molecular dynamics studies of simple sorbates in …

MOLECULAR DYNAMICS STUDIESOF SIMPLE SORBATES IN ZEOLITES

SUDESHNA KAR

DEPARTMENT OF CHEMISTRY

INDIAN INSTITUTE OF TECHNOLOGY, DELHI

INDIA

AUGUST 2001

c© Indian Institute of Technology New Delhi - 2001.

All rights reserved.

MOLECULAR DYNAMICS STUDIESOF SIMPLE SORBATES IN ZEOLITES

BY

SUDESHNA KAR

DEPARTMENT OF CHEMISTRY

Submitted

in fulfillment of the requirements of the degree of

Doctor of Philosophy

to the

INDIAN INSTITUTE OF TECHNOLOGY, DELHI

INDIA

AUGUST 2001

Certificate

This is to certify that the thesis titled “Molecular Dynamics Studies of Simple

Sorbates in Zeolites” being submitted by Sudeshna Kar to the Department of Chemistry,

Indian Institute of Technology, Delhi, for the award of the degree of Doctor of Philosophy, is

a record of bona-fide research work carried out by her under my guidance and supervision. In

my opinion, the thesis has reached the standards fulfilling the requirements of the regulations

relating to the degree.

The results contained in this thesis have not been submitted to any other university or

institute for the award of any degree or diploma.

Dr. C. ChakravartyAssistant Professor

Department of ChemistryIndian Institute of Technology

New Delhi - 110 016

i

To my mother

iii

Acknowledgements

A student owes a lot to his/her teachers. I would like to express my deep sense of gratitude

and respect to all my teachers, past and present, who have contributed immensely at various

stages of my life. This is a humble effort on my part to thank them all and specially my

supervisor, Dr Charusita Chakravarty, with whom I had the good fortune to work

with and gain a lot. Her untiring effort, valuable advice and guidance, deep insight into

problems, imaginative ideas, constant encouragement, constructive criticism and zeal for

work has always inspired and motivated me. I will cherish this association for the rest of

my life.

I would like to thank Prof. M.N.Gupta, Head, Department of Chemistry for providing

me with the necessary facilities. I am grateful to Dr A.Ramanan, Dr A.K.Ganguly,

Dr N.D.Kurur, Prof. B. Jayaram, Dr D. Bandyopadhyay, Dr R. Chatterjee and

Prof. S. Banerjee for their helpful suggestions and advice.

I am grateful to my labmates: Pooja, who was always there when I needed her and

Anirban, whose presence always brought a smile to my face.

I express deep sense of gratitude to my dear friend Shweta for her selfless support and

constant company through the highs and lows of my stay at IITD. We really had good time

together. I will miss you!!

I would also like to thank my friends who have made my stay at IIT-D memorable-

Arindam, Sanchita, Subroto, Sampriya, Jagat, Rupali, Gunjan, Ruchi and Preeti

. I am also immensely thankful to Achintya for extending help whenever I asked for

it. It is indeed my pleasure to acknowledge the help, suggestions and advice of Urvashi

v

Bharghava(M.Tech), Uma. B.(M.Tech) and Dr Sumantra Dutta Roy; Department of

Computer Science, IIT-Delhi. I am indebted to my friends at IIT-K specially Dr Biswajit

Maity, Snehashish Chowdhury and Sandip Paul for their kind hospitality and help

during my stay at IIT-K.

I am extremely grateful to Prof. R. Ramaswamy, Dean, School of Physical Sciences,

JNU for helping me in many ways. The interactions with his research group helped a lot

in looking at my work from new perspectives. I also take this opportunity to thank JNU

authorities for allowing me to use their library facilities which has proved invaluable to me

many times.

I am indebted to Dr J.K.Deb(Kaku), Kakima, Mishmi and Ankur for giving me

a ‘home away from home’ and patiently listening and solving many of my problems. I will

always remember the time we spent together. I am also grateful to Prof. B.P.Pal and his

family for many suggestions and advice.

I would like to thank Indian National Science Academy and Indian Institute of

Technology, Delhi for providing financial support.

Words fail me in expressing my deep gratitude towards all my family members for

constantly encouraging and supporting me. I respectfully remember Dadubhai, Dida,

Behalar Dadu, Baba and Mesho who have always been a source of inspiration to me.

It is a futile attempt on my part to express my deep love and respect for my mother, Mrs

Rita Kar, to whom I owe everything I have achieved today. This thesis is dedicated to her.

I am also immensely grateful to my husband, Dr Manas Kumar Ghosh and my sister,

Debolina Kar, for always being there and for making everything so easy and comfortable

for me while taking all the trouble themselves. I really treasure you!!

New Delhi Sudeshna Kar

vi

Abstract

Zeolites are three-dimensional, microporous, inorganic crystalline polymers which can acco-

modate mobile cations and neutral molecules in the pore spaces. They form a very important

category of industrial adsorbents and shape-selective molecular sieves. The physisorption

properties of simple atomic and molecular sorbates in zeolites can be understood on the

basis of just two parameters, their van der Waal radius and polarizability. In this thesis,

microcanonical ensemble molecular dynamics simulation has been used to explore several

aspects of sorption and diffusion of such simple Lennard-Jones sorbates in zeolites. The aim

has been to show how computer simulations of such simple systems can provide considerable

insight into unusual diffusional properties that emerge as a consequence of confinement of

fluids in zeolites. In addition to adding to our understanding of diffusion phenomenon, such

simulations illustrate methodological issues of relevance to simulations of fluids in porous

media.

The first chapter contains an introduction to the field of zeolites. Structural features

of zeolites from the perspective of their molecular sieve properties are reviewed. Some key

conceptual aspects of sorption and diffusion of molecular sorbates in zeolites are discussed

followed by a brief review of the recent developments in computer simulations of sorption

and diffusion in zeolites. The final section of the chapter contains the scope and chapter

plan of the thesis.

The primary input in any computer simulation is the potential energy surface (PES).

vii

viii Abstract

The parametrisation of potential energy surface for physisorbed molecules in zeolites is

discussed in Chapter 2. The exact parametric form of the PES used for the simple sorbates

studied in this work and its range of applicability is described in detail.

Chapter 3 discusses the computational techniques used in this work. The primary

simulation technique, molecular dynamics (MD), is described with reference to the MD

algorithm as implemented in my simulations. The calculation of the various quantities

related to diffusional behaviour are discussed. A discussion of the estimation of chemical

potential by the Widom particle insertion method is also given. Instantaneous normal mode

analysis and its usefulness as an indicator of short-time dynamical information is reviewed.

Chapter 4 applies instantaneous normal mode analysis to understand the correlated

changes in diffusivity and in the PES that occur as a result of variations in sorbate size.

From simple geometrical considerations, it would appear that the self-diffusion coefficient,

D, will be maximum for very small sorbates and will decrease with increasing sorbates

size. However, under certain circumstances, a nonmonotonic increase in D with increasing

sorbate size is observed for sizes close to the minimum channel width. This anomalous

peak in transport properties is termed the levitation effect. In this chapter we perform in-

stantaneous normal mode analysis, in conjunction with MD simulations, for Lennard-Jones

sorbates of variable size and polarizability diffusing in NaY zeolite. The size-dependence

of the diffusivity, including the anomalous levitation peak, is reflected in various proper-

ties of the instantaneous normal mode spectrum, such as the fraction of imaginary modes

and the Einstein frequency. The existence of clear signatures of the levitation effect in the

instantaneous normal mode properties indicates a close connection between the anomalous

diffusivity peak and the curvature distribution of the potential energy surface. My work

shows that an INM analysis provides a very simple simulation test for the levitation ef-

fect since the anomalous peak in the diffusivity is strongly correlated with the fraction of

imaginary frequency modes and the mean curvature of the potential energy surface. This

Abstract ix

work on instantaneous normal mode analysis of the levitation effect leads one to expect

that the INM spectra can be used provide important clues to qualitative changes in diffu-

sional dynamics of sorbates in porous media, specially when the system dynamics is largely

controlled by the topography of the potential energy surface.

Our results on estimation of Henry’s constants and isosteric heats of sorption at infinite

dilution for Lennard-Jones sorbates in zeolites are presented in Chapter 5. Henry’s constants

are obtained by an integration method which corresponds to the infinite dilution limit of the

Widom particle insertion method for determining the chemical potential. Isosteric heats

of sorption at infinite dilution are calculated using MD simulations as well as from the

temperature dependence of the Henry’s constant. The systems studied are: (i) CH4 in

silicalite (ii) Ar, Kr and Xe in Na-Y and (iii) variable size Lennard-Jones sorbates in Na-

Y. The first two sets of sorbate-zeolite systems provide a way to test the accuracy of the

potential energy surfaces used in this thesis in predicting sorption and related properties.

The third set of simulations highlights an important difference between molecular dynamics

simulations and insertion techniques when applied to adsorbates in porous media since the

MD techniques sample the dynamically connected pore space whereas insertion techniques

sample all available, low potential energy pore regions.

Diffusional anisotropy is a phenomenon which is characteristic of sorbates in porous crys-

taline solids because the structural properties of confining solids are direction-dependent.

This is expected to result in inequality of the three principal components of the diffusion

tensor. A more unusual effect is an interdependence of the components of the diffusion ten-

sor due to geometrical correlations which arise as a result of special features of the channel

network and have been studied in detail in ZSM-5. I have studied the diffusional anisotropy

of several Lennard-Jones sorbates in all-silica compositional variants of three zeolites: ZSM-

5, ZSM-11 and ferrierite. All of them are low porosity zeolites but only ZSM-5 and ZSM-11

can show diffusion along the z-direction due to correlated motions along the x- and y- di-

x Abstract

rections. The most unexpected result that emerges from my study of diffusional anisotropy

in ZSM-5 and ZSM-11 is the very slow crossover from ballistic to diffusional motion and the

existence of a protracted period of subdiffusional motion along the correlated z-direction.

The results also suggest that the extent of subdiffusional behaviour will be partly deter-

mined by the size and polarizability of the sorbate, with helium showing the least tendency

to subdiffusional behaviour. An important methodological point which emerges from our

study of diffusional anisotropy is the importance of checking the applicability of the Einstein

relations and the possibility of deviations from Fickian behaviour when fluids are confined

in nanoporous media.

Chapter 7 highlights the various results and conclusions arrived at in the chapters 4 to

6 and outlines the implications for future computational and experimantal work.

Contents

1 Introduction 1

1.1 Structures of Zeolites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Chemical Compositions of Zeolites and Zeolite-type materials . . . . 4

1.1.2 Aspects of Zeolite Structure relevant to Sorption and Molecular Sieve

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.3 Examples of Zeolite Structures . . . . . . . . . . . . . . . . . . . . . 8

1.2 Sorption and Diffusion in Zeolites . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Sorption Isotherms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.2 Transport and Self-Diffusivities . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Effect of Anisotropic Porous Media on Diffusivity . . . . . . . . . . . 15

1.2.4 Molecular Picture of Diffusion . . . . . . . . . . . . . . . . . . . . . . 15

1.2.5 Length and Time Scales for Diffusional Processes . . . . . . . . . . . 17

1.3 Computer Simulations of Sorbates in Zeolites . . . . . . . . . . . . . . . . . 17

1.3.1 Potential Energy Surfaces and Simulation Methods . . . . . . . . . . 19

1.3.2 Computer Simulations of Sorption Properties . . . . . . . . . . . . . 21

1.3.3 Computer Simulations of Sorbate Diffusion . . . . . . . . . . . . . . 22

1.4 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Potential Energy Surface 35

1

2.1 Interactions between Lattice Atoms . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Sorbate-Sorbate Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.1 Rare Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.2 Alkanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.3 Aromatic Hydrocarbons . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Sorbate-Zeolite Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 Potential Energy Surface of Lennard-Jones Sorbates in Zeolites . . . . . . . 45

2.4.1 Lennard-Jones Sorbates . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4.2 Rigid Lattice Approximation . . . . . . . . . . . . . . . . . . . . . . 46

2.4.3 Neglect of Electrostatic multipolar and Induction Interactions . . . . 47

2.4.4 Potential Energy Function . . . . . . . . . . . . . . . . . . . . . . . . 48

3 Computational Methods 49

3.1 Molecular Dynamics Simulations of Lennard-Jones Sorbates in Zeolites . . . 50

3.1.1 Integration of Newton’s Laws of Motion . . . . . . . . . . . . . . . . 50

3.1.2 Potential Energy Surface and Force Calculation . . . . . . . . . . . . 54

3.1.3 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 55

3.1.4 Choice of Input Configurations and Velocities . . . . . . . . . . . . . 57

3.1.5 Choice of the Time-step . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.1.6 Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.1.7 Production Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Estimation of Diffusional Properties . . . . . . . . . . . . . . . . . . . . . . 62

3.2.1 Ballistic to Diffusional Crossover . . . . . . . . . . . . . . . . . . . . 63

3.2.2 Diffusion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.3 Time-correlation Functions . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 The Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Instantaneous Normal Mode Analysis . . . . . . . . . . . . . . . . . . . . . 70

2

3.4.1 INM Analysis of Liquid State Dynamics . . . . . . . . . . . . . . . . 70

3.4.2 Significant features of the INM Spectra of Atomic fluids . . . . . . . 72

3.4.3 Extension of INM Analysis to Fluids Adsorbed in Zeolites . . . . . . 76

4 The Levitation Effect 81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.1 Zeolite Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.2 Potential Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.3 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . 83

4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.1 Signatures of the Levitation Peak in the INM Spectra . . . . . . . . 86

4.3.2 INM Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.3 Velocity Autocorrelation Function . . . . . . . . . . . . . . . . . . . 94

4.3.4 Sorbate Polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3.5 Ballistic to Diffusional Crossover . . . . . . . . . . . . . . . . . . . . 97

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Estimation of Henry’s constant 103

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2 Potential Energy Surface and Computational Details . . . . . . . . . . . . . 110

5.2.1 Zeolite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2.2 Potential Energy Surface . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.3 Molecular Dynamics Simulations . . . . . . . . . . . . . . . . . . . . 113

5.2.4 Evaluation of Henry’s constants by the Integration Method . . . . . 113

5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.3.1 Methane in ZSM-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3

5.3.2 Lennard-Jones Sorbates in Na-Y Zeolite . . . . . . . . . . . . . . . . 116

5.3.3 Rare Gases in Na-Y Zeolite . . . . . . . . . . . . . . . . . . . . . . . 121

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6 Diffusional Anisotropy 125

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2.1 Zeolite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2.2 Potential Energy Surface . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2.3 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3 Sorbates in ZSM-5: Results and Discussion . . . . . . . . . . . . . . . . . . 140

6.3.1 Diffusional and Ballistic Regimes . . . . . . . . . . . . . . . . . . . . 140

6.3.2 The Velocity Autocorrelation Function . . . . . . . . . . . . . . . . . 147

6.3.3 Instantaneous Normal Mode Analysis . . . . . . . . . . . . . . . . . 149

6.4 Comparison of Sorbate Diffusion in ZSM-5, ZSM-11 and Ferrierite: Results

and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7 Conclusions 157

4

List of Figures

1.1 Structure of Zeolite A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2 Structure of Zeolite Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Structure of Ferrierite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Structure of ZSM-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.5 Structure of ZSM-11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.6 Structure of Theta-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1 Crossover from ballistic to diffusional motion. . . . . . . . . . . . . . . . . . 85

4.2 Variation of (a) self-diffusivity,D, and (b) fraction of imaginary modes,Fimag,

with 1/σ2SS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3 Variation of (a) guest-host potential energy, 〈Ugh〉, and (b) frequency, ω, with

1/σ2SS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4 INM spectra in the linear and anomalous regimes. . . . . . . . . . . . . . . 91

4.5 Schematic diagram showing the interactions of sorbate atoms of different

sizes with the oxygen atoms of the 12-ring window of Na-Y zeolite. . . . . . 92

4.6 Normal mode frequencies of a single Lennard-Jones sorbate molecule located

in the centre of a 12-ring window of zeolite Na-Y. . . . . . . . . . . . . . . . 93

4.7 Comparison of velocity autocorrelation function using MD and INM ap-

proaches for three sorbate sizes. . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.8 Variation of (a) τn and (b) τm, obtained from INM and MD, with 1/σ2SS . . 96

5

4.9 Variation of (a) self-diffusivity, D (b) fraction of imaginary modes, Fimag,

and (c) Einstein frequency, ωE , with sorbate polarizability. . . . . . . . . . 98

4.10 Crossover from ballistic to diffusional motion as a function of sorbate size. . 99

5.1 Experimental adsorption isotherms in ZSM-5 of (a) methane, (b) n-butane

and (c) iso-butane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Contour plots of a single methane molecule in x- and y- channels of ZSM-5. 111

5.3 Potential energy surface of a xenon atom in Na-Y in the x = a/4 plane. . . 115

5.4 Temperature dependence of Henry’s constant. . . . . . . . . . . . . . . . . . 117

5.5 Comparison of the average values of the sorbate-zeolite potential energy ob-

tained by different computational methods. . . . . . . . . . . . . . . . . . . 118

5.6 Variation in computed values of Henry’s constant with effective radius of

α-cage of Na-Y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1 Schematic diagrams showing channel connectivity patterns in (a) ZSM-5 and

(b) ZSM-11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2 Contour plots of the potential energy surface of a single (a) xenon and (b)

helium atom in ZSM-5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.3 Plot of ∆2x(t) and ∆2z(t) as a function of time on (a) logarithmic and (b)

standard scales for Ar in ZSM-11 using order-N algorithm. . . . . . . . . . . 139

6.4 Log10-log10 plot of mean square displacements as a function of time in the x,

y and z-directions for CH4 in ZSM-5. . . . . . . . . . . . . . . . . . . . . . . 141

6.5 Log10-log10 plot comparing mean square displacements in the z-direction for

different Lennard-Jones sorbates in ZSM-5. . . . . . . . . . . . . . . . . . . 146

6.6 Short-time behaviour of time-correlation functions for (a) neon and (b) xenon

in ZSM-5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6

6.7 Plots of exponents of mean square displacements vs Imin along (a) x-, (b) y-

and (c) z- directions for argon in ZSM-5, ZSM-11 and ferrierite. . . . . . . . 154

7

List of Tables

1.1 Examples of zeolites and zeolite-type materials . . . . . . . . . . . . . . . . 6

1.2 Approximate dimensions of n-ring windows . . . . . . . . . . . . . . . . . . 9

1.3 Summary of structural features of some zeolites . . . . . . . . . . . . . . . . 10

1.4 Some experimental techniques for studying diffusion in micropores . . . . . 18

2.1 IR frequencies(in cm−1) of Silica Sodalite . . . . . . . . . . . . . . . . . . . 39

2.2 Potential parameters for simple Lennard-Jones sorbates . . . . . . . . . . . 46

4.1 Potential energy parameters used for Lennard-Jones sorbates in Na-Y. . . . 84

5.1 Potential energy parameters for rare gases in Na-Y. . . . . . . . . . . . . . . 112

5.2 Isosteric heats of sorption, 〈Usz〉, and the excess free energy, µex, for rare

gases in Na-Y zeolite obtained by different computational methods. . . . . . 122

6.1 Lennard-Jones parameters for the sorbate-sorbate and sorbate-oxygen inter-

actions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2 Convergence tests for CH4 in ZSM-5 at different concentrations. . . . . . . 136

6.3 Molecular dynamics simulation parameters used for different Lennard-Jones

sorbates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.4 Molecular dynamics simulation parameters used for argon in the zeolites

ZSM-11 and ferrierite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

9

6.5 The exponent of the time dependence of the mean square displacement for

Lennard-Jones sorbates in ZSM-5. . . . . . . . . . . . . . . . . . . . . . . . 142

6.6 Ballistic to diffusional crossover times for Lennard-Jones sorbates in ZSM-5. 144

6.7 Diffusion coefficients of different Lennard-Jones sorbates in ZSM-5 at a tem-

perature of 300K at different concentrations. . . . . . . . . . . . . . . . . . 145

6.8 Key features of the instantaneous normal mode spectra for Lennard- Jones

sorbates in ZSM-5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.9 The exponents of the time-dependence of the mean-square displacements of

argon in ZSM-5, ZSM-11 and ferrierite. . . . . . . . . . . . . . . . . . . . . 151

6.10 Diffusion coefficients of argon in ZSM-5, ZSM-11 and ferrierite at 300K. . . 152

10

Chapter 1

Introduction

Zeolites are three-dimensional, microporous, inorganic crystalline polymers which can ac-

comodate mobile cations and neutral molecules in the pore spaces [1]-[6]. The ability of

zeolites to act as rigid, crystalline sponges that can reversibly absorb large amounts of

vapours, liquids and gases is one of their most characteristic features. Naturally occurring

zeolites, first discovered in 1756, are aluminosilicate minerals in which the sorbed species

is water. On heating, the water molecules in the pore spaces desorb and the resulting ap-

pearance of “boiling stones” (from Greek: zeo≡ “to boil”; lithos≡ “stone”) is responsible

for the name for this class of minerals. The pore volume in zeolites ranges between 25 to

50% and the microporous nature implies a very large internal surface area. The ordered,

crystalline structures imply that the micropores in zeolites have very well-defined dimen-

sions and connectivities. In addition, the framework composition can be tuned to enhance

chemical selectivity i.e. the tendency to sorb organophilic or hydrophilic species. These

features combine to ensure that zeolites form a very important category of cheap, highly re-

producible, size and shape selective molecular sieves and adsorbents which find widespread

use in industry. For example, zeolites are used in O2 extraction from air, drying of natural

gas and hydrocarbon separations. The mobile cations in zeolites can be exchanged thus

allowing zeolites to act as very efficient ion exchangers which can be used for heavy metal

1

2 Chapter 1: Introduction

removal and recovery from industrial waste and as detergent builders. In addition to their

distinctive sorption properties, the zeolitic framework typically contains a large number of

Lewis acid sites making them very powerful, shape selective catalysts, widely used in the

petrochemical industry. For example, the synthetic high silica zeolite ZSM-5 has numerous

uses such as catalytic isomerisation of C8 aromatics to produce isomerically pure xylenes,

the conversion of methanol to gasoline and alkylation of benzene to ethylbenzene. The three

properties of zeolites which are responsible for their enormous industrial usage are therefore

their sorption, ion-exchange and catalytic properties. All three functions stem from the un-

usual structural properties of zeolites. The first section of this chapter, therefore, contains

a brief overview of structural features of zeolites from the perspective of their molecular

sieve properties.

In this work, we focus on computer simulations of sorption and diffusion of atoms and

simple molecules in zeolites. The sorption process is fundamental to the characterisation of

zeolites as porous materials and molecular sieves and is important in understanding their

ion-exchange and catalytic activities[1]-[6]. The sorption process is similar to bulk absorp-

tion since it may be regarded as a process of filling the void volume of the sorbent by the

sorbate molecules. However, unlike bulk absorption and like surface adsorption, it is domi-

nated by surface effects due to the large intracrystalline surface area of microporous solids

with pore sizes of less than 20A . As in the case of surface adsorption, sorption processes

may be classified as physisorption or chemisorption, depending on the extent to which the

electronic charge clouds of the sorbate and sorbent are modified[7]. Since chemisorption and

catalytic activity are closely connected, one typically restricts discussion of sorption to cases

of physisorption where the sorbate is bound to the sorbent by electrostatic, induction or dis-

persion interactions. Such sorption processes are reversible and exothermic, so that a rise in

temperature is sufficient to desorb the sorbates without any change in the molecular identity

of the sorbents. In such situations, the microporous sorbent or host lattice essentially serves

Chapter 1: Introduction 3

as a low potential energy trap that can house the sorbate or guest molecules. While molecu-

lar size and shape will determine the equilibrium sorption isotherms in different zeolites, the

use of zeolites as molecular sieves for separation processes requires an understanding of the

transport properties, specially the diffusivities, of different sorbates. Diffusion in zeolites is

therefore a subject of practical relevance. In addition, it holds considerable interest from

a conceptual and theoretical perspective since such ordered, microporous solids with their

diverse void-space topologies provide very elegant illustrations of the effect of confinement

on static and dynamical properties of fluids. The following section of this chapter outlines

some of the key conceptual aspects of sorption and diffusion of molecular sorbates in zeolite

lattices.

To understand how modifications at the atomic level in either sorbates or the host

zeolite lattice can affect sorption, diffusion and catalytic functions, it is essential to employ

computer simulations. Computer simulations provide a means to understand how variations

in microscopic properties at the atomic level affect structural, thermodynamic and transport

properties at the macroscopic level [8]-[12]. Such simulation techniques complement both

theory and experiment, by providing quantitative or semiquantitative predictions and by

acting essentially as numerical or computer experiments for systems or conditions which

may not be easily realisable in a laboratory. This is specially true of complex guest-host

systems for which analytical models are unavailable and an unambiguous molecular level

interpretation of complex experimental data is often difficult to achieve. For this reason,

computer simulations currently play an important role in rational design of zeolite catalysts

and related crystal engineering approaches [13]-[17]. Section 1.3 of this chapter contains

a short review of highlighting some of the more striking recent developments in computer

simulations of sorption and diffusion in zeolites.

The final section of this chapter outlines the scope of this thesis in the context of recent

work on computer simulations of sorption and diffusion of zeolites. An overview of the


material contained in this thesis and the chapter plan is provided here.

1.1 Structures of Zeolites

1.1.1 Chemical Compositions of Zeolites and Zeolite-type materials

From a mineralogical point of view, zeolites are defined as porous aluminosilicates with the

general chemical formula[4, 5] :

Mm/z[mAlO2 · nSiO2] · qH2O

where M is an exchangeable cation of charge +z and q is the number of sorbed water

molecules since naturally occurring zeolites are usually found in their hydrated form. The

aluminosilicate network is anionic and the ratio of one Al/Si atoms to two framework O

atoms indicates that zeolites belong to the more general structural category of tectosili-

cates. Tectosilicates are built up of AlO4 and SiO4 tetrahedra in such a manner that each

TO4 (T=Si or Al) unit shares its four oxygens with four other tetrahedra. Note that the

framework atoms are shown in square brackets. The tectosilicates include a wide range of

naturally occurring minerals, such as feldspars, feldspathoids and crystalline silicas. The

zeolites may be identified as tectosilicates with large pore volumes. The International Ze-

olite Association quantifies the pore volume criterion by classifying as zeolites only those

materials which have a framework density of less than 21.0 T-atoms per 1000A3 [18].

The chemical selectivity of zeolite molecular sieves can be very dramatically altered by

varying the Si/Al ratio. Clearly as the Si content is increased, the framework will become less

anionic and the number of exchangeable cations will decrease. Concomitantly, there will be

a decrease in the Lewis acid character of the zeolite, an increase in the hydrophobic character

of the sieve and a rise in thermal stability. Zeolites are classified as low, intermediate and

high silica zeolites depending on whether Si/Al ratios are below 1.5, between 2 and 5 or

above 10. The minimum Si/Al ratio for a zeolite is 1 since Lowenstein’s rule precludes


existence of Al-O-Al bonds in tectosilicates and therefore implies that the maximum Al

content must correspond to a situation in which Si and Al alternate on the tetrahedral

sites[5]. The maximum Si/Al ratio is infinity and is found in the pure silica molecular sieves

which are thermodynamically metastable, highly porous allotropes of SiO2.

A number of zeolite-type materials have been synthesised which share as the common

feature the presence of an extended, 3-dimensional anionic network of TO4 tetrahedra but

vary in the identity of the tetrahedral atoms, the presence of framework hydroxyl groups,

hydroxo bridges and non-aqueous ligands. The extended formula for zeolite-type materials

may be written as [5]:

MxM′yNz[TmT′n . . . O2(m+n+...)−ε(OH)2ε](OH)br(aq)p · qQ

where M and M′ are exchangeable and non-exchangeable metal cations, N denotes non-

metallic cations (generally removable on heating), (aq) corresponds to chemically bonded

water or other strongly held ligands of the T-atoms, Q are sorbate molecules and (OH) and

(OH)br are framework and bridging hydroxyl groups respectively. The framework atoms

are shown in square brackets. Clearly an enormous variety of zeolite materials can be gen-

erated by appropriate chemical substitutions (see Table 1.1 for some examples). However,

the crucial structural factor which governs the shape selective sorbent and catalyst func-

tions of zeolites is the topology of the anionic framework characterised by the connectivity

pattern and spatial disposition of the T-O-T bonds. Accordingly, zeolites are primarily

classified into structure types based on distinctive framework topologies. Zeolites belonging

to the same structure type may vary significantly in chemical composition but share the

same 3-dimensional framework structure with relatively minor variations in crystallographic

properties.


Table 1.1: Examples of zeolites and zeolite-type materials

The two types of tetrahedral atoms forming the anionic network are denoted by T and T′.

Unit cell compositions of specific materials are given in brackets

Name T T′ Examples References

Zeolites Al Si Zeolite NaY (Na48Al48Si144O384) [19]

Gallosilicates Ga Si Gallozeolite-L (K10.3Ga10.3Si25.7O72) [20]

Aluminium Al P AlPO4-11 (Al20P20O80) [21]

Phosphates SAPO-35 (Al27P21.6Si5.4O108) [22]

MAPO-43 (Mg2.2[Al5.8P8O32]· 2.3 (C3H7)2NH2) [23]

Beryllosilicates Be Si Lovdarite (Na12K4[Be8Si28O72]· 18H2O) [24]


1.1.2 Aspects of Zeolite Structure relevant to Sorption and Molecular

Sieve Properties

The aspects of zeolite structure which play a crucial role in determining the sorption and

sieve properties of zeolites are: (i) the dimensions and shapes of the void spaces which deter-

mine the net pore space; (ii) the aperture dimensions which regulate entry and diffusional

pathways of sorbate molecules in the void spaces and (iii) the connectivity of void spaces

to result in diffusional channels of widely varying topologies.

The primary building unit in zeolite-like frameworks is the TO4 tetrahedron. These

TO4 tetrahedra can be linked to form finite secondary building units (SBUs), containing

upto 16 T atoms, which are derived assuming that the entire framework is made up of only

one type of SBUs. To date, 20 SBUs are known which has increased from the 16 known in

1992[25]. Typical examples of SBUs include the 4, 5, 6 and 8 ring structures containing the

corresponding numbers of T and O atoms. More complex SBUs include the cube, hexagonal

and octagonal prisms which are essentially double ring structures. Since the infinite zeolite

framework is constructed from linked SBUs, the void spaces are typically polyhedral. The

dimensions and shapes of the polyhedra determine the shape and size of sorbate molecules as

well as the sorption capacity. A very wide range of polyhedral units occur in zeolites which

are normally classified according to the number of polygonal faces [1]. The cube or 6-hedron

is the smallest such unit but cannot normally accomodate sorbed species. The next largest

unit is the hexagonal prism or 8-hedron which has an approximate dimension of 2.3A in

the plane of the 6-ring. The Type I 14-hedron corresponds to a truncated octahedron and

has an approximate free dimension of 6.6A for the inscribed sphere. The largest polyhedral

units are the Type I and Type II 26-hedrons which occur, for example, in Zeolites A and

Y respectively. While some of the polygonal faces will be too small to allow passage of

sorbates, others will act as apertures or windows for the channel systems in the zeolites.

