulation calculation [27, 122, 123]. However, this technique suffers one lack of generality since it demands that the space in which the system evolves contains at least one invariant direction to accommodate continuous variations of the volume. This is obviously the case not only for the empty space used to determine the equilibrium properties of bulk fluids (the initial aim of this technique) but also for perfect cylinders [27] or slit [37] pores, which enabled the use of this powerful technique for the determination of coexistence diagrams in such simples geometries. However, this technique has not yet been applied to realistic pores presenting some disorders at length scale from a few angstroms to a few nanometers, except in the special case of random porous media in the limit of very low density (a few percent) [124, 125]. The aim of this work is to propose a new extension of this technique to a new class of material shapes. This new extension of the Gibbs ensemble technique [126, 127] can handle any porous material which can be described by an external potential ψ pore (r, z), where ◭

Fig. 10.5 Local density in the chemically undulated pore as a function of the reduced radial and axial coordinates, at a reduced temperature of 0.75, for the gas-like state (upper panel), bridge-like state (central panel), and liquid-like state (lower panel)

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r and z are the radial and axial coordinates in a cylindrical representation of space. For comparison, in the same cylindrical representation of space (natural for a long pore), the original technique proposed by Panagiotopoulos could deal only with an external potential ψ pore (r). The new technique has been validated in the perfectly cylindrical pore. In the Gibbs ensemble Monte Carlo (GEMC) simulation technique [122, 123], the complete system is made of two parts, each describing a possible confined fluid state (for instance, one of the branches of the adsorption–desorption isotherms). This technique describes the possible coexistence between both fluid states, which means that they can exchange particles and volume. The usual GEMC method deals with systems presenting translation invariance, and volume variations associated with volume exchange are performed along this invariant direction. In our generalized method, tubular pores are considered, and each simulation box contains a dihedron representing the system. Volume exchange induces volume variations in each simulation box, which are performed by angular variations, like in isobaric–isothermal ensemble. Let us now consider two such systems in contact, able to exchange energy, particles, and volume through variations of their dihedral angle. The usual derivation for thermal, chemical, and mechanical equilibrium shows that the two subsystems tend to equilibrate their temperature by energy exchange; they equilibrate their chemical potential by particle exchange and their generalized pressure  by volume exchange. The numerical method we propose to calculate the phase coexistence will take these three equilibria into account, as in the Gibbs ensemble Monte Carlo method. Figure 10.6 shows the complete system consisting of two dihedra labeled 1 and 2. Both parts of the system are supposed to be in thermal contact with a reservoir at temperature T. It is also supposed that the two parts of the system can exchange particles and volume even though they are not in direct contact. In a real macroscopic system, both phases would be in contact at their interface, but the explicit introduction of such an interface (and finite size effects) is likely to be avoided in a simulation study of coexistence. The Monte Carlo acceptance probabilities associated with the trial moves corresponding to the different possible exchanges are given elsewhere [126, 127].

10.5.2 Simulation Results The complete coexistence diagram is obtained as follows. Starting from two initial configurations representative of two given pseudo-phases, the GEMC algorithm is run until equilibrium is reached. At low temperature, such equilibration was always possible in our system. The temperature is then increased, and the GEMC algorithm is run to reach a new equilibrium. This allows constructing the whole diagram up to singular points. Their nature is twofold. First, one has pseudo-critical points where both coexistence branches merge continuously (the energetic barrier vanishes and then the coexisting local minima tend to merge into a single one). Second, one has

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Fig. 10.6 Schematic representation of the elementary trial steps for the proposed generalized Gibbs ensemble Monte Carlo method (see text)

discontinuities in the branches, which correspond to the limit of stability of one of the two pseudo-phases. In this case, the unstable pseudo-phase jumps irreversibly into a third one, and the system reaches a new equilibrium. The consistency of the procedure is confirmed by the fact that the different branches of the coexistence diagram are reproducible with different initial conditions. For the perfectly cylindrical pore, for each reduced temperature, the points corresponding to the gas-like and liquid-like pseudo-phases have been reported on the so-called coexistence diagram (Fig. 10.7a, triangles). The bulk Lennard-Jones coexistence curve from Refs. [27] and [122] is also reported (crosses). As a general

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Fig. 10.7 Generalized GEMC coexistence diagrams for argon confined in the four-pore models of Fig. 10.1. The bulk phase diagram from Refs. [27, 122] is given for comparison. Each pseudo-phase is sketched at the bottom of the curves and characterized by a color. Identical symbols are used for coexisting pseudo-phases

1.4 1.2

bulk

gas

1.0

liquid

0.8 a) pore cyl 0.6 0.4 1.0

reduced temperature kT/εAr-Ar

0.8 0.6

b) pore geom

0.4

c) pore chem I

1.0 0.8 0.6 0.4

d) pore chem II

1.0

0.8

0.6

0.4 0.0

0.2

0.4

0.6

reduced density N σAr-Ar

0.8

3/V

comment, one can see that confinement strongly affects the thermodynamics of the adsorbed fluid. The gas-like phase is denser, due to adsorption. The liquid-like phase is less dense, partly due to hard core effect of the pore wall and accessible volume definition. And finally, and most importantly, the reduced pseudo-critical tempera-

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ture of the fluid is lowered by confinement effects (1.3 in bulk, 1.05 in cylindrical pore of reduced radius 4). The introduction of geometric deformations in pore geom is a priori expected to change the coexistence diagram. The results are given in Fig. 10.7b (squares). The comparison to the fluid behavior in the perfectly cylindrical pore (triangles) shows no difference. The geometric deformation up to 25% of the external potential does not change the thermodynamics of the adsorbed fluid, in partial contradiction with what is commonly admitted. It is however reminded that this pore does not take (on purpose) into account the possible modification of the external potential well due to real pore constriction. In the chemically disordered pore chem I, the external potential is obtained by a 25% modulation of the amplitude of the perfectly cylindrical potential. As previously, for each temperature, the total density is calculated and given as a function of temperature (coexistence diagram, shown in Fig. 10.7c, exhibiting three subdiagrams: gas–bridge: up triangles; bridge–liquid: down triangles; and gas–liquid: circles). As can be seen by comparison with the results in the perfectly cylindrical pore, the chemical undulation strongly modifies the thermodynamic behavior. A new state of intermediate density (around 0.3), the so-called bridge phase [94–97], has appeared, and consequently, two pseudo-critical temperatures are now visible associated with the two subdiagrams corresponding to gas–bridge and bridge–liquid coexistences. The intersection point is a triple point (coexistence of the three states). Contrary to the previous case, the coexistence may not be stable against perturbations. For instance, below the triple point, the gas–bridge and the bridge–liquid coexistences are metastable compared to the gas–liquid coexistence, because the bridge state is of higher grand potential than the corresponding gas and liquid states [70]. This means that these gas–bridge and bridge–liquid coexistences are possible, but limited to the bridge state lifetime. After a while or a perturbation, these coexistences turn into gas–liquid coexistence. On the contrary, above the triple point, the gas–liquid coexistence is metastable and splits into gas–bridge or bridge–liquid coexistence depending on the total average density of the system. Note that in this case, the nucleation barrier is not so high since a long Monte Carlo run allows to observe the breakup of the gas–liquid coexistence. The gas–liquid coexistence is then actually unstable above the intersection point (nucleation barrier very low). When increasing the number of chemical heterogeneities (pore chem II), the number of possible coexistences increases significantly (up to 10 coexistences in pore chem II, see Fig. 10.7d). Four of them end up in pseudo-critical points, while six are limited by the stability of one of the local minima. As can be seen, the external envelope of the diagram does not dramatically change with the number of heterogeneities. However, an increasing number of pseudo-phases of intermediate densities appear and progressively fill up the diagram, which drastically depart from the simple capillary condensation picture.

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10.6 Conclusion In conclusion, this molecular simulation approach has shown that the purely geometric deformations in tubular pores have weak consequences for fluid properties. On the contrary, modulation of the fluid–wall interaction intensity along the pore axis (due to chemical heterogeneities for instance) has strong consequences. Obviously, in real systems, pore constrictions are accompanied by modulation of fluid–wall interaction, which is responsible for the modification of adsorbed fluid properties. Considering the importance of the modulation of the fluid–wall interaction intensity, this study focuses on its effect on adsorbed fluid properties. Several local minima or pseudo-phases appear in the free energy, giving rise to complex adsorption and coexistence properties which drastically depart from the simple capillary condensation picture. Dissymmetric hysteresis loops are observed, showing subloops (scanning curves) which correspond to various paths in the free energy landscape. Adsorption or desorption is shown to proceed in several steps (jumps from one local minimum to another), questioning the validity of the thermodynamic integration procedure along the external envelope of the hysteresis. Concerning the coexistence diagram, it is shown that its envelope is not strongly affected by the chemical heterogeneities, whereas its inner part becomes rapidly more and more complex with an increasing number of pseudo-phases. Such heterogeneous cylindrical pores are expected to be realistic models for MCM-41 mesoporous silica. This work shows that in such a case, the adsorption–desorption hysteresis and the coexistence diagram are expected to be filled by local minima, in agreement with experimental observations. A theoretical coarse-grained approach developed by Kierlik et al. [15, 16, 45, 107] in disordered systems arrives at similar conclusions. Note that our work is complementary since it is based on a molecular approach and thus gives a molecular picture of the local minima. Furthermore, it focuses on realistic chemical heterogeneities disregarding interconnections and thus constitutes a good model to describe independent-pore systems like MCM-41 or porous silicon. Acknowledgments The simulations were performed locally on workstations purchased, thanks to grants from Région Centre (France). The Centre de Ressources Informatique de Haute Normandie (CRIHAN, St Etienne du Rouvray, France) and the Institut de Développement des Ressources en Informatique Scientifique (IDRIS-CNRS, Orsay, France) are gratefully acknowledged.

