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Multiscale Theoretical and Computational
Modeling of the
Synthesis, Structure and Performance of
Functional Carbon Materials
Samir Hemant Mushrif
Department of Chemical Engineering
McGill University, Montréal
November 2009
Doctor of Philosophy
A thesis submitted to McGill University in partial fulfilment of
the requirements of the degree of Doctor of Philosophy
© Samir Hemant Mushrif 2009
II
Dedicated to,
My wife Shivangi, My daughter Ananyaa
and
My family …
III
CONTRIBUTIONS OF THE AUTHOR
The author chooses the manuscript based thesis option according to the
guidelines for thesis preparation given by the Faculty of Graduate and
Post doctoral Studies of McGill University. Contents of chapters 2-6 of the
present thesis are adopted or revised from articles published in or to be
submitted to scientific journals under the normal supervision of the
author’s research supervisor Prof. Alejandro D. Rey. All the theoretical,
computational and experimental work is done by the author, except in
chapter 3, where Mr. Halil Tekinalp performed the fibre spinning and
helped the author perform stabilization, carbonization and activation of
the fibre. The articles included in chapters 2-6 are also written by the
author.
The role of Prof. Gilles H. Peslherbe, who is also a co-author of articles in
chapters 3-5, is similar to that of a co-supervisor.
IV
ACKNOWLEDGEMENTS
I would like to take this opportunity to thank my supervisor Prof.
Alejandro D. Rey for his invaluable support and guidance throughout the
course of this research. Prof. Rey is a perfect role model for any student
who aims to make a career in research. His hard-work and dedication
towards research and teaching always inspired me. Working under his
supervision also taught me how to survive and prosper in the competitive
world of engineering and scientific research. It was an honour working
with you Prof. Rey! Alongside Prof. Rey, I would also like to thank
Barbara for her care, love and warmth and for making me feel a part of
their family.
I would also like to express my sincere gratitude to Prof. Gilles H.
Peslherbe for his guidance on molecular modeling methods. Prof.
Peslherbe’s molecular modeling class taught me the basic concepts of
molecular modeling without which I would not have been able to do this
research. Prof. Peslherbe’s incredible knowledge of first—principles
calculations, which allowed him to provide a constructive criticism on my
work, is deeply appreciated.
I would also like to thank my past and current labmates, Dr.
Benjamin Wincure, Dr. Gino de Luca, Prof. Dae Kun Hwang, Prof. Susanta
Das, Dr. Majid Ghiass, Prof. Ezequiel Soule, Mojdeh Golmohammadi,
Gaurav Gupta, Ghoncheh Rasouli, Yogesh Murugesan, Moeed Shahamat,
Paul Phillips, Alexandre Proulx for many useful discussions and for their
support and friendship. A special thanks to Dr. Nasser Abukhdeir and
Raymond Chang for their help with the computer cluster.
A big thank you is also due to all my friends in Montreal, including
Parmeet, Ashish, Naveen, Rao-Archana, Shree and family, Pankaj Kumar
and family, Thiru, Vinayak, Mani, Mohan, Nancy and everyone in the
V
Montreal Marathi Mandal gang, for making my stay a wonderful and
memorable experience.
Financial support provided by the National Science Foundation
(via the Centre for Advanced Engineering Fibres and Films), the Natural
Sciences and Engineering Research Council of Canada and the Eugenie
Ulmer Lamothe fund (in the form of post-graduate scholarships) is greatly
appreciated. Computational resources were provided by the Réseau
québécois de calcul de haute performance (RQCHP) and I would like to
thank Jacques Richer, Michel Beland, Daniel Stubbs, Huizhong LU and
Richard Lefebvre for the excellent technical support.
Last, but not the least, I would like to thank my better half, my wife,
Shivangi and my family in India for their unconditional love, support,
encouragement and belief. I would not have been able to come so far
without them.
VI
ABSTRACT
Functional carbon based/supported materials, including those
doped with transition metal, are widely applied in hydrogen mediated
catalysis and are currently being designed for hydrogen storage
applications. This thesis focuses on acquiring a fundamental
understanding and quantitative characterization of: (i) the chemistry of
their synthesis procedure, (ii) their microstructure and chemical
composition and (iii) their functionality, using multiscale modeling and
simulation methodologies. Palladium and palladium(II) acetylacetonate
are the transition metal and its precursor of interest, respectively.
A first principles modeling approach consisting of the
planewave pseudopotential implementation of the Kohn—Sham density
functional theory, combined with the Car—Parrinello molecular
dynamics, is implemented to model the palladium doping step in the
synthesis of carbon based/supported material and its interaction with
hydrogen. The electronic structure is analyzed using the electron
localization function and, when required, the hydrogen interaction
dynamics are accelerated and the energetics are computed using the
metadynamics technique. Palladium pseudopotentials are tested and
validated for their use in a hydrocarbon environment by successfully
computing the experimentally observed crystal structure of palladium(II)
acetylacetonate. Long standing hypotheses related to the palladium
doping process are confirmed and new fundamental insights about its
molecular chemistry are revealed. The dynamics, mechanism and energy
landscape and barriers of hydrogen adsorption and migration on and
desorption from the carbon based/supported palladium clusters are
reported for the first time.
VII
The effects of palladium doping and of the synthesis procedure on
the pore structure of palladium doped activated carbon fibers are
quantified by applying novel statistical mechanical based methods to the
experimental physisorption isotherms. The drawbacks of the conventional
adsorption based pore structure analysis methods are demonstrated.
Since the functionality of carbon materials is strongly dependant on their
microstructure, a thermodynamics poromechanics based model, to
explain and predict the adsorption induced deformations in their
microstructure, is developed and validated. A molecular level explanation
for the experimentally observed unique trend in the deformation is also
provided.
In summary, this thesis provides a comprehensive scientific
investigation of the functional carbon based/supported materials by
integrating: (i) the first principles modeling of palladium doping and
hydrogen interaction, (ii) pore structure calculations extracted from the
experimental physisorption data, and (iii) statistical
mechanical continuum level modeling of adsorption induced
deformation.
VIII
RÉSUMÉ
Les matériaux fonctionnels à base de carbone ou composés d'une
matrice de carbone, incluant les matériaux dopés par des métaux de
transition, sont largement utilisés dans le domaine de la catalyse par
l'hydrogène et sont présentement développés pour des applications dans
l'emmagasinage d'hydrogène. Le but premier de ce travail de thèse est
d'acquérir des connaissances fondamentales et des détails quantitatifs sur :
(i) la chimie de leur mode de synthèse, (ii) leur microstructure et leur
composition chimique et (iii) leur fonctionnalité, en utilisant des
modélisations à différentes échelles ainsi que des méthodologies de
simulation. Le palladium et le palladium (II) acétyl acétonate sont
respectivement le métal de transition et son précurseur d'intérêt.
Une approche de modelage ab initio consistant en une implantation
d'une onde planaire pseudo-potentielle dans la théorie de densité
fonctionnelle de Kohn-Sham, combinée à une dynamique moléculaire de
Car-Parrinello, a été mise en application afin de modéliser l'étape de
dopage par le palladium dans la synthèse de matériaux à base de carbone
ou composés d'une matrice de carbone et ses interactions avec les atomes
d'hydrogène. La structure électronique est analysée à l'aide de la fonction
de localization par électron et, lorsque nécessaire, les interactions
dynamiques de l'hydrogène ont été accélérées et le bilan énergétique des
interactions a été calculé en utilisant des techniques méta-dynamiques. Les
pseudo-potentiels de palladium sont testés et validés quant à leur
utilisation dans un environnement d'hydrocarbures par des simulations
corroborant la stucture crystalline du palladium (II) acétyl acétonate
observée expérimentalement. Des hypothèses expérimentales de longue
date sur le dopage par le palladium ont été confirmés et de nouvelles
considérations fondamentales en lien avec leur chimie moléculaire ont été
IX
révélées. La cinétique, les mécanismes et les barrières et empreintes
énergétiques de l'adsorption de l'hydrogène ainsi que sa migration et sa
désorption du complexe de palladium et de matériel à base de carbone ou
composé d'une matrice de carbone ont été mis de l'avant pour une
première fois.
Les effets du dopage par le palladium et de la procédure de
synthèse sur la structure poreuse de fibres de carbones activées par le
dopage par le palladium sont quantifiés par une application d'une
nouvelle méthode mécanique statistique sur les isothermes de
physisorption expérimentales. Les inconvénients des méthodes
conventionnelles d'analyse de structures poreuses basées sur l'adsorption
sont démontrées. Puisque les fonctionnalités des matériaux à base de
carbone dépendent fortement de leur microstructure, un modèle basé sur
la thermo-poro-mécanique, servant à expliquer et prédire l'adsorption
induite par les déformations au niveau de la microstructure, est développé
et validé. Une explication au niveau moléculaires des tendances observées
expérimentalement quant à la déformation est également proposée.
En résumé, ce travail de thèse propose une étude scientifique
compréhensible des matériaux fonctionnels à base de carbone ou
composés d'une matrice de carbone en intégrant : (i) un modelage ab initio
du dopage par le palladium et de l'interaction avec les atomes
d'hydrogène, (ii) un calcul de la structure des pores basé sur les données
issues des études expérimentales sur la physisorption, et (iii) un modelage
statistique au niveau du continuum mécanique de l'adsorption induite par
la déformation.
X
TABLE OF CONTENTS
CONTRIBUTIONS OF THE AUTHOR .......................................................... IIIACKNOWLEDGEMENTS................................................................................ IVABSTRACT ......................................................................................................... VIRÉSUMÉ ............................................................................................................VIIITABLE OF CONTENTS ......................................................................................XLIST OF FIGURES............................................................................................XIVLIST OF TABLES..............................................................................................XXI1 INTRODUCTION AND GENERAL LITERATURE REVIEW .............. 1
1.1 General Introduction ........................................................................... 11.2 Carbon Materials.................................................................................. 3
1.2.1 Structure, properties and applications...................................... 31.3 Carbon Materials in Catalysis and Hydrogen Storage ................... 7
1.3.1 Carbon Precursors or Raw Materials ........................................ 91.3.2 Methods of preparation and metal doping .............................. 91.3.3 Effect of metal loading on carbon support ............................. 12
1.4 Structural and Functional Issues...................................................... 131.4.1 Experimental Investigations..................................................... 161.4.2 Theoretical Investigations......................................................... 181.4.3 Need for further research.......................................................... 21
1.5 Motivation and Objectives................................................................ 221.6 Thesis Scope, Methodology and Organization.............................. 24
1.6.1 Chapter 2 ..................................................................................... 251.6.2 Chapters 3 and 4......................................................................... 261.6.3 Chapter 5 ..................................................................................... 281.6.4 Chapter 6 ..................................................................................... 281.6.5 Chapter 7 ..................................................................................... 291.6.6 Appendix..................................................................................... 29
1.7 References............................................................................................ 312 EFFECT OF METAL SALT ON THE PORE STRUCTURE EVOLUTION OF PITCH-BASED ACTIVATED CARBON FIBERS........... 43
2.1 Summary ............................................................................................. 432.2 Introduction and literature survey .................................................. 442.3 Experimental....................................................................................... 472.4 Adsorption Isotherm Analysis Methods ........................................ 48
2.4.1 Traditional methods of isotherm analysis .............................. 482.4.2 Chi-theory ................................................................................... 482.4.3 Adsorption Potential Distribution........................................... 542.4.4 Density functional theory ......................................................... 56
2.5 Results and Discussion...................................................................... 562.5.1 Adsorption Isotherm Analysis ................................................. 56
XI
2.5.2 Effect of Pd on pore structure evolution................................. 642.6 Conclusions......................................................................................... 662.7 References............................................................................................ 67
3 FIRST-PRINCIPLES CALCULATIONS OF THE PALLADIUM(II) ACETYLACETONTE CRYSTAL STRUCTURE ............................................ 74
3.1 Summary ............................................................................................. 743.2 Introduction ........................................................................................ 743.3 Computational Methods ................................................................... 763.4 Results and Discussion...................................................................... 79
3.4.1 Crystal Structure ........................................................................ 793.4.2 ELF Analysis ............................................................................... 83
3.5 Conclusions......................................................................................... 873.6 References............................................................................................ 88
4 TOWARDS UNDERSTANDING PALLADIUM DOPING OF CARBON SUPPORTS: A FIRST-PRINCIPLES MOLECULAR DYNAMICS INVESTIGATION .............................................................................................. 91
4.1 Summary ............................................................................................. 914.2 Introduction ........................................................................................ 924.3 Results and Discussion...................................................................... 934.4 Conclusions....................................................................................... 1014.5 Supporting Information .................................................................. 102
4.5.1 Computational Details ............................................................ 1024.5.1.1 Simulation system set-up.................................................... 1024.5.1.2 Computational Methods ..................................................... 104
4.5.2 Additional Results ................................................................... 1064.5.2.1 ELF analysis of palladium-oxygen interactions .............. 106
4.5.3 Additional Relevant Information (Experimental and Simulation): ............................................................................................... 107
4.6 References.......................................................................................... 1105 THE DYNAMICS AND ENERGETICS OF HYDROGEN ADSORPTION, DESORPTION AND ITS MIGRATION ON A CARBON SUPPORTED PALLADIUM CLUSTER........................................................ 115
5.1 Summary ........................................................................................... 1155.2 Introduction ...................................................................................... 1165.3 System set-up and Simulation details ........................................... 1225.4 Results and Discussion.................................................................... 1275.5 Conclusions....................................................................................... 1415.6 References.......................................................................................... 142
6 AN INTEGRATED MODEL FOR ADSORPTION-INDUCED STRAIN IN MICROPOROUS SOLIDS ......................................................................... 150
6.1 Summary ........................................................................................... 1506.2 Introduction ...................................................................................... 1516.3 Model Development ........................................................................ 157
XII
6.3.1 Mechanics of porous adsorbent ............................................. 1586.3.2 Chemical potential of the adsorbate...................................... 1656.3.3 Calculating adsorption-induced strain ................................. 168
6.4 Results and discussion .................................................................... 1706.5 Conclusions....................................................................................... 1776.6 Supporting Information .................................................................. 1786.7 References.......................................................................................... 180
7 CONCLUSIONS....................................................................................... 1867.1 General conclusions......................................................................... 186
7.1.1 Introduction .............................................................................. 1867.1.2 Pore structure computation and analysis (Chapter 2)........ 1867.1.3 Crystal structure calculations of palladium(II) acetylacetonate (Chapter 3) .................................................................... 1877.1.4 Palladium doping of carbon support (Chapter 4) ............... 1887.1.5 Hydrogen interaction with carbon supported palladium cluster (Chapter 5).................................................................................... 1887.1.6 Adsorption induced deformation in carbon materials (Chapter 6)................................................................................................. 189
7.2 Original contributions to knowledge ............................................ 1897.3 Recommendations for future work ............................................... 191
A APPENDIX: MOLECULAR MODELING METHODS....................... 194A.1 Introduction ...................................................................................... 194A.2 Molecular Modeling methods ........................................................ 196
A.2.1 Molecular Mechanics............................................................... 198A.2.2 Electronic Structure Calculations........................................... 205
A.2.2.1 Electronic Structure of Atom and Wave-particle duality205
A.2.2.2 Postulates of Quantum Mechanics ................................ 206A.2.2.3 Schrödinger Equation...................................................... 207A.2.2.4 Solution for Hydrogen atom and Approximate Solution for Helium 210A.2.2.5 Linear Combination of Atomic Orbitals (LCAO)........ 214A.2.2.6 Hartree-Fock Calculations .............................................. 216A.2.2.7 Semi-empirical methods ................................................. 218A.2.2.8 Post Hartree-Fock............................................................. 219
A.2.3 Density Functional Theory (DFT).......................................... 221A.2.3.1 Origin and formulation of DFT...................................... 221A.2.3.2 Kohn-Sham formulation ................................................. 223A.2.3.3 Local density approximation and Generalized Gradient Approximation ..................................................................................... 226
A.2.4 Planewave-pseudopotential Methods .................................. 229A.2.5 Optimization techniques......................................................... 241
A.2.5.1 Molecular Dynamics algorithm ..................................... 244
XIII
A.2.5.2 Nose-Hoover thermostat................................................. 247A.2.6 Car-Parrinello Molecular Dynamics...................................... 250
A.2.6.1 Car-Parrinello Scheme..................................................... 252A.2.7 Metadynamics .......................................................................... 257
A.2.7.1 Concept.............................................................................. 259A.2.7.2 Extended Car-Parrinello Lagrangian for metadynamics
260A.2.8 Electronic Structure analysis methods.................................. 263
A.2.8.1 Electron Localization Function (ELF)............................ 264A.3 References.......................................................................................... 266
XIV
LIST OF FIGURES
Figure 1.1: Four basic building blocks of material science and research.
Adapted from Allen and Thomas3. .......................................................... 2
Figure 1.2: Different forms of sp2 type carbon materials. (a) graphite, (b)
buckyball, (c) Nanotube, (d) mesophase carbon and (e) isotropic
carbon............................................................................................................ 6
Figure 1.3: Method of preparation of activated carbons and activated
carbon fibres............................................................................................... 11
Figure 1.4: Thesis summary and organization............................................... 30
Figure 2.1: Chi-theory representation of adsorption isotherms. – ACFs
from pure pitch, × – ACFs from palladium acetylacetonate containing
pitch. The straight line indicates the chi-theory predicted isotherm
and the experimental isotherm is shown using symbols. ................... 57
Figure 2.2: Pore size distribution for ACFs prepared from pure pitch, at
burn-off values of 34% ( ), 55% ( ) and 80% (···)........................... 58
Figure 2.3: Pore size distribution for ACFs prepared from Pd-containing
pitch, at burn-off values of 20% ( ), 45% ( ), 65% (···) and 85% (
·). .................................................................................................................. 59
Figure 2.4: Adsorption potential distribution for ACFs prepared from pure
pitch, at burn-off values of 34% ( ), 55% (···) and 80% ( ). ......... 60
Figure 2.5: Adsorption potential distribution for ACFs prepared from Pd-
containing pitch, at burn-off values of 20% ( ), 45% (···), 65% ( ·)
and 85% ( ). .......................................................................................... 61
Figure 2.6: Pore size calculations using BET ( ), BJH ( ) and chi-theory
( ). Solid lines represent ACFs prepared from pure pitch and dotted
lines represent ACFs prepared from palladium acetylacetonate
containing pitch. ........................................................................................ 62
XV
Figure 2.7: Total pore volumes calculated using adsorption isotherm ( ),
NLDFT ( ) and chi-theory ( ). Solid lines represent ACFs prepared
from pure pitch and dotted lines represent ACFs prepared from
palladium acetylacetonate containing pitch. ........................................ 63
Figure 2.8: BET ( ) and chi-theory ( ) total surface area and chi-theory
external surface area ( ). Solid lines represent ACFs prepared from
pure pitch and dotted lines represent ACFs prepared from palladium
acetylacetonate containing pitch............................................................. 63
Figure 2.9: Micropore [t-plot ( ) and NLDFT ( )] and mesopore volumes
[BJH ( ) and NLDFT ( )] as a function of activation. Solid lines
represent ACFs prepared from pure pitch and dotted lines represent
ACFs prepared from palladium acetylacetonate containing pitch.... 64
Figure 3.1: Packing of palladium(II) acetylacetonate in the crystal lattice
and labeling of atoms in the molecule. .................................................. 76
Figure 3.2: Relative energy vs. lattice volume. The filled squares and
circles are the DFT computed energies and the lines represent the
equation of state fit.................................................................................... 78
Figure 3.3: Experimental and computed parameters quantifying the non-
planar geometry of the palladium(II) acetylacetonate molecule. (a)
Angle between the Pd1-O1-O2-O1-O2 and O1-O2-C1-C2-C3-C4-C5
mean planes, (b) the distance between two parallel O1-O2-C1-C2-C3-
C4-C5 planes and (c) distance between the most nucleophillic carbon
C3 and Pd cation of two neighbor palladium(II) acetylacetonate
molecules.................................................................................................... 81
Figure 3.4: ELF isosurfaces (G+LDA) at isovalues of (a) 0.86, (b) 0.65, (c)
0.77 and (d) 0.815. Only the C2, C3, C4, O1, O2 and Pd atoms are
shown. The Pd atom in (a) belongs to the palladium(II)
acetylacetonate molecule while the Pd atom shown in (b), (c) and (d)
is the nearest Pd atom of the neighboring palladium(II)
XVI
acetylacetonate molecule. Oxygen atoms shown in red, carbon atoms
in blue and palladium atoms in brown. The dashed-line connects
atomic cores. .............................................................................................. 85
Figure 3.5: ELF isosurfaces at isovalues of (a) 0.703, (b) 0.727 (G+LDA), (c)
0.720 and (d) 0.744 (TM+PBE). The color convention is same as in Fig.
3.4................................................................................................................. 86
Figure 4.1:(a) MD trajectory of the average distance between Pd and O
atoms of each acetylacetonate ligand of the Pd acac molecule; (b) and
(c) The ELF isosurface at an isovalue of 0.8 showing the covalent
linkage between the acetylacetonate ligands and the chrysene
molecule; (d) the decomposition of Pd acac in the presence of
chrysene molecule; (e) and (f) The ELF isosurfaces at an isovalue of
0.8 showing the modified bonding structure of the acetylacetonate
ligands after they get covalently bonded with the chrysene molecule.
