Modelling Hydrogen Storage in Novel

Nanomaterials

Wei Xian Lim

Thesis submitted for the degree of

Doctor of Philosophy

in

Applied Mathematics

at

The University of Adelaide

(Faculty of Engineering, Computer and Mathematical Sciences)

School of Mathematical Sciences

October 2, 2015

Contents

Abstract x

Signed Statement xii

Acknowledgements xiv

List of Publications xv

1 Introduction 1

1.1 Hydrogen storage . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Metal-organic frameworks . . . . . . . . . . . . . . . . . . . . 3

1.3 Porous aromatic frameworks . . . . . . . . . . . . . . . . . . . 6

1.4 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Analytical formulations of nanospace in porous materials 9

2.1 Molecular interactions with building blocks . . . . . . . . . . . 10

2.1.1 Interaction with a point . . . . . . . . . . . . . . . . . 13

2.1.2 Interaction with a line . . . . . . . . . . . . . . . . . . 13

2.1.3 Interaction with a plane . . . . . . . . . . . . . . . . . 15

2.1.4 Interaction with a ring . . . . . . . . . . . . . . . . . . 16

i

2.1.5 Interaction with a spherical surface . . . . . . . . . . . 19

2.1.6 Interaction with an infinite cylindrical surface . . . . . 21

2.2 Gas adsorption model . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Materials represented by building blocks . . . . . . . . . . . . 27

2.3.1 A carbon atom . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2 Polyacetylene . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Graphene sheet . . . . . . . . . . . . . . . . . . . . . . 31

2.3.4 Benzene ring . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.5 Fullerene . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.6 Carbon nanotube . . . . . . . . . . . . . . . . . . . . 35

2.4 Example of interactions with porous materials . . . . . . . . . 38

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Beryllium based metal-organic frameworks 42

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Metal-organic frameworks performance at room temperature . 43

3.3 Modelling methodology . . . . . . . . . . . . . . . . . . . . . . 48

3.3.1 TIMTAM formulation . . . . . . . . . . . . . . . . . . 49

3.3.2 Thermodynamic energy optimisation . . . . . . . . . . 53

3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Pore size analysis . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.1 Fractional free volume . . . . . . . . . . . . . . . . . . 58

3.5.2 Optimal storage and delivery conditions . . . . . . . . 60

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ii

4 Porous aromatic frameworks 70

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Porous aromatic frameworks . . . . . . . . . . . . . . 72

4.2.2 Lithiated porous aromatic frameworks . . . . . . . . . 74

4.2.3 Impregnated porous aromatic frameworks . . . . . . . 75

4.2.4 Parameter values . . . . . . . . . . . . . . . . . . . . . 76

4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 78

4.3.1 Gravimetric and volumetric uptake . . . . . . . . . . . 81

4.3.2 Potential energy . . . . . . . . . . . . . . . . . . . . . 87

4.3.3 Free volume for adsorption . . . . . . . . . . . . . . . 90

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Summary 95

5.1 Analytical formulations of nanospace in porous materials . . . 96

5.2 Beryllium based metal-organic frameworks . . . . . . . . . . . 97

5.3 Porous aromatic frameworks . . . . . . . . . . . . . . . . . . . 98

6 Appendix 101

6.1 Evaluation of Equations in Chapter 2 . . . . . . . . . . . . . . 101

6.1.1 Interaction with a line . . . . . . . . . . . . . . . . . . 101

6.1.2 Interaction with a plane . . . . . . . . . . . . . . . . . 102

6.1.3 Interaction with the top or bottom of a ring . . . . . . 103

iii

List of Figures

1.1 The schematic representation of MOFs. . . . . . . . . . . . . . 4

1.2 Structure of (a) PAF-301 and (b) PAF-302. . . . . . . . . . . 7

2.1 Atomic representation of structures that can be described by

building blocks. An atom can be described by a point (a),

benzene by a ring (b), fullerene as a spherical surface (c),

carbon chain by a line (d), graphene sheets by planes (e), and

carbon nanotube by a cylindrical surface (f). . . . . . . . . . . 10

2.3 Structure of polyacetylene. . . . . . . . . . . . . . . . . . . . . 30

2.8 Contour plots of H2 interacting the outer walls of a fullerene. . 36

2.9 Contour plots of H2 interacting with the inner walls of a fullerene. 36

2.10 Contour plots of H2 interacting with the inner walls of an

infinite carbon nanotube. . . . . . . . . . . . . . . . . . . . . . 37

2.11 Contour plots of H2 interacting with the outer walls of an

infinite carbon nanotube. . . . . . . . . . . . . . . . . . . . . . 38

iv

3.1 Total H2 uptake at room temperature and high pressures (more

than 35 bar) for MOFs with and without open metal sites.

(a) H2 uptake versus heat of adsorption; (b) H2 uptake versus

BET surface area; (c) H2 uptake vs pore volume. . . . . . . . 46

3.2 Structure of Be-BTB. The spheres and cylinders represent

ideal building blocks for adsorption cavities in the structure. . 49

3.3 Total H2 uptake at 77 K and 298 K for TIMTAM, GCMC

[1] and experimental data [2]. Solid lines show the density

of compressed H2 gas in a tank at 77 K and 298 K. Circles

represent experimental data, squares represent GCMC model

results and dotted lines represent TIMTAM predictions for

Be-BTB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Potential energy for the ideal Be-BTB building blocks con-

sisting of spherical cavities (red dashed line) and cylindrical

Be-ring cavities (blue solid line) that were constructed for the

TIMTAM approximation. Shaded area represents heat of ad-

sorption measured experimentally [2]. . . . . . . . . . . . . . . 57

3.5 Fractional free volume for adsorption (Vad/V ) at 77 K and

298 K within Be-ring cylindrical cavity building block. The

yellow cylinder illustrates the variation in pore size around the

Be-ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 The optimum gravimetric H2 uptake at 233 K for a pressure-

swing adsorption process. . . . . . . . . . . . . . . . . . . . . . 63

3.7 The optimum gravimetric H2 uptake at 12 bar for a temperature-

swing adsorption process. . . . . . . . . . . . . . . . . . . . . . 64

v

3.8 H2 gravimetric uptake with respect to temperature and pres-

sure. (a) H2 uptake for Be-BTB. (b) H2 uptake for MOF-5.

(c) H2 uptake for MOF-177. The yellow diamond and green

circle denote the TEO optimised desorption and adsorption

conditions, respectively, that maximises the net energy. The

bar on the right describes the value gravimetric uptake in wt%. 66

4.3 Gravimetric uptake for (a) PAF-302 and (b) PAF-303 with

respect to pressure at 77 K and 298 K. The plots shows the

comparison between our results (solid lines) with simulation

(crosses) and experimental results (circles) from Lan et al. [3]

and simulation (dotted line) from Konstas et al. [4]. . . . . . . 81

4.4 Gravimetric and volumetric uptakes for Li-PAF-302 and Li-

PAF-303. The figures show gravimetric uptake comparison

plots for (a) 77 K and (b) 298 K and volumetric uptake com-

parison plots for (c) 77 K and (d) 298 K respectively where the

blue and red lines represent PAF-302 and PAF-303. The solid,

dashed, and dotted lines represents the bare PAFs, 2%Li-

PAFs, and 5%Li-PAFs. . . . . . . . . . . . . . . . . . . . . . . 83

4.5 Gravimetric uptake comparison plots for (a) 77 K and (b)

298 K. Volumetric uptake comparison plots for (c) 77 K and

(d) 298 K. The blue, red and green lines represent uptakes by

PAF-302 and PAF-303, lithiated PAF-302 and PAF-303, and

fullerene impregnated lithiated PAF-303. . . . . . . . . . . . 84

vi

4.6 Potential energy for Li-PAF-302 in (a) and (b), Li-PAF-303 in

(c) and (d), and C60@Li-PAF-303 in (e) and (f) with respect

to the distance from cavity centre. The contour plots on the

right depict the depth of the potential energy with varying Li

atoms and distance from cavity centre. . . . . . . . . . . . . . 88

4.7 Fractional free volume for adsorption (%) calculated at (a)

77 K and (b) 298 K with varying cavity size. The dimensions

for available ligands are depicted in the molecular diagrams. . 90

vii

List of Tables

2.1 Numerical values of various parameters (C-H2 denotes the in-

teraction between a hydrogen molecule with a carbon atom,

and C-H2 denotes the interaction between a hydrogen molecule

with a hydrogen atom). . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Data for hydrogen gas sorption for metal-organic frameworks

with and without open metal sites at 298 K. . . . . . . . . . . 45

3.2 Coefficient of determination (R2) for the correlation of total

H2 uptake with heat of adsorption, BET surface area and pore

volume at high pressure (more than 35 bar). . . . . . . . . . . 47

3.3 Lennard-Jones parameter values. . . . . . . . . . . . . . . . . 51

3.4 Numerical values of constants used. . . . . . . . . . . . . . . . 52

3.5 Values of constants used to calculate optimum storage (ads)

and delivery (des) conditions within constraints of U.S. DOE

delivery conditions (5 – 12 bar, 233 – 358 K). . . . . . . . . . 55

3.6 Maximum energy generation at optimised storage (ads) and

delivery (des) conditions restricted to DOE operating range

for a pressure-swing only cycle. . . . . . . . . . . . . . . . . . 62

viii

3.7 Maximum energy generation at optimised storage (ads) and

delivery (des) conditions restricted to the DOE operating range

for a temperature-swing only cycle. . . . . . . . . . . . . . . . 64

3.8 Maximum energy generation at optimised storage (ads) and

delivery (des) conditions restricted to the DOE operating range

for a combined pressure-swing and temperature-swing cycle. . 67

4.1 Van der Waals force field parameters between H2 and PAF. . . 77

4.2 Constants used for the potential energy calculation. . . . . . . 79

4.3 Gravimetric Uptake (wt%) at 100 bar, 77 K and 298 K. . . . . 86

4.4 Volumetric Uptake (g/L) at 100 bar, 77 K and 298 K. . . . . . 87

ix

Abstract

Gas storage using nanomaterials has been researched as possible enhance-

ment of gas tanks in fuel cell vehicles. The structures of nanomaterials and

its interaction with gases are often explored using computer simulations and

experiments which are both time consuming and expensive. In this thesis,

we overcome these problems by performing these investigations through a

simplified mathematical modelling approach.

In this approach, we first develop simple solutions to calculate the in-

teraction energies between hydrogen gas and the materials using symmetric

building blocks to represent the cavity of the structure. The gas uptake in

the nanomaterial can then be calculated using these solutions to identify the

quantity of gas stored in the adsorbed and bulk states. We also introduce a

novel method, the thermodynamic energy optimisation (TEO) model, to cal-

culate the energy produced by a hydrogen fuel cell coupled with a materials

storage device.

In this thesis these models are used to explore beryllium linked with ben-

zene tribenzoate (Be-BTB) and porous aromatic frameworks (PAF). The

models are able to identify reasons why these materials have demonstrated

potential for gas storage and suggest ways to improve and optimise the struc-

x

tures.

Our investigation into Be-BTB reveals that the beryllium rings contribute

strongly to the hydrogen interaction with the framework. We propose that

beryllium rings of 10 A at 298 K and 15 A at 77 K will optimise the fractional

free volume for adsorption within the material. Investigations using the

TEO method demonstrate that current high performing MOFs are unable

to outperform gas tanks for fuel cell vehicles. To improve uptake capacity

further improvements are required to decrease specific heat capacity and heat

of adsorption while also ensuring that the material possesses optimal cavity

sizes to maximise the fractional free volume.

Another application of the mathematical model is undertaken on PAFs,

in particular PAF-302 and PAF-303. Using analytical methods, three pos-

sible modifications on PAFs are adopted to investigate their effects on gas

uptake; (i) fullerene impregnation, (ii) lithium doping, and (iii) a combina-

tion of methods (i) and (ii). Results show that lithiation strengthens the

interaction energy whilst fullerene impregnation doubles the number of at-

tractive surfaces. The final results indicate that 8%Li-PAF-303 provides the

highest gravimetric uptake at 77 K and 298 K, and 8%Li-PAF-302 provides

the highest volumetric uptake at 77 K and 298 K.

xi

Signed Statement

I certify that this work contains no material which has been accepted for the

award of any other degree or diploma in my name in any university or other

tertiary institution and, to the best of my knowledge and belief, contains

no material previously published or written by another person, except where

due reference has been made in the text.

In addition, I certify that no part of this work will, in the future, be used in

a submission in my name for any other degree or diploma in any university

or other tertiary institution without the prior approval of the University of

Adelaide and where applicable, any partner institution responsible for the

joint award of this degree.

I give consent to this copy of my thesis, when deposited in the University

Library, being made available for loan and photocopying, subject to the pro-

visions of the Copyright Act 1968. The author acknowledges that copyright

of published works contained within this thesis resides with the copyright

holder(s) of those works.

I also give permission for the digital version of my thesis to be made avail-

xii

able on the web, via the University’s digital research repository, the Library

catalogue, and also through web search engines, unless permission has been

granted by the University to restrict access for a period of time.

SIGNED: ....................... DATE: .......................

xiii

Acknowledgements

I would like to thank Prof. Jim Hill for encouraging me to pursue a PhD

and for his guidance throughout my candidature. I would also like to thank

Dr. Barry Cox for assisting me with curly mathematical problems and for his

warm welcome whenever I drop by his office unannounced with questions. In

addition, I would like to thank Dr. Aaron Thornton for being a driving force

for my publications. Working with him has taught me critical thinking and

to explore new ideas.

I wish to express gratitude to Robbie for all his love, support and advice

with the editing of my thesis. I would like to also thank my friends in

the maths office for keeping me sane with their wonderful and unforgettable

company: Kate, Stephen, Rhys, Alice, Jo, Adam, Chris, Lyron, Ty, David,

Jess, Paul, Farah and Ewan. Special mention to Prof. Nigel Bean for his

advice during my PhD blues.

Finally I would also like to thank my colleagues at CSIRO including

Dr. Anita Hill and Dr. Matt Hill for their contributions and the opportunity

to collaborate and work with them.

This thesis is dedicated to my family.

xiv

List of Publications

1. K. Konstas, J. W. Taylor, A. W. Thornton, C. M. Doherty, W. X. Lim,

T. J. Bastow, D. F. Kennedy, C. D. Wood, B. J. Cox, J. M. Hill,

A. J. Hill, and M. R. Hill, “Lithiated porous aromatic frameworks with

exceptional gas storage capacity”, Angewandte Chemie (International

ed. in English), vol. 51, pp. 6639–6642, 2012.

2. W.-X. Lim, A. W. Thornton, A. J. Hill, B. J. Cox, J. M. Hill, and

M. R. Hill, “High performance hydrogen storage from Be-BTB metal-

organic framework at room temperature”, Langmuir, vol. 29, pp. 8524–

8533, 2013.

3. W.-X. Lim and A. W. Thornton, “Analytical representations of regular-

shaped nanostructures for gas storage applications”, ANZIAM Journal,

vol. 57, pp. 43–61.

xv

Chapter 1

Introduction

1.1 Hydrogen storage

Safe and dependable storage of hydrogen is a key technological challenge for

the development of hydrogen fuel cell vehicles. Environmentally, hydrogen

fuel cells are considered to be an imperative because they generate electricity

with emissions of water and heat as compared to carbon dioxide emissions

from the use of fossil fuels and biofuels. In order for the technology to

become competitive with existing vehicle fuels in terms of cost and driving

range, at least 5 kg of hydrogen is required to be stored. In order to achieve

this, present gas storage technology requires either very high pressure or

cryogenics to satisfy space requirements [5, 6].

There are serious restrictions with the use of very high pressure and/or

cryogenic technologies due to complicated technical requirements and cost.

Furthermore, the energy penalties associated with compression and/or cool-

ing of the hydrogen dramatically diminish the environmental benefits of hy-

1

drogen powered vehicles. The latest Department of Energy Annual Merit

Review of the Hydrogen Storage Sub-Program [7] concludes that physical

sorbents are the leading candidates for hydrogen storage media, with low

storage capacity and loss of usable hydrogen as the key remaining challenges.

A gas in its adsorbed state can achieve far greater densities than in the

bulk gas state. The use of novel nanostructured materials as adsorbents

within a storage tank therefore offers the potential to increase the storage

density through chemisorption or physisorption processes. Chemisorption

refers to the formation of hydride chemical bonds within a material such

as in metal hydrides; whereas physisorption relies on weaker van der Waals

interactions between gas molecules and the internal surfaces of a porous

material, such as in zeolites and activated carbons. The interaction strength

of physisorbed gas molecules is primarily governed by local dipole moments

induced in the hydrogen molecule by vacant point charges and the atomic

structure of the material [8].

The temperature at which hydrogen is adsorbed and desorbed is a key pa-

rameter in the overall material performance. Temperature is embodied in the

enthalpy of adsorption, which is the energy associated with the adsorption

process. Theoretical calculations show that the ideal enthalpy of adsorption

at room temperature is in the range of 15–25 kJ mol−1 [9]. Higher values of

the enthalpy of adsorption are typically found in chemisorbents, which re-

quire heating to release stored hydrogen with only a fraction of the hydrogen

released due to the strength of the chemical bond. The ability to repeatedly

cycle these chemisorbent materials is limited due to significant changes in

the volume of the unit cell during the filling and emptying processes.