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126. Puibasset J (2005) Phase coexistence in heterogeneous porous media a new extension to Gibbs Ensemble Monte Carlo simulation method. J Chem Phys 122: 134710 127. Puibasset J (2005) Capillary condensation in a geometrically and a chemically heterogeneous pore a molecular simulation study. J Phys Chem B 109: 4700–4706

Chapter 11

Structuring and Behaviour of Water in Nanochannels and Confined Spaces Martin F. Chaplin

Abstract The structure of bulk liquid water is dominated by its ability to form networks of directed hydrogen bonds. Although this is also true for water in confined spaces, there are additional conflicting consequences of the extensive surface and the fit within the available space. A relatively large proportion of water molecules in confined spaces occupy the interface and their interactions with the cavity surface may govern their ability to form hydrogen-bonded networks with each other. The physical properties and state of the contained water may vary widely from its bulk properties and show great dependence on the molecular characteristics of the cavity surface and the degree of confinement, as well as temperature and pressure. Apparently small changes in the surfaces or the confinement dimensions may bring about substantial changes in these properties.

11.1 Introduction Nanoconfined water is found widespread in nature in granular and porous material and around and within macromolecules, supramolecular structures and gels, but it is only relatively recently that its structuring and properties has been subject to more thorough investigation. Cavities impose properties on liquids contained within them. They possess an extensive surface for their volume such that surface interactions may dominate to a wide-ranging extent. Even if such interactions are very weak, the properties of the contained material may still be dominated by the properties of the interfacial liquid due to its relatively large surface area. Larger volumes involve the interactions between this interfacial structure and the more central bulk liquid phase. The shape and diameter of the cavity determines the way in which contained molecules can interact with each other, particularly where the space is highly restrictive. Where the diameter is of the same order of magnitude as a single M.F. Chaplin (B) London South Bank University,Borough Road, London SE1 0AA e-mail: [email protected] L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_11,  C Springer Science+Business Media B.V. 2010

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molecule or just a few molecules, its dimension can be particularly influential and the structuring of the contained liquid tends to fluctuate widely with relatively small changes in the distance between surfaces or the physical conditions [1]. The presence of water as the liquid in the cavities adds considerable complexity to the system. Water structuring is driven by molecules associated through hydrogen bonding wherever possible. There is often conflict between orientational ordering (low enthalpy and entropy tending to produce low density or frozen water) and the bulk liquid state (higher enthalpy and entropy). The strength of interactions between the cavity surface and water may assist or oppose the hydrogen-bonding interactions between the water molecules within the surface layers. Additionally, variation of the spacing between any interaction sites on the cavity surface determines the ease for water–to-water binding to occur and its consequent binding energy. Thus, the distribution and the organization of the several layers of water molecules are affected by the surface. The adsorption of water to a bare surface involves continuous changes in its structuring as the surface becomes covered and further water layers are added. This can give rise to substantial hysteresis in structure and properties on its removal [2].

11.2 Bulk Properties of Water Although liquid water is a common and familiar liquid, its bulk properties are not predictable with confidence. Water is one of the smallest molecules known but often acts as though it is part of a loose network. Individual water molecules are held together by hydrogen bonding as variably sized dynamic clusters, which constantly rearrange, dissociate and associate. For neighbouring water molecules, there is an energetic conflict between anisotropic hydrogen bonding with its more negative enthalpy and low entropy and more isotropic van der Waals interactions with higher enthalpy and entropy. Mostly, hydrogen bonding prevails, at ambient temperatures, with the majority of the water molecules forming at least two hydrogen bonds and many forming four, approximately tetrahedrally arranged, hydrogen bonds [3, 4]. Close non-hydrogen-bonded water molecules have weak van der Waals interactions with each other but, importantly, may also interact with existing hydrogen bonds by forming bifurcated (shared) bonds. Such bonding may allow it to substitute for and replace the original hydrogen bond, thus releasing one of the molecules concerned (Fig. 11.1). This continual molecular “dance” reduces the lifetimes of individual hydrogen bonds to the order of picoseconds and results in their average strength being reduced. The hydrogen bonding between water molecules is cooperative with equal numbers of donor and acceptor hydrogen bonds being energetically preferred over the situation where there are different numbers of each. At low temperatures and, particularly when the density of water is also low, there are fewer water molecules free to form bifurcated bonds. Hydrogen bonds last longer under such conditions and appear to be stronger. This results in greater tetrahedral clustering, which may lead to an ice-like state.

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Fig. 11.1 (a) The approach of a non-bonded water molecule to a water molecule with four hydrogen bonds, (b) the formation of a bifurcated hydrogen bond and (c) the replacement of the original hydrogen-bonding partner by the incoming water molecule. This is an important route for the exchange of hydrogen bonding in the liquid phase but becomes less important when the water density is low or the hydrogen bonding is more ice-like

11.3 Interfacial Properties of Water 11.3.1 Interactions There are five types of interaction possible between water and a solid surface: electrostatic, dispersion, induction, repulsion and hydrogen bonding [5]. The first four are known as van der Waals forces and strongly affect the first layer of water in contact with the surface, with far less effect on more distant layers. Both strong electrostatics and hydrogen bonding can significantly affect the interactions and orientations between water molecules somewhat deeper towards the bulk. The electrostatics exerts an orienting force on the water molecules, which is often in competition with the hydrogen bonding. Hydrogen bonding also affects the molecular orientation but additionally can increase or reduce the strength of neighbouring hydrogen bonds by their cooperativity or anticooperativity [6]. Water surfaces of macrosized droplets or cavities have a charge distribution across them and there is no reason to suppose that surfaces around nanosized droplets or cavities do not have this property. Thus the gas–liquid surface of water is generally negatively charged above a pH of about 3 due to the absorption of hydroxide ions [7].

11.3.2 Non-polar Surfaces In the absence of polar or hydrogen-bonding interactions with the cavity surface, hydrogen bonding within the surface water is stronger than in the bulk [8] due to the reduced competition from neighbouring water molecules, lower anticooperativity and compensation for the increased chemical potential on the necessary loss of some bonding. Therefore, surface water has considerable structuring, decreasing the diffusion of some molecules whilst allowing other molecules on the very outside

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more freedom to move [9]. van der Waals interactions, between the cavity and aqueous surfaces, are individually weak but the total interaction may involve a number of atoms if the cavity is narrow or has pronounced curvature. The surface of liquid water will be similar to that at the liquid–gas interface except that confinement prevents movement of the water molecules towards the solid surface and van der Waals interactions loosely hold the outer molecules in preferred positions and orientations relative to the surface atoms.

11.3.3 Polar Surfaces and Hysteresis Polar and hydrogen-bonding interactions with the surface may vary between those that are very much weaker than typical water–water hydrogen bonds (which are ∼22 kJ mol−1 ) and those that are very much stronger. The organization and freedom of movement of the water molecules involved will vary accordingly. If the binding is strong, such as at a glass surface, then the enthalpy of adsorption will be high and the entropy of adsorption will be low. However, if the adsorbed layer consists of far fewer molecules than those required to form a monolayer, then the lack of sideways water–water interactions will lower the average enthalpy and raise the average entropy of adsorption, with water molecules skipping between binding positions. Surface interactions vary with the amount of water that is bound, both in two and subsequently three dimensions. Further hydrogen bonding to additional water will alter the energetics, geometry and orientation of the water molecules initially bound to the surfaces. Thus, if there are surface interactions, there will generally be a hysteresis in the absorption and desorption processes. Also, surface effects can propagate through several surface layers to affect relatively deeply held molecules. On adsorption to a clean surface, as the layer increases to a monolayer, water–water interactions become more important and add to the binding enthalpy of the firstbound water. Also, the clustering relaxes to form more favoured orientations and the molecules possess less entropy due to their reduced freedom of sideways movement. With further adsorption above the initial monolayer, water–water interactions become more important, eventually tending towards their domination over the bulk properties. When the reverse process occurs during drying, the water–water interactions dominate down to low surface coverage, and the relaxed three-dimensional, hydrogen-bonded network is harder to break than it was to form, giving rise to the noted hysteresis (Fig. 11.2). The thickness of the absorbed layer on polar surfaces depends on the surface interactions but has been estimated to be about two or three water layers using enthalpic determination or NMR and up to about six or so (1.5 nm) in many experimental and computer modelling studies [10]. Once surface water has adsorbed, cavities may fill with water due to the additional surface tension effects caused by the tight curvature of the surfaces and the resultant high Laplace pressure that this produces.