The maximum free diameters of such windows will be found when the windows are planar.


Table 1.2 gives the approximate free dimensions of n-ring windows[1]. An n-ring window

will contain n tetrahedral atoms and n- bridging oxygens, with the free dimension being

calculated by taking the diameters of the framework oxygens to be 2.8A. It can be seen

that windows with n ≤ 6 will allow only water molecules and cations to penetrate. The

polyhedral void spaces in zeolites can be joined to form one, two and three-dimensional

channel systems. The dimensionality of a channel system is defined as one, two or three

depending on whether a sorbate molecule can move in all directions, in a plane or in a

line through the zeolite pore spaces, given some initial location of the sorbate. In some

cases, the sorbate size will determine the dimensionality of the channel system. There are

some systems also in which interpenetrating but unconnected channel systems exist. These

aperture dimensions and void space dimensionality considerations apply to cation migration

in the zeolite framework, as well as to sorbate diffusion; however, cation sizes are typically

much smaller than those of neutral atoms and molecules and consequently they can move

through much more restricted channel systems.

1.1.3 Examples of Zeolite Structures

The structural aspects discussed above are illustrated with reference to six zeolite systems

which are of special interest from the point of view of this work. These zeolites are listed in

Table 1.3, together with their characteristic structural features such as unit cell composition,

restricting window size, channel directions, framework density and corresponding space

groups. As discussed above, a wide range of chemical compositions can be found for a given

structure type with slightly different crystallographic parameters and atom positions. This

discussion has been restricted to only the pure silica variants, as shown in Table 1.3, since

this is sufficient to discuss geometric and topological aspects of each structure. Further

crystallographic information is provided in the relevant chapters.

The discussion may be started with two widely used, high-porosity zeolites, Zeolites


Table 1.2: Approximate dimensions of n-ring windows

A n-ring window in a zeolite will contain n oxygen and n tetrahedral atoms[1]

n Diameter Zeolite Typical molecules

(A ) accepted

4 1.15 Chabazite water, cations

5 1.96 Mordenite water, cations

6 2.8 Sodalite hydrogen

8 4.5 Zeolite A ethylamine, diborane

10 6.3 Ferrierite argon, propane

12 8.0 Zeolite Y xenon, benzene

A and Y, with three-dimensional channel systems. Zeolite A is a highly porous structure

where the framework is formed by stacking 26-hedrons of type I to create cubic array of

relatively large cages, of 11.2A diameter, interconnected through eight membered oxygen

windows of free diameter 4.3A. A ball and stick model of Zeolite A is shown in Figure

1.1. Each large cage is connected to six similar adjacent cages by eight-ring windows. At

each of the eight corners of the large cage, there are six-membered oxygen rings which give

access to the smaller “sodalite” cages. The total pore volume of Zeolite A is approximately

50%; however, the 6-membered oxygen ring windows cannot be penetrated by any sorbates

larger than water and the accessible pore volume is therefore considerably reduced for most

sorbates. Zeolite Y is another highly porous zeolite having similar framework density as

Zeolite A and consequently almost the same pore volume of approximately 50%. Zeolite Y

is, however, formed by linking 26-hedrons of Type II to give 3-dimensional array of sodalite

units interconnected through hexagonal prisms. The sodalite units have an approximate


Table 1.3: Summary of structural features of some zeolites

Both the number of bridging oxygens and the effective size of the windows defining the

zeolitic channel systems are given. For elliptical windows, the lengths of both the major and

minor axes are given. Angular brackets indicate that channels exist in crystallographically

equivalent directions. Framework densities are given in terms of number of tetrahedral

atoms per unit volume.

Zeolite Unit cell Restricting Channel Effective Framework Space Ref.

Composition window directions window size Density Group

(A) (T/1000A3)

Zeolite A Si192O384 8 < 100 > 4.3 12.9 Fm3c [26]

Zeolite Y Si192O384 12 < 111 > 8.0 12.7 Fd3m [19]

ZSM-5 Si96O196 10 [010] 5.7×5.1 17.9 Pnma [27]

10 [100] 10.8

ZSM-11 Si96O192 10 < 100 > 5.4×5.3 17.7 I4m2 [28]

Ferrierite Si36O72 10 [010] 5.4×4.2 17.7 Pnnm [29]

8 [100] 4.8×3.5

Theta-1 Si24O48 10 [001] 5.5×4.4 19.7 Cmc21 [30]


diameter of 4.4A. Ten such sodalite units, also called β-cages, interconnect to give large

cages of diameter of about 11.8A called α-cages. Each α-cage is tetrahedrally connected to

four other α-cages through twelve membered oxygen rings of approximate diameter of 8A.

Figure 1.2 depicts the detailed three-dimensional framework structure of Zeolite Y. From

the above discussion, it is evident that though Zeolite A and Zeolite Y have very similar

pore volumes, their molecular sieve properties are markedly different. This arises due to

dissimilar basic building blocks in the two zeolites which interconnect differently to form

interconnecting channels with different aperture dimensions.

Ferrierite, ZSM-5 and ZSM-11 are all low-porosity zeolites with comparatively high

framework density. Ferrierite is an example of a two-dimensional zeolite where the pore

system consists of intersecting 10-ring straight channels parallel to y-direction and 8-ring

straight channels parallel to the x-direction, as shown in Figures 1.3(a) and 1.3(b) respec-

tively. ZSM-5 is another example of a zeolite with a two dimensional channel system. The

10-ring straight channels parallel to the y-direction have elliptical cross-sections with major

and minor axes of 5.7A and 5.1A respectively. The 10-ring sinusoidal channels parallel to the

x-direction have circular cross-sections of 10.8A diameter. An interesting feature of ZSM-5

is that though it has a two-dimensional channel system like Ferrierite, diffusion of guest

molecules occurs along the z-direction as well. This is possible because the sinusoidal chan-

nels have an oscillating component along the z-direction though the channel system travels

along the x-direction, as shown in Figure 1.4. This type of motion along the z-direction

due to correlated motion along the x- and y- directions is a consequence of the geometry of

the zeolite and is known as geometrical correlation. Geometrical correlation can also be ob-

served in ZSM-11 which has 10-ring nearly cylindrical straight channels running parallel to

x- and y- directions, as shown in Figure 1.5. The channel dimensions are provided in Table

1.3. This low porosity zeolite has two types of intersections: small intersections which build

cavities of about 1.5 times the size of the channel and large intersections, which are like


short channels along the z-direction, as the distance between the centre of the intersecting

channels ≈ 5A .

An example of a unidimensional pore zeolite system is provided by Theta-1. Figure 1.6

shows a view down the z-axis of this zeolite in which the 10-ring channel windows can be

clearly seen. It is a low porosity zeolite with a framework density of 19.7 T/1000A3

1.2 Sorption and Diffusion in Zeolites

1.2.1 Sorption Isotherms

The equilibrium aspects of the adsorption process are captured by sorption or adsorption

isotherms which portray the intracrystalline concentration of the sorbate, Cs, in equilib-

rium with the concentration of the sorbate in the fluid or gas phase, Cg, at constant

temperature[1]-[6]. At equilibrium, the chemical potential of the sorbate or guest is the

same in the adsorbed and gas phases. If the adsorbed guest molecules are regarded as

forming an ideal solid solution with the sorbent and the sorbate gas is assumed to be ideal,

one expects Henry’s law to be obeyed

Cs = KhCg (1.1)

where the coefficient of proportionality is Henry’s constant, Kh. Non-ideality of the sorbate

gas can be accounted for relatively simply by replacing the gas pressure with the fugacity

when calculating Cg. In the infinite dilution limit, all sorption isotherms show this linear

behaviour. However, as concentrations increase, deviations from ideality are seen, chiefly

due to sorbate-sorbate interactions and heterogeneity of sorption sites resulting in a fairly

wide spectrum of qualitatively different adsorption isotherms. Thermodynamic analysis

of adsorption isotherms yields information about sorption energetics as well as on free

energy and entropic aspects of intracrystalline sorbates. The information on energetics

of sorption processes provides crucial information on the nature and strength of sorbate-


sorbent interactions from the point of view of constructing potential energy surfaces for use

in computer simulations of sorbates in zeolites.

1.2.2 Transport and Self-Diffusivities

From a macroscopic perspective, diffusion corresponds to the process by which spatial vari-

ations in concentration are removed in the absence of any fluid flow. Diffusion processes

are integral to the attainment of sorption equilibria. Diffusion in a porous medium, such as

a zeolite, may be regarded as a special case of binary diffusion (intermixing of two species)

where the two species have very different mobilities[6]. Framework vibrational frequencies

of zeolites are typically so large, that at normal temperatures and pressures, it may be re-

garded as essentially static. Therefore, the zeolitic framework can be regarded as providing

a periodic, confining potential regulating sorbate diffusion.

In common with bulk diffusion, one may distinguish between transport and self-diffusivities

of sorbates in zeolites. Fick’s first law of diffusion states that the macroscopic sorbate flux,

J , must be linearly proportional to the concentration gradient i.e.

J = −Dt∇Cs (1.2)

where the constant of proportionality is termed the transport diffusivity, Dt[7]. While the

standard form of Fick’s law assumes Dt to be concentration independent, this is frequently

not the case at high concentrations. From a microscopic point of view, the removal of

concentration gradients by diffusive processes takes place due to random thermal motion of

the molecules. This type of motion will be present even at equilibrium when no macroscopic

concentration gradients or fluxes are present. Such motion of particles under the action

of random forces exerted by the molecules constituting the medium is termed Brownian

motion. It can be shown that, in three dimensions, the mean square displacement of a

Brownian particle will be a linear function of time, with the coefficient of proportionality

given by 6Ds where Ds is the self-diffusivity. While under certain conditions, discussed


below for adsorption cases, the transport and self-diffusivities will be the same, this is not

necessarily true for all concentrations and systems[6].

To derive the relationship between the transport and self-diffusivities of fluids in porous

media, one must consider a steady-state situation in which the thermodynamic force driving

the sorbates to move against the concentration gradient is counterbalanced by the frictional

drag due to the medium. The frictional force, F = αv, where α is the coefficient of friction

and v is the net particle velocity. The macroscopic particle flux, J, is given by the product

of the net particle velocity and concentration and therefore may be written as

J = (Cs/α)F (1.3)

The thermodynamic force is equal to the gradient of the chemical potential of the sorbate

inside the sorbent, denoted by µs. Therefore, in one-dimension, we can write

J = (Cs/α)∂µs

∂x(1.4)

At equilibrium, the chemical potentials of the sorbate in the gas and adsorbed phases must

be the same. Therefore

µs = µ◦g + RT ln f (1.5)

where µ◦g is the chemical potential of the gas in its standard state and f is the fugacity.

Combining equations 1.4 and 1.5 , we get

J =RT

α

∂(ln f)∂(ln Cs)

∂Cs

∂x(1.6)

Comparing the above expression with that for Fick’s first law(equation 1.2), one may derive

the Darken relation

Dt(Cs) = D0∂(ln f)∂(lnCs)

(1.7)

where D0 = RT/α is the corrected diffusivity which does not depend directly on concen-

tration and may be considered as a measure of molecular mobility. At low pressures, the


fugacity can be replaced by the pressure, and the factor of ∂(ln p)/∂(ln Cs) is then just the

slope of the adsorption isotherm in logarithmic coordinates. As pressure approaches zero,

all systems will enter the Henry’s law regime for which this factor is unity and the corrected

and transport diffusivities are identical. In the limit that sorbate concentrations are very

small, the sorbate-sorbate interactions will play no role in determining molecular mobility

and the corrected and transport diffusivities will be the same.

1.2.3 Effect of Anisotropic Porous Media on Diffusivity

Throughout the above discussion, we have assumed that diffusivity is a scalar quantity

characterising the sorbate mobility. This is true in liquids, in random porous media and

in ordered porous media of cubic symmetry. If the confining medium or framework is

anisotropic, then the flux vector will have unequal components in the x, y and z directions[6].

In this case, Fick’s law must be written as

J = D · ∇Cs (1.8)

where D is the diffusion tensor. A component, Dαβ, of the diffusion tensor determines the

contribution to the flux in direction α due to the concentration gradient in direction β.

By choosing the frame of reference properly, one can define the principal directions such

that off-diagonal elements of the diffusion tensor are all zero. For cubic, tetragonal and

orthorhombic zeolites, the choice of principal directions is obvious since they must coincide

with the crystallographic axes.

1.2.4 Molecular Picture of Diffusion

At finite temperatures, atoms constituting the medium will have a mean kinetic energy.

Provided the temperature and/or atomic masses are sufficiently high, quantum effects can

be ignored and the atoms can be regarded as classical particles obeying Newton’s laws of

motion. Even though the classical equations of motion are deterministic, the large number


of intermolecular and molecule-wall collisions in a typical many-particle system will ensure

that the overall trajectory of the system through phase space is chaotic. The molecules may

then be described as being in continuous, random motion. If the position, ri, of a single,

tagged molecule is followed as a function of time t, the theory of Brownian motion predicts

the following Einstein relation:

∆2r(t) = 〈|ri(t)− ri(0)|2〉 = 2dDt (1.9)

where d is the dimensionality of the space through which the molecule moves. Note that

〈. . .〉 indicate an ensemble average. Thus motion through the unidimensional channel system

of a zeolite, such as Theta-1, will result in a mean square displacement that grows as 2Dt,

rather than 6Dt, as expected for diffusion through a three-dimensional channel system.

The Einstein relation given above is clearly not consistent with Newton’s second law which

states that

∂2ri

∂t2=

Fi

m(1.10)

where Fi is the instantaneous force acting on the particle. Therefore the linear growth of

the mean square displacement must be a feature of the system on a time scale that is very

long compared to the time scales associated with molecular motions such as rotations and

vibrations. On short time scales, the system is expected to show ballistic behaviour with

the mean square displacement growing quadratically with time. A simple one-dimensional

Langevin model for a particle subject to frictional drag due to the medium and random

stochastic forces predicts that the average mean square displacement will behave as:

∆2r(t) = 2kBTB[t−mB(1− e−t/mB)] (1.11)

where B is the sorbate mobility, m is the sorbate mass, T is the temperature and kB is the

Boltzmann constant. The ballistic and diffusional limits will occur as t → 0 and t → ∞

respectively. In a real molecular system, the time and nature of the ballistic to diffusional

crossover can provide information on the potential energy landscape of the system.


1.2.5 Length and Time Scales for Diffusional Processes

In a microporous medium, the diffusional mechanism is complex and involves processes

on different length and time scales[6]. In addition to sorbate-sorbate interactions, random

collisions with the walls of the confining micropores play an important role, specially at

low concentrations. An important difference between diffusion in solids and diffusion in

liquids is the role of hopping events which involve jumps between low potential energy

adsorption sites. These elementary processes connect adsorption sites separated spatially

by distances of the order of A; therefore, equation(1.9) is not expected to be applicable. Over

a time period that is long compared with the timescale of the hopping motion, one expects

diffusional behaviour to be applicable. This motion may be regarded as intracrystalline

self-diffusion and has a length and time scale which is determined by the size of the zeolite

crystals. Most zeolite type materials are prepared in microcrystalline form, with crystallites

of 10-100 µm diameters. In practical applications, zeolites are typically in pelletised form

with spatial inhomogeneities on length scales of 10−5m. Thus macroscopic measurements

of transport diffusivity operate on length scales much greater than 1 µm and time scales of

the order of seconds. The diffusivities from such measurements will not be the same as the

intracrystalline self diffusivities because of the effect of crystallite surfaces and extended

crystal defects. Table 1.4 summarises the length and time scales associated with some

common experimental techniques used to study diffusion in zeolites. It can be seen that the

pulsed field gradient (PFG) NMR technique is the only experimental method which has the

correct time and length scale to probe intracrystalline diffusion.

1.3 Computer Simulations of Sorbates in Zeolites

The purpose of a computer simulation is to provide a representative collection of microstates

of an atomic system from some well-defined statistical mechanical ensemble that can be used


Table 1.4: Length and time scales associated with different experimental techniques for

studying diffusion in micropores[6]

Method Subject Length scale Time scale

Tracer Technique Self-Diffusion ≥ µ m ≥ min

Pulsed Field Gra- Self-Diffusion µ m to mm ms...s

dient NMR

Transient and Transport ≥ µ m ≥ s

Steady state Diffusion

measurements

Neutron Elementary ≤ nm ≤ 10−8 s

Scattering processes

NMR(line shape, Elementary ≈ A 10−10..

Relaxation processes ...10−1 s


to obtain appropriate equilibrium averages, fluctuations and time-correlation functions [8]-

[12]. Simulations generally contain of the order of 102 to 106 atoms which is much smaller

than Avogadro’s number. However, the use of periodic boundary conditions allows for the

simulation of macroscopic properties with reasonable accuracy. Most computer simulation

methods in current use assume that the atoms can be treated as classical particles obeying

Newton’s laws of motion.

1.3.1 Potential Energy Surfaces and Simulation Methods

To perform a computer simulation it is essential to be able to write the system Hamiltonian

in terms of the microscopic variables i.e. the positions and momenta of the N atoms in

the simulation cell. While the kinetic energy can be written trivially in terms of particle

momenta, the specification of the potential energy is more complex and contains effectively

all the information of chemical interest. To compute the potential energy, it is necessary to

make a Born-Oppenheimer separation of electronic and nuclear motion, leading to the defi-

nition of an electronic potential energy surface (PES) which controls the atomic or nuclear

motion[7, 31, 32]. The PES is the ground state electronic energy of the system constructed

as a function of the atomic configurations. It is therefore a recipe for calculating the po-

tential energy of a collection of atoms and constitutes the fundamental microscopic level

input in all computer simulations. Two approaches to constructing the PES are available.

The parametric potential route assumes that the PES can be constructed as a sum of pair

(or at most three- and four- body) interactions between atoms[33, 34]. These few-body

potentials are derived from independent spectroscopic and quantum chemical studies of

dimers and small clusters. The ab initio electronic structure route, in contrast, calculates

the PES using quantum chemical approaches to compute the ground electronic state of

the many-body system during the course of a simulation[35]-[38]. The second approach is

much more computationally demanding and is necessary only when the valence electrons


are strongly delocalised, or chemical reactions can take place. In the context of zeolites, ab

initio methods are required for studying catalytic processes but not sorption and diffusion

of non-reactive physisorbed molecules and atoms [39]-[41]. Since the focus of this thesis is

sorption and diffusion of simple sorbates in zeolites, we restrict ourselves in this study to

simulation studies based on parametric potentials.

Classical Monte Carlo techniques generate the equilibrium statistical mechanical prop-

erties of a system, usually within the canonical ensemble[11, 12]. Static properties of the

system such as the average potential energy and structural quantities such as the radial dis-

tribution functions can be obtained from a Monte Carlo simulation though generalisation to

other ensembles is possible. For an N -atom system, such Monte Carlo techniques construct

a random walk through the 3N -dimensional configuration space with a bias such that for

sufficiently long walks, a set of configurations, x, distributed according to the Boltzmann

formula e−βU(x) is generated. If N is sufficiently large and suitable boundary conditions

are applied, then equilibrium properties of the bulk system can be generated. Unlike MD

simulations, Monte Carlo simulations do not provide dynamical information. However, the

stochastic approach offers considerable flexibility in the design of efficient strategies for sam-

pling configuration space. Consequently, in the context of zeolites, Monte Carlo simulations

are preferentially used for sorbates with many conformational degrees of freedom, such as

long-chain alkanes.

Classical molecular dynamics, instead of using a random walk approach, involves nu-

merical integration of the classical equations of motion to generate the trajectory of the

N -particle system through phase space[11, 12]. Provided certain conditions are satisfied

(ergodicity, sufficient length of trajectory and the like), the phase space points sampled

during the course of an MD run provide information on both the equilibrium and dynam-

ical statistical mechanical quantities. Thus, unlike MC, MD methods can provide time-

correlation functions related to transport properties and spectroscopy. However, MD can


prove inefficient relative to MC for systems with multiple time scales. Most simulation stud-

ies of diffusional behaviour in zeolites rely on microcanonical MD studies. An alternative

approach, applicable when activated hops between well-defined adsorption sites provide the

diffusional mechanism, is to use transition state theory to calculate rates of elementary pro-

cesses and a kinetic Monte Carlo scheme to compute overall diffusional behaviour [42]-[46].

While this is a less general and accurate computational scheme than a full MD simulation,

it is very useful for studying transport processes which occur on very long time scales.

1.3.2 Computer Simulations of Sorption Properties

As discussed above, the chief quantity of interest in sorption studies is the adsorption

isotherm. One of the pioneers of the use of grand canonical Monte Carlo (GCMC) methods

to simulate adsorption isotherms was Rowlinson who used it to study Xe and CH4 adsorption

in zeolites X and Y[47]. Similar studies have been carried out for a number of other systems;

for example, CH4 in ZSM-5, Xe, Ar and CH4 in Zeolite NaA and CH4-CO2 mixtures in

micropores[48]-[56]. While the grand canonical ensemble is often the most convenient for

adsorption studies, MD and MC simulations implemented in other ensembles have also been

successfully used to study adsorption.

Standard Monte Carlo methods are adequate for small, near spherical sorbates but they

are grossly inefficient for most large hydrocarbons of interest to the petrochemical industry.

Simulations for such systems could only be carried out once more sophisticated sampling

techniques, like Configuration Bias Monte Carlo methods, were available[12]. CBMC meth-

ods have since been used to study sorption behaviour of a variety of linear and branched

chain alkanes in zeolites. The simulations of linear alkanes in zeolites provide an illustra-

tion of how simulations can predict novel behaviour and thereby stimulate new experimen-

tal work [57]-[61]. Smit and Maesen, based on CBMC simulations, predicted that while

n-alkanes, CmH2m+2, with m < 5 or m > 10 would have simple Langmuir isotherms,


n-hexane and n-heptane would have kinked or stepped isotherms as a result of commensu-

rate freezing of these long chain molecules in the zig-zag channels of ZSM-5. Subsequent

experimental and theoretical work has confirmed this prediction.

Quantum sorbates are an interesting, though limited, class of guest molecules for which

the classical particle approximation breaks down and quantum simulation techniques, such

as path integral Monte Carlo, must be employed. Examples of quantum sorbates are 3He,

4He, H2, D2 and Ne. The zeolitic frameworks in which He adsorption has been studied

experimentally and computationally indicate interesting differences in sorption behaviour

of the two helium isotopes [62]-[66]. An interesting prediction is that such systems may

perhaps show quantum sieving behaviour such that light molecules are excluded from the

host lattice in favour of heavier isotopomers due to zero-point energy effects.

1.3.3 Computer Simulations of Sorbate Diffusion

Computer simulations of diffusion of simple sorbates in zeolites provide a very useful ap-

proach to studying the effect of confinement and pore topology on the transport properties.

The simplicity of the systems allows for very accurate simulations under a very wide range

of macroscopic conditions. For many such systems, specially rare gases in zeolites, compari-

son with experimental data is feasible. We highlight only some of the more striking variants

of diffusional behaviour induced by confinement, such as the levitation effect, single-file

diffusion and the role of geometrical correlations. Simple geometrical arguments predict a

monotonic decrease in diffusivity with sorbate size; the levitation effect, on the contrary,

predicts a peak in the diffusivity for sorbate sizes close to the aperture dimensions of the

channel system[67]-[73]. Single-file diffusion is an extreme case of one-dimensional diffu-

sion observed in unidimensional pore systems, such as those seen in zeolites Theta-1 and

AlPO4-5, and predicted to occur in ion-channels [74]-[87]. The diffusants in single file sys-

tems cannot pass each other at high concentrations and the effect of a “string of strung


pearls” is to alter the time-dependence of the mean square displacements of the particles.

Recent PFG-NMR experiments and simulations indicate that such single-file diffusion may

be observed over limited time periods of observation but in most realistic systems, Fick-

ian behaviour is likely to be restored in the very long-time limit. Geometrical correlations

arise in pore topologies where motion in a certain direction is possible only by a series of

correlated hops through the channel system and result in removing the independence of

the three principal elements of the diffusion tensor. Such effects have been studied in some

detail for the ZSM-5 system[88]-[91]. Molecular sorbates present a more complex set of

diffusion-related behaviour since they contain vibrational or rotational modes which can

couple to the translational or diffusional modes [92]-[96]. A particularly well-studied set

of systems is that of the linear and branched alkanes in a number of zeolites which are

also of practical importance [97]-[104]. Binary mixtures of such alkanes show diffusional

properties which cannot necessarily be predicted on the basis of properties of the pure

components [45, 49, 50, 89, 103, 105]. The smaller alkenes, with their possibilities for geo-

metrical isomerism, are also of interest from the point of view of understanding the effects

of molecular shape on transport properties [80, 106]. Benzene in the large pore Na-X and

Na-Y zeolites is very well-studied [14, 17, 42, 92],[107]-[115]. Since considerable 2H NMR

and quasielastic neutron scattering data is available for this system, detailed comparisons

between simulations, theory and experiment on aspects such as orientational randomisa-

tion are possible[109, 116, 117]. The C7 and C8 aromatics are also the focus of current

research[118, 119, 123].

1.4 Scope of the Thesis

In this thesis, we use molecular dynamics and related computational techniques to study

unusual diffusional properties of simple fluids that arise as a result of confinement in a crys-

talline nanoporous material. We define simple fluids or simple sorbates as those composed


of one-component systems where the constituent atoms or molecules can be characterised

simply by their van der Waals radius and polarizability. The rare gases are obvious mem-

bers of this class. In addition to the rare gases, a number of spherical top molecules of

practical significance fall in this category e.g. CH4, CF4, CCl4, SF6, SnCl4 and SnBr4. The

sorbates are assumed to be classical particles obeying Newton’s laws of motion. Previous

studies indicate that this is true even for the helium isotopes when adsorbed in ZSM-5 at

temperatures of 300K[64].

Sorption behaviour of simple sorbates in zeolites is useful for establishing the role of

molecular size and polarizability on the thermodynamic and transport properties of sorbates

in microporous solids [1, 2]. The van der Waal radius of the sorbate determines its geometric

properties in relation to the sorbent and the polarizability determines the strength of the

sorbate-zeolite interactions. The adsorption isotherms and diffusional properties of a large

number of simple, unreactive molecules in zeolites are routinely mapped out in the process

of characterising the pore volumes and molecular sieve properties of zeolites. The sorption

properties of such simple sorbates are of significance in a number of industrial processes

such as purification of gas streams and removal of radioactive gases as a by-product of

nuclear reactions. The recent popularity of 129Xe NMR as a technique for probing surfaces

and microporous materials has added to the interest in the sorption properties of rare gases

[124]-[126]. Recent work indicates that 3He can act as an NMR-active probe of adsorbate

structure. One can expect that 129Xe and 3He NMR spectra will give complementary

information on adsorbate structure given the large size and polarizability differences [125,

127, 128]. In addition to the practical importance of sorption processes in zeolite science

and technology, simple sorbates in zeolites provide very elegant model systems to study and

understand a number of aspects of diffusion, as discussed in Sections 1.2 and 1.3.

In this work, I use microcanonical ensemble molecular dynamics simulations to explore

several aspects of sorption and diffusion of simple sorbates in zeolites. Chapters 2 and 3 of


this thesis provide the necessary background on potential energy surfaces and computational

methods. Chapters 4 to 6 contain the simulation results. Chapter 7 provides a concluding

summary of the work. A brief description of the contents of Chapter 2 to 6 is given below.

The parametrisation of potential energy surfaces (PESs) for physisorbed molecules in

zeolites is discussed in Chapter 2. The exact parametric form of the PES used for the simple

sorbates studied in this work and its range of applicability is described in detail.

Chapter 3 discusses the computational techniques used in this work. The primary simu-

lation technique, molecular dynamics (MD), is described with reference to the MD algorithm

as implemented in my simulations. I also discuss the calculation of the various quantities

related to diffusional behaviour, such as the self-diffusivities, velocity autocorrelation func-

tions and ballistic to diffusional cross-over times. Since the chemical potential of the system

controls the distribution of the sorbate between the gas and crystalline phases, a brief dis-

cussion is given in this chapter regarding the estimation of the chemical potential by the

Widom particle insertion method. A brief review of instantaneous normal mode analysis

and its usefulness as an indicator of short-time dynamical information is also contained in

this chapter.

Chapter 4 applies instantaneous normal mode analysis to understand the correlated

changes in diffusivity and in the PES that occur as a result of variations in sorbate size.

From simple geometrical considerations, it would appear that the self-diffusion coefficient,

D, will be maximum for very small sorbates and will decrease with increasing sorbates size.

However, under certain circumstances, a nonmonotonic increase in D with increasing sorbate

size is observed for sizes close to the minimum channel width. This anomalous peak in

transport properties is termed the levitation effect. In this chapter we perform instantaneous

normal mode analysis, in conjunction with MD simulations, for Lennard-Jones sorbates

of variable size and polarizability diffusing in NaY zeolite. The size-dependence of the

diffusivity, including the anomalous levitation peak, is reflected in various properties of


the instantaneous normal mode spectrum, such as the fraction of imaginary modes and

the Einstein frequency. The existence of clear signatures of the levitation effect in the

instantaneous normal mode properties indicates a close connection between the anomalous

diffusivity peak and the curvature distribution of the potential energy surface.

Our results on estimation of Henry’s constants and isosteric heats of sorption at infinite

dilution for Lennard-Jones sorbates in zeolites are presented in Chapter 5. Henry’s constants

are obtained by an integration method which corresponds to the infinite dilution limit of

the Widom particle insertion method for determining the chemical potential. Isosteric

heats of sorption at infinite dilution are calculated using MD simulations as well from the

temperature dependence of the Henry’s constant. The systems studied are: (i) CH4 in

silicalite (ii) Ar, Kr and Xe in Na-Y and (iii) variable size Lennard-Jones sorbates in Na-

Y. The first two sets of sorbate-zeolite systems provide a way to test the accuracy of the

potential energy surfaces used in this thesis in predicting sorption and related properties.

The third set of simulations highlights an important difference between molecular dynamics

simulations and insertion techniques when applied to adsorbates in porous media. The

MD approach samples the low potential energy regions which form part of the dynamically

connected pore space. In contrast, the Henry’s constants are evaluated from the infinite

dilution limit of the Widom particle insertion method and therefore can, in principle, sample

all available, low potential energy pore regions regardless of dynamical connectivity. We

show that the difference in the results obtained by the two approaches depends significantly

on sorbate size.

Chapter 6 presents the results of molecular dynamics simulations of diffusional anisotropy

exhibited by simple Lennard-Jones sorbates in zeolites. Diffusional anisotropy in porous

crystalline solids occurs because the structural properties of the solids are direction-dependent.


tensor. A more unusual effect is an interdependence of the components of the diffusion


tensor due to geometrical correlations which arise as a result of special features of the chan-

nel network. The main focus of my study is sorbate behaviour in ZSM-5 since this is the

zeolite for which the most extensive studies of diffusional anisotropy exist in the literature.