2

These inherent limitations of chemisorbents have led to an intense interest

in physisorbents. Hydrogen storage in physisorbents has been studied using

porous materials that have large internal surface areas such as activated

carbon, nanotubes and zeolites [8, 10, 11]. However, a design proposal for

the three-dimensional porous structures by Hoskins and Robson [12] in 1990

and subsequently the first reported synthesis of metal-organic frameworks

(MOFs) by Li et al. [13] in 1999 has generated considerable interest in the

study of MOFs. Besides having all of the advantages of physisorbents such

as being able to maintain structural stability on adsorption cycling, having

ordered crystalline structures, being highly porous with tunable pore sizes

and providing well-defined hydrogen adsorption sites, MOFs possess one of

the highest surface areas and hydrogen capacities for any physisorbent.

1.2 Metal-organic frameworks

MOFs are nanoporous polymeric materials comprising metal atoms or clus-

ters linked periodically by organic ligands. A schematic representation for

a MOF pore is shown in Figure 1.1. There are currently more than 37,000

documented metal-organic frameworks (MOFs) in the Cambridge Structure

Database (CSD) and that are traditionally built using exploratory methods

until the implementation of reticular synthesis in 1998 by Li et al. [14] to pro-

duce MOF-2. By choosing the appropriate secondary building units (SBUs),

a highly ordered and tunable structure can be created. The success of MOF

creation through reticular synthesis has allowed for the design of periodic

structures that have high surface area and porosity which are properties that

3

are suitable for gas storage purposes.

Metal-organic framework

Metal ion/ cluster

Organic ligands

Figure 1.1: The schematic representation of MOFs.

Computer simulations have also been used to design MOFs, with Wilmer

et al. [15] in 2012 generating more than 130,000 hypothetical MOFs using

modular building blocks from known MOFs. The advantage of using com-

puter simulation is that the properties of these MOFs can be calculated to

simulate gas adsorption isotherms without the need for experiments. The

paper calculates the surface area, porosity, pore-size distribution and powder

X-ray diffraction pattern of the hypothetical MOFs to identify structures

that are suitable for methane storage. The simulated methane adsorption

isotherms closely match other published simulation and experimental results.

The simulated storage results revealed a MOF candidate with record capac-

ity, which was later confirmed and synthesised.

Although the simulation method applied by Wilmer et al. [15] allows for

4

successful screening of numerous MOFs, the method requires very extensive

computing power and time. A far more computationally efficient alternative

is to use analytical mathematical models to design MOFs. One example of

the application of the modelling method is given by Thornton et al. [16] for

MOF-177 impregnated by magnesium-decorated fullerenes. The adsorption

isotherm of a series of isoreticular MOFs and MOF-177 impregnated with

fullerenes were evaluated and compared with experimental results to establish

the validity of the model. This model is then used to predict the adsorption

isotherm for MOF-177 impregnated with a magnesium-decorated fullerene.

A current obstacle for MOFs as the leading candidates for hydrogen stor-

age is the difficulty of meeting the 2015 gravimetric and volumetric target set

by the U.S. Department of Energy (DOE) of 5.5 wt% and 40 g/L, with the

ultimate targets of 7.5 wt% and 70 g/L. These hydrogen storage systems also

have to meet the 2017 target delivery and operating temperatures of 233–

358 K and 233–333 K, and the range of delivery pressure of 5–12 bar [17].

Currently, the highest total gravimetric uptake at room temperature by a

MOF is 2.55 wt% by PCN-68 (or Cu3(ptei)) [18] at 100 bar, which is less

than half of the DOE target. Calculations by Bhatia and Myers [9] indicate

that to achieve the target gravimetric uptake at an adsorption and desorption

pressure of 30 and 1.5 bar, the ideal enthalpy of H2 adsorption is required

to be in the range 15–25 kJ mol−1. However, most of the isosteric heat of

adsorption reported for MOFs are in the range of 5–12 kJ mol−1.

The main factors in optimising the storage performance of physisorbents

are maximising the internal surface area, which is known to increase the stor-

age capacity, [19] and the control of pore architecture and surface chemistry

5

in order to maximise adsorption enthalpy [20]. The impact of surface area,

pore volume and heat of adsorption on the performance of H2 adsorption in

MOFs at room temperature will be presented in Chapter 3.

1.3 Porous aromatic frameworks

Although MOFs have a high surface area and a porous framework that con-

tribute greatly to the efficiency of gas storage, these same factors also cause

low thermal and chemical stability [21]. Such difficulties may be addressed by

replacing coordinate covalent bonds with strong covalent bonds, as is done in

covalent organic frameworks (COFs) [22, 23], porous organic polymers and

more recently, porous aromatic frameworks (PAFs).

Ben et al. [24] first conceptualised the structure of PAF from the struc-

turally stable diamonds in 2009 which comprise only carbon and hydrogen

with the carbon-carbon (C-C) bonds in the diamond structure replaced with

phenyl rings. The replacement of each of the C-C bond by one (PAF-301)

or two (PAF-302) phenyl rings increased the BrunauerEmmettTeller (BET)

surface area to 1880 m2g or 5640 m2g respectively. This produces a structure

which is as physicochemically stable as COFs and porous organic polymers,

but has a large surface area that is comparable to MOFs, and has the addi-

tional advantage of having a lower crystal density. The structure of PAF-301

and PAF-302 are shown in Figure 1.2. However, the H2 uptake in PAFs still

needs to be improved to achieve the U.S. DOE target. Lan et al. [3] report

that their experimental isotherms of H2 uptake in PAF-302 do not achieve

more than 3 wt% at 150 K and 50 bar.

6

In this thesis we discuss possible methods that may increase the adsorp-

tion enthalpy of PAFs. The first method involves impregnation of PAF cavi-

ties with fullerenes (C60) and the second method entails doping lithium (Li)

atoms above the organic units. Previous work in the literature has shown that

C60 impregnation provides more attractive sites for H2 adsorption [25, 26].

It is also known that Li doping creates strong polarisation effects caused by

the charge transfer from H2 and Li atoms, producing a strong binding inter-

action [27]. This results in a significantly higher H2 uptake as compared to

non-doping as reported in several papers [26, 28, 29, 30]. In Chapter 4, the

outcomes of C60 impregnation and/or Li doping in PAFs are observed.

a b

Figure 1.2: Structure of (a) PAF-301 and (b) PAF-302.

1.4 Thesis structure

This thesis comprises six chapters, with Chapter 1 containing the background

for hydrogen adsorption and a discussion on some of the latest materials that

have high potential for hydrogen storage. In the next chapter we present the

7

methodology developed here to use building blocks such as a point, line,

plane, sphere, and cylinder, to represent the cavities in nanostructures to

calculate the interaction energy with H2. Following this in the same chapter,

we discuss the methodology used to calculate the gas uptake in nanostruc-

tures and its implementation in a model to calculate energy production from

a materials-based fuel tank.

In Chapter 3, the methods introduced in Chapter 2 are used to investigate

the structure of benzene tribenzoate (Be-BTB) and to compare the energy

produced by theoretical fuel cells coupled with a MOF storage device. Fol-

lowing this, Chapter 4 compares three techniques that are aimed to improve

gas adsorption in PAFs: C60 encapsulation, Li doping in PAFs and the com-

bination of the two techniques. Finally, in Chapter 5 we summarise the work

in this thesis. Some brief concluding remarks on the methods and materials

studied here are provided and also some points of interest for future research

in this area are presented.

8

Chapter 2

Analytical formulations of

nanospace in porous materials

In this chapter we model both simple and complicated geometries of nano-

materials using idealised building blocks that describe the interactions with

simple and elegant analytical calculations. Section 2.1 introduces the ana-

lytic representations of the van der Waals interaction, which is the primary

force responsible for gas adsorption, between a hydrogen molecule and various

building blocks. These building blocks are represented by standard geometri-

cal shapes such as points, rings, lines, planes, spheres and cylinders as shown

in Figure 2.1. At first sight such a simplified modeling approach may seem

geometrically unrealistic, but in similar situations it has been shown to pro-

vide the major contribution to the interaction energy of the actual structure,

confirmed by either independent experimental results or numerical simula-

tion [10, 31, 32]. The model used to study gas adsorption is introduced in

Section 2.2 and in Section 2.3 we will provide some examples of how the

9

building blocks can be incorporated. The interaction between a hydrogen

molecule and a carbon atom, polyacetylene, graphene sheet, benzene ring,

fullerene and carbon nanotube will be discussed independently.

a

b

c

d e f

Figure 2.1: Atomic representation of structures that can be described bybuilding blocks. An atom can be described by a point (a), benzene by a ring(b), fullerene as a spherical surface (c), carbon chain by a line (d), graphenesheets by planes (e), and carbon nanotube by a cylindrical surface (f).

2.1 Molecular interactions with

building blocks

In this section, we first introduce methods that can be used to describe

the van der Waals energy that exists between a nanostructure J and an

atom P . If the locations every atom j in structure J and a single atom P

are known and defined by coordinate positions, then we can evaluate the

total interaction energy U by calculating the sum of the individual atomic

10

interactions between each j and P , thus

U =∑j∈J

Φ(ρjP ), (2.1.1)

where Φ(ρ) is the potential energy function and ρjP is the distance between

atoms j and P .

If J is a large structure or the location of its atoms are not precisely

known, the discrete method can be replaced by the continuum approxima-

tion provided that the geometry is reasonably simple. Examples include

cylindrical surfaces for nanotubes and spherical surfaces for fullerenes. Using

this method, we assume that the atoms on the surface are uniformly dis-

tributed so that we can perform a continuous integration over the surface to

obtain the total interaction energy

U = ηJ

∫SJ

Φ(ρ) dSJ , (2.1.2)

where ηJ denotes the atomic density (number of atoms/surface area) on

surface J and ρ represents the distance between P and the infinitesimal

surface element dSJ .

The van der Waals interaction energy between two non-bonded atoms

can be described using the 6-12 Lennard-Jones potential [33]

Φ(ρ) = 4ε

[−(σ

ρ

)6

+

(σ

ρ

)12]

(2.1.3)

= −C1

ρ6+C2

ρ12,

11

where C1 = 4εσ6 and C2 = 4εσ12. The distance between the atoms is denoted

by ρ, σ is the atomic distance when the potential energy is zero, and ε

is the value of the energy when the atoms are at the equilibrium distance,

ρ0 = 21/6σ. The negative and positive terms describe the balance of attractive

and repulsive forces experienced by the two atoms of interest.

The interatomic potential parameters σ and ε for a different species of

atoms are determined using the empirical Lorentz-Berthelot mixing rules

where ε12 =√ε1ε2 and σ12 = (σ1 + σ2)/2. The long range electrostatic

forces which also contribute to the atomic interaction are found to have

negligible effects on the total hydrogen uptake [20], and are not included in

our model. The values for ε and σ for various elements may be taken from

generic force fields such as the Dreiding force field [34] and the universal

force field (UFF)[35], derived from first principles with quantum mechanics

fundamentals or fitted to experimental data.

The interaction between two non-bonded atoms can be extended to the

interaction of a hydrogen molecule with nanostructures that can be quite

complicated and difficult to represent geometrically. Here we introduce the

idea of ‘building blocks’ to represent nanostructures. Due to the symmetric

nature of the building blocks chosen, the interaction energy between an arbi-

trary point and these building blocks can be modelled using the continuum

approximation.

The van der Waals interaction energy between an atom and a nanostruc-

ture is modelled by substituting Eq. 2.1.3 into Eq. 2.1.2 to obtain

U = ηJ (−C1R3 + C2R6) , (2.1.4)

12

where

Rn =

∫SJ

1

ρ2ndSJ (2.1.5)

for n = 3 and 6. In the following subsections, we present the interaction

energy between an arbitrary point and the various building blocks using this

approach.

2.1.1 Interaction with a point

In this subsection we examine the interaction between P and a structure J

which contains only one element j. We denote the position of P by (xp, yp, zp)

and j by (xj, yj, zj). Thus the distance between P and j is given by

ρ =[(xp − xj)2 + (yp − yj)2 + (zp + zj)

2] 1

2 . (2.1.6)

The Lennard-Jones potential between the two atoms can then be obtained

by substituting Eq. 2.1.6 into Eq. 2.1.3.

2.1.2 Interaction with a line

Using a two-dimensional coordinate system, if a line L (which can be finite

or infinite) lies on the y-axis, the points on L can be defined parametrically

by (0, yL). Assuming that atom P is located at (g, 0), the distance between

P and a point on L is given by

ρ =[g2 + y2L

] 12 . (2.1.7)

13

The total potential energy for atom P interacting with L, ULP is evaluated

by firstly solving Eq. 2.1.5 for Rn given that the line element is dyL to obtain

Rn =

∫ ∞−∞

(g2 + y2L

)−ndyL

= g1−2nB

(1

2, n− 1

2

). (2.1.8)

The full derivation for the equation above is provided in Appendix 6.1.1.

Rn is then substituted into Eq. 2.1.4 to yield

ULP = ηπ

(−3C1

8g5+

63C2

256g11

), (2.1.9)

where η is the mean atomic density of the line and B(x, y) is the beta function

such that

B (x, y) =Γ(x)Γ(y)

Γ(x+ y)(2.1.10)

and

Γ(m

2

)=

(m− 2)!!√π

2(m−1)/2 , (2.1.11)

where m is a positive integer and !! denotes the double factorial.

If the coordinates are known such that P = (xp, yp, zp) and two points

located on the line L are L1 = (x1, y1, z1) and L2 = (x2, y2, z2), the parameter

g in the previous equation can be replaced by the shortest distance between

14

P and L given by

g =|(L1 − P )× (L2 − L1)|

|(L2 − L1)|. (2.1.12)

2.1.3 Interaction with a plane

We now consider the interaction of a plane S with a point P using a three-

dimensional coordinate system. If S lies on the yz-plane and P lies on the

x-axis, then S = (0, ys, zs) where -∞ < ys, zs <∞, and P = (g, 0, 0), where

g is the perpendicular distance of P from S. The distance between P and a

point on S is

ρ =[g2 + y2s + z2s

] 12 . (2.1.13)

Given that the area element of the plane is dysdzs, we substitute Eq. 2.1.13

into Eq. 2.1.5 to obtain

Rn =

∫ ∞−∞

∫ ∞−∞

(g2 + y2s + z2s

)−ndysdzs

= g2−2nB

(n− 1

2,1

2

)B

(n− 1,

1

2

). (2.1.14)

The derivation of this equation is provided in Appendix 6.1.2.

The interaction energy between P and S is then calculated by substituting

Eq. 2.1.14 into Eq. 2.1.5. This is given by

UPS = πη

(−C1

2g4+

C2

5g10

), (2.1.15)

15

where η is the mean atomic density for the plane which is calculated by

dividing the number of atoms on the sheet by the surface area.

As shown in the previous subsection, we can replace the parameter g

with the shortest distance from P to S if coordinates are provided. If P =

(xp, yp, zp) and three noncollinear points located on S are S1 = (x1, y1, z1),

S2 = (x2, y2, z2) and S3 = (x3, y3, z3), then the shortest distance between P

and S is defined as

g = n · (P − S), (2.1.16)

where n is the unit normal for S and is given by

n =(S2 − S1)× (S3 − S1)

|(S2 − S1)× (S3 − S1)|. (2.1.17)

2.1.4 Interaction with a ring

The interaction of a point with a ring can be categorised into two cases: i)

the point interacts with the ring from the side, and ii) the point interacts

with the ring from an off-side position. These two cases are examined in

turn.

Point P located at the side of the ring

The first case discusses the scenario of a point interacting with a ring from

the side. Using a two-dimensional coordinate system, if the centre of the ring

of radius q is located on the origin, the coordinates of a point on the ring is

Q = (q sin θ, q cos θ). If point P is located at (0, g), the distance between P

16

and Q is

ρ =

[(q − g)2 + 4gq sin2 θ

2

] 12

. (2.1.18)

Given that the line element is q dθ, we solve Eq. 2.1.5 to obtain

Rn =

∫ 2π

0

q

[(q − g)2 + 4gq sin2 θ

2

]−ndθ. (2.1.19)

Through the bisection of the interval of integration and the substitution of

t = sin2(θ/2), we obtain

Rn =q

(q − g)2n

∫ 1

0

t−12 (1− t)−

12

[1 +

4gqt

(q − g)2

]−ndt, (2.1.20)

where the integral is of the standard hypergeometric form,

∫ 1

0

xb−1(1− x)c−b−1(1− zx)−a dx = B(b, c− b) 2F1(a, b; c; z) (2.1.21)

Substituting the standard hypergeometric form into Rn produces

Rn =πq

(q − g)2nF

(n,

1

2; 1;− 4gq

(q − g)2

). (2.1.22)

Finally, we substitute Eq. 2.1.22 into Eq. 2.1.4 to calculate the interaction

energy between P and Q.

UPQ = πηq

[− C1

(q − g)6F

(3,

1

2; 1;− 4gq

(q − g)2

)+

C2

(q − g)12F

(6,

1

2; 1;− 4gq

(q − g)2

)]. (2.1.23)

17

To provide some variation on the location of point P for this case, we can

consider the case where P lies on the yz-plane and therefore has the coordi-

nates (0, yP , zP ). To do this we substitute g =√y2P + z2P into Eq. 2.1.23.

Point P located at the top or bottom of the ring

The second case discusses the scenario where a point is interacting with a

ring from its top or bottom. Using a three-dimensional coordinate system,

if the centre of a ring of radius q is located on the origin, the coordinates of

the points on the ring are Q = (q cos θ, q sin θ, 0). If the location of point P

is (xp, yp, zp), the distance between P and Q is

ρ = [β − 2αq cos(θ − θ0)]12 , (2.1.24)

where β = q2 + x2p + y2p + z2p , α =√x2p + y2p and θ0 = arctan

(ypxp

).

To solve for Rn, similar calculations as used by Tran-Duc et al. [36] can

be used to obtain

Rn =

∫ 2π

0

q [β − 2αq cos(θ − θ0)]−n dθ

=2πq

(β − αq)n 2F1

(n,

1

2; 1;

4αq

2αq − β

). (2.1.25)

The full derivation of this can be found in Appendix 6.1.3.