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Fig. 11.2 The typical hysteresis of adsorption–desorption, of liquid water from the gas phase, on polar cavities is shown. It is harder to remove the water once it has adsorbed

11.4 Effect of Confinement on the Properties of Water The most obvious property change from that of bulk water is that the confined water has a larger surface area for the number of water molecules present. As the surface properties of water are known to change within its interface and the interface extends a nanometre or so into the bulk, most of the water molecules in a nanosized droplet may be affected even without taking into consideration the more direct interactions with the surface of the containing cavity. Thus, the water molecules in cavities cannot be treated as though they were “bulk” water molecules since their properties are substantially modified. As the surface restricts molecular access, the outer water molecules are held by fewer hydrogen bonds to other water molecules. However, there is less chance that hydrogen-bonded water molecules are displaced by non-hydrogen-bonded water via bifurcation-promoted exchange due to the lower water density and anisotropy. The remaining hydrogen bonds therefore last longer. Their individual strength may be stronger because of this or, if the water molecules are isolated, may be weaker due to a lack of cooperativity with their neighbours. Water molecules entering a nanopore must lose some water–water hydrogen bonds but gain van der Waals, and possibly hydrogen bonding, interactions with the confining surface. The narrower the pores are, the more important these enthalpic effects become until there is the extreme case of a single chain of water molecules down a narrow pore (Fig. 11.3a). Also, in the absence of interactions with the

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Fig. 11.3 (a) A one-dimensional strand of water molecule within a (6,6) carbon nanotube of dia meter 0.81 nm. Each water molecule has two cooperative hydrogen bonds plus multiple van der Waals interactions from the carbon nanotube. Such organized water chains make rapid proton conductors due to the Grotthuss effect [58]. (b) A square ice nanotube formed within a (14,0) carbon nanotube of diameter 1.11 nm. Each water molecule has four cooperative hydrogen bonds, which are more bent than in hexagonal ice. They also have multiple van der Waals interactions from the carbon atoms

cavity surface, water molecules at the interface will gain rotational and, possibly, translational entropy. Generally the reduction in total hydrogen bonding inside a cavity gives rise to a lower dielectric but the lack of competing water molecules or environmental fluctuations causes a lower diffusion coefficient coupled with a higher viscosity. The enhanced strength of the hydrogen bonding lowers the density due to its

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tetrahedrality. The difficulty in finding space for crystallization lowers the nucleation temperature and this may give rise to cubic ice formation rather than the normal hexagonal ice as it has lower surface energy. Water can be supercooled, with care, down to about −38◦ C, where it exists as an extension of normal liquid water showing the non-Arrhenius viscosity behaviour of a so-called fragile liquid. Liquid supercooled water also exists between about 136 K and 160 K [11] showing the normal Arrhenius viscosity behaviour of a so-called strong liquid due to long-range density ordering. Normally, however, there is an inaccessible region between 160 K and 235 K where no bulk liquid water exists and where a fragile-to-strong transition is expected to occur. As confined systems can lead to freezing points below that achievable with bulk water, they have been used to investigate the properties of liquid water in this inaccessible region. Although such work is of great interest, it does, however, seem to be very unlikely that the behaviour of confined liquid water can be taken as necessarily representative of the behaviour of highly supercooled bulk water any more than the properties of the surface of water can be representative of those of bulk water.

11.5 Effect of the Surface on the Properties of the Confined Water 11.5.1 Interfacial Water Water molecules that lie next to the cavity surface are always affected by van der Waals interactions with it, in addition to possible hydrogen-bonding interactions dependent on the surface composition. Although non-polar van der Waals interactions are small, they may be multiple in narrow pores. For example, the water molecules in the one-dimensional chain shown in Fig. 11.3a have about 36 carbon atoms as nearest neighbours. Interfacial polar and/or hydrogen bond interactions may be weaker or stronger than water-to-water interactions, giving rise to a wide range of possible consequences depending not only on these interactions but also on their surface two-dimensional distribution in relation to possible surface and spatial organization of water molecules as governed by hydrogen bonding. This difference between various surface environments is fundamental to the biological activity of unique biomolecules such as the nucleic acids and proteins. For the most favourable interactions with tetrahedrally hydrogen-bonded water molecules, the surface should have not only similarly spaced polar interactions to the preferred water–water separation but also the ability to both donate and accept hydrogen bonds [12]. It may be, however, that the surface so encourages such hydrogen bonding that the hydrated surface becomes very hydrophobic as it has no capacity for further hydrogen bonding with a second aqueous layer. This occurs on Ru(0001) surfaces between 140 and 160 K, where the water molecules form a hexagonal hydrogen-bonded monolayer network with each water molecule donating a hydrogen to a metal atom [13].

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Water may be contained within well-established cavities, such as the spherical polyoxomolybdate cluster {(Mo)Mo5 }12 (Mo2 spacer)30 nanodrop, which may contain an almost perfectly tetrahedrally hydrogen-bonded cluster of a hundred water molecules [14]. The structure of the inner surface of this cavity can be varied to give different aqueous structuring of the contained water. Although normally the water molecules within the centre of confined droplets would be expected to have more hydrogen bonds and, therefore, have higher viscosity and diffuse more slowly, the hydrogen bonding within the surface layers may be disrupted and oriented such that there is a detachment of the absorbed layer from the innermost bulk-like water. Under these circumstances, the more central water molecules behave more similarly to water in non-polar cavities and are more able to readily diffuse relative to the surface water.

11.5.2 Hydrophobic Confinement Much theoretical and experimental work has concerned water in carbon nanotubes. These are molecules made up of a hexagonal lattice of carbon atoms rolled up into cylinders of diameters of a nanometre or so. Such structures are often thought of as hydrophobic cylinders but that assumption can underestimate the importance of the carbon atoms in the surface. These provide a high number of van der Waals interactions in an orderly hexagonal lattice arrangement that varies between the different types of nanotubes and with their diameters. Although uncharged, these surfaces provide extensive π-electron orbitals that can interact with and be polarized by the water molecules to cause effects significantly different from that expected of a flat and uniform hydrophobic cylinder. Small non-polar cavities may be filled with highly structured water, with the filling sensitive to the size of the cavity and the interactions with the cavity wall. In modelling studies, penetration of water into these pores is modulated by small changes in the polarity of the wall and the dimensions of the cavity [15, 16]. These properties also control the wet–dry transition [17] and the flow rate of the water through the carbon nanotubes’ cylindrical cavities [18]. The surface of pH-neutral water next to several hydrophobic surfaces, such as R and polystyrene, is negative like that at the water–gas interface. This charge Teflon has been attributed to the absorption of hydroxide ions, due to the orientation of the water molecules at the aqueous surface [19]. In a narrow cylindrical hydrophobic pore, there may be room only for a single chain of water molecules down the centre (Fig. 11.3a). Such a one-dimensionally ordered water molecule chain is somewhat self-aligned to give an almost all-trans zigzag arrangement. Under this highly constricted condition, there is less freedom of movement and the rotational and diffusional entropies return towards their bulk values [20]. In the one-dimensional strand formed within the (6,6) carbon nanotube, all the water molecules have two quite strong hydrogen bonds with the molecular configuration determined by the confinement rather than any interaction with the pore surface [21].

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As the radius of carbon nanotubes increases up to 1.1 nm, a second layer of water develops above the surface layer causing the interactions between the water molecules to become more dominant and the interactions of the water with the surface to become weaker [22]. The diffusion rate along the centre of carbon nanotubes wide enough to have several surface layers is greater than that in bulk water but diffusion in other directions is slower [23]. Recently, it has become clear that water can flow rapidly through some carbon nanotubes. This has been shown to be due to the depletion region at the tube–water interface and the smoothness of the confining walls. If the walls are rougher or possess a higher hydrophilicity, giving a stickier surface, such rapid flow is not seen [24]. Also, the effect is highly sensitive to the interatomic spacing in the tube surface [25], which slows down the flow rate if it coincides with the optimum hydrogen-bonded distance between water molecules. Water structuring tends to dominate the surface properties at more extensive hydrophobic surfaces with the presence of a layer of depleted density governing its high slip. The aqueous interface also possesses a negative charge [26] and a low viscosity. Nanobubbles are often found at such surfaces and these may appear to be unstable with a highly dynamic, if two dimensional, character [27]. Different studies give widely different values of the thickness of this density-depleted layer as between 0.2 and 5 nm. In some cases, nanobubbles have been found on hydrophobic surfaces up to 30 nm in height and 100–300 nm in diameter. They appear to be rather more stable than expected from consideration of the Laplace equation, which may be due to their surface charge preventing collapse or their flattened structure and hence greater radius of curvature, in turn due to van der Waals interactions across the low-density gap. Such nanobubbles have been shown to have some, but not total, dependence on the presence of dissolved gases.

11.5.3 Hydrophilic Confinement It has been known for some time that water will condense from the gas phase onto glass to give at least four layers of absorbed water [28]. Water next to silica is highly structured, dependent on the surface, and can be distinguished from more distant water by infrared [29] and X-ray [30] spectroscopy. Even the second neighbours to the surface are considerably affected due to the competition between the surface water and the effects of confinement [31]. When the confinement is less than about 2 nm wide, the viscosity may increase considerably to over a million times that of bulk water, but this viscosity increase is not noticed if the confining surfaces are hydrophobic [32, 33]. The surface hydroxyls on the silica pore walls hydrogen bond to the water molecules and control the distortion of the hydrogen bond network and its surface dynamics [34]. Water within the pore of alumina gel has been studied by high-resolution quasielastic neutron scattering technique at, and below, room temperature. Surface water is found to be only allowed local motion, with the remaining water still possessing lower translational diffusion than bulk water but unchanged rotational diffusion

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[35]. The wide variety of cages and channels within microporous aluminosilicates adsorb water via hydrogen-bonding and polar interactions that depend on the cations present and the charges on the surface atoms. Surface cations may shift their positions in response to changes in the water’s hydrogen-bonding network, resulting in their relaxation into more energetically favoured placements [36]. Neutron diffraction has been used to determine the three-dimensional structure of water in Vycor, a porous silica with pores of about 4 nm diameter. The number of hydrogen bonds per water molecule was found reduced to about 2.2, differing from about 3.6 in bulk water, and with the number of neighbouring water molecules also significantly reduced [37]. In addition, there is a contraction within the second shell of water molecules. When water is more strongly bound to silica surfaces, there is less difference between the zero-point energy of its isoptomers. This is the reason why there is somewhat less isotopic enrichment than might be expected of the surface water with respect to deuterated water [38]. As with narrow non-polar cylindrical pores, water molecules may form onedimensional chains in strongly polar pores. If the interfacial interactions are close in strength and spacing to bulk water interactions, the confined water behaviour may possibly behave similarly to bulk water. Thus, it is often assumed that the properties of water in polar cavities mimic those expected in bulk water. Although this assumption is not always justified, it has been of particular use when the conditions of temperature and pressure are beyond those reachable for bulk liquid water. Thus, supercooled heavy water (D2 O) contained in 1.5-nm, 1D cylindrical pores of the nanoporous silica gave a density minimum at 210 K [39]. Bulk supercooled water cannot be obtained under these conditions, but this value has proved likely to be a good indicator for the density minimum of supercooled bulk D2 O.