I also compare some key aspects of diffusional anisotropy in ZSM-5 with the diffusional

behaviour of sorbates in ZSM-11 and siliceous ferrierite. ZSM-11 and ferrierite are both

low-porosity zeolites but differ in that ZSM-11, like ZSM-5, shows geometrical correlations

whereas ferrierite can only form simple two or one-dimensional channel systems. The de-

pendence of the extent of anisotropy on sorbate size and polarizability is illustrated using

Lennard-Jones parametrisations for helium, neon, argon, methane and xenon. In addition

to calculating the diagonal elements of the diffusion tensor, I also examine the anisotropy

in several related quantities such as the velocity autocorrelation function, the ballistic to

diffusional crossover times and the instantaneous normal mode spectra.


Figure 1.1: Structure of Zeolite A

Ball and stick model of unit cell of Zeolite A. The silicon atoms are indicated by the yellow

balls and the oxygen atoms by the red balls. Four eight-ring windows are visible which have

cubic arrangement


(a)

(b) FAUJASITE

Figure 1.2: Structure of Zeolite Y

(a) A Ball and stick model showing framework structure of unit cell of Zeolite Y, the yellow

balls indicating silicon atoms and the red balls indicating oxygen atoms. The central cavity

is the 12-ring window of Zeolite Y (b) The tetrahedral arrangement of smaller sodalite or β-

cages gives rise to the large α-cages. Each α-cage is connected to four other α-cages through

12-ring oxygen window. The smaller β-cages interconnect through hexagonal prisms. The

yellow dots indicate the positions of the framework cations


(a)

(b)

Figure 1.3: Structure of Ferrierite

(a) Wire-frame structure of siliceous ferrierite and the 10-ring straight channels parallel

to the y-direction (b) Wire-frame structure of siliceous ferrierite showing 8-ring channels

parallel to the x-direction. The yellow balls indicate the silicon atoms while the red balls

represent the oxygen atoms of the framework structure in the above figures


(a)

(b)

Figure 1.4: Structure of ZSM-5

(a) Wire-frame structure of ZSM-5 with 10-ring sinusoidal channels along the x-direction (b)

A wireframe model of ZSM-5 showing 10-ring straight channels parallel the y-direction.The

yellow balls indicate the silicon atoms while the red balls represent the oxygen atoms of the

framework structure in both the figures


Figure 1.5: Structure of ZSM-11

Wire-frame structure of tetragonal ZSM-11 showing 10-ring channels running parallel to

the y-direction. The yellow balls indicate the silicon atoms and the red balls represent the

oxygen atoms of the framework structure in the above figure of ZSM-11


Figure 1.6: Structure of Theta-1

Unidimensional wire-frame structure of theta-1 with 10-ring non-intersecting channels par-

allel to the z-direction. The yellow balls indicate the silicon atoms and the red balls represent

the oxygen atoms of the framework structure of Theta-1

Chapter 2

Potential Energy Surface

The potential energy surface(PES) describes the interatomic interactions and constitutes

the microscopic input in any computer simulation. Since very realistic PESs are difficult

to construct as well as computationally demanding, it is usually necessary to introduce

physically reasonable approximations[11, 12, 32, 33]. The quality of the approximate PES

will determine the type of questions that can be addressed by a simulation and the predictive

or explanatory value of the simulation results. In this chapter, the general features of

parametric PESs available in the literature to model physisorption of non-polar sorbates in

zeolites are first described and then the PES used in this work is discussed.

Assuming that sorbates are physisorbed in the zeolite pore spaces, the total parametric

potential for the sorbate-zeolite system, U , can be represented as

U = Ulat + Usz + Uss (2.1)

where Ulat represents the potential energy of interaction between the lattice or the frame-

work atoms of zeolite, Usz represents the sorbate-zeolite interaction energy while Uss repre-

sents the sorbate-sorbate interaction energy. Section 2.1 discusses the nature of interactions

between lattice atoms and the typical parametric forms used for Ulat. Sections 2.2 and 2.3

discuss the functional forms used to represent sorbate-sorbate and sorbate-zeolite interac-

35

36 Chapter 2: Potential Energy Surface

tions respectively, with reference to three types of non-polar sorbates: rare gases, alkanes

and aromatic hydrocarbons. These three sections outline the general features of each type

of interaction and the factors which determine the functional form of the PES using some

of the force-fields available in the literature as illustration. The last section of this chapter

then discusses the specific functional form of the PES as used in this work. The nature of

the approximations implicit in the PES and its range of applicability are elucidated.

2.1 Interactions between Lattice Atoms

The three-dimensional framework of a zeolite is based on covalent linkages between the

tetrahedral (T) atoms, such as Si and Al, and oxygen atoms. The T-O bonds are very rigid

and directional in character, as expected in the case of covalent bonds. The typically large

difference in electronegativity between the T and O atoms, however, lends a significant

degree of ionic character to the T-O bonds with a fairly high negative partial charge on the

framework oxygen atoms. In the case of all-silica zeolites, the overall framework is neutral

but in the case of tetrahedral atoms carrying a formal charge of +3 or less, the framework

will carry a net negative charge that must be compensated by the presence of framework

cations. Such framework cations are bound by coulombic interactions and are mobile. In

the case of large polarizable anions and cations, polarisation and related many-body effects

may also play a significant role in determining the potential energy of the crystalline lat-

tice. Given the complexity of the interatomic interactions which determine the structure

and stability of the zeolite lattice, there is no unique parametric representation of the lattice

potential energy function. Most parametric representations choose as a starting point either

an ionic model or a valence bond one. The ionic model represents the zeolite as a collection

of charged species interacting via short and long-range forces[15, 99, 106, 118, 119, 129]. In

contrast the valence bond models depict the lattice atoms as bound by short-range two and

three-body forces[15, 130, 131]. The parameters that enter into the potential energy terms

Chapter 2: Potential Energy Surface 37

are obtained by a variety of methods, such as quantum mechanical calculations on small

molecules which mimic the zeolite structure. The lattice potentials are tested by compar-

ison of calculated IR spectra and lattice geometries with experimental spectroscopic and

crystallographic data[129]. Different functional forms and parametrisations are appropriate

depending on the nature of the zeolite and the physical problem of interest[132].

As an illustration of the manner in which Ulat is computed in simulations, a widely used

parametric representation of the zeolite lattice potential energy function, based on the ionic

model, developed by Catlow and co-workers for all-silica zeolites is discussed[99, 118]. They

partition the total lattice potential energy, Ulat, as:

Ulat = Ubuck + Uthree−body + Ucoul (2.2)

Ubuck is a short-range two-body Buckingham function for interaction of T-O(T=Si,Al) and

O-O interactions. Uthree−body is a three-body O-Si-O anharmonic potential chosen to de-

scribe bond angle-bending forces. Ucoul represents long-range electrostatic interactions.

The Ubuck terms sums over all unique Si-O and O-O bonds. The pair potential, ubuck(rij),

for each type of bond is represented as follows

ubuck(rij) = Aij exp(−rij/ρij) rij < r1 (2.3)

= Bijr5ij + Cijr

4ij + Dijr

3ij + Eijr

2ij + Fijrij + Gij r1 < rij < r2 (2.4)

= Hijr3ij + Iijr

2ij + Jijrij + Kij r2 < rij < r3 (2.5)

= −Lij/r6ij r3 < rij < rc (2.6)

The function is splined at r1, r2 and r3 so that the potential has continuous first and second

derivatives, and a minimum at r2. An exponential repulsion term is used at very close range

which shifts to a polynomial form at intermediate distances. For r values greater than r3

the potential is attractive, rather than repulsive and is represented by an r−6 dispersion

term. The spherical cut-off distance employed during the simulations is denoted by rc.


To model the O-Si-O bond angle-bending forces, the following three-body potential was

used:

Uthree−body =∑

angles

(1/4)AijkB2ijk exp(−rij/ρ1) exp(−rik/ρ2) (2.7)

where

Aijk = Kijk/2(θ0 − π)2 (2.8)

and

Bijk = (θ0 − π)2 − (θ − π)2 (2.9)

Kijk is the three-body spring constant and θ0 is the equilibrium bond angle (109◦28′ in this

case) and θ is the calculated bond angle.

Electrostatic interactions are modelled by placing electric charges on the framework

atoms which result in the Coulombic term which sums over unique pairs of partial charges

as[33]:

Ucoul = (1/2)∑

i,j,i6=j

qiqj/rij (2.10)

where qi and qj represent the electric charges on the ith and the jth framework atom sepa-

rated by a distance rij . The coulombic interactions are long-ranged and may become more

important than the Lennard-Jones terms. The charges on the framework atoms depend on

the Si/Al ratio and are assigned by optimizing the computed structure. For ZSM-5, the

partial charges were taken as +2e on the Si atoms and -1e on the oxygen atoms.

Since Catlow et al were interested in studying all-silica zeolites which are essentially

polymorphic forms of silica, the potential energy parameters were taken to be the same as

those developed by Vessal et al in the study of SiO2[99]. They were fitted to reproduce

the structural and elastic properties of α-quartz and were shown to provide a good de-

scription of the dynamical processes underlying the melting of β-crystoballite. Alternative

parametrisations of Ulat have been developed. For example, Nicholas at al have developed a

force field for silica sodalite which is designed to reproduce the IR spectrum and radial dis-


Table 2.1: IR frequencies(in cm−1) of Silica Sodalite

Experimental frequencies as well as those obtained from molecular dynamics(MD) and

normal mode analysis(NMA) using the PES of ref[130], are shown. The ring vibrational

frequency could not be assigned from the MD data.

MD NMA expt

ring . . . 302 289

O-Si-O 456 481 450

Si-Osym 776 796 787

Si-Oasym 1106 1108 1107

tribution function of silica sodalite. As can be seen from Table 2.1, the agreement between

the calculated and experimental values is very good. A simple computationally convenient

model potential has been proposed by Demontis et al in which the potential energy of a

system of atoms is expanded in a power series around equilibrium distances of single po-

tential functions between particles[15]; retaining terms to second order is then equivalent

to an equilibrium normal mode analysis. Better accuracy can be obtained by retaining

terms upto third order for the Al-O, Si-O and O-O contacts and upto fifth order for M-O

(M=Na+, Ca2+ etc). While this model provides a simple way to incorporate the effect of

lattice vibrations on sorbate dynamics, it is not accurate enough for structural studies of

aluminosilicates.


2.2 Sorbate-Sorbate Interactions

2.2.1 Rare Gases

The approximation of pairwise additivity is an excellent one in the case of rare-gases and

therefore one can write

Uss(r) =N∑

i=1

N∑

j=1,j<i

vpair(rij) (2.11)

where r is a 3N -dimensional vector representing the positions of all the rare gas atoms.

The dominant contributions to the pair potential are from the short-range repulsion and

long-range dispersion interactions. When two atoms are brought very close to one another,

there is a large increase in the potential energy of the system as the orbitals of the atoms

overlap. Since this contribution is due to the interaction between electrons with the same

spin, therefore the short range repulsive forces are also known as exchange or overlap forces.

Quantum mechanical calculations suggest an exponential form to model this short-range

two-body contribution. However, an inverse twelfth power term is found to be reasonable

for rare-gases and is also found to be computationally cheap. The dispersion interaction

arises due to correlations between instantaneous multipoles which are generated due to

fluctuations in the electron clouds of two atoms and can, in general, be written as a sum of

contributions[33]:

Udisp = −C6

r6− C8

r8− C10

r10− .... (2.12)

where the first term is the most important contribution. The minus signs indicate that

the dispersion terms are all attractive. The simplest parametrisation of the rare gas pair

potential is therefore provided by the 6-12 Lennard-Jones (LJ) potential which has the form

[8]-[12], [33]

vpair(r) = 4ε[(σ

r)12 − (

σ

r)6] (2.13)


where ε is the LJ energy parameter and σ, is the LJ size parameter. ε represents the

minimum energy of the sorbate-sorbate interaction and σ represents the distance between

the interacting sorbates when the energy of interaction between them is zero. The two

parameters, ε and σ, therefore index the two characteristic properties of Lennard-Jones

sorbates i.e. their polarizability and size respectively.

2.2.2 Alkanes

In the case of molecular sorbates, it is necessary to subdivide the sorbate-sorbate interaction

into intra- and inter-molecular contributions:

Uss = Uintra + Uinter (2.14)

The potential used by Catlow and co-workers[17, 97, 99, 104, 119] is used as an example

of the type of parametric potentials used for the sorbate-sorbate potential energy. The

intramolecular interaction is due to atoms belonging to the same sorbate molecule and may

be further subdivided as follows:

Uintra = Ustretch + Ubend + Utor + Ucoul (2.15)

The Ustretch and Ubend terms reflect changes in the potential energy due to deviation of bond

lengths and bond angles from their equilibrium values. Since these deviations are typically

small for hydrocarbons, harmonic potentials are sufficient. For individual bond stretches

the contribution may be written as:

ustretch = (1/2)ks(r − ro)2 (2.16)

where ro is the equilibrium bondlength and the force constant, ks, will necessarily be differ-

ent for C-H and C-C bonds. Similarly, contribution due to distortion of a bond angle can

be written as

ubend = (1/2)kb(θ − θo)2 (2.17)


where θo is the equilibrium bond angle and kb is parametrised to reflect differences between

H-C-H, C-C-C and C-C-H bond angles. For alkanes, proper modeling of the molecule

requires representation of the torsional interaction associated with conformational changes

involving four atoms which are in different planes. The functional form of the torsional

potential associated with an individual dihedral angle, φ, is taken as

utor = kt(1− cos(nφ)) (2.18)

where kt is a four-body force constant and n is an integer.

The intermolecular sorbate-sorbate interaction may be partitioned as:

Uinter = ULJ + Ucoul (2.19)

The first term represents the net effect of the short-range repulsion and dispersion inter-

actions and is typically modelled by a pair additive Lennard-Jones potential[99, 118]. The

interaction energy, uLJ(rij) between atoms i and j separated by a distance rij is

u(rij) =Bij

r12ij

− Cij

r6ij

(2.20)

The above representation is often used in place of the ε, σ representation given in equation

(2.13), since the coefficients of the dispersion and repulsion terms are often optimised inde-

pendently during parameter fitting. The Ucoul term comes from electrostatic interactions

between partial charges located on the atoms. The partial charges assigned to the sorbate

molecules depends on the experimentally determined moments of the sorbate molecules.

For hydrocarbons, bond polarities are very small and the Coulombic term is unimportant.

2.2.3 Aromatic Hydrocarbons

Detailed simulation studies are currently being carried out on sorption and diffusion of aro-

matic hydrocarbons in a number of zeolites since aromatic alkylation is one of the main

applications of zeolite catalysts[108, 109, 120, 121, 122]. The sorbate-sorbate interaction


for aromatic hydrocarbons is complicated, in comparison to alkanes, because of the pres-

ence of planar π-electron systems. Modelling the interaction between the aromatic rings

is crucial when simulating molecular crystals and liquids. However, the interactions be-

tween aromatic ring systems is less crucial for aromatic hydrocarbons adsorbed in zeolites

since the intracrystalline sorbate densities are generally small. As in the case of simple

alkanes, the sorbate-sorbate potential energy is partitioned into intra- and intermolecular

contributions[106, 118]. The intramolecular contribution is subdivided into contributions

due to stretches (two-body terms), bends (three-body terms), torsions (four-body terms) as

well as Coulombic interactions. In this parametrisation, the main difference between alkanes

and aromatic hydrocarbons is the necessity for treating hydrogens and carbons belonging

to the ring and to the side chains separately. For example, in the case of xylene, the ring

and methyl hydrogens are assigned partial charges of +0.153e− and -0.110e− respectively.

The intermolecular sorbate-sorbate interaction is parametrised as indicated in equations

(2.19) and (2.20). The Coulomb term is usually much more significant for aromatics than

for alkanes.

2.3 Sorbate-Zeolite Interactions

The physical contributions to the sorbate-zeolite interaction energy come from short-range

repulsion-dispersion interactions as well as long range electrostatic multipolar and inductive

interactions. The short-range repulsion and dispersion interactions will be present for both

polar and non-polar sorbates in all types of zeolites. The electrostatic multipolar interaction

will be very important only for sorbates possessing dipole or quadrupole moments but will

be absent for non-polar sorbates such as rare gases and saturated hydrocarbons. The

induction interaction will also be universally present for all types of sorbates since the

presence of the anionic zeolite framework and the mobile cations will tend to distort the

electronic charge distributions of the adsorbed molecules. However, the importance of the


induction interaction will vary significantly depending on the nature of the sorbate and

sorbent.

Before discussing the parametric forms used for Usz in simulation studies, we first con-

sider an important assumption that is made for the majority of currently-used sorbate-

zeolite force-fields. The assumption is based on the pioneering work of Kiselev and co-

workers who suggested that it is reasonable to assume that the silicon and aluminium ions

of the zeolite structure do not interact directly with the sorbate atoms[133]-[135]. It was

assumed that the influence of these two atoms is manifested through the effective charge and

polarizability of the oxygen ions which depend on the Si/Al ratio. This approximation can

be justified on two counts: Firstly, the silicon and aluminium ions are completely shielded

by oxygen ions and secondly their polarisability is very small and hence their contribu-

tion to the energy of dispersion interaction can be neglected. The Kiselev model therefore

assumes that the lattice tetrahedral atoms are effectively invisible to the sorbate atoms.

When computing the sorbate-zeolite interaction energy, this is equivalent to ascribing zero

charge, van der Waals radius and polarizability to the tetrahedral atoms. The Kiselev

approximation has been used in numerous simulations without giving rise to any major

discrepancies. However, recent work indicates that for very small sorbates, such as helium

atoms, the approximation may break down[64]. Moreover, ab initio computations indicate

that the ionic radii of the framework Si/Al atoms may be sufficiently large that they may

protrude into the intracrystalline void space and reduce the pore volume[17].

A commonly used parametric form for the sorbate-zeolite interaction potential takes

into account the short-range repulsion, dispersion and electrostatic multipolar interactions

by using a sum of Lennard-Jones and Coulombic interactions

Usz =Nz∑

i=1

N∑

j=1

qiqj

rij+ 4εij

[(σij

rij

)12

−(

σij

rij

)6]

(2.21)

The partial charges can be initially assigned on the basis of the electrostatic charge distri-

bution of the bare zeolite and the isolated sorbate. The LJ interaction parameters between


unlike atoms can be obtained from the parameters of the like pairs using mixing rules. For

example, the Lorentz-Berthelot mixing rule gives the collision diameter σAB for the A-B

interaction as the arithmetic mean of the values for the two pure species and the well-depth

εAB by the geometric mean :

σAB = (σAA + σBB)/2 (2.22)

εAB =√

εAAεBB (2.23)

While the above simple prescription will give physically reasonable values for the parameters

in equation (2.21), it is common to reparametrise by fitting simulation results to adsorption

data such as isosteric heats of sorption.

For rare gases or spherical top molecules, the electrostatic multipolar interaction is either

absent or negligible. In such cases the induction interaction may be significant and can be

accounted for by a functional form of the type[132]:

Usz = −N∑

j=1

αjE2(rj)2

+Nz∑

i=1

N∑

j=1

4εij

[(σij

rij

)12

−(

σij

rij

)6]

(2.24)

where αj is the polarizability of the j-th atom and E(rj) is the electric field strength at the

position rj of the j-th atom. The electric field strength is calculated on the basis of the

partial charges assigned to the lattice atoms.

2.4 Potential Energy Surface of Lennard-Jones Sorbates in

Zeolites

2.4.1 Lennard-Jones Sorbates

In this section, we discuss the potential energy function used to study different aspects

of sorbate diffusion in zeolites in our work. As discussed in the introduction, we restrict

ourselves to simple, non-polar sorbates which can be characterised by their size and po-

larizability. While the rare gases form the most obvious representatives of this class of


Table 2.2: Potential parameters for simple Lennard-Jones sorbates

εSS(in kJ/mol) and σSS(in A) correspond to the Lennard-Jones well-depth and size param-

eters respectively for the sorbate-sorbate interaction

Sorbates He Ne Ar Kr Xe CH4 CF4 CCl4 SF6 SnBr4

εSS 0.085 0.28 1.183 1.36 3.437 1.23 1.27 2.72 1.67 3.87

σSS 2.28 2.85 3.35 3.827 3.85 3.73 4.70 5.881 5.51 6.666

sorbates, a number of spherical top molecules can be considered to fall in this category

as a first approximation, unless there is strong coupling of the intramolecular degrees of

freedom to the lattice vibrations. We therefore denote atomic and spherical top molecules

as Lennard-Jones sorbates. Table 2.2 lists the LJ well depth and size parameters for several

such atoms and molecules. A number of the molecules are of considerable practical interest.

The sorbate-sorbate interaction potential for such systems will be given by equations (2.11)

and (2.13).

2.4.2 Rigid Lattice Approximation

In all our studies, we assume that the zeolite lattice is rigid and immobile. The zeolite

framework is therefore assumed to exert a periodic confining potential which regulates the

motion of the sorbate molecules but the atoms of the zeolite lattice are assumed to be

stationary. The physical basis for this approximation is the large separation between the

time scales of the lattice vibrations and the sorbate diffusional modes at temperatures of the

order of 300K and at moderate sorbate loadings. For example, Table 2.1 shows that peaks

in the IR spectrum of silica sodalite occur above 200 cm−1. In contrast our instantaneous

normal mode analysis, discussed in Chapter 4, shows the frequencies associated with xenon

confined within zeolite lattice to be of the order of 10 cm−1. While some low frequency


phonon modes can and do couple to sorbate modes, this is unlikely to play a major role

except in very special situations. This is substantiated by previous studies comparing

diffusional behaviour of methane in rigid and flexible zeolite frameworks indicating that the

inclusion of lattice vibrations has a relatively small effect on diffusion constants [136]-[138].

Studies on sorbates in other zeolites, including Na-A and Na-Y, indicate that the effect of

lattice vibrations on diffusional properties is relatively small in cation-free zeolites.

While the presence of lattice vibrations assists the equilibration process in the NVE

ensemble simulations, it is by no means essential provided equilibration times are sufficiently

long [91]. Moreover, our primary interest in this work is to study the effect of the confining

zeolite lattice on sorbate dynamics. Significant qualitative trends in diffusional behaviour

of simple sorbates will not be altered by the inclusion of a flexible, as opposed to rigid,

framework in the simulations. In this context, several recent works on unusual diffusional

properties of sorbates in zeolites which assume a rigid lattice may be cited, including studies

on the levitation effect[67]-[73] , single-file diffusion[74]-[82] and diffusional anisotropy [88]-

[91], which use the rigid lattice assumption. Therefore, we feel that for an exploratory study

of the type presented in this work, the rigid lattice approximation is adequate provided

appropriate care is taken with regard to equilibration.

2.4.3 Neglect of Electrostatic multipolar and Induction Interactions

In the case of rare gases, the electrostatic interaction is absent. For the spherical top

molecules, the multipoles are of very high order and therefore may be neglected. The

importance of inductive interactions depends both on the polarizability of the sorbates as

well as on the nature of the zeolite. In all-silica zeolite analogues with small channels,

the electric field does not vary much across the channel and as a consequence induction

interactions can be neglected. In cation-containing zeolites, the cations create an intense

local electric field giving rise to strong sorption sites and a relatively large contribution of


the induction energy to the total potential energy[139]. Even so, the contribution due to

the induction interaction is generally much smaller than that due to dispersion-repulsion

interactions in the case of Lennard-Jones sorbates. For example, switching off the induction

contribution for the rare gases in the Kiselev potential discussed in NaY results in only a

20% change in the computed isosteric heat of sorption(see Section 5.3.3)

2.4.4 Potential Energy Function

The rigid lattice approximation is equivalent to assuming that Ulat makes a constant con-

tribution to the potential energy of the sorbate zeolite system and therefore need not be

explicitly considered when performing a simulation i.e.

U = Usz + Uss (2.25)

The sorbate-sorbate potential is given as a pair-additive Lennard-Jones potential, as in

equations (2.11) and (2.13). The functional form of the sorbate-zeolite potential is based

on the Kiselev model. Since the induction interactions are not considered in our study,

equation (2.24) may be re-written as:

Usz =∑

l

nl∑

j=1

4εlS

N∑

i=1

(σ12lS

r12ij

− σ6lS

r6ij

)(2.26)

where l numbers the types of atoms present other than tetrahedral atoms such as Si or Al,

nl is the number of lattice atoms of type l, N is the total number of sorbate atoms, rij

is the distance between sorbate atom i and lattice atom j and εlS and σlS are Lennard-

Jones parameters for interaction between lattice atom of type l and the sorbate S. Given

the LJ parameters for the l-l and S-S interactions, the appropriate values of εlS and σlS

can be computed using the Lorentz-Berthelot combination rules: (i) εlS =√

εllεSS and (ii)

σlS = 0.5(σll + σSS) [33]. In the case of all-silica zeolites, the lattice oxygen atoms are

considered and therefore only the O-O and S-S self-interaction parameters are needed.

Chapter 3

Computational Methods

Molecular Dynamics (MD) simulations in the microcanonical (NVE) ensemble provide ac-

cess to equilibrium static and dynamic properties of molecular systems, provided the con-

stituent atoms can be treated as classical particles. Consequently, MD has been the primary

simulation method used in this study of sorption in zeolites. The MD algorithm, as imple-

mented in our simulation code, is described in Section 3.1. Factors influencing the choice of

simulation parameters as well as convergence issues are discussed in this section but com-

putational details regarding specific systems are provided in the relevant chapters. Some

macroscopic properties, such as the internal energy, are readily obtained from a molecular

dynamics simulation. Others, such as the diffusion coefficient and the free energy, must be

evaluated by special techniques or may prove to be relatively difficult to estimate accurately.

Since the primary focus of this study is the diffusion of sorbates in microporous environ-

ments, Section 3.2 considers in some detail the estimation of self-diffusion constants and

related quantities, such as ballistic to diffusional crossover times, from molecular dynamics

simulations. The excess chemical potential, which governs the distribution of the sorbate

between the vapour and solid phases, plays a central role in adsorption studies and cannot be

directly obtained from an MD simulation. Section 3.3 discusses the frequently used Widom

approach to measure the chemical potential in the context of adsorption studies. One of the

49

50 Chapter 3: Computational Methods

recent developments in understanding liquid state dynamics is instantaneous normal mode

analysis (INM) as a tool for understanding the short-time dynamical behaviour. In this

thesis, INM analysis is extended to fluids in confining media. Section 3.4 present a brief

review of the aspects of INM analysis of interest from the point of view of this work.

3.1 Molecular Dynamics Simulations of Lennard-Jones Sor-

bates in Zeolites

3.1.1 Integration of Newton’s Laws of Motion

In a Molecular Dynamics simulation, the trajectory of a set of N atoms is followed by

integrating the corresponding equations of motion. In the microcanonical ensemble, where

the total energy E and volume V are kept constant, the equations of motion correspond

to Newton’s laws of motion. Consequently, positions and momenta sampled along the MD

trajectory can be used to calculate both static and dynamic properties of the system. In

contrast, in other ensembles, the classical equations of motion must be modified, typically

using extended Lagrangian approaches, and therefore the dynamical information is no longer

reliable though static quantities can be obtained[11, 12, 140, 141, 142]. The discussion here

is restricted to the MD algorithm as formulated in the microcanonical ensemble.

Consider a system of N particles of mass m with positions and momenta given by

the 3N -dimensional vectors, r = {x1, . . . , x3N}, and p = {p1, . . . , p3N}, respectively. The

coordinate system is assumed to be Cartesian. The Hamiltonian, H, for the system is

written as:

H =3N∑

i=1

p2i

2m+ U(r) (3.1)

where U(r) is the potential energy function describing the interactions of the N -particles.

Chapter 3: Computational Methods 51

Hamilton’s equations of motion are then written as

∂pi

∂t= −∂H

∂xi(3.2)

∂xi

∂t=

∂H∂pi

(3.3)

One can rewrite equation (3.2) as Newton’s second law of motion:

m∂2xi

∂t2= −∂U

∂xi(3.4)

where −∂U/∂xi corresponds to the force acting along coordinate xi. Thus to obtain the

classical trajectory for a system of N particles, one must integrate either 6N coupled first-

order differential equations or 3N coupled second-order ordinary differential equations.

A large number of numerical methods exist for the solution of coupled ordinary differen-

tial equations since these arise in a number of different areas of physics and engineering[143].

The appropriate choice for MD simulations depends on some special considerations. In

virtually all atomic and molecular systems, the time taken to compute the forces is compu-

tationally much more demanding than the time taken to integrate the equations of motion.

Since the goal of an MD simulation is to provide an adequate sampling of configurations

from the microcanonical ensemble, it is essential to run trajectories that are long compared

to molecular time scales of vibration or rotation. Therefore, algorithms which are stable

and have good long-term energy and momentum conservation properties are preferable to

algorithms which have excellent accuracy for short time scales but relatively less reliable

long-term conservation properties. It is also essential that the algorithms be formulated in

such a manner that they retain the time-reversibility of the equations of motion. It turns

out that one of the earliest algorithms used for MD simulations, the Verlet algorithm, is

one of the simplest and best algorithms for MD purposes and is the algorithm used in this

work for simulating atomic and pseudo-atomic sorbates in zeolites[144]. In most present-

day simulations, it is useful, though not always essential, to go beyond the Verlet algorithm

and consider multiple-time step methods only when there are a large number of different


time scales present in the physical system e.g widely different frequencies for intramoelcu-

lar and intermolecular vibrations[145, 146]. For simulations of guest molecules in zeolites,

this becomes necessary when we allow vibrational motion of the lattice since the diffusional

time scales associated with the sorbates and the vibrational frequencies of the lattice differ

typically by an order of magnitude. Since in our work, the zeolite lattice is assumed to be

rigid (see Section 2.4.2), the use of multiple-step methods is not warranted.

The Verlet Algorithm is based on the Taylor expansion of the particle configuration,

r(t) about the current time t and is derived below. To propagate the system trajectory by

a time step h, we can write the position at time t + h as:

r(t + h) = r(t) + hv(t) +12!

h2a(t) +13!

h3b(t) +O(h4) (3.5)

where the velocity and acceleration vectors, v and a respectively, are defined as:

v =

(∂x1

∂t, . . . ,

∂xi

∂t, . . . ,

∂xN

∂t

)(3.6)

a =

(∂2x1

∂t2, . . . ,

∂2xi

∂t2, . . . ,

∂2xN

∂t2

)(3.7)

The vector b is clearly a similar set of third derivatives. Since the equations of motion are

time reversible, we can write the particle positions at time t− h as:

r(t− h) = r(t)− hv(t) +12!

h2a(t)− 13!

h3b(t) +O(h4) (3.8)

From equation (3.4), we know that the forces acting on the particles at time t, denoted by

the vector F(t), can be computed from the gradient of the potential function, U(r(t)), at

time t. Therefore, by adding equations (3.5) and (3.8), we get,

r(t + h) = 2r(t)− r(t− h) + h2F(t)m

+O(h4) (3.9)

where the estimation of the new position involves a truncation error of the order h4. This

formulation of the Verlet algorithm is time-reversible, simple to implement, accurate and

stable. To initiate the trajectory, the positions at two successive time points are required.