We then substitute Eq. 2.1.25 into Eq. 2.1.4 to calculate the interaction

18

between P and Q, given by

UPQ = 2πηq

[− C1

(β − αq)3F

(3,

1

2; 1;

4αq

2αq − β

)+

C2

(β − αq)6F

(6,

1

2; 1;

4αq

2αq − β

)]. (2.1.26)

Note that in both cases, η is the mean atomic density of the ring which

is calculated by dividing the number of atoms by the circumference of the

ring.

2.1.5 Interaction with a spherical surface

For a sphere S of radius t, we assume that the centre of S is at the origin O.

Thus a point on the surface of the sphere is defined parametrically by

(t sin θ cosφ, t sin θ sinφ, t cos θ). (2.1.27)

For convenience we position the point P on the z-axis a distance g from the

centre of S. P is therefore located at (0, 0, g) and the distance between P

and S is

ρ =[t2 sin2 θ + (t cos θ − g)2

] 12 . (2.1.28)

Given that the area element is t2 sin θ dφ dθ, the equation for Rn is calcu-

lated using similar methods by Cox et al. [37] and is given by

19

Rn =

∫ π

0

∫ π

−πt2 sin θ

[t2 sin2 θ + (t cos θ − g)2

]−ndφ dθ

=2πt2∫ π

0

sin θ[t2 − 2tg cos θ + g2

]−ndθ.

Using the substitution m = t2− 2tg cos θ+ g2, the equation can be rewritten

to become

Rn =tπ

g

∫ (t+g)2

(t−g)2m−n dm

=tπ

g(1− n)

[1

(t+ g)2(n−1)− 1

(t− g)2(n−1)

].

Substituting Rn into Eq. 2.1.4 provides us with

UPS =πηt

g

{C1

2

[1

(g + t)4− 1

(g − t)4

]− C2

5

[1

(g + t)10− 1

(g − t)10

]},

(2.1.29)

where UPS is the potential energy of P interacting with S. The parameter

η is the mean surface density for the sphere which is calculated by dividing

the number of atoms by the surface area of the sphere.

Interactions inside and outside a sphere

Two cases for the interaction of P with a sphere are presented here: i) P is

located inside the sphere, and ii) P is located outside the sphere. To account

for the position of P inside and outside the sphere, we redefine the position

of P to be located on the yz-plane such that the position of P = (0, yp, zp).

20

To calculate the interaction energy between P and S, g =√y2p + z2p is

substituted into Eq. 2.1.29. If P is inside the sphere, then the distance of P

from the centre of S has to be smaller than its radius such that g < t. If P

is located outside the sphere, then g > t.

2.1.6 Interaction with an infinite cylindrical surface

The final building block that we describe is the cylindrical surface. We

assume that point P is located on the x-axis at (g, 0, 0) and the cylindrical

surface C is defined to be (c cos θ, c sin θ, zc) where c is the radius of the

cylinder. Two cases are presented here: i) point P is located inside the

cylinder and ii) point P is located outside the cylinder.

Point P located inside cylinder

When P is inside the cylinder (in other words, when g < c), the distance

from P to C is given by

ρ =[(c cos θ − g)2 + c2 sin2 θ + z2c

] 12

=

[(c− g)2 + z2c + 4cg sin2

(θ

2

)] 12

. (2.1.30)

The area element is given by c dθ dzc which is used to solve for Rn. Fol-

21

lowing similar calculations by Cox et al. [37], Rn is given by

Rn =

∫ ∞−∞

∫ π

−π

c[(c− g)2 + z2c + 4cg sin2

(θ2

)]n dθ dzc=c

∫ π

−π

1

α2n−1 dθ

∫ π2

−π2

cos2n−2 ψ dψ, (2.1.31)

where α2 = (c − g)2 + 4cg sin2(θ/2). Substituting t = sin2(θ/2) into the

equation will give us

Rn =2c

(c− g)2n−1B

(n− 1

2,1

2

)∫ 1

0

t−12 (1− t)−

12

(1 +

4cgt

(c− g)2

) 12−n

dt .

(2.1.32)

The integral is now in fundamental form for the usual hypergeometric

function as given in Eq. 2.1.21. Using the usual power series expansion for

the hypergeometric function, we rewrite Rn as

Rn =2πc

(c− g)2n−1B

(n− 1

2,1

2

)F

(n− 1

2,1

2; 1;− 4cg

(c− g)2

)=

2π2

Γ(2n− 1)(2c)2n−2

∞∑m=0

(Γ(2n+ 2m− 1)gm

Γ(n+m)m!(4c)m

)2

. (2.1.33)

Finally, we can express the potential energy between C and P as

UCP =π2η

192(−C1R3 + C2R6) , (2.1.34)

22

where

R3 =1

c4

∞∑m=0

((2m+ 4)!gm

(m+ 2)!m!(4c)m

)2

(2.1.35)

and

R6 =1

9676800c10

∞∑m=0

((2m+ 10)!gm

(m+ 5)!m!(4c)m

)2

. (2.1.36)

Point P located outside cylinder

The second case is when point P is outside the cylinder, or when g > c. In

this case, the distance from P to C is

ρ =[(g − c cos θ)2 + c2 sin2 θ + z2c

] 12

=

[(g − c)2 + z2c + 4cg sin2

(θ

2

)] 12

. (2.1.37)

We perform the similar calculations as above to solve for the integral

Rn =

∫ ∞−∞

∫ π

−π

c[(g − c)2 + z2c + 4cg sin2

(θ2

)]n dθ dzc (2.1.38)

to yield

Rn =2cπ

(g − c)2n−1B

(n− 1

2,1

2

)F

(n− 1

2,1

2; 1;− 4cg

(g − c)2

)=

4cπ2

Γ(2n− 1)(2g)2n−1

∞∑m=0

(Γ(2n+ 2m− 1)cm

Γ(n+m)m!(4g)m

)2

. (2.1.39)

23

Thus the potential energy between C and P is

UCP =cπ2η

192(−C1R3 + C2R6) , (2.1.40)

where

R3 =1

g5

∞∑m=0

((2m+ 4)!cm

(m+ 2)!m!(4g)m

)2

(2.1.41)

and

R6 =1

9676800g11

∞∑m=0

((2m+ 10)!cm

(m+ 5)!m!(4g)m

)2

. (2.1.42)

In both cases, η is the mean atomic density for the cylinder which is calculated

by dividing the number of atoms by the surface area of the cylinder.

If the location of P can be written as (0, y6, z6) then the potential energy

inside the cylinder by substituting g =√y26 + z26 into Eq. 2.1.34. To calculate

the potential energy outside the cylinder, we substitute g =√y26 + z26 into

Eq. 2.1.40.

2.2 Gas adsorption model

In this section, we will discuss the model used to investigate the hydrogen

adsorption performance and effect of pore size in nanostructures. We fol-

low the Topologically Integrated Mathematical Thermodynamic Adsorption

Model (TIMTAM) approach by Thornton et al. [32] which assumes ideal

building blocks to represent the cavity of the structure and uses this to cal-

24

culate the potential energy interactions between the gas and the adsorbate

discussed in Section 2.1.

Based on the paper by Walton and Snurr [38], adsorption can be domi-

nated by the pore filling or multilayered formation mechanisms. To ensure

that both types of adsorption are considered in our model, we make the as-

sumptions that the gas molecules can exist as a mixture of adsorbed and bulk

gas inside the adsorbent. The probability that a molecule is adsorbed on the

surface of the cavity can be expressed as 1− exp(U/RT ) when its potential

energy (U) is larger than its kinetic energy, calculated from RT , the ideal gas

constant (R) and the temperature (T ). Consequently, the probability that

the molecule exists as bulk gas phase is exp(U/RT ). In this formulation,

pore filling and multilayer mechanisms are not distinguishable, although the

ratio of bulk phase over adsorbed phase is an approximate indicator.

As indicated in the previous chapter, free volume (or pore volume) is

explored here for its role in adsorbent performance. Total free volume (Vf ),

is comprised of the volume available for adsorbed phase (Vad) and the volume

available for bulk gas phase that remains within each of the ideal building

blocks (Vbulk). Within the building block, Vad is calculated by integrating the

probability of adsorption over the total free volume, and Vbulk is calculated

by integrating the probability that the molecule remains as bulk gas over the

total free volume. The formulae for Vad and Vbulk are given by

Vad =

∫Vf

(1− exp

[U(ρ)

RT

])dρ, (2.2.1)

25

and

Vbulk =

∫Vf

(exp

[U(ρ)

RT

])dρ, (2.2.2)

where R is the ideal gas constant and T is the temperature.

The total number of gas molecules in the cavity is calculated using the

method described by Thornton et. al [16] where Vad and Vbulk are combined

with the appropriate equations of state. The number of molecules in bulk

gas state (nbulk) is obtained by solving the simplified van der Waals equation

of state given by

P

(Vbulknbulk

− C)

= RT. (2.2.3)

In this equation, P is the pressure and C = RTc/(8Pc) is the occupied vol-

ume calculated from critical parameters, Tc and Pc which are the critical

temperature and pressure of the gas respectively.

The number of molecules in the adsorbed state (nad) is calculated by

solving a modified version of the Dieterici equation of state, given by

P

(Vadnad− C

)= RT exp

(−QRT

). (2.2.4)

In this equation, Q is the heat of adsorption based on the equation described

by Everett and Powl [39],

Q = |Umin|+ αRT, (2.2.5)

where Umin is the minimum potential energy and α represents the excess

26

thermal energy in the adsorbed phase, fixed at 0.5 [39]. The α constant

may vary for different materials and therefore the model should be used

as an insightful fit rather than a complete prediction. By rearrangement,

Eq. 2.2.4 is identical to the Boltzmann distribution law where local density

is proportional to exp(-U/kT )[40, 41].

The gravimetric uptake, G, is determined using the total number of

molecules that are calculated from Eq. 2.2.3 and Eq. 2.2.4. This is given

by

G =nm

nm+M× 100, (2.2.6)

where n is the total number of molecules in the cavity (nad +nbulk), m is the

mass of the gas molecule and M is the mass of a unit cell of the adsorbent.

2.3 Materials represented by building blocks

In this section, we present some case studies of the interaction of a hydro-

gen atom with nanostructures represented by building blocks that consist

mainly of carbon (C) and hydrogen (H) atoms. Our case studies include a

carbon atom which is represented with a point, a polyacetylene with a line,

a graphene sheet with a plane, a benzene with a ring, fullerene with a sphere

and finally, a carbon nanotube with a cylinder. It should be noted that the

hydrogen molecule interacting with the nanostructures is modelled as a sin-

gle point. The calculations are done using the algebraic package Maple [42]

27

and the figures are produced using MATLAB [43].

The parameters used for the calculations are provided in Table 2.1. In this

table, the attractive and repulsive constants C1 and C2 are calculated using

the Lennard-Jones parameter values reported in the papers by Thornton et

al. [16] and Rappe et al. [35]

Table 2.1: Numerical values of various parameters (C-H2 denotes the inter-action between a hydrogen molecule with a carbon atom, and C-H2 denotesthe interaction between a hydrogen molecule with a hydrogen atom).

Parameter Description Value

rC radius of C ring (benzene) 1.4 ArH radius of H ring (benzene) 2.48 At radius of fullerene (C60) 3.55 Ac radius of (10,10) carbon nanotube 6.784 Aη2 mean atomic density of C and H in 0.719 A−1

polyacetyleneη3 mean atomic density of graphene 0.382 A−2

η4C mean atomic density of C ring (benzene) 0.682 A−1

η4H mean atomic density of H ring (benzene) 0.385 A−1

η5 mean atomic density of fullerene 0.379 A−2

η6 mean atomic density of carbon nanotube 0.381 A−2

C1 C−H2 attractive constant C-H2 23.58 eVA6

C1 H−H2 attractive constant H-H2 6.10 eVA6

C2 C−H2 repulsive constant C-H2 36776.87 eVA12

C2 H−H2 repulsive constant H-H2 3806.11 eVA12

28

2.3.1 A carbon atom

In this section, we use the model described in Subsection 2.1.1 to calculate

the potential energy between a carbon (C) and a hydrogen molecule (H2).

We assume that C is located at the origin O and H2 is located at (0, y1, z1).

Figures 2.2(a) and 2.2(b) shows the contour plots for the interaction potential

with the carbon atom located in the centre of the circle in Figure 2.2(a). The

hydrogen molecule experiences the strongest interaction energy at a distance

of 3.8 A from the carbon atom. This is represented by the dark blue areas

in the figures where the minimum potential energy occurs.

(a) (b)

Figure 2.2: Contour plots of the potential energy of a hydrogen moleculeinteracting with a carbon atom in (a) two-dimensions, and (b) three-dimensional space.

2.3.2 Polyacetylene

This section describes the interaction of a hydrogen atom interacting with

polyacetylene, which is an organic polymer with the unit (C2H2)n repeated

29

to create a long polymer chain. In this case, the hydrogen molecule is given a

location of (0, y2, z2). The atoms in the polyacetylene is modelled as a single

line and is located on the x-axis.

The total potential energy between the hydrogen molecule and polyacety-

lene is calculated by adding together the potential energies for H2 interacting

with the hydrogen line and carbon line to yield

Utot = UC + UH , (2.3.1)

where UC is the potential energy for H with the carbon line and UH is the

potential energy for H with the hydrogen line. We assume that the carbon

and hydrogen atoms lie on the same line to simplify calculations. Note that

the calculation relies on the bold assumption that the carbon and hydrogen

atoms are located on the same line. A more rigorous mathematical model is

required to model the line curvature and branches.

Figure 2.3: Structure of polyacetylene.

The mean line density, η2, is calculated using the length of a particular

section of the polyacetylene, C11H13 (as shown in Figure 2.3) to be 12.027 A.

The mean line density for carbon and hydrogen for the polyacetylene is then

obtained by dividing the corresponding number of atoms to its length to

30

obtain 11/13.305 = 0.719 A−1 and 13/13.305 = 0.849 A−1.

(a) (b)

Figure 2.4: Contour plots of the potential energy of H2 interacting with apolyacetylene in (a) two-dimensions, and (b) three-dimensions.

The potential energy between the hydrogen atom and the polyacetylene

line is shown in Figures 2.4(a) and 2.4(b). The line is located in the middle of

the circle found in Figure 2.4(a) and the blue areas in both figures represents

the minimum potential energy. We calculate that the H2 is most stable at a

distance of 3.4 A from the polyacetylene.

2.3.3 Graphene sheet

This section presents the case of a hydrogen molecule interacting with a sheet

of graphite. The mean surface density for a graphene sheet, η3 is 4√

3/(9γ2)

where γ is the carbon-carbon bond length. For a sheet of graphene, γ =

1.42 A and therefore η3 = 0.382 A−2. The parameters C1 and C2 for the

hydrogen-carbon interactions can be found in Table 2.1.

The graphene sheet is modelled as a plane. To visualise this, three random

31

coordinates which lies on the plane are selected: (1, 2, 4), (2, 4, 5) and (3, 5, 7)

and a coordinate for H2 is assigned: (0, y3, z3). The potential energy between

a hydrogen molecule and a graphite sheet is calculated using Eq. 2.1.15 and

Eq. 2.1.16.

The two and three-dimensional interaction between H2 and the graphene

sheet is presented in Figures 2.5(a) and 2.5(b). We can observe from these

figures that the minimum potential energy is located at the dark blue section

of the contour plots. The hydrogen atom is stable when its distance from the

graphene sheet is approximately 11.3 A.

(a) (b)

Figure 2.5: Contour plots of the potential energy of H2 interacting with agraphene sheet in (a) two-dimensions, and (b) three-dimensions.

2.3.4 Benzene ring

For the interaction of H2 with a benzene ring (C6H6), the centre of the

benzene ring lies on the origin O. The benzene ring consists of two rings,

one to represent the hydrogen atoms and the other to represent the carbon

32

atoms. The radius and mean surface density of the hydrogen ring, rH and

η4H is 2.48 A and 0.385 A−1 and the radius and mean surface density of the

carbon ring, rC and η4C is 1.4 A and 0.682 A−1. The parameters C1 and C2

for the C-H2 and H-H2 interactions can be found in Table 2.1.

The total potential energy for H2 interacting with a benzene ring can

be calculated by summing the potential energies for H2 interacting with the

hydrogen ring and carbon ring following Eq. 2.3.1, where UC is the potential

energy for H2 with the carbon ring and UH is the potential energy for H2

with the hydrogen ring.

Two different outcomes which corresponds to two different locations of

H2 when it interacts with the benzene ring are introduced. The first case

presented here is H2 interacting with the benzene ring from the side where

we assume that the position of the hydrogen atom is at (0, y4, z4). To obtain

the potential energy from the interaction, g =√z24 + y24 is substituted into

Eq. 2.1.23. The two-dimensional contour plot for this interaction is presented

in Figure 2.6(a) and the three-dimensional contour plot in Figure 2.6(b). The

figures show that the hydrogen atom has the strongest interaction with the

benzene ring at 5.4 A from the centre of the benzene ring.

For the case of H2 interacting with the benzene ring from the top (or

bottom), we assume that the position of H2 is at (4, y4, z4) and apply this

into Eq. 2.1.25 and 2.1.26 to investigate their interaction energy. The two-

dimensional and three-dimensional contour plots of the potential energy func-

tion is presented in Figures 2.7(a) and 2.7(b). The hydrogen molecule is at

its most stable position at y4 = 0 and z4 = 2.6 as shown by the blue area

of the figures. The red area of the figures show the repulsive force that the

33

(a) (b)

Figure 2.6: Contour plots of the potential energy of H2 interacting with abenzene ring from the side in (a) two-dimensions, and (b) three-dimensions.

hydrogen atom experiences in the centre of the benzene ring.