Blue circles indicate C, red indicates O and brown indicates Pd....... 95
Figure 4.2:(a) MD trajectory of the average distance between Pd and the
six nearest C atoms of the two neighboring chrysene molecules; (b)
the ELF isosurface at an isovalue of 0.8 showing the bonding
interaction between Pd and a carbon atom of the chrysene molecule;
the ELF- isosurface surrounding the Pd atom of (c) an intact
Pd acac molecule and of (d) the Pd acac molecule that is
decomposed in the presence of chrysene and whose Pd atom is
bonded with the chrysene molecule; spin density isosurfaces of Pd
atom (e) in Fig. 4.2.c and (f) in Fig. 4.2.d................................................ 98
Figure 4.3:(a) The ELF isosurfaces at an isovalue of 0.8, showing the
covalent cross-linking bonding between the two neighboring
chrysene molecules due to the interaction of one of the chrysene
molecules with Pd acac; (b) the breaking of the resonance structure
of the chrysene molecule due to its interaction with Pd acac and the
XVII
new bonding structure; (c) the ELF isosurfaces at an isovalue of 0.8
showing the new bonding structure in the chrysene molecule shown
in (b). ........................................................................................................... 99
Figure 4.4:MD trajectories of (a) cross linking bonds between the chrysene
molecules and of (b) the bonds between the acetylacetonate ligands
and the chrysene molecule. ................................................................... 100
Figure 4.5:(a) The simulation cell containing chrysene and palladium (II)
acetylacetonate molecules and (b) ELF isosurfaces at an isovalue of
0.8 showing no “pre-existing” intermolecular interactions. ............. 103
Figure 4.6:Fictitious electronic kinetic energy vs. time during the CPMD
production run. ....................................................................................... 105
Figure 4.7:ELF contour plots (a) for palladium (II) acetylacetonate
decomposed in the presence of chrysene molecule and (b) for an
intact palladium (II) acetylacetonate molecule. .................................. 107
Figure 5.1:Coronene supported Pd4 cluster and the interacting H2
molecule. The collective variables for the metadynamics simulation
are also shown. Carbon atoms are shown in blue, palladium atoms in
brown and hydrogen atoms in white................................................... 127
Figure 5.2:MD trajectories of (a) the distance of two H atoms from the tip
atom of the Pd4 cluster and of (b) the distance between the two H
atoms. The inset of (a) shows the MD snapshots of two H atoms
before (as an intact H2 molecule) and after getting dissociatively
chemisorbed on the Pd cluster. ............................................................. 128
Figure 5.3:The three dimensional free energy surface reconstructed from
the metadynamics simulation of the system with fixed Pd
coordinates. S1, S2, and S3 indicate the key minima in the free energy
surface and the images displayed below the plot are the snapshots of
the system at corresponding values of the collective variables. The
color coding of the atoms is the same as that of Fig.5.1..................... 131
XVIII
Figure 5.4:The three dimensional free energy surface reconstructed from
the metadynamics simulation of a system with the Pd4 cluster
partially saturated with 3 H atoms. S1 – S6 indicate the key minima in
the free energy surface and the images displayed below the plot are
the snapshots of the system at corresponding values of the collective
variables. The color coding of the atoms is the same as that of Fig.5.1,
however, the H atoms included in the collective variables’ definitions
are coded in red color. ............................................................................ 135
Figure 5.5:The three dimensional free energy surface reconstructed from
the metadynamics simulation of a system with the Pd4 cluster
partially saturated with 3 H atoms. The coordinates of the 3 H atoms
are fixed. S1 – S5 indicate the key minima in the free energy surface
and the images displayed below the plot are the snapshots of the
system at corresponding values of the collective variables. The color
coding of the atoms is the same as that of Fig.5.4. ............................. 138
Figure 6.1:Schematic of adsorption-induced strain in microporous
adsorbents. A typical trend observed in microporous adsorbents
where the adsorbent first contracts and then expands...................... 156
Figure 6.2:A Porous adsorbent continuum consisting of; (i) the solid
matrix and (ii) the pore space (adapted from Coussy, 2004). It is also
referred to as skeleton at some places in the text. .............................. 157
Figure 6.3:An illustration of adsorption isotherms when the adsorbent
undergoes deformation, as shown in Figure 6.1, (full line) and when
the adsorbent is prevented from deformation (dashed line). ........... 163
Figure 6.4:Flowchart of the procedure for calculating the adsorption-
induced strain. The corresponding explanatory text is given in the
Appendix.................................................................................................. 169
Figure 6.5:(a) CO2 Adsorption isotherms and (b) CO2 adsorption-induced
strain data (adapted from Yakovlev et al., 2005) at 243 K ( ), 273 K
XIX
( ), and 293 K ( ). The adsorbent is microporous activated carbon
material. .................................................................................................... 171
Figure 6.6:Porosity change in the deformed adsorbent calculated using
equation (6.16) and experimental strain data10 at 243 K ( ), 273 K ( ),
and 293 K ( ). ......................................................................................... 173
Figure 6.7: diff (calculated using equation 6.18, section 2.1) as a function of
the amount of gas adsorbed; at 243 K ( ), 273 K ( ), and 293 K ( ).
.................................................................................................................... 174
Figure 6.8: diff calculated using equation (6.26) as a function of the
amount of gas adsorbed at 273 K ( ) and at 293 K ( ). The filled
symbols indicate calculated diff using equation (6.18) and
experimental adsorption isotherm (as described in section 2.1). ..... 175
Figure 6.9:Predicted CO2 adsorption-induced strain in microporous
activated carbon adsorbent at 243 K ( ), 273 K ( ), and 293 K ( ).
Filled symbols indicate the experimental data from Yakovlev et al.,
2005............................................................................................................ 177
Figure 6.10:The objective function 2~
diff
diffdiff
plotted against the
distance between the trial and experimental strain curves at 273 K (as
illustrated in the inset)............................................................................ 180
Figure A.1:Computational modeling methods at different length and time
scales. ........................................................................................................ 197
Figure A.2:An illustration of energy terms in molecular mechanics
(Adapted from Frank Jensen 7). ............................................................ 198
Figure A.3:The wavefunction of the system under the nuclear potential
and under the pseudopotential and AE pseudo pseudoZ r ........ 235
Figure A.4:An illustration of a 1-dimensional potential energy surface of a
system. ...................................................................................................... 243
XX
Figure A.5:The Velocity Verlet Molecular dynamics algorithm. .............. 245
Figure A.6:Variation of different energies during the Car-Parrinello
molecular dynamics run of bulk silicon. Adapted from Pastore and
Smargiass 75 .............................................................................................. 256
Figure A.7:An illustration showing a large energy barrier for the system to
go from a state A to the more stable state B. ....................................... 259
Figure A.8:The system initially placed in well A goes to the global
minimum in well C after filling up the energy surface. Adapted from
Laio and Gervasio 82................................................................................ 260
Figure A.9:A 2-D illustration showing the difference in the topology of the
electron localization function isosurfaces for ethane, ethylene and
ethyne........................................................................................................ 266
XXI
LIST OF TABLES
Table 3.1: Bond lengths and angles (atom labels are defined in Figure 1) of
the optimized geometry of palladium(II) acetylacetonate molecule in
the crystal lattice.a ..................................................................................... 82
Table A.1:A list of few common force fields in molecular modeling 8..... 201
1
1 INTRODUCTION AND GENERAL LITERATURE REVIEW
1.1 General Introduction Chemical engineering and material science are two different, yet
very closely related disciplines. One of the strongest links between
material science and chemical engineering is the use of an array of
materials in the construction of reactors, separators and ancillary
equipments1 in any chemical industry. However, the relation is much
deeply rooted since chemical engineers are also directly involved in the
production of the materials used by them. The four building blocks of
materials, as shown in Fig. 1.1, are properties, structure, performance and
processing. These four building blocks are inter—related and thus
combine material science research with chemical engineering at a
fundamental level, i.e., the processing and synthesis of materials. Also
from the applications perspective, the role of materials in chemical
engineering spans a wide range including catalytic materials, membranes,
bioactive materials, electrode materials, coating materials, adsorbents, to
name a few. The inter—relationship between materials and chemical
engineering has simulated extensive materials related research in chemical
engineering community in the past.2
2
Figure 1.1: Four basic building blocks of material science and research. Adapted from Allen and Thomas.3
Materials relevant to chemical engineering can be classified, based
on their applications, into two main categories; viz., structural and
functional. Structural materials are required to have excellent mechanical
and thermal properties; however, the properties of functional materials
may vary depending upon the application. For example, catalytic
materials need a large surface area, membranes require specific
permeability and electrode materials require specific electrical and
chemical properties. These structural and functional materials can also be
classified based on their chemistry such as metals, ceramics, plastics and
polymers, composites, carbons, zeolites etc. Carbon based materials are
probably the oldest of all since carbon, in the form of charcoal, is been
used from prehistoric times. The present thesis focuses on functional
carbon—based materials, particularly for catalysis and hydrogen storage
and adsorption. The following sub—sections (1.1—1.3) of this chapter
(i) give an overview of different carbon—based materials, their
chemistry and applications,
(ii) discuss the necessary details of synthesis, processing and
performance of carbon materials for catalysis and hydrogen
storage and adsorption,
Structure
PropertiesProcessing
Performance
3
(iii) critically summarize the research related to these materials,
including some of the key recent advances and
(iv) identify the areas that need further investigation.
Based on this information, section 1.4 defines the objectives of the present
thesis and section 1.5 discusses its overall scope, organization and
implemented methodology.
1.2 Carbon MaterialsCarbon is the most abundant element in the universe after
hydrogen and helium. Not only it is used as household and industrial fuel
in the form of coal but is also of great importance as coke in metallurgy, as
graphitic carbon in electrodes, as graphite in nuclear industry, as carbon
black in tyres and printing inks, as carbon fibers in aerospace and sports,
as activated carbons, nanotubes, activated carbon fibers in adsorption,
separation and catalysis, to name a few. The following subsection briefly
describes structures, properties and applications of some of these carbon
based materials that are currently being investigated.
1.2.1 Structure, properties and applicationsCarbon can exhibit different types of orbital hybridization, sp, sp2
and sp3, thus giving rise to a variety of carbonaceous structures. The most
precious form of carbon, diamond, consists of carbon with sp3 type
hybridization. It is extremely hard due to its purely covalent bonds and
highly localized electrons. A large family of carbonaceous materials,
particularly those that are currently being researched and are of interest to
this thesis, exhibits sp2 type bonding. One of the basic building blocks of
these materials, including graphitic carbon, carbon fibers, porous carbons,
carbon composites, fullerenes, is the hexagonal carbon ring. A number of
such rings connected to each other in a plane form a layer, sometimes
called as graphene sheet. The delocalized electrons in these rings impart
4
the graphene sheet a good electrical conductivity along the layer. The size
of this layer, its agglomeration, interconnectivity, geometry, stacking and
the presence of non—carbon elements vary, thus giving rise to different
types of carbon materials (with different structural and functional
properties) in the family of sp2 bonded carbons.
Graphitic carbon consists of large sized layers of hexagonal carbon
rings stacked together. The layered structure imparts strong orientation
and anisotropy. Its excellent thermal resistance, heat conductance and
chemical inertness make it suitable for use in refractories and break
linings, lubricants, batteries, carbon brushes, crucibles, etc. Fullerenes are
again a special type of carbon materials entirely made up of carbon where
the hexagonal (and pentagonal) rings form different shapes such as
hollow sphere, ellipsoid or cylinder. The spherical fullerenes are called
buckyballs and the cylindrical ones are called nanotubes. These carbon
materials, particularly nanotubes, have garnered huge attention in the
recent past due to their exceptional mechanical, thermal and electrical
(even medicinal) properties. These materials are also envisioned for usage
in armoury, space elevator, solar cells, superconductors, displays,
hydrogen storage and buckypaper.
The other types of carbon materials in the sp2 family, which are not
as perfectly structured as graphite or fullerenes, but still exhibit good
mechanical properties are the mesophase carbon fibers and composites.
Carbon rings are also the building blocks of these materials, however, the
size of the sheet they form in these types of materials is relatively smaller
and there may also be present some non—aromatic chain like carbons to
connect these rings. They are not as perfectly stacked as in graphite,
however, they show anisotropy due to preferred orientation and
positioning, owing to good structural properties. These types of materials
are currently being used in aerospace, sports, automobile and military
5
applications and they are much less expensive and easy to manufacture in
bulk. The least structured carbon materials of this family are the isotropic
carbon based materials. As mentioned before, they also consist of carbon
ring sheets, however, the size of the sheet is too small to form any
anisotropy. Hence, these materials do not possess the mechanical
properties similar to mesophase carbons. However, their functionality
arises from their porous structure and high surface area which allows the
material to adsorb gases and liquids. Hence, these materials are used
extensively for separation and purification purposes. The functionality of
these carbonaceous materials is also enhanced by the addition of an
external component. Figure 1.2 shows the different forms of carbons
described above.
6
Figure 1.2: Different forms of sp2 type carbon materials. (a) graphite, (b) buckyball, (c) Nanotube, (d) mesophase carbon and (e) isotropic carbon.
(a) (b)
(c) (d)
(e)
Pores Inside
7
1.3 Carbon Materials in Catalysis and Hydrogen StorageFunctional carbonaceous materials, comprising of isotropic sp2 type
carbons, are widely used as adsorbents for drinking water, gas, and
waste—water purification. However, these isotropic carbons are also used
in catalysis as catalysts themselves or as catalyst supports.4 Some of the
benefits of using these carbons as a catalyst support are4
Resistant to both acidic and basic media
Stable at high temperatures
Pore structure can be tailored
Chemical properties can be modified
Less expensive than conventional catalyst supports
They are also potential materials for hydrogen storage and fuel cells.5, 6
Since all the above mentioned applications are based on adsorption of
different species, the microstructure and surface chemistry of these
materials govern their functionality. The present thesis focuses on these
functional carbon materials for catalysis and hydrogen storage
applications and a brief overview of these materials is as follows:
(a) Carbon black:7 Carbon black is an amorphous form of carbon
consisting of imperfect graphitic structures arranged randomly. It contains
about 99% of carbon and rest being hydrogen, sulphur, nitrogen, oxygen
and others. Carbon black is used as a catalyst directly where it is mixed
with the reactants to form a slurry or it is also used as a support material
for catalysts. The surface area varies from 150 to 500 m2/gm and the
porous structure consists of mostly large meso and macropores
(micropore: < 2 nm, mesopore: 2-5 nm and macropore: > 5 nm). Carbon
black is usually prepared by pyrolysis of hydrocarbons.
(b) Activated Carbon:7-9 Activated carbon is also an amorphous form of
carbon. However, the microstructure of activated carbon consists of large
number of micropores and a very high surface area (~ 1000-1500 m2/gm).
8
Activated carbons are prepared from wood, nut shells, coal, petroleum
coke and pitches etc. The thermal activation of this material is a two stage
process where first it is heated in an inert atmosphere to remove volatile
matter and to enrich the carbon content and then heated in the presence of
an oxidizing gas to remove carbon and to create the porous structure;
2solid gas 2 gasC CO CO . If the interaction of activated carbon with
an external species is only physical then adsorption is dominated by the
molecular volume of the external species and the surface area, pore size,
pore volume and distribution of the activated carbons. However, when
the activated carbon is loaded with catalytically active metals like Cu, Ni,
Pd or Pt, they become chemically active. The activated carbons in this case
also act as a support material for the metals. It has to be noted that metal
loading not only enhances the chemical activity of the carbon materials
but may also alter the microstructure of carbons.
(c) Activated carbon fibers:4, 7, 10-12 Activated carbon fibers have similar
microstructural properties to those of activated carbons and can similarly
be loaded with catalytically active metals. However, they are structurally
more stable, ensuring stable physical and chemical properties than
activated carbons and, hence, are more promising.
(d) Carbon nanotubes and fullerenes:13 As mentioned before, carbon
nanotubes and fullerenes exhibit exceptional mechanical, thermal and
electrical properties and are exception to the rest of the sp2 type functional
carbon materials due to their structured form. They have excellent
adsorption properties and are envisioned as potential hydrogen storage
materials. Similar to activated carbons, they can also be decorated with
catalytically active metals to make them function as catalyst support
materials and hence are commanding a lot of research attention.
Other than fullerenes and nanotubes, activated carbons and
activated carbon fibers (often referred collectively as active carbons or just
9
as carbons henceforth in the thesis) are the two most widely employed
and researched carbon materials for catalytic and hydrogen storage
purposes and the final microstructure and chemistry of these carbon—
based materials govern their functionality. The precursor or the raw
material used for the preparation of these carbons and their methods of
preparation, thus, play a key role in altering their properties.
1.3.1 Carbon Precursors or Raw Materials Activated carbons and activated carbon fibers are prepared using a
variety of precursor materials including carbon containing species like
natural gas, benzene, volatile products of coal, petroleum pitches,
bituminous coals, wood, polyacrylonitrile, rayon, phenolic resin etc.
Depending upon the precursor, the method of preparation of the material
varies. In the case of gas phase preparation, the formation of carbons
results due to the nucleation during pyrolysis or chemical vapour
deposition of carbon precursor on an inert material or during the heat
treatment in the presence of a catalytically active metal. In liquid phase
reactions, the formation of functional carbons takes place by carbonization
at high temperatures depending upon the type of the precursor. In solid
phase carbon formation, the high temperature thermal decomposition
results in the formation of carbons.4 The following subsection describes
the methods of (i) preparation of active carbons from petroleum, pitch—
based and polymer based precursors since these precursors are the most
dominant of all types in industrial applications and in academic research
and (ii) metal doping of these active carbons.
1.3.2 Methods of preparation and metal dopingThe method of preparation of active carbons is depicted in Fig. 1.3.
The first stage, though not necessarily employed in all types of carbon
materials, is the pre-processing stage where the carbon precursor material
10
undergoes thermal processes at 300-400°C in an inert atmosphere.11 The
second step in the preparation of activated carbon fibers is the spinning
step where fibers are spun using either melt-spinning, centrifugal
spinning or jet spinning. This is followed by the stabilization step where
the spun fibers are oxidized in the presence of air. Stabilization is needed
to ensure that the spun fibers do not change their shape or structure
during the succeeding carbonization and activation process.14 The
carbonization and activation steps are common for both, active carbons
and active carbon fibers. In carbonization, the material is heat treated in an
inert atmosphere up to 1000°C. In this step, a chain of thermal reactions
take place, thereby eliminating the non—carbon species from the material
and enriching it in carbon. If carbonization is continued further up to
3000°C, smaller aromatic rings condense to form larger rings, thereby
graphitizing the carbon. During the activation step, pore structure is
created due to removal of carbon by heating it in the presence of steam or
CO2. Activation may also sometime include chemical activation where the
porous carbon is heated in the presence of oxidising agents like HNO3,
H2SO4 or H3PO4 to introduce surface oxygen groups.7
As mentioned above, in catalytic applications, active carbons are
often loaded with catalytically active metals. Broadly classifying, there are
two different ways of doping the carbons with metal. The first one is to
impregnate the carbons using an aqueous solution of metal precursors and
the other method is to mix the metal precursor in the carbon precursor
before the preparation of the carbons. The former method has been
practised traditionally,4 while, the later is relatively newer and has only
recently been practised and researched.10, 11, 15-19 The two common
impregnation methods are incipient—wetness impregnation and excess—
solution impregnation. In wetness—impregnation, the active carbon
support is wetted with the metal precursor solution drop by drop and the
11
support is then dried to remove the solvent. In the excess—solution
method, as the name suggests, a slurry of active carbon support and metal
precursor is formed and the impregnated carbon support is removed by
filtration. The impregnation and drying step is usually followed by the
reduction of the metal precursor. It has to be noted that the catalyst
performance depends on the amount of loaded metal and there exists an
optimum amount above which the performance may decline.7
The other, more recent, method of metal doping is to mix the metal
precursor with the carbon precursor (pitch and polymeric). This method
has been mostly employed to activated carbon fibers where the fibers are
spun from the precursor mixture and are then stabilized, carbonized and
activated.11 Since the metal and carbon precursors undergo these different
stages of fiber preparation together, their interactions are more complex to
investigate in this case.
Figure 1.3: Method of preparation of activated carbons and activated carbon fibres.
Carbon Precursor
Activated Carbons
Carbonization Fiber Spinning
Activated Carbon Fibers
Activation Stabilization
Pre-processing
Carbonization
Activation
12
1.3.3 Effect of metal loading on carbon supportAs mentioned above, the presence of metal on a carbon support
(active carbons) enhances the catalytic activity of the carbon support.
However, the microstructure and chemical composition of the carbon
support itself also plays an important role in governing their
functionality.20 It is known that the loading of metal, using impregnation,
alters the microstructure7, 21 and the chemical composition of the carbon
support.4, 22-24 The nature of the interaction between the support and the
metal precursor also regulates the metal dispersion on the support.4, 7
Some specific examples are as follows. Charry et al.25 showed using two
different metal precursors that the activity of the carbon supported
catalyst is different in both the cases and it was suggested that the
difference in the activity is due to the difference in the microstructure of
the support and due to the difference in the metal dispersion. Escalon et
al.21 demonstrated that Re loading on an activated carbon supported
catalyst results in the decrease in surface area and porosity. A sharp
decline was observed after a certain Re loading. Coloma et al.26 showed
that the carbon surface gets oxidised after impregnation with aqueous
solution of H2PtCl6. Sepulveda et al.23 confirmed the same and also
demonstrated different Pt dispersion for oxidised and non—oxidised
carbon support, concluding that metal dispersion depends on metal—
precursor and support interaction. Pradoburguete et al.27 also showed that
significant migration of Pt takes place on a non—functionalized carbon
support, thus causing sintering and that the resistance to sintering
increases with increased functionalization. Solar et al.28 showed that the
catalyst uptake by carbon support is determined by the support—
precursor interaction.