The velocity does not appear explicitly and is not needed for computing trajectories. The

velocity is necessary, however, for calculating the kinetic energy and hence the total energy

of the system. To obtain a symmetrised, time-reversible expression for the velocity, one can

subtract equation (3.8) from equation (3.5), to obtain

r(t + h)− r(t− h) = 2hv(t) +O(h3) (3.10)

and rearrange to give

v(t) =r(t + h)− r(t− h)

2h+O(h2). (3.11)

Since the velocity is obtained as a difference of two large quantities, the numerical impre-

cision introduced is relatively large. Moreover, the truncation errors are of different orders

for the velocities and positions, which may prove problematic when monitoring the level of

energy conservation and computing the system temperature.

To overcome the disadvantages associated with the velocity computation in the simple

Verlet algorithm, the velocity Verlet formulation of the algorithm was developed[147]. This

algorithm stores positions, velocities and accelerations all at the same time t and minimizes

round-off errors. The equation for propagating the particle positions is written as

r(t + h) = r(t) + hv(t) + h2F(t)2m

+O(h3) (3.12)

To obtain the expression for the velocity, the velocity at time t+0.5h is written as a Taylor

expansion about t:

v(t + 0.5h) = v(t) +h

2F(t)m

+O(h2) (3.13)

Since equation (3.12) gives the positions, and therefore forces, at time t + h, we can also

write

v(t + 0.5h) = v(t + h)− h

2F(t + h)

m+O(h2) (3.14)

Equating the RHSs of equations (3.13) and (3.14) and rearranging, we obtain

v(t + h) = v(t) + h[F(t) + F(t + h)]

2m+O(h2) (3.15)


The velocity Verlet Algorithm involves two stages with a force evaluation in between. In

the first stage, the new positions at time (t + h) are calculated using equation(3.12). Then

forces and accelerations at time (t + h) are computed and then velocity at time (t + h) is

calculated using equation (3.15). The kinetic energy at time (t + h) can be calculated at

this stage. The velocity is now calculated as a sum, rather than a difference, of two large

quantities which reduces the error relative to equation (3.11). Initialisation of the velocity

Verlet algorithm requires specification of both initial velocities and positions at some initial

time, t = 0. Numerical stability, convenience and simplicity make this algorithm the most

widely used one to date. In the MD simulations discussed in this work, the velocity Verlet

algorithm is used to propagate the trajectory of the sorbate molecules through the pore

spaces of the rigid zeolite lattice.

3.1.2 Potential Energy Surface and Force Calculation

The functional form of the potential energy surface used in our simulations is discussed

in Section 2.4. Within the rigid lattice approximation, the total potential energy can be

written as the sum of contributions from the sorbate-sorbate, Uss, and sorbate-zeolite, Usz,

interactions with the expressions for the sorbate-sorbate and sorbate-zeolite contributions

given by equations (2.11) and (2.26) respectively. In the absence of induction interactions,

both Uss and Usz are pair-additive potentials which simplifies the calculation of the forces.

The force acting on a sorbate atom i in the x-direction is then given by:

Fxi = −∂U

∂xi=

N∑

j=1,i 6=j

24εSS

r2ij

[2

(σSS

rij

)12

−(

σSS

rij

)6](xi − xj)

+∑

l

nl∑

k=1

24εlS

r2ik

[2

(σlS

rik

)12

−(

σlS

rik

)6](xi − xk) (3.16)

where j indexes the sorbate atoms, l indexes the types of lattice atoms and k indexes

the lattice atoms of each type. The total number of sorbate atoms is N . The number

of lattice atoms of type l is nl. The Lennard-Jones parameters for the sorbate-sorbate


interactions and the sorbate-lattice atom l interactions are identified by the subscripts SS

and lS respectively.

3.1.3 Periodic Boundary Conditions

In computer simulations, a relatively small, finite size system is used to mimic the macro-

scopic or bulk system by employing periodic boundary conditions. The simulation volume

is taken to be in the shape of a regular, space-filling polyhedron, such as a cube or a rhombic

dodecahedron. In the case of crystalline solids, such as zeolites, the unit cell of the zeolite

is a space-filling three-dimensional shape and the simulation cell is typically composed of

an integral number of multiples of the unit cell. If a simulation cell is taken to be a single

unit cell, then periodic boundary conditions imply that a particle i at position ri generates

image particles at ri + n1a + n2b + n3c where a, b and c represent the translation vectors

lying along three adjacent edges of the unit cell and (n1, n2, n3) represent the set of all

possible translations of the unit cell. In case the simulation cell is an integral multiple of

the unit cell, the lengths of the vectors a, b and c must be appropriately modified. While

there is, in principle, no restriction on the shape of the unit cell, in practice, the choice of

unit cells of orthorhombic, tetragonal or cubic symmetry makes the programming of the

periodic boundary conditions particularly simple. The Cartesian x, y and z axes can be

assumed to lie parallel to the crystallographic axes a, b and c respectively and the origin

of the Cartesian coordinate system can be placed at one vertex of the unit cell. As can be

seen from Section 1.1 of the introduction, all the zeolites studied in this work belong to the

cubic, orthorhombic or tetragonal space groups and therefore share this simplifying feature.

Special care must be taken when estimating dynamic properties in the presence of peri-

odic boundary conditions as these are properties which require measurement of the distance

moved by the particles of the system with time. In such cases, periodic boundary conditions

should not be imposed on the particles as then the actual distance moved by the particles


cannot be known due to the restriction imposed on it. This is the case for measurement

of the diffusion coefficient of sorbates in zeolites for which the mean-square distance moved

by the particles has to be computed as a function of time. In such cases, it is necessary to

keep account of the number of time the particle crosses the length of the simulation cell.

The imposition of periodic boundary conditions reduces the number of particles in the

system for which the positions and momenta have to be tracked but it still remains necessary

to consider the interactions between a particle in the central simulation cell with all the

(N − 1) other particles inside it as well as an infinite number of periodic images of these

N particles. If the range of the pair potential is short enough that the interaction between

any two particles dies off in a distance that is less than the size of the simulation cell, then

it is possible to make use of the minimum image convention to replace the infinite number

of images by just one set of images. The minimum image approximation assumes that

each particle interacts with only those particles which lie within a neighbouring volume

centred on the particle itself with dimensions identical to those of the central simulation

box. For example, in the case of a cubic simulation cell of edge length L, this may be stated

mathematically as follows: given particles i and j at positions ri and rj respectively, the

distance rij between the particles is given by rij = min{|ri − rj + Ln|} where n is the set of

possible translations. Thus each particle interacts with only the N−1 particles contained in

a cube of edge length L centred on the particle of interest. The applicability of the minimum

image convention depends crucially on the range of the interparticle interaction. If the pair

potential varies as r−n and n is more than the dimensionality of the system, the minimum

image convention may be applied. In the case of zeolites, the minimum image convention

may certainly be applied for the short-range repulsion and dispersion forces, encapsulated

in the form of Lennard-Jones terms. However, the electrostatic multipolar and inductive

contributions are typically long-range interactions; for example, in case of charged ions U(r)

varies as r−1 and, for dipolar molecules, U(r) varies as r−3. In such cases, the minimum


image convention cannot be applied and alternatives such as Ewald summation must be

used. In our simulations, the interactions were all sufficiently short-ranged that the use of

Ewald sums was unnecessary. The short-range of the dispersion and repulsion terms also

allowed for a further simplification to speed up the computations since a spherical cut-off

distance could be employed for the pair interactions i.e. the pair interaction was assumed

to be zero if the distance between a particle and its nearest image was greater than a cut-off

radius rc. The cut-off radius was typically chosen to be slightly less than half the shortest

edge length of the simulation cell.

3.1.4 Choice of Input Configurations and Velocities

In order to initiate the system trajectory, it is necessary to specify a suitable set of input

conditions. For example, initialising the velocity Verlet algorithm requires specification of

positions and velocities at time t = 0. In principle, the choice of initial conditions should

not affect the final results provided the simulation run length is long enough. In the limit

that the simulation run length approaches infinity, the system eventually settles down to the

equilibrium state and during the relaxation period, the system forgets the initial state. In

practice, however, simulations are of finite length and hence a physically reasonable choice

of initial conditions must be made. A poor choice of initial conditions results in statisti-

cal inefficiency of the simulation as then the system is placed in an irrelevant part of the

phase space. In the case of sorbate-zeolite systems, the positions of the lattice atoms can

be obtained from crystallographic data. Since the pore spaces in the zeolite lattice are eas-

ily identified, at low sorbate concentrations, we generated initial positions for the sorbate

particles by randomly placing them in the pore spaces and equilibrating the system at a

relatively high temperature. To generate input configurations at lower temperatures, the

system was slowly cooled starting from a well equilibrated high temperature run at approx-

imately 600 to 700K. To generate input configurations when the sorbate concentration was


high, sorbate particles were systematically added one at a time at a high temperature and

the system was allowed to equilibrate. High temperature equilibration was followed by slow

cooling to the temperature of interest.

The initial velocities were sampled from a Maxwell distribution at the particular temper-

ature of interest. The Maxwell-Boltzmann distribution giving the probability of observing

particle i with velocity in the x-direction, vxi is given by[7]

ρ(vxi) =

(mi

2πkBT

) 12

exp

(−1

2mi

v2xi

kBT

)(3.17)

at a temperature T . Similar equations can be obtained for all the velocities of all the

particles in the three Cartesian directions. Since the distribution of velocities is a product

of simple Gaussian distributions, random sampling can be easily performed using a Gaussian

random number generator.

3.1.5 Choice of the Time-step

Time integration algorithms for MD simulations are obtained by discretising Newton’s equa-

tions of motion for the system[11, 12]. The discretisation error will depend on the magni-

tude of the time step h. The value of h determines the accuracy of the computed trajectory.

There is no hard and fast rule to determine the value of the time-step for any particular

system, but the choice of h is important as it affects the accuracy of the computed proper-

ties as well as the statistical efficiency of the simulation. A very small time step will lead

to a very accurate trajectory but the sampling of phase space may be computationally very

inefficient i.e. a large amount of computer time may be required to generate a trajectory

that is sufficiently long that a reasonable sampling of microstates from the microcanonical

ensemble is obtained. A large time step on the other hand, may lead to significant lack

of energy conservation and may introduce significant numerical instabilities. Ideally, one

should aim for a correct balance between the two extremes so that the computed trajectory

follows the actual trajectory as closely as possible and, at the same time, the computer time


required to generate run lengths long enough to be ergodic is not excessive. In this work,

we have followed the general rule that the time-step should be small enough to conserve the

total system energy to the fourth significant figure.

3.1.6 Equilibration

The initial input configuration does not necessarily correspond to an equilibrium state of

the system. Having specified the initial conditions in the initialization stage, it is therefore

necessary to run the system trajectory for some time so that the system is allowed to ‘relax’

to the equilibrium state over a number of time-steps. During this equilibration period,

various properties of the system, such as the temperature, pressure and the total potential

and kinetic energies, must be monitored. Typically, during the equilibration period, the

running averages corresponding to the above observables will show a steady upward or

downward drift. This is because a physical quantity, A, generally approaches its equilibrium

value, Ae, exponentially with time i.e [11]

A(t) = Ae + C exp(− t

τ) (3.18)

where A(t) is the value of the physical quantity at time t. The important quantity to observe

here is the relaxation time τ . If τ is small compared to the MD simulation run, then we

observe A(t) converging to its equilibrium value Ae and we can make a direct measurement

of the property from the MD run. If however τ is large compared to the simulation run,

then A(t) does not converge to Ae and a reliable equilibrium value of A cannot be extracted

from the run. It is not possible to specify a priori the duration of the period of time for

which equilibration must be carried out since it depends on the system, the choice of input

configurations, the properties of interest and the temperature. However, the attainment of

equilibrium in a simulation is easily detected since the observables of interest then show

fluctuations about a steady mean value.

To promote a system to equilibrium in an MD simulation, energy is either added or


removed from the system until it reaches the desired value as indicated by the temperature

of the system. In our simulations, this was done by periodically rescaling the velocities

during the equilibration period. The logic of the velocity scaling approach is as follows.

Using the equipartition theorem for a classical system, one can relate the temperature

directly to the average kinetic energy, 〈K〉, of the system i.e.

〈K〉 = 1.5NkBT ∗ (3.19)

where T ∗ is the desired temperature. The instantaneous temperature, Tinst, of the system

is defined by the equation:

1.5NkBTinst = 0.53N∑

i=1

mv2i (3.20)

To make the instantaneous temperature equal to the desired temperature T ∗, we can mul-

tiply all the velocities by a scaling factor β such that

1.5NkBT ∗ = 0.53N∑

i=1

m(βvi)2. (3.21)

Clearly the scaling factor β must be

β =

√3NkBT ∗∑

i mv2i

(3.22)

Repeated rescaling of velocities every 10 to 100 time steps results in equilibration of the

system at the temperature of interest. Newton’s equations of motion are not followed during

velocity scaling and the total energy of the system is not conserved. Therefore configurations

should not be sampled from the equilibration period of a run since the microstates do not

belong to the microcanonical ensemble.

From a microscopic point of view, the attainment of equilibrium in a simulation proceeds

by randomisation of velocities by collisional processes and the redistribution of particles in

such a way as to sample regions of configuration space accessible at the temperature of

interest. At densities typical of liquids, equilibration of an MD simulation is facile. For

simulations of sorbates in zeolites, sorbate-sorbate and sorbate-lattice collisions will both


play a role in assisting equilibration. Clearly, a vibrating lattice will assist the randomisation

and equilibration process. Since our simulations are conducted in a rigid lattice framework,

it is necessary to pay special attention to equilibration and use relatively long equilibration

times. In Chapters 4 to 6, we have specified both the equilibration and production times

used for the simulations wherever relevant.

3.1.7 Production Runs

At the end of the equilibration period, the accumulators for the ensemble averages are reset

to zero and then the production phase of the MD simulation run begins. No velocity scaling

is carried out during the production phase and Newton’s laws of motion are therefore obeyed.

Energy is conserved to an accuracy dictated by the size of the time step, as discussed in

Section 3.1.5. The positions and velocities sampled from the production phase of an MD run

belong to the microcanonical ensemble and can be used to calculate observables of interest.

Since the observables take the form of ensemble averages, it is necessary to ensure that the

production run is of sufficient length that an acceptable statistical error is obtained for the

quantities of interest. The different types of properties that can be obtained from molecular

dynamics simulations and their ease of estimation are summarised below.

• Mechanical quantities are those that may be defined as functions of the positions

and momenta of the particles constituting the system at any instant of time. The

kinetic energy, temperature, pressure and potential energy may all be written as simple

ensemble averages of mechanical quantities and are estimated with a fair degree of

accuracy from relatively short MD runs. Fluctuations in these quantities may also

correspond to observables, such as the specific heat, and are generally more susceptible

to statistical error. An important mechanical property from the point of view of our

work is the instantaneous normal mode spectrum which is discussed in some detail

in Section 3.4. In our work, the run lengths for the simulations were sufficiently long


that simple mechanical averages converged to within 1% error.

• Time-correlation functions reflect the correlation between the value of an observable

A at some time t with the value of a second observable B at some time t′ where both A

and B are mechanical quantities. Time correlation functions are related to significant

dynamical properties of a system such as its diffusivity. Since the diffusion coefficient

plays a particularly important role in understanding sorbate mobility in zeolites, Sec-

tion 3.2 discusses the time correlation functions and related time-dependent quantities

that are relevant for estimating the self-diffusion constant from an MD simulation.

• Thermodynamic quantities such as the entropy, the free energy or the chemical poten-

tial cannot be directly obtained from a molecular dynamics simulation. Such quan-

tities are measures of the extent of phase space volume available to the system and

require special estimation methods. Section 3.3 discusses a commonly used approach

for measuring the chemical potential of a substance since the chemical potential plays

a crucial role in determining the distribution of the sorbate between the gas and solid

phases in an adsorption experiment.

3.2 Estimation of Diffusional Properties

Section 1.2 discussed the different types of diffusivity that may be defined for a sorbate

moving in a porous medium i.e. the transport, corrected and self diffusivities. The self-

diffusivity is the measure of mobility which emerges most naturally from a molecular picture

of diffusion and is therefore most straightforward to estimate from a molecular dynamics

simulation. As can be seen from the Einstein relation given in equation (1.9), the self-

diffusivity can be measured from the slope of the mean square displacement, denoted by

∆2r(t) = 〈|r(t)− r(0)|2〉 (3.23)


plotted against the elapsed time t. Since an MD run provides r(t), the calculation of

the diffusivity is simple, in principle. In practice, attention must be paid to a couple of

aspects. The first is the determination of the time of onset of diffusional behaviour, which

is discussed in Section 3.2.1. The second aspect concerns the run lengths necessary to obtain

an accurate value of the diffusion coefficient. Since the diffusion coefficient is a consequence

of the long-time dynamical behaviour of the systems, the MD run lengths to estimate it

with reasonable accuracy are much longer than those required to estimate simple mechanical

averages. Moreover, deviations from true diffusional behaviour and diffusional anisotropy

are additional aspects of importance when studying diffusion in zeolites. These points are

discussed in Section 3.2.2. An alternative to the Einstein relation for measuring the diffusion

coefficient is provided by the Green-Kubo relation between the velocity autocorrelation

function and the diffusion coefficient which is discussed in Section 3.2.3.

3.2.1 Ballistic to Diffusional Crossover

If the mean square displacement of a tagged particle is monitored over a time t that is

small compared with the mean collision time in the medium, then the random motion

characteristic of diffusional processes is not observed. Instead the particle moves under the

action of the forces due to the surrounding particles and is said to exhibit ballistic behaviour.

From Newton’s laws of motion, it is expected that in the ballistic phase the mean square

displacement will grow as t2 where t is the elapsed time. This can also be seen by taking

the t → 0 limit of equation (1.11) which is derived from a simple Langevin model i.e.

∆2r(t) = 2kBTB[t−mB(1− exp(−t/mB))] (3.24)

→ 2kBTB[t−mB(1− 1 +t

mB− t2

2m2B2+ . . .)] as t → 0 (3.25)

= (kBT/m)t2 (3.26)

= 〈v2〉t2 (3.27)


where 〈v2〉 is the mean square velocity at temperature T and B is the mobility coefficient. As

T increases, the particle undergoes collisions with other particles resulting in randomisation

of both its position and velocity. Once the randomisation process is complete, the particle

motion is said to be in the diffusional regime. The diffusional limit of the Langevin equation

is

∆2r(t) → (2kBTB)t as t →∞ (3.28)

The time, τc, required for crossover from the ballistic to diffusional regime will be related to

the interaction of the particle with the medium. In many cases, notably those discussed in

Chapter 6 of this work, there will not be a unique crossover point but an extended crossover

period.

During the course of an MD simulation, the mean squared displacement (MSD), ∆2r(t),

is calculated as

∆2r(t) = 〈|r(t)− r(0)|2〉 (3.29)

= 1/NN∑

i=1

1(tmax − t)

∫ tmax−t

0[ri(t + τ)− ri(τ)]2dτ (3.30)

where tmax is the duration of the simulation and i indexes the sorbate atoms. A log-

log plot of the mean-squared displacement, ∆2r(t), versus time, t, usually shows a fairly

distinct change of slope as the system crosses over from the initial ballistic regime for which

∆2r(t) ∝ t2 to the diffusional regime with ∆2r(t) ∝ t [136]. In our work, we have recorded,

in addition, the mean square displacements in the three Cartesian directions which are

equivalent, in the case of our sorbate-zeolite systems, to the three crystallographic directions.

3.2.2 Diffusion Coefficients

In the diffusional regime, the Einstein relation defines the diffusion coefficient, D, to be

〈|r(t)− r(0)|2〉 = 6Dt (3.31)


where r(t) is the 3N -dimensional position vector for the sorbate atoms at time t. The

direction-dependent diffusion coefficient in the x-direction, Dx, is defined as

∆2x(t) = 〈|x(t)− x(0)|2〉 = 2Dxt (3.32)

where x is an N -dimensional vector. The definitions of Dx and Dy are analogous. For

most sorbate-zeolite systems studied by us, the time period till approximately 1 ps could be

definitely classed as in the ballistic regime while beyond 10 ps, the system could be classed

as in the diffusional regime. A least-squares fitting procedure was used to obtain straight

line fits in the two regions using the expression:

ln〈|x(t)− x(0)|2〉 = ln(2Dx) + nx ln t (3.33)

and its analogues in the y and z directions. The three-dimensional generalisation is

ln〈|r(t)− r(0)|2〉 = ln(6D) + n ln t. (3.34)

Ideally the values of nx, ny, nz and n should be 2 and 1 in the ballistic and diffusional

regions respectively but in practice deviations from integer power dependence are observed

due to the effects of the confining potential [136]. Since such deviations from unity contain

interesting information on the degree of sub-diffusional character, we have used equations

(3.33) and (3.34) to fit our data rather than equations (3.31) and (3.32) which enforce an

integer power law dependence. The intercept of the plot of ln |x(t) − x(0)|2 against ln t is

taken to be 2Dx in the one-dimensional case and 6D in the three-dimensional case. The

ballistic to diffusional crossover time, τc, is taken to be the point of intersection of the

straight line fits to the data in the ballistic and diffusional regimes.

Since sorbate-zeolite systems are typically studied at quite low densities, fairly long

run lengths must be used to compute the diffusion coefficient accurately. The low sorbate

densities imply that equilibration is somewhat slow- this is typically not a problem when

evaluating simple thermodynamic averages but may prove to be so when evaluating the


slope of the mean square displacement. For long runs, a straightforward implementation of

equation (3.31) which records the MSD at equal time intervals proves to be fairly inefficient.

Therefore, for long runs lengths, an order-N algorithm was used to compute the mean-square

displacement as a function of time [12]. The order-N algorithm increases the time intervals

at which the MSD is evaluated with increasing time; thus it is able to capture the rapid

changes associated with shift from ballistic to diffusional motion as well as the long time

scale behaviour of the MSD. Errors in the diffusion coefficient are best estimated by using

block averaging[148]. However, since our models were well-studied systems, we preferred to

handle the problem by testing convergence using variable run lengths and reporting results

based on fairly long runs. A special situation arises for high symmetry zeolites, belonging

to the cubic or tetragonal space-groups, when evaluating the Dx, Dy and Dz coefficients.

Symmetry dictates that all three or at least two of the directional diffusion coefficients

must be identical- in practice, small inequalities and some dependence on initial conditions

may occur because, at low sorbate concentrations, the distribution of sorbates within the

simulation cell may not be completely isotropic. When presenting our results in Chapters

4, 5 and 6 we have discussed the potential sources of error with respect to specific systems.

3.2.3 Time-correlation Functions

The velocity autocorrelation function is defined as

Cvv(t) =〈v(t) · v(0)〉〈v2(0)〉 (3.35)

and is related to the diffusion coefficient, D, by the relation:

D = (1/3)∫ ∞

0Cvv(t)dt. (3.36)

We have, however, not used the integral of the velocity autocorrelation function to compute

the diffusion coefficient since the relation using the mean square displacement as a function

of time proves to be computationally more efficient. The directional analogues are defined


as:

Cvxvx(t) =〈vx(t) · vx(0)〉

〈v2x(0)〉 (3.37)

Cvyvy(t) =〈vy(t) · vy(0)〉

〈v2y(0)〉 (3.38)

Cvzvz(t) =〈vz(t) · vz(0)〉

〈v2z(0)〉 (3.39)

The structure of the velocity autocorrelation function can provide information on the short

and intermediate time dynamics of the system. The time, τn, at which Cvv(t) first turns

negative represents the average time at which the sorbate first encounters a repulsive barrier

and is often closely related to the crossover time from ballistic to diffusional motion. The

position of the first minimum in Cvv(t) indicates the average time at which the sorbate

is likely to reverse its direction of motion. As collisions with the wall and other sorbates

increase, the correlation function decays to zero.

3.3 The Chemical Potential

A thermodynamics text would define the chemical potential of a substance J in a mixture

as the partial molar free energy at constant temperature, pressure and composition i.e.

µJ =

(∂G

∂nJ

)

T,P,n′(3.40)

where nJ is the number of moles of J and the subscript n′ indicates that the number of

moles of all other substances in the mixture are held constant. At constant temperature

and volume,

µJ =

(∂A

∂nJ

)

T,V,n′. (3.41)

The Widom particle insertion method for estimating the chemical potential relies on a con-

venient computational interpretation of the above equation. In the case of a pure substance,

the compositional variables can be ignored, and for large N , the chemical potential can be


approximated in the canonical ensemble as

µ ≈ A(N + 1, V, T )−A(N, V, T ) (3.42)

The canonical partition function has been defined as

QNV T = (QidNV T /V N )ZNV T

= (QidNV T /V N )

∫dr1dr2 . . . drN exp(−βU(r)) (3.43)

= QidNV T

∫dr1dr2 . . . drN exp(−βU(r; L)) (3.44)

where L = V 1/3 and r = r/L are the scaled particle coordinates. The chemical potential

can be then written as a sum of the ideal and excess contributions i.e. µ = µid + µex The

ideal contribution is

µid = −kBT ln

(V

Λ3(N + 1)

)(3.45)

The excess contribution can be written as

µex = −kBT ln

(Z(N+1)V T

ZNV T

)(3.46)

= −kBT ln

(∫dr1 . . . drNdrN+1 exp(−βU(rN+1;L))∫

dr1 . . . drN exp(−βU(rN ; L))

)(3.47)

Writing the potential energy function of the ensemble containing N+1 particles as U(rN+1; L) =

U(rN ; L)+∆U(rN+1; L) where ∆U(rN+1; L) is the interaction of the N +1-th particle with

the other N particles. The ratio of the configurational partition functions as written in

the above equation may then be written as a Metropolis Monte Carlo average in the NVT

ensemble such that

µex = −kBT ln

⟨∫drN+1 exp(−β∆U(rN+1;L))

⟩(3.48)

To evaluate this integral, we can imagine inserting the N + 1-th particle as a test particle

in a typical configuration sampled from the NVT ensemble. The interaction of the test

particle with the remaining particles can be computed to obtain ∆U ; however, the test


particle by definition does not alter the Metropolis sampling in the NVT ensemble. Since

the average of an integral over the position of the test particle is required, it is simplest

to use a brute force approach and distribute the test particle positions uniformly over the

volume of the unit cell. However, positions corresponding to very high values of ∆U will

clearly make a negligible contribution to the integral and smarter schemes which account

for this non-uniformity can be devised.

The above derivation of the Widom method for determining the chemical potential is a

general one applicable to pure substances as well as mixtures. The method works specially

well for fluids at relatively low densities for which test particle insertions do not have a

high probability of generating repulsive overlaps. In the case of dilute solutions, the Widom

particle insertion method takes a particularly simple form. In the infinite dilution limit,

solute-solute interactions will not exist. One can then think of the solute molecule as being

a test particle inserted in the solvent. If the difference in chemical potential of the solute in

the solvent and in the ideal gas phase is denoted by µex, one can rewrite equation (3.47) as

µex = −kBT ln

(∫dr1 . . . drNdrsolute exp(−βU(rN , rsolute;L))∫

dr1 . . . drN exp(−βU(rN ; L))

)(3.49)

The integral in the denominator of the expression within brackets then represents an inte-

gral over accessible configurations of the solvent under the given temperature, volume and

density conditions. A gas or liquid adsorbed in a zeolite can be regarded as a solid solution.

The partition function of the zeolite lattice would then correspond to the denominator in the

above equation. The rigid lattice approximation is equivalent to stating that displacements

about the equilibrium configuration are so small that one can assume that the equilibrium

geometry is the only one that contributes significantly to the partition function. In this

case, we need to evaluate only the numerator by integrating over all possible locations of

the sorbate in the rigid lattice. In that case, equation (3.49) can be further simplified as:

µex = −kBT ln

(∫drsolute exp(−βUsz(r))

V

)(3.50)


At infinite dilution, the equilibrium constant describing the partitioning of the sorbate

between the gas and crystalline phases is termed Henry’s constant, Kh (see Section 1.2.1);

therefore Kh = exp(−βµex). As mentioned in Section 1.2.1, Henry’s constant is readily

obtained from the low pressure regime of the experimental adsorption isotherm and therefore

provides a convenient quantity for comparison of experimental and simulation results.

3.4 Instantaneous Normal Mode Analysis

3.4.1 INM Analysis of Liquid State Dynamics

The instantantaneous normal mode approach to dynamics and solvation in the liquid state

has been developed in the past decade by Stratt, Keyes and co-workers. The instantaneous

normal mode (INM) spectrum is obtained as the set of normal mode frequencies associated

with configurations sampled from some suitable ensemble. Since at finite temperatures

the configurations will not correspond exactly to local minima on the potential energy

surface (PES), the INM spectrum will have real and imaginary branches indicating the

extent to which positive and negative curvature regions of the PES respectively are sampled

by the system. The INM frequencies will be related to the short-time dynamics since

for sufficiently small displacements and therefore for sufficiently small times, a quadratic

expansion of the potential about any reference configuration will be adequate. This has

motivated the development of INM analysis as a tool to understand liquid state dynamics

and solvation in the ultrafast or short-time regime [149, 150]. Translational and rotational

velocity autocorrelation functions for molecular liquids can be reproduced for time scales

of less than a picosecond from INM data [151]. The fraction of imaginary modes can be

correlated with the self-diffusion constant. The degree of delocalisation of the imaginary

branch modes can be correlated with the onset of glassy behaviour [152].

While INM analysis to obtain short-time scale features must be correct in the limit that


the observed time displacement approaches zero, the extension of INM analysis to longer

time scales requires additional assumptions. Based on Zwanzig’s model of self-diffusion

in which a liquid hops between local minima on the PES with a lifetime in each minima

described by some survival time distribution, Keyes and co-workers have derived long-

time dynamical properties, such as the diffusion coefficient, from the INM spectrum [153]-

[156]. The Lyapunov spectra of Lennard-Jones liquids can also be derived from the INM

spectrum with the aid of a reasonable estimate of the decorrelation time [157]. Since

connecting the INM data to such long-time averaged dynamical quantities as the diffusion

constants or Lyapunov spectra requires additional assumptions about the nature of liquid-

state dynamics, its range of validity is still subject to discussion [158]. Despite this caveat,

simulations on a wide range of systems, including atomic clusters, molecular liquids, liquid

metals and ionic melts, have indicated that the INM spectrum is a useful indicator of

dynamical behaviour [159]-[165].