11.5.4 The Fragile-to-Strong Transition Many studies have investigated phase changes, fragile-to-strong transitions and glassification within the confined environment, often involving highly polar nanoporous silica and non-polar carbon nanotubes. MCM-41 mesoporous silica is manufactured with highly controlled cylindrical channels and very narrow pore size distributions. The surface interacts strongly with water [40], which is much distorted but still seems to show similarity to bulk water in its properties. The fragile-to-strong transition has been found to be at about 225 K in 1.4-nm cylindrical silica pores at ambient pressure [41] and at about this temperature in a fully hydrated Na vermiculite clay [42]. In double-walled, 1.6-nm-diameter carbon nanotubes, the fragileto-strong transition lies about 35 K lower than in the nanoporous silica [43]. Clearly the extra flexibility allowed by the small interactions at the non-polar surface of the carbon nanotubes prevents the fragile-to-strong transition unless the molecular movements are somewhat reduced at lower temperatures. Water adsorbed on Vycor silica glass has been studied by a combination of calorimetric, diffraction, high-resolution quasi-elastic and inelastic neutron scattering

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where it shows a glass transition at 165 K and a liquid–liquid transition, from a low-density to a high-density liquid, at 240 K [44]. This unusual behaviour, compared with bulk water, where the glass transition temperature is generally thought to be at 136 K and any liquid–liquid transition to be at much lower temperature and higher pressures, is thought due to less cooperative hydrogen bonding, caused by spatial restriction, weakening the hydrogen bond strength.

11.5.5 Phase Changes Confinement may cause raised or lowered melting points, depending on the surface and the space available. Generally, it might be expected that the raised boiling point and depressed freezing point may be inversely related to the capillary diameter due to the Gibbs–Thomson effect [45]. For example, cubic ice may form in 4.2-nmdiameter cylindrical silica pores at 232 K [46]. However, surface interactions and the confinement diameter may cause opposite effects, such as the ice that forms at 400 K in very narrow pores (0.6–1.0 nm diameter) in porous glass [47]. Water freezes when confined between flat surfaces when the water is able to form the optimally ordered tetrahedrally placed hydrogen bonds required. The confined water undergoes significant changes in density and viscosity as the surface separation is varied. For example in a model, water goes from liquid to solid and back again as the space decreases from 0.58 to 0.53 to 0.47 nm between interacting but non-hydrogenbonding surfaces [48]. If unable to form the required ordered structures, it may still undergo wide fluctuations in density as it attempts to optimize its structure within the restrictions of the confinement. If the confined water does not interact, or interacts quite weakly, with the surface, its entropy is raised relative to the bulk due to its extra degrees of rotational and translational freedom. The freezing point would be expected to be lower; for example, water forms a glass at 240 K plus some hexagonal ice at 243 K in 2-nm hydrophobic pores in activated carbon [49]. However, if it interacts strongly with the surface, its entropy is lower than that of bulk water and, therefore, the free energy difference between the liquid and solid phases is reduced and the freezing temperature should be higher. This difference may be counteracted if the surface interactions cause distortion to the hydrogen-bonded network which then cannot easily form crystalline ice. This occurs in 3-nm-diameter silica pores where glassification occurs at 193 K, whereas wider (10 nm diameter) pores allow the formation of cubic ice [49]. The disruptive ability of silica surfaces on ice formation has been investigated using NMR and differential scanning calorimetry on melting confined ice. Crystalline ice crystallites are found away from the surface but the water within about 0.5 nm from the surface remains liquid or glassy depending on the temperature [50]. A similar conclusion was reached using dielectric relaxation spectroscopy [51]. When ice does form at strongly binding or atomically rough confined surfaces, there may be no room for expansion and a high-density amorphous ice may result [52]. Different ices may form within interacting confined spaces, with a high-density confined ice (1.2 g cm−3 ) formed from two ice helices and crystallizing at ambient

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temperature in AlPO4 -5 zeolite 0.73-nm-diameter micropores [53]. Both cubic and amorphous ices can form at strongly binding silica surfaces at low temperatures [31]. However rather than forming ice within confinement, liquid water in confined spaces may leave the confinement on freezing, to crystallize freely external to the cavity, so long as they can easily be replaced by the gas phase under the ambient pressure [54]. Ice forming in regular but extremely confined non-interacting spaces may be unlike normal ice phases but determined by the confinement and the tendency of the water molecules to form four hydrogen bonds, even if required to be somewhat distorted. The form of the ice has a close dependence on the diameter of the nanotube and the pressure used but their hydrogen-bonding capacity is almost or completely satisfied, but with some of the hydrogen bonds somewhat bent. Four kinds of ice nanotubes have been found in carbon nanotube simulations at low temperatures (240 K). These consist of square (Fig. 11.3b), pentagonal, hexagonal and heptagonal nanotubes of water molecules, all with two donor and two acceptor hydrogen bonds, formed within carbon nanotubes of diameters increasing from 1.11 to 1.42 nm. Conversion between these ices can be achieved by increasing the pressure such that the square ice converts to the pentagonal ice at about 200 MPa at about 275 K [55]. Six phases of high-density ice have been reported in nanotubes of diameter 1.35–1.9 nm [56]. More recently, a phase diagram of water within carbon nanotubes has been proposed for tube diameters up to 1.7 nm. This suggests that there may be at least nine phases possible within the cylindrical space, including those found by X-ray diffraction and by simulation. The global maximum freezing point is found to be for “square” ice in 1.08-nm-diameter nanotubes at 290 K [57].

11.6 Conclusions The properties of water in confined spaces are determined by the conflict between its energetic minimization by forming hydrogen bonds to itself and its interactions with the cavity surface. The structure of bulk liquid water is dominated by its ability to form directed hydrogen bonds, but when confined, this must be tempered by the very large effect of the surface, whether interacting or not. The physical properties and state of the contained water may vary widely and show great dependence on the molecular characteristics of the cavity surface and the degree of confinement, as well as temperature and pressure. As such, the properties of the confined water are difficult to predict and may be very different from those of bulk water.

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30. Fouzri A, Dorbez-Sridi R, Oumezzine M et al. (2001) Water confined in silica gel at room temperature X-ray diffraction study. Int. J. Inorg. Mat. 3:1315–1317 31. Fouzri A, Dorbez-Sridi R, Missaoui A et al. (2002) Water–silica gel interactions X-ray diffraction study at room and low temperature. Biomol. Eng. 19:207–210 32. Goertz MP, Houston JE, Zhu X-Y (2007) Hydrophilicity and the viscosity of interfacial water. Langmuir 23:5491–5497 33. Li T-D, Gao J, Szoszkiewicz R et al. (2007) Structured and viscous water in subnanometer gaps. Phys. Rev. B 75:115415 34. Takahara S, Kittak S, Mori T et al. (2005) Neutron scattering study on dynamics of water molecules confined in MCM-41. Adsorption 11:479–483 35. Mukhopadhyay R, Mitra S, Pillai KT et al. (2002) Dynamics of confined water in porous alumina: neutron-scattering study. Appl. Phys. A 74:S1314–S1316 36. Crupi V, Majolino D, Longo F et al. (2006) FTIR/ATR study of water encapsulated in Na-A and Mg-exchanged A-zeolites. Vib. Spectrosc. 42:375–380 37. Thompson H, Soper AK, Ricci MA et al. (2007) The three dimensional structure of water confined in nanoporous vycor glass. J. Phys. Chem. B 111:5610–5620 38. Richard T, Mercury L, Massault M et al. (2007) Experimental study of D/H isotopic fractionation factor of water adsorbed on porous silica tubes. Geochim. Cosmochim. Acta 71: 1159–1169 39. Liu D, Zhang Y, Chen C-C et al. (2007) Observation of the density minimum in deeply supercooled confined water. Proc. Nat. Acad. Sci. USA 104:9570–9574 40. Yamaguchi T, Yoshida K, Smirnov P et al. (2007) Structure and dynamic properties of liquids confined in MCM-41 mesopores. Eur. Phys. J. Spec. Top. 141:19–27 41. Liu L, Chen S-H, Faraone A et al. (2005) Pressure dependence of fragile-to-strong transition and a possible second critical point in supercooled confined water. Phys. Rev. Lett. 95:117802 42. Swenson J, Bergman R, Longeville S (2002) Experimental support for a dynamic transition of confined water. J. Non-Cryst. Solids 307–310:573–578 43. Chu X-Q, Kolesnikov AI, Moravsky AP et al. (2007) Observation of a dynamic crossover in water confined in double-wall carbon nanotubes. Phys. Rev. E 76:021505 44. Zanotti J-M, Bellissent-Funel M-C, Kolesnikov AI (2007) Phase transitions of interfacial water at 165 and 240 K. Connections to bulk water physics and protein dynamics. Eur. Phys. J. Spec. Top. 141:227–233 45. Liu Z, Muldrew K, Wan RG et al. (2003) Measurement of freezing point depression of water in glass capillaries and the associated ice front shape. Phys. Rev. E 67:061602 46. Christenson HK (2001) Confinement effects on freezing and melting. J. Phys.: Condens. Matter 13:R95–R133 47. Venzel BI, Egorov EA, Zhizhenkov VV et al. (1985) Determination of the melting point of ice in porous glass in relation to the size of the pores. J. Eng. Phys. Thermophys. 48:346–350 48. Zangi R (2004) Water confined to a slab geometry: A review of recent computer simulation studies. J. Phys.: Condens. Matter 16:S5371–S5388 49. Yamaguchi T, Hashi H, Kittaka S (2006) X-ray diffraction study of water confined in activated carbon pores over a temperature range of 228–298 K. J. Mol. Liquids 129:57–62 50. Rault J, Neffati R, Judeinstein P (2003) Melting of ice in porous glass: why water and solvents confined in small pores do not crystallize? Eur. Phys. J. B 36:627–637 51. Sinha G, Leys J, Wübbenhorst M et al. (2007) Dielectric spectroscopy of water confined between Aerosil nanoparticles and in Vycor nanoporous glass. Int. J. Thermophys 28: 616–628