In the case of mixing the metal and carbon precursor before the
carbon preparation, the effect of metal loading on the microstructure and
13
on chemical composition of active carbons is believed to be more
pronounced.10, 11, 19, 29 The magnified effect of metal mixing, however, can
be used to tailor the microstructure to make it suitable for specific
applications. Some examples are as follows. Basova et al.30 suggested,
using the SEM pictures, that when the precursor pitch is mixed with
palladium, cobalt and silver precursors, the metal particles tunnel through
the carbon support during the preparation stages and alter the pore
structure of the support. It was also observed that different metal
precursors result in different surface areas and pore volumes in the carbon
support,30 thus, suggesting that metal precursor—carbon precursor
interactions play a role in governing the microstructure of the support. It
was also shown that the chemical composition of the carbon precursor
changes with the addition of metal precursors, thereby changing the
chemical composition of the carbon support.29
1.4 Structural and Functional IssuesThough the microstructure, chemical composition and hence the
functionality of the catalytically active carbon material are significantly
affected due to metal loading and the metal precursor and carbon support
(or precursor) interaction governs the dispersion of metal particles on the
carbon support, as discussed section 1.2.3, only limited efforts are directed
towards understanding the underlying reasons behind these phenomena,
particularly, the exact nature of interaction between the carbon support (or
precursor) and metal precursor. In order to leverage these studies to
control and tailor the microstructure and surface chemistry of active
carbon supported catalytic materials for their application in
hydrogenation catalysis and hydrogen adsorption, it is of utmost
importance to understand this interaction. Also it needs to be emphasized
that when the effect of metal-carbon interactions on the microstructural
changes in catalytic materials is investigated, it is extremely important to
14
correctly quantify the microstructural changes. Most often these changes
are quantified in terms of pore structure and surface area analysis using
physisorption experiments. Hence, accurate theoretical models and
mathematical treatments should be applied to the experimental
physisorption isotherms.
Controlling and tailoring the microstructure and chemical
composition of the active carbon materials is undoubtedly important and
can be achieved by developing a better understanding of its relationship
with the methods of preparation. Similarly, it is also necessary to have a
fundamental understanding of the relationship between the structural and
chemical properties and the functionality of the material (cf. Fig. 1.1). A
detailed and fundamental understanding of the functional mechanism of
active carbon materials in catalysis and hydrogen storage is also required
to be able to optimize the functionality with respect to their structural and
chemical properties. The functionality of these materials is primarily
based on adsorption phenomenon. The metal—loaded carbon catalyst
provides a platform of active sites for the reactants to get adsorbed and
then to react and form products. There is a direct relationship between the
microstructure and chemical composition of the carbon catalytic material
and its functionality due to the virtue of adsorption, since adsorption is
dependant on these properties. Several mechanistic steps occur
sequentially on the catalytic carbon material during the adsorption of
active species. They are either associated with the active species (often
referred as an adsorbate) or with the catalytic carbon material (often
referred as an adsorbent). One of the key steps related to the adsorbate is
the spillover/migration effect. Spillover is nothing but the transport of
adsorbed species from one site to the other on a catalyst material which
would not adsorb the species at the first place under the same
conditions.31 This way the adsorption site with higher adsorption
15
capability is available for any new species to get adsorbed again. The
spillover/migration phenomenon which received the most attention5, 31-45
is that of H2 since catalytic hydrogenation is one of the most important
processes in chemical industries and hydrogen spillover is also envisioned
as an important mechanistic step in enhanced hydrogen adsorption
capacities of carbon based materials.5, 33, 35, 37, 39, 42, 46 Hence, if the active
carbon supported catalytic material has to be employed for catalytic
hydrogenation and hydrogen storage purposes, a clear and definitive
understanding of the spillover/migration process is required.
Adsorption studies (on carbon materials), either mechanistically
motivated or for microstructure analysis purpose, with or without
spillover, are mostly performed assuming the adsorbent to be
mechanically inert.47 However, there exists enough experimental evidence
to suggest that adsorption of active species on carbon materials induces
mechanical deformations in the adsorbent itself48-54 and those may alter
the adsorption properties and hence the functionality of the adsorbent.54-58
Understanding the potential of adsorption induced mechanical
deformations to affect the structure and performance of carbon materials
and active carbon supported catalysts, it is significantly important to
investigate the science behind them.
To summarize, given the importance (i) of the mechanism of metal
loading on an active carbon support, (ii) of quantifying the effect of metal
loading on the microstructure and the chemical composition of the
support, (iii) of better understanding the H2 adsorption and
spillover/migration mechanism on the metal loaded active carbon
material and (iv) of the mechanical deformations in these active carbon
materials, the following sub-sections discuss some of the key experimental
and theoretical investigations in these areas and identify the specific needs
for further research.
16
1.4.1 Experimental InvestigationsThough the metal precursor active carbon support (or precursor)
interaction has been recognized as a key factor governing the
microstructure and chemical composition of the carbon supported
catalytic material, it received comparatively much less attention from the
researchers because the exact chemical composition of the carbon support
or precursor in the carbon supported catalytic material is complex and
hence extremely difficult to comprehend. It also has to be noted that the
complex interplay of different factors affecting these interactions and
limitation of experimentally accessible length and time scales also act as a
hindrance in the experimental investigations. Though few, Toebes et al.59
and Serp et al.20 have summarized the important experimental
investigations in this field. Mojet et al.60 showed, using XAFS studies, that
during the loading of palladium on carbon fiber support, there exists a
strong metal-carbon interaction, much stronger than the metal-
interaction. Simonov et al.,61 while studying the palladium carbon
interaction during the doping of graphitic carbon materials using
palladium precursor, observed that palladium reduces in the presence of
graphitic carbon and that this reduction leads to the formation of broad
sizes of palladium particles. Choi et al.,62 while depositing silver and
platinum on carbon nanotubes, also showed the reduction of metal
precursors due to the carbon support. Tamai et al.63 were the first
investigators to prepare carbon supported catalyst by mixing the noble
metal precursors and carbon precursors before the formation of active
carbons and they noticed that metal precursors undergo thermal
decomposition at lower temperatures in the presence of carbons. Edie et
al.10, 29, 30 also showed that palladium, cobalt and silver precursors, when
mixed with the active carbon precursor pitch, alter the chemical
composition of the pitch even at 500 K, thereby suggesting a chemical
17
reaction between the two species. They also suggested that in the case of
metal acetylacetonate complexes, the metal separates from the
acetylacetonate ligands after mixing with the carbon precursor and gets
attached to the carbons in the pitch. Wu et al.64 also observed that the
distance between the aromatic stacks in the carbon precursor mixed with
the metal precursor is less than that of the pure carbon precursor,
indicating metal precursor induced cross linking in the carbons. Benthem
et al.,65 using the experimental electron energy-loss spectroscopy, showed
the presence of enhanced -type bonding behaviour in carbons near the
metal particles, again suggesting cross-linking in the carbons. It can be
noted that though all the above mentioned investigations indicate a
chemical interaction between the carbon support (or a precursor) and the
metal precursor, none of them sheds light on molecular level details of this
interaction and its mechanism.
Experimental investigations of hydrogen adsorption and
spillover/migration also face the similar difficulty as that of the
investigations of metal-precursor and carbon support (or precursor)
interactions. The investigations, including some of the most recent ones,
are usually directed towards investigating the presence of adsorbed and
spilled over monoatomic or diatomic hydrogens on supported carbons.43,
66-68 These are not discussed here in detail. The very few experimental
investigations directed towards understanding the mechanism of
hydrogen adsorption and spillover/migration are mostly done by Yang et
al.39, 40, 42, 69 They demonstrated that an improved contact between the
metal particles and the carbon support enhances the spillover/migration
process. They also suggested that the spillover/migration process is not
only dependant on the splitting of hydrogen on the metal and its
transportation to the active carbon support but is also controlled by the
reception capacity of the supporting carbon material.
18
Experimental investigations in the field of mechanical deformations
in carbon materials due to the adsorption of active species were never
directed towards understanding the underlying mechanism and hence are
not discussed here.
1.4.2 Theoretical Investigations A large number of theoretical studies, at molecular level, have been
performed to investigate the interaction of individual metal atoms and
metal clusters with supporting carbons.70-76 These studies use
polyaromatic hydrocarbon molecules as a model for the carbon support.
Philpott and Kawazoe75 performed density functional theory
computations of transition metal structures sandwiched between aromatic
hydrocarbon molecules and they observed that the presence of metal
atoms on the edge of the aromatic carbons bent the aromatic molecules.
Similar work was also performed by Labéguerie et al.72, using one
dimensional palladium chains sandwiched between aromatic
hydrocarbons, and they showed a significant electron transfer from the -
system of the aromatics to the metal chains causing a strong interaction
between them. Kandalam et al.71, when investigated the geometry of
cobalt and its dimer attached to a polyaromatic hydrocarbon using density
functional theory, observed that the metal atom prefers to occupy the edge
sites of the ring instead of being at the centre of the ring. Density
functional study of a relatively bigger, 9-atom palladium, cluster with
stacked polyaromatic hydrocarbons and with carbon nanotubes was
performed by Duca et al.70 Their calculations revealed that the strong
interaction between carbon and palladium distorts the palladium cluster’s
geometry. The curvature in carbon nanotubes was believed to enhance the
metal carbon interaction. The molecular dynamics simulations of Sanz-
Navarro et al.76 using molecular mechanics methods studied the
interaction between a Pt100 cluster and carbon platelets. They found out
19
that the attachment of this Pt cluster to the carbon platelets results in the
detachment of a few Pt atoms from the cluster and the cluster geometry
gets rearranged. It was also observed that the average Pt-Pt bond length
enlarges and this is supposed to play an important role in the enhanced
catalytic activity. It has to be noted that since all these studies are directed
towards attaining molecular level details of the interactions in the final
product i.e. active carbon supported metal catalysts, they were unable to
shed any light on the molecular level details of the synthesis procedure of
these materials which are crucial in controlling the interactions,
microstructure and chemical composition of this final product.
Molecular level investigations directed towards understanding
hydrogen adsorption and its spillover/migration on carbon materials,
metal clusters and active carbon supported metal clusters are performed
by different researchers using ab initio methods.34, 36, 37, 45, 69, 77-99 Some of
the key findings are as follows:
1. Number of molecular hydrogens adsorbed on to a carbon supported
metal atom increases with lesser filled d-orbitals90 of the metal.
2. Hydrogen molecules adsorbed in the interlayer space of graphene
sheets is significantly dependant on the spacing and there exists an
optimum layer spacing77, 82 for maximum hydrogen storage.
3. Diffusion of hydrogen atoms on a Pd surface adsorbed on the MgO
surface and on bare Pd clusters is associated with a low activation
energy barrier.45, 79
4. The Ti and Pd doped carbon nanotubes dissociatively adsorb the first
H2 molecule with a negligible energy barrier and the subsequent
adsorptions are molecular with elongated H-H bonds.83, 96
5. If an H atom is chemisorbed on the carbon material (nanotubes or
graphene sheet) then its diffusion becomes energetically very difficult
since it requires breaking the C-H covalent bond.36, 37
20
Du et al.,84 Fedorov et al.,86 Cheng et al.,37 and Chen et al.,36 are the only
investigators, to the best of the author’s knowledge, to have attempted a
mechanistic study of hydrogen adsorption and spillover on a combined
metal support (carbon and non carbon) system. However, they did not
model the dynamics of the process and the computations were performed
with a preset mechanism of the adsorption and spillover processes.
Additionally, Du et al.84 and Fedorov et al.86 perform the computations on
a single metal atom and on a metal surface respectively without taking
into account the possibility of pre-existing hydrogens on the metal. Cheng
et al.37 and Chen et al.36 perform the computations with metal clusters;
however, the spillover is modelled by bringing the pre-saturated (with
hydrogen) metal cluster near the support arbitrarily so as to make the
hydrogen atoms, in between the metal cluster and support, to spillover.
This may not be realistic since the metal cluster is in contact with the
support (carbon) even before it comes in contact with hydrogen and hence
the spillover will never occur from the metal cluster surface that is in
contact with the support.
The two most significant and valuable theoretical investigations
directed towards understanding the mechanism of and predicting
adsorption induced deformations, particularly in carbon based functional
materials for catalysis and hydrogen storage (and sequestration) are: (i)
the recent work of Ravikovitch and Neimark;100 however, in this work any
simple relation to predict the adsorption induced deformation was not
established and the deformation in the adsorbent was assumed to be
entirely due to the deformation of pore space, thereby neglecting the
change in the volume of the solid active carbon matrix. (ii) Jakubov and
Mainwaring101 developed a model based on the relation between the
adsorption induced deformation and the difference in the chemical
potentials of an adsorbate, when adsorbed on a deformed and on an
21
undeformed adsorbent but they also make a similar assumption of the
deformation in pore space only. When they calculate the difference in the
chemical potentials using the difference in adsorption isotherms, on a
deformed and on an undeformed adsorbent, the magnitude of the
percentage difference in the isotherms is three orders higher than that of
the deformation.
1.4.3 Need for further research There exists a knowledge gap (i) between the synthesis procedure
of metal loaded active carbon materials and their final microstructure and
chemical composition and (ii) between the microstructure and chemical
composition of these materials and their functionality in catalysis and
hydrogen storage. The candidate believes that bridging the existing gaps
between the synthesis, the structure and the performance of these carbon
based materials is possible when a clear understanding of the individual
components is obtained. Based on the above discussed literature review of
the synthesis, structure and performance of these materials, and in the
view of limitations of experimental investigations due to their complex
compositions and time and length scales, following specific areas are
identified for further theoretical research and development:
1. Molecular level understanding of the synthesis procedure of metal
loaded active carbon supported materials.
2. Quantifying the effect of metal loading on the microstructure and
chemical composition of the carbon support.
3. An atomic level understanding of the dynamics, mechanism and
energetics of hydrogen adsorption and spillover/migration on these
materials.
4. A molecular level understanding of adsorption induced effects on
these materials and an ability to predict those.
5. Developing a toolbox for (1)-(4).
22
1.5 Motivation and Objectives The use of hydrogen as a fuel faces the biggest challenge on the
front of hydrogen storage. Insufficient progress in the development of
suitable storage and delivery system is a major barrier for commercial
applications. Storing hydrogen as compressed gas or in liquefied form
requires very high pressures or very low temperatures and inputs of
significant energy. The safety of this storage mode is also a concern.
Adsorption in the form of metal hydrides results in low weight percentage
storage and presents difficulties in getting the hydrogen desorbed. Hence,
adsorption in carbon materials still remains the most promising option
and hence needs to be explored further. It is also important to note that
there exists a wide inconsistency and non reproducibility in the
experimentally reported hydrogen storage capabilities of carbon
materials102-105 due to large variations (i) in the methods of preparation of
samples, (ii) in the raw material used to prepare these carbons and (iii) in
the measurement techniques. Despite of these discrepancies, it is now
apparently clear that carbon nanotubes, activated carbons and carbon
fibers, in their pure form, do not meet the required criterion for hydrogen
to be used for automotive applications.33 Hence the future research needs
to be concentrated on the structural and chemical modifications of these
materials based on their impact on the storage capacity and on the
fundamental adsorption mechanism. To focus these efforts in a unified
direction might be a daunting task if the history of, and scientific reasons
for, disparities in experimental research are taken into account.
Similar challenges are also faced in the development of carbon
materials as supports in hydrogenation catalysis. In spite of their higher
resistance to deactivation by deposition of coke and their better
performance in hydrodesulphurization and hydrodeoxygenation106, 107
than that of the conventional alumina support, they not fully exploited
23
commercially. In addition, experimental investigations directed towards
evaluating the performance of these materials face difficulty due to
complex, variable and unknown composition of the feed material
(petroleum based carbon precursor). The differences in raw materials and
methods of preparation and surface modifications and the complex
composition of the raw material of active carbon supported catalyst
material hinder the experimental research in the synthesis procedure of
these materials.
Theory, modeling and simulation approaches, including molecular
level modeling, though not completely free of, are much less prone to the
inconsistency and non-reproducibility issues. Appropriate simplifications
and assumptions may be needed to establish a suitable structure
representing the active carbon materials. Along with the well established
carbon structures, theoretical research in the field of functional carbon
materials can also be implemented to investigate novel materials and
structures. As mentioned above, time and length scales not accessible by
experiments can also be investigated. The fundamental knowledge
obtained using theory, modeling and simulation not only can complement
but can also serve as a valuable input to further experimental research and
investigations in the development of carbon materials. Thus, realizing the
importance and relevance of theoretical investigations in carbon materials
for catalysis and hydrogen storage, the key issues as identified in section
1.2 and 1.3 are the focus of this thesis. The unifying theme of the present
thesis is to implement appropriate theoretical methods to address issues
related to structure, performance, properties and synthesis of functional
carbon based materials. The specific objectives are as follows:
1. Investigate the different methods used to calculate and quantify the
pore structure of carbon materials and discuss the related issues.
Implement the most appropriate methods to quantitatively
24
understand the effect of metal doping on the pore structure and its
evolution in active carbon materials.
2. Identify and validate an appropriate molecular modeling
methodology to study the interaction between the metal precursor
and carbon precursor which affects the microstructure and the
chemical composition of the metal doped active carbon material.
Implement the tool and quantitatively understand the molecular
level details of this interaction using an appropriate model for the
carbon and metal precursor.
3. Identify and validate an appropriate molecular modeling
methodology to investigate hydrogen adsorption and its transfer on
metal loaded active carbon material. Implement the tool to
mechanistically study the process, its dynamics and energetics.
4. Develop a mathematical model to investigate the mechanism of and
to predict the adsorption induced deformations in active carbon
materials. Validate the model with available experimental data.
5. Demonstrate the potential and applicability of the above modeling
approaches to further advance the theoretical research in active
carbon based material, even beyond the scope of this thesis.
1.6 Thesis Scope, Methodology and Organization Given the variety of functionally active carbon materials, their
structural, processing, precursor and synthesis method differences, and
possible different functional mechanisms, it is unrealistic to provide an
insight, using theoretical tools, into the structure, synthesis and
functionality of these materials and their inter relationships which will be
universally applicable to all the carbon materials. The limited but scattered
experimental data further adds to the difficulty since the choice of
theoretical methods to be used is not only dependant on the physics of the
system and on the phenomenon under investigation but is also dependant
25
on the type of the experimental data available for validation. Hence, the
scope of the present thesis is defined based on the physics of the system
and of the phenomenon under investigation, on the computational needs
of an appropriate theoretical method and on the material for which
reliable experimental data is available. The scope and organization of this
thesis are described below to prepare the reader for more detailed
discussions in chapters 2-6. A flow-chart containing a detailed description
of this thesis is shown in Fig. 1.4.
1.6.1 Chapter 2The objective of quantitatively characterizing the effect of metal
doping on the microstructure of active carbon materials is dealt in chapter
2. Activated carbon fibers were used as the carbon material and the metal
of choice is palladium, using palladium(II) acetylacetonate as a precursor.
Petroleum pitch is used as the carbon-precursor material and metal
doping is performed by initially mixing the metal and carbon precursor
before the fiber formation. The reasons for the choice of materials and
experimental methods are as follows:
The pore structure calculations require experimental adsorption data.
The active research collaboration of author’s thesis supervisor Prof.
Rey with Prof. Emeritus Dan D. Edie at Clemson University made it
feasible for the candidate to obtain this experimental data by
conducting experiments in Dr. Edie’s research laboratory at Clemson
University.
Prof. Emeritus Edie’s current research focuses on petroleum pitch
based activated carbon fibers and hence it was possible for the author
to prepare active carbon fibers and to perform adsorption experiments,
with the help of Mr. Halil Tekinalp in Prof. Emeritus Edie’s research
group.
26
The choice of metal as palladium is motivated by its catalytic activity
and by the initial reports108 from Prof. Emeritus Edie’s lab suggesting
that palladium loaded activated carbon fibers can store an order of
magnitude more hydrogen than the fibers without palladium.
Palladium(II) acetylacetonate is one of the most widely used
precursors for palladium.
Some of the most recent and scientific pore structure analysis methods like
the non local density functional theory and the chi-theory are
implemented and are contrasted against the traditional methods like BET
and BJH. The difference in the pore structure and its evolution in
palladium containing fibers and in pure fibers is demonstrated
quantitatively and the drawbacks of the conventional adsorption analysis
methods are demonstrated in this chapter. The necessary details of these
theoretical methods and the solution techniques are also given in this
chapter.
1.6.2 Chapters 3 and 4The chosen molecular modeling method to investigate the
interaction between the metal precursor, palladium(II) acetylacetonate,
and the carbon precursor, affecting the microstructure and chemical
composition of the metal doped activated carbon fiber, is ab initio
molecular dynamics. Ab initio methods are required since chemical
reactions involving transition metal species need to be modelled and
molecular dynamics need to be incorporated since these interactions take
place at finite temperatures. The Kohn-Sham implementation of the
density functional theory is used for performing ab initio calculations.