In this thesis, I consider the INM spectrum as an equilibrium property of the system

which contains information on the short-time system dynamics. This view of INM anal-

ysis is particularly convenient for systems or phenomena for which a reliable dynamical

simulation method does not exist. For example, for quantum many-body systems, path

integral methods provide a way to simulate static but not dynamic properties. Since the

INM spectrum is an equilibrium quantity it can be computed for a quantum system and

is relevant since there is no reliable simulation method for many-body quantum dynam-

ics [166, 167]. In the case of classical systems, for many problems such as adsorption or

phase transitions, ensembles other than the microcanonical are convenient [11, 12]. While

molecular dynamics schemes can be set up in other ensembles, such as the canonical or the

isothermal-isobaric, the interpretation of the simulation dynamics in terms of the physical

motion of the molecules can become tricky because of the fictitious dynamics associated

with macroscopic variables such as the temperature and pressure. The INM spectrum,


however, can be computed in any ensemble and may therefore be useful in such situations.

In addition, since the INM spectrum is a measure of the curvature of the potential energy

landscape, it can be a useful property to study for fluids confined in porous media since

confinement substantially affects the potential energy landscape. In Section 3.4.2, I briefly

summarise the key results on INM analysis as applied to atomic fluids. Extension of INM

analysis to fluids in porous media is discussed in Section 3.4.3.

3.4.2 Significant features of the INM Spectra of Atomic fluids

An instantaneous normal mode analysis is performed by expanding the potential energy

function to second order in the displacement, r(t) − r(0), where r(0) is the initial config-

uration of the N -particle system at time t = 0 and r(t) is the configuration after a short

interval of time t. The short-time classical Hamiltonian can then be written as

H ≈3N∑

i=1

mi

2

(dri

dt

)2

+U(r(0))−F′ · (r(t)−r(0))+0.5(r(t)−r(0))T ·D′ · (r(t)−r(0)) (3.51)

where F′ is the vector representing the forces acting on the system at t = 0 and D′ is

the second-derivative matrix or Hessian of the potential evaluated at r(0). Converting to

mass-weighted coordinates, zi =√

miri,

H ≈3N∑

i=1

12

(dzi

dt

)2

+ U(r(0))−F · (z(t)− z(0)) + 0.5(z(t)− z(0))T ·D · (z(t)− z(0)) (3.52)

where the derivative matrices F and D are constructed with respect to the mass-weighted

coordinates. The Hessian matrix can be diagonalised to give the eigenvalues, {ω2α, α =

1, 3N}, and eigenvectors W(r(0)). The eigenvalues correspond to the squares of the normal

mode frequencies. As mentioned in the introduction since an instantaneous configuration

need not necessarily (in fact at finite temperature is almost never) exactly at a minimum

in the PES, therefore there will always be a set of imaginary frequency normal modes.

Conventionally the imaginary branch is depicted on the negative frequency axis.


The INM spectrum or the normalised INM density of states is obtained by averaging

the INM frequencies over a set of configurations sampled from the equilibrium distribution

in any ensemble. Mathematically, it may be represented as

ρ(ω) =⟨

(1/3N)3N∑

α=1

δ(ω − ωα)⟩

(3.53)

If the system has any collective translational or rotational modes, as in the case of a cluster

or a liquid, they are removed from this distribution. For rare gas atoms moving within a

rigid zeolite framework, such zero-frequency modes are absent and therefore no modes need

to be explicitly removed from the INM distribution. Several features of the INM spectrum

can be correlated with the dynamical behaviour of the system particularly in the short-time

limit. For example, the fraction of imaginary modes, Fimag, indicates the extent to which

the system samples regions of negative curvature, including barrier and shoulder regions

of the PES. The Einstein frequency, ωE , is defined as: ω2E =

∫ω2ρ(ω) dω where ρ(ω) is

the normalised INM spectrum. Since ω2E = 〈U ′′〉/m(3N − 3) where 〈U ′′〉 is the ensemble

average of the trace of the Hessian D’ the Einstein frequency is a measure of the average

force constant, 〈U ′′〉, of the system. ωE can be decomposed into real, ωR, and imaginary,

ωI , components such that

ω2E = (1− Fimag)ω2

R + Fimagω2I (3.54)

where Fimag is the fraction of imaginary frequencies. An interesting quantity associated with

INM analysis is the participation ratio which measures the number of atoms participating in

that normal mode. An eigenvector representing the α-th instantaneous normal mode associ-

ated with some configuration is denoted by qα = {Wαj , j = 1, 3N} = {Wαix,Wαiy,Wαiz, i =

1, N} where in the second notation the Cartesian coordinates of each atom i are explicitly

indicated. The participation ratio is then defined as

Pα =1(∑N

k=1(W 2α,kx + W 2

α,ky + W 2α,kz)2

) (3.55)


In the normalization convention defined above, for a completely delocalized mode all the

components of the eigenvector will equal ±1/√

3N and Pα = N . For a localised mode

involving only one atom, Pα = 1. This convention is useful if one wishes to identify the

number of atoms participating in a given mode and has been used to compare the relative

degree of delocalisation of INMs in different frequency ranges for a number of liquids and

glasses. A participation ratio distribution, P (ω)dω, representing the average participation

ratio associated with INMs lying in the frequency range ω to ω + dω can be calculated

by ensemble averaging. It should be noted, however, that to rigorously identify the truly

delocalised modes which extend essentially over the entire simulation cell, it is necessary to

do a finite-size scaling study and normalise the participation ratio as P ′α = Pα/N . With

this normalisation, the true delocalised modes will have P ′α of the order of unity regardless

of system size whereas the modes involving localised clusters of atoms will have P ′α → 0 as

N →∞.

The properties discussed above are features of the equilibrium INM spectrum and can

be defined in any ensemble and for both quantum and classical systems. Unlike in many of

the studies of INMs in liquids, in this study we do not attempt to quantitatively reproduce

the diffusion constant, which is a long-time averaged property, from the INM spectrum.

However, for a classical system, it is possible to further analyse the short-time dynamics in

terms of the kinematics of the normal mode coordinates. By using the INM modes, W (r(0))

for some configuration, r(0), one can rewrite the Hamiltonian as:

H ≈ U(r(0)) +3N∑

α=1

(12

(dqα

dt

)2

+12ω2

αq2α − fαqα

)(3.56)

where the summation extends over all normal modes, indexed by α. The normal mode α is

defined as

qα(t, r(0)) =∑

i

Wαi{zi(t)− zi(0)} (3.57)

where the summation extends over all 3N atomic displacements. The force, fα, acting along


the normal mode direction qα is given by

fα(r(0)) =∑

j

WiαFi (3.58)

where i sums over the 3N atomic displacements. To remove the term linear in {qα} from

the Hamiltonian, one can define shifted normal mode coordinates, xα = qα− (fα/ω2α), such

that

H ≈ U(r(0)) +3N∑

α=1

(12

(dxα

dt

)2

+12ω2

αx2α − (f2

α/2ω2α)

)(3.59)

where the subscript α serves to distinguish the shifted normal mode corrdinate from the

Cartesian coordinates xi. |fα/ω2α| can be interpreted as the displacement of the normal

mode coordinate from its nearest extremum: minimum in the case of stable modes with

ω2α > 0 and maximum in the case of unstable modes with ωα < 0. For the stable modes,

if the harmonic picture were applicable, the canonical ensemble average of |f/ω2| will be

proportional to√

kBT/ω and the slope of log |f/ω2| versus log ω should be -1. Alternatively,

if f were constant with ω, then the corresponding slope should be -2. We term the ratio

|f/ω2| as the harmonicity ratio. For water, both the stable and unstable modes show a

linear dependence with slopes of -1.44 and -1.9 respectively [159]. The time evolution of the

shifted normal modes and the corresponding velocities are given by:

xα(t) = xα(0) cos(ωαt) + (vα(0)/ωα) sin(ωαt)

vα(t) = vα(0) cos(ωαt)− ωαxα(0) sin(ωαt)

Using these relations for the time evolution and assuming a Maxwell-Boltzmann distribution

of velocities at t = 0, Stratt et al have shown that the velocity autocorrelation function is

given by:

Cvv(t) =∫

ρ(ω) cos(ωt)dω

= 1− 〈ω2〉2!

t2 +〈ω4〉4!

t4 + . . . (3.60)


where 〈ωn〉 =∫

ωnρ(ω)dω. For simple nonpolar and dipolar liquids, the short time be-

haviour of the translational autocorrelation function was well reproduced by the above

formula provided only stable modes were used for evaluating the integral.

3.4.3 Extension of INM Analysis to Fluids Adsorbed in Zeolites

Confinement of a liquid in a porous medium will modify the dynamical behaviour, the

extent of modification depending on factors such as the pore volume, strength of inter-

action between adsorbed fluid and adsorbent, concentration of adsorbate and geometrical

features of the adsorbing host. The modification in the short-time dynamics and some as-

pects of the changed potential energy landscape due to confinement should be accompanied

by corresponding changes in the INM spectra. Initial work on extending INM analysis to

confined fluids done in our group, tested the usefuless of INM spectra for obtaining dynam-

ical information for this class of systems using both Monte Carlo and molecular dynamics

simulations of rare gases in zeolites[168]. While both classical and quantum sorbates were

considered, only the work on classical sorbates is relevant to my work. For classical systems,

the MD technique provides accurate dynamical information in the microcanonical ensemble

and comparisons between the “exact” behaviour and various INM quantities can be made.

In this section, I first review the main results obtained from the previous INM study of

atomic sorbates in zeolites. I then indicate how I have extended INM analysis to confined

fluids in my work. In particular, the formalism associated with obtaining INM spectra for

fluids in anisotropic confining media is discussed.

In previous work on INM analysis of atomic sorbates in zeolites, xenon adsorption in

all-silica polymorphs of four zeolites: silicalite (ZSM-5), faujasite (Zeolite Y), mordenite and

Na-A (Zeolite A), was examined. In contrast to the collective diffusional dynamics exhibited

by bulk liquids, short-time dynamics in a zeolite, particularly at low sorbate concentrations,

is largely ballistic and controlled by the sorbate-wall interactions. A number of interesting


features of the INM spectrum emerged in this low concentration regime. The high fraction

of imaginary frequencies reflected the curvature distribution of the confining medium and

did not necessarily correspond to barrier crossing or diffusional modes. The fraction of

imaginary modes increased with temperature and decreased with increase in the strength of

sorbate-sorbent binding. A correlation was found between the fraction of imaginary modes

and the long-time averaged diffusion coefficient though no simple proportionality relations of

the type observed in ionic melts could be deduced. The participation ratios were very close

to unity indicating the complete absence of any delocalised or collective modes and reflecting

the ballistic character of the short-time dynamics. Qualitative changes in the location of

xenon atoms in the α-cages for zeolites A and Y with temperature were reflected in the

temperature-dependence of the fraction of imaginary modes and the Einstein frequency.

The average potential energy, however, appeared to be insensitive to such changes in the

location of xenon atoms. Two INM features which were found to behave differently for

xenon in zeolite, particularly at low concentrations, when compared with liquids, were

the harmonicity ratio and the short-time velocity autocorrelation function computed from

the INM spectrum. While Cvv(t) obtained from INM and MD must agree as t → 0,

the time period for which the agreement survives was found to be strongly dependent on

concentration and temperature. In general, the agreement improved if simulation conditions

are chosen so as to reduce the fraction of imaginary modes. The plot of |fα/ω2α| versus ωα

on a log10 / log10 plot for the stable INM branch did not show the expected slope of -1 for a

perfectly harmonic system but an approximate slope of -1.9. With increasing concentration,

the emergence of delocalised, collective motions in the short-time dynamics is clearly shown

by the rise in the participation ratio of the low frequency modes. Based on this study, it

was concluded that instantaneous normal mode analysis would provide a useful diagnostic

for simulation studies of adsorption in random and ordered porous media.

In my work, I have extended instantaneous normal mode analysis to study the levitation


effect as well as diffusional anisotropy. The INM study of the levitation effect was based

on the key INM results described in Section 3.4.2 and the results are discussed in Section

4.3. The extension of the INM formalism to anisotropic zeolites is described below. The

application of this formalism to diffusional anisotropy is discussed in Chapter 6.

To define the components of the INM spectrum along the three Cartesian directions,

an eigenvector representing the α-th normal mode associated with some configuration is

denoted by

qα =N∑

j=1

∑

β

Wk,jβ |ξjβ〉 (3.61)

where ξjβ = zjβ(t) − zjβ(0) is the mass-weighted displacement of the j-th atom in the

β-direction with β = x, y or z. In the case of orthorhombic zeolites, projections along

Cartesian directions are equivalent to projections along the crystallographic axes. We wish

to separate the INM density of states into the direction-dependent densities of states ρβ(ω)

such that ρβ(ω) indicates the probability of finding a mode with frequency ω with a pro-

jection in the β-direction. Following earlier approaches for computing translational and

rotational components of INMs in molecular liquids, we define the projection matrix

P βkl = 〈qk|pβ pβ |ql〉 (3.62)

where

pβ|ql〉 =N∑

j=1

Wl,jβξjβ (3.63)

and the diagonal elements of the projection matrix are:

P βkk =

N∑

j=1

W 2k,jβ . (3.64)

Since P xkk + P y

kk + P zkk = 1, we can write

ρ(ω) = ρx(ω) + ρy(ω) + ρz(ω) (3.65)

where the projection of the INM density of states in the β-direction yields:

ρβ(ω) =⟨

(1/3N)3N∑

k=1

P βkkδ(ω − ωk(r))

⟩. (3.66)


Each of the direction-dependent INM densities of states will have a fraction of imaginary

modes, Fimag,β and given our definition, Fimag = Fimag,x + Fimag,y + Fimag,z.

Chapter 4

The Levitation Effect

4.1 Introduction

Zeolites display a wide range of pore sizes, channel geometries and network topologies and

a given zeolite will act as a sieve for sorbates with dimensions smaller than the narrowest

portion of the connecting channels or windows. A simple geometric picture of diffusional

behaviour would also lead one to expect that the self-diffusion coefficient, D, will be max-

imum for very small sorbates and will decrease with increase in sorbate size till the size

cut-off determined by the zeolite channel structure is reached. An analogy with gas phase

diffusion, appropriate for low sorbate concentrations, leads one to predict that the diffusiv-

ity, D, will vary as 1/σ2 where σ is the diameter of the sorbate [6]. At higher concentrations,

models based on size-dependence of diffusivity in liquids are more appropriate and lead to

the prediction that D will vary as 1/σ [6]. Previous work using molecular dynamics (MD)

simulations of sorbates in zeolites show that at low concentrations the 1/σ2-dependence is

commonly observed. However, in many zeolites, for a small range of sorbate sizes which

lie just below the minimum channel width, an increase in D with σ is observed rather

than a 1/σ2 decrease. This anomalous peak in transport properties has been termed the

levitation effect[67]-[73]. The levitation effect has been demonstrated in MD simulations of

81

82 Chapter 4: The Levitation Effect

Lennard-Jones sorbates in a number of zeolites, such as Na-Y, Na-A, ZSM-5 and VPI[67]-

[73]. Since the effect generally becomes more pronounced as temperature or concentration

are decreased, it would appear to be strongly correlated with the potential energy surface

imposed on the sorbate by the confining medium. A detailed study of adsorption sites and

minimum energy pathways of Lennard-Jones sorbates in Na-Y indicates that the onset of

the levitation effect can be correlated with the availability of additional absorption sites in

the windows of the sodalite cages [73]. It is reasonable to expect that if the levitation effect

is strongly determined by the nature of the PES, it should be associated with signatures

in the INM spectrum. As discussed in Section 3.4, the INM spectrum is obtained as the

set of normal mode frequencies associated with configurations sampled from some suitable

ensemble. Therefore, the INM spectrum closely reflects the curvature distribution of the

PES as sampled by the system and has been shown to be a good predictor of the short-time

dynamical properties of bulk and confined fluids. In this chapter, I consider the levitation

effect for Lennard-Jones sorbates of variable size and polarizability diffusing in Na-Y zeolite.

Dynamical information from molecular dynamics simulations is compared with predictions

based on instantaneous normal mode analysis. The INM spectrum is shown to carry several

striking signatures of the levitation effect. Aspects of molecular dynamics simulations and

instantaneous normal mode analysis relevant for this chapter have been discussed in Chap-

ter 3. Computational Details are given in Section 4.2. Results and discussion are presented

in Section 4.3 and conclusions in Section 4.4.

4.2 Computational Details

4.2.1 Zeolite Structure

Zeolite Na-Y belongs to the space group Fd3m with lattice parameter 24.85A and unit

cell composition Na48Si144Al480384 [19]. As discussed in Section 1.1, it is a high-porosity

Chapter 4: The Levitation Effect 83

zeolite with large α-cages approximately 11.8A in diameter, in addition to the smaller β-

or sodalite cages. Each α-cage is tetrahedrally connected to four other α-cages by windows

of approximately 8A diameter formed by 12 oxygen atoms. Since only water, cations and

very small sorbates, such as He and Ne, are able to enter the β-cages, diffusional behaviour

of molecular sorbates in Na-Y refers to diffusion through the tetrahedral lattice of α-cages.

The large pore size of the 12-ring window therefore determines the upper limit of sorbate

size which can be adsorbed by Na-Y.

4.2.2 Potential Energy Surfaces

The functional form of the potential energy surface for Lennard-Jones sorbates in Na-Y is

described in Section 2.4. The parameters used are identical to those employed in previous

studies of the levitation effect in Na-Y zeolite[70, 73]. In the case of Na-Y zeolite, the LJ

parameters for the O-O, Na-Na and S-S interactions (see Table 4.1) where O, Na and S

refer to lattice oxygen, sodium and sorbate atoms respectively are required. The appropriate

values of εlS and σlS can be computed using the Lorentz-Berthelot combination rules: (i)

εlS =√

εllεSS and (ii) σlS = 0.5(σll + σSS) where l is either a lattice oxygen or sodium

atom[33]. Since our interest is in understanding the size-dependence of the self-diffusion

coefficient, LJ parameters for O-O and Na-Na interactions are kept constant while varying

σSS , the sorbate size parameter. At a given value of εSS , σSS is varied between 2.67A and

7A and the σOS and σNaS values are correspondingly altered. Note that εSS and σSS values

for xenon are 4.1A and 221K respectively [34].

4.2.3 Molecular Dynamics Simulations

Molecular dynamics (MD) simulations were carried out in the microcanonical (NVE) en-

semble using the velocity Verlet algorithm [11, 12]. Initial velocities were sampled from a

Maxwell-Boltzmann distribution corresponding to some preset temperature and then tem-


Table 4.1: Potential energy parameters used for Lennard-Jones sorbates in Na-Y.

type σ ε

(A ) (kJ mol−1)

Xe-Xe 4.1 1.8378

O-O 2.5447 1.2891

Na-Na 3.369378 0.0392

perature scaling was carried out during the equilibration period. In keeping with previous

studies [70], the preset temperature was fixed at 190K; the mean temperatures in all the

runs were within ±5K of this value. A single unit cell of Na-Y zeolite was taken as the

simulation cell. Eight Lennard-Jones sorbate atoms of mass 131 amu were loaded in each

simulation cell. Cubic periodic boundary conditions were imposed. A spherical cut-off ra-

dius of 12A was employed for sorbate-sorbate and sorbate-zeolite interactions. A timestep

of 800 a.u(19.2fs) was found to conserve energy to better than the third significant figure for

a runlength of 192ns and was used for all the simulations. Runlengths ranged from 1.92ns

to 192ns with an equilibration period of 0.50ns. A rigid zeolite framework was assumed in

our simulations to ensure consistency with earlier studies of the levitation effect [11-13]. In-

stantaneous normal modes were calculated at intervals of 100 timesteps. The INM spectra

results from MD runs at the average temperature of the run, Tr, coincided with canonical

MC results at the same temperature within the error bars of the simulation [168].

The mean squared displacement, ∆2r(t), as defined in equation (3.31), was monitored

as a function of time. Figure 4.1 shows the log-log plots for ∆2r(t) versus t for xenon

(σSS=4.1A and εSS=221K) in Na-Y which shows clearly the transition from ballistic to

diffusional motion. In all the cases of Lennard-Jones sorbates in Na-Y zeolite studied


Figure 4.1: Crossover from ballistic to diffusional motion.

The crossover is seen in the log-log plot of ∆2r(t) (A2) versus t(picoseconds) for σSS=4.1,

ε=221K, and T=190K. The symbol ‘s’ indicates the values of the slope of the two lines in

the ballistic and the diffusional regimes.

0.0001

0.01

1

100

10000

1e+06

1e+08

1e+10

1e+12

0.01 0.1 1 10 100 1000 10000 100000 1e+06

∆2r(t) / A2

t/ps

diffusional(s=1.0)

ballistic(s=1.9)


here, the time period till approximately 1 ps can be definitely classed as in the ballistic

regime while beyond 50 ps, the system is definitely in the diffusional regime. It can be

seen that there is a short crossover region rather than a unique crossover time. To obtain

a reasonable approximation to the crossover time, τc, a least-square fitting procedure was

used to obtain straight line fits in the two regions and the point of intersection of the two

lines was taken to be τc. The slope in the diffusional regime was very close to one in all

the cases studied and therefore equation (3.34) could be applied to obtain the self-diffusion

coefficient. Convergence of the diffusion coefficient was tested with respect to run lengths

by computing D for each sorbate size and polarizability for run lengths of 1.92, 19.2 and

192ns. Reported values of D correspond to those calculated from the runs of longest length.

Error bars on D for these runs are of the order of ±10%.

4.3 Results and Discussion

4.3.1 Signatures of the Levitation Peak in the INM Spectra

The levitation peak has been defined as an anomalous peak in the values of various transport

coefficients, primarily the self diffusion constant, D, as a function of sorbate size. In this

section, we consider two significant properties which characterise the INM spectrum: the

fraction of imaginary modes, Fimag and the Einstein frequency, ωE , as a function of sorbate

size. Figure 4.2(a) shows the variation of the self-diffusivity, D, as a function of 1/σ2SS for

εSS=221K and reproduces the earlier results of Yashonath and Santikary [70]. The position

of the levitation peak at σSS=6A is the same in the two studies. Differences observed in the

values of D between their work and ours may be attributed to statistical errors. It may be

noted that the earlier study used much shorter run lengths of 2.6ns. Figure 4.2(b) shows the

fraction of imaginary modes,Fimag, as a function of 1/σ2SS . It is observed that the fraction of

imaginary modes essentially mirrors the behaviour of the diffusion coefficient as a function


Figure 4.2: Variation of (a) self-diffusivity,D, and (b) fraction of imaginary modes,Fimag,

with 1/σ2SS .

0.050.10.150.20.250.30.350.40.450.50.55 0.02 0.04 0.06 0.08 0.1 0.12 0.14D(m2 s-1 ) 1/σss2(Ao-2)(a)Diffusion Coefficient σ=2.67Aoσ=4.1Aoσ=6Ao

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.02 0.04 0.06 0.08 0.1 0.12 0.14

F imag

1/σss2(Ao-2)

(b)Fraction of imaginary modesσ=2.67Ao

σ=5.5Ao


of sorbate size. An anomalous peak in Fimag is seen at σSS=5.5A which may be compared

with the peak position at 6A in Figure 4.2(a). Thus the INM spectrum contains a very

clear signature of the levitation effect and confirms previous observations that the fraction

of imaginary modes and the self-diffusivity coefficient are closely correlated. Moreover, the

position of the levitation peak is clearly related to the extent of negative curvature of the

potential energy surface, rather than to the magnitude of the guest-host binding energy.

This may be seen by comparing Figure 4.2(a) with Figure 4.3(a), which shows that the

average guest-host potential energy, Ugh, decreases smoothly with 1/σ2SS and is clearly

uncorrelated with the levitation peak. To support the correlation between the levitation

effect and the curvature distribution of the PES, Figure 4.3(b) shows the variation of the

Einstein frequency, ωE , with sorbate size since ωE is proportional to the average curvature

of the PES. Also shown are the average frequencies of the real and imaginary branches of

the INM spectrum. All three frequencies show a positive correlation with each other. It is

notable that all the three frequencies show a trough as a function of 1/σ2SS at the position

of the anomalous peak in D. The trough is more pronounced for ωE and ωR than ωI . Thus

the anomalous peak is associated with a minimum value for the average force constant of

the system and a maximum value for the fraction of imaginary modes.

4.3.2 INM Spectra

In this section, the actual INM spectral distributions, rather than properties averaged over

the INM distribution are considered. Figure 4.4(a) shows INM spectra for three values of

σSS which lie in the linear regime. All three INM spectra are qualitatively similar though

the imaginary branch intensity decreases with increasing σSS . As σSS increases, the peak of

the real branch shifts to higher frequencies. Figure 4.4(b) shows the INM spectrum for three

sorbate sizes in the anomalous regime. The onset of the anomalous rise in D is signalled

by the formation of a shoulder in the real branch of the INM spectrum at 5.5A. Further


Figure 4.3: Variation of (a) guest-host potential energy, 〈Ugh〉, and (b) frequency, ω, with

1/σ2SS .

All three frequencies i.e Einstein frequency ωE , the average real frequency ωR, and average

imaginary frequency ωI are plotted against 1/σ2SS . εSS is fixed at 221 K and temperatures

for all the runs are held at 190±5 K.

-50-45-40-35-30-25-20-15-10-5 0.02 0.04 0.06 0.08 0.1 0.12 0.14<U gh> 1/σss2(Ao-2)(a)Potential energy due to guest-host interactions

10

15

20

25

0.02 0.04 0.06 0.08 0.1 0.12 0.14

ω /

cm�

-1

1/σss2(Ao-2)

(b) Frequency(b) Frequency

ωEωRωI


increase in sorbate size leads to a broadening of the shoulder and then the formation of a

second peak, as seen in the case of σSS=6.3A. The position of this second peak moves to

higher frequencies as the sorbate size increases.

The formation of the second peak can be explained using the fact that as the system

parameters are tuned to move into the anomalous diffusional regime, a second adsorption

site opens up at the centre of the 12-ring window in addition to sites within the α-cage [73].

This can be better understood from Figure 4.5. Figure 4.5(a) shows the situation when the

sorbate size is small compared to the window dimension of the zeolite. The sorbate then

interacts favourably with only a few oxygen atoms of the 12-ring window of zeolite Na-Y.

However, when the sorbate dimension is comparable to the window dimension, as shown in

Figure 4.5(b), the sorbate interacts favourably with almost all oxygens of the ring leading

to significant lowering of sorbate-zeolite interaction energy near the window. Since the 12-

ring window site controls the transition rate from one α-cage to another, a lowering in the

potential energy of the sorbate in this region reduces the activation energy for the diffusion

process and results in increased diffusivity and levitation peak. Since the sorbate atom can

then occupy distinct cage and window sites, in the anomalous regime the INM spectrum

becomes a double-peaked structure. To demonstrate this connection, we have computed

the three normal mode frequencies of a Lennard-Jones sorbate located at the centre of one

of the 12-ring windows. Figure 4.6 shows the frequencies of the out-of-plane and doubly

degenerate in-plane vibrations as a function of sorbate size. It is seen that as we move into

the anomalous regime, the degenerate in-plane frequency as well as the mean frequency

of vibration of the sorbate atom in the window-cage increases sharply indicating stronger

localization of the sorbate atoms in the window-site. The positions of the second peak in the

INM spectrum for σSS=6.3A and 6.8A coincide with the frequency of the in-plane doubly

degenerate vibration, as shown in Figure 4.4.


Figure 4.4: INM spectra in the linear and anomalous regimes.

Instantaneous normal mode spectra for different sorbate sizes in the (a)linear regime and

(b) anomalous regime. εSS is fixed at 221 K and temperatures for all the runs were kept at

190±5 K.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-20 -10 0 10 20 30 40 50 60

ρ(ω)

ω/cm-1

2.673.484.1

(a)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-20 -10 0 10 20 30 40 50 60

ρ(ω)

ω/cm-1

5.56.36.8

(b)


Figure 4.5: Schematic diagram showing the interactions of sorbate atoms of different sizes

with the oxygen atoms of the 12-ring window of Na-Y zeolite.

when (a) it is of small size compared to the window dimension and when (b) it is comparable

to the window dimension.


Figure 4.6: Normal mode frequencies of a single Lennard-Jones sorbate molecule located in

the centre of a 12-ring window of zeolite Na-Y.

The normal mode frequencies are shown as a function of sorbate size, σSS for εSS=221K.

The frequencies of the doubly-degenerate in-plane sorbate vibration and the non-degenerate

out-of-plane vibration are shown separately. Also shown is the mean of the three sorbate

frequencies.

-20

-10

0

10

20

30

40

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

ω /

cm�

-1

σSS / Αο

degeneratenondegenerate

mean


4.3.3 Velocity Autocorrelation Function

The strong signatures of the levitation effect seen in the INM spectrum, indicate that

a more detailed investigation of the short-time dynamical information contained in the

spectrum is warranted. Previous studies show that the short-time behaviour of the velocity

autocorrelation function or vacf, Cvv(t), is well-reproduced from INM data, particularly at

low temperatures [168]. Figures 4.7(a), 4.7(b) and 4.7(c) show the behaviour of the Cvv(t)

function for sorbate atoms of sizes 2.67, 4.1 and 6A respectively, derived both from the

INM spectrum as well as calculated from an MD run. The short-time behaviour of the

autocorrelation function is well reproduced by the INM result. The agreement between

INM and MD curves is better in the anomalous region than in the linear regime. To

quantify this comparison between the INM and MD derived Cvv(t) curves, we consider two

quantities: (i) τn, the time at which the vacf curve crosses the abscissa for the first time and

(ii) τm, the location of first minimum in the vacf curve. The behaviour of these quantities

based on the MD results are discussed and are compared with the INM predictions. Figure

4.8(a) compares τn values derived from the INM expression with the molecular dynamics

result. The magnitude of τn provides an estimate of the probable time, on average, that a

molecule first encounters a repulsive barrier. From the molecular dynamics results, we see

that as sorbate size increases, τn initially decreases. At the low concentrations studied in

our simulations, this may be attributed to the increasing frequency of collision of a large

sorbate with the walls of the cage. This trend in τn is reversed in the anomalous regime

and a peak is seen at about 6A. This signature of the levitation peak in the τn values

indicates that for favourable sorbate sizes, the system dynamics alters so as to reduce the

frequency of repulsive encounters. A comparison of the INM and MD values of τn shows

that the agreement in the anomalous region is very good which is not surprising in view

of the evidence that the dynamics in this regime is strongly dominated by the topography

of the potential energy surface. However, for small sorbate sizes in the linear regime, the


Figure 4.7: Comparison of velocity autocorrelation function using MD and INM approaches

for three sorbate sizes.

The comparison is shown for the σSS values of (a) 2.67A (b) 4.1A and (c) 6A. The parameter

εSS is held fixed at 221K.

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Cvv

(t)

t/ps

(a)(a)(a)

INMMD

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Cvv

(t)

t/ps

(b)

INMMD

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

Cvv

(t)

t/ps

(c)(c)(c)

INMMD


Figure 4.8: Variation of (a) τn and (b) τm, obtained from INM and MD, with 1/σ2SS .

τn is the time at which the Cvv(t) curve first crosses the x-axis and τm is the time at which

the first minimum in the Cvv(t) curve occurs. εSS is fixed at 221K.