(a) (b)

Figure 2.7: Contour plots of the potential energy of H2 interacting with abenzene ring from the top in (a) two-dimensions, and (b) three-dimensions.

34

2.3.5 Fullerene

In this section, two cases of the interaction between a hydrogen atom with

a fullerene (C60) are demonstrated: (i) H2 is located outside C60 and (ii) H2

is located inside C60. For both cases, H2 is given a location of (0, y5, z5) and

the centre of the fullerene lies on the origin O. Eq. 2.1.29 is used to generate

a potential energy surface along a two dimensional slice. The mean surface

density for a fullerene, η5 is 60/(4πt2) = 0.379 A−2 where t = 3.55 A is the

radius of the fullerene. The parameters C1 and C2 for the hydrogen-carbon

interactions can be found in Table 2.1.

The two and three-dimensional portrait of the interaction energy between

H2 and C60 for the first case is presented in Figures 2.8(a) and 2.8(b). The

figures show that minimum potential occurs when H2 is located 6.9 A from

the centre of C60 as represented by the dark blue areas.

The second case for this interaction is when H2 is interacting with C60

from the inside. Figures 2.9(a) and 2.9(b) shows the two and three-dimensional

plots of the potential energy of this interaction. The minimum potential en-

ergy in this case occurs in the centre of C60 due to its small size. If H is

interacting with a larger sphere, the minimum potential energy will be lo-

cated further from the centre and nearer to the walls of the sphere.

2.3.6 Carbon nanotube

In this section, we model the interaction between H2 and a semi-infinite

(10,10) carbon nanotube (CNT) of radius c = 6.784 A. The mean surface den-

35

(a) (b)

Figure 2.8: Contour plots of H2 interacting the outer walls of a fullerene.

(a) (b)

Figure 2.9: Contour plots of H2 interacting with the inner walls of a fullerene.

sity for a CNT is the same as for a graphene. Therefore, η6 = 4√

3/(9γ2) =

0.381 A−2 where γ = 1.421 A is the carbon-carbon length. The value of the

parameters C1 and C2 for the hydrogen-carbon interactions can be found in

Table 2.1.

Two different cases of H2 interacting with the CNT are presented here.

For the case where H2 is located inside the CNT, we assume that H2 is

36

located at (0, y6, z6) and the centre of the CNT lies on the x-axis. The

potential energy between H2 and the CNT is calculated using Eq. 2.1.34 and is

represented by the two and three-dimensional contour plots in Figures 2.10(a)

and 2.10(b). The dark blue areas represent the equilibrium distance, 3.33 A

for H2 from the centre of the CNT. As the H2 molecule moves towards the

centre of the nanotube, the strength of the potential energy is reduced.

(a) (b)

Figure 2.10: Contour plots of H2 interacting with the inner walls of an infinitecarbon nanotube.

The second variation to H2 interacting with the CNT is demonstrated

here with H2 interacting outside an infinite CNT. The potential energy for

this interaction is calculated using Eq. 2.1.40. Figures 2.11(a) and 2.11(b)

provides the two and three-dimensional contour plots of the potential energy

for this interaction. As shown in the dark blue sections of the figures, H2 is

at equilibrium distance at 10.2 A from the centre of the CNT.

37

(a) (b)

Figure 2.11: Contour plots of H2 interacting with the outer walls of an infinitecarbon nanotube.

2.4 Example of interactions with porous ma-

terials

In this section, we present some examples of how we can implement the

idea of building blocks discussed in the previous sections. An example of

how nanostructures can be represented by planes and rings is given in the

paper by Tran-Duc et al. [44]. This paper investigated the adsorption of

polycyclic aromatic hydrocarbons (PAHs) or in particular, coronene (C24H12)

onto a graphite surface using both the discrete and continuous approach.

The coronene is modelled as four circular rings and the graphite surface

as a plane. The equation of the potential energy between the two for the

continuous approach is based on Eq. 2.1.15 and 2.1.26.

Comparisons of results from the discrete and continuous method show

that the continuous method provides results that are as accurate as the dis-

crete method. The energy profiles for the interaction between two structures

38

are also provided for different values of the vertical distance between C24H12

and the graphene sheet (Z). The different values used are Z ≥ 7.5 A when

it is far from the graphite surface, 3.9 A< Z < 7.5 A when it is at an in-

termediate distance and Z ≤ 3.9 A when it is near the graphite plane. The

authors concluded that the most stable configuration for coronene molecule

when Z ≥ 7.5 A is when it is perpendicular to the graphene sheet. At

3.9 A< Z < 7.5 A, a tilted configuration is preferred and at Z ≤ 3.9 A, the

minimum potential energy occurs when it is parallel to the plane.

The spherical model is used in the paper by Thornton et al. [16] where the

gas uptake for three types of MOFs are predicted; MOF-177, MOF-177 im-

pregnated with C60 fullerenes (C60@MOF) and MOF-177 impregnated with

magnesium-decorated fullerenes (Mg-C60@MOF). The paper first verified the

model, which is based on Eq. 2.1.29 with other experimental and simula-

tion results. The authors reported that the model accurately portrays the

observed effects of temperature, pressure and cavity size on hydrogen up-

take. The model is then used to predict the hydrogen and methane uptake

for MOF-177 and the proposed structures C60@MOF and Mg-C60@MOF by

first calculating the potential energy within the cavity. In C60@MOF and

Mg-C60@MOF, the fullerene and magnesium decorated fullerene is assumed

to be located in the middle of the MOF structure and is modelled as a sphere.

The potential energy distribution within the structure is calculated by adding

the potential energy between the gas and the MOF structure with the po-

tential energy between the gas and the fullerene or magnesium decorated

fullerene. The authors in [16] concluded that Mg-C60@MOF has a greater

potential energy compared to the other structures and therefore is able to

39

adsorb gases more efficiently.

2.5 Summary

In this chapter, we determine analytical potential energy models which de-

scribe the interactions between an atom and nanostructures made of sim-

ple building blocks. Various idealised building blocks are discussed, such

as points, lines, planes, rings, spheres and cylinders. Case studies of these

models provide the potential energy distribution between a hydrogen atom

and the various building blocks represented by a carbon atom, polyacety-

lene, graphene sheet, benzene ring, fullerene and carbon nanotube. These

analytical models can be combined to represent more complicated struc-

tures. Examples of structures that represent a combination of the analytical

models to determine the total potential energy between the interacting struc-

tures [16, 44] are discussed. Describing complicated structures using idealised

building blocks allows us to simplify the model so that calculations can be

done easily with lower computing requirements while maintaining a high level

of accuracy.

The analytical method presented here is to approximate the interactions

between the atoms on the building blocks and the gas molecules, particu-

larly for those structures that have uniformly distributed atoms. Even for

non-uniformly distributed atoms, this technique provides an average approxi-

mation as shown by Thornton et al. [16]. In that paper, MOF-177 is modelled

as a sphere by smearing the atoms of the structure, which are a mixture of

zinc, carbon, oxygen and hydrogen, over the surface of a sphere, and the

40

adsorption isotherms from the analytical model agree well with experimental

isotherms.

We have only described some idealised building blocks that are regu-

lar in shape. However, fortunately for porous materials that are irregularly

shaped, their pores can still be represented using cylindrical, spherical or

slit-shaped porosity, since these are used for all pore size characterisation

methods which includes permporometry, thermoporometry, mercury intru-

sion, positron annihilation lifetime spectroscopy and gas adsorption and (or)

desorption methods [45].

This chapter only explores the use of the 6-12 Lennard-Jones potential

function to describe the interaction energies between these nanostructures.

Further work in this area can be explored using other types of potential

function such as the Morse potential, which is discussed in Subsection 4.2.

41

Chapter 3

Beryllium based metal-organic

frameworks

3.1 Introduction

Metal-organic frameworks (MOFs) comprise metal atoms or clusters linked

periodically by organic molecules to establish an array where each atom forms

part of an internal surface. MOFs have delivered the highest surface areas

and hydrogen storage capacities for any physisorbent and are shown to be a

promising material for gas storage [46]. Exposed metal sites [47, 48], pore

sizes [49] and ligand chemistries [50, 51] have been found to be the most

effective routes for increasing the hydrogen enthalpy of adsorption within

MOFs.

The MOF adsorbent that is amongst the top adsorbent for gravimetric hy-

drogen storage capacity at room temperature is the first structurally charac-

terised beryllium-based framework, Be12(OH)12(1,3,5-benzenetribenzoate)4

42

or Be-BTB (BTB = benzene tribenzoate). Be-BTB has a Brunauer-Emmett-

Teller (BET) [52] surface area of 4400 m2g−1, and can adsorb 2.3 wt% hydro-

gen at 298 K and 100 bar [2]. In the following subsection, the performance

of Be-BTB is compared to other high performing MOFs.

3.2 Metal-organic frameworks performance at

room temperature

In this subsection, we compare the characteristics of high performing MOFs

using data from Suh et al.[8] with a focus on room temperature adsorption.

MOFs with pore volume and excess gravimetric uptake at 298 K and pres-

sures more than 35 bar are used for the comparison. The total gravimetric

uptake, Utot is calculated using the method employed by Frost et al. [53],

Utot = Uex + Vgρ (3.2.1)

where Uex is the excess gravimetric uptake, Vg is the pore volume and ρ is

the density of the ambient gas phase for H2.

Information on the MOFs are listed in Table 3.1, where they are cat-

egorised into two groups: MOFs with open metal sites and MOFs without

open metal sites. The lines of best fit for MOFs with and without open metal

sites are shown in Figure 3.1. MOFs with open metal sites are shown to have

better adsorption capability compared to MOFs without open metal sites at

low pressures (≤ 1 bar) [54]. Figures 3.1(a), (b) and (c) show that this is

also true at higher pressures with the exception of Be-BTB (labelled as ‘m’)

43

which has the highest H2 uptake amongst MOFs without open metal sites.

To explain the high performing nature of Be-BTB, we examine the paper

by Frost et al. [53] which studied the relationship between heat of adsorp-

tion, pore volume and surface area with total uptake at room temperature

and various pressures. They found that these relationships depend on the

strength of the interaction between H2 and the porous material. For weak

adsorption, the uptake correlates with the pore volume at all pressures. For

a stronger adsorption, the uptake depends on the pressure such that the up-

take correlates with the heat of adsorption at low pressure (0–10 bar), surface

area at intermediate pressure (10–35 bar), and pore volume at high pressure

(more than 35 bar). This is verified through an analysis of the MOFs in

Table 3.1 using the software R [55]. In Figure 3.1, the lines of best fit for

MOFs with and without open metal sites are shown. The coefficient of de-

termination of the total H2 uptake with heat of adsorption, BET surface area

and pore volume are reported in Table 3.2.

For MOFs with open metal sites at 298 K and high pressures, the coef-

ficient of determinant (R2) calculated for the heat of adsorption, Brunauer-

Emmett-Teller (BET) surface area and pore volume are 0.23%, 91% and 83%

respectively. This means that with stronger adsorption, up to 91% and 83%

of the variability of total uptake can be described by the BET surface area

and pore volume, respectively. Similarly to the findings of Frost et al. [53],

the heat of adsorption does not play a strong role in high pressures at room

temperature for MOFs with open metal sites.

For MOFs without open metal sites, the R2 for the heat of adsorption,

44

Table 3.1: Data for hydrogen gas sorption for metal-organic frameworks withand without open metal sites at 298 K.

MOF label BET i Vporeii Qst iii Uptakeiv

(m2 g−1) (cm3 g−1) (kJ mol−1) (wt %)

With open metal sitesCr3OF(BDC)3 [56] a 1.900 10.00 1.656Ni(dhtp)2 [57] b 0.410 0.534Cu(peip) [58] c 1560 0.696 6.63 1.025Sm2Zn3(oxdc)6 [59] d 719 0.310 0.628Cr3OF(BTC)2 [56] e 1.000 6.30 0.745Cr3OF(ntc)1.5 [60] f 0.120 0.084Cu2(TCM) [61] g 0.310 6.65 0.437Cu2(bdcppi) [62] h 2300 1.080 7.1 0.934Cu3(btei) [18] i 3000 1.360 6.36 1.656Cu3(ntei) [18] j 4000 1.630 6.22 1.968Cu3(ptei) [18] k 5109 2.130 6.09 2.551Mn3[(Mn4Cl)3(BTT)8]2 [63] l 2100 0.795 10.1 1.521

Without open metal sitesBe12(OH)12(BTB)4 [2] m 4030 1.480 5.5 2.298Co(HBTC)(4,4’-bpy) [64] n 887 0.540 7.0 1.276Co3(NDC)3(dabco) [65] o 1502 0.820 1.005Ni(HTBC)(4,4’-bpy) [64] p 1590 0.810 8.8 1.674Cu(hfipbb)(h2hfipb)0.5 [66] q 0.116 14.7 1.046Zn(MeIM)2 [67] r 1630 0.350 4.5 0.301Zn(NDC)(bpe)0.5 [68] s 0.200 0.406Zn4O(dcbBn)3 [69] t 396 0.130 1.031Zn5O(dcdEt)3 [69] u 502 0.200 1.198Zn5O(TCBPA)2 [70] v 3670 1.520 7.05 1.604Zn6(BTB)4(4,4’-bipy)3 [71] w 4043 0.170 4.62 0.520Cd3(bpdc)3 [72] x 0.190 1.255

i BET surface areaii Pore volumeiii Isosteric heativ Total gravimetric uptake

45

4 6 8 10 12 140

0.5

1

1.5

2

2.5

Qst (kJ/mol)

Tota

l H

ydro

gen U

pta

ke (

wt.%

)

a

c

e

g

h

i

j

k

l

m

n

p

q

r

v

w

with open metal sites

without open metal sites

(a)

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5

c

d

h

i

j

k

l

m

n

o

p

r

tu

v

w

BET surface area (m2/g)

Tota

l H

ydro

gen U

pta

ke (

wt.%

)

with open metal sites

without open metal sites

(b)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

a

e

f

h

i

k

l

m

n

p

b

c

d

g

j

oq

rs

tu

v

w

x

Pore Volume (cm3/g)

Tota

l H

ydro

gen U

pta

ke (

wt.%

)

with open metal sites

without open metal sites

(c)

Figure 3.1: Total H2 uptake at room temperature and high pressures (morethan 35 bar) for MOFs with and without open metal sites. (a) H2 uptakeversus heat of adsorption; (b) H2 uptake versus BET surface area; (c) H2

uptake vs pore volume.

BET surface area and pore volume are 1.43%, 7.22% and 52.63% respectively

which matches the results from Frost et al. [53]. This indicates that for weak

adsorption, the pore volume describes up to 52.63% of the variability of total

uptake. We therefore conclude that the Be-BTB’s superior uptake at room

temperature is partly due to its high pore volume. Investigation into the

46

Table 3.2: Coefficient of determination (R2) for the correlation of total H2

uptake with heat of adsorption, BET surface area and pore volume at highpressure (more than 35 bar).

Material R2 for MOFs with R2 for MOFs withoutcharacteristics open metal sites (%) open metal sites (%)

Heat of adsorption 0.23 1.43BET surface area 90.99 7.22Pore volume 83.43 52.63

nature of Be-BTB’s pore volume is presented in Section 3.5.

Another way of increasing gravimetric hydrogen adsorption in MOFs is to

use lighter metals to reduce the weight of the structure [25]. This approach

was investigated by Ferey et al. [73], which synthesized M(OH)(O2C-C6H4-

CO2)(M = Al3+, Cr3+), known as MIL-53. Results of their investigation

into the capabilities of MIL-53 show that using Al3+ over Cr3+ provides an

additional hydrogen storage amount of 0.7 wt% at 77 K and 1.6 MPa. This

leads us to believe that the light Be2+ metal ions in Be-BTB contributes to

the high performance of Be-BTB.

Hydrogen adsorbents are often compared with reference to their volu-

metric or gravimetric capacity alone. However, to completely assess their

potential to outperform current technologies, an analysis of their cyclability

(i.e. adsorption/desorption or store/release abilities) must also be consid-

ered. The assessment must incorporate the energy required to regenerate

the material whether that be by temperature-swing, pressure-swing, or a

combination of both. As each MOF exhibits its own unique isotherm behav-

ior, the optimal cycle conditions are also unique. Hence, here we adopt a

47

thermodynamic energy optimisation (TEO) model to determine the optimal

cycle conditions for an empty compressed gas tank and a tank filled with

Be-BTB, MOF-5 (or IRMOF-1), and MOF-177.

Our objectives for this chapter are to explain the high performance of

Be-BTB as a room temperature adsorbent, placing into context the prevail-

ing views on the contributions of surface area and bond enthalpy to room

temperature performance. We also examine the pore architecture within

Be-BTB to understand its role in both hydrogen storage and cyclability.

3.3 Modelling methodology

When H2 gas is inside the Be-BTB framework, it is acted upon by inter-

molecular van der Waals forces with every atom comprising the Be-BTB.

This phenomenon may be accurately modeled using the 6-12 Lennard-Jones

potential function [33] introduced in Section 2.1 where the intermolecular po-

tential energy between two non-bonded atoms of interest (framework atom

and H2 molecule) is given by Eq. 2.1.3.

The H2 adsorption within the Be-BTB framework is modelled using an

analytical approach called the Topolotically Integrated Mathematical Ther-

modynamic Adsorption Model (TIMTAM) [32]. This approach offers im-

mediate access to the complete isotherm profiles for any temperature and

pressure and is compatible with the TEO model to determine the feasibility

of a hydrogen fuel cell coupled with a MOF storage device in comparison to

a compressed gas tank.