Periodic boundary conditions are implemented to simulate the bulk
conditions and planewaves are used as Kohn-Sham orbitals for the density
functional theory calculations. Since all-electron calculations using
planewaves as orbitals become too expensive, the planewave-
27
pseudopotential implementation is used. The Carr-Parrinello scheme is
used to perform the ab initio molecular dynamics calculations. For the
complete background information of these methods and the technical
reasons behind their choice, the reader is referred to the Appendix. To be
able to perform accurate ab-initio calculations, it is of utmost importance
to have a validated and tested pseudopotential. The available
pseudopotentials for palladium were not tested in a hydrocarbon
environment prior to this work. Chapter 3 reports the results of the
validation and testing of palladium pseudopotentials by performing
palladium(II) acetylacetonate crystal structure calculations. Additionally, a
significant experimental hypothesis about the molecular structure of
palladium(II) acetylacetonate in crystal structure is also verified.
The reasons behind the choice of metal precursor as palladium(II)
acetylacetonate and carbon precursor as petroleum pitch are explained in
section 1.5.2. However, the composition of petroleum pitch is extremely
complex and hence it is required to mimic this carbon precursor with a
model hydrocarbon compound for molecular level simulations. Edie et
al.,29 using gas chromatography-mass spectrometry, identified chrysene as
one of the five major components of the pitch and they also demonstrated
that palladium(II) acetylacetonate when mixed with the pitch for the
preparation of the fiber, chemically reacts with chrysene. Polyaromatic
hydrocarbons like chrysene are also very often used in molecular level
investigations to mimic the hexagonal ring containing carbon structures.
Because of these two reasons, chrysene is used as a model compound to
study the interaction of metal precursor with carbons. The results of metal
precursor-carbon precursor interactions are reported in chapter 4 and the
numerical aspects and simulation details of the Carr-Parrinello molecular
dynamics simulations of palladium (II) acetylacetonate and chrysene are
also discussed. The simulation results not only validate the long standing
28
experimental hypothesis of chemical interaction between the metal
precursor and carbon but also reveal the atomic level details of the
interactions. The scope of the present thesis is limited to the initial
interactions between the two species and the proposed methodology and
simulation scheme can be used in the future to investigate each and every
individual step of the synthesis procedure (as discussed in section 1.2.2).
1.6.3 Chapter 5The dynamics of hydrogen adsorption on palladium loaded active
carbon support are reported in chapter 5. Coronene is used as a model
polyaromatic hydrocarbon for the carbon support because it is a relatively
large molecule with compact structure and has a central carbon ring, very
similar to graphene or nanotubes, to support the metal cluster. The choice
of the size of metal cluster is governed by computational and theoretical
constraints and is discussed in detail in chapter 5. The dynamics are
investigated using Carr-Parrinello molecular dynamics. The dynamics,
when required, are accelerated and the energetics are calculated using the
metadynamics technique and the required background and mathematical
details of metadynamics are described in the appendix. Though
experimental investigations have shown the existence of palladium
hydride,66, 67 computational constraints prevented to extend the scope of
this work to incorporate the effect of palladium hydride. The details are
again discussed in chapter 5. The simulation results reveal, for the first
time, the atomic level dynamics of hydrogen adsorption and
spillover/migration on a carbon supported palladium cluster.
1.6.4 Chapter 6A novel multiscale theoretical tool to investigate and predict the
adsorption induced deformation mechanism in carbon materials is
developed. The scope of this work is limited to physical adsorption
29
because there is no experimental data available to validate and
parameterize the model for dissociative adsorption. However, it is shown
that the model can be extended to take dissociative adsorption into
account, provided the experimental input is available. This is a
continuum mesoscopic level model (instead of molecular) and hence can
be implemented to study deformations in different types of carbon
materials. The carbon material studied in this work is activated carbon
and is selected due to the availability of the experimental data. The
modeling effort suggests a possible mechanism for the deformation and is
shown to successfully predict the deformations after validation. All the
necessary details of the model are also discussed in this chapter.
1.6.5 Chapter 7Chapter 7 reports the main conclusions of the present thesis and
summarizes the main accomplishments and contributions to knowledge.
Recommendations for future work are also suggested.
1.6.6 AppendixThe background and specific details of molecular modeling
techniques, their relevance to the research performed in this thesis and
their literature review are provided in the Appendix. Emphasis is given on
novel ab-initio, molecular dynamics, accelerated molecular dynamics and
energy reconstruction techniques used in the present thesis. Since atomic
level and ab-initio (electronic structure) modeling (i) is an integral part of
this thesis and (ii) is rarely practised in chemical engineering, a detailed
overview and necessary background of the electronic structure and
atomistic level modeling methods, including those that are used in this
thesis, are also provided in this section.
30
Figure 1.4: Thesis summary and organization.
31
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43
2 EFFECT OF METAL SALT ON THE PORE STRUCTURE EVOLUTION OF PITCH-BASED ACTIVATED CARBON FIBERS
2.1 Summary
The effect of palladium(II) acetylacetonate on the pore structure evolution
of isotropic petroleum pitch-based activated carbon fibers (ACFs) is
characterized by comparing the pore structure evolution of ACFs,
prepared from pure pitch and from palladium(II) acetylacetonate
containing pitch. The pore structure was interpreted by applying chi-
theory, BET, BJH, t-plots, adsorption potential distribution (APD) and
non-local density functional theory (NLDFT) to experimental N2
adsorption isotherms. Pore size and pore volume calculations from chi-
theory are in agreement with those from APD and NLDFT, respectively;
whereas, those from BET, BJH and t-plot are not. However, chi-theory
underestimates the total surface area. The validated porosity and surface
area results, pore size distribution and APD were then studied as a
function of burn-off value. The pore structure evolution analysis of both
types of ACFs showed that addition of palladium(II) acetylacetonate to the
pitch, prior to fiber formation, causes (i) formation of macropores, (ii)
small increase in microporosity during early stages of activation and (iii)
increased mesoporosity at burn-off values greater than 60%. The
presented data and analysis provide a new understanding on the porous
44
structure of novel pitch-based activated carbon adsorbents and potential
hydrogen storage materials.
2.2 Introduction and literature survey
Activated carbon fibers (ACFs) are highly porous carbon materials
that have excellent adsorption properties for a wide range of substances.
ACFs obtained from different precursor materials like PAN, rayon,
phenolic resin and petroleum pitch are used in separation and purification
applications.1-5 Their ability to be drawn into sheet, cloth and felt gives
them an advantage over activated carbon powder, particularly for high
volume applications. Different methods are employed to modify the pore
structure of an ACF to make it suitable for a particular application;
addition of an organometallic salt to the fiber precursor6-10 is one of these
methods and is the topic of the present paper. A comparative study of the
effect of addition of different metal salts and their mixtures, prior to fiber
formation, on the pore structure of petroleum pitch based ACFs has been
performed6, 7, 11 using Brunauer-Emmett-Teller (BET) surface area analysis,
Horvath-Kawazoe (H-K) and Barrett-Joyner-Halenda (BJH) pore volumes
and scanning electron microscopy (SEM) analysis. It was found that the
porosity of the ACFs is directly affected by the composition of metal
salt(s).11 Based on: (1) the GC-MS analysis of pure pitch and metal salt
containing pitch, (2) the large body of SEM observations of as-spun fibers
made from Ag-, Pd- and Co-salt containing pitches, and (3) XRD patterns
of the fibers, it was concluded12 that metal salts react with the
hydrocarbons in the pitch and the spun fiber contains pure metal particles.
In case of fibers prepared from palladium acetylacetonate containing
pitch, size distribution of palladium particles, based on a large number of
SEM observations, has revealed that 65-85% of the particles are less than
200 Å, 10-20% are in the range of 200-500 Å and the remainder are larger
45
than 500 Å.11 These observations also show the tunneling of large Pd
particles within the fiber core during the activation process. Thus, the
presence of Pd particles could lead to a broad pore size distribution if the
random tunneling of metal particles of all sizes (5 Å – 1000 Å) in the fiber
core happens during activation. Recent hydrogen adsorption
experiments13 on the ACFs prepared from isotropic petroleum pitch
mixed with palladium(II) acetylacetonate salt have shown that the fibers
prepared from palladium salt containing pitch exhibit an order of
magnitude higher hydrogen adsorption capacity than the ACFs prepared
from pure pitch. However, hydrogen uptake was considerably less after
repetitive cycling.13 Pd-hydride formation was not the only reason for
high hydrogen adsorption capacity13 and hence the dissociation of
hydrogen molecule into individual atoms and subsequent adsorption into
the surrounding carbon matrix is suspected.14, 15
All the applications of ACFs, including hydrogen storage, are
directly affected by their porous structure and since the porous structure
is created during the activation stage of fiber preparation, an
unambiguous understanding of the effect of addition of an organometallic
salt on the pore structure evolution during the activation process is
crucial. Knowledge of pore structure evolution and organometallic salt-
precursor chemistry will eventually lead to a methodology to control the
pore structure of the ACFs and to optimize their applications.
Different methods are used to quantify the pore structure in terms
of physical quantities such as surface areas, pore sizes, pore volumes and
pore size or adsorption energy distributions.6, 8-10, 16-18 These methods
include adsorption (physisorption) isotherm analysis, mercury
porosimetry, X-ray analysis and NMR, to name a few.19, 20 Adsorption
isotherm is perhaps the most popular of all these methods. In case of
physisorption analysis, the physical quantities are estimated by applying
46
theoretical models and mathematical treatment to the experimental signal,
i.e. the adsorption isotherm.16-18 Not all adsorption models are capable of
calculating the physical quantities independently due to limitations in
their underlying theory. Different adsorption models may estimate
different values for the same physical quantity. Adsorption isotherm
analysis using any one method may neither be complete nor be correct.18
Some of the isotherm analysis methods are also restricted to specific types
of porous materials. Following observations support these statements:
The BJH method21 calculates pore volume and pore radius in mesopore
ranges (20-50 Å) but its applicability is questionable for micropores (<
20 Å).
Classical theories like (HK)22, Dubinin-Radushkievich (DR)23 and t-
plot24, 25 can be used in the micropore range, using low pressure
adsorption data, but they are not applicable in the mesopore range.
Quantities that are obtainable from HK, BJH, DR methods are pore
radius and pore volume. Pore radius calculation is however dependent
on BET.
The only methods that can independently calculate surface area are
BET and chi-theory.
Density functional theory (DFT)26 is by far the most scientific methods
of all and has almost become a standard for calculating pore size
distribution but it also needs specific interaction parameters that
depend on the adsorbate and adsorbent and surface area and pore size
calculations using DFT needs BET.
Chi-theory27-33 has the theoretical basis for calculating pore volume,
pore size and surface area but has not been quantitatively validated
with real life, heterogeneous, micro- and mesoporous adsorbents.
Hence, to extract reasonable conclusions from the interpreted
results of an experimental isotherm, limitations and appropriateness of
47
each of the methods need to be understood and a reasonable agreement
from different methods needs to be obtained. This work focuses on (i)
identifying and validating appropriate adsorption isotherm analysis
methods for adsorbents like ACFs that could exhibit wide range of
porosity and on (ii) quantitative evaluation of the effect of palladium
acetylacetonate addition on the evolution of pore structure of ACFs, using
these methods. Though the petroleum pitch and ACF preparation method
used in this work are exactly similar to those used by Basova et al.,11, 12 this
work focuses on the pore structure evolution of ACFs prepared from
palladium(II) acetylacetonate containing pitch and performs a detailed
adsorption isotherm analysis using non-local density functional theory
(NL-DFT), adsorption potential distribution (APD) and chi-theory, along
with the aforementioned traditional methods.
The organization of this paper is as follows. Section 2.3 describes
the experimental fiber preparation method and adsorption isotherm
measurements. Since the theoretical background behind adsorption
analysis methods like BET, BJH, t-plots, and NLDFT can be found in the
literature16-18, 25, 34-38, only the theoretical background and details of chi-
theory and APD, respectively, are described in section 2.4. We present the
results of the adsorption analysis methods in section 2.5 and we analyze
those results to extract information on the effect of palladium(II)
acetylacetonate on the evolution of pore structure of ACFs. We conclude
our findings in section 2.6.
2.3 Experimental
The ACFs used in this work are prepared from the isotropic
petroleum pitch supplied by Chungnam National University in Daejeon,
Korea. Palladium(II) acetylacetonate is supplied by Alfa Aesar Company,
USA. Ground palladium salt particles of sizes less than 38 m are mixed
48
with isotropic pitch so as to have 1 wt% of palladium in the melt spun
fibers. Pure and metal-salt containing pitch is then melt spun into fiber
form and the fibers are then stabilized by heating in air at 0.5 °C/min. to
265 °C and keeping them at that temperature for 13-15 hrs. The stabilized
fibers are then carbonized by placing them in an inert atmosphere at 1000
°C for 1 hr. The final activation process takes place in the presence of CO2
at 900 °C. Longer time periods give higher burn-off values i.e. greater
extents of activation and vice versa. Activation in the presence of CO2
creates a porous structure in the fiber through the reaction:
COCOC 22 (2.1)
More detailed procedure of the fiber preparation can be found in the
paper by Lee et al.39
ACFs were degassed for 12 hrs. and N2 adsorption experiments
were carried out at 77.35 K using Micromeritics ASAP 2020.
2.4 Adsorption Isotherm Analysis Methods
2.4.1 Traditional methods of isotherm analysis
Total surface area was calculated using BET. Micropore and
mesopore volumes were calculated using t-plot and BJH method,
respectively, and the total pore volume was calculated using single point
adsorption volume at relative pressure of 0.975. The pore sizes were
calculated using BET and BJH methods.
2.4.2 Chi-theory
The chi-plot representation of the adsorption isotherm29 is a way to
develop an analytical expression for the standard plot.31-33 This method
does not use any different standard curve or an empirical correlation for
porosity calculations. Given the theoretical basis of the chi theory, it is a
promising new tool for surface area and porosity analysis of energetically
49
heterogeneous micro- and mesoporous adsorbents. This section presents
a brief but sufficient discussion of the chi-theory and its underlying
principles from the surface area and porosity calculations perspective;
further specific mathematical details, detailed description of the physical
significance of all the terms, and underlying quantum mechanical
considerations can be found in the literature.33 According to the chi-theory
the adsorption isotherm is29:
Um
sad fA
An (2.2)
where the definition and physical significance of the symbols are:
U : Unit Step or Heaviside function,
adn : Amount of gas adsorbed,
ms AA , : Total and molar surface area respectively,
f : Correction factor in chi-theory to account for difference between hard
sphere model and Lennard-Jones 6-12 potential (f = 1.84),
c ,
= 0lnln PP ,
kTEa
c ln , Ea: adsorption energy of the first molecule to be adsorbed.
Equation (2.2) considers the adsorbent to be homogeneous, since it
has a single value for Ea. For adsorbents that are energetically
heterogeneous, with patches of different adsorption energies, equation
(2.2) was modified to the following discrete form:
50
iii
m
isad fA
An U, (2.3)
Differentiating equation (2.3) twice gives
iic
m
isad
fAAn
,,
2
2
(2.4)
where the delta functions, ci , arise from the discreetness of the
distribution. The above expression (2.4) gives the distribution of
adsorption energies as a sum of (delta) functions. Considering the
complex composition of the precursor pitch and the presence of
palladium, the ACFs are expected to have a chemically heterogeneous
surface. It should be noted here that chi-plots of N2 adsorption on the
ACFs (Fig. 2.1) do not show distinct straight line regions (indicating
discrete values of adsorption energies), but a smooth transition is
observed. Therefore, it is more relevant to have a distribution of Ea and
hence equation (2.4) must be modified accordingly. This feature was
accounted by taking a normal distribution of adsorption energy, instead of
sum of functions31-33:
2
2
2
2
2exp
2 c
c
cm
sad
fAAn (2.5)
where nad+ is the amount of material adsorbed and will continue to be
adsorbed if there is no restriction on the pore size, c corresponds to the
mean value of adsorption energy and c is the deviation parameter in the
51
distribution function. Integrating equation (2.5) twice gives the total
amount of gas that would have been adsorbed under no pore size
restrictions. To include pore size restriction effects, the chi-theory
proceeds as follows. The value of at which the standard curve for
adsorption is terminated ( p ) is a measure of pore size. Hence a normal
distribution of p (needed to represent pore size distribution) is taken
into account:
22
2
22
2
2exp
2p
m
punad
V
Vn (2.6)
where nunad is the amount of unadsorbed gas, p is the mean value of
where the standard curve would be terminated and 2 is the combined
energy and pore size deviation parameter. Integrating twice equation (2.5)
minus equation (2.6) will give an analytical expression for the adsorption
isotherm that takes into account the energy and pore size distribution:
yp
m
p
c
c
cm
sad dx
x
V
VxfA
Adyn 2
2
2
22
2
2exp
22exp
2
(2.7)
The solution of this equation can be expressed in the following functional
format:
2,,,, pm
pcc
m
sad V
VfAA
n ZZ (2.8)
52
where the function Z is defined later in equation (2.12). If the parameters
of equation (2.8), 2and,,,,, pm
pcc
m
s
VV
fAA , are fitted using the
experimental adsorption isotherm one would obtain the surface area and
microporosity information of the adsorbent. Technical details can be
found in pages 176-182 of the reference33 while the physical significance of
the model parameters has been defined below equation (2.2).
Mesoporous adsorbents will have a sharp increase in adsorption
isotherm due to capillary condensation and the microporous analysis will
not be appropriate. Also it would be inappropriate to use either micro- or
mesopore analysis for samples that have micro- and mesoporosity.
Furthermore, to take into account mesoporosity and capillary
condensation, the microporosity and mesoporosity analysis were
combined into one formulation.33 The combined micro- and mesopore
equation for a single energy of adsorption and a single pore size is
pm
ppp
m
sad V
Vp
fAAn UUU (2.9)
where p is fractional amount that is in the pores and (1-p) is the fraction
that is adsorbed on the external surface. If normal distributions for
adsorption energy and pore size are assumed then the corresponding
integral equation for the adsorption isotherm (equivalent to equation (2.7)
for the micropore analysis) is:
53
yp
m
s
c
c
cm
sad dx
x
fApAx
fAA
dyn 22
2
22
2
2exp
22exp
2
dxx
VV p
m
p22
2
2 2exp
21 (2.10)
The solution of equation (2.10) will give the combined micro- and
mesoporous equation [equation (2.11)], analogous to equation (2.9), but
with distributions of adsorption energy and pore size:
2
-erf1
2,,,,
2
p2
JHGn pccad ZZ (2.11)
where the definition of the symbols is:
2erf1
22exp
2,, 2
2
zyxyx
zyxzzyxZ (2.12)
G = fAA ms (2.13)
H = pG (2.14)
J = mp VV (2.15)
Parameters G, H and J are related to the surface area, micropore volume
and mesopore volume respectively, as can be seen in equations 2.13, 2.14
and 2.15. The equations used to calculate the porosity information from
these parameters are as follows:
ms GfAA (2.16)
JHVV pmp (2.17)
54
teRT
Vr pmgl
p
2 (2.18)
mexternal fAHGA (2.19)
For further details on equations (2.16-2.19), the reader is referred to
the reference.33 With an embedded distribution of adsorption energy and
of pore size and with the inclusion of mesoporosity analysis, the chi-
theory is a promising tool for porosity analysis of heterogeneous
adsorbents like ACFs, where a clear demarcation of the adsorbent as
microporous or mesoporous can not be made. The parameters in Equation
(2.11) ( JHG pcc ,,,,,,, 2 ) are fitted for the experimental
adsorption data. Parameter fitting was done by minimizing the square of
error in the calculated and experimental adsorption isotherm (expressed
as mmoles of N2 adsorbed/gm of adsorbent). When the relative change in
the sum of squares of error was less than 10-6 for consecutive five
iterations, the minimization was considered complete. A graphical
representation of the experimental and chi-theory predicted isotherm is
used to get the initial guesses right.
2.4.3 Adsorption Potential Distribution
Adsorption in pores occurs gradually and as the pressure increases,
adsorption takes place at lower adsorption potential sites.40-47
Experimental adsorption data is fitted using cubic splines and the
adsorption potential distribution (APD) is calculated by taking the
55
numerical derivative of the adsorption isotherms with respect to the
adsorption potential:42
0ln PPRTGA (2.20)
dAdn
APD ad (2.21)
The magnitude of APD expresses the structural and energetic
heterogeneity of the adsorbent.43-45, 47, 48 Kruk et al.49 simulated adsorption
isotherms for homogeneous model (single and multiple) pore sizes using
non-local density functional theory and when they plotted the APD for
those isotherms, it was observed that the lowest APD peak was dictated
by the pore size. They observed that the position of the peak moved to a
lower potential with an increase in the pore size. Similar results were also
observed recently by Calleja et al.,50 where the model adsorption
isotherms for different pore sizes were generated using grand canonical
Monte-Carlo (GCMC) simulations. For a pore size range of 12-22 Å, the
energy distribution from GCMC matched very well with the APD and the
shift in the lowest adsorption potential peak (that is governed by the pore
size) was also present. Kruk et al.49 also applied this method to real life
carbonaceous adsorbents and observed that although the surface
heterogeneity present in those adsorbents widened the adsorption
potential distribution peaks, their position on the adsorption potential axis
was in agreement with that from the simulated isotherms of homogeneous
pores.
Given the model independent nature of APD and its justified
validity,49, 50 it is used in this work to support the conclusions drawn from
the chi-theory and from the pore size distribution results.
56
2.4.4 Density functional theory
DFT is a statistical mechanic technique in which the density of the
adsorbate molecules is adjusted as a function of distance from the
adsorbent surface, to minimize the free energy.26, 35 The difference between
local and non-local DFT is that the former does not take into account the
short range correlations and assumes the fluid to be uniform whereas, the
latter takes into account those short range interactions and can thus
predict the variations in density near the adsorbent surface accurately.