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.02 0.04 0.06 0.08 0.1 0.12 0.14

τ n (

ps)

1/σSS2 (Aο)-2

(b)(b)

(a)

INMMD

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0.02 0.04 0.06 0.08 0.1 0.12 0.14

τ m (

ps)

�

1/σSS2 (Aο)-2

(b)(b)

INMMD


size-dependent trend in τn is not correctly predicted by INM analysis.

The time scale on which a particle reverses its velocity as a result of repulsive encounters

is indexed by τm, which is shown as a function of sorbate size in Figure 4.8(b). τm values

are always larger than τn, which leads to poorer agreement between the MD result and the

INM prediction. For times larger than τm, one expects that the INM predictions will not

be very reliable.

4.3.4 Sorbate Polarizability

It is expected that as sorbate polarizability is increased by tuning the parameter εSS , the

levitation peak will become more prominent as the effect of the confining potential will

be accentuated. However, for large εSS , the sorbate may be essentially trapped at an

adsorption site with a self-diffusion coefficient that is virtually zero. This behaviour parallels

the effect of decreasing temperature on the levitation peak [73]. In our simulations, εSS was

varied between 221K and 442K. For εSS=442K, the values of D were very close to zero. A

comparison of the relative values of D, Fimag and ωE at εSS values of 221K and 287.3K is

shown in Figure 4.9.

4.3.5 Ballistic to Diffusional Crossover

It has been observed from previous studies [136] that τc, the approximate crossover time

from the ballistic to diffusional regime, is influenced by the nature of the potential energy

landscape. The crossover times for εSS = 221K and 287.3K as a function of 1/σ2SS are

shown in Figure 4.10. Note that, as in the earlier studies, the τc values also show an

anomalous peak. More interestingly, the τc values for εSS=221K are very close to the τn

values shown in Figure 4.8(a). Since the crossover region corresponds to the regime in

which the velocity gets randomised as a result of collisional process, this correspondence is

quite natural. Note that for the low sorbate concentrations studied here, these collisonal


Figure 4.9: Variation of (a) self-diffusivity, D (b) fraction of imaginary modes, Fimag, and

(c) Einstein frequency, ωE , with sorbate polarizability.

The plots give the variation with 1/σ2SS for εSS values of 221K and 287.3K.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.02 0.04 0.06 0.08 0.1 0.12 0.14

D /

m

� 2s-1

1/σSS2 (Aο)-2

(c)(c)

(a)

ε=221Kε=287.3K

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.02 0.04 0.06 0.08 0.1 0.12 0.14

Fim

ag

1/σSS2 (Aο)-2

(b)(b)

ε=221Kε=287.3K

16

18

20

22

24

26

0.02 0.04 0.06 0.08 0.1 0.12 0.14

ωE (

cm�

-1)

1/σSS2 (Aο)-2

(c)(c)

(a) (b)

(c)

ε=221Kε=287.3K


Figure 4.10: Crossover from ballistic to diffusional motion as a function of sorbate size.

The crossover times, τc, from ballistic to diffusional motion as a function of 1/σ2SS are ob-

served for εSS values of 221K and 287.3K. Temperatures for all the runs are held at 190±5K.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.02 0.04 0.06 0.08 0.1 0.12 0.14

τ c (

ps)

1/σSS2 (Aο)-2

ε=221Kε=287.3K

processes must correspond to sorbate-wall collisons rather than sorbate-sorbate collisions.

4.4 Conclusions

I have performed an instantaneous normal mode analysis of the levitation effect for monoatomic

Lennard-Jones sorbates adsorbed in Na-Y zeolite. The central conclusion of this study is

that the INM spectrum carries several distinct signatures of the levitation effect. The

fraction of imaginary modes mirrors the trend in the diffusion coefficient as a function of

sorbate size and shows an anomalous levitation peak. Thus the self-diffusivity constant can

be correlated with the extent to which the system samples regions of negative curvature.

The Einstein frequency, as a function of sorbate size, shows a minimum at the position of

the levitation peak. The qualitative shape of the INM spectrum changes in the anomalous

regime, reflecting the availability of 12-ring window sites for adsorption, in addition to the


α-cage sites. The velocity autocorrelation functions of the sorbate are well-reproduced from

INM data for short time scales of one picosecond, particularly in the anomalous regime.

An initial increase in sorbate polarizability accentuates the levitation peak though for large

polarizabilities the diffusion constant is negligible for all sorbate sizes. The time of crossover

from ballistic to diffusional motion can be approximately predicted from INM spectra since

it is found to be similar to the time at which the velocity autocorrelation function first turns

negative.

The instantaneous normal mode analysis of the levitation effect in this work leads one

to expect that the INM spectra can be used provide important clues to qualitative changes

in diffusional dynamics of sorbates in porous media. INM analysis is expected to be par-

ticularly useful when the system dynamics is largely controlled by the topography of the

potential energy surface. Since the INM spectrum can be defined in any ensemble, it

is therefore worthwhile to couple INM analysis with ensembles and techniques which are

more convenient for studying adsorption e.g. the grand-canonical Monte Carlo methods.

Qualitative changes in the INM spectra would then indicate significant shifts in the sys-

tem dynamics which can then be studied more accurately using microcanonical molecular

dynamics. Moreover, the INM spectrum could be used to provide insight into the role of

various factors such as concentration, nature and strength of confining potential and sorbate

properties on diffusional dynamics in porous media.

From an experimental point of view, the existence of the anomalous levitation peak im-

plies that the monotonic dependence of the diffusivity on sorbate size is not always expected

for sorbates in zeolites, even when the corresponding heats of sorption show monotonic be-

haviour. This can be manipulated to promote efficiency of passage of certain sorbate sizes

by choosing aperture dimensions, σw, and sorbate sizes, σSZ , in such a manner that the

window size parameter, γ, defined as

γ = 2.21/6 σSZ

σw(4.1)


is approximately 1. σSZ is the Lennard-Jones size parameter for sorbate-lattice atom in-

teractions where Z=O, Na or Ca and σw is the distance between diagonally placed oxygen

atoms of the windows of the zeolites. Since σw is different for Zeolite Y and Zeolite A,

they show levitation peak at different sorbate sizes [70]. The work presented in this chap-

ter shows that a very simple simulation test for enhancement of diffusivity by tuning the

window-size parameter is possible by applying INM analysis since the anomalous peak in

D is strongly correlated with the mean curvature of the potential energy surface.

Chapter 5

Estimation of Henry’s constant

5.1 Introduction

The amount of a guest molecule taken up into a host zeolite depends upon the equilibrium

pressure, the temperature, the nature of the guest molecule and the complexity of the zeolite

structure[3]. A common way of analysing some of these parameters is to plot the amount

of guest sorbed as a function of pressure at a fixed temperature. This produces a sorption

isotherm which can be repeated at different temperatures to compare molecular capacities

of a zeolite over the temperature range studied. Adsorption isotherms summarise much of

the basic information on sorption capacities and molecular sieve properties of zeolites. It is

also useful to plot isobars, which record the amount of sorbate taken up by the zeolite as

a function of pressure, to demonstrate the drying capabilities of zeolites. Isosteres showing

the change in pressure with temperature when the amount of sorbate taken up by the zeolite

is kept constant may also be plotted. Sorption data, in conjunction with NMR, XRD and

neutron-diffraction measurements, provide quite detailed information as to the amount and

location of sorbates within the zeolite lattice. For example, argon, nitrogen and oxygen

sorption data suggest that these species occupy 755A3 per unit cell in Na-A, which closely

corresponds to the volume of the α-cages calculated from crystallographic data. Water,

103

104 Chapter 5: Estimation of Henry’s Constant

however, occupies a volume of 833A3 per unit cell approaching the calculated filling of

all void space in the A structure(926A3)[3]. This implies that some water molecules are

in the β-cages, in line with other experimental evidence. Of course this information also

demonstrates that nitrogen does not easily pass into the β-cages, providing one reason

why nitrogen surface measurements are difficult to reconcile with zeolite sorption. Similar

calculations can be applied to other molecules, such as argon, carbon-dioxide, ammonia

and methane, taken up by Na-X. It is seen that only water penetrates into the β-cages thus

filling the ‘total void’ available which is close to 50% of the zeolite structure in faujasitic

zeolites. As will be seen later in this chapter, this distinction between total and accessible

pore volume is of considerable relevance from a simulation point of view.

The simplest model isotherm is the Langmuir isotherm which gives the fractional occu-

pancy of sorption sites, θ, as

θ =Kp

1 + Kp(5.1)

where p is the pressure of the sorbate gas in equilibrium with the adsorbed phase and K

is the equilibrium constant. The derivation of the Langmuir isotherm assumes monolayer

coverage, equivalence of all sorption sites and negligible influence of sorbate-sorbate inter-

actions on sorbate site occupancy[1, 7]. Even though these assumptions are not strictly

obeyed for zeolitic adsorption, sorption isotherms for many sorbates can be made to fit the

Langmuir model with some modifications. Adsorption isotherms for methane and n-butane

in ZSM-5, shown in Figures 5.1(a) and (b) respectively, are examples of systems which

conform to the Langmuir model[53]. When deviations from Langmuir-type behaviour are

seen, they provide useful information on sorbate packing and phase transitions. For exam-

ple, the inflection in the adsorption isotherms for iso-butane in ZSM-5, shown in Figure

5.1(c)[53], can be related to the preferential adsorption of the branched alkane at the chan-

nel intersections[58, 61, 103]. The inflections in the adsorption isotherms of n-hexane and

n-heptane can be related to the commensurate freezing of these molecules in the zig-zag

Chapter 5: Estimation of Henry’s Constant 105

channels of ZSM-5[59, 60].

Information on the thermodynamics of sorption can be derived from calorimetric mea-

surements [56, 169] as well as from the adsorption data [49, 50, 134]. Regardless of the

overall shape of the adsorption isotherm, a Henry’s law regime can always be identified at

low pressures. As the pressure, P , of the sorbate gas in equilibrium with the sorbent ap-

proaches zero, there is essentially a linear relationship between the concentrations in the gas

and adsorbed phases, denoted by Cg and Cs respectively. The constant of proportionality

is termed the Henry’s constant, Kh, and is defined by the relation

Cs = KhCg (5.2)

In experimental studies the adsorption isotherm is often expressed as the pressure p of the

adsorbed gas in equilibrium with an intracrystalline sorbate concentration of Cs. In this

case, the Henry’s law limit to the isotherm is often expressed as

Kp =Cs

p(5.3)

Since for a perfect gas p = CgRT , one has for this case

Kh = KpRT. (5.4)

Kp or Kh as a function of temperature can be obtained from the low pressure regime of

adsorption isotherms and used to determine the enthalpy and internal energy of adsorption,

denoted by ∆Hads and ∆Uads respectively, using the relations:

∆Hads = RT 2 d ln Kp

dT(5.5)

∆Uads = RT 2 d ln Kh

dT(5.6)

Thus Henry’s constants and isosteric heats of sorption, qiso = −∆Hads, at infinite dilution

can be obtained relatively easily in experimental studies of sorption equilibria in zeolites

[1]. Kh and Kp can be related to the Helmholtz and Gibbs free energy changes, denoted by


Figure 5.1: Experimental adsorption isotherms in ZSM-5 of (a) methane, (b) n-butane and

(c) iso-butane.

0

0.5

1

1.5

2

2.5

3

0 500 1000 1500 2000 2500

Am

ount

Ads

orbe

d(m

ol/k

g)

�

Pressure(kPa)

3.8

34.8

79.6

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 20 40 60 80 100 120

Am

ount

ads

orbe

d(m

ol/k

g)

�

Pressure(kPa)

3.834.879.6

(b)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 20 40 60 80 100 120

Am

ount

ads

orbe

d(m

ol/k

g)

�

Pressure(kPa)

3.8

34.8

79.6

(c)


∆Aads and ∆Gads respectively, associated with adsorption by the relations:

∆Aads = −RT lnKh (5.7)

∆Gads = −RT ln Kp (5.8)

We have referred to the the excess adsorption free energy per molecule of sorbate in the

solid phase as the excess chemical potential µex.

At intermediate and high gas phase pressures, the adsorption isotherms will not typi-

cally show the linear behaviour given by equations (5.2) and (5.3). However, by comparing

adsorption isotherms at different temperatures, one can obtain the variation in the equilib-

rium pressure p of the adsorbate gas, at a given intracrystalline concentration, as a function

of temperature. ∆Hads may then be determined from Clapeyron-Clausius equation

(∂ ln p

∂T

)ns

= −(∆Hads

RT 2

)=

qiso

RT 2(5.9)

where qiso is the isosteric heat of sorption since the term on the l.h.s is the slope of the

isostere, plotted as ln p against T for a constant uptake, ns, of guest species in a fixed

weight of zeolite. The dimensionless equilibrium constant, K, for partition of the guest

between the zeolite and the external phase is

K =as

ag=

Csγs

Cgγg(5.10)

where a denotes activity, C is the concentration, γ is the activity coefficient and the sub-

scripts g and s denote the gas and solid phase respectively. Clearly with increasing concen-

tration, deviations of the activity coefficients from unity are expected. The standard free

energy, internal energy and entropy of sorption are then given by:

∆Aads = −RT lnK (5.11)

∆Uads = RT 2 d ln K

dT(5.12)

∆Sads = R ln K + RTd ln K

dT(5.13)


The heat capacity per mole of the sorbed guest, Cs, for a given uptake may be used to

obtain the entropy of the sorbate in the intracrystalline phase using the relation

Cs = T(∂Ss

∂T

)ns

. (5.14)

Therefore calorimetric and adsorption isotherm data, in conjunction with equations (5.9) to

(5.13), can be used to obtain information on both energetic and entropic aspects of sorption.

Thermodynamic sorption properties can be estimated from Monte Carlo and molecular

dynamics simulations. A comparison of calculated and experimental sorption properties

can provide very useful information on the sorbate-zeolite potential energy surface. From a

simulation point of view, the simplest quantity to estimate is the isosteric heat of sorption

which may be written as:

Q = −〈Usz〉 − 〈Uss〉+ RT (5.15)

where Usz and Uss are the sorbate-zeolite and sorbate-sorbate interaction potentials. The

contribution from the sorbate-sorbate potential energy function, Uss, is negligible at low

concentrations. The excess chemical potential must be evaluated by more elaborate free

energy estimation techniques, such as the Widom particle insertion method described in

Section 3.3. If µex is evaluated at a given sorbate concentration and temperature, the

corresponding pressure of the sorbate in the gas phase can be determined. In the infinite

dilution limit, the problem is considerably simplified because

µex = −(1/β) ln Kh, (5.16)

and Henry’s constant, Kh, can be expressed simply as the ratio of the partition functions

of a single sorbate particle in the adsorbed and gas phases:

Kh = (1/V )∫

dr exp(−βUsz(r)) (5.17)

where V is the volume of the unit cell of the zeolite, β = 1/kBT , T is the temperature

and Usz(r) is the sorbate-zeolite potential energy as a function of the position r of the


sorbate particle in the unit cell [102, 133, 134, 135, 172]. If the sorbate is treated as

a structureless, Lennard-Jones particle, the above integral is three-dimensional and can

be evaluated by a number of different quadrature schemes. If necessary, the expression

and quadrature schemes can be modified appropriately to take into account intramolecular

degrees of freedom. The temperature derivative of lnKh is given by:

∂(lnKh)∂β

= −∫

drUsz(r) exp(−βUsz(r))∫dr exp(−βUsz(r))

= −〈Usz〉 (5.18)

where the angular brackets denote the appropriate ensemble average from a molecular dy-

namics (MD) or Monte Carlo (MC) simulation.

In this chapter, the estimation of Henry’s constants and the related heat of sorption at

infinite dilution are considered for a number of different sorbate-zeolite systems. One of the

motivations for this study was to benchmark the accuracy of the potential energy surfaces

used in this thesis which are described in Chapter 2. With this end in view, the methane

in ZSM-5 and rare gases in zeolite Na-Y systems were studied. A second motivation was

to study the size dependence of Henry’s constants for Lennard-Jones sorbates in Na-Y ze-

olite. As discussed in the previous chapter, the levitation peak in the diffusivity correlates

with several features of the equilibrium INM distribution but not with the isosteric heat of

sorption. Here we compute the Henry’s constant and the corresponding heat of sorption

for the same set of Lennard-Jones sorbates in Na-Y. By calculating the heat of sorption by

two different routes, an important methodological point is illustrated. The first approach,

referred to subsequently as the integration method, computes Kh as a function of tempera-

ture and uses equation (5.15) to evaluate 〈Usz〉 assuming it to be temperature independent.

The second approach computes 〈Usz〉 as a simple simulation average from an MD run at

the temperature of interest. The first method is typical of insertion methods, such as the

Widom method for evaluating the chemical potential and grand canonical Monte Carlo

simulations[11, 12]. Such insertion methods will sample all available, low potential energy

pore regions unless artificial restrictions are applied. In contrast, microcanonical ensemble


MD simulations sample the dynamically connected pore regions. We show that the differ-

ence in the results obtained by the two approaches depends significantly on sorbate size

and, for a certain range in the sorbate size parameter, will be sensitive to both temperature

and the MD run lengths. The chapter plan is as follows: The potential energy surface

and computational details are given in Section 5.2. The simulation results are discussed in

Section 5.3. Conclusions are contained in Section 5.4.

5.2 Potential Energy Surface and Computational Details

5.2.1 Zeolite Structures

In this chapter, I consider sorption in Na-Y and ZSM-5. The key structural features of both

zeolites have been discussed in Section 1.1. Crystallographic details of the Na-Y structure

are given in Section 4.2.1.

The unit cell of ZSM-5 contains 96 silicon atoms and 196 oxygen atoms. The positions of

the framework atoms were taken from the crystallographic data for the orthorhombic form

[27, 170]. ZSM-5 belongs to the Pnma space group and has lattice parameters of a = 20.07A,

b = 19.92A and c = 13.42A. Figure 5.2 shows contour plots of the potential energy of a

single methane molecule in ZSM-5 in the planes y = 4.98A and x = 10A respectively,

using the potential energy parameters given below. The zigzag channels running parallel

to the x-axis with a circular cross section of 5.4A radius can be clearly seen in the planes

y = b/4 and y = 3b/4. The straight channels run parallel to the y-axis having elliptical

cross sections, with major and minor axes of 5.7A and 5.1A respectively, and are bisected

by the planes x = 0 and x = a/2.


Figure 5.2: Contour plots of a single methane molecule in x- and y- channels of ZSM-5.

The plots show (a) zig-zag channels running along the x-direction in the plane y = 4.98A

and (b) straight channels running along the y-direction in the plane x = 10A. Contour lines

give the potential energy in kJ mol−1.

-18 -17 -16 -15 -14 -13 -12 -10 -8 0

0 2 4 6 8 10 12 14 16 18 20

X/Ao

0

2

4

6

8

10

12

14

Z/Ao

(a)

-18 -17 -13 -12 10

0 2 4 6 8 10 12 14 16 18 20

Y/Ao

0

2

4

6

8

10

12

14

Z/Ao

(b)


Table 5.1: Potential energy parameters for rare gases in Na-Y.

type σ ε ε

(A ) (K) (kJ mol−1)

Xe-Xe 3.88 277 2.299

Xe-O 3.32 185 1.539

Xe-Na 3.73 32.36 0.269

Ar-Ar 3.42 102 0.852

Ar-O 3.09 119.70 0.995

Ar-Na 3.34 26.70 0.222

Kr-Kr 3.53 188 1.566

Kr-O 3.14 160.69 1.336

Kr-Na 3.42 34.04 0.283

5.2.2 Potential Energy Surface

The functional form of the potential energy surface for all the systems is based on the Kiselev

model described in Chapter 2 which includes short-range repulsion and dispersion terms and

excludes induction contribution(see Section 2.4 for details). The relevant potential energy

parameters for CH4 in ZSM-5 are εSS=135K, σSS=3.73A , εOO=120K and σOO=2.70A [89].

The potential energy function for variable size Lennard-Jones sorbates in Na-Y is identical

to that used in previous studies of the levitation effect [70, 171] and tabulated in Chapter

4. The potential energy parameters for Ar, Kr and Xe interactions with the lattice atoms

of Na-Y are taken from ref.[135] and listed in Table 5.1.


5.2.3 Molecular Dynamics Simulations

All Molecular Dynamics (MD) simulations were carried out in the microcanonical (NVE)

ensemble using the velocity Verlet algorithm [11, 12]. Initial velocities were sampled from a

Maxwell-Boltzmann distribution corresponding to a preset temperature and then temper-

ature scaling was done during the equilibration period. Orthorhombic periodic boundary

conditions were imposed. A spherical cut-off radius of 12A was employed for sorbate-sorbate

and sorbate-zeolite interactions. A rigid zeolite framework was assumed in our simulations.

Methane in ZSM-5

The temperature of the simulation study was fixed at 300K and the actual temperature

of the MD run was 317K. The simulation cell contained two unit cells of ZSM-5 along the

z-direction with a concentration of one methane molecule per unit cell. The mass of each

sorbate molecule was taken to be 76 amu. The time-step for the system was chosen to

be 220 a.u(5.3fs) which conserved energy to better than the third significant figure. The

run-length of the simulation was 5.3ns with a equilibration period of 5.3ns.

Variable size Lennard-Jones sorbates in Na-Y

The mean temperatures of all the runs were kept at 190±5K. A single unit cell of Na-

Y zeolite was taken as the simulation cell. Eight Lennard-Jones sorbate atoms of mass

131 amu were loaded in the α-cages of the simulation cell; sorbate-sorbate interactions are

negligible at this concentration. A timestep of 800 a.u(19.2fs) and a runlength of 192 ns

was used in all the simulations with an equilibration period of 0.50ns.

5.2.4 Evaluation of Henry’s constants by the Integration Method

To estimate Henry’s constant for simple, spherical sorbates, it is necessary to evaluate

the three-dimensional integral defined in equation (5.17) numerically. A simple, three-

dimensional equispaced quadrature scheme was adopted for this purpose. The potential

was evaluated on a uniformly spaced 50 × 50 × 50 grid spanning the entire unit cell. For


each sorbate size, Kh was evaluated at five temperatures (150K, 190K, 250K, 300K and

400K) For all sorbate sizes, the dependence of lnKh on β was well approximated by a

straight line, the slope of which was taken as −〈Usz〉.

In the case of methane in ZSM-5, the results for 〈Usz〉 obtained by an unrestricted

integration over the entire unit cell volume agreed well with results from the MD simulation

and the results are discussed in Section 5.3.1 . In the case of zeolite Na-Y, however, there

are two types of cages, α- and β-, the relative occupancy of which will depend on the

sorbate size, and hence there is a serious discrepancy between the results obtained by the

above-mentioned two methods which will be discussed further in Section 5.3.2.

Figure 5.3 shows a contour plot of the Usz potential of a xenon atom with σSS=4.1A

for the x = a/4 = 9.32A plane through the unit cell. Both the sodalite and the α-cage are

low potential energy regions but for this sorbate size, the β-cages are not accessible from

the α-cages. In previous works, the sorbates have been assumed to be restricted to α-cages

with effective radii of 8.7A [47, 133, 135]. We have therefore evaluated Henry’s constants

with and without imposing this effective radius restriction.

5.3 Results and Discussion

5.3.1 Methane in ZSM-5

For methane in ZSM-5, ln Kh was calculated at the temperatures 200K, 225K, 250K, 300K

and 350K using the integration method. Figure 5.4 shows the straight line plot of ln Kh

as a function of 1000/T, the slope of which gives the isosteric heat of sorption as 3.48

Kcal/mol. Molecular dynamics simulation at a sorbate concentration of one molecule per

unit cell yielded a value of 3.23 Kcal/mol. The difference in the values of isosteric heat of

sorption by the above two methods may be attributed to the fact that in calculation by

the integration method, isosteric heat of sorption has been assumed to be constant over the


Figure 5.3: Potential energy surface of a xenon atom in Na-Y in the x = a/4 plane.

The contour lines are plotted at intervals of 5 kJ/mole. The low potential en-

ergy regions corresponding to the large α- and much smaller β-cages can be seen.

-30 -25 -20 -15 -10 -5 0

0 5 10 15 20

Y/Ao

0

5

10

15

20

Z/Ao


temperature range of study which is not strictly true. The discrepancy is unlikely to be due

to the length of the MD run as it has been seen that isosteric heat of sorption shows good

convergence for quite short runlengths. The isosteric heat of sorption reported here is lower

than the previously reported theoretical values. The previously reported simulated values

of isosteric heat of sorption range from 3.56Kcal/mol to 4.33Kcal/mol[102, 172] and differ

from the values reported here due to the different potential energy parameters used in the

simulation runs. In this work, the potential energy parameters were taken from ref.[89] in

order to ensure consistency with previously reported diffusion studies of methane in ZSM-5.

These parameters have been used to study diffusional anisotropy of methane in ZSM-5 in

Chapter 6. The experimental values range from 4.33Kcal/mol to 4.78Kcal/mol[53, 134] and

again do not agree quantitatively with the simulation values. This may be due to either

neglect of induction interactions in the simulations or due to inadequate parameterisation of

the potential energy surface. The induction interaction contributes very little to the total

energy of interaction in ZSM-5 compared to faujasite, due to the absence of framework

cations in the ZSM-5. Therefore, it would appear that better calibration of the potential

energy parameters for the short-range dispersion/repulsion terms is required.

5.3.2 Lennard-Jones Sorbates in Na-Y Zeolite

Figure 5.5 compares 〈Usz〉 as a function of sorbate size obtained by three different methods:

(i) from MD simulations at 190K; (ii)from the temperature dependence of the Henry’s

constant, Kh, where Kh was evaluated by an unrestricted integration over the unit cell

and (iii)from the temperature dependence of the Henry’s constant, Kh, where Kh was

evaluated by a restricted integration in which the test particle is contained within an α-

cage of radius 8.7A. Also shown is lnKh at 190K. The 〈Usz〉 values obtained by the three

methods agree for the three largest sorbate sizes of 6.3, 6.8 and 7A. In this size range, the

sodalite cage is a high energy site and is therefore unoccupied at the temperatures studied.


Figure 5.4: Temperature dependence of Henry’s constant.

Plot of calculated value of lnKh against 1000/T for methane in ZSM-5.

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

2.5 3 3.5 4 4.5 5

ln K

h

1000/T(K-1)

For sorbate sizes lying between 4.1A and 7A , the results of the restricted integration and

MD methods agree but those obtained by unrestricted integration differ by over 50% . This

indicates that while the β-cage is a low energy site for such sorbates, the small size of the

6-membered oxygen ring window effectively prohibits migration from α- to β- cages. For the

three smallest sorbates, the results obtained by the three methods do not agree. In order to

understand this, Figure 5.6 shows the variation in lnKh = −µex/kBT as the effective radius,

Reff , of the α-cage is varied from 5A to 14A . For the intermediate size sorbates, lnKh

is essentially constant till Reff becomes large enough that the sodalite cage is included in

the integration volume; at this point it rises sharply to a new value which remains stable

as Reff is further increased. Clearly the region between the α- and β- cages is a very high

potential energy region with zero Boltzmann weight contribution to the Henry’s constant.

For the small sorbates, this is no longer true and the intervening regions have a small, but

non-negligible probability of occupancy. The potential energy barrier separating the α- and

β-cages for such sorbates can therefore be surmounted as temperature is raised. A short

MD run or low temperature will result in quasi-ergodic behaviour which effectively restricts


Figure 5.5: Comparison of the average values of the sorbate-zeolite potential energy obtained

by different computational methods.

Variation in average sorbate-zeolite potential energy, 〈Usz〉(kJ/mol), with 1/σ2SS(A−2).

The corresponding values of the sorbate size parameter, σSS (A) are shown on the up-

per horizontal axis. Three 〈Usz〉 curves are shown. The MD simulations are at a tem-

perature of 190K. The other two lines are obtained from the temperature dependence of

the Henry’s constant, Kh, one considering unrestricted integration over the unit cell and

the other with restricted integration in which the test particle is contained in an α-cage

with an effective radius of 8.7A. Also shown is − ln Kh at 190K as a function of 1/σ2SS .

-70

-60

-50

-40

-30

-20

-10

0

0.02 0.04 0.06 0.08 0.1 0.12 0.141/σss

2(Ao-2)

σss (Ao)

7.07 5.00 4.08 3.54 3.16 2.89 2.67

<Usz>(MD)<Usz>(full int)

<Usz>(res.int.)-lnKh(190K)


Figure 5.6: Variation in computed values of Henry’s constant with effective radius of α-cage

of Na-Y.

Variation in computed values of lnKh with the effective radius, Reff (A), of the α-cage for

sorbate of sizes 6A, 4.1A, 3.48A, 3.07A and 2.67A.

0

2

4

6

8

10

12

14

16

5 6 7 8 9 10 11 12 13 14

lnK

� h

Reff / Ao

σ=4.1 Ao

σ=3.48 Ao

σ=3.07 Ao

σ=2.67 Ao


occupancy to the α-cages. Artificially restricting such sorbates to the α-cage is clearly

inappropriate and the possibility of activated diffusion via the β-cages must be considered

at higher temperatures. For such small sorbates, vibrations of the zeolite lattice may also

assist in crossing the 6-membered oxygen ring window. Moreover, when comparing with

experiments performed under equilibrium conditions, there may be discrepancies between

MD and experimental results. We plan to explore these issues in future work on MD

simulations of small sorbates.

Interestingly, the unrestricted integration gives rise to a peak in − lnKh and 〈Usz〉

at 6.3A and a minimum at 5.5A. The levitation peak shows up as a peak in the diffusion

coefficient at 6A. The fraction of imaginary modes in the instantaneous normal mode spectra

peaks at about 5.5A. At first sight, this appears to be a correlation between the levitation

peak and the sorbate size for which the chemical potential is minimum; however, this is a

spurious correlation, because the MD results which provide a correct measure of sorbate

mobility do not show any such non-monotonic behaviour in the 〈Usz〉. The minimum in

the µex/kBT = − lnKh obtained by the unrestricted integration arises because two factors

contribute to the low potential energy: (i) Both α- and β-cages represent low energy sites,

unlike for the larger sorbates and (ii) the relatively large sorbate size optimises sorbate-cage

interactions compared to the small sorbate sizes. Since the α-cages via the 12-membered

oxygen ring windows provide the three-dimensional channel structure of the zeolite, in

the absence of diffusion between the α- and β- cages, the unrestricted integration is not

meaningful and this minimum in the chemical potential is not a physically observable effect.

The levitation peak arises when as sorbate size is increased, in addition to sorption sites

within the α-cage, additional sites open up at the center of the 12-membered oxygen ring

window. The levitation peak therefore has a different origin from the peak structure seen

in the chemical potential and the isosteric heat of sorption for the unrestricted integration

method.