48

3.3.1 TIMTAM formulation

The TIMTAM approach [32] uses applied mathematical modeling to repre-

sent the structure of H2 and Be-BTB with assumed ideal building blocks

that comprise continuous surfaces and volumes for adsorption. TIMTAM is

a phenomenological model based on approximations that have proved use-

ful in investigating the effect of pore size [16, 32]. Figure 3.2 shows the

Be-BTB structure as an ideal composition of cylindrical rings, which repre-

sent the [Be12(OH)12]12+ rings (or Be-ring), and spheres which represents the

connected BTB3− ligands comprised mostly of carbon and hydrogen atoms.

B B

C A

Figure 3.2: Structure of Be-BTB. The spheres and cylinders represent idealbuilding blocks for adsorption cavities in the structure.

At first sight such a simplified modeling approach may seem complicated

geometrically, but in similar situations it has been shown to provide the major

contribution to the interaction energy of the actual structure, confirmed by

either independent experimental results or computational modeling [31, 10,

32]. Calculations that involves the TIMTAM approach are evaluated using

the algebraic computer package Maple [42] and the figures are produced using

MATLAB [43].

49

In principle, the total potential energy within the system, Utot(ρ), is ob-

tained by a summation of the potential energy for the interaction between

H2 (as a point) and a sphere, with the potential energy for the interaction

between H2 and an infinite cylindrical ring. The derivation of these equations

are shown in Eq. 2.1.29 and Eq. 2.1.34 of Section 2.2.

To model the potential energy for an infinite cylinder interacting with H2

molecules, we use mathematical models based on works by Cox et al. [31]

and Thornton et al. [32] to obtain

U1(ρ) =3∑i=1

ηi(−AiR3 +BiR6), (3.3.1)

Rj =8π2r21

(2r1)2j(2j − 2)!

∞∑s=1

[ρs(2j + 2s− 2)!

(4r1)ss!(j + s− 1)!

]2,

where ρ is the distance between the H2 molecule and the center of the cavity,

ηi denotes the mean atomic surface densities of atom i on the surface of the

cylinder with radius r1.

In order to make the mathematics tractable we assume that the interac-

tion is accurately approximated by the potential energy of an infinite cylinder.

However, when determining the gravimetric uptake, the volume of the cylin-

der is calculated by multiplying the length of the beryllium-ring (Be-ring), h,

by the area of the base, 2πr1. The radius of the Be-ring is measured from the

atomistic structure using Materials Studio [74] and the radius of the sphere

is adjusted to match the total free volume of the atomistic cell. The values

of the parameters are given in Table 3.4.

The potential energy within a sphere of radius r2, based on the model by

50

Cox et al. [37] and Thornton et al. [32], is given by the expression

U2(ρ) =4∑j=2

ηj(−AjQ6 +BjQ12), (3.3.2)

Qk =2r2π

ρ(2− k)

[1

(ρ+ r2)k−2− 1

(ρ− r2)k−2

].

The parameter values for the framework atoms are taken from the uni-

versal force fields (UFF) [75] and the H2 guest molecule is treated as a single-

point interaction according to van den Berg et al. [76]. Table 3.3 provides

the parameter values used for the potential energy calculations.

Table 3.3: Lennard-Jones parameter values.

σ ε Mass(A) (K) (g mol−1)

H2 2.96 36.73 2.02Be 2.45 42.77 9.01C 3.43 52.84 12.01H 2.57 22.14 1.01O 3.12 30.19 16.00

To investigate the free volume within Be-BTB, we refer to the methods

introduced in Section 2.2. The total free volume Vf is calculated by summing

the volume available for the adsorbed phase Vad and the volume available for

bulk gas Vbulk for both the Be-ring and the sphere. Derivations of Vad and

Vbulk are explained by Eq. 2.2.1 and Eq. 2.2.2. The formulae for Vad and Vbulk

51

for the Be-ring are given by

Vad =

∫ ρ1

0

2πρh

(1− exp

[U

RT

])dρ,

Vbulk =

∫ ρ1

0

2πρh

(exp

[U

RT

])dρ, (3.3.3)

and the formulae for the sphere are given by

Vad =

∫ ρ2

0

4πρ2(

1− exp

[U

RT

])dρ,

Vbulk =

∫ ρ2

0

4πρ2(

exp

[U

RT

])dρ, (3.3.4)

where h is the height of the cylinder, ρ1 is the radial boundary of the free

volume and ρ2 is the value of the radius where potential is zero for the cylinder

and sphere [16].

Table 3.4: Numerical values of constants used.

Description Parameters Values

Radius of cylinder r1 (A) 6.1Height of cylinder h (A) 4.967Radius of sphere r2 (A) 12.85H2 critical temperature Tc (K) 33.16H2 critical pressure Pc (atm) 12.8H2 molecular mass m (g mol−1) 2.02Total unit cell mass M (g mol−1) 8215.68

The calculations for the gravimetric uptake by Be-BTB can be performed

by first calculating the total number of H2 molecules in the cavity using the

52

appropriate equations of state (Eq. 2.2.3 and Eq. 2.2.4). The total gravi-

metric uptake can then be calculated by substituting the number of H2 into

Eq. 2.2.6. In this equation, m is the mass of a hydrogen molecule and M is

the mass of a unit cell of the Be-BTB cavity. The parameter values used to

calculate the number of molecules and total uptake are given in Table 3.4.

3.3.2 Thermodynamic energy optimisation

The optimum storage and delivery condition for a hydrogen fuel cell coupled

with a MOF storage device can be determined using the thermodynamic

energy optimisation (TEO) model. We do this by referring to the work by

Lin et al. [77] which combines temperature and pressure-swing processes to

optimise regeneration conditions for materials based carbon dioxide capture

and storage in coal-fired power stations. The energy required to capture and

store carbon dioxide can be separated into three main components: energy

required to (i) heat the material, (ii) supply the heat of desorption (equivalent

to the heat of adsorption), and (iii) pressurise the gas for transport and

storage.

Here we create a similar model to calculate the optimal storage and deliv-

ery conditions for a MOF device delivering H2 to a fuel cell considering only

the energy produced by the pressure and/or temperature-swing adsorption

cycle, ignoring any other factors such as insulation, tank mass and heating.

The net energy produced by a fuel cell powered by a MOF device can be

thought as the energy produced from the gas minus the energy required to

store and deliver the gas to the fuel cell. This is described using the following

53

equation,

Enet =EfcWc − V |Pads − Pdes|

− Cp|Tads − Tdes| −QWc, (3.3.5)

where the subscript ads denotes storage condition (adsorbed) and des denotes

delivery condition (desorbed).

The first term in Eq. 3.3.5, EfcWc calculates the total energy produced

by the fuel cell where Efc is the amount of energy per mol of hydrogen which

can be provided as electrical energy (calculated to be 237.1 kJ mol−1 using

the Gibbs free energy function [78]). The parameter Wc denotes the working

capacity of the material and is given by

Wc = nads − ndes. (3.3.6)

Wc describes the difference in amount of H2 adsorbed at pressure Pads and

Pdes and temperature Tads and Tdes.

The second term, V |Pads − Pdes| determines the energy required to in-

crease the pressure and the third term, Cp|Tads−Tdes| determines the energy

required to heat the material from storage condition to desorption condition.

V and Cp denote the volume of a unit cell of adsorbant and specific heat

capacity respectively. The last term, QWc represents the energy required to

remove a mol of gas where Q denotes the heat of adsorption of the adsorbant,

respectively.

The parameters used to calculate the optimum storage and delivery con-

ditions for a MOF-5, MOF-177 and Be-BTB storage device are given in

54

Table 3.5. For the case of a compressed H2 tank, the volume of the tank is

2.2 L [79]. We used the specific heat capacity for a H3BTB ligand to repre-

sent the heat capacity for Be-BTB. The specific heat capacity are obtained

from the paper by Mu and Walton [80].

Table 3.5: Values of constants used to calculate optimum storage (ads) anddelivery (des) conditions within constraints of U.S. DOE delivery conditions(5 – 12 bar, 233 – 358 K).

Parameter MOF-5 Be-BTB MOF-177

V (×10−26 m3) 1.74 3.22 3.57Cp (kJ (mol K)−1) 0.2 0.6 0.55Q (kJ mol−1) -3.8[81] -5 [2] -4.4[51]

3.4 Results and discussion

To ensure that the mathematical modeling approaches describe the properties

of the Be-BTB structure accurately, the gravimetric uptake at 77 K and

298 K are compared to the atomistic simulation reported by Lim et al [1]

which was done using the Grand-Canonical Monte Carlo (GCMC) algorithm.

Our results are also compared to the experimental data reported by Sumida

et al. [2] who first synthesised Be-BTB. The experimental data has been

converted from excess to total uptake using the method outlined by Frost

et al. [53]. Total uptake is of interest here for the DOE requirements and

comparison with compressed tank [67].

55

The GCMC simulation and TIMTAM continuum modeling results are

demonstrated to be in good agreement with the experimental results as shown

in Figure 3.3. For the 77 K data, Be-BTB remarkably stores up to four times

more H2 than the compressed gas tank (solid lines). At room temperature,

the GCMC results slightly underestimate the experimental uptake, possibly

due to interaction energy cut-offs chosen for efficient computation resulting

in close to compressed gas phase.

0 20 40 60 80 1000

2

4

6

8

10

Gra

vim

etr

ic U

pta

ke (

wt%

)

0 20 40 60 80 1000

10

20

30

40

Volu

metr

ic U

pta

ke (

g/L

)

Pressure (Atm)

77K

298K

Figure 3.3: Total H2 uptake at 77 K and 298 K for TIMTAM, GCMC [1] andexperimental data [2]. Solid lines show the density of compressed H2 gas in atank at 77 K and 298 K. Circles represent experimental data, squares repre-sent GCMC model results and dotted lines represent TIMTAM predictionsfor Be-BTB.

56

3.5 Pore size analysis

In this section, we investigate the pore size distribution (PSD) of Be-BTB.

The pore size distribution is a critical property to examine because the

strength of adsorption of gases onto the structure varies with the size of

the pores [82]. PSD analysis by Lim et al. [1] using the Geometric Accessi-

ble approach and the Cavity Energetic Sizing Algorithm indicates that the

dominant pore size of Be-BTB ranges from ∼ 8–13 A which includes our

estimated Be-ring cavity size. As a H2 molecule has a diameter of 2.958 A,

a pore size larger than 9 A leads to the ideal condition to allow for confined

adsorption [15].

-15 -10 -5 0 5 10 15-7

-6

-5

-4

-3

-2

-1

0

1

Distance from cavity center (Å)

Pote

ntia

l energ

y for

adsorp

tio

n (

kJ/m

ol)

Figure 3.4: Potential energy for the ideal Be-BTB building blocks consistingof spherical cavities (red dashed line) and cylindrical Be-ring cavities (bluesolid line) that were constructed for the TIMTAM approximation. Shadedarea represents heat of adsorption measured experimentally [2].

57

The potential energy distribution for the interaction between H2 with the

ring and spherical building blocks are presented in Figure 3.4. The H2 inter-

action with the Be-ring has a deeper potential energy minimum compared

to interaction with the spherical cavity. In other words, the gas experiences

a stronger interaction with the Be-ring compared to the spherical cavity of

the Be-BTB. Thus as H2 is adsorbed into Be-BTB, it will first prefer to be

adsorbed at the minimum potential energy of the Be-ring at a distance of

2.6 A from the center of the Be-ring cavity, after which it will start to adsorb

onto the surface of the spherical cavity at a distance of 10 A from the center

of the spherical cavity.

Further analysis shows that the heat of adsorption for the Be-ring is

predicted to be 4.8 kJ mol−1, which is consistent with the measured heat of

adsorption reported as 4.5–5.5 kJ mol−1 [2].

3.5.1 Fractional free volume

To understand the effect of the pore size on the adsorption capacity of the

Be-BTB, a fractional free volume analysis is done for the Be-ring cavity. The

fractional free volume for adsorption describes the proportion of volume of

H2 gas in the cavity that is in the adsorbed state as compared to bulk gaseous

state (Vad/V ).

Figure 3.5 shows the fractional free volume for adsorption at 77 K and

298 K within the Be-ring. At 298 K the size of the Be-ring cylindrical cavity

and its surrounding configuration (9 A) is very close to the optimal size. Up

to 23% of the free volume within the Be-ring cavity is able to store H2 gas in

its adsorbed state. At 77 K, gas adsorption is at an optimum level when the

58

effective pore size is 15 A, which is due to the decreased importance of kinetic

energy over inter-atomic potential energy. A larger cavity size allows more

gas molecules to be adsorbed at various and larger distances and therefore

encourages multiple adsorption layers.

4 6 8 10 12 14 16 18 2010

20

30

40

50

Cavity Size (Å)

Fra

ctional F

ree V

olu

me for

Adsorp

tion (

%)

77K

298K

Figure 3.5: Fractional free volume for adsorption (Vad/V ) at 77 K and 298 Kwithin Be-ring cylindrical cavity building block. The yellow cylinder illus-trates the variation in pore size around the Be-ring.

The study by Wang et al. [11] which investigated the adsorption capa-

bilities of carbon slit pores and carbon nanotubes indicated that the density

of hydrogen is not liquid-like at room temperature. Thus the amount of

adsorbed H2 is usually smaller than the amount of bulk H2 gas at room tem-

perature. A similar relationship can be observed in Figure 3.5 which shows

that the fractional free volume for adsorption at room temperature is lower

59

than at 77 K inside the cylindrical Be-ring cavity of Be-BTB. The effects

of van der Waals interaction on H2 are therefore less pronounced at room

temperature than at 77 K.

3.5.2 Optimal storage and delivery conditions

In this section, the optimal cycle condition for a fuel cell coupled with a

Be-BTB storage device is compared a MOF-5 device, a MOF-177 device

and a compressed H2 tank fuel cell, considering only the pressure and/or

temperature adsorption process and ignoring other factors that contribute to

the production of energy. The energy produced by these fuel cells is analysed

using the TEO function. This function optimises the operating pressure and

temperature within the range of delivery pressure of 5 – 12 bar, and delivery

and operating temperatures of 233 – 358 K and 233 – 333 K, respectively,

as set by the 2017 U.S. DOE target [5]. The adsorption pressure is set

to 100 bar for the pressure-swing (Tables 3.6 and 3.8) and 12 bar for the

temperature-swing (Table 3.7) process so that the systems can be compared

effectively.

A study on densified MOF-177 pellets by Dailly and Poirier [83] revealed

a decrease in available pore volume of around 24% compared with the known

crystal pore volume. To account for this in our calculations, the hydrogen

uptake is decreased by 24%. We would like to note here that total gravi-

metric uptake is considered instead of excess gravimetric uptake to allow

for comparison with the 2017 U.S. DOE targets which use total gravimetric

uptake.

The number of H2 molecules at varying pressures and temperatures for

60

Be-BTB and MOF-5 are calculated using the TIMTAM model described in

the previous subsection. The TIMTAM parameters for MOF-5 are obtained

from Thornton et al. [16]. For MOF-177, the number of H2 molecules at

varying pressures and temperatures have been calculated from the modi-

fied Dubinin-Astakhov model [84] using the fitted parameters from Poirier

and Dailly [85]. The number of H2 molecules in a compressed gas tank is

calculated using the ideal gas law matching the standard for engineering as-

sessments [79].

Pressure-swing adsorption cycle

A pressure-swing only adsorption cycle is considered with the operating tem-

perature fixed to the DOE fuel cell standards. Results of this analysis are

reported in Table 3.6, showing that the H2 tank is the most efficient at pro-

ducing energy compared to the MOF devices, particularly for energy per mass

at 34.93 kWh/kg. Amongst the MOF-based fuel cells, Be-BTB produces the

most energy per mass and volume, followed by MOF-177 and MOF-5. Our

calculations show that 233 K is the optimum temperature for the pressure-

swing for all three MOF devices, with adsorption pressure of 100 bar and

desorption pressure of 5 bar.

Figure 3.6 provides a visual comparison of their optimal working capaci-

ties for a pressure-swing only process utilizing Be-BTB, MOF-5 or MOF-177.

A Be-BTB device clearly outperforms the other MOF-based fuel cells because

of the higher gravimetric and volumetric uptake at all pressures. In addition,

a Be-BTB fuel cell provides the largest working capacity (as shown by the

61

red arrow to the right of the figure), followed by a MOF-177 (purple arrow)

and MOF-5 (blue arrow) fuel cell.

Table 3.6: Maximum energy generation at optimised storage (ads) and deliv-ery (des) conditions restricted to DOE operating range for a pressure-swingonly cycle.

Parameter Tank a MOF- Be- MOF-5 BTB 177

Pads (bar) 100 100 100 100Pdes (bar) 5 5 5 5Tads (K) 233 233 233 233Tdes (K) 233 233 233 233Etot (kWh/L) 0.35 0.07 0.32 0.24Etot (kWh/kg) 34.93 0.12 0.75 0.57

a Compressed H2 gas

Temperature-swing adsorption cycle

Analysis for a temperature-swing only adsorption process for the compressed

gas tank and MOF-based fuel cells show that for a temperature-swing fuel

cell, the optimum operating pressure and adsorption temperature is 12 bar

and 233 K. The optimum desorption temperature is 302 K for a H2 tank and

358 K for MOF-5, Be-BTB and MOF-177.

The analysis from the TEO model presented in Table 3.7 shows that

the Be-BTB fuel cell produces the most energy per volume followed by the

H2 tank and MOF-177 (both at 0.01 kWh/L) and lastly by a MOF-5 fuel

62

Desorption

Pressure

Wo

rkin

g c

ap

acity

Adsorption

Pressure

Gra

vim

etr

ic U

pta

ke

(w

t.%

) a

Pressure (bar)

Figure 3.6: The optimum gravimetric H2 uptake at 233 K for a pressure-swingadsorption process.

cell. The H2 tank produces the most energy per mass at 35.21 kWh/kg.