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52. Churaev NV, Setzer MJ, Kiseleva OA et al. (2007) On the thermodynamic equilibrium between ice and electrolyte solutions in the conditions of confined geometry. Colloids Surfaces A: Physicochem. Eng. Aspects 300:327–334 53. Floquet N, Coulomb JP, Dufau N et al. (2005) Confined water in mesoporous MCM-41 and nanoporous AlPO4 -5: structure and dynamics. Adsorption 11:139–144 54. Fan J-G, Zhao Y-P (2008) Freezing a water droplet on an aligned Si nanorod array substrate. Nanotechnology 19:155707 55. Koga K, Gao GT, Tanaka H et al. (2002) How does water freeze inside carbon nanotubes? Physica A 314:462–469 56. Bai J, Wang J, Zeng XC (2006) Multiwalled ice helixes and ice nanotubes. Proc. Nat. Acad. Sci. USA 103:19664–19667 57. Takaiwa D, Hatano I, Koga K et al. (2008) Phase diagram of water in carbon nanotubes. Proc. Nat. Acad. Sci. USA 105:39–43 58. Gileadi E, Kirowa-Eisner E (2006) Electrolytic conductivity—the hopping mechanism of the proton and beyond. Electrochim. Acta 51:6003–6011

Chapter 12

Freezing and Melting in Nanopores Kyunghee Lee, Guiduk Yu, Euntaek Woo, Soohwan Hwang, and Kyusoon Shin

Abstract In two-dimensionally confined spaces such as nanopores or nanotubes, freezing and melting occurs differently from the bulk phase transition. While bulk materials are crystallized mainly via crystal growth, crystallization in nanopores is dominated by nucleation, and the growth of the crystal is restricted due to the imposed spatial constraint. Different crystallization mechanisms result in different crystal structures and physical properties. Under nanoscopic cylindrical confinement, the crystals are oriented to a certain favorable direction with nucleation being dominant, and the crystal orientation can shift upon the variation of dominant crystallization mechanism. The melting temperatures (Tm ) are also influenced by the reduced dimension of crystal and interfacial interaction between crystallizable components and their environment in nanopores.

12.1 Introduction Materials, regardless of being organic or inorganic, exhibit size-dependent behavior on the nanoscale. The freezing/melting behavior, together with crystal orientation, is influenced by the imposed geometric constraint. Since the crystallization behavior determines several important properties like optical, electronic, and magnetic properties, studies in crystallization/melting under one-, two-, and three-dimensional confinements have been of importance both in scientific research and in industrial application. Freezing and melting behavior in nanopores can be determined by the degree of confinement, interfacial interaction between the crystalline material and the pore wall, and geometric anisotropy. As materials are confined to the limited space of the nanopores, crystal growth is restricted, and the crystallization process is more

K. Shin (B) School of Chemical and Biological Engineering, Seoul National University, Seoul, South Korea e-mail: [email protected] L.J. Dunne, G. Manos (eds.), Adsorption and Phase Behaviour in Nanochannels and Nanotubes, DOI 10.1007/978-90-481-2481-7_12,  C Springer Science+Business Media B.V. 2010

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affected by nucleation rather than growth. Moreover, the confined geometry affects the grain size as well as the orientation of crystallite. Especially, the decreased crystal size and increased interfacial area influence melting temperature. So far, metals and crystalline polymers in nanopores or crystallizable nanodomains in block copolymers have been mainly employed to investigate the influence of nanoscopic cylindrical confinement on crystallization kinetics, orientation, and melting behavior. The crystallization kinetics under nanoscopic cylindrical confinement is discussed in Section 12.2, and several results on the crystal orientation under the geometric constraint are given in Section 12.3. The melting behavior in nanopores is summarized in Section 12.4.

12.2 Crystallization Behavior in Nanopores 12.2.1 Nucleation-Dominant Crystallization Crystallization in nanopores usually occurs at much lower temperatures than in bulk. The crystallization behavior of polyethylene (PE) is affected by nanoscopic geometric confinement [1]. The Tc of PE in nanoscopic cylindrical pore drastically decreases as compared to that in bulk due to geometric restriction. It is intriguing that the enlargement of degree of supercooling is influenced by the degree of nanoconfinement. Crystallization in smaller pores takes place in a wider range of temperature, but still lower than that in bulk; crystallization in larger pore diameters occurs in a narrower range of Tc . Including crystalline polymers, the Tc s of most materials are observed to be lowered under nanoscopic confinement. However, few materials are reported to exhibit an exceptional behavior by staying in crystallized state at higher temperature than Tc in bulk [2–4]. Ar (argon) in narrow space, for instance, forms clusters (locally crystallized) above Tc of bulk (Tc = 83 K) due to the strong attractiveness between the adsorbate (Ar) and the pore wall. In this section, however, we will mainly discuss the general phenomena in T under nanoscopic confinement: the larger degree of supercooling. Polyvinylidene fluoride (PVDF) under nanoscopic cylindrical confinement also exhibits larger reduction in Tc [5]. The cooling thermograms (solid line) in Fig. 12.1(a) show that the degree of surpercooling of PVDF depends on the pore diameters. As the nanopore becomes smaller, Tc tends to decrease and the Tc range becomes broader. Figure 12.1(b) represents analogous behavior of linear PE in cylindrical nanopores with PVDF; crystallization occurs in lower, broader temperature ranges than in bulk (~117◦ C) [1]. Crystallization rate of PE as a function of the crystallization temperature (Tc ) or the degree of supercooling ( T) at a range of pore diameters is shown in Fig. 12.3(a). In addition to the decrease in Tc (or the increase in T), the remarkable variation in the slope of the plot of lnt1/2 versus Tc (or T) is also observed near d ~ 50 nm. The slope is related to the type of nucleation mechanism, and it will be covered in detail in Section 12.2.2. The crystallization of not only polymers but also metals is influenced by the geometric

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Fig. 12.1 The crystallization behavior of polymers and metals in cylindrical nanopores examined by thermal analysis. DSC cooling thermograms of (a) PVDF nanorods with d = 35 nm, 400 nm, and bulk PVDF (solid line: cooling curve; dashed line: heating curve) [5], Reprinted with permission from Physical Review Letters 97, 027801 (2006). Copyright 2006 American Physical Society. (b) linear PE in cylindrical nanopores with the diameters from 15 to 110 nm and in bulk [1], Reprinted with permission from Physical Review Letters 98, 136103 (2007). Copyright 2007 American Physical Society. (c) Bi nanorods and bulk Bi [6]. The numbers on the curves imply the diameters of nanorods Reprinted with permission from Chemical Physics Letters 444, 130 (2007). Copyright 2007 Elsevier.

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constraint. Figure 12.1(c) shows the cooling thermogram of Bi in various diameters of cylindrical nanopores [6]. As observed in crystalline polymers, the onset point of crystallization (Tc ) gets lower with the decrease in the pore diameter. Larger T (or the reduction of Tc ), the general behavior under nanoscopic confinement, is ascribed to the fact that nucleation is dominant and the growth of crystal is relatively suppressed. As materials are confined in nanopores, the crystal growth is mostly perturbed by the limited volume of the nanopores. Apart from the spatial restriction, the huge interfacial area per unit volume is also believed to result in nucleation-dominant crystallization. Variation in T on the basis of the geometric structure of materials, furthermore, provides the kinetic background of crystal formation. A more detailed discussion about crystallization mechanism, dependent on geometric condition, is given in the following section.

12.2.2 Alteration of Crystallization Kinetics Under Nanoconfinement As an effort to understand crystallization kinetics in detail in nanoscopically confined spaces, crystalline organic polymers, rather than inorganic materials, have been widely studied. Since crystallization of inorganic materials occurs too fast, investigation of isothermal crystallization behavior for those materials has been difficult. Literatures on detailed kinetics, therefore, mainly discuss polymers such as crystalline homopolymers or crystalline–amorphous block copolymers to comprehend the isothermal crystallization behavior [1, 7, 5, 8–23]. Experimental evidence supports that the crystallization of polymers under cylindrical confinement is mostly dominated by nucleation rather than crystal growth as described in Section 12.1. Also, the dominant nucleation is reported to alter the variation of the pore diameter. In order to discuss the crystallization mechanism, Avrami constants are often considered from the equation given by [24] 1 − Xc = exp (− Ktn )

(12.1)

where Xc is the relative crystallinity at a specific time t, and n and K are the Avrami constants. Crystalline–amorphous diblock copolymers, for example, have been widely used for studying the crystallization behavior under nanoscopic confinement. As a crystalline polymer is confined to the nanodomain of diblock copolymers, n is evaluated to be smaller than that in bulk. Above all, the crystallization behavior of crystalline nanodomain of sphere-forming block copolymer is usually considered to follow the first-order kinetics (n = 1) [16, 17]. The n is related to the type of crystal nucleation and crystal growth geometry. Since nuclei exist in each isolated nanodomain, and the growth of crystal is restricted within the nanodomain, it leads to the decrease of n. The overall crystallization mechanism is, therefore, dominated by nucleation due to spatial restriction on crystal growth. Similar to the observation in block copolymer, the isothermal crystallization kinetics of polyethylene (PE) in cylindrical nanopores is also analyzed with Avrami theory (Fig. 12.2(c)) [1]. The