The zeolite Na-Y structure provides a particularly simple illustration of how the dis-

tinction between available and dynamically connected pore volumes is sensitive to sorbate

size and temperature. Similar effect has also been noticed in ZSM-5 where a levitation peak

had been previously predicted for LJ sorbates at 2.25A [72]. Our analysis of the potential

energy surface for LJ sorbates indicates that for sorbate size parameters of 2A and less, the

low potential energy regions of the zeolite have a significantly different topography [173].

This may be due to the passage of very small sorbates through the zeolite structure or

because of the assumption of complete shielding of Si atoms by oxygen tetrahedra fails for

such unphysically small sorbate sizes.

5.3.3 Rare Gases in Na-Y Zeolite

Having considered the general case of Lennard-Jones sorbates, sorption of Ar, Kr and Xe

in Na-Y is studied for which a comparison with experiment is possible. Table 5.2 compares

the results for µex and 〈Usz〉 using restricted and unrestricted integration methods (labelled

(b) and (c) respectively) with the theoretical results of Kiselev and with experimental data

(given under column headings (a) and (d) respectively). The restricted integration method

used in this work and Kiselev’s approach are equivalent and the differences arise from the

neglect of induction interactions in our calculations. For example, in the case of xenon, the

induction effects modify µex and < Usz > by 20.6% and 16.12% respectively. The unre-

stricted integration method gives values for µex and 〈Usz〉 which are clearly unacceptably

low when compared with experiment. Since the sorbate diameters of Ar, Kr and Xe are

3.42A , 3.53A and 3.88A respectively[135] , all three can fit in the β-cages but none of them

can penetrate the 6-membered oxygen ring window with a diameter of 2.8A. Consequently

the unrestricted integration method is inapplicable. The agreement between Kiselev’s re-

sults and experiment is good since the potential energy parameters derived in ref.[135] are

based on fitting to experimental data for Na-Y, K-X and Na-X zeolites.


Table 5.2: Isosteric heats of sorption, 〈Usz〉, and the excess free energy, µex, for rare gases

in Na-Y zeolite obtained by different computational methods.

Isosteric heats of sorption, 〈Usz〉, and the excess free energy, µex, for rare gases in Na-

Y zeolite obtained by the following methods : (a) calculations described in ref.[135] (b)

restricted and (c) unrestricted integration methods to obtain Kh and (d) experimental

results from ref.[135].

Sorbate µex 〈Usz〉

(kJ mol−1) (kJ mol−1)

(a) (b) (c) (a) (b) (c) (d)

Xenon -12.54 -9.95 -23.58 -17.25 -14.47 -32.43 -18.0

Krypton -8.26 -6.40 -14.37 -12.42 -10.23 -21.78 -12.80

Argon -5.06 -3.52 -7.91 -8.50 -6.65 -13.89 -3.14


5.4 Conclusions

A comparison of the experimental and theoretical results for the isosteric heat of sorption at

infinite dilution for methane in ZSM-5 and rare gases in Na-Y indicates that the potential

energy surfaces used are of reasonable accuracy. Since the difference between experimental

and computed isosteric heats of sorption is of the order of 15 to 20%, we can expect our

simulation results to correctly reflect all qualitative features. The results on variable size

Lennard-Jones sorbates in Na-Y shows that Henry’s constants and the excess free energy

show a non-monotonic behaviour when unrestricted integration is carried out which is how-

ever not shown during restricted integration and molecular dynamics simulation. It also

highlights an important difference between molecular dynamics simulations and insertion

techniques when applied to adsorbates in porous media. MD in the microcanonical en-

semble mimics the physical dynamics of the system and therefore samples the dynamically

connected pore space. Therefore MD sampling should, in general, be the appropriate one

for comparison with experiment. Discrepancies can arise only in cases where the MD tra-

jectory is quasi-ergodic on the time scale of the simulations. In such cases, pore volumes

accessible on experimental time scales will appear to be significantly larger than those ob-

tained from the MD run. In this work, the simplest example of an insertion technique-

the infinite dilution limit of the Widom method for estimating the chemical potential, has

been illustrated. Other insertion techniques include grand canonical Monte Carlo methods.

Unless artificial restrictions are applied, such methods will sample the total available pore

volume rather than the dynamically connected pore volume. Such artificial restrictions are

easy to implement in the case of simple systems of the type studied here but will clearly be

difficult to apply for more complex random porous media, such as Vycor.

Chapter 6

Diffusional Anisotropy

6.1 Introduction

A disordered phase, such as a liquid or fluid, is truly isotropic in the sense that all properties

are direction independent. This is not the case with crystalline solids for which there

is clearly a dependence of properties on spatial directions relative to the crystallographic

axes. In zeolites, one would expect this anisotropy to manifest itself in the diffusion tensor

associated with sorbate motion through the pore spaces (see Sections 1.2 and 3.2). If the

zeolite has cubic symmetry, then the components of the diffusion tensor in the principal

directions are equal and there will be no diffusional anisotropy. The majority of zeolites,

however, belong to non-cubic symmetry groups and, for such systems, the diffusivities in the

three principal directions will be unequal. In extreme cases, motion in a particular direction

may be completely suppressed leading to one- or two-dimensional channel systems. In this

chapter, I will restrict myself to the study of zeolites belonging to tetragonal or orthorhombic

space groups. Such zeolites show diffusional anisotropy, as manifested in the direction-

dependent diffusivities, but have the simplifying feature that the three crystallographic

directions can be taken as parallel to the three Cartesian axes.

Inequality of the direction-dependent diffusivities is only one possible manifestation of

125

126 Chapter 6: Diffusional Anisotropy

the effect of anisotropy of the confining porous solid on the dynamics of the adsorbed fluids.

An interesting effect that can arise is the interdependence of the directional diffusivities

due to special features of the channel network. Such geometrical correlations are said to

occur when diffusion in a particular direction is possible as a result of correlated motions in

orthogonal directions. In zeolites, this may be expected to occur when the channel system

is known to be two dimensional but the overall network topology is such that diffusion in

all three directions takes place; the displacement in the third direction must then take place

as a result of correlated moves through the channel system. Examples of zeolites in which

this is known to occur are ZSM-5, ZSM-11 and gismondine.

Diffusional anisotropy of sorbates will be determined not just by the crystal structure of

the confining zeolite but also by the properties of the sorbate. For simple sorbates, both the

size and the polarizability will play an important role. The geometrical consequences of size

are easy to predict. For example, from the channel properties of ferrierite, as summarised

in Table 1.3, it can be seen that ferrierite will act as a two-dimensional channel system for

argon atoms, with a sorbate diameter of 3.4A, but as a one-dimensional channel system for

xenon which has a sorbate diameter of 4.1A. Increasing polarizability will also play a role

in enhancing the effects of the crystalline lattice since it will determine the strength of the

sorbate-zeolite interaction energy.

In this chapter, I examine the diffusional anisotropy exhibited by simple Lennard-Jones

sorbates in zeolites using molecular dynamics simulations. The main focus of my study

is sorbate behaviour in ZSM-5 (also referred to as silicalite-1) since this is the zeolite for

which the most extensive studies of diffusional anisotropy exist in the literature. I also

compare some key aspects of diffusional anisotropy in ZSM-5 with the diffusional behaviour

of sorbates in ZSM-11 and siliceous ferrierite. Both ZSM-11 and ferrierite are low-porosity

zeolites but differ in that ZSM-11, like ZSM-5, shows geometrical correlations whereas fer-

rierite can only form simple two or one-dimensional channel systems. The dependence of

Chapter 6: Diffusional Anisotropy 127

the extent of anisotropy on sorbate size and polarizability is illustrated using Lennard-Jones

parametrisations for helium, neon, argon, methane and xenon. In addition to calculating

the diagonal elements of the diffusion tensor, I also examine the anisotropy in several re-

lated quantities such as the velocity autocorrelation function and the ballistic to diffusional

crossover times. As discussed in Chapters 3 and 4, instantaneous normal mode (INM)

analysis has been recently shown to be very useful for understanding short-time dynamical

behaviour of sorbates in zeolites. In this chapter, INM analysis is extended to study dif-

fusional behaviour in anisotropic zeolites. Determining the effects of crystalline anisotropy

on a range of diffusion-related properties, including the INM spectrum, is a novel feature

of my work.

We consider the pure silica analogue of the industrially important zeolite ZSM-5. ZSM-5

belongs to the orthorhombic Pnma space group and is therefore expected to induce diffu-

sional anisotropy. As discussed in Section 1.1.3, ZSM-5 contains two types of interconnected

channels: straight channels parallel to the y-direction and zig-zag channels parallel to the

x-direction. Figure 6.1(a) shows that the channel connectivity in ZSM-5 is such that at a

channel intersection the sorbate can move in one of the four directions in the x, y- channel

system and sorbate diffusion along the z-direction is only possible by alternation of the

sorbate between straight and zig-zag channels. Thus diffusion in the z-direction depends

on the diffusion coefficients in the x and y directions and results in a geometry-induced

correlation between the components of the diffusion tensor. To understand the effect of

such geometrical correlations on the diffusional anisotropy, Karger developed a simple and

elegant Markovian random walk model for diffusion in ZSM-5 which predicts the following

relationship between the diffusion coefficients, Dx, Dy and Dz, in the x, y and z directions

[88]:

c2

Dz=

a2

Dx+

b2

Dy(6.1)

where a, b and c are the unit cell dimensions. Based on the random walk model, Karger


made several related predictions of the effect of geometrical correlations on the direction-

dependent diffusivities. For example, the anisotropy parameter, A, which indexes the rate

of diffusion in the z-direction as a result of correlated motions in the x- and y- directions is

defined as

A = (Dx + Dy)/2Dz (6.2)

and is predicted to have a value greater than 4.4[88, 89]. The randomisation parameter, β,

is defined as:

β =c2/Dz

a2/Dx + b2/Dy(6.3)

A value of unity for the β parameter indicates that the basic assumption of Karger’s model

is exactly obeyed i.e. the probability of a sorbate to move in any one of four directions

on reaching a channel intersection is the same. β > 1 indicates a tendency for the sor-

bate to continue in a channel of the same type whereas β < 1 indicates a tendency to

alternate between straight and zig-zag channels. More elaborate models which take into

account other sources of correlations, such as vacancy correlations due to concentrations

significantly greater than zero and kinetic correlations due to incomplete randomisation,

have subsequently been developed [90, 94]. For xenon, methane, ethane and propane com-

parison of predictions based on random walk methods with pulsed field gradient NMR and

molecular dynamics (MD) results indicate a fair degree of consistency[89, 98, 178, 179].

Simulation work on diffusion of sorbates in ZSM-11 and siliceous ferrierite is much more

limited[29, 80, 100] than that available for ZSM-5[43, 49, 51, 53, 58, 59, 64, 72, 88, 89]. While

some simulations of linear alkanes and small alkenes are available, no detailed studies of

diffusional anisotropy have been carried out to date. I have studied the diffusional behaviour

of argon in ZSM-11 and ferrierite as a first step in extending systematic studies of diffusional

anisotropy to zeolites other than ZSM-5.

The chapter is organised as follows. Computational details are given in Section 6.2.

Results of simulations of a range of sorbates in ZSM-5 are discussed in Section 6.3 while those


from simulations of argon in ZSM-11 and ferrierite are given in Section 6.4. Conclusions

are presented in Section 6.5.

6.2 Computational Details

6.2.1 Zeolite Structures

The unit cell of ZSM-5 (silicalite-1) contains 96 silicon atoms and 196 oxygen atoms. The

positions of the framework atoms were taken from the crystallographic data for the or-

thorhombic form [27, 170]. All-silica ZSM-5 belongs to the Pnma space group and has

lattice parameters of a = 20.07A, b = 19.92A and c = 13.42A. The straight channels paral-

lel to the y-axis have elliptical cross sections, with major and minor axes of 5.7A and 5.1A

respectively, and are bisected by the planes x = 0 and x = a/2. The zigzag channels run

parallel to the x-axis with a circular cross section of 5.4A radius. Figures 5.2(a) and 5.2(b)

show the shapes of the two types of channels using contour plots to represent the potential

energy of a CH4 molecule in ZSM-5 in the planes x = 4.98A and y = 10A respectively.

Figure 6.1(a) shows the channel interconnections in ZSM-5 indicating the possibility

of three-dimensional diffusion in this zeolite as a result of correlated moves through the

channel system.

The unit cell of ZSM-11 contains 96 silicon atoms and 192 oxygen atoms. The data for

the crystallographic positions of the framework atoms are taken from the database of zeolite

structures[18]. ZSM-11 belongs to the I4m2 space group and is tetragonal with the lattice

parameters a = b = 20.067A and c = 13.411A [28] . It has straight channels running along

x- and y- directions. Channel cross-sections are almost circular with major and minor axes

of 5.4A and 5.3A respectively. The connectivity between the channels is shown in Figure

6.1(b). It can be seen that net displacement along the z-direction is possible, in addition to

displacements along the two channel directions. A more detailed structural analysis shows


Figure 6.1: Schematic diagrams showing channel connectivity patterns in (a) ZSM-5 and

(b) ZSM-11.


that there are two types of intersections between these channels: small intersections, which

build cavities of about 1.5-times the size of the channel: and large intersections, which

are more like channels along z-direction with distance between the intersecting channels

amounting to ≈ 5A [80].

The unit cell of siliceous ferrierite contains 36 silicon atoms and 72 oxygen atoms.

Crystallographic data for the positions of the framework atoms of the zeolite ferrierite is

taken from the database of zeolite structures[18]. The data corresponds to the orthorhombic

form of siliceous ferrierite belonging to the space group Pnnm having the lattice parameters

a = 14.07025A , b = 7.41971A and c = 18.720A. Ferrierite has a two-dimensional channel

system. Straight 10-ring channels run parallel to the y-direction with elliptical cross-sections

characterised by major and minor axes of 5.4A and 4.2A respectively. The second set of

channels consists of 8-ring straight channels parallel to the x-direction with elliptical cross-

sections characterised by major and minor axes of 4.8A and 3.5A respectively. Sorbates

with diameters greater than 3.5A cannot penetrate the 8-ring channels.

6.2.2 Potential Energy Surface

The functional form of the potential energy surface, based on the Kiselev model is discussed

in Section 2.4. Since all-silica compositional variants of ZSM-5, ZSM-11 and ferrierite are

considered in our study, we use the same set of potential energy parameters for the three sets

of sorbate-zeolite systems. Table 6.1 shows the Lennard-Jones parameters for the sorbate-

sorbate as well as sorbate-framework oxygen interaction used in this work. Parameters

for neon, argon and xenon are taken from ref.[174]. The methane parameters are those

used in a recent study of diffusional anisotropy [89]. The parameters for helium are taken

from ref.[64]. Note that CH4 has a very similar εSS value to Ar and a very similar σSS

value to that of Xe. Figure 6.2 shows contour plots of the potential energy as a function

of the location of a single rare gas atom in the ZSM-5 framework. The steeper repulsive


Figure 6.2: Contour plots of the potential energy surface of a single (a) xenon and (b)

helium atom in ZSM-5.

The plots are constructed in the xz-plane with y-coordinate fixed at 4.98A. Contour lines

give the potential energy in kJ mol−1.

-30 -25 -20 -10 0

0 2 4 6 8 10 12 14 16 18 20X/Ao

02468101214

Z/Ao

(a)

-5 -3 -2 -1 0

0 2 4 6 8 10 12 14 16 18 20X/Ao

02468101214

Z/Ao

(b)


Table 6.1: Lennard-Jones parameters for the sorbate-sorbate and sorbate-oxygen interac-

tions.

Sorbate εSS σSS εOS σOS

(kJ mol−1) (A) (kJ mol−1) (A)

He 0.085 2.28 0.426 2.62

Ne 0.28 2.85 0.529 2.78

Ar 1.183 3.35 1.028 3.03

CH4 1.23 3.73 1.108 3.214

Xe 3.437 3.85 1.737 3.28

walls and reduced dimensions of the channels for the larger xenon atom when compared to

helium are obvious. In the case of helium, additional very small pores can be seen which are

isolated from the channel system and are artifacts of the potential energy surface. These

unphysical pore spaces arise because the assumption of complete shielding of the sorbate

from direct interaction with framework silicon atoms breaks down when the sorbate size

is small. The effect is small for helium and MD dynamics is not affected provided the

simulation is initiated with the sorbates located in the channel regions. For very small

sorbate sizes, however, the distortion of the channel geometry is substantial. In previous

work on the levitation effect in ZSM-5, sorbates with Lennard-Jones size parameters as

small as 1.5A were considered [72]. Our analysis of contour plots of such small sorbates

indicates that the channel structure of ZSM-5 is severely distorted in such cases and we

have therefore not considered any sorbates with σSS less than 2.28A in this study.


6.2.3 Molecular Dynamics

Molecular Dynamics (MD) simulations were carried out in the microcanonical (NVE) en-

semble using the velocity Verlet algorithm [11, 12]. The simulation program developed by

us has been tested against results available in the literature and used in previous work

[168, 171]. Initial velocities were sampled from a Maxwell-Boltzmann distribution corre-

sponding to some preset temperature and then temperature scaling was carried out during

the equilibration period. The reference temperature was taken to be 300K; actual temper-

atures during runs were within ±20 K of this value. Concentrations ranging from 2 to 24

sorbates per unit cell were studied for ZSM-5. In case of ZSM-11 and ferrierite, concentra-

tions of 12 and 4 sorbates per unit cell respectively were considered. Orthorhombic periodic

boundary conditions were imposed. The simulation cell size for each zeolite was determined

by taking the appropriate number of unit cells in each direction such that the overall edge

lengths were in the range of 20A to 28A. Thus for ZSM-5 and ZSM-11, the simulation cells

contained two unit cells along the z-axis. In the case of ferrierite, two unit cells along the

x-direction and three along the y-direction were taken. A spherical cut-off radius of 12A

was imposed for sorbate-sorbate and sorbate-zeolite interactions. The guest-host potential

energy with these simulation cell dimensions was found to be converged to better than 2%

[168, 171]. The zeolite framework was assumed to be rigid. The time step for each system

was chosen to ensure energy conservation to better than the third significant figure. Instan-

taneous normal modes were calculated at intervals of 100 timesteps. Since a rigid zeolite

lattice was used in the simulations, it was necessary to take appropriate care with regard

to equilibration. To ensure the latter, an equilibration protocol was followed in which the

system is first thermalised at a high temperature of 500 K and then cooled to 300K in steps

of 50K, ensuring thermalisation at intermediate temperatures.

As mentioned in Sections 3.1 and 3.2, the statistical errors in quantities such as the

diffusion coefficient, which are both time-dependent and expressed in terms of a mean


square deviation, are relatively much greater than for simple averages, such as the mean

potential energy [11, 12]. A detailed study of the error in the diffusion coefficient of sorbates

in zeolites depends on the nature of the system and the underlying potential energy surface

[148]. Since results reported in the literature vary significantly in terms of runlengths,

equilibration protocols and potential energy parametrisations, we have performed a fairly

detailed set of convergence tests. The results for CH4 in ZSM-5 are summarised in Table

6.2. We have considered five different concentrations of 2, 8, 12, 16 and 24 sorbates per unit

cell as well as three different sets of equilibration times, tequil, and production run lengths,

tprod, for each concentration. As expected, a simple statistical average, such as the guest-

host interaction energy, is very well converged even for the smallest concentration and run

length. The sharp rise in 〈Usz〉 on going from a concentration of 16/u.c. to 24/u.c. indicates

that a concentration of 24 atoms per unit cell exceeds the maximum packing density for

sorbates of this size [174]. Therefore, for CH4 and Xe, we have not considered the results

for concentrations above 16/u.c. Convergence of the D values with increasing equilibration

and production times is much slower, as expected for a transport property, specially at the

lowest concentration of 2/u.c. This is expected given the slow thermalisation for very low

sorbate concentrations in a rigid lattice in the NVE ensemble. A comparison with the results

of Jost et al who used an NVT ensemble with weak coupling to the thermal bath shows

that the agreement between their results and ours is much better at higher concentrations

[89]. Based on previous work [148, 168, 171], we estimate an error of ±10% for the diffusion

coefficients obtained from the longest runs. We also show the convergence behaviour for

the exponents, nx, ny, nz and n in the diffusional regime (see equations (3.33) and (3.34)).

The convergence behaviour of nx, ny and n is more robust than that of nz. As mentioned

in the introduction, the motion in the z-direction is a consequence of correlated motions

along the x- and y- directions. The variation in the nz values may therefore be due to two

reasons: (i) slower diffusional motion along the z-axis and (ii) slow restoration of Fickian


Table 6.2: Convergence tests for CH4 in ZSM-5 at different concentrations.

The concentration is measured in sorbates per unit cell. A time step of 5.3 fs was used for

all the runs. Temperature scaling was carried out during the equilibration period, tequil,

and switched off during the production run of length, tprod.

Conc tprod tequil T 〈Usz〉〈Uss〉 Dx Dy Dz D Diffusional

(ns) (ns) (K) (kJ mol−1) (10−8 m2s−1) nx ny nz n

2 5.3 5.3 305 -15.76 -0.09 0.62 0.60 0.19 0.43 0.89 1.05 0.83 0.99

5.3 26.4 301 -15.75 -0.09 0.49 0.74 0.13 0.45 0.95 0.96 0.94 0.98

26.4 26.4 305 -15.76 -0.09 0.49 0.56 0.15 0.39 0.95 1.02 0.91 0.99

8 5.3 5.3 305 -15.35 -0.49 0.49 0.63 0.23 0.43 0.93 0.99 0.81 0.96

5.3 26.4 299 -15.37 -0.49 0.47 0.63 0.23 0.43 0.94 0.97 0.80 0.95

26.4 26.4 298 -15.38 -0.49 0.48 0.59 0.20 0.41 0.93 0.99 0.83 0.90

12 5.3 5.3 303 -15.25 -0.78 0.36 0.51 0.15 0.34 0.91 0.94 0.84 0.92

5.3 26.4 299 -15.26 -0.78 0.31 0.49 0.22 0.32 0.95 0.95 0.74 0.93

26.4 26.4 299 -15.26 -0.79 0.30 0.47 0.16 0.31 0.95 0.96 0.82 0.94

16 5.3 5.3 300 -15.14 -1.04 0.22 0.36 0.19 0.23 0.89 0.96 0.68 0.92

5.3 26.4 304 -15.12 -1.04 0.26 0.39 0.15 0.25 0.89 0.98 0.77 0.93

52.8 26.4 317 -15.08 -1.04 0.20 0.41 0.12 0.24 0.95 0.96 0.81 0.95

24 5.3 5.3 291 -4.57 -0.98 0.15 0.14 0.15 0.11 0.74 0.98 0.55 0.89

5.3 26.4 299 -4.51 -0.96 0.08 0.14 0.15 0.10 0.97 0.96 0.59 0.92

52.8 26.4 298 -4.51 -0.97 0.08 0.14 0.09 0.09 0.94 0.95 0.69 0.93


behaviour in the very long-time limit. These points are discussed further in Section 6.3. It

should be noted, however, that despite a relatively higher variation in nz compared to nx

and ny, the nz exponent is always lower than the exponents in the x and y directions.

Convergence tests, similar to those reported above for CH4 and Xe, were carried out for

the other four sorbates. The results reported here are for the longest run length simulations

for each concentration. The corresponding simulation parameters are summarised in Table

6.3. Based on our convergence tests, we expect our diffusion coefficients to be correct to

within ±10 to ±20% for the longest runs. When comparing diffusional behaviour in ZSM-5,

ZSM-11 and ferrierite, we considered only Ar as a sorbate since Ar is the largest sorbate

that is able to traverse the entire channel network in all three zeolites. The MD simulation

parameters for Ar in ZSM-11 and ferrierite were determined on the basis of the convergence

tests in ZSM-5 and are listed in Table 6.4. The determination of the exponents nx, ny, nz

and n in the diffusional regime is an issue of special relevance here since one of the main

conclusions of this chapter is that the presence of geometrical correlations can result in

subdiffusional behaviour (n < 1) for fairly extended periods of time. This situation does

not arise in liquids, or even in all sorbate-zeolite systems, and therefore has not received

much attention in the literature. Figures 6.3(a) and (b) show distribution of points at

which ∆2x(t) and ∆2z(t) as a function of time is sampled by the order-N algorithm, on a

logarithmic and on a standard scale respectively, for Ar in ZSM-11. The order-N algorithm

distributes the points unequally with a large number of points concentrated in the short-

time regime where fluctuations in the velocity autocorrelation function are relatively large.

This is convenient since it allows for an accurate representation of the behaviour during

crossover from the ballistic to diffusional regimes. In the case that the slope of the MSD

is time-independent in the diffusional regime, a simple linear least squares fit of the data

to equation (3.34) causes no problems. This is the case for sorbates in Na-Y zeolite, as

illustrated in Figure 4.1. The situation in zeolites showing geometrical correlations is,


Table 6.3: Molecular dynamics simulation parameters used for different Lennard-Jones

sorbates.

Sorbate mass Time step Conc tequil tprod T 〈Usz〉〈Uss〉

(amu) (fs) (sorbates/u.c.) (ns) (ns) (K) (kJ mol−1)

He 4 0.24 2 1.20 2.40 318 -2.40 0.002

12 1.20 1.20 309 -2.31 0.02

16 1.20 2.40 301 -2.33 0.03

24 1.20 2.40 304 -2.31 0.06

Ne 20 0.48 2 4.80 19.20 299 -3.89 -0.007

12 2.40 2.40 308 -3.76 -0.03

16 2.40 4.80 299 -3.76 -0.04

24 2.40 4.80 306 -3.64 -0.03

Ar 40 1.20 2 11.76 0.24 294 -11.47 -0.08

12 6.00 6.00 296 -11.08 -0.60

16 6.00 12.00 285 -11.07 -0.82

24 6.00 12.00 288 -10.95 -1.20

CH4 16 5.30 2 26.4 26.4 305 -15.76 -0.09

8 26.4 26.4 298 -15.38 -0.49

12 26.4 26.4 299 -15.26 -0.79

16 26.4 52.8 317 -15.08 -1.04

Xe 131 12.0 2 116.40 3.60 305 -27.97 -0.30

12 60.00 60.0 304 -26.61 -2.93

16 60.00 120.00 313 -26.35 -4.00


Figure 6.3: Plot of ∆2x(t) and ∆2z(t) as a function of time on (a) logarithmic and (b)

standard scales for Ar in ZSM-11 using order-N algorithm.

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

0.001 0.01 0.1 1 10 100 1000 10000 100000

∆2 (t)/

(A

o )2

t/ps

∆2x(t)∆2z(t)

(a)

0

1000

2000

3000

4000

5000

6000

7000

0 1000 2000 3000 4000 5000 6000

∆2 (t)/

(A

o )2

t/ps

∆2x(t)∆2z(t)

(b)


Table 6.4: Molecular dynamics simulation parameters used for argon in the zeolites ZSM-11

and ferrierite.

Zeolite mass Time step Conc tequil tprod T 〈Usz〉〈Uss〉

amu (fs) (sorbates/u.c.) (ns) (ns) (K) (kJ mol−1)

ZSM-11 40 1.20 12 6.00 12.00 320 -10.94 -0.60

Ferrierite 40 1.20 4 6.00 12.00 283 -12.87 -0.39

however, somewhat different. As can be seen from Figure 6.3(a), the slope of the ∆2z(t)

versus t plot on a logarithmic scale has a small but significant time-dependence. In such a

case, the unequal distribution of points due to the order-N algorithm may result in a slightly

erroneous value of the exponent if an unweighted linear least squares fit is used. One route

to correct for this, followed here, is to remove some of the points close to the crossover

region to generate an equispaced distribution of points, as has been done for the results

shown in Tables 6.9 and 6.10. Alternatively, a weighted least squares fitting procedure can

be used. In the case of ZSM-11, the difference in the nx values is less than 1% but it is

somewhat greater for nz.

6.3 Sorbates in ZSM-5: Results and Discussion

6.3.1 Diffusional and Ballistic Regimes

The separation between the ballistic and diffusional regimes is most clearly shown in log-

log plots of the mean square displacement against time. As an illustration, Figure 6.4


Figure 6.4: Log10-log10 plot of mean square displacements as a function of time in the x, y

and z-directions for CH4 in ZSM-5.

The simulations were performed at 300K and at a concentration of 12 particles per unit

cell.

0.00010.001

0.010.1

110

1001000

100001000001e+06

0.1 1 100 10000

∆2 (t) / �

(Ao )2

t/ps

∆2x(t)∆2y(t)∆2z(t)

compares the direction-dependent mean square displacements, ∆2x(t), ∆2y(t) and ∆2z(t),

against time for CH4 in ZSM-5. The slopes for all three curves in the ballistic region from 0

to approximately 0.5ps are very close to 2. The transition to the diffusional regime is marked

by a crossover period rather than a sharp crossover time, τc. More interestingly, the slopes

in the diffusional regime are close to unity but are by no means identical — the nx and ny

values are 0.95 and 0.96 respectively while nz is 0.82. The net mean square displacement,

∆2r(t) grows as t0.94. Thus, sorbate motion along the z-direction, corresponding to the

c-axis of the unit cell, is distinctly subdiffusional. From Table 6.5, it can be seen that this

subdiffusional behaviour for the z-displacement is present for all the sorbates studied here,

though it is least pronounced for helium.

Since the greater tendency to subdiffusional behaviour along the z-direction is a very

distinct feature of diffusional anisotropy that emerges from our study, it is worthwhile to

examine it more carefully. Previous studies of the time dependence of the MSDs of sorbates


Table 6.5: The exponent of the time dependence of the mean square displacement for

Lennard-Jones sorbates in ZSM-5.