Amongst the MOF-based fuel cells, Be-BTB produces the most energy per

mass, followed by MOF-177 and MOF-5.

Figure 3.7 provides a visual comparison between the three MOFs for the

temperature-swing only cycle. This figure shows that Be-BTB has the highest

gravimetric and volumetric uptake at all temperatures between 233 - 358 K,

followed by MOF-177 and MOF-5. In addition, Be-BTB (denoted by the red

arrow to the right of the figure) has the largest range of working capacity,

followed by MOF-177 (purple arrow) and MOF-5 (blue arrow). Comparisons

between Figure 3.6 and Figure 3.7 shows that the pressure-swing adsorption

63

Table 3.7: Maximum energy generation at optimised storage (ads) and deliv-ery (des) conditions restricted to the DOE operating range for a temperature-swing only cycle.

Parameter Tank a MOF- Be- MOF-5 BTB 177

Pads (bar) 12 12 12 12Pdes (bar) 12 12 12 12Tads (K) 233 233 233 233Tdes (K) 302 358 358 358Etot (kWh/L) 0.01 0.004 0.04 0.01Etot (kWh/kg) 35.21 0.007 0.10 0.02

a Compressed H2 gas

b

Desorption

Pressure

Work

ing c

ap

acity

Gra

vim

etr

ic U

pta

ke (

wt.%

)

Temperature (K)

Adsorption

Pressure

Figure 3.7: The optimum gravimetric H2 uptake at 12 bar for a temperature-swing adsorption process.

64

process produces more energy than the temperature-swing adsorption process

due to the larger working capacity of the former.

Combined pressure and temperature-swing cycle

The optimum desorption pressure and adsorption temperature are calcu-

lated to be 5 bar and 233 K for all four systems when analyzing a com-

bined pressure-swing and temperature-swing adsorption process. Our analy-

sis presented in Table 3.8 shows that the H2 tank produces the most energy

compared to the MOF-based fuel cells, particularly for energy per mass at

34.9 kWh/kg. Amongst the MOF-based fuel cells, Be-BTB produces the

most energy for both mass and volume, followed by MOF-177 and MOF-5.

Figure 3.8(a), (b) and (c) displays the contour plots of the gravimetric

uptake of Be-BTB, MOF-5 and MOF-177 with respect to temperature and

pressure at the standard operating conditions. The contour plots show that

the gas uptake for a Be-BTB fuel cell is higher than that of MOF-5 and

MOF-177 at any combination of temperature and pressure.

Overall, when constrained to the U.S DOE operating pressure and tem-

perature, the H2 tank produces substantially more energy per mass for all

three types of adsorption processes compared to the MOF-based fuel cells.

The H2 tank also produces more energy per volume except for the temperature-

swing only cycle where the Be-BTB fuel cell has a higher energy output.

Amongst the MOF-based fuel cells, the Be-BTB fuel cell generates the most

energy per volume and mass, followed by MOF-177 and MOF-5.

The 2017 U.S. DOE target for energy produced by onboard H2 storage

65

a b

c

Figure 3.8: H2 gravimetric uptake with respect to temperature and pressure.(a) H2 uptake for Be-BTB. (b) H2 uptake for MOF-5. (c) H2 uptake forMOF-177. The yellow diamond and green circle denote the TEO optimiseddesorption and adsorption conditions, respectively, that maximises the netenergy. The bar on the right describes the value gravimetric uptake in wt%.

systems is 1.3 kWh/L for volumetric capacity and 1.8 kWh/kg for gravimetric

capacity. Our results show that only a compressed gas tank satisfies the U.S.

DOE requirement for the gravimetric capacity. The H2 tank does not satisfy

the requirements for the volumetric capacity, and the MOF-based fuel cells

also do not reach the requirements for either the gravimetric or volumetric

capacities using any of the adsorption processes. Overall, there is still much

improvement in MOF performance required to meet the DOE targets.

66

Table 3.8: Maximum energy generation at optimised storage (ads) and de-livery (des) conditions restricted to the DOE operating range for a combinedpressure-swing and temperature-swing cycle.

Parameter Tank a MOF- Be- MOF-5 BTB 177

Pads (bar) 100 100 100 100Pdes (bar) 5 5 5 5Tads (K) 233 233 233 233Tdes (K) 302 358 358 358Etot (kWh/L) 0.35 0.07 0.34 0.25Etot (kWh/kg) 34.93 0.12 0.80 0.58

a Compressed H2 gas

3.6 Conclusion

In this chapter, we explore hydrogen adsorption by Be-BTB using the Topo-

logically Integrated Mathematical Thermodynamic Adsorption Model (TIM-

TAM). The surfaces and volumes of adsorption for Be-BTB are represented

using ideal building blocks. The model confirms that the Be-BTB structure,

represented by ideal building blocks of cylinders and spheres, replicates the

H2 gravimetric uptake from experimental and simulation results from the

literature.

The continuum model allows exploration of the available parameter land-

scape for optimising H2 uptake. Our calculations using the TIMTAM model

show that a H2 molecule experiences a stronger interaction with the Be-ring

cavity as compared to the spherical cavity of the Be-BTB. At 298 K, the size

of the Be-ring cavity is at the optimum size to encourage efficient adsorption

of gas.

67

Comparisons of Be-BTB with MOFs with and without open metal sites at

room temperature confirm the findings of Frost et al. [20] that the uptake of

H2 is correlated with pore volume for weak adsorption (MOFs without metal

sites), and surface area and pore volume for stronger adsorption (MOFs with

metal sites).

We also described a new approach, the thermodynamic energy optimiza-

tion (TEO) model to calculate the energy output from MOF-based fuel cells.

This model is based on the Gibbs free energy function [78] and the work-

ing capacity of the MOF [77] using three different adsorption processes: (i)

pressure-swing only, (ii) temperature-swing only and the (iii) combined pres-

sure and temperature-swing process. The energy output from a tank filled

with Be-BTB, MOF-5 and MOF-177 is benchmarked against an empty com-

pressed gas tank.

Our results show that an adsorption process that relies on the three

aforementioned processes is insufficient for the compressed tank and MOF-

based fuel cells to achieve the 2017 U.S. DOE volumetric capacity target of

1.3 kWh/L. For the gravimetric capacity, only the compressed tank fulfil the

U.S. DOE gravimetric capacity of 1.8 kWh/kg. Amongst the MOF-based

fuel cells, the Be-BTB fuel cell provides the most energy per volume and per

mass, followed by MOF-177 and MOF-5. We attribute this to the superior

pore effect which the Be-BTB has over MOF-5 and MOF-177. Our analysis

using the TEO model suggests that there is still much improvement in MOF

performance required to meet the DOE targets.

An investigation using the TEO model provides us with key insights

into how the performance of MOF-based fuel cells can be improved. From

68

Eq. 3.3.5, we can identify parameters that can be adjusted to optimise the

energy produced by a MOF-based fuel cell. For the materials-based fuel

cell to produce more energy, it needs to have a larger working capacity so

that the hydrogen adsorption is not significantly affected by the pressure and

temperature at adsorption and desorption conditions.

Our model also indicates that a material with smaller specific heat ca-

pacity and heat of adsorption will improve performance. A smaller heat

of adsorption will improve gas adsorption by strengthening the interaction

energy between the gas and material.

A quick observation of the model may suggest that decreasing the unit cell

volume of the material would improve the net energy, however its effect on

the net energy produced is complicated by its contribution in the derivation

of the number of gas molecule in the cavity. We recommend instead to

determine the appropriate cavity size to optimise the fractional free volume

as demonstrated in Figure 3.5. Finally, we propose that further analyses

should be done to determine the parameters that would maximise the net

energy produced by a fuel cell powered by a MOF device.

69

Chapter 4

Porous aromatic frameworks

4.1 Introduction

Porous aromatic frameworks (PAFs) are presented in this chapter as an alter-

native method to MOFs for storing gas. The first synthesised PAF, PAF-302

[24] demonstrated physicochemical stability and a large BET surface area of

5640 m2 g−1. It also possesses exceptional uptake abilities with an excess

hydrogen uptake of 7 wt% at 77 K and 48 bar. The general structure of

PAFs is based on that of diamonds, where the carbon-carbon (C-C) bonds

are replaced with phenyl rings. The structures of PAF-301, PAF-302 and

PAF-303 are presented in Figure 4.1. PAF-301 has one phenyl ring con-

necting the carbon atoms, while PAF-302 and PAF-303 have two and three

phenyl rings as the organic linkers.

Recent investigations on PAFs involve methods to improve its interaction

with gases to increase overall gas storage capacity. Due to its low heat of

70

a b

c

Figure 4.1: PAF structures and their associated organic linkers for (a) PAF-301, (b) PAF-302 and (c) PAF-303.

adsorption, for example 4.6 kJ mol−1 for hydrogen in PAF-302, storage at

ambient conditions (1 bar at 298 K) is limited [9]. In 2011, Babarao et al.

[86] designed functionalised PAFs that improved CO2 adsorption at ambient

conditions. The paper uses computer simulation to show that PAFs function-

alised with tetrohydrofuran-like ether groups have the highest improvement

in CO2 adsorption and selectivity over CH4, H2 and N2.

Other methods that are known to increase gas adsorption are fullerene

impregnation and metallation. Thornton et al. [16] demonstrated through

71

mathematical modelling that MOF-177 impregnated with magnesium-decorated

fullerenes would achieve a higher hydrogen adsorption compared to a an

empty MOF-177 at high pressures.

In this chapter, we investigate the performance of Li-doped PAFs (de-

noted as Li-PAFs) and PAFs impregnated with fullerenes (denoted as C60@PAFs)

to determine if these modifications will improve the hydrogen uptake.

4.2 Methodology

4.2.1 Porous aromatic frameworks

a b

Figure 4.2: Geometric representation of PAF.

To understand and predict the performance of PAF, a continuum ap-

proach is adopted where the internal surface of the PAF is represented as

a packing of spheres (shown in Figure 4.2) and the H2 as a single point.

72

The advantage of this approach is that it allows us to rapidly explore the

large parameter landscape consequently revealing important characteristics

to enhance gas uptake.

In this chapter, we explore a different method of modelling the inter-

actions between H2 and PAFs using the Morse potential energy instead of

the Lennard-Jones potential. The quantum mechanics calculations done by

Mendoza et al. [87] using the Morse potential reported better representations

of the potential energies for the interaction between H2 and covalent-organic

frameworks (COFs), metallated COFs, MOFs and metallated MOFs.

The equation for the Morse potential is given by

U(ρ) = D[a2 exp(−2bρ)− 2a exp(−bρ)

](4.2.1)

a = exp(γ

2

), b =

γ

2re,

where D is the well depth, γ is the stiffness (or the force constant) and re is

the equilibrium distance between the H2 and atoms on the structure.

The potential energy between H2 and PAF is calculated by integrating

the Morse force field over the continuous spherical surface as demonstrated

in Section 2.1.5. The atoms on the surface of the spherical cavity is assumed

to be uniformly distributed. We derive the total potential energy as a sum

of the individual potential energies of the atoms on the PAF framework and

the H2 molecule to be

Utot(rd) =∑i

ηi

∫U(rd) dS, (4.2.2)

where dS is the surface element of the PAF structure. The mean atomic

73

density of atom i is denoted by ηi and is calculated by dividing the number

of atoms by the surface area. The interaction distance rd is given by

rd = r2 − 2rρ cos θ + ρ2, (4.2.3)

where ρ is the distance between the centre of the H2 molecule and the centre

of the spherical cavity.

Substituting Eq. 4.2.1 into Eq. 4.2.2 provides the total interaction energy

between H2 and a PAF,

Utot(ρ) =3∑i=1

πrDiηiai2b2i ρ

(Q2 −Q1), (4.2.4)

Qm = 8 exp(−bicm)(bicm + 1)− ai exp(−2bicm)(2bicm + 1),

ai = exp(γi

2

), bi =

γi2rei

, c1 = |r − ρ|, c2 = r + ρ,

where the subscript i = 1, 2 and 3 denotes H2 interacting with the resonant

coordinated carbon, tetrahedral coordinated carbon and the hydrogen atoms

located on the PAF structure respectively.

4.2.2 Lithiated porous aromatic frameworks

A higher capacity of H2 uptake can be achieved by doping lithium (Li) atoms

above the aromatic rings in the PAF structure. The model to calculate the

potential energy experienced by H2 when interacting with lithiated PAFs (Li-

PAF) is similar to Eq. 4.2.4 with some minor changes. The summation in

Eq. 4.2.4 is modified to include the interaction between H2 and lithium atoms

(denoted by i = 4). The lithium atom is assumed to be located approximately

74

1.4 A above the phenyl ring and thus, the radius r is adjusted to r − 1.4.

Thus the total interaction energy for a H2 interacting with Li-doped PAFs is

given by

Utot(ρ) =4∑i=1

π(r − 1.4)Diηiai2b2i ρ

(Q2 −Q1), (4.2.5)

Qm = 8 exp(−bicm)(bicm + 1)− ai exp(−2bicm)(2bicm + 1),

ai = exp(γi

2

), bi =

γi2rei

, (4.2.6)

c1 = |(r − 2.4)− ρ|, c2 = (r − 2.4) + ρ.

4.2.3 Impregnated porous aromatic frameworks

In this section, we describe the interaction of H2 and a fullerene impreg-

nated PAF, denoted by C60@PAF. This impregnation is expected to provide

a stronger well depth from the overlapping potential energies created by the

interaction between H2 and C60, and H2 and PAF. The fullerene is repre-

sented by a spherical building block and is located in the centre of the PAF

cavity. The modelling for fullerene impregnation is only done for PAF-303

as PAF-302 cavities are too small to accommodate a C60 [3].

The potential energy between H2 and the fullerene is calculated using the

following equation,

U(ρ) =πrD5η5a5

2b25ρ(Q2 −Q1), (4.2.7)

Qm = 8 exp(−b5cm)(b5cm + 1)− a5 exp(−2b5cm)(2b5cm + 1),

a5 = exp(γ5

2

), b5 =

γ52re5

, c1 = |r − ρ|, c2 = r + ρ,

75

where the subscript i = 5 represents a fullerene. The total potential energy

between H2 and a C60@PAF is calculated by summing Eq. 4.2.4 and Eq. 4.2.7.

In addition to the interaction between H2 and C60@PAF-303, we will also

investigate the interaction between H2 with a Li-doped PAF-303 impregnated

with C60, denoted by C60@Li-PAF-303, which can be modelled by summing

Eq. 4.2.5 and Eq. 4.2.7. All calculations are done using the software program

Maple [42] with figures produced using MATLAB [43].

4.2.4 Parameter values

The radius for the spherical building blocks representing the family of porous

aromatic frameworks can be systemically derived assuming that their ligand

lengths are equivalent to their radii. This is done using the following equation

r = 5.82 + 4.34(nb − 1), (4.2.8)

where nb is the number of benzene rings in a ligand, which is 32 for PAF-302

and 48 for PAF-303. The equation is derived using a linear function based

on the length of a PAF-301 ligand, which is measured to be 5.82 A using

Materials Studio [74].

Materials Studio is also used to measure the free volume of a PAF cavity,

where the volume of the sphere representing the cavity is made to equal

the PAF free volume. To do this, we multiply the volume of the spherical

building block by ns. Mathematically, ns is represented by the formula

ns =3Vf4πρ30

, (4.2.9)

76

where Vf is the free volume for a unit cell of PAF and ρ0 is the distance from

the centre of the cavity when the value of the potential energy is zero. Using

the formula above, the value of ns for PAF-302 is 5.19 and 4.50 for PAF-303.

The well depth, stiffness parameter and equilibrium distance between H2

and atoms on the PAF structures are obtained from [29] and [3] and are

presented in Table 4.1.

Table 4.1: Van der Waals force field parameters between H2 and PAF.

Atom types Subscript D(kcal/mol) re(A) γ Ref.

H Aa· · ·C Rb 1 0.0892 3.240 11.600 [3]H A· · ·C 3c 2 0.0620 3.240 11.006 [3]H A· · ·H d 3 0.0124 3.201 12.003 [3]H A· · ·Li 4 1.5970 1.994 7.94 [29]H A· · ·C C60

e 5 0.1008 3.120 12.006 [29]

a H A denotes H in a H2 moleculeb C R denotes the resonant coordinated C in PAFc C 3 denotes the tetrahedral coordinated C in PAFd H denotes H in PAFe C C60 denotes C in C60

The number of Li atoms in the PAF structure is assumed to be limited

by the number of aromatic rings available for the attachment of Li atoms.

Hence, the maximum number of lithium atoms allowed for lithiation in PAF-

302 is 32 and for PAF-303 is 48. The proportion of Li atoms in a PAF unit

cell is calculated using the equation

φ =nLmL × 100

nLmL +M, (4.2.10)

77

where nL is the number of Li in the unit cell, mL is the mass of a Li atom,

and M is the mass of the PAF. Table 4.2 provides the parameter values used

in our calculations. Note that if φ is x%, the lithiated PAF is denoted as

x%Li-PAF.

To predict hydrogen uptakes, our model adopts a dual equation of state

approach where the total free volume Vf contains a ratio of the bulk gas

phase (Vbulk) and the adsorbed phase (Vad), discussed in Section 2.2 (refer to

Eq. 2.2.1 and Eq. 2.2.2). Furthermore, the total number of H2 in the cavity

is calculated using the appropriate equations of states using Eq. 2.2.3 and

Eq. 2.2.4.