Mathematical and physical details can be found in the literature.25, 34-38
The pore size distribution was calculated using non-local DFT module of
the Autosorb software supplied by Quantachrome instruments.
2.5 Results and Discussion
2.5.1 Adsorption Isotherm Analysis
Figure 2.1 shows the chi-plot representation of the isotherms for
ACFs prepared from pure pitch and from Pd-salt-containing pitch. All the
adsorption isotherms are similar to Type-I51, and therefore reveal the
presence of micropores. Though there is no sharp rise in adsorption
isotherm like Type IV/V51, the presence of mesopores can not be
completely denied based on a visual analysis of the adsorption isotherm.
Adsorption isotherms for ACFs prepared from pure pitch show negligible
adsorption after the relative pressure of 0.4 ( 0.08) but for the ACFs
prepared from Pd-salt-containing pitch, a slight increase is observed. SEM
pictures of as-spun fibres have shown11, 12 that Pd particles as large as 500
to 1000 Å exist and their migration during the activation process could
cause large macropores and thus multilayer adsorption.
57
Figure 2.1: Chi-theory representation of adsorption isotherms. – ACFs from pure pitch and × – ACFs from palladium acetylacetonate containing pitch at different burn-off values. The straight line indicates the chi-theory predicted isotherm and the experimental isotherm is shown using symbols.
Pore size distribution (PSD) curves for pure pitch and Pd-
containing ACFs are shown in Fig. 2.2 and 2.3, respectively. For burn-off
values of 34% and 55%, micro and mesopores are formed simultaneously
but with further increase in activation it can be seen that an additional
peak in PSD at ~15 Å arises. Also it can be noticed that there is a
significant rise in the volume fraction of mesopores. The PSD curve for
80% activated ACFs is lower than that of 55% activated ACFs, in the
region of 4-10 Å and it indicates the widening of smaller micropores.
Similar trends in the evolution of pore structure are observed for ACFs
prepared from Pd-salt-containing pitch (Fig. 2.3 –20%, 45% and 65% burn-
off) except that the additional peak at 15 Å starts appearing from 20%
activation and it becomes significant at 45% activation. With further
increase in activation it can be observed that for Pd-containing ACFs the
formation of larger pores becomes more significant and small peaks in the
mesopore region, that start appearing at 65% activation, become dominant
with increase in activation. The PSD curve for 85% activation is drastically
different from the rest of the curves since micropores less than 10 Å have
58
widened significantly and mesopores are now dominant. Also the PSD
curve shows a slight increase in the macropore region, which is due to the
tunneling of large Pd particles, as mentioned above. It can be observed
that the presence of metal slightly enhances the formation of small
micropores but it also adds to the formation of larger pores. Higher
activation gives higher porosity but due to the presence of Pd, the
formation of meso and macropores is catalyzed and the desirable increase
in microporosity is not achieved.
Figure 2.2: Pore size distribution for ACFs prepared from pure pitch, at burn-off values of 34% ( ), 55% ( ) and 80% (···).
59
Figure 2.3: Pore size distribution for ACFs prepared from Pd-containing pitch, at burn-off values of 20% ( ), 45% ( ), 65% (···) and 85% ( ·).
Figures 2.4 and 2.5 show adsorption potential distributions for
ACFs prepared from pure pitch and from Pd-salt-containing pitch,
respectively. Due to the unavailability of extremely low pressure
adsorption data, an adsorption potential distribution peak for monolayer
formation is not seen and the observable peaks in the APD plot are a sign
of the pore filling process. The APD plot for pure pitch ACFs with a burn-
off value of 34% has a prominent peak at ~2400 J/mole and a small crest at
~1675 J/mole. ACFs with a burn-off value of 55 % have peaks at similar
locations but now the lower adsorption potential peak is the prominent
one. ACFs with a burn-off value of 80% (Fig. 2.4) have only one peak at
~1050 J/mole and the peak at the higher adsorption potential changes to a
shoulder. Based on the adsorption potential distributions of simulated
isotherms of adsorbents with multiple pore sizes,49, 50 the location of APD
peaks for pore sizes of 20 Å and 50 Å are ~3000 J/mole and ~1000 J/mole
respectively. The shift of the prominent adsorption potential peak from
2500 J/mole to 1675 J/mole indicates the dominance of the pore widening
process over the formation of new pores. At the burn-off value of 80%,
60
there exists only one adsorption potential peak that is shifted to ~1000
J/mole. Complete absence of any peak at a higher adsorption potential
indicates that new pores are not formed and that only widening of existent
pores is taking place.
For ACFs prepared from Pd-salt-containing pitch (Fig. 2.5), apart
from having two peaks similar to that of ACFs from pure pitch, there is
also a steep rise in the APD at an extremely low adsorption potential and
this indicates the occurrence of multilayer adsorption on macropores at
higher relative pressures (> 0.4). For Pd-containing ACFs, abscissa
locations of the peaks remain unchanged and the ordinate value increases.
At 65% burn-off, though the peak at lower adsorption potential is the only
peak where as the crest at higher adsorption potential turns to a shoulder,
the peaks do not shift on the adsorption potential axis. The position of the
prominent peak on adsorption potential axis is unchanged for Pd-
containing ACF with burn-off value of 85% when compared to that of the
pure pitch ACF, with the burn-of value of 80%.
Figure 2.4: Adsorption potential distribution for ACFs prepared from pure pitch, at burn-off values of 34% ( ), 55% (···) and 80% ( ).
61
Figure 2.5: Adsorption potential distribution for ACFs prepared from Pd-containing pitch, at burn-off values of 20% ( ), 45% (···), 65% ( ·)and 85% ( ).
Figure 2.6 shows the pore size calculation results of chi-theory
compared to those from BJH and BET. Estimated pore sizes from the APD
plots point out an increase in pore size, from 20 Å to 50 Å, with increase in
activation and the chi-theory results are in quantitative agreement with
APD. The DFT pore size distribution also clearly suggests the increase in
pore size with increase in activation as was observed with both, the chi-
theory and APD. However, BET and BJH pore size calculation fail to
capture the trend. Figure 2.7 shows the pore volumes calculated using
chi-theory, DFT and directly from adsorption isotherm at relative pressure
~ 0.98. Pore volume results from all the methods show the same trend.
Though the trend in pore volume in obvious, a quantitative agreement of
chi-theory with other methods demonstrates its ability to correctly predict
this physical quantity. Figure 2.8 shows the surface area calculated using
the chi-theory and BET; the chi-theory greatly underestimates the surface
area. The difference in the BET surface area of ACFs from pure pitch and
from palladium(II) acetylacetonate containing pitch does not differ
significantly and hence it is difficult to comment on the difference in
porosity, based on BET surface area results. The external surface area
62
values also appear to be underestimated but the trend in external surface
area calculations captures the formation of large macropores in Pd-
containing fibres and it also demonstrates that at high activation, it
increases significantly. This is also in agreement with the APD plots that
showed steep rise at an extremely low adsorption potential, indicating
multilayer surface adsorption in large macropores. We expect the chi-
theory surface area results to improve if low pressure adsorption isotherm
data ( < -2) is available. Figure 2.9 shows the micropore and mesopore
volumes using NLDFT, BJH and t-plot method. It can be clearly seen that
t-plot micropore volumes fail to provide a correct estimate at higher
activation, when mesoporosity increases. Together, BJH and t-plot pore
volume analysis, give an impression that past 60% burn-off, mesoporosity
increases at the cost of microporosity. This contradicts the NLDFT results
that demonstrate that both micro and mesoporosity increase with increase
in activation but mesoporosity development dominates at later activation
stages.
Figure 2.6: Pore size calculations using BET ( ), BJH ( ) and chi-theory ( ). Solid lines represent ACFs prepared from pure pitch and dotted lines represent ACFs prepared from palladium acetylacetonate containing pitch.
63
Figure 2.7: Total pore volumes calculated using adsorption isotherm ( ), NLDFT ( ) and chi-theory ( ). Solid lines represent ACFs prepared from pure pitch and dotted lines represent ACFs prepared from palladium(II) acetylacetonate containing pitch.
Figure 2.8: BET ( ) and chi-theory ( ) total surface area and chi-theory external surface area ( ). Solid lines represent ACFs prepared from pure pitch and dotted lines represent ACFs prepared from palladium(II) acetylacetonate containing pitch.
64
Figure 2.9: Micropore [t-plot ( ) and NLDFT ( )] and mesopore volumes [BJH ( ) and NLDFT ( )] as a function of activation. Solid lines represent ACFs prepared from pure pitch and dotted lines represent ACFs prepared from palladium(II) acetylacetonate containing pitch.
In partial summary, adsorption isotherm analysis based on pore
size and micro and/or mesopore volume information may give
misleading information about the porosity unless validated using other
techniques like chi-theory, APD and DFT. A significant effort was made to
get an agreement between different analysis methods (i.e. chi-theory,
NLDFT, APD and BET) which resulted in reliable estimates of the physical
quantities that will lead to a correct interpretation of the effect of
palladium(II) acetylacetonate on porosity evolution.
2.5.2 Effect of Pd on pore structure evolution
The ACFs studied in this work are complex adsorbents due to the
following reasons: (i) presence of a wide range of pore size, (ii) presence of
a wide range of palladium particles that could alter the fiber structure and
(iii) the chemical heterogeneity of the ACFs. A large increase in the
external surface area of the palladium containing ACFs over that of the
ACFs prepared from pure pitch supports the SEM based suggestion11, 12
65
that the movement of large metal particles during the process of activation
creates large macropores. Analysis of pore size distribution, adsorption
potential distribution, pore size, pore volume and surface area results at
different burn-off values demonstrates the evolution of pore structure in
the ACFs during the process of activation. Formation of smaller
micropores, formation of larger micropores and mesopores and
conversion of smaller micro- and mesopores into larger pores are the three
different phenomena happening during the process of activation. In the
initial stage of activation, formation of smaller pores dominates and then
with a further increase in the activation time, the formation of larger
micro- and mesopores increases remarkably. Further increase in burn-off
causes the smaller pores to widen. This phenomenon is not significant at
lower burn-off values.
Pd containing ACFs show slight increase in the microporosity at
lower burn-offs but with higher burn-offs, Pd enhances the formation of
large micro-and mesopores. The difference in the micropore and total pore
volume of the ACFs, prepared from pure pitch and from Pd containing
pitch, at lower burn-offs suggests that Pd particles of sizes less than 20 Å
are few in number, whereas the large sized particles tunnel through the
fiber to create large macropores. Electron microscopy studies of these
fibers, before activation, show that finely dispersed Pd particles (30-50 Å)
remain in the fiber and they do not agglomerate till the carbonization
temperature is below 800 °C.52 The in situ XRD studies also showed that
for the carbonization (1000°C) and activation (900°C and CO2) conditions
that are used for ACFs in this work, sintering of Pd particles occurred and
it increased with increase in activation.52 The chi-theory results and APD
calculations, which reveal the presence of large macropores even at 20%
burn-off, suggest the agglomeration of 30-50 Å Pd particles, to form larger
particles, during the carbonization process. This agglomeration continues
66
with increase in activation. At higher burn-off value, the activation reaches
to the bulk of the fiber and hence increases the agglomeration and
eventually formation of more macropores. In addition, less than 40% of
the carbon is left at burn-offs greater than 60% and Pd particles less than
20 Å become relatively more significant. The enhancement in the
formation of larger micropores and mesopores, in Pd containing ACFs,
could be due to the agglomeration of these particles.
2.6 Conclusions
N2 adsorption isotherms on the ACFs prepared from isotropic
petroleum pitch, with and without palladium(II) acetylacetonate, were
analyzed using BET, BJH, NLDFT, APD and chi-theory. Pore volume
calculations using chi-theory and NLDFT and pore size calculations using
chi-theory and APD are in good quantitative agreement. Surface area is
greatly underestimated by chi-theory but the trend in external surface area
is in agreement with the interpretation from APD and with the previous
literature.12, 52 Interpretation of pore structure using only the traditional
BJH, BET and t-plot methods could lead to misleading information.
Collective analysis of adsorption isotherms using the above mentioned
methods lead us to following conclusions about the effect of addition of
palladium acetylacetonate on the pore structure evolution of ACFs:
1. Pd containing ACFs exhibit formation of large macropores that are not
observed in ACFs prepared from pure pitch and increased activation
further catalyzes this effect.
2. Slight increase in the microporosity is observed for Pd containing
ACFs.
3. Pd containing ACFs show significant increase in the formation of large
micropores and mesopores with increased activation (burn-off value >
60%).
67
4. Very high activation of ACFs (> 50-60 % born-off value) leads to the
widening of existing small micropores and this effect gets catalyzed in
the Pd containing fibers.
An explanation, based on the adsorption isotherm analysis and on
the experimental evidence of presence of Pd particles and their tunneling,
was provided for the observed difference in the pore structure. We believe
that further investigation into the metal salt carbon precursor chemistry at
different stages of fiber preparation, when combined with the pore
structure evolution analysis in this work, can lead to a way to control the
pore structure of these ACFs. These results provide new microstructural
evolution information that is necessary for the use and optimization of
pitch-based ACFs for adsorption and separation applications. In addition,
the potential use of these ACFs in hydrogen storage application requires a
fundamental understanding and characterization of the underlying
microporosity.
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42. Jaroniec, M.; Choma, J., Characterization of Geometrical and
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adsorption potential distribution and pore volume distribution for
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Energetic and Structural Heterogeneities of Activated Carbons. Langmuir
1988, 4, (4), 911-917.
45. Kruk, M.; Jaroniec, M.; Gadkaree, K. P., Determination of the
specific surface area and the pore size of microporous carbons from
adsorption potential distributions. Langmuir 1999, 15, (4), 1442-1448.
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link between the adsorption potential distribution and energetic
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Tekinalp, H.; Edie, D. D., Effect of Pd on Hydrogen Adsorption Capacity
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74
3 FIRST-PRINCIPLES CALCULATIONS OF THE PALLADIUM(II) ACETYLACETONTE CRYSTAL STRUCTURE
3.1 Summary
The geometry of palladium(II) acetylacetonate in a monoclinic crystal
lattice is calculated using the planewave-pseudopotential implementation
of density-functional theory. Both the Troullier-Martin pseudopotential
with the generalized gradient Perdew-Burke-Ernzerhof approximation
and the Goedecker pseudopotential with the local density approximation
are employed. The non-planar, step-like, structure of the molecule
observed experimentally is successfully reproduced. A topological
analysis of the Electron Localization Function suggests a weak interaction
between a Pd cation and the nearest carbon atom of the neighboring
molecule of the closed-shell, non electron-sharing type, presumably of
electrostatic or dispersive nature, and possibly responsible for the bending
of the palladium(II) acetylacetonate salt molecule in the crystal structure.
3.2 IntroductionThe crystal structure of palladium(II) acetylacetonate was first
reported by Knyazewa et al.1 as a monoclinic crystal lattice (a = 10.835 Å, b
= 5.148 Å, c = 10.125 Å, = 93.19°) with space group P21/n and 2 planar
molecules per unit cell. As a result of packing, a weak intermolecular
interaction between the Pd atom and the C3 atom of a neighboring
75
molecule (cf. Fig. 3.1 for the definition of atom labels) was anticipated, but
it was concluded that the intermolecular interaction was not strong
enough to make the molecule bend. The structure of copper(II)
acetylacetonate was, however, reported by Lebrun et al.2 to involve a step-
like structure with the molecule bent with an angle of 7.05° between the
Cu-O plane and the 2, 4-pentanedionate plane. The experimental data of
copper(II) acetylacetonate2 prompted Hamid et al.3 to revisit the
palladium(II) acetylacetonate structure; the bond lengths and bond angles
were not found to differ significantly from those previously reported1 but
the non-planar geometry of the molecule was revealed, with an angle of
4.33 (The original paper by Hamid et al.3 reports an angle of 3.4°, but the
structure provided by the authors in the Cambridge Structural Database4
has an angle of 4.33°.) between the Pd-O1-O2-O3-O4 and the O1-O2-C1-
C2-C3-C4-C5 mean planes (cf. Fig. 3.1 for a definition of atom labels). The
distance between the two parallel 2,4-pentanedionate ligand planes was
reported to be 0.212 Å and the intermolecular distance between the C3 and
Pd atoms to be 3.31 Å while the lattice parameters were reported to be a =
9.9119 Å, b = 5.2232 Å, c = 10.3877 Å, = 95.807°. The molecular bending
was attributed to an intermolecular interaction between the Pd cation and
the closest, most nucleophillic, atom (C3) of the neighboring molecule.
Restricted Hartree-Fock computations of the isolated palladium(II)
acetylacetonate molecular geometry by Burton et al.,5 using non-local
pseudopotentials and atom-centered Gaussian basis functions, did not
reveal any deviation from planarity, presumably because crystal packing
and the resulting intermolecular interactions were not considered in the
calculation. Lewis et al.6 calculated the electronic structure and optical
spectra of Ni (II), Pd (II) and Pt (II) acetylacetonates, using semiempirical
methods, for the experimental planar geometry of Knyazewa et al.1 for Pd
and the experimental geometries of Horrocks et al.7 for Ni and Pt, and
76
inferred from their calculations that the intramolecular metal-ligand
covalency increased with increase in atomic number. This would result in
a weaker interaction between the metal atom and the C3 atom of a
neighboring molecule, and could explain the smaller bend observed in Pd
(II) acetylacetonate3 compared to that in Cu (II) acetylacetonate.
In this Letter, the palladium(II) acetylacetonate crystal structure is
calculated from first-principles, paying particular attention to the
geometry of the individual molecules. Results are then compared to
experimental data3 for the palladium(II) acetylacetonate molecule in the
crystal lattice. The electronic structure is then analyzed in terms of the
electron localization function (ELF)8-13 in order to gain insight into the
structural results.
Figure 3.1: Packing of palladium(II) acetylacetonate in the crystal lattice and labelling of atoms in the molecule.
3.3 Computational MethodsCalculations were performed with the CPMD software, version
3.11.1,14 which provides an implementation of the planewave-
pseudopotential Kohn-Sham formulation of density-functional theory
77
(DFT). Both the Goedecker pseudopotential15, 16 with the local density
approximation17 (G+LDA) and the Troullier-Martins pseudopotential18
with the Perdew-Burke-Ernzerhof generalized gradient approximation19
(TM+PBE) were used. These pseudopotentials have been used
previously20-22 for modeling Pd-containing systems, and no attempt has
been made to generate a new pseudopotential with a different exchange-
correlation functional. Since only the Bravais lattice type and one out of 32
point groups can be specified in CPMD, a monoclinic unit cell with 2
molecules in each unit cell and a crystallographic point group symmetry
C2h (2/m) were specified, and periodic boundary conditions were applied.
Only the -point was used for integration over the Brillouin zone in
reciprocal space. The optimum planewave energy cut-offs were
determined for both pseudopotential-functional combinations by
wavefunction optimizations; the total energy converged at a 170 Ryd. cut-
off for G+LDA and at a 100 Ryd. cut-off for TM+PBE (not surprisingly, the
Goedecker pseudopotential needs a higher cut-off since it is a harder
pseudopotential). There are two Pd cations per unit cell, each with two
unpaired electrons, and the multiplicity of the system could be either 1 or
5; a multiplicity of 5 gave the lowest energy and hence was used
throughout. The electronic structure was further investigated by an ELF
topological analysis, which was performed with the VMD software,23
version 1.8.6.
The ELF, first introduced by Becke et al.,8 is an effective tool to
classify the nature of the interactions between atoms and molecules.9-13
Silvi et al.13 examined the spatial arrangement of the ELF local maxima
(attractors) to classify chemical bonds as covalent or ionic and to
characterize multiple bonds. Inspection of the localization domain, the
spatial region bounded by a closed ELF isosurface, also provides insight
into the electronic structure. At low values of the ELF, there is only one
78
localization domain, containing all the attractors. As the ELF value is
increased, the localization domain splits into a number of irreducible and
reducible domains, containing one and multiple attractors, respectively,
until all the domains become irreducible. The reduction of the reducible
localization domains gives rise to distinguishable valence basins. The
synaptic order of a valence basin, i.e. the number of atomic core basins in
contact with the valence basin, is also used to characterize the chemical
interaction as electron-sharing or non electron-sharing.12 The hierarchy of
bifurcations of the valence basins with increasing ELF isovalue can also be
related to the relative electronegativities of atoms in molecules, and hence
can be used to identify the most electronegative atom in a molecule.9
Figure 3.2: Relative energy vs. lattice volume. The filled squares and circles are the DFT computed energies and the lines represent the equation of state fit.
79
3.4 Results and Discussion
3.4.1 Crystal Structure Figure 3.1 shows the packing arrangement of palladium(II)
acetylacetonate molecules in a monoclinic crystal lattice. Simultaneous
geometry and cell size optimization allows the lowest energy structure for
optimum lattice parameters to be obtained, but variable cell calculations
with a given number of planewaves (which depends on the energy cut-
off) results in a change of the effective cut-off (with a smaller cell size, one
obtains a higher effective cut-off and vice-versa). This effect could be
minimized by converging the stress-tensor with respect to the energy cut-
off, but the stress tensor converges extremely slowly. This problem was
then circumvented by first determining optimal lattice parameters. The
angle and ratios acab and were kept fixed at the experimental values of
95.807°, 0.527 and 1.048, respectively, and a series of energy calculations
were performed at different values of the lattice parameter a (at a constant
energy cut-off). The relative energies were fitted as a function of lattice
volume using the Murnaghan equation of state,24 and optimum lattice
parameters were chosen as those minimizing the energy (Fig. 3.2). We
note that, though it is correct to determine lattice parameters at a fixed
energy cut-off, the energy cut-off needs to be well converged (which is the
case in this work) to avoid jumps in the plot of energy vs. volume.25, 26 The
optimum lattice parameter a is 10.42 Å and 10.60 Å for G+LDA and
TM+PBE, respectively, which compares well to the 100 K experimental
lattice parameter of 9.9119 Å3 (we note that the computed results
correspond to a 0 K temperature). It should be noted that the equation-of-
state curve is relatively flat near the minimum and that the crystal energy
is not far from optimum at the experimental value of the lattice parameter.