The exponents were calculated in the ballistic and diffusional regimes . The exponents

nx, ny, nz and n corresponds to the ∆2x(t), ∆2y(t), ∆2z(t) and ∆2r(t) displacements

respectively. Simulation parameters, as well as the mean temperature and potential energy,

are given in Table 6.3

Sorbate Conc. Ballistic Diffusional

(sorbates/u.c.) nx ny nz n nx ny nz n

He 2 1.93 1.94 1.92 1.93 0.99 0.93 0.93 0.96

12 1.91 1.92 1.90 1.91 0.98 1.02 0.87 0.99

16 1.91 1.92 1.90 1.91 1.00 1.03 0.93 1.01

24 1.90 1.90 1.89 1.90 1.01 1.00 0.90 1.00

Ne 2 1.91 1.91 1.90 1.91 0.98 1.01 0.81 1.00

12 1.93 1.94 1.93 1.90 0.96 1.09 0.83 1.02

16 1.93 1.93 1.92 1.93 1.00 0.95 0.89 0.96

24 1.91 1.91 1.91 1.91 0.94 0.99 0.90 0.96

Ar 2 1.96 1.96 1.95 1.96 0.99 1.02 0.79 1.00

12 1.94 1.94 1.94 1.94 0.92 0.96 0.73 0.92

16 1.93 1.93 1.93 1.93 0.93 0.98 0.78 0.94

24 1.90 1.89 1.89 1.90 0.88 0.93 0.70 0.87

CH4 2 1.95 1.95 1.93 1.94 0.95 1.02 0.91 0.99

8 1.93 1.93 1.92 1.93 0.93 0.99 0.83 0.90

12 1.90 1.90 1.90 1.90 0.95 0.96 0.82 0.94

16 1.90 1.90 1.90 1.90 0.95 0.96 0.81 0.95

Xe 2 1.94 1.93 1.91 1.93 0.91 0.93 0.76 0.91

12 1.89 1.89 1.89 1.89 0.92 0.97 0.78 0.94

16 1.87 1.86 1.87 1.87 0.89 1.01 0.76 0.98


in zeolites have not considered anisotropic effects. The diffusion of CH4 in ZSM-5 has,

however, been reported as being subdiffusional with an exponent of 0.78 at a temperature

of 298K [136]. We find, however, that the exponent for the total MSD is 0.94. The dif-

ferences between their study and ours may stem from different potential energy surfaces

as well as much shorter run lengths(0.2 ns as opposed to 2.6 ns). Our study finds dis-

tinctly subdiffusional behaviour only along the z-direction. The nz exponent for different

run lengths is fairly similar and therefore is unlikely to be an artifact of a simulation that

is too short. It would therefore appear that this type of subdiffusional motion is a conse-

quence of the correlated nature of diffusion along the z-axis which can take place only if the

particle alternately diffuses through straight and zig-zag channel segments. However, the

geometry of the channel network is clearly not the only determining factor since the very

small, light and weakly bound helium atom does not display such pronounced diffusional

anisotropy. It is also clear that the subdiffusional behaviour is more pronounced at higher

concentrations indicating that avoided crossings of sorbates in the z-direction may play a

role, as in the case of single-file diffusion. It would appear therefore that a combination

of the potential energy landscape within a channel, the packing density as well as the ge-

ometrical connectivity is necessary to produce this subdiffusional behaviour. The results

for the diffusional anisotropy behaviour of the various Lennard-Jones sorbates in ZSM-5

are taken from ref.[175]. Based on this, one can predict that similar direction-dependent

subdiffusional behaviour will be present in other two-dimensional networked channel zeo-

lites, such as ZSM-11, and the effect will be attenuated with increasing temperature and

accentuated by increasing concentration. To test this prediction, we have compared the

diffusional behaviour of Ar in ZSM-5, ZSM-11 and ferrierite (see Section 6.4). In future

work, it will be of interest to examine if there is a slow approach to Fickian behaviour in

the long-time limit, as has been suggested in the case of single-file systems [176].

The crossover time from ballistic to diffusional motion has been computed as the point


Table 6.6: Ballistic to diffusional crossover times for Lennard-Jones sorbates in ZSM-5.

Ballistic to diffusional crossover times (in ps) are computed at a temperature of 300K and

concentration of 12 particles per unit cell.

Sorbate τcx τcy τcz τc

He 0.29 0.31 0.11 0.23

Ne 0.40 0.28 0.21 0.26

Ar 0.29 0.31 0.24 0.26

CH4 0.24 0.37 0.23 0.26

Xe 0.20 0.35 0.18 0.21

of intersection of the straight line fits to the MSDs in the two regions. The results, for

12 particles per unit cell, are given in Table 6.6. The crossover times are found to be

fairly similar for all the systems and the most notable feature is that the crossover time

for the MSD in the z-direction is distinctly lower than the crossover times in the other two

directions. There is, however, some ambiguity in the exact location of τc because of the

extended nature of the crossover region. An examination of Figure 6.5 also indicates that

the above method of locating τc will generally result in values which are lower than the time

at which the MSD begins to show a deviation from ballistic behaviour on a log-log plot.

Intuitively our results for a reduced value for τcz, compared to τcx or τcy, do appear to be

reasonable since the range of ballistic motion in the z-direction is strongly confined by the

channel walls.

Table 6.7 shows the direction-dependent diffusion coefficients as well as the anisotropy

parameter, A, the randomisation parameter, β, and the ratio Dy/Dx. As expected, the

diffusion coefficients decrease with increasing binding energy and size of the sorbate. The


Table 6.7: Diffusion coefficients of different Lennard-Jones sorbates in ZSM-5 at a temper-

ature of 300K at different concentrations.

Sorbate Conc. Usz Uss Dx Dy Dz D A β Dy/Dx

(sorbates/u.c.) (kJ mol−1) (10−8 m2 s−1)

He 2 -2.40 0.002 7.08 11.21 1.65 6.61 5.54 1.18 1.58

12 -2.31 0.02 6.28 7.22 1.97 5.10 3.43 0.76 1.15

16 -2.33 0.03 5.19 6.03 1.55 4.21 3.62 0.81 1.16

24 -2.31 0.06 4.29 5.74 1.57 3.13 3.19 0.70 1.34

Ne 2 -3.89 -0.007 2.29 2.55 1.02 1.86 2.37 0.53 1.11

12 -3.76 -0.03 1.89 1.60 0.80 1.33 2.18 0.49 0.85

16 -3.76 -0.03 1.38 2.07 0.58 1.32 2.97 0.64 1.50

24 -3.64 -0.03 1.32 1.36 0.45 1.03 2.98 0.67 1.03

Ar 2 -11.47 -0.09 0.76 0.94 0.38 0.65 2.24 0.50 1.24

12 -11.08 -0.60 0.66 0.73 0.31 0.55 2.24 0.50 1.11

16 -11.07 -0.82 0.44 0.47 0.26 0.37 1.75 0.39 1.07

24 -10.95 -1.20 0.22 0.19 0.22 0.19 0.93 0.21 0.86

CH4 2 -15.76 -0.09 0.49 0.56 0.15 0.39 3.50 0.78 1.14

8 -15.38 -0.49 0.48 0.59 0.20 0.41 2.68 0.60 1.23

12 -15.27 -0.79 0.30 0.47 0.16 0.31 2.41 0.51 1.57

16 -15.08 -1.04 0.20 0.41 0.12 0.24 2.54 0.50 2.05

Xe 2 -27.97 -0.30 0.11 0.16 0.05 0.10 2.70 0.59 1.45

12 -26.61 -2.93 0.14 0.29 0.12 0.17 1.79 0.35 2.07

16 -26.35 -4.00 0.06 0.18 0.05 0.09 2.40 0.40 3.00


Figure 6.5: Log10-log10 plot comparing mean square displacements in the z-direction for

different Lennard-Jones sorbates in ZSM-5.

The temperature of the simulations are fixed at 300K and a concentration of 12 particles

per unit cell is considered for each type of sorbate.

1e-06

0.0001

0.01

1

100

10000

1e+06

0.01 1 100 10000

∆2 (t)/

(A

o )2

t/ps

HeNeAr

CH4Xe

Dy/Dx ratio is most sensitive to the specific nature of the sorbate. This must reflect

differences in the straight and zig-zag channel architecture experienced by the different

diffusing particles. Increasing concentration results in lowering the diffusion coefficients,

as well as the anisotropy and randomisation parameters. The assumption of complete

randomisation which underlies equation (6.1) appears to be most closely obeyed by the He

sorbate which is entirely expected given the small size, weak binding and high diffusion

coefficients for this system. For all concentrations of helium in ZSM-5, β ≈ 1 and A ≈

4. The simple random walk model of Karger predicts β = 1 and A > 4.4. Clearly the

motion of the helium atoms is very rapidly randomised in the channels so that at the

channel intersection the probability of moving in any four of the available channel segments

is essentially equal. The other four sorbates, on the other hand, have β values very close

to 0.5, indicating a propensity for the sorbates to alternate between straight and zig-zag

channels on reaching an intersection presumably due to greater influence of the local nature


of the potential energy surface. The anisotropy parameter, A, is approximately two for the

larger sorbates, instead of ≈ 4 for helium. The results for β and A are consistent with our

results for the exponent of the time dependence which also indicate that helium is clearly

a sorbate which closely obeys the assumptions underlying the simple random walk model.

It is important at this point to make an explicit comparison with previous work on

diffusional anisotropy. Only two Lennard-Jones sorbates, Xe and CH4, have been studied

from the point of view of understanding diffusional anisotropy in ZSM-5. Our potential

energy parameters for CH4 are identical to those used by Jost et al. Their results are:

Dx=0.5, Dy=0.6, Dz=0.1 and D=0.4 in units of 10−8m2s−1 which may be compared with

our results in Table 6.7. The agreement is reasonable given that the values quoted from

ref[89] have been read of the graph and therefore are approximate. Results for xenon show

poorer agreement partly because of small differences in potential energy parameters. In

general we find that Dz values are overestimated and the β parameters underestimated

when compared with previous work. This may be due to differences in run lengths. A more

important reason may be that I have allowed the exponent of the time dependence of the

MSD to deviate from unity whereas in past work, a linear dependence on time has been

assumed.

6.3.2 The Velocity Autocorrelation Function

We have looked at the short-time behaviour of the velocity autocorrelation function, Cvv(t),

and its directional counterparts, Cvxvx(t), Cvyvy(t), and Cvzvz(t). Figure 6.6 shows the four

correlation functions for neon and xenon in ZSM-5. Cvxvx(t) and Cvyvy(t) are indistinguish-

able on the scale of the plots. Cvzvz(t) differs from its counterparts in the x and y directions

in having a much deeper first minimum. In the case of xenon, Cvzvz(t) also has a pronounced

second maximum (also seen in methane and argon) which is absent for helium and neon.

All the four velocity autocorrelation functions for each system are almost identical till a


Figure 6.6: Short-time behaviour of time-correlation functions for (a) neon and (b) xenon

in ZSM-5.

The temperature of simulations is fixed at 300K the concentration being 12 particles per

unit cell

-0.4-0.2

00.20.40.60.8

11.2

0 0.5 1 1.5 2 2.5 3

C(t

)

�

t/ps

<v(0).v(t)> <vx(0)vx(t)><vy(0)vy(t)><vz(0)vz(t)>

(a)

-0.4-0.2

00.20.40.60.8

1

0 0.5 1 1.5 2 2.5 3

C(t

)

�

t/ps

<v(0).v(t)><vx(0)vx(t)><vy(0)vy(t)><vz(0)vz(t)>

(b)


Table 6.8: Key features of the instantaneous normal mode spectra for Lennard- Jones

sorbates in ZSM-5.

The concentration and temperature were fixed at 12 particles per unit cell and 300 K

respectively.

Sorbate Fimag Fimag,x Fimag,y Fimag,z ωE (cm−1)

He 0.50 0.16 0.16 0.17 67.7

Ne 0.46 0.15 0.15 0.15 41.3

Ar 0.39 0.13 0.13 0.13 34.4

CH4 0.33 0.12 0.11 0.11 25.4

Xe 0.28 0.10 0.09 0.09 22.9

time τn when they first turn negative. While the location of the first minimum occurs for

smaller times for the more mobile sorbates, the location of τn for four larger sorbates is

between 0.4 and 0.5 ps. Earlier work on diffusion in faujasite indicated that τn is strongly

correlated with the ballistic to diffusional crossover time, τc [171]. In ZSM-5, however, τn

is found to be significantly larger than the τc values which lie between 0.2 and 0.4 ps.

6.3.3 Instantaneous Normal Mode Analysis

Table 6.8 summarizes the key features of the INM spectra. As expected on the basis of

previous studies, there is a strong correlation between the diffusion coefficient, D, the frac-

tion of imaginary modes, Fimag, and the Einstein frequency, ωE . The fraction of imaginary

modes for the projections of the INM density of states on the x, y and z axes is essentially

identical. This is consistent with the isotropy in the short-time dynamics that is displayed

by the velocity autocorrelation functions. Based on our previous work, the time scales for


which the INM approach can be expected to be useful can be estimated from the value of

τn, the time at which the velocity autocorrelation function first turns negative. Since this

is of the order of 0.5ps, and at thermal velocities at 300K corresponds to distances of a few

Angstrom, it is not surprising that the INM spectra are not sensitive to the geometrical

connectivity of the ZSM-5 lattice which manifests its anisotropy over longer length scales.

Moreover, the INM spectrum can be thought of as displaying the local curvature of the

PES as sampled by the system. A diffusional property, such as the levitation effect, which

is closely connected with the curvature distribution of the potential will be mirrored by

changes in the INM spectrum but long length and time scale properties such as geometrical

correlations will not be reflected in the INM spectrum. The anomalous levitation peak for

sorbates in ZSM-5 has been observed for sorbates in the size range from 1.5A to 2.2A; as

discussed in Section 6.2.2, we believe the Kiselev potential energy parametrisation to be

somewhat problematic in this size regime and we have therefore not studied the levitation

effect in this zeolite.

6.4 Comparison of Sorbate Diffusion in ZSM-5, ZSM-11 and

Ferrierite: Results and Discussion

The results of our detailed study of diffusional anisotropy of sorbates in ZSM-5 revealed two

unexpected aspects of diffusional anisotropy which are related to the presence of geometrical

correlations. These two aspects are: (a) slow crossover from ballistic to diffusional motion

in the direction showing correlated displacements when compared to the directions showing

direct displacements and (b) subdiffusional behaviour in the correlated direction which

persists over fairly long time scales of the order of 1ns. To test whether these features will

persist in other zeolites, I compare Ar diffusion in ZSM-5, ZSM-11 and ferrierite. ZSM-11,

like ZSM-5, will show correlated motion along the z-direction. Ferrierite, on the other hand,


Table 6.9: The exponents of the time-dependence of the mean-square displacements of argon

in ZSM-5, ZSM-11 and ferrierite.

The exponents in the diffusional regime were determined by sampling the mean square

displacement at equispaced points in time.

Zeolite Conc. Ballistic Diffusional

(sorbates/u.c.) nx ny nz n nx ny nz n

ZSM-5 12 1.94 1.94 1.94 1.94 0.93 0.95 0.86 0.93

ZSM-11 12 1.94 1.94 1.93 1.94 0.97 0.98 0.88 0.96

Ferrierite 4 1.95 1.94 - 1.93 0.92 0.98 - 0.97

will allow diffusion in the x- and y-directions, but not in the z-direction.

Table 6.9 compares the exponents for the time dependence of the mean square displace-

ment of Ar in ZSM-5, ZSM-11 and ferrierite in both the ballistic and diffusional regimes at

comparable concentrations. The subdiffusional behaviour along the correlated z-direction

is evident in ZSM-11 as well as ZSM-5. In the case of ferrierite, the nx coefficient is found

to be somewhat smaller than ny though not as small as the values of nz seen in ZSM-5 and

ZSM-11. This may be attributed to the fact that the sorbate diameter and the minor axis

of the 8-ring window are very similar and therefore sorbate motion along the x-direction

is somewhat constrained. Table 6.10 shows the average sorbate-sorbate and sorbate-zeolite

potential energies as well as diffusion coeffients for Ar in ZSM-5, ZSM-11 and Ferrierite.

The small difference between the results for ZSM-5 shown in Table 6.7 and Table 6.10


Table 6.10: Diffusion coefficients of argon in ZSM-5, ZSM-11 and ferrierite at 300K.

Zeolite Conc. Uss Usz Dx Dy Dz D Dy/Dx

(sorbates/u.c.) (kJ mol−1) (10−8 m2 s−1)

ZSM-5 12 -0.60 -11.08 0.63 0.76 0.23 0.53 1.21

ZSM-11 12 -0.60 -10.94 0.70 0.71 0.22 0.54 1.01

Ferrierite 4 -0.39 -12.87 0.12 0.62 - 0.38 5.17

is due to the use of equispaced time points when evaluating the MSD in Table 6.10. The

differences (of 4-5%) lie well-within the estimated error bars of ±10% for Dx and Dy. The

differences in the values in the two tables is noticeable for Dz and lies outside the estimated

error bars of ±4% as for all the Lennard-Jones sorbates studied in this work, the Dz value

shows a slow crossover, as can be seen in Figure 6.5, which is not seen in case of displacements

along the x- and y- directions. Interestingly the structural similarity of ZSM-5 and ZSM-11

is reflected in the values of 〈Uss〉, 〈Usz〉 as well as the diffusion coefficients. The anisotropy

in the diffusion coefficients is very large in the case of Ar since it has to essentially squeeze

its way through the 8-ring channels parallel to the x-axis.

To examine the crossover behaviour when going from the ballistic to the diffusional

regime, I have used the following procedure. The exponent of the time-dependence of the

directional MSDs are estimated by a linear least-squares fit of the data to the equations

(3.33) and (3.34). The ∆2r(t) regime for which t lies between Imin and Imax is considered. I

fix Imax to be half of the total length of the production run. Imin was varied from 5ps to 50ps


and the values nx, ny, nz in the three zeolites were plotted as functions of Imin. Figures 6.7

(a), (b) and (c) show the results. For argon in ZSM-5 and ZSM-11 systems, nx settles down

fairly quickly to an asymptotic value with fluctuations of ±0.01. However, an anomalous

and slow rise of nx values is observed in case of argon in ferrierite because the minor axis

of the elliptical channels of ferrierite is comparable to the van der Waal’s diameter of argon

and this results in constrained motion along the x direction. The exponent ny stabilizes

to an asymptotic value fairly quickly for argon in all the zeolite systems studied. The nz

exponent in ZSM-5 and ZSM-11 shows a much slower rise and does not attain a plateau

value in 50ps.

6.5 Conclusions

In this chapter, I have presented my work on the anisotropy in diffusional and related

dynamical properties of Lennard-Jones sorbates in three different zeolites: ZSM-5, ZSM-11

and siliceous ferrierite.

The bulk of my work is focussed on diffusional anisotropy in ZSM-5 since sorbate be-

haviour in this zeolite, including the effects of geometrical correlations on directional dif-

fusivities, is very well studied. Using molecular dynamics simulations I have studied the

diffusional behaviour of five spherical sorbates of varying size and polarizability in ZSM-

5. Helium, the smallest and most weakly bound sorbate, complies most closely with the

behaviour expected on the basis of the simple random walk model of Karger with a ran-

domisation parameter close to 1 and an anisotropy parameter close to 4. The larger and

more strongly bound sorbates (Ne, Ar, CH4 and Xe) show significant deviations from this

model. The diffusion of these particles along the z-direction is distinctly subdiffusional

with the mean square displacement growing as ≈ t0.8. The randomisation parameters for

all these systems are close to 0.5 and the anisotropy parameters are all close to 2. The

relative rates of diffusion along the straight and zig-zag channels are more sensitive to the


Figure 6.7: Plots of exponents of mean square displacements vs Imin along (a) x-, (b) y-

and (c) z- directions for argon in ZSM-5, ZSM-11 and ferrierite.

0.8

0.85

0.9

0.95

1

5 10 15 20 25 30 35 40 45 50

n� x

Imin/ps

ZSM-5ZSM-11

Ferrierite

(a)

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

5 10 15 20 25 30 35 40 45 50

n� y

Imin/ps

ZSM-5ZSM-11

Ferrierite

(b)

0.8

0.85

0.9

0.95

1

5 10 15 20 25 30 35 40 45 50

n� z

Imin/ps

ZSM-5ZSM-11

(c)


nature of the sorbate than the anisotropy and randomisation parameters. For all five sor-

bates, the subdiffusional behaviour along the z-direction as well as deviations from the

predictions of the random walk model, are more pronounced at higher concentrations. The

subdiffusional motion along the z-direction appears to be a consequence of a combination

of factors: (i) the geometrical connectivity of the ZSM-5 channel network (ii) the nature of

the potential energy landscape seen by a sorbate located within the channel spaces and (iii)

the packing density. Based on our results, one would expect such subdiffusional behaviour

in specific directions for sorbates in other zeolites which can display geometrical correlations

in diffusional behaviour. The PES dependence would imply that such subdiffusional be-

haviour would become less prominent with increasing temperature. The anisotropy in the

short-time dynamics has been examined by studying the velocity autocorrelation functions

and the instantaneous normal mode spectra. For very short-times of less than 0.5 ps, the

velocity autocorrelation function and its directional analogues are virtually identical but di-

vergences are seen by times of the order of 1ps. The motion along the z-direction is clearly

more correlated than along the x- or y-directions. The instantaneous normal mode spectra

show the expected correlation between the diffusion coefficient, the Einstein frequency and

the fraction of imaginary modes. There is no significant anisotropy in the INM spectra

which is consistent with the behaviour of the velocity autocorrelation functions for short

time scales.

Two unexpected aspects of diffusional anisotropy which are related to the presence of

geometrical correlations emerged from the study of sorbates in ZSM-5. These are: (a) slow

crossover from ballistic to diffusional motion in the direction showing correlated displace-

ments when compared to the directions showing direct displacements and (b) subdiffusional

behaviour in the correlated direction which persists over fairly long time scales of the order

of 1ns. These two features were shown to persist for Ar diffusion in ZSM-11 which also has

a channel topology that gives rise to geometrical correlations but not in ferrierite which is


a simple two-dimensional channel system,

The results suggest that there are several aspects to diffusional anisotropy, in addi-

tion to the relative magnitudes of the direction-dependent diffusion constants, which may

be worth exploring further. These include the extent of subdiffusional behaviour and the

short-time dynamics. The qualitative difference in behaviour of helium and xenon as sor-

bates in ZSM-5 may be of interest experimentally since 129Xe NMR spectroscopy is widely

used and 3He NMR has been suggested as a probe in porous media [128]. One can also

predict that similar direction-dependent subdiffusional behaviour will be present in other

two-dimensional networked channel zeolites with geometrical correlations, such as gismon-

dine, and that the effect will be attenuated with increasing temperature and accentuated

by increasing concentration. Our simulations indicate that the deviations from Fickian be-

haviour, as indicated by exponents less than unity in the long-time regime, persist for times

as large as 12ns for Argon as sorbate. This is a very long time scale that may potentially

be detectable experimentally. Moreover, since the effect seems to be due to the nature of

the channel topology, it is unlikely to be an artifact due to the inaccurate parametrisation

of the potential energy surface or the rigid lattice approximation. On theoretical grounds,

one expects to see true subdiffusional behaviour only in fractal porous media; therefore in a

zeolite, the Fickian behaviour should eventually be restored. In future work, it will also be

of interest to examine this very slow approach to Fickian behaviour in the long-time limit.

Chapter 7

Conclusions

This thesis presents the results from molecular dynamics studies of simple atomic and molec-

ular sorbates in zeolites. My primary aim has been to show how computer simulations of

such relatively simple systems can provide considerable insight into unusual diffusional

properties that emerge as a consequence of confinement of a fluid in a crystalline porous

medium, with particular reference to the levitation effect and diffusional anisotropy. In

addition to adding to our understanding of diffusion phenomenon, some significant method-

ological points of relevance to simulations of fluids in porous media emerge from my work.

The levitation effect is an anomalous peak in the diffusivity as a function of sorbate size

seen in sorbates with dimensions close to the minimum channel width, and represents a

deviation from the monotonic decrease in diffusivity with sorbate size predicted on simple

geometrical grounds. In my study, the levitation effect is illustrated using Lennard-Jones

sorbates adsorbed in Na-Y zeolite. The levitation peak is shown to be uncorrelated with

either the isosteric heats of sorption or the chemical potential of the sorbate in the zeo-

lite. However, the application of instantaneous normal mode analysis to understand the

relationship between the diffusivity behaviour and the underlying potential energy surface

leads to the central conclusion that the presence of the levitation peak is closely correlated

with the curvature distribution of the sorbate-zeolite potential energy surface. As a conse-

157

158 Chapter 7: Conclusions

quence, the INM spectrum carries several distinct signatures of the levitation effect. The

fraction of imaginary modes mirrors the trend in the diffusion coefficient as a function of

sorbate size and shows an anomalous levitation peak. Thus the self-diffusivity constant can

be correlated with the extent to which the system samples regions of negative curvature.

The Einstein frequency, as a function of sorbate size, shows a minimum at the position of

the levitation peak. The qualitative shape of the INM spectrum changes in the anomalous

regime, reflecting the availability of 12-ring window sites for adsorption, in addition to the

α-cage sites. The velocity autocorrelation functions of the sorbate are well-reproduced from

INM data for short time scales of one picosecond, particularly in the anomalous regime.

An initial increase in sorbate polarizability accentuates the levitation peak though for large

polarizabilities the diffusion constant is negligible for all sorbate sizes. The time of crossover

from ballistic to diffusional motion can be approximately predicted from INM spectra since

it is found to be similar to the time at which the velocity autocorrelation function first turns

negative. From an experimental point of view, it is of interest to have a simple diagnostic

test of whether there is an enhancement in the intracrystalline diffusivity of a particular

sorbate as a result of a favourable matching of the sorbate size with the aperture dimensions

of the zeolite. My work shows that an INM analysis provides a very simple simulation test

for the levitation effect since the anomalous peak in the diffusivity is strongly correlated

with the fraction of imaginary frequency modes and the mean curvature of the potential

energy surface.

Diffusional anisotropy is a phenomenon which is characteristic of sorbates in porous crys-

taline solids because the structural properties of confining solids are direction-dependent.


tensor. A more unusual effect is an interdependence of the components of the diffusion ten-

sor due to geometrical correlations which arise as a result of special features of the channel

network and have been studied in detail in ZSM-5. I have studied the diffusional anisotropy

Chapter 7: Conclusions 159

of several Lennard-Jones sorbates in all-silica compositional variants of three zeolites: ZSM-

5, ZSM-11 and ferrierite. All three are low-porosity zeolites belonging to non-cubic space

groups and therefore must have unequal directional components of diffusivity. In addition,

ZSM-5 and ZSM-11 show geometrical correlation effects. Both ZSM-5 and ZSM-11 have

channels which run along the a and b crystallographic axes but no channels which run

parallel to the c, or Cartesian z-axis. However, diffusion in the z-direction is possible as a

result of correlated motions along the x and y-directions. The most unexpected result that

emerges from my study of diffusional anisotropy is the very slow crossover from ballistic to

diffusional motion and the existence of a protracted period of subdiffusional motion along

the correlated or z-direction. This anisotropic behaviour of the mean square displacement

as a function of time, as a consequence of geometrical correlations, appears not to have been

anticipated in the literature. The results also suggest that the extent of subdiffusional be-

haviour will be partly determined by the size and polarizability of the sorbate, with helium

showing the least tendency to subdiffusional behaviour.

In addition to the effect of geometrical correlations on diffusional behaviour, I have

looked at the diffusional behaviour of Lennard-Jones sorbates in ZSM-5 in detail. Helium,

the smallest and most weakly bound sorbate, complies most closely with the behaviour

expected on the basis of the simple random walk model of Karger with a randomisation

parameter close to 1 and an anisotropy parameter close to 4. The larger and more strongly

bound sorbates (Ne, Ar, CH4 and Xe) show significant deviations from this model. The

diffusion of these particles along the z-direction is distinctly subdiffusional with the mean

square displacement growing as ≈ t0.8. The randomisation parameters for all these systems

are close to 0.5 and the anisotropy parameters are all close to 2. The relative rates of

diffusion along the straight and zig-zag channels are more sensitive to the nature of the

sorbate than the anisotropy and randomisation parameters. For all five sorbates, the sub-

diffusional behaviour along the z-direction as well as deviations from the predictions of the


random walk model, are more pronounced at higher concentrations. The subdiffusional

motion along the z-direction appears to be a consequence of a combination of factors: (i)

the geometrical connectivity of the ZSM-5 channel network (ii) the nature of the potential

energy landscape seen by a sorbate located within the channel spaces and (iii) the packing

density. The instantaneous normal mode spectra show the expected correlation between

the diffusion coefficient, the Einstein frequency and the fraction of imaginary modes. There

is no significant anisotropy in the INM spectra which is consistent with the behaviour of

the velocity autocorrelation functions for short time scales.

An important methodological point which emerges from this work is the applicability of

instantaneous normal mode analysis to fluids adsorbed in porous media. The instantaneous

normal mode analysis of the levitation effect in this work leads one to expect that the INM

spectra can be used provide important clues to qualitative changes in diffusional dynamics

of sorbates in porous media, specially when the system dynamics is largely controlled by the

topography of the potential energy surface. Since the INM spectrum can be defined in any

ensemble, it is therefore worthwhile to couple INM analysis with ensembles and techniques

which are more convenient for studying adsorption e.g. the grand-canonical Monte Carlo

methods. Qualitative changes in the INM spectra would then indicate significant shifts

in the system dynamics which can then be studied more accurately using microcanonical

molecular dynamics.

Our work on estimation of Henry’s constants highlights an important difference be-

tween molecular dynamics simulations and insertion techniques when applied to adsorbates

in porous media. MD in the microcanonical ensemble mimics the physical dynamics of

the system and therefore samples the dynamically connected pore space. Therefore MD

sampling should, in general, be the appropriate one for comparison with experiment. Dis-

crepancies can arise only in cases where the MD trajectory is quasi-ergodic on the time scale

of the simulations. In such cases, pore volumes accessible on experimental time scales will

Chapter 7: Conclusions 161

appear to be significantly larger than those obtained from the MD run. In this work, the

simplest example of an insertion technique- the infinite dilution limit of the Widom method

for estimating the chemical potential, has been illustrated. Other insertion techniques in-

clude grand canonical Monte Carlo methods. Unless artificial restrictions are applied, such

methods will sample the total available pore volume rather than the dynamically connected

pore volume. Such artificial restrictions are easy to implement in the case of simple systems

of the type studied here but will clearly be difficult to apply for more complex random

porous media, such as Vycor.

A final methodological point which emerges from our study of diffusional anisotropy is

the importance of checking the applicability of the Einstein relations and the possibility of

deviations from Fickian behaviour when fluids are confined in nanoporous media. In many

instances, as in the case of ZSM-5 and ZSM-11, deviations may be seen which persist on

time scales long enough to be potentially observable using pulsed-field gradient NMR.

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182

BIODATA OF THE AUTHOR

Name Sudeshna Kar

Date of birth 26th October, 1972

Sex Female

Nationality Indian

Permanent H2/53, Shakuntala Park,

address Calcutta-700 061

Education MSc(Chemistry), First Class, Calcutta University

BSc(Chem. Hons), First Class, Calcutta University

Publications

1. Kar, S.; Chakravarty, C. Instantaneous normal mode analysis of the levitation

effect in zeolites; J. Phys. Chem. B 2000, 104, 709.

2. Kar, S.; Chakravarty, C. Computational evaluation of Henry’s constants and

isosteric heats of sorption for Lennard-Jones sorbates in Na-Y zeolite

Mol. Phys. 2001, 99, 1517.

3. Kar, S.; Chakravarty, C. Diffusional anisotropy of simple sorbates in Silicalite;

J. Phys. Chem. A 2001, 105, 5785.

4 Kar, S. and Chakravarty, C. Diffusion of simple sorbates in silicalite: effect of

anisotropic frameworks and geometrical correlations; Proceedings of the 13th

International Zeolite Conference, Montpellier , France, 8th-13th July, 2001

5. Kar, S. and Chakravarty, C. Diffusional behaviour of simple sorbates in zeolites:

effect of anisotropic frameworks and geometrical correlations; Chem. Phys. Lett.

(in press)

6. Kar, S. and Chakravarty, C. Wandering through molecular mazes:

unusual diffusional behaviour of fluids contained in zeolites (in preparation)

184

molecular dynamics studies of simple sorbates in …

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