An approximation of the gravimetric uptake G in the material is provided

by Eq. 2.2.5. The total mass M in the equation is replaced with M1 to repre-

sent the total mass of PAF-302 and M2 for PAF-303. Finally, the volumetric

uptake in a material is calculated as

v =nm

V, (4.2.11)

where n and m are the number and mass of the gas molecules, and V is

the total volume of the cavity which can be calculated using the density of

PAF-302 and PAF-303.

4.3 Results and Discussion

In this section, we present the results of our analysis of H2 interacting with

a variety of bare and modified PAF-302 and PAF-303. We denote a fullerene

78

Table 4.2: Constants used for the potential energy calculation.

Mass (g/mol)H mH 1.01C mC 12.01H2 mH2 2.02Li mL 6.94C60 mF 720.60PAF-302 M1 2531.28PAF-303 M2 3747.87

Mean atomic density (A−2)Ca η1 0.1542Ha η2 0.0987Lia η3 0.00631×nLcCb η4 0.3789

Radius (A)PAF-302 10.16PAF-303 14.50C60 3.55

Density (g/cm3)PAF-302 [3] 0.3150PAF-303 [3] 0.1611

Critical H2 temperature (K) Tc 33.16Critical H2 pressure (atm) Pc 12.80

a Atoms on the PAF surfaceb Atoms on the fullerene surfacec nL denotes the number of Li atoms on the PAF surface.

79

impregnated PAF as C60@PAF, lithiated PAF as Li-PAF and fullerene im-

pregnated lithiated PAF as C60@Li-PAF. The level of lithiation of PAF is

denoted by x%Li-PAF, where x is the level of lithiation calculated using

Eq. 4.2.10.

The ability for our model to predict H2 uptake in these theoretical PAFs

is demonstrated by verifying the PAF-302 and PAF-303 H2 uptake with ex-

perimental data and simulation results from Lan et al. [3]. To simulate the

H2 uptake in PAFs, Lan et al. used the multiscale simulation method, which

is a combination of first-principles calculations and Grand Canonical Monte

Carlo (GCMC) simulation. Previous experimental results for PAF-302 done

by the group was successfully replicated using the simulation method. In

addition, our results are also compared to the GCMC simulation results for

PAF-302 at 77 K by Konstas et al. [4], who investigated the technique of

lithiation of PAFs and its effect on H2 uptake.

In Figure 4.3(a), our model is validated against the simulation results of

Lan et al. (denoted by crosses) at both 77 K and 298 K. The simulated and

experimental uptakes for PAF-302 are approximately twice the 2015 DOE

target of 5.5 wt% at 77 K. Figure 4.3(b) displays PAF-303 isotherm at 77 K

and 298 K generated by our model (solid lines) which matches well against

simulation results by Lan et al. but with a slight overestimation at 77 K.

These comparisons show that our model is able to provide an estimate of H2

uptakes for both PAF-302 and PAF-303 that is comparable with simulation

and experimental results.

80

0 20 40 60 80 1000

2

4

6

8

10

12

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

2017 DOE target

77K

298 K

(a)

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

2017 DOE target

77K

298 K

(b)

Figure 4.3: Gravimetric uptake for (a) PAF-302 and (b) PAF-303 with re-spect to pressure at 77 K and 298 K. The plots shows the comparison betweenour results (solid lines) with simulation (crosses) and experimental results(circles) from Lan et al. [3] and simulation (dotted line) from Konstas et al.[4].

4.3.1 Gravimetric and volumetric uptake

In this section, we use the verified model to predict the gravimetric and

volumetric uptakes of the three varieties of modified PAFs: i) lithiation in

PAFs, (ii) fullerene impregnated PAFs and (iii) lithiated fullerene impreg-

nated PAFs. A comparison between the effect on uptakes on these modifica-

tions will also be presented. Note that as discussed in Subsection 4.2.4, the

maximum number of lithium atoms allowed for lithiation in PAF-302 and

PAF-303 are 32 and 48. This translates to a maximum of 8wt% lithiation in

both PAFs.

81

Lithiation in PAFs

Here we discuss the gravimetric and volumetric uptakes of PAF-302, PAF-

303, Li-PAF-302 and Li-PAF-303 which are shown in Figure 4.4. A compar-

ison of the volumetric uptakes of the first two materials shows that PAF-302

is superior to PAF-303 at all pressures at both 77 K and 298 K. For gravimet-

ric uptake, PAF-303 outperforms PAF-302 at 298 K at all pressures except

below 15 bar. At 77 K, PAF-303 outperforms PAF-302 at all pressures.

The effects of lithiation in PAFs is well illustrated in the figure. Here the

uptake in bare PAFs is compared to the uptakes in 2%Li-PAF and 5%Li-PAF.

The figure shows that lithiation in both PAF-302 and PAF-303 improves

gravimetric and volumetric uptake at 77 K and 298 K. This result not only

validates the report by Rao et al. [26] that Li doping enhances H2 uptake

at ambient temperature, but also predicts that uptakes are also enhanced at

77 K.

Gravimetric and volumetric uptakes in both PAF-302 and PAF-303 are

maximised at 8wt% lithiation. Our calculation shows that at 100 bar, 8%Li-

PAF-302 is able to store up to 11 wt% and 5.2wt% at 77 K and 298 K. In

addition, the volumetric uptake for 8%Li-PAF-302 at 100 bar is capable of

reaching 18.3 g/L at 298 K and 39 g/L at 77 K.

In PAF-303, 8wt% lithiation increases the gravimetric storage capacities

to 17.96 wt% and 6.5 wt% at 100 bar at both 77 K and 298 K. Volumetric

uptake is increased to 32.5 g/L and 11.8 g/L at 77 K and 298 K at 100 bar.

A comparison between lithiation in PAF-302 and PAF-303 at both 77 K

and 298 K and 100 bar shows that 8%Li-PAF-303 has superior gravimetric

uptake compared to 8%Li-PAF-202. However 8%Li-PAF-202 provides better

82

volumetric uptake compared to 8%Li-PAF-302.

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

2%Li-PAF-302

5%Li-PAF-302

PAF-303

2%Li-PAF-303

5%Li-PAF-303

(a)

0 20 40 60 80 1000

1

2

3

4

5

6

Pressure (bar)G

ravim

etr

ic U

pta

ke (

wt%

)

PAF-302

2%Li-PAF-302

5%Li-PAF-302

PAF-303

2%Li-PAF-303

5%Li-PAF-303

(b)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

Pressure (bar)

Volu

metr

ic U

pta

ke (

wt%

)

PAF-302

2%Li-PAF-302

5%Li-PAF-302

PAF-303

2%Li-PAF-303

5%Li-PAF-303

(c)

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

Pressure (bar)

Volu

metr

ic U

pta

ke (

g/L

)

PAF302

2wt%Li-PAF302

5wt%Li-PAF302

PAF303

2wt%Li-PAF303

5wt%Li-PAF303

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

Pressure (bar)

Volu

metr

ic U

pta

ke (

wt%

)

PAF-302

2%Li-PAF-302

5%Li-PAF-302

PAF-303

2%Li-PAF-303

5%Li-PAF-303

(d)

Figure 4.4: Gravimetric and volumetric uptakes for Li-PAF-302 and Li-PAF-303. The figures show gravimetric uptake comparison plots for (a) 77 K and(b) 298 K and volumetric uptake comparison plots for (c) 77 K and (d) 298 Krespectively where the blue and red lines represent PAF-302 and PAF-303.The solid, dashed, and dotted lines represents the bare PAFs, 2%Li-PAFs,and 5%Li-PAFs.

83

Fullerene impregnation in PAFs and lithiated PAFs

Here we present the effects of fullerene impregnation on PAFs and lithiated

PAFs. We first compare the uptakes of C60@PAF-303 with PAF-303 and

PAF-302 as shown in Figure 4.5.

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF303

C60@5wt%Li-PAF303

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF303

C60@5wt%Li-PAF303

(a)

0 20 40 60 80 1000

1

2

3

4

5

6

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF-303

C60@5wt%Li-PAF-303

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF303

C60@5wt%Li-PAF303

(b)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

Pressure (bar)

Volu

metr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF-303

C60@5wt%Li-PAF303

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

Pressure (bar)

Volu

metr

ic U

pta

ke (

g/L

)

PAF302

5wt%Li-PAF302

PAF303

5wt%Li-PAF303

C60@PAF303

C60@5wt%Li-PAF303

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF303

C60@5wt%Li-PAF303

(c)

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

Pressure (bar)

Volu

metr

ic U

pta

ke (

g/L

)

PAF302

5wt%Li-PAF302

PAF303

5wt%Li-PAF303

C60@PAF303

C60@5wt%Li-PAF303

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF303

C60@5wt%Li-PAF3030 20 40 60 80 100

0

2

4

6

8

10

12

14

16

18

Pressure (bar)

Gra

vim

etr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF303

C60@5wt%Li-PAF303

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

Pressure (bar)

Volu

metr

ic U

pta

ke (

wt%

)

PAF-302

5%Li-PAF-302

PAF-303

5%Li-PAF-303

C60@PAF-303

C60@5wt%Li-PAF303

(d)

Figure 4.5: Gravimetric uptake comparison plots for (a) 77 K and (b) 298 K.Volumetric uptake comparison plots for (c) 77 K and (d) 298 K. The blue,red and green lines represent uptakes by PAF-302 and PAF-303, lithiatedPAF-302 and PAF-303, and fullerene impregnated lithiated PAF-303.

84

At 77 K, the gravimetric uptake of C60@PAF-303 significantly outper-

forms both PAF-303 and PAF-302 at all pressures except at 95–100 bar

where it matches the uptake by PAF-303. For volumetric uptake, C60@PAF-

303 outperforms PAF-303 at all pressures. However PAF-302 outperforms

C60@PAF-303 from 20–100 bar.

At 298 K, the gravimetric uptake of C60@PAF-303 is not significantly

different from PAF-303 but they both outperform PAF-302. The reverse

is true for volumetric uptake: C60@PAF-303 is not significantly different

from PAF-302 but they both outperform PAF-303. This is consistent with

the results reported by Rao et al. [26] and Mulfort et al. [88] that C60

impregnation reduces the gravimetric uptake and increases volumetric uptake

at room temperature.

We next examine the effect of impregnation of lithiated PAF-303. Fig-

ure 4.5(a) shows that the gravimetric uptake at 77 K by C60@5%Li-PAF-303

exceeds the uptake by 5%Li-PAF-303 only at lower to medium pressures

(0–55 bar). At 298 K, the gravimetric uptake of both C60@5%Li-PAF-303

and 5%Li-PAF-303 are similar at lower pressures with the latter outper-

forming the former at higher pressures. Finally, the volumetric capacity of

C60@5%Li-PAF-303 at both 77 K and 298 K are poorer than the performance

of 5%Li-PAF-302.

Summary

In conclusion, our calculations show that at 100 bar, lithiated PAF-303 has

the highest gravimetric uptake compared to the other materials at both 77 K

and 298 K. It is important to note that at 100 bar, 8%Li-PAF-303 and

85

C60@7%Li-PAF-303 exceeds the 2015 DOE target of 5.5 wt% at 77 K and

298 K. The summary of the gravimetric uptakes at 100 bar for both 77 K

and 298 K are tabulated in Table 4.3.

Table 4.3: Gravimetric Uptake (wt%) at 100 bar, 77 K and 298 K.

PAF % Lithiation 77 K 298 K

0 10.65 2.92PAF-302 2% 10.89 3.29

5% 11.08 4.188% 11.04 5.11

0 16.18 4.81PAF-303 2% 16.83 5.06

5% 17.64 5.708% 17.91 6.52

0 16.00 4.57C60@PAF-303 2% 16.24 4.71

5% 16.70 5.307% 16.88 5.96

We also conclude that the best performing material for volumetric uptake

amongst the material presented here is lithiated PAF-302 at both 77 K and

298 K. In addition, 8%Li-PAF-302 is close to achieving the volumetric target

of 40 g/L at 77 K but is still far from achieving it at 298 K. The summary

of the volumetric uptakes at 100 bar for both 77 K and 298 K are tabulated

in Table 4.4. Further analyses on the impact of lithiation and impregnation

are presented in the following subsections.

86

Table 4.4: Volumetric Uptake (g/L) at 100 bar, 77 K and 298 K.

PAF % Lithiation 77 K 298 K

0 34.74 9.53PAF-302 2% 36.16 10.94

5% 37.92 14.308% 39.04 18.02

0 27.02 8.04PAF-303 2% 28.79 8.62

5% 30.99 10.028% 32.45 11.81

0 31.86 9.10C60@PAF-303 2% 32.99 9.58

5% 34.96 11.097% 36.06 12.74

4.3.2 Potential energy

The key to optimisation is to efficiently utilise the amount of space within

the structure available for adsorption. This depends on the potential energy

landscape, where regions of strong potential energy capture gas into the ad-

sorbed phase while regions of weak potential energy contain the unadsorbed

bulk gas phase. Figure 4.6(a) and (c) displays the potential energy of the in-

teraction between H2 and the bare and lithiated PAFs. Figure 4.6(e) presents

the potential energy for the interaction between H2 and bare and fullerene

impregnated Li-PAF-303. Their corresponding contour plots are displayed

in Figure 4.6(b), (d) and (f) where the varying potential energies are plotted

against the proportion of Li atoms and distance from cavity centre.

Using geometry optimisation [4], the Li atoms are assumed to be located

87

5.5 6 6.5 7 7.5 8-4

-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-302

2%Li-PAF-302

5%Li-PAF-302

(a)

5.5 6 6.5 7 7.5 8-4

-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-302

2%Li-PAF-302

5%Li-PAF-302

5.5 6 6.5 7 7.5 8-4

-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-302

2%Li-PAF-302

5%Li-PAF-302

6 7 8 9 10 11 12-3

-2

-1

0

1

2

3

Distance from cavity centre (Å)

Pro

port

ion o

f Li ato

ms (

wt%

)

PAF-303

C60@PAF-303

C60@2%Li-PAF-303

C60@5%Li-PAF-303

(b)

9 10 11 12-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-303

2%Li-PAF-303

5%Li-PAF-303

(c)

9 10 11 12-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-303

2%Li-PAF-303

5%Li-PAF-303

9 10 11 12-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-303

2%Li-PAF-303

5%Li-PAF-303

6 7 8 9 10 11 12-3

-2

-1

0

1

2

3

Distance from cavity centre (Å)

Pro

port

ion o

f Li ato

ms (

wt%

)

PAF-303

C60@PAF-303

C60@2%Li-PAF-303

C60@5%Li-PAF-303

(d)

6 7 8 9 10 11 12-3

-2

-1

0

1

2

3

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-303

C60@PAF-303

C60@2%Li-PAF-303

C60@5%Li-PAF-303

(e)

9 10 11 12-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-303

2%Li-PAF-303

5%Li-PAF-303

9 10 11 12-3

-2

-1

0

1

2

Distance from cavity centre (Å)

Pote

ntial energ

y (

kJ/m

ol)

PAF-303

2%Li-PAF-303

5%Li-PAF-303

6 7 8 9 10 11 12-3

-2

-1

0

1

2

3

Distance from cavity centre (Å)

Pro

port

ion o

f Li ato

ms (

wt%

)

PAF-303

C60@PAF-303

C60@2%Li-PAF-303

C60@5%Li-PAF-303

(f)

Figure 4.6: Potential energy for Li-PAF-302 in (a) and (b), Li-PAF-303 in(c) and (d), and C60@Li-PAF-303 in (e) and (f) with respect to the distancefrom cavity centre. The contour plots on the right depict the depth of thepotential energy with varying Li atoms and distance from cavity centre.

88

at a distance of approximately 1.4 A from the aromatic rings. The lithiation

of frameworks have been experimented with in the literature as there exists a

strong interaction between H2 and Li which is caused by strong polarising ef-

fects from the transfer of charges from H2 to Li [27]. Figure 4.6 demonstrates

this effect, showing the growing strength of the potential between a hydrogen

molecule and the structure as lithium atoms are added to the framework.

The potential energy of H2 interacting with PAF-302 is slightly stronger

than with PAF-303 due to its smaller cavity size, causing a slight overlap

of the potential energy between the hydrogen and the PAF framework. The

interaction between H2 and lithiated PAF-302 also produces a stronger po-

tential energy compared to with lithiated PAF-303. Figure 4.6(a) shows that

the interaction between H2 and 5%Li-PAF-302 provides the strongest poten-

tial energy well depth of -3.5 kJ/mol at a distance of 7.15 A from the centre

of the cavity.

The fullerene impregnation of both PAF-303 and Li-PAF-303 produces

two potential energy minimums when interacting with H2 as represented by

the blue, green and red lines in Figure 4.6(e). Due to the large cavity size

of PAF-303, the potential energy created from H2 interacting with C60 and

the walls of PAF-303 does not overlap. This provides additional evidence

for Yaghi and Rowsell’s [25] theory that impregnation increases the number

of attraction sites for H2 adsorption. For the interaction with C60@5%Li-

PAF-303, two distinct minimum potential energies of -2.08 kJ mol−1 and

-2.51 kJ mol−1 exist at 6.4 A and 11.5 A from the centre of the cavity.

Our model suggests that impregnation of lithiated PAF-303 improves

the strength of the well depth located closer to its walls. However, the

89

improvement is not enough to make it a better material for gas adsorption

because the minimum potential energy of 5%Li-PAF-302 is 39% deeper than

that of C60@5%Li-PAF-303.