Figure 3.3 shows selected experimental and computed geometrical
parameters of palladium(II) acetylacetonate in the crystal lattice. The angle
80
between the Pd1-O1-O2-O1-O2 and O1-O2-C1-C2-C3-C4-C5 mean planes
is 4.42° for G+LDA and 6.42° for TM+PBE. For comparison, geometry
optimization of an isolated (triplet) palladium acetylacetonate molecule
gives an angle between the mean planes of 0.61° for G+LDA and 0.86° for
TM+PBE. The largest deviation of the C3 atom from the mean O1-O2-C1-
C2-C3-C4-C5 plane and the root mean square deviation observed
experimentally3 are 0.030 Å, and 0.016 Å, while values of 0.027 Å and
0.019 Å are obtained with G+LDA and values of 0.014 Å and 0.014 Å are
obtained with TM+PBE, respectively (not shown). G+LDA seems to
perform slightly better than TM+PBE in predicting these properties. As for
bond angles and bond lengths for the molecule (selected ones are collected
in Table 3.1) both pseudopotential-functional combinations give similar
results, but closer inspection reveals that TM+PBE predicts bond angles in
better agreement with experimental data, whereas G+LDA predicts bond
lengths in closer agreement with experimental data. In general, bond
lengths are better reproduced than bond angles.
81
Figure 3.3: Experimental and computed parameters quantifying the non-planar geometry of the palladium(II) acetylacetonate molecule. (a) Angle between the Pd1-O1-O2-O1-O2 and O1-O2-C1-C2-C3-C4-C5 mean planes, (b) the distance between two parallel O1-O2-C1-C2-C3-C4-C5 planes and (c) distance between the most nucleophillic carbon C3 and Pd cation of two neighbour palladium(II) acetylacetonate molecules.
82
Table 3.1: Bond lengths and angles (atom labels are defined in Fig. 3.1) of the optimized geometry of palladium(II) acetylacetonate molecule in the crystal lattice.a
Bonds/Angles Experimentalb G+LDA TM+PBE
Pd1-O2 1.9815 2.047 2.081
Pd1-O1 1.9837 2.243 2.33
O2-C4 1.276 1.269 1.282
O1-C2 1.275 1.261 1.274
C5-C4 1.5 1.502 1.545
C4-C3 1.394 1.403 1.417
C3-C2 1.396 1.423 1.426
C2-C1 1.5 1.566 1.572
O2-Pd1-O1 95.19 85.41 83.57
C2-O1-Pd1 122.44 126.87 126.54
C4-O2-Pd1 122.56 129.63 131.27
O1-C2-C3 126.5 130.52 127.04
C4-C3-C2 126.63 125.97 126.77
C1-C2-C3 118.54 112.76 115.58
C3-C4-C5 118.52 114.8 115.37
a Bond lengths in Å and bond angles in degrees.b From Reference 3
83
The better performance of the Goedecker pseudopotential in
predicting the bend in the molecule (and the palladium-oxygen bond
lengths) may be attributed to its semi-core nature. Inclusion of the semi-
core (Pd 4s and 4p) electrons in the explicit DFT treatment of the electronic
structure improves (i) the description of the electron density around the
Pd cation, (ii) the description of the positively charged ions and (iii) the
transferability of the pseudopotential.16, 27 Since both pseudopotentials
used in this work were not developed explicitly for the molecule
investigated here, transferability of the pseudopotential is an important
aspect. One would have expected TM+PBE to yield better results since it is
a well known fact that PBE describes weak interactions better than LDA,
but the choice of pseudopotential is apparently determinant in the present
case. As for the improvement in predicting ligand bond angles with
TM+PBE over G+LDA, it could be attributed to the better performance of
PBE over LDA, as the semi-core nature of the Pd pseudopotential, which
possibly improves the performance of G+LDA in calculating the
palladium-oxygen bond lengths and the molecular bend due to the Pd-C3
interaction, may play a lesser role in this case. The O1-Pd1-O2 angle
exhibits the largest deviation from experimental data in this work (as well
as in the work of Burton et al.5 for the isolated molecule).
3.4.2 ELF Analysis
Figure 3.4 shows ELF isosurfaces for part of the optimized crystal
structure (with G+LDA). The presence of disynaptic attractors between
neighbouring carbon atoms and between carbon and oxygen atoms
reflects the intra-ligand covalent bonding. An attractor is also found close
to the line joining Pd and O cores, but (i) the closer location of the attractor
to the oxygen core (Fig. 3.4a), (ii) the disappearance of the monosynaptic
basin of Pd with increasing value of the ELF (Fig. 3.4d) and (iii) the
84
spherical distribution of attractors surrounding the Pd core (at a higher
ELF isovalue), all point out to the ionic nature of the bonding
interaction.11, 12 However, none of the aforementioned ELF features of
covalent or ionic bonding are observed for the interaction between the Pd
and C3 atoms. The absence of any attractor on the line joining the atomic
cores and of any disynaptic valence basin suggests a weak interaction of
the closed-shell, non-electron sharing type, most likely of electrostatic or
dispersive nature. The same is found in electronic structures calculated
with TM+PBE (not shown).
85
Figure 3.4: ELF isosurfaces (G+LDA) at isovalues of (a) 0.86, (b) 0.65, (c) 0.77 and (d) 0.815. Only the C2, C3, C4, O1, O2 and Pd atoms are shown. The Pd atom in (a) belongs to the palladium(II) acetylacetonate molecule while the Pd atom shown in (b), (c) and (d) is the nearest Pd atom of the neighbouring palladium(II) acetylacetonate molecule. Oxygen atoms shown in red, carbon atoms in blue and palladium atoms in brown. The dashed-line connects atomic cores.
86
Figure 3.5: ELF isosurfaces at isovalues of (a) 0.703, (b) 0.727 (G+LDA), (c) 0.720 and (d) 0.744 (TM+PBE). The color convention is same as in Fig. 3.4.
The ELF isosurfaces, calculated using both G+LDA and TM+PBE
are compared in Fig. 3.5, to investigate the possible differences in the
interaction between the Pd and C3 atoms. Figure 3.5 clearly shows that the
bifurcation between the ELF basins occurs at the highest ELF value for the
C3 atom, confirming that it is the most electronegative atom of the ligand.
Careful observation of the TM+PBE ELF isosurfaces shows a localization
of electrons near the C3 atom on the side of the neighbour-molecule Pd
cation (Fig. 3.5c), which could be a reflection of a dispersion interaction
between the Pd and C3 atoms that polarizes the C3 electron localization
towards the Pd cation. This interaction is not observed with G+LDA. Also,
the bifurcation of valence basins occurs at a higher ELF value for TM+PBE
than for G+LDA: the ELF valence basin surrounding the Pd atom vanishes
even before the isovalue reaches 0.5 (the ELF value for a homogeneous
87
electron gas) with TM+PBE, whereas for G+LDA, a reducible
monosynaptic basin remains till an isovalue of ~0.80 is reached (not
shown). The greater localization of electrons around the Pd atom in
G+LDA calculations may result in a screening effect in the interaction
between the electronegative carbon and the Pd cation, an effect missing in
TM+PBE calculations (electron density isosurfaces, not shown, also show
greater electron density around the Pd atom with G+LDA than with
TM+PBE). To summarize, (i) a slightly higher electronegativity, (ii) the
presence of greater electron localization near C3 and (iii) the absence of
screening effect due to electron localization around Pd, all provide
support for a stronger interaction in TM+PBE calculations, and thus a
larger bend in the molecule.
3.5 Conclusions
The molecular structure of palladium(II) acetylacetonate was
calculated in a monoclinic crystal lattice. The non-planar step-like
geometry of the molecule has been successfully computed and structural
results are in good agreement with experimental data. The Goedecker
pseudopotential together with the local density approximation (LDA)
functional predicts a bending of the molecule in better agreement with
experimental data than the Troullier-Martins pseudopotential with the
Perdew-Burke-Ernzerhof (PBE) functional, but the latter performs better
in predicting the bond angles of the ligand. A topological analysis of the
Electron Localization Function (ELF) suggest a weak interaction between
the Pd atom and the nearest carbon atom of the neighboring molecule, of
the closed-shell, non-electron-sharing type and presumably of electrostatic
or dispersive origin, related to the bend observed in the salt molecule in
the crystal structure. A detailed analysis of this interaction also explains
88
why a larger molecular bend is observed using the Troullier-Martins
pseudopotential with PBE.
3.6 References
1. Knyazeva, A. N.; Shugam, E. A.; Shkol’nikova, L. M., Crystal
chemical data regarding intracomplex compounds of beta diketones.
Journal of Structural Chemistry 1971, 11, 875-876.
2. Lebrun, P. C.; Lyon, W. D.; Kuska, H. A., Crystal-Structure of
Bis(2,4-Pentanedionato)Copper(II). Journal of Crystallographic and
Spectroscopic Research 1986, 16, (6), 889-893.
3. Hamid, M.; Zeller, M.; Hunter, A. D.; Mazhar, M.; Tahir, A. A.,
Redetermination of bis(2,4-pentanedionato)-palladium(II). Acta
Crystallographica Section E-Structure Reports Online 2005, 61, M2181-M2183.
4. Allen, F. H., The Cambridge Structural Database: a quarter of a
million crystal structures and rising. Acta Crystallographica Section B-
Structural Science 2002, 58, 380-388.
5. Burton, N. A.; Hillier, I. H.; Guest, M. F.; Kendrick, J.,
Pseudopotential Calculations of the Geometry and Ionization Energies of
Palladium(Ii) Acetylacetonate. Chemical Physics Letters 1989, 155, (2), 195-
198.
6. Lewis, F. D.; Salvi, G. D.; Kanis, D. R.; Ratner, M. A., Electronic-
Structure and Spectroscopy of Nickel(Ii), Palladium(Ii), and Platinum(Ii)
Acetylacetonate Complexes. Inorganic Chemistry 1993, 32, (7), 1251-1258.
7. Horrocks, W. D.; Templeto.Dh; Zalkin, A., Crystal and Molecular
Structure of Bis(2,4-Pentanedionato)Bis(Pyridine N-Oxide)Nickel(2)
Ni(C5h702)2(C5h5no)2. Inorganic Chemistry 1968, 7, (8), 1552-1557.
8. Becke, A. D.; Edgecombe, K. E., A Simple Measure of Electron
Localization in Atomic and Molecular-Systems. Journal of Chemical Physics
1990, 92, (9), 5397-5403.
89
9. Fuster, F.; Sevin, A.; Silvi, B., Determination of substitutional sites
in heterocycles from the topological analysis of the electron localization
function (ELF). Journal of Computational Chemistry 2000, 21, (7), 509-514.
10. Kohout, M.; Wagner, F. R.; Grin, Y., Electron localization function
for transition-metal compounds. Theoretical Chemistry Accounts 2002, 108,
(3), 150-156.
11. Savin, A.; Nesper, R.; Wengert, S.; Fassler, T. F., ELF: The electron
localization function. Angewandte Chemie-International Edition in English
1997, 36, (17), 1809-1832.
12. Silvi, B.; Fourre, I.; Alikhani, M. E., The topological analysis of the
electron localization function. A key for a position space representation of
chemical bonds. Monatshefte Fur Chemie 2005, 136, (6), 855-879.
13. Silvi, B.; Savin, A., Classification of Chemical-Bonds Based on
Topological Analysis of Electron Localization Functions. Nature 1994, 371,
(6499), 683-686.
14. CPMD Copyright IBM Corp. 1990-2006, Copyright MPI für
Festkörperforschung Stuttgart 1997-2001.
15. Goedecker, S.; Teter, M.; Hutter, J., Separable dual-space Gaussian
pseudopotentials. Physical Review B 1996, 54, (3), 1703-1710.
16. Hartwigsen, C.; Goedecker, S.; Hutter, J., Relativistic separable
dual-space Gaussian pseudopotentials from H to Rn. Physical Review B
1998, 58, (7), 3641-3662.
17. Kohn, W.; Sham, L. J., Self-Consistent Equations Including
Exchange and Correlation Effects. Physical Review 1965, 140, (4A), 1133-
1137.
18. Troullier, N.; Martins, J. L., Efficient Pseudopotentials for Plane-
Wave Calculations. Physical Review B 1991, 43, (3), 1993-2006.
90
19. Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized gradient
approximation made simple. Physical Review Letters 1996, 77, (18), 3865-
3868.
20. Caputo, R.; Alavi, A., Where do the H atoms reside in PdHx
systems? Molecular Physics 2003, 101, (11), 1781-1787.
21. Li, C. B.; Li, M. K.; Liu, F. Q.; Fan, X. J., Ab initio calculation of the
electronic and mechanical properties of transition metals and their
nitrides. Modern Physics Letters B 2004, 18, (7-8), 281-289.
22. Rogan, J.; Garcia, G.; Valdivia, J. A.; Orellana, W.; Romero, A. H.;
Ramirez, R.; Kiwi, M., Small Pd clusters: A comparison of
phenomenological and ab initio approaches. Physical Review B 2005, 72,
(11), 115421-1-6.
23. Humphrey, W.; Dalke, A.; Schulten, K., VMD: Visual molecular
dynamics. Journal of Molecular Graphics 1996, 14, (1), 33-38.
24. Murnaghan, F. D., The compressibility of media under extreme
pressures. Proceedings of the National Academy of Sciences of the United States
of America 1944, 30, 244-247.
25. Bernasconi, M.; Chiarotti, G. L.; Focher, P.; Scandolo, S.; Tosatti, E.;
Parrinello, M., First-Principle Constant-Pressure Molecular-Dynamics.
Journal of Physics and Chemistry of Solids 1995, 56, (3-4), 501-505.
26. Dacosta, P. G.; Nielsen, O. H.; Kunc, K., Stress Theorem in the
Determination of Static Equilibrium by the Density Functional Method.
Journal of Physics C-Solid State Physics 1986, 19, (17), 3163-3172.
27. Goedecker, S.; Maschke, K., Transferability of Pseudopotentials.
Physical Review A 1992, 45, (1), 88-93.
91
4 TOWARDS UNDERSTANDING PALLADIUM DOPING OF CARBON SUPPORTS: A FIRST-PRINCIPLES MOLECULAR DYNAMICS INVESTIGATION
4.1 Summary
The interaction between palladium precursors and aromatic carbon
materials is of fundamental importance to the synthesis of
carbon supported palladium catalysts and to the synthesis of
palladium doped carbon fibers for hydrogen storage. Though
experimental studies suggest that the palladium precursor decomposes in
the presence of aromatic carbon and that the carbon structure is
chemically modified in the presence of the palladium precursor, a
molecular level understanding of the underlying chemistry has yet to be
provided. First principles molecular dynamics simulations are performed
for a mixture of chrysene, a model polyaromatic carbon compound, and
palladium(II) acetylacetonate, a palladium complex often used as a
palladium precursor and the details of the electronic structure of the
mixture along the trajectory are analyzed with the electron localization
function. The simulation results show that the palladium(II)
acetylacetonate decomposes into two acetylacetonate ligands in the
presence of chrysene molecules, with one of the acetylacetonate ligands
carrying palladium. The acetylacetonate ligand chrysene covalent
92
interaction results in the loss of conjugation in the chrysene molecule and
in turn gives rise to cross-linking in the neighbouring aromatic molecules.
The simulations not only confirm the premise that chemical interactions
take place in the palladium precursor carbon system but it also reveals the
underlying molecular chemistry of those interactions.
4.2 Introduction
Palladium (Pd) loaded carbon supports are commonly used as
hydrogenation and combustion catalysts1 and are promising materials for
carbon nanostructure growth catalysts2 and hydrogen storage.3, 4 These
carbon supports are porous materials, with pores of size ranging from a
few angstroms to nanometers, embedded in the matrix of aromatic sheets
and ribbons. Pd doping is either carried out on an already prepared and
pre-shaped carbon support5 (where a precursor is deposited on the carbon
support and is then reduced by H2 to produce Pd) or during the
preparation of the carbon support,6 in which a Pd precursor is mixed with
the carbon support precursor, such as petroleum pitch, and the Pd then
separates from the precursor complex in a subsequent thermal process. In
both cases, to different though significant extents, the underlying
chemistry between the Pd precursor and the aromatic carbons affects the
resulting microstructure, the chemical structure and hence the
performance of Pd loaded carbon materials.7-9 In the case of a pre existing
carbon support, experimental studies suggest significant interaction
between the Pd precursor and the aromatic carbons of the support and
partial reduction of Pd, even before exposure to H2.9 In the latter case,
experimental studies reveal that the Pd precursor, when mixed with the
aromatic precursor pitch, interacts with the aromatic carbons, decomposes
and chemically modifies the aromatic precursor pitch.7, 10 However, due to
uncertainties in the exact chemical composition of the carbons and due to
93
the complex interplay of various factors in the synthesis of Pd doped
carbon materials, it is extremely difficult to experimentally determine the
precise nature of the Pd precursor carbon chemistry and the consequences
on the resulting molecular and micro structure of both moieties.
To obtain a molecular level insight into the chemistry of Pd
precursors with aromatic carbons and to shed light into the experimental
results,7, 9 first principles molecular dynamics (MD) simulations are
performed for Pd(II) acetylacetonate (Pd acac), one of the commonly used
Pd precursors, and a model polyaromatic hydrocarbon, chrysene, at 500 K,
using the Car-Parrinello molecular dynamics scheme.11 The simulation cell
contained 5 chrysene and 2 Pd-acac molecules. The planewave
pseudopotential implementation of density functional theory is employed
with the Troullier-Martin pseudopotentials12 and the Perdew-Burke-
Ernzerhof generalized gradient approximation,13 an approach that has
been validated in our previous work.14 The electron localization function15,
16 (ELF) is used to characterize the nature of the chemical interactions.
Further details of the simulation are provided in section 4.5.
4.3 Results and Discussion
Figure 4.1a shows the average Pd–oxygen bond distance for each
acetylacetonate ligand of the Pd acac molecule along a trajectory. The
Pd acac molecule breaks into two fragments, each containing an
acetylacetonate ligand, while the Pd remains attached to one of the ligands
(cf. Fig. 4.1d). However, the individual acetylacetonate ligands remain
intact. Since Pd acac by itself does not decompose at 500 K,17, 18 the
decomposition results from the interaction of the acetylacetonate ligands
with the carbons in the chrysene molecule. The presence of disynaptic ELF
attractors between the carbon atoms of the acetylacetonate ligands and
those of the chrysene molecule (cf. Fig. 4.1b and 4.1c) reveals a covalent,
94
electron-sharing bonding interaction between the atoms, which disrupts
the conjugated system in the acetylacetonate ligands and is accompanied
by the release of Pd from the acetylacetonate ligand (cf. Figure 4.1e and
4.1f). The decomposition of the Pd acac molecule observed in our
simulations is consistent with reported experimental data.18, 19
95
Figure 4.1: (a) MD trajectory of the average distance between Pd and O atoms of each acetylacetonate ligand of the Pd acac molecule; (b) and (c) The ELF isosurface at an isovalue of 0.8 showing the covalent linkage between the acetylacetonate ligands and the chrysene molecule; (d) the decomposition of Pd acac in the presence of chrysene molecule; (e) and (f) The ELF isosurfaces at an isovalue of 0.8 showing the modified bonding structure of the acetylacetonate ligands after they get covalently bonded with the chrysene molecule. Blue circles indicate C, red indicates O and brown indicates Pd.
(d)
(e) (f)
CH3
CH3
O
O
Pd
O
O
CH3
CH3
+
O
O
CH3
CH3
CH3
+Pd
O
CCH3
CH3
CH
3
O
(b) (c)
Bond
Distance(Å)
Time (picosecond)
(a)
96
Figure 4.2a shows the average distance of Pd, attached to one of the
acetylacetonate ligands, from the six nearest carbon atoms of the two
neighbouring chrysene molecules along a trajectory. It can be seen that the
Pd initially attached to one chrysene molecule separates from it to attach
to the other chrysene molecule. Though this displacement of Pd from one
chrysene molecule to the other is mostly a consequence of the movement
of the acetylacetonate ligand, snapshots along the trajectory reveal that Pd
tends to stay away from the centre of the carbon ring of the chrysene
molecules, contrary to what is typically observed in cation complexes.