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Fig. 12.2 Analysis of crystallization kinetics of polymers under nanoscopic confinement by Avrami theory. (a) Avrami constants of linear PE in nanopores and in bulk are obtained from isothermal crystallization experiment [1]. Reprinted with permission from Physical Review Letters 98, 136103 (2007). Copyright 2007 American Physical Society. Decrease in n and significant increase in K implicate that the overall crystallization is dominated by nucleation in nanoscopic cylindrical pore. By computer simulation, Avrami exponent n (b) and Avrami constant K (c) of a polymer in rigid nanopores are analyzed when the wall is slippery (•) or sticky (•) [25] Reprinted with permission from Soft Matter 4, 540 (2008)). Copyright 2008 Royal Society of Chemistry.

exponent n (n = 1.6–1.9) gets smaller than that in bulk (n = 2.4) with the decrease in pore diameter. It is striking that K is approximately six orders of magnitude larger in nanopores than that in bulk. Since the K is proportional to the concentration of nuclei and the nucleation rate, it is interpreted that crystallization in nanopore is governed by nucleation rather than crystal growth. Nucleation-dominant crystallization under nanoscopic confinement was also supported by molecular simulation studies. Figure 12.2(b) and (c) shows the crystallization kinetics of polymers simulated by Ma et al. [25]. In their simulation, the Avrami exponent n, coincidently with the experimental studies, decreases with the decrease in nanopore size; it is even closer to 1 in the smallest pore (first-order kinetics observed in sphere-forming diblock copolymer) (Fig. 12.2(b)). In addition to the

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effect of pore diameter, surface property and crystallization temperature exert influence over the Avrami constants n and K. Figure 12.2(c) exhibits that the Avrami constant K increases with the decrease in the pore diameter and decreases with the increase in temperature. However, the non-consistent behavior of n and K is observed above a certain temperature, and the extraordinary behavior is attributed to a transition in nucleation behavior and the preferential orientation in nanopores as temperature increases. Depending on the type of crystal nucleation, nucleation is classified into two different mechanisms: homogeneous nucleation and heterogeneous nucleation. Homogeneous nucleation is initiated without any assistance of foreign objects. On the contrary, heterogeneous nucleation is originated with the help of foreign objects. According to the classical homogeneous nucleation theory, the relation between t1/2 and Tc is described by [26] 2

ln t1/2

o 1 32γ 2 γe Tm = const.+ o 2 k( Hm ρc ) Tc T 2

(12.2)

where t1/2 is the time taken for crystallinity to reach half of the equilibrium value at each Tc . H◦ m is the bulk free enthalpy of the crystal and ρ c is density of the crystal. γ and γ e are the lateral and fold surface free energies of crystal, respectively. When t1/2 of PE in cylindrical nanopores is plotted as a function of 1/T T2 , the slope varies significantly near d ~ 50 nm in Fig. 12.3(b) [1]. The slope is observed to be steep in the larger nanopores (d > 50 nm) and gradual in the smaller nanopores (d < 50 nm). Especially, in larger nanopores (d ~ 62 nm and 110 nm), the surface energy products are calculated from the slopes to be 8.2 × 10–6 J3 /m6 and 6.0 × 10–6 J3 /m6 for d ~ 62 nm and d ~ 110 nm, respectively. The numerical analysis on crystallization kinetics can be compared to an isolated homogeneous nucleation system, polymer droplet, studied by Gornick et al. [9]. According to their study, the γ 2 γ e value of PE droplets estimated from isothermal analysis is explained by the fact that crystallization rate strongly depends on the crystallization temperature. Furthermore, the surface energy product of PE droplets is analogous to the values of PE in larger nanopores. The temperature-dependent steep slope observed in larger pores, therefore, is interpreted to follow homogeneous nucleation mechanism. On the contrary, the crystallization behavior in smaller nanopores (15–48 nm) shows much weaker temperature dependence. This sudden shift of crystallization behavior is an obvious indication that the dominant nucleation mechanism alters the diameter of nanopore (or variation of pore diameter). Figure 12.3(c) and (d) shows the schematic illustration of nucleation mechanisms in the different pore diameters: homogeneous nucleation and heterogeneous nucleation, respectively. In larger nanopores (d > 50 nm for PE), the nucleation is believed to originate from the density fluctuation (Fig. 12.3(b)). In the meantime, the formation of nuclei in smaller nanopores (d < 50 nm for PE) is conjectured to be influenced by pore wall, which suggests that heterogeneous nucleation is more dominant in smaller nanopores (Fig. 12.3(d)).

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Fig. 12.3 Alteration of nucleation mechanism of PE on the variation of the degree of confinement. (a) The reciprocal crystallization halftime (1/t1/2 ) as a function of Tc (lower axis) and T (upper axis) [1]. Reprinted with permission from Physical Review Letters 98, 136103 (2007). Copyright 2007 American Physical Society. (b) Crystallization halftime (t1/2 ) as a function of supercooling (1/Tc T2 ) with the pore diameters 15–110 nm [1]. The strong temperature dependence in larger nanopores (d > 50 nm) implies that homogeneous nucleation dominates over heterogeneous nucleation. The symbols in the upper right in (a) and (b) imply the diameters of nanopores. The schematic illustrations of homogeneous nucleation in larger nanopores (d > 50 nm) and heterogeneous nucleation in smaller nanopores (d < 50 nm) are given in (c) and (d), respectively. The crystallization in larger nanopores is conjectured to be mainly driven by the density fluctuation of the polymeric melt. In the meantime, the nucleation in smaller pores is thought to be mostly influenced by the pore wall surface

Also, calculations on critical nucleus size (l∗ ) give some more insight into the nucleation mechanism [26]. The dimension of critical nucleus is given by l∗ =

4γe Tm o Tρ

Hm c

(12.3)

The l∗ s of PE crystals in the larger nanopores (d > 50 nm) are calculated to be on the order of several nanometers. But, l∗ in the smaller nanopores (d < 50 nm) is evaluated to be smaller than 1 nm, which is not physically acceptable [1]. In other words, the l∗ in smaller nanopores is too small to be the nuclei originating from homogeneous nucleation; the classical homogeneous nucleation is no longer valid in smaller nanopores (d < 50 nm for PE). Therefore, heterogeneous nucleation, induced by pore wall surface, is regarded to dominate in smaller nanopores.

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12.3 Preferential Crystal Orientation in Nanopores In nanoscopic cylindrical pores, the crystal growth is restricted by geometric constraints, and the crystals are preferentially oriented along the pore axes. It is thought that two-dimensional confinement determines crystal orientation even though crystallization is dominated by nucleation. According to experimental observations, the orientation of crystallites in nanopores is affected by geometric confinement, attraction between the material and the pore, and experimental conditions such as Tc and the application of external field. In the following, we will discuss the crystal structures of small organic molecules, polymers, and metals in cylindrical nanopores. The crystal orientation of polyvinylidene fluoride (PVDF) was investigated by Steinhart et al., as shown in Fig. 12.4(a) [7]. The polymer chains in PVDF crystallite are perpendicularly oriented to the long axes of nanopores, which is the direction of the lowest curvature. The crystal orientation is also determined by kinetics selection mechanisms under two-dimensional confinement [5]. In a separated nanoconfined system, the PVDF crystals are randomly nucleated by homogeneous nucleation and subsequently grow in a certain direction due to the geometric restriction. The lamellae with direction are oriented to the statistically frequent direction while the orientation to other directions is limited (Fig. 12.4(b), left). When nanostructures are connected with PVDF bulk film, the PVDF crystals are thought to first grow with spherulitic form in bulk and the crystallites with direction are uniaxially aligned to the pore axis (Fig. 12.4(b), right). Similarly, it is reported that PE crystals are oriented in a certain favorable direction in cylindrical nanopores [23]. As shown in the ψ-dependent XRD patterns in Fig. 6.6 the b-axis of PE crystallite is oriented parallel to the long axis of pore. It is also found that the degree of orientation increases with the decrease in the pore diameter. Since severely confined geometry restricts the direction of crystal orientation, crystallites in smaller pores orient in a more anisotropic manner. The crystal orientation of small molecules is mostly investigated by computer simulations. In the case of medium-length n-alkanes, the crystal orientation is inflicted by the lattice structure of porous Si template [27]. As crystallographic direction of the pore axes corresponds to direction, n-alkane molecules are aligned perpendicular to the long axis of the pore. The crystallographic orientation relying on the lattice of porous material is also observed in linear 1-alcohol [28]. The 1-alcohol molecules are preferentially oriented perpendicular to the pore axes, which coincides with direction of porous Si. Particularly 1-alcohol molecules interact themselves via hydrogen bonding; therefore, the alcohol molecules, differently from n-alkanes, form tail-to-tail bilayered assembly in the radial direction. Under nanoscopic cylindrical confinement, the interfacial interaction between the crystallite and the pore wall can also be influential in the determination of the crystal orientation. Most of the studies regarding interfacial interaction are proposed by computer simulation. Figure 12.4(c) shows the change in the crystal orientation for polymer chains from parallel to perpendicular direction, depending on the surface property of the pore wall [25]. On a neutrally repulsive wall, namely slippery nanopores, the polymer chains (lattice units) are uniformly aligned parallel to the

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Increase ofTc Fig. 12.4 Preferential orientation of crystals along the pore axes. (a) The schematic crystal structure of PVDF in nanotubes [7]. Reprinted with permission from Macromolecues 36, 3646 (2003). Copyright 2003 American Chemical Society. (b) The direction of the crystallites is preferentially oriented parallel to the pore wall in separated nanopores (left); direction is favored in the nanostructures connected to the bulk (right) [5]. In separated nanopores, the growth in other directions (, l = 0) is statistically limited (middle). Reprinted with permission from Physical Review Letters 97, 027801 (2006). Copyright 2006 American Physical Society. (c) In a simulation study, the crystal orientation is shifted from parallel to perpendicular to the pore wall (from left to right) on the variation of the surface stickiness of the pore wall (from slippery to sticky) [25]. Reprinted with permission from Soft Matter 4, 540 (2008)). Copyright 2008 Royal Society of Chemistry. (d) The lamellae normal of PEO crystals in the nanodomains of PS-b-PEO diblock copolymer orient from random to perpendicular to the rod axis on the increase in Tc [22]. Reprinted with permission from Macromolecues 34, 6649 (2001). Copyright 2001 American Chemical Society.