4.3.3 Free volume for adsorption

A critical adsorption optimisation factor is to consider the ligand size trans-

lated to the cavity size. In this section, we present the efficiency of the various

PAFs in adsorbing H2 as adsorbed gas using the fractional free volume for

adsorption. This is illustrated in Figure 4.7 at both 77 K and 298 K, accom-

panied with ligand sizes for PAF-301, PAF-302 and PAF-303. The fractional

free volume describes the ratio of adsorbed H2 to bulk gaseous H2 within the

total free volume of the cavity.

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

Cavity Size (Å)

Fra

ctional F

ree V

olu

me f

or

Adsorp

tion (

%)

PAF

2% Li-PAF

5% Li-PAF

C60@PAF

C60@2%Li-PAF

C60@5%Li-PAF

(a)

0 5 10 15 20 25 300

5

10

15

20

25

30

35

Cavity Size (Å)

Fra

ctional F

ree V

olu

me f

or

Adsorp

tion (

%)

PAF

2% Li-PAF

5% Li-PAF

C60@PAF

C60@2%Li-PAF

C60@5%Li-PAF

(b)

Figure 4.7: Fractional free volume for adsorption (%) calculated at (a) 77 Kand (b) 298 K with varying cavity size. The dimensions for available ligandsare depicted in the molecular diagrams.

90

The figure demonstrates that the volume of adsorbed H2 increases as

lithium atoms are added to the framework. At 77 K (Figure 4.7(a)), the

optimum cavity size for Li-PAFs is 17.6 A. At this cavity size, 28% of the

free volume of 5%Li-PAF contains hydrogen in the dense adsorbed phase,

which is 5.8% more than the fractional free volume of the bare PAF. The

optimum cavity size for impregnated Li-PAFs at 77 K is 24.6 A where the

fractional free volume for C60@5%Li-PAF is 34%, which is 87% and 15%

more than a bare PAF and C60@PAF, respectively.

There is an improvement in the storage of hydrogen gas in its adsorbed

state at 298 K in impregnated and lithiated PAFs as displayed in Fig-

ure 4.7(b). The cavity size for PAF-301 and PAF-302 offers the correct

porosity to exploit the maximum benefits of lithiation (13–15 A). The frac-

tional free volume is 20.3% and 19.7% at the optimum cavity sizes for Li-PAF

at 13–15 A and C60@Li-PAF at 20–22 A.

An examination of the fractional free volume ratio at the PAF-302 cavity

size at 20.3 A shows that PAF-302 is able to store hydrogen in its adsorbed

state in 23.5% and 10% of its cavity volume at 77 K and 298 K. The frac-

tional free volume ratio for the cavity size of PAF-303 of 29 A is 14% and

5% at 77 K and 298 K. Our model suggests that lithiation and impregna-

tion significantly increases the ratios with C60@5%Li-PAF-303 providing the

highest adsorption ratio of 28% at 77 K. At 298 K, 5%Li-PAF-302 improves

the adsorption by 50% to allow 15% of its cavity volume to store adsorbed hy-

drogen. If we could fit a fullerene into a PAF-302 cavity, C60@5%Li-PAF-302

would provide the highest fractional free volume of 18% at 298 K.

91

4.4 Conclusion

In this chapter we investigate hydrogen storage with three different PAF

structures; (i) lithiated PAF-302 and PAF-303, (ii) fullerene impregnated

PAF-303 and (iii) lithiated fullerene impregnated PAF-303. Lithiation is

demonstrated in PAF-302 and PAF-303 but only PAF-303 is used to model

fullerene impregnation because its cavity is large enough to accommodate a

fullerene.

The interaction energies between H2 and these structures are derived from

the Morse potential using a continuous approximation method. Gravimetric

uptake results are validated against simulation and experimental results in

the literature before being used to provide predictions of gravimetric and vol-

umetric uptakes for modified PAFs. Further analyses on potential energy and

fractional free volume are also presented to provide a better understanding

of the impact of the modifications on the PAFs.

Our model indicates that gravimetric uptake and volumetric uptake is

enhanced with lithiation of PAF-302 and PAF-303 at both 77 K and room

temperature. Compared to PAF-302 at 100 bar, 8% lithiation in PAF-302

increases the gravimetric capacity by 3.5% at 77 K and 43% at 298 K and the

volumetric uptake increases by 11% at 77 K and 47% at 298 K. An increase

of lithiation to 8% in PAF-303 at 100 bar will increase gravimetric uptake

by 10% at 77 K and 26% at 298 K, while volumetric capacities increases by

17% at 77 K and 32% at 298 K.

Fullerene impregnation in PAF-303 provides mixed outcomes for gravi-

metric capacity with increased uptake at 77 K and low to medium pressures

and a slight decline at room temperature. Volumetric uptake results show

92

that fullerene impregnation improves uptake for PAF-303 by 15% at 77 K

and 12% at 298 K.

Finally, fullerene impregnation of lithiated PAF-303 shows some improve-

ment in gravimetric and volumetric uptake when compared to bare PAF-303.

Despite this, Li-PAF-303 outperforms C60@Li-PAF-303. At 100 bar, 8%Li-

PAF-303 provides the highest gravimetric storage of 17.9 wt% and 6.5 wt%

at 77 K and 298 K. The material with the highest volumetric capacity is

8%Li-PAF-302 which at 100 bar is able to store 39 g/L at 77 K and 18 g/L

at 298 K.

An examination of the potential energy of H2 interacting with PAFs shows

that lithiation of PAFs result in a stronger potential energy and fullerene

impregnation in PAF-303 creates an additional attraction site. Our results

also show that the potential energies in PAF-302 and Li-PAF-302 are slightly

deeper than PAF-303 and Li-PAF-303. This is explained by the smaller

cavity size of PAF-302 which causes overlapping of potential energies between

H2 and both PAF-302 and Li-PAF-302.

A comparison of the fractional free volume for adsorption between bare,

2%Li and 5%Li-PAFs and their fullerene impregnated versions reveal that

the adsorption of H2 is enhanced by impregnated and lithiated PAFs. Our

calculations show that 5%Li-PAF-302 possesses the largest fractional free

volume at 298 K, where 15% of the cavity volume is able to store H2 in its

adsorbed state. The most adsorbent material at 77 K is C60@5%Li-PAF-303

with a fractional free volume ratio of 28%. Using our model, the ideal cavity

sizes to maximise the fractional free volume for Li-PAFs are 17.6 A and 14 A

at 77 K and 298 K. For C60@Li-PAF, the ideal cavity sizes at 77 K and

93

298 K are 24.6 A and 21 A.

94

Chapter 5

Summary

This thesis outlines mathematical models to assess the efficiency of hydro-

gen storage in various adsorbents. This is in contrast to the more popular

computer simulations and experiments done in research which are far more

time consuming and expensive. Our research shows that hydrogen storage

is affected by pore size, surface area, operating conditions and increased at-

traction sites in the adsorbent. The theories and models presented here are

aimed at contributing towards the discovery of better performing adsorbents

applicable not only for hydrogen but also for other gases.

Two types of potential energies are used to model the interaction between

the hydrogen molecule and porous material; the Lennard-Jones and Morse

potentials. With the assumption that the porous materials have uniform

atomic densities, calculations are simplified by using building blocks to model

the complicated shapes of these materials.

Once the mathematical models are verified against simulation and ex-

perimental results, the performance of the porous materials for hydrogen

95

storage is explored. This involves examining the pore size, surface area and

appropriate operating conditions for the material. Modifications in adsor-

bents to increase attraction site are also investigated. In addition, we extend

our models in our research in Be-BTB to include calculations to optimise

hydrogen storage and delivery in hydrogen fuel cells coupled with a storage

device based on MOF-5, MOF-177 and Be-BTB. The work in this thesis are

summarised in the following sections.

5.1 Analytical formulations of nanospace in

porous materials

In Section 2, models are presented to demonstrate the interaction between a

hydrogen molecule and various nanomaterials represented by simple building

blocks. These models assume that the atoms on the surface of the nano-

materials are uniformly distributed to enable the use of continuous models.

Continuous models are used instead of discrete models to provide simpler

analytical expressions and reduce computation time. Applications using the

interaction models are demonstrated using simple examples such as a car-

bon atom (point), polyacetylene (line), graphene sheet (plane), benzene ring

(circle), fullerene (sphere) and carbon nanotube (cylinder).

The gas adsorption model is also introduced in this chapter to enable the

investigation of hydrogen uptake in adsorbents. Within the adsorbent, gas

molecules are assumed to exist in both adsorbed and bulk gas states. The

gravimetric uptake in an adsorbent can then be assessed through calculations

involving the sum of the bulk and adsorbed gas in the cavity.

96

This chapter only explores the use of the 6-12 Lennard-Jones potential

function to describe the potential energies of H2 interacting with the various

nanostructures. Further work in this area can be explored using other types

of potential function such as the Morse potential.

5.2 Beryllium based metal-organic frameworks

The high performing beryllium based metal-organic frameworks are examined

in Chapter 3. To do this, the Lennard-Jones potential energy is used to model

the interaction between hydrogen atoms and the structure. The geometry of

Be-BTB is simplified by combining cylindrical and spherical building blocks

to represent the beryllium ring and the rest of the internal cavity.

Calculations of the gravimetric uptake using our model is successfully

verified with simulation and experimental data at 77 K and 298 K. The

predicted heat of adsorption of 4.8 kJ mol−1 is also proven to be consistent

with the measured heat of adsorption reported in Sumida et al. [2]. An

analysis of the results show that the hydrogen interaction with the beryllium

ring is stronger compared to its interaction with the spherical cavity. In

addition, the size of the beryllium ring is determined to be the optimal size

required to maximise hydrogen adsorption.

This chapter also explores the usage of MOF-based storage in fuel cells

through the development and use of the novel thermodynamic energy opti-

misation model. The model provides the optimal storage and delivery condi-

tions for hydrogen fuel cells coupled with a variety of material-based storage

tanks within the requirements set by the U.S. Department of Energy. The

97

energy produced by these fuel cells is then calculated using three different

adsorption cycles: the pressure-swing, temperature-swing, and the combined

pressure and temperature-swing cycle. The storage tanks examined here are

the compressed H2 storage tank and MOF-based storage tanks made of three

different materials: Be-BTB, MOF-5 and MOF-177.

Our analysis shows that for volumetric capacity, the various storage tanks

do not produce enough energy to achieve the 2017 U.S. DOE target of

1.3 kWh/L for all three adsorption processes. For the gravimetric capac-

ity, only the compressed tank fulfills the DOE target of 1.8 kWh/kg using

all three adsorption processes.

Amongst the MOF-based storage tanks, the Be-BTB tank provides the

best performance for both gravimetric and volumetric capacity, followed by

MOF-177 and MOF-5 for all three adsorption processes. This outcome is

attributed to Be-BTB’s superior pore structure over MOF-5 and MOF-177.

Ultimately the TEO model suggests that there is still a gap in MOF perfor-

mance that needs to be bridged to meet the DOE targets.

5.3 Porous aromatic frameworks

In Chapter 4, the performance of porous aromatic frameworks as an adsor-

bent for hydrogen gas is examined. The unique structure of PAFs are mod-

elled using spherical building blocks on which the assumption of a uniform

distribution of PAF atoms. PAF-302 and PAF-303 are primarily discussed

here and their interaction with hydrogen is modelled using the Morse poten-

tial function. In addition, methods proposed by the literature to improve

98

overall gas storage capacities of adsorbents are applied to our model. These

modifications include lithiation, fullerene impregnation and the combination

of lithiation and fullerene impregnation.

Validation of hydrogen uptakes in PAF-302 and PAF-303 are performed

against the experimental data and simulation results from Lan et al. [3] and

Konstas et al. [4] at both 77 K and 298 K. The model is then used to pre-

dict the gravimetric and volumetric uptakes of lithiated PAF-302, lithiated

PAF-303 and impregnated PAF-303, followed by the gravimetric uptakes of

fullerene impregnated PAF-303 and fullerene impregnated, lithiated PAF-

303. Studies on fullerene impregnation in PAF-302 are not done as PAF-302

cavities are too small to fit a fullerene.

Our results show that lithiation in PAF-302 and PAF-303 enhances both

gravimetric and volumetric uptakes at 77 K and 298 K. This is explained by

the overlap in potential energy resulting in a deeper potential energy between

the hydrogen and PAF framework. Even though the fullerene impregnation

of PAF-303 does not result in an overlap of potential energy, the number of

attraction site is doubled, thus improving storage capabilities.

An investigation into the fractional free volume for adsorption reveals

that the adsorption of H2 is enhanced in impregnated and lithiated PAFs.

The size of the cavity affects the adsorption ratio with our results showing

that the cavity size for PAF-301 and PAF-302 offers the correct porosity

to exploit the maximum benefits of lithiation. The material that provides

the highest adsorption ratio at 77 K is C60@5%Li-PAF-303 where 28% of its

cavity volume is able to store adsorbed hydrogen. If a fullerene could fit into

PAF-302, C60@5%Li-PAF-302 would provide the highest adsorption ratio at

99

298 K with 18% of its cavity volume able to store adsorbed hydrogen.

Finally, a comparison amongst all three modified PAF materials of their

gravimetric uptake ability shows that the 8%Li-PAF-303 provides the best

performance of up to 17.9 wt% at 77 K and 6.5 wt% at room temperature.

This achievement means that 8%Li-PAF-303 exceed the 2015 DOE gravi-

metric target of 5.5 wt% at 77 K and 298 K.

When comparing volumetric storage capacities of these materials, 8%Li-

PAF-302 provides the best performance where it is capable of storing 39 g/L

at 77 K and 18.3 g/L at 298 K. Thus 8%Li-PAF-302 is close to achieving

the 2015 DOE volumetric target of 40 g/L at 77 K, however it is far from

meeting this at room temperature.

Finally we would like to add that a thermodynamic energy optimisation

analysis could be a possible direction for future research on the theoretical

PAFs to discover if they would be able to meet the DOE targets. This

analysis was not included in Chapter 4 as there are currently no records of

the specific heat capacity of the theoretical PAF ligands and their heat of

adsorptions, which are both necessary to perform the study.

100

Chapter 6

Appendix

6.1 Evaluation of Equations in Chapter 2

6.1.1 Interaction with a line

The derivation of Rn which is used to determine the total potential energy

for atom P interacting with L in Section 2.1.2 is obtained by solving

Rn =

∫ ∞−∞

(g2 + y2L

)−ndyL

Application of the transformation yL = gtan(θ) will give

Rn =

∫ π2

−π2

(g2 + g2 tan2 θ

)−ng sec2 θ dθ

= g1−2n∫ π

2

−π2

cos2n−2θ dθ.

101

As cos is a symmetric function, the integral can be rewritten as

Rn = 2g1−2n∫ π

2

0

cos2n−2θ dθ.

The form of this integral can then be rewritten as a beta function shown

below

∫ π2

0

sinpθ cosqθ dθ =1

2B

(p+ 1

2,q + 1

2

), (6.1.1)

where p = 0 and q = 2n− 2. Therefore,

Rn = g1−2nB

(1

2, n− 1

2

).

6.1.2 Interaction with a plane

The potential energy between a point P which lies on the x-axis and a plane

S which lies on the yz-plane can be derived by solving for Rn given by

Rn =

∫ ∞−∞

∫ ∞−∞

(g2 + y2s + z2s

)−ndys dzs.

Using the substitution ys =√g2 + z2s tan θ, the equation above becomes

Rn =

∫ ∞−∞

∫ π2

−π2

(g2 + z2s

) 12−n

cos2n−2 θ dθ dzs.

As cos is a symmetric function, the integral can be rewritten as

Rn = 2

∫ ∞−∞

∫ π2

0

(g2 + z2s

) 12−n

cos2n−2 θ dθ dzs.

102

As shown in the previous subsection, the integral can be rewritten as a beta

function shown in Eq. 6.1.1 where p = 0 and q = 2n− 2 to give

Rn = B

(1

2, n− 1

2

) ∫ ∞−∞

(g2 + z2s

) 12−n

dzs.

Upon substituting zs = g tan θ into Rn, the equation becomes

Rn = B

(1

2, n− 1

2

)∫ π2

−π2

[g2(1 + tan2 θ)

] 12−ng sec2θ dθ

= g2−2nB

(1

2, n− 1

2

)2

∫ π2

0

cos2n−3 θ dθ.

Note that the integral can again be rewritten as a beta function using Eq. 6.1.1

with p = 0 and q = 2n− 3 to obtain

Rn = g2−2nB

(1

2, n− 1

2

)B

(1

2, n− 1

).

6.1.3 Interaction with the top or bottom of a ring

The derivation of Rn for the case of a point interacting with a ring (from the

top or bottom) is obtained by solving

Rn =

∫ 2π

0

q [β − 2αq cos(θ − θ0)]−n dθ.

As the integral is a periodic function of θ of period 2π, its limits can be

changed from (0, 2π) to (θ0, 2π + θ0). Therefore θ0 can be omitted from the

integral. Using the fundamental Pythagorean trigonometric identity, Rn can

103

be written as

Rn = q

∫ 2π

0

[β − 2αq cos θ]−n dθ

= q

∫ 2π

0

[β − 2αq

(1− 2sin2 θ

2

)]−ndθ.

Upon substituting t = sin2 θ2, the equation becomes

Rn =2q

(β − 2αq)n

∫ 1

0

[1− 4αqt

2αq − β

]−nt−

12 (1− t)−

12 dt,

where the integral is the standard hypergeometric function given in Eq. 2.1.21.

Therefore,

Rn =2q

(β − 2αq)nB

(1

2,1

2

)F

(n,

1

2; 1;

4αq

2αq − β

)=

2πq

(β − 2αq)nF

(n,

1

2; 1;

4αq

2αq − β

).

104

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