The average Pd (nearest) carbon distance lies in the range 2.5 3.0 Å and
Pd locates at the corner of the carbon ring of the chrysene molecule. The
ELF topological analysis of the interaction of Pd and the nearest chrysene
carbon atom, at a distance of 2.1 Å from Pd, indicates the presence of an
attractor between the two atoms (cf. Fig. 4.2b) and thus suggests a covalent
interaction. Though the valence basin surrounding Pd vanishes at an ELF
isovalue of 0.52, the reduction of the ELF localization domains at lower
isovalues shows that it is a disynaptic attractor. Proximity of this attractor
to the carbon atom might be due to the fact that the carbon to which the
Pd atom is bonded is the most electronegative atom of the neighboring
chrysene molecule (since the bifurcation between the ELF basins occurs at
the highest ELF value for the most electronegative carbon atom).20 ELF- 21
isosurfaces (cf. Fig. 4.2d) and spin density isosurfaces (cf Fig. 4.2f) for Pd
in the Pd acac interacting with chrysene differ from those for Pd in an
isolated Pd acac molecule (cf. Fig. 4.2c and 4.2e), which is a reflection of
the Pd-carbon interaction in the former. The spin dependant ELF analysis
also indicates a lower localization of the spin electrons for the
Pd acac chrysene mixture (cf. Fig. 4.2d vs. Fig. 4.2c), suggesting the
reduction of Pd metal when Pd acac interacts with chrysene. The
simulation results are in agreement with experimental results: (i) X-ray
97
absorption fine structure spectroscopy studies of Pd on carbon fibrils20
reveal an average Pd–carbon distance of 2.6 Å and a distance of 2.2 Å
between Pd and the closest carbon and (ii) very small Pd particles are
observed in close contact with the carbon fibrils after the reduction of Pd,22
and this, along with other experimental evidences,9 suggest a very strong
Pd–carbon interaction, much stronger than the metal interaction that
anchors Pd to the carbon support.
98
Figure 4.2: (a) MD trajectory of the average distance between Pd and the six nearest C atoms of the two neighboring chrysene molecules; (b) the ELF isosurface at an isovalue of 0.8 showing the bonding interaction between Pd and a carbon atom of the chrysene molecule; the ELF-isosurface surrounding the Pd atom of (c) an intact Pd acac molecule and of (d) the Pd acac molecule that is decomposed in the presence of chrysene and whose Pd atom is bonded with the chrysene molecule; spin density isosurfaces of Pd atom (e) in Fig. 4.2.c and (f) in Fig. 4.2.d.
Time (picosecond)
(a)Bo
ndDistance(Å)
(b)
(f)(e)(d)(c)
99
Figure 4.3: (a) The ELF isosurfaces at an isovalue of 0.8, showing the covalent cross-linking bonding between the two neighboring chrysene molecules due to the interaction of one of the chrysene molecules with Pd acac; (b) the breaking of the resonance structure of the chrysene molecule due to its interaction with Pd acac and the new bonding structure; (c) the ELF isosurfaces at an isovalue of 0.8 showing the new bonding structure in the chrysene molecule shown in (b).
Figure 4.3a shows ELF isosurfaces for the two chrysene molecules
next to the Pd acac molecule. The presence of ELF disynaptic attractors
reveal covalent cross-linking between the two chrysene molecules. Since
acetylacetonate ligands covalently attach to the neighbouring chrysene
molecule, the conjugation in the chrysene molecule is broken, and the
chrysene carbon atoms form covalent cross linkage with the neighbouring
chrysene molecule (cf. Fig. 4.3b and 4.3c) to satisfy their valency.
CH3
CH3
Pd-
CH3
CH3
CH3
CH3
(a)
(b)
(c)
100
Figure 4.4: Molecular dynamics trajectories of (a) cross linking bonds between the chrysene molecules and of (b) the bonds between the acetylacetonate ligands and the chrysene molecule.
Figure 4.4 shows the ligand chrysene and chrysene chrysene cross-
linking bond distances along a trajectory; as the acetylacetonate ligand
carrying the Pd atom detaches from the chrysene molecule at 1.7 ps, the
cross linking between the neighboring chrysene molecules disappears.
Since the cross linking in the chrysene molecules is induced by the
ligand chrysene covalent interaction, as the ligand chrysene bond breaks,
the corresponding cross linking bond also breaks. However, the cross-
linking induced by the other acetylacetonate ligand persists throughout
the simulation time since the ligand chrysene bonding remains intact. The
⁄⁄
1.7 ps
Ligand Chrysene Bonding Interaction
Chrysene Chrysene Cross-Linking
(a)
(b)
Time (picosecond)
Bond
Distance(Å)
101
simulation results are in agreement with experimental electron energy-
loss spectroscopy studies,23 which suggest that, upon mixing of the
polyaromatic hydrocarbons in the petroleum pitch with Pd acac, the -
type bonding behaviour increases in the vicinity of Pd. As sp2-type
carbons in the neighbouring chrysene molecules change hybridization to
sp3 due to the observed covalent cross linking, -type bonding increases in
the vicinity of Pd.
4.4 Conclusions
In conclusion, experimental investigations of Pd doping of carbon
supported catalysts suggest a significant interaction between the Pd
precursor and carbon in the initial stage and in the first attempt to
precisely understand the chemistry between the two species, our
first principles MD simulations of Pd acac and chrysene show (i) the
dissociation of the metal complex into two acetylacetonate ligands in the
presence of chrysenes and simultaneous (ii) covalent cross linking in
chrysenes, induced by the covalent bonding of the acetylacetonate ligands
to the chrysene. Pd remains attached to one of the acetylacetonate ligands
in the decomposed metal complex and also develops a bonded interaction
with a carbon atom of the chrysene molecule. Motivated by the fact that
the challenge of synthesis of Pd doped carbon supported materials with a
controlled microstructure and chemical composition needs to be
confronted by understanding the molecular interactions of Pd and carbon
species during the Pd doping process of the carbon support, the present
paper validates some of the crucial experimental hypotheses related to
these interactions and also sheds light into the fundamental chemistry
behind the doping process.
102
4.5 Supporting Information
4.5.1 Computational Details
4.5.1.1 Simulation system set-up
A cubic simulation cell of 13Å each side with periodic boundary
conditions was used. Figure 4.5a shows the simulation cell with 5
chrysene molecules and 2 palladium (II) acetylacetonate molecules, giving
a density of 1.35 g/cc. The distance between individual molecules was
fixed arbitrarily, but a topological analysis of the electron localization
function (ELF) of the initial system (following geometry optimization) was
performed to make sure that there were no “pre-existing” intermolecular
interactions within the simulation cell (cf. Fig. 4.5b). Chrysene was chosen
as a model polyaromatic hydrocarbon in this study because Edie and
coworkers,7 in their investigation of the effect of metal precursor–carbon
chemistry on the microstructure of palladium loaded carbon fibers have
established that chrysene was one of the major components of the pitch
that they used to prepare the carbon support. Furthermore, it was
observed that when palladium(II) acetylacetonate was mixed with the
pitch, the concentration of chrysene in the pitch decreased significantly,
suggesting that palladium(II) acetylacetonate is chemically interacting
with the chrysene molecules in the pitch at the initial mixing stage itself.
Based on this observation, it was suggested that this palladium complex–
aromatic carbon chemistry governs the microstructure and the chemical
composition of the final fibers. However, the exact nature of the
interaction between palladium(II) acetylacetonate and chrysene remained
unclear.
The molecular dynamics simulations were performed at 500 K, the
temperature at which palladium(II) acetylacetonate is mixed with the
carbon-support-precursor pitch24 in the synthesis of palladium doped
103
carbon supports, especially carbon fibers. Also, the simulation
temperature is lower than the temperature at which palladium(II)
acetylacetonate thermally decomposes (by itself) and it is within ±50 K of
the temperature at which the prepared and preshaped carbon support,
while impregnating, is heated (in an inert atmosphere and in the presence
of H2 for reduction of palladium), when contacted with a palladium
precursor.25
The simulation system was limited to two palladium(II)
acetylacetonate molecules due to the computational cost constraints and
the multiplicity issue. A multiplicity of 5 was chosen, as it is the
lowest energy spin state of the system made up of two palladium cations.
Increasing number of palladium(II) acetylacetonate molecules would
result in a factorial increase of possible multiplicities, in turn increasing
the number of computations and possibly leading to convergence
problems.
Figure 4.5: (a) The simulation cell containing chrysene and palladium (II) acetylacetonate molecules and (b) ELF isosurfaces at an isovalue of 0.8 showing no “pre-existing” intermolecular interactions.
(a) (b)
104
4.5.1.2 Computational Methods
Calculations were performed using the CPMD software, version
3.11.1,26 which provides an implementation of the first-principles Car-
Parrinello molecular dynamics (MD) scheme. The first principles
calculations were performed using the planewave pseudopotential
implementation of the Kohn-Sham formulation of density functional
theory. The Troullier-Martins pseudopotentials12 with the Perdew-Burke-
Ernzerhof generalized gradient approximation,13 which have been
validated in our previous work,14 were used. Only the -point was used
for integration over the Brillouin zone in reciprocal space. The planewave
energy cut-off for the pseudopotential was determined by inspection of
the variation of the energy of the system with energy cut-off. A cut-off of
100 Ryd. Seemed to produce a converged energy, and was thus used
thereafter. To check for spin contamination in our calculations, even
though it is in principle not an issue in density functional theory, S2
was calculated using the procedure given by Wang et al.27 (as CPMD
calculates the total integrated value of the spin density and not S2 ).
Since the calculated S2 value (6.35) was found to be within 10% of the
exact value (6.00) for a system with quintet multiplicity, spin
contamination appears to be negligible in our calculations.
Temperature control was achieved using the Nosé-Hoover
thermostat. The frequency for the ionic thermostat was 1800 cm-1
(characteristic of a C–C bond vibration frequency) and that for the electron
thermostat was 10000 cm-1. The fictitious electron mass was taken as 600
a.u. Short MD runs were performed without the thermostat to obtain an
approximate value around which the electronic kinetic energy oscillates; a
value of the electronic kinetic energy of 0.14 a.u. was used in the
simulations. The MD time step used in the simulation was 0.0964 fs. The
system was first equilibrated for 0.25 ps (to have the Kohn-Sham energy
105
oscillate around a mean value) and then a production run of 7 ps was
performed. Figure 4.6 shows the variation of the fictitious electronic
kinetic energy of the system during the CPMD run; the fictitious electronic
kinetic energy oscillates around the mean value of 0.14 a.u., confirming
that the electrons do not “heat up” in the presence of the “hot” nuclei and
the system remains in the Born-Oppenheimer ground state. A Born-
Oppenheimer molecular dynamics (BOMD) simulation of 1 ps, with a
time step of 0.482 fs, was also performed for the equilibrated system and
an excellent agreement between the CPMD and BOMD bond trajectories
was found. Due to the extremely high computational cost of BOMD
simulations for this system (36000 CPU hours for 1 ps), the CPMD scheme
was used thereafter (computational cost of 7700 CPU hours for 1 ps).
Figure 4.6: Fictitious electronic kinetic energy vs. time during the CPMD production run.
106
The electron localization function (ELF), introduced by Becke and
Edgecombe15 was used as a tool to characterize the nature of the
interactions between atoms and molecules along the MD trajectory. Silvi
and Savin16 proposed a topological analysis of the ELF to classify chemical
bonds as covalent or ionic and to characterize multiple bonds. Inspection
of the localization domain, the spatial region bounded by a closed ELF
isosurface, also provides insight into the electronic structure. At low
values of the ELF, there is only one localization domain, containing all the
attractors (local maxima). As the ELF value is increased, the localization
domain splits into a number of irreducible and reducible domains,
containing one and multiple attractors, respectively, until all the domains
become irreducible. The reduction of the reducible localization domains
gives rise to distinguishable valence basins. The synaptic order of a
valence basin, i.e. the number of atomic core basins in contact with the
valence basin, is also used to characterize the chemical interaction as
electron-sharing or non electron-sharing.28 The hierarchy of bifurcations of
the valence basins with increasing ELF isovalue can also be related to the
relative electronegativities of atoms in molecules, and hence can be used
to identify the most electronegative atom in a molecule.20 Kohout and
Savin21 also provided a method to calculate the ELF separately for each
spin density. Spin dependent ELF calculations provide information about
the localization of unpaired electrons (e.g. the two unpaired electrons of
palladium in our simulation system). The reader is referred to section
A.2.8.1 of the appendix for further details.
4.5.2 Additional Results
4.5.2.1 ELF analysis of palladium-oxygen interactions
The variation of the palladium–oxygen bond distances in the
palladium(II) acetylacetonate molecule along the MD trajectory shows that
107
palladium remains associated with one of the acetylacetonate moiety and
detaches from the other. The ELF topology of the decomposed
palladium(II) acetylacetonate was also compared to that of an intact
palladium(II) acetylacetonate molecule. Figure 4.7 shows contour plots
(these are not the actual contour plots but we constructed ELF isosurfaces
of different isovalues ranging from 0.2 to 1 and took a thin volume slice in
the plane of palladium acetylacetonate) of the ELF for both palladium(II)
acetylacetonate molecules. The ELF results support the earlier conclusions
that palladium detaches from one acetylacetonate ligand and remains
associated with the other. The position of the lone pairs of the oxygens
bonded to palladium is similar to that for an intact palladium(II)
acetylacetonate molecule, while that for the oxygens not having a bonded
interaction with palladium reveal completely different.
(a) (b)
Figure 4.7: ELF contour plots (a) for palladium(II) acetylacetonate decomposed in the presence of chrysene molecule and (b) for an intact palladium (II) acetylacetonate molecule.
4.5.3 Additional Relevant Information (Experimental and Simulation):As mentioned before, palladium is incorporated on a carbon
support for a variety of reasons, including catalysis and hydrogen storage.
The conventional method of palladium doping on the carbon support is
Lone pairs of electrons
Bonding electrons
108
to impregnate the palladium precursor on the carbon support, followed by
hydrogen reduction. The chemistry of palladium precursors and carbon
affects the electronic structure of the active sites (since it depends on the
palladium-carbon interactions), the number of active sites (since a weak
palladium-carbon interaction leads to sintering of palladium particles) and
the microstructure of the carbon support (uncontrolled dispersion of
palladium particles may clog the desired pores in the support). The other,
more recent, method for palladium doping is to mix palladium precursor
with the carbon support precursor before the preparation of the carbon
support. After mixing, the carbon material, containing the palladium
precursor, undergoes carbonization and activation at temperatures as high
as 1250 K.24 Analysing the effect of palladium precursor–carbon chemistry
on the (i) electronic structure of active sites, (ii) the dispersion of
palladium particles and (iii) the pore structure of the carbon support is
relatively more difficult in this case. Though there is a significant body of
literature related to both types of carbon supported palladium materials,
there are very few investigations related to the chemistry involved in the
synthesis of these materials. In this section, additional experimental and
simulation results that support our findings are reviewed.
(i) Tribolet et al.29 prepared palladium loaded carbon nanofibers (grown
on sintered metal fibers) and active carbons by the adsorption of
palladium precursor on both carbon supports, and heating the
samples to 400 K in the presence of He and H2. They observed
smaller palladium particles and better dispersion of palladium in the
case of carbon nanofibers. They concluded that there exists a strong
palladium-carbon interaction, stronger than cation interaction,
that prevents the sintering of palladium particles.
(ii) Kim et al.30 prepared monodispersed palladium nanoparticles using
palladium(II) acetylacetonate and trioctylphosphine, and showed
109
using FT-IR spectroscopy that palladium(II) acetylacetonate
decomposes by the acetylacetonate ligands getting separated from
the palladium center.
(iii) Simulation studies31, 32 of palladium atoms sandwiched between
polyaromatic hydrocarbons have shown that palladium does not
tend to locate at the centre of the carbon ring, but rather attaches to
the peripheral carbon atoms of the aromatic ring. However, there are
no simulation studies of the interaction of palladium precursors with
aromatic carbons, to the best of our knowledge.
(iv) Tamai et al.33 were pioneers in mixing palladium(II) acetylacetonate
with a petroleum pitch to prepare palladium loaded pitch based
carbon fibers for 1-hexene hydrogenation catalysis. They noticed the
appearance of a palladium peak in the XRD profiles at temperatures
higher than 900 K.
(v) Edie et al.6, 7, 34 have also performed studies in the field of
palladium loaded pitch based activated carbon fibers. They used
palladium(II) acetylacetonate as the precursor for palladium and the
aromatic pitch is mixed with the palladium complex at 500 K. They
have provided substantial evidence of the presence of a chemical
interaction between palladium(II) acetylacetonate and aromatic
carbons in the pitch at this temperature. They identified five major
polyaromatic hydrocarbons in the precursor pitch (including
chrysene) using gas chromatography–mass spectrometry, and they
noticed that after mixing this pitch with palladium(II) acetylacetonate
at 500 K, the concentration of all the polyaromatic hydrocarbons in
the pitch decrease by up to 20%. They also hypothesized that
palladium separates from the acetylacetonate ligands at this stage
and attaches to the hydrocarbons.
110
(vi) Wu et al.10 studied the same material as in (v), and reported reduced
distances between the parallel stacks of polyaromatic carbons in
palladium(II) acetylacetonate doped activated carbon fibers, thereby
indicating an enhanced interaction and cross linking between the
aromatic carbons in the palladium containing carbon support.
(vii) Recently, Okabe et al.35 prepared palladium loaded molecular
sieving carbons (MSC) by mixing polyamic acid with palladium(II)
acetylacetonate. The mixture was then heated to 573 K and then
carbonized at temperatures as high as 1273 K. The thermogravimetric
analysis indicated that palladium(II) acetylacetonate decomposes at
473 K in the presence of polyamic acid during the mixing stage.
However, in their XRD analysis they did not observe a peak for pure
palladium till the carbonization temperature reaches 873 K.
4.6 References
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A.; Schlogl, R.; Milroy, D.; Jackson, S. D.; Torres, D.; Sautet, P.,
Understanding Palladium Hydrogenation Catalysts: When the Nature of
the Reactive Molecule Controls the Nature of the Catalyst Active Phase.
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2. Lai, C.; Guo, Q. H.; Wu, X. F.; Reneker, D. H.; Hou, H., Growth of
carbon nanostructures on carbonized electrospun nanofibers with
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3. Amorim, C.; Keane, M. A., Palladium supported on structured and
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4. Lachawiec, A. J.; Qi, G. S.; Yang, R. T., Hydrogen storage in
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5. Augustine, R. L., Heterogeneous Catalysis for the Synthetic Chemist Marcel
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7. Basova, Y. V.; Edie, D. D., Precursor chemistry effects on particle
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10. Wu, X.; Gallego, N. C.; Contescu, C. I.; Tekinalp, H.; Bhat, V. V.;
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14. Mushrif, S. H.; Rey, A. D.; Peslherbe, G. H., First-principles
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16. Silvi, B.; Savin, A., Classification of Chemical-Bonds Based on
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17. Our CPMD simulations of an isolated palladium (II)
acetylacetonate molecule at 500 K do not show any decomposition of the
molecule.
18. Semyannikov, P. P.; Grankin, V. M.; Igumenov, I. K.; Bykov, A. F.,
Mechanism of Thermal-Decomposition of Palladium Beta-Diketonates
Vapor on Hot Surface. Journal De Physique Iv 1995, 5, (C5), 205-211.
19. Dal Santo, V.; Sordelli, L.; Dossi, C.; Recchia, S.; Fonda, E.; Vlaic, G.;
Psaro, R., Characterization of Pd/MgO catalysts: Role of organometallic
precursor-surface interactions. Journal of Catalysis 2001, 198, (2), 296-308.
20. Fuster, F.; Sevin, A.; Silvi, B., Determination of substitutional sites
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function (ELF). Journal of Computational Chemistry 2000, 21, (7), 509-514.
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22. Mojet, B. L.; Hoogenraad, M. S.; van Dillen, A. J.; Geus, J. W.;
Koningsberger, D. C., Coordination of palladium on carbon fibrils as
determined by XAFS spectroscopy. Journal of the Chemical Society-Faraday
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23. Benthem, K. V.; Wu, X.; Pennycook, S. J.; Contescu, C. I.; Gallego,
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Ryu, S. K., Preparation and characterization of trilobal activated carbon
fibers. Carbon 2003, 41, (13), 2573-2584.
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and pretreatment on the adsorption and absorption behavior of supported
Pd catalysts. Applied Catalysis a-General 1998, 173, (2), 137-144.
26. CPMD Copyright IBM Corp. 1990-2006, Copyright MPI für
Festkörperforschung Stuttgart 1997-2001.
27. Wang, J. H.; Becke, A. D.; Smith, V. H., Evaluation of [S-2] in
Restricted, Unrestricted Hartree-Fock, and Density-Functional Based
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electron localization function. A key for a position space representation of
chemical bonds. Monatshefte Fur Chemie 2005, 136, (6), 855-879.
29. Tribolet, P.; Kiwi-Minsker, L., Palladium on carbon nanofibers
grown on metallic filters as novel structured catalyst. Catalysis Today 2005,
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30. Kim, S. W.; Park, J.; Jang, Y.; Chung, Y.; Hwang, S.; Hyeon, T.; Kim,
Y. W., Synthesis of monodisperse palladium nanoparticles. Nano Letters
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115
5 THE DYNAMICS AND ENERGETICS OF HYDROGEN ADSORPTION, DESORPTION AND ITS MIGRATION ON A CARBON SUPPORTED PALLADIUM CLUSTER
5.1 Summary
The functionality of metal-doped carbon materials in catalytic and
hydrogen storage applications is governed by the characteristics of their
interaction with hydrogen. The dynamics of hydrogen chemisorption, its
migration and desorption are studied using a model system of Pd4 cluster
supported on a coronene molecule. Molecular simulations are performed
using first-principles molecular dynamics. The longer time scale events
are accelerated using the metadynamics technique and a continuous
energy surface of hydrogen interaction with the carbon-supported metal
cluster is constructed. The dynamics and energetics of (i) dissociative
chemisorption of diatomic hydrogen on a carbon-supported Pd cluster, (ii)
the transport of atomic hydrogen on the cluster, towards the carbon
support and (iii) the associative desorption in a molecular form are
simultaneously reported for the first time in this paper. It is found that the
dissociative chemisorption of hydrogen on the cluster is associated with
an insignificant energy barrier and takes place instantaneously at room
temperature. However, the migration of atomic hydrogen from the tip of
the tetrahedral Pd4 cluster to its sides is associated with an energy barrier
116
of ~6 KJ/mole. This energy barrier, however, gets reduced when the
cluster is partially saturated with pre-existing hydrogens adsorbed on its
sides. The energy barrier for the subsequent migration of the atomic
hydrogen, from the sides of the cluster towards the carbon support, is
however not affected significantly due to the presence of the pre-existing
hydrogens. The transport of atomic hydrogen from the tip of the cluster
towards the carbon support results in moving the system towards lower
energy levels on the energy surface. The associative desorption of two
hydrogen atoms on the cluster to form a diatomic hydrogen molecule is an
endothermic process and is associated with a small energy barrier. We
believe that the dynamical studies of hydrogen interaction with carbon-
supported Pd cluster, performed in this paper, are in the right direction to
elucidate the mechanism of hydrogen interaction with a Pd-doped-carbon-
supported catalytic material and the findings reveal some crucial details
which can be leveraged to develop a better understanding of hydrogen
spillover on such an important functional material.