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pore axes. The nucleation of the polymer chains, oriented parallel to the pore axes, is formed by the entropy effect under the confined environment; that is, the decrease in conformational entropy in confined space leads to the parallel orientation. The nucleation is initiated near the pore wall by the parallel-stretched chains; the parallel growth is dominant in slippery nanopores. By contrast, in the nanopores with sticky walls, the polymer chains orient perpendicular to the pore wall. The crystallization rate of perpendicularly oriented crystal is faster than that of parallelly oriented crystals due to the sticky property of the walls; retarding effect induces perpendicular orientation in sticky nanopores. The crystal orientation in nanopores can, therefore, change on the variation of the surface property of the pore wall. In addition, the favorable direction of crystal orientation under nanoconfinement can be altered by experimental conditions such as crystallization temperature or application of external field. In the case of cylinder-forming

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Fig. 12.5 The crystal orientation of Bi in cylindrical nanopores. (a) The schematic diagram of XRD setup [6]. The crystal orientation can be changed (b) by the application of electric field or (c) by thermal recrystallization. (d) Crystal orientation of Bi nanorods with different rod diameters examined by XRD from the setup of (a) [6]. The numbers above the XRD pattern imply nanorod diameters. (The first row: XRD patterns after electrodeposition. The second row: XRD patterns after recrystallization). Reprinted with permission from Chemical Physics Letters 444, 130 (2007). Copyright 2007 Elsevier.

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polystyrene-b-poly(ethylene oxide) (PS-b-PEO) diblock copolymer, the crystal orientation is dependent on the crystallization temperature [22]. Figure 12.4(d) shows self-organized orientation of PS-b-PEO with different crystallization temperatures. At low temperatures (Tc ≤ –30◦ C), the c-axes of the PEO crystals are randomly oriented under cylindrical nanoconfinement. Above Tc = –30◦ C, the c-axes of crystal orientation is rather tilted to the pore axis, â (the x-direction). With the further increase in Tc (Tc ≥ 2◦ C) the tilt angle also increases, and then the orientation of the c-axes PEO crystals gets more perpendicular to the pore axis. External field can also determine the preferential orientation. The orientation of Bi crystals is characterized by XRD [6]. The setup of XRD measurement to observe the crystal structure is given in Fig. 12.5(a). Parts (b) and (c) of Fig. 12.5 are the schematic illustrations of the crystal orientation of Bi as electrodeposited and recrystallized under twodimensional confinement, respectively. This difference in Bi crystal structures is verified by the XRD patterns. Figure 12.5(d) is the XRD patterns of Bi nanorods acquired by electrodeposition and recrystallization with various pore sizes. The distinctive difference in the orientation of Bi crystals between electrodeposited and recrystallized is the (012) diffraction plane. The (012) plane normal of Bi crystals under electric field is inclined to the direction of 40–45◦ from the pore axis (Fig. 12.5(b)) while the Bi crystals by recrystallization are oriented parallel to nanopores (Fig. 12.5(c)). The above results can be interpreted that the crystal orientation is governed by the crystallization mechanism: field-induced and nucleationdominant crystallization mechanisms. Furthermore, the degree of orientation for Bi crystals both as deposited and recrystallized (regardless of crystallization mechanism) increases on the decrease in pore diameter due to geometric confinement.

12.4 Melting Behavior in Nanopores 12.4.1 Melting Temperature Depression In previous sections, we dealt with the crystallization behavior and crystalline structure under nanoscopic cylindrical confinement. Since the crystal is formed mainly by nucleation in such a small space, the dimension of crystal is reduced relying on the degree of nanoscopic confinement. The decreased crystal size and accompanying increment of interfacial area result in the reduction of the melting temperature (Tm ). Gibbs–Thomson equation is widely employed to explain the melting point depression [29]. Gibbs–Thomson equation depending on the diameter of nanopores (d) is given by [30, 31] o o o − Tm = 4σ Tm /(d Hm ρc )

Tm = Tm

(12.4)

where T◦ m denotes the equilibrium melting temperature of the crystal with infinite sizes, ρ c is the density of the crystal, and H◦ m is the heat of fusion per unit mass. The degree of melting temperature depression ( Tm ) indicates the difference between the equilibrium melting point and the size-dependent melting point (Tm ).

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(a) (b)

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Fig. 12.6 Size-dependent Tm of the nanorods. (a) The degree of melting temperature depression ( Tm ) of organic molecules in nanopores is reciprocally proportional to the nanorod diameter for cis-decalin (◦), trans-decalin (•), cyclohexane (∇), benzene (), chlorobenzene ( ), naphthalene (), and heptanes () [30]. Reprinted with permission from Journal of Chemical Physics 93, 9002 (1990). Copyright 1990 American Institute of Physics. (b) and (c) Heating thermograms of PE [23] and Bi [6] in nanopores, respectively. (In (c), solid line: as-deposited Bi nanorods. Dotted line: thermally recrystallized Bi nanorods) Reprinted with permission from Macromolecues 40, 6617 (2007). Copyright 2007 American Chemical Society. Reprinted with permission from Chemical Physics Letters 444, 130 (2007). Copyright 2007 Elsevier.

Equation (12.4) predicts that Tm is inversely proportional to the pore diameter (d). Figure 12.6(a) shows the relation between the degree of melting point depression and the pore diameters for small organic molecules, i.e., cis-decalin, trans-decalin, cyclohexane, benzene, chlorobenzene, naphthalene, and heptanes [30]. With the decrease in pore diameters, the Tm s of small organic molecules decrease due to the reduction of small crystal size. In addition to small organic molecules, the sizedependent melting behavior is also experimentally observed in metallic nanowires under constrained environment [32–35]. The melting peaks of indium (In) [33], zinc (Zn) [34], and bismuth (Bi) [35] nanowires shift to the lower temperatures with the decrease in pore diameter. Normally in bulk, the melting peak appears to be narrower due to uniform crystal distribution. In the meantime, the lower and broader melting peak under nanoscopic cylindrical confinement supports that the crystal is

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smaller than bulk with broader size distribution due to random nucleation in the crystal formation. The depression of T m for semi-crystalline polymer (PE) in cylindrical nanopores is also analyzed with Gibbs–Thomson equation [23]. Assuming the rectangularshaped crystal, the Gibbs–Thomson equation can be written as

Tm =

o Tm

   2 σ1 σ2 σ3 1− + + o L1 L2 L3 ρc Hm

(12.5)

where σ 1 , σ 2 , and σ 3 are the specific surface energies of the crystallite, and L1 , L2 , and L3 are the dimensions of the crystallite. The subscripts stand for the three axial directions of the crystallites in lamella form. As the degree of confinement becomes more severe (decrease in the pore diameter from 220 to 15 nm), the melting peaks move to lower temperatures because of the reduction in crystallite size (Fig. 12.6(b)). Interestingly, two different peaks are observed for the crystallites grown in the pores of d ~ 30 nm. This result implicates that there exist different crystallites grown in different crystallization kinetics. According to the explanation about crystallization kinetics in nanopores (see Section 12.2.2), the dominant crystallization mechanism changes from homogeneous nucleation to heterogeneous nucleation with the decrease in the pore diameter. For this reason, different melting behaviors are anticipated to come up upon the alteration of the pore diameter. In addition to the degree of confinement, the melting behavior in nanopores is also affected by the crystal structure. As discussed in Section 12.3, Bi nanorods obtained from two different crystallization mechanisms, electrodeposition and recrystallization, exhibit differences in crystal orientation [6]. The heating thermograms of Bi nanorods are plotted in Fig. 12.6(c) where solid line and dotted line represent the heating scans for electrodeposited and recrystallized Bi nanorods, respectively. Obviously, the melting peaks tend to shift to lower temperatures with the decrease in the pore diameter. Furthermore, a slight difference in Tm is observed between electrodeposited (solid line) and recrystallized nanorods (dotted line); the Tm of recrystallized crystallites was slightly lower than that of electrodeposited ones. It can be attributed to the difference in the crystal size or interfacial energy originating from different orientations by the different crystallization conditions. Even though most research reported that the melting temperature is depressed, some reports suggest that the melting temperature can elevate with the decrease in the dimension of materials. The increase in melting temperature, for instance, is observed for palmitic acid (PA) confined in nanopores [36]. Due to the strong chemical affinity between PA molecules and the pore wall, the PA molecules are chemically adsorbed on the pore wall and subsequently trapped in nanopores with physisorption over the adsorbed PA molecules. Accordingly, the solid-like PA (the trapped PA) exists in cylindrical nanopores even above the bulk Tm .

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12.4.2 Size-Dependent Heat of Fusion The size-dependent melting behavior is studied about the heat of fusion ( Hm ) as well as the melting point (Tm ). The reduction of Hm is clearly observed in Sn spherical nanoparticles by Lai et al. [37]. They assumed that liquid surface layer exists on the surface of nanocrystals, and the surface layer was studied to behave differently from bulk-like core region in their melting. Based on the assumption, they explained the depression of H◦ m quantitatively. The size-dependent Hm for metal nanoparticles is expressed with the radius of nanoparticle (r) by  tl  3 o 1−

Hm = Hm r

(12.6)

where H◦ m is the heat of fusion in bulk and tl is the critical thickness of the liquid layer. The experimental observation of Hm as a function of the radius (r) of Sn is shown in Fig. 12.7(a). The Hm is remarkably decreased as the radius of nanoparticles decreases; Hm , hence, is linearly proportional to the reciprocal radius of nanoparticles. The Hm of small organic molecules confined in nanoscopic cylindrical pores is plotted in Fig. 12.7(b) [30]. As the pore diameter gets smaller, the Hm decreases in accordance with reduction in crystal size. Hui and coworkers [38] have investigated the melting behavior of metallic nanowires on the basis of computer simulation. The melting temperature analyzed by computer simulation is also predicted to decrease with the decrease in the diameter of metallic nanowires as in the experimental observation of metallic nanoparticles (Eq. (12.4)).

(a)

(b)

Fig. 12.7 Size-dependent heat of fusion of nanocrystals. The heat of fusion of (a) Sn spherical nanoparticles [37]. Reprinted with permission from Physical Review Letters 77, 99 (1996). Copyright 1996 American Physical Society. and (b) small organic molecules in porous glasses [30] is depressed as the size of nanocrystal becomes smaller. (cis-decalin (◦), trans-decalin (•), cyclohexane (∇), benzene (), chlorobenzene ( ), naphthalene (♦), and heptanes  in (b)). Reprinted with permission from Journal of Chemical Physics 93, 9002 (1990). Copyright 1990 American Institute of Physics.

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The decrease in heat of fusion due to nanoscopic cylindrical confinement also appears in crystalline polymers. The crystallinity of polymer is usually evaluated from the heat of fusion. The crystallinity X c is defined as Xc (%) =

Hm × 100 o

Hm

(12.7)

It is interesting that the crystallinity of linear PE in nanopores (from 15 to 110 nm in pore diameter) is less than 50% [23]. In d ~ 15 nm pore, the crystallinity of PE is even less than 30%, which is much smaller than that in bulk (71.6%). The decrease in crystallinity can be ascribed to the reduction of H m with the decrease in the crystal size. As the degree of confinement gets severe (the decrease in the pore diameter), the growth of crystals is, therefore, more perturbed by the limited volume and curved geometry, leading to reduction in crystallinity and Tm as well.

References 1. Woo E, Huh J, Jeong YG et al. (2007) From homogeneous nucleation to heterogeneous nucleation of chain molecules under nanoscopic cylindrical confinement. Phys Rev Lett 98:136103 2. Coasne B, Jain SK, Naamar L et al. (2007) Freezing of argon in ordered and disordered porous carbon. Phys Rev B 76:085416 3. Slovák J, Koga K, Tanaka H (1999) Confined water in hydrophobic nanopores: Dynamics of freezing into bilayer ice. Phys Rev E 60:5833–5840 4. Brovchenko I, Oleinikova A (2007) Water in nanopore: III. Surface phase transitions of water on hydrophilic surfaces. J Phys Chem C 111:15716–15725 5. Steinhart M, Göring P, Dernaika H et al. (2006) Coherent kinetic control over crystal orientation in macroscopic ensembles of polymer nanorods and nanotubes. Phys Rev Lett 97:027801 6. Noh KW, Woo E, Shin K (2007) Alteration of crystal structure of bismuth confined in cylindrical nanopores. Chem Phys Lett 444:130–134 7. Steinhart M, Senz S, Wehrspohn RB et al. (2003) Curvature-directed crystallization of poly(vinylidene difluoride) in nanotube walls. Macromolecules 36:3646–3651 8. Cormia RL, Price FP, Turnbull D (1962) Kinetics of crystal nucleation in polyethylene. J Chem Phys 37:1333–1340 9. Gornick F, Ross GS, and Frolen LJ (1967) Crystal nucleation in polyethylene: The droplet experiment. J Polym Sci Part C 18:79–91 10. Sommer J-U, Reiter G (2003) Polymer Crystallization. Springer, Heidelberg 11. Despotopoulou MM, Frank CW, Miller RD et al. (1996) Kinetics of chain organization in ultrathin poly(di-n-hexylsilane) films. Macromolecules 29:5797–5804 12. Sun Y-S, Chung T-M, Li Y-J et al. (2006) Crystalline polymers in nanoscale 1D spatial confinement. Macromolecules 39:5782–5788 13. Zhu L, Cheng SZD, Calhoun BH et al. (2000) Crystallization temperature-dependent crystal orientations within nanoscale confined lamellae of a self-assembled crystalline-amorphous diblock copolymer. J Am Chem Soc 122:5957–5967 14. Zhu L, Calhoun BH, Ge Q et al. (2001) Initial-stage growth controlled crystal orientations in nanoconfined lamellae of a self-assembled crystalline-amorphous diblock copolymer. Macromolecules 34:1244–1251 15. Hu Z, Baralia G, Bayot V et al. (2005) Nanoscale control of polymer crystallization by nanoimprint lithography. Nano Lett 5:1738–1743

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Chapter 13

Elasticity Theory for Graphene Membranes Juan Atalaya, Andreas Isacsson, Jari M. Kinaret, and Ener Salinas

Abstract Starting from an atomistic approach we have derived a hierarchy of successively more simplified continuum elasticity descriptions for modeling the mechanical properties of suspended graphene sheets. We find that already for deflections of the order of 0.5 Å a theory that correctly accounts for nonlinearities is necessary and that for many purposes a set of coupled Duffing-type equations may be used to accurately describe the dynamics of graphene membranes. The descriptions are validated by applying them to square graphene-based resonators with clamped edges and studying numerically their mechanical responses. Both static and dynamic responses are treated, and we find good agreement with recent experimental findings.

13.1 Introduction Single-walled carbon nanotubes which are a central theme in other chapters of this book are rolled-up graphene sheets whose properties we still need to understand. Recent progresses in fabricating graphene, a monolayer of graphite, have considerably boosted attention in this material [1]. Graphene is a genuine two-dimensional material composed of carbon atoms arranged in a honeycomb lattice [2]. Its exceptionally large mechanical strength [3], its ability to sustain large electrical currents, its linear dispersion relation at low energies, ballistic transport, etc. [4–5], have made graphene one of the hottest research topics in the field of solid state physics. Applications related to adsorption phenomena, such as the design of a cantilever based on the binding of molecules and atoms to a graphene sheet, would certainly be possible to investigate with the theory to be developed in this chapter. Yet, in another particular field, namely nanoelectromechanical systems (NEMS), graphene-based mechanical resonators were recently demonstrated [6–8], and theoretical work indicates a E. Salinas (B) London South Bank University, Borough Road, London SE1 0AA, UK e-mail: [email protected]

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strong coupling between deformation and intrinsic electronic properties of graphene [9–10]. Most research on graphene has hitherto focused on its electronic properties, and less attention has been directed to the mechanical properties. For modeling NEMS, a reliable and efficient description of the mechanical response of nanocarbons to external forces is essential [11–12]. While continuum elasticity theory has been applied successfully to the study of the mechanical properties of nanotubes for a long time [13–16], it has only recently been applied to graphene membranes [17–23]. However, a good understanding of the latter is very important for the development of graphene-based NEMS. In this work, we present an elasticity theory for graphene membranes. This theory predicts a nonlinear response of graphene when it has clamped boundaries as happens in drum-like graphene resonators. This nonlinear response has been observed experimentally [3, 6, 8]; thus, a nonlinear model is necessary to accurately model the mechanics of graphene in the context of NEMS applications.

13.2 Elasticity Theory for Graphene In this section we develop a general elasticity theory for graphene membranes, where large displacements are taken into account. In Section 13.2.1, we present a simple interatomic force field model for the potential energy of graphene sp2 bonds. This potential energy is then used to obtain the elastic energy for the graphene seen as a continuous medium. In Section 13.2.2, we summarize some geometrical relations of two-dimensional manifolds which will be used later for the formulation of the elasticity theory. In Section 13.2.3, we formulate the general elasticity equations for graphene membranes.

13.2.1 Interatomic Force Field Model for Graphene sp2 Bonds Graphene is a two-dimensional crystal of carbon atoms which are bonded by sp2 hybridized bonds. In this type of hybridization each carbon atom, in the absence of defects, has three nearest neighbors. The most accurate description of the bond deformations can be obtained from ab initio calculations such as density functional theory. However, the computational effort involved in these fundamental calculations scales exponentially with the size of the problem and typically we are not able to handle relevant sizes (1 µm) of graphene sheets with several millions of atoms. An alternative approach is to use an empirical formula for the potential energy of these bonds and then use a mechanistic approach to find the trajectory or position of each atom. This approach is called Molecular Dynamics (MD) [20]. The models used to evaluate the potential energy are called interatomic force field models or valence force field models. These models contain terms for the stretching and bending energies of the bonds. If we are interested in deformations which are much longer than the lattice constant or the characteristic distance between atoms (the

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long-wavelength limit), we can replace the atomic layout by a continuous medium. The theory that describes the mechanics of this medium is called elasticity theory. Elasticity theory is a further improvement in the modeling of graphene and other nanostructures because it allows to model relevant size structures which may be cumbersome to study with atomistic formulations. Here we base the continuum elasticity theory on a simple valence force field model which was first formulated by Keating [24–25] for semiconductors and then extended to graphene by Lobo [26]. The potential energy for graphene bond deformation is given by

Usp2

Nat  β Nat  α  2  1 2 + r ¯ = · r ¯ − a ij ij 0 2 2 i=1 j 4a20 i=1 j