5.2 Introduction
Hydrogen is involved in a large number of catalytic reactions like
hydrogenation, hydrocracking, hydrodesulphurization, hydrocarbon
synthesis, to name a few. A significantly important class of catalytic
material in hydrogen involving reactions consist of transition metal
clusters anchored on a support, particularly a carbon based support.
Activated carbons, activated carbon fibers, carbon nanotubes and
graphenes, made up of sp2-type carbons, have gained significant
popularity in recent times as a carbon support. One of the key phenomena
involved in hydrogen related catalytic reactions is the interaction of
hydrogen with the carbon supported transition metal catalyst. If
understanding catalysis at a molecular level is the ultimate goal of
117
catalytic chemistry, then obtaining detailed molecular level information of
the interaction of hydrogen with the catalyst material is an important
milestone in achieving this goal. The nature of hydrogen bonding
(physisorption or chemisorption) and the possibility of its dissociation and
migration on the catalyst material are the key factors governing the
functionality of these materials.1-3 Hence the course of interaction of
hydrogen with the transition metal-doped carbon supported materials
garnered a significant attention from the researchers in catalysis
community.3-8 However, particularly after (i) the sp2 type carbon based
materials like nanotubes, activated carbon and activated carbon fibers
have been recognized as potential hydrogen storage materials9-20 and (ii)
transition metal doping has demonstrated an increase in the hydrogen
storage capacities of such materials,21-29 the interaction of hydrogen with
these materials has become the centre of attraction for hydrogen storage
researchers as well. In the absence of transition metal, hydrogen only
physisorbs on the carbon materials in the molecular form10, 11, 13, 14, 16, 17, 19,
20, 22 and it is believed that the addition of transition metal to carbons alters
this mode of interaction by diatomic hydrogen getting dissociatively
chemisorbed on the transition metal clusters, hydrogen getting absorbed
as metal hydride and atomic hydrogen possibly migrating on the cluster
and on to the carbon support.4, 5, 7, 8, 18, 20, 24, 26-28, 30-34 This migration of
hydrogen on the metal cluster and on to the support is also known as
spillover. Though of great importance, the interaction mechanism of
hydrogen with metal-doped carbon materials is mostly studied by
interpreting experimental data for a combination of sequential steps in the
entire catalysis process and the deficiency of isolated studies of hydrogen
interaction with these materials has left some doubts about its mechanism
and dynamics, the existence of spillover, the energetics of the process and
118
the process of desorption of monoatomic hydrogen in the form of
diatomic hydrogen molecule.
The few key experimental investigations, exclusively studying the
interaction of hydrogen with metal-doped carbon supported materials, are
done by Mitchel et al.,7 Contescu et al.30, 35 and Amorim et al.4 Mitchel et
al., using inelastic neutron scattering, identified two forms of hydrogen
atoms on the platinum-doped carbon support : (i) H atoms at the edge
sites of the graphite and (ii) weakly bound layer of mobile H atoms on the
carbon surface. Using the same experimental method, Contescu et al.35
also identified the presence of C-H bonds on the activated carbon fiber
doped with palladium and, using in situ X-ray diffraction,30 they also
demonstrated the presence of palladium hydride. Amorim et al.4 also
revealed the simultaneous presence of spillover hydrogen (C-H bond),
chemisorbed hydrogen and hydrogen in hydride phase in the Pd
supported activated carbon, graphite and nano-fibers. It has to be noted
that all the above mentioned valuable experimental investigations
successfully probed the nature of hydrogen present in these materials,
however, they do not directly reveal the mechanism of the formation of
these different forms of hydrogen on these materials. The only
experimental investigation, to the best of our knowledge, directed towards
understanding the mechanism is done by Yang et al.26, 33, 36. They
demonstrated that an improved contact between the metal particles and
the carbon support enhances the spillover of atomic hydrogen from the
metal cluster to the carbon support. They also suggested that the spillover
process is not only dependant on splitting of hydrogen on the metal and
its transportation to the active carbon support but is also controlled by the
reception capacity of the supported carbon material.
There are numerous theoretical, particularly molecular level,
investigations as well to understand the interaction of hydrogen with
119
transition metal clusters, carbon materials and transition metal doped
carbon materials. The key findings are as follows:
1. Cabria et al.12 and Aga et al.10 found that the adsorption of
molecular hydrogen is dependant on the layer spacing between the
graphene sheets (equivalent to pore size) and that there exists an optimum
layer spacing for maximum molecular hydrogen adsorption. As far as
adsorption of atomic hydrogen on carbon is concerned, Yang et al.37
showed that the binding energy depends on the geometry of the carbon
support and its hydrogen occupancy and Casolov et al.38 computed an
energy barrier of up to 20 KJ/mol for the binding of atomic hydrogen to
graphene. Chen et al.31 and Cheng et al.5 showed that the diffusion of H
atom chemisorbed on carbon nanotubes or graphene is energetically very
difficult since it requires the breaking of C-H covalent bond.
2. Ferreira et al.39 showed that a tilted hydrogen molecule is the
precursor state before getting dissociatively adsorbed on a Pd surface and
in the case of interaction of hydrogen with a Pd cluster, Matsura et al.34
demonstrated that the vertices of the cluster are the initial point of
interaction. Zhou et al.8 and Bartczak et al.40 suggested that the diffusion
of hydrogen atoms adsorbed on an unsupported Pd cluster is associated
with a low energy barrier. Caputo et al.41 also showed that the octahedral
site is more favourable when hydrogen is absorbed in Pd as Pd-hydride.
3. Theoretical investigations of hydrogen interaction with metal atom
(Ti, B, Ni, Pt, Li, Pd, Sc, V) doped carbon systems23, 25, 27-29, 42 revealed an
increased hydrogen storage (in atomic and molecular form) capacities due
to metal doping. Yildirim et al.27 and Dag et al.22 showed that the Ti, Pt
and Pd atom doped carbon nanotubes dissociatively adsorb the first H2
molecule with negligible energy barrier and the subsequent adsorptions
are molecular with elongated H-H bonds. Kiran et al.42 also suggested that
the number of molecular hydrogens adsorbed on to a carbon supported
120
metal atom increases with lesser filled d-orbitals of the metal. Guo et al.43
showed that the desorption energy barrier of the hydrogen adsorbed on
Pd-doped carbon nanotubes can be reduced with deformation in the
carbon nanotube. The computations of Fedorov et al.32 suspected that the
migration of hydrogen from Pt to carbon surface is associated with a
significant energy barrier.
The theoretical studies related to molecular hydrogen adsorption
on pure carbon systems were motivated by some of the initially reported
high hydrogen storage capacities of carbon materials9 and hence are
physically too far and simple to shed any light on hydrogen interaction
with metal doped carbon materials. Similarly, the studies related to metal
atom doped carbon materials were also done in the search of high
hydrogen storage materials and were mostly restricted to the adsorption
of hydrogen (in atomic and molecular form) surrounding the metal atom
and hence could not comment on its migration and dynamics. The
behaviour of hydrogen with sub-nano and nano sized metal clusters may
also be different that with a single metal atom. The adsorption, absorption,
energetics and migration of hydrogen on unsupported metal structures
and the energetics, adsorption and migration of monoatomic hydrogen on
carbon surfaces were investigated separately in most of the above
mentioned theoretical studies, thus again dividing the system into two
parts (metal and carbons) and missing the system as whole. The two
papers that recently attempted to study the hydrogen interaction
procedure with a carbon supported metal cluster are those by Cheng et
al.5 and Chen et al. 31 However, they did not model the dynamics of the
process and the computations were performed with a preset mechanism
of adsorption and spillover process. The spillover of hydrogen was
modelled by bringing a pre-saturated (with hydrogen) Pt cluster near the
carbon support arbitrarily so as to make the hydrogen atoms, in between
121
the metal cluster and support, to spillover. This may not be realistic since
the metal cluster is in contact with the support (carbon) even before it
comes in contact with hydrogen and hence the spillover will never occur
from the metal cluster surface that is in contact with the support.
To summarize, the experimental investigations targeted towards
understanding the hydrogen interaction procedure with metal doped
carbon supported materials are limited due to difficulty in isolating the
interaction from the complex interplay of different factors and due to
limitations on the accessible time and length scales. Theoretical
investigations either assume the system to be too simple to be compared
to a realistic metal doped carbon supported system or the course of
hydrogen interaction is simplified with a preset (and sometimes
unphysical) mechanism. Additionally, to the best of our knowledge, none
of the theoretical studies of hydrogen interaction with metal-doped carbon
supported materials have attempted to model and reveal the dynamics of
the entire interaction process at a finite temperature. In the present paper,
for the first time, we have attempted to model simultaneously the finite
temperature dynamics and energetics of the interaction of hydrogen with
a carbon supported Pd cluster. The dynamics are modelled using the Car-
Parrinello scheme44 and the energy barriers in the process are overcome,
the dynamics are accelerated and the energy surface is reconstructed using
the metadynamics scheme.45-48 The necessary and sufficient details of the
molecular modelling methodologies and of the simulation system are
discussed in section 5.3. The findings of the computations are discussed in
the results and discussion section (5.4) of the paper and are compared and
contrasted with previous experimental and theoretical results. We
conclude the key contributions in section 5.5.
122
5.3 System set-up and Simulation details
A cubic simulation cell of 16 Å each side is used with a system
containing a coronene molecule, a Pd4 cluster and a hydrogen molecule.
Since the carbon support is usually made up of sp2 type carbons, the
polycyclic aromatic hydrocarbon coronene is used here as a model carbon
support.7, 49 The geometry of the molecule is optimized before using it in
the simulation cell. Small Pd clusters prefer high spin states and several,
very close in energy states are possible.50 The convergence of single point
calculations is often difficult. As the cluster size increases, the HOMO-
LUMO gap decreases and the convergence of single point calculations is
only possible if fractional occupation is taken into account.50 If the free
energy functional approach51 is implemented for single point calculations
of a system with fractional occupation number, the electronic structure at
a particular nuclear configuration is completely independent from that of
the previous configuration. This makes the implementation of the Car-
Parrinello scheme44 impossible and Born-Oppenheimer molecular
dynamics need to be performed, thereby increasing the computational
cost. In order to be able to implement the Car-Parrinello scheme, we chose
to work with a small Pd cluster of 4 atoms. The lowest energy state for this
cluster, according to the literature50 and our calculations, is with a
multiplicity of 3. The optimized tetrahedral geometry is adapted from
Nava et al.50 The coronene supported Pd4 cluster is prepared by
comparing the energies (after geometry optimization without putting any
symmetry constraint on the Pd4 cluster) of the system with the Pd4
tetrahedron at various distances from the centre of the plane of coronene
molecule. The distance of ~2 Å gave the minimum energy. After getting
the minimum energy coronene supported Pd4 cluster, a hydrogen
molecule was placed 5 Å above the tip of the tetrahedron and the system
geometry was reoptimized (cf. Figure 5.1).
123
All the calculations in this paper are performed using the CPMD
software, version 3.13.2,52 which provides an implementation of the first-
principles Car-Parrinello molecular dynamics scheme.44 The first-
principles calculations are performed using the planewave
pseudopotential implementation of the Kohn-Sham density functional
theory.53 The Goedecker pseudopotential54, 55 with the local density
approximation,53 which has been validated in our previous work,56 is
used. Only the -point is used for integration over the Brillouin zone in the
reciprocal space. The implemented planewave energy cut-off for the
pseudopotential, as determined by converging the energy, is 170 Ryd. All
the simulations are run at 300 K and the temperature control is achieved
using the Nosé-Hoover thermostat. The frequency for the ionic thermostat
is 1800 cm-1 and that for the electron thermostat is 10000 cm-1. The
fictitious electron mass in the Car-Parrinello scheme is taken as 800 a.u.
Short molecular dynamics run are performed without the thermostat to
obtain an approximate value around which the fictitious electronic kinetic
energy in the Car-Parrinello scheme oscillates and based on this
observation, a value of 0.016 a.u. is chosen. The molecular dynamics
timestep of 0.0964 fs is used in the simulations. Energies, including the
fictitious electronic kinetic energy, are monitored to ascertain that the
system does not deviate from the Born-Oppenheimer surface during the
molecular dynamics run. Molecular dynamics trajectories are visualized
using the VMD software.57
Car-Parrinello scheme, even after reducing the computational cost
of ab initio molecular dynamics, may not access time scales of more than a
few picoseconds in practically available computer time and resources. The
timescale for a chemical event to happen increase with the increase in the
energy barrier associated with it and, based on previous literature,
believing the existence of energy barriers associated with the course of
124
interaction of hydrogen with metal-doped carbon supported materials, we
implement the metadynamics technique45-48 to accelerate the dynamics
and to reconstruct the energy surface as a function of the coordinates of
interest, during the course of the interaction. The metadynamics
technique, as described by Laio and Gervasio,46 is based on the principle
of filling up the energy well with potentials, to help the system overcome
the energy barriers. In metadynamics, the potentials dropped to fill the
energy well are tracked and the energy surface is then reconstructed using
these potentials. The metadynamics technique is implemented by
extending the Car-Parrinello Lagrangian as45
21£ £ ,2MTD CP cv cv cv cv cv cv cv
cv cvm k R t (5.1)
where £CP is the Car-Parrinello Lagrangian and is a vector of the
collective variables (coordinates of interest) that form the energy well to be
filled or that form the energy surface of interest. The first term is the
kinetic energy of the collective variables, the second term is the harmonic
restraining potential and the last term is the Gaussian-type potential that
fills the energy surface. It is given as2
2, exp2
i
i
tt
cv MTD it t i i
t H tt tw
(5.2)
where the parameter iH t represents the height of the added potential
and it and itw , together, represent the width of the Gaussian
potential. The metadynamics technique is coded in the CPMD software,
version 3.13.2 and for further mathematical and conceptual details, the
reader is referred to the recent paper by Laio and Gervasio.46
In the present paper, the distance between the two hydrogen atoms
in the diatomic hydrogen molecule interacting with the Pd cluster-
Coronene system is chosen as one collective variable while the other
125
collective variable is chosen as the average coordination number of both
the hydrogen atoms with the carbons in coronene. The coordination
number is defined as follows:6
012
1
0
1
1
Carbon
H jn
Hj H j
dd
CNdd
(5.3)
where the reference distance 0d is chosen such that the magnitude of the
difference between the minimum and the maximum coordination number
is the largest. We did not perform an exclusive numerical analysis of the
different parameters of the metadynamics technique in order to assess the
precision and accuracy of the method in developing the free energy
surface for the specific system investigated in this paper, however, the tips
and guidelines provided by Ensing et al.,58 Laio et al.59 and by Schreiner et
al.60 are followed and are implemented. The Gaussian width parameter
is taken as ¼th of the fluctuation of the collective coordinate with the
smallest amplitude of oscillation. The oscillations of the collective
coordinates are calculated by running a sample metadynamics run
without adding Gaussian potentials. Parameter w is calculated using the
criterionmax cv cvw . The height of the potential is kept fixed at ~
0.27 KJ/mol. The metadynamics time step to add the Gaussian potential is
adjusted in such a way that the following criterion is satisfied
1.5it t (5.4)
An additional criterion that the potential be also added if the time addt
given by the following equation is passed since the last metadynamics
step, is also applied.
126
1.5 cvadd
B
mtk T
w (5.5)
Analogous to the original Car-Parrinello scheme, the dynamics of the
collective variables are separated from the ionic and fictitious electronic
motion by choosing an appropriate value for the fictitious mass cvm of the
collective variables. If the fictitious mass cvm is large then the dynamics of
the collective variables will be slow and thus can be separated from the
ionic dynamics. The dynamics of the collective variables are also
dependant on cvk . It has been shown that the extra term in metadynamics
introduces an additional frequency for the motion of collective variables
as cv cvk m 58. As suggested by Schreiner et al.60 and Ensing et al.,58 the
choice of cvk and cvm are made in such a way that during a sample
metadynamics run of 20 fs without addition of potentials, and cv cv move
close to each other. The temperature of the collective variables is set to 300
K (same as the physical temperature of the system) and is controlled in a
window of ± 200 K using velocity rescaling.
127
Figure 5.1: Coronene supported Pd4 cluster and the interacting H2molecule. The collective variables for the metadynamics simulation are also shown. Carbon atoms are shown in blue, palladium atoms in brown and hydrogen atoms in white.
5.4 Results and Discussion
A regular Car-Parrinello molecular dynamics simulation (without
implementing metadynamics) at 300 K is initially run on the system
shown in Fig. 5.1. Figure 5.2 (a) shows the molecular dynamics trajectories
of the distance between the Pd atom of the tip of the tetrahedral Pd4
cluster and the two H atoms in the H2 molecule. It can be seen that the H2
molecule gets chemisorbed on the Pd4 cluster in less than 0.2 ps of the
simulation time. Similarly, Fig. 5.2 (b) shows the MD trajectory of the H-H
bond in the H2 molecule, as it is getting attached to the Pd4 cluster. It can
be seen that the adsorption of hydrogen on the Pd4 cluster tip is
simultaneously accompanied by the dissociation of the H-H bond. When
hydrogen chemisorbs on the cluster at 0.16 ps, the H-H bond distance
1Distance
CVH H
H –
C C
oordination No.
CV – 2
128
Figure 5.2: MD trajectories of (a) the distance of two H atoms from the tip atom of the Pd4 cluster and of (b) the distance between the two H atoms. The inset of (a) shows the MD snapshots of two H atoms before (as an intact H2 molecule) and after getting dissociatively chemisorbed on the Pd cluster.
129
changes from ~0.75 Å to ~0.9 Å (with wide oscillations unlike a bonded H2
molecule), thus causing dissociative chemisorption of hydrogen. The
dynamics of such a dissociative chemisorption of H2 are reported for the
first time in this paper. The process is associated with a negligible energy
barrier and the Kohn-Sham energy variation along the molecular
dynamics trajectory shows that the adsorption energy is approximately 50
KJ/mol. This value of H2 chemisorption on a coronene supported Pd4
cluster is in agreement with that on the unsupported cluster, as calculated
in the previous theoretical work of Zhou et al.8 The chemisorption energy
value on a unsupported Pd4 cluster reported by Matsura et al.34 was,
however, ~36 KJ/mol. It is worth mentioning that Matsura et al.34 and
Zhou et al.8 calculated the above mentioned chemisorption energies by
performing geometry optimizations of the system at two different states
and since the H-H bond distance observed by them was 0.82 Å and 0.85 Å
respectively (slightly higher than the H-H bond distance in H2 molecule),
they term this chemisorbed state as a state with “stretched” or
“weakened” H-H bond. In contrast, the H-H bond MD trajectory in Fig.
5.2 (b) suggests the dissociation of the H2 molecule. A non-electron
sharing interaction may exist though.
The CPMD simulation at 300 K is run up to 7 ps (not shown here)
after the hydrogen gets dissociatively chemisorbed, however, no chemical
event is observed in the system. This suggests that further migration of the
dissociated H atoms is linked with some energy barrier. Hence, to
accelerate the dynamics of the process and to compute the associated
energy barriers, metadynamics is implemented. The collective variable for
the metadynamics are shown in Fig. 5.1 as well. The starting point of the
metadynamics simulation is two H atoms dissociatively chemisorbed on
the tip of the coronene supported Pd4 cluster. Running the metadynamics
simulations results in an initial spillover of the chemisorbed H atoms on
130
the sides of the tetrahedral Pd4 cluster (will be discussed later). However,
further addition of potentials in the H-H collective variable space during
the metadynamics, not only results in the migration of the H atoms but it
also disintegrates the Pd4 cluster in the direction of the movement of H
atoms. Since, the H atoms are chemically bonded to the Pd atoms of the
cluster, their movement in metadynamics causes the Pd atoms to be
dragged along with the H atoms, instead of breaking the Pd-H bond. As a
consequence of this initial observation, all the metadynamics simulations
reported henceforth in this paper are run under the constraint of fixed
coordinates for the Pd atoms in the Pd4 cluster. We believe that a larger Pd
cluster will not disintegrate in this fashion though.
Figure 5.3 shows the reconstructed free energy surface as a function
of the above described collective variables, after the metadynamics
simulations are run on the system with 2 H atoms chemisorbed on the tip
of the Pd4 cluster supported on coronene and with fixed Pd coordinates.
The simulation system snapshots at the key landmarks in the energy
surface are also shown. The reconstructed energy surface does not appear
as smooth as one would expect and the possibility that it is a numerical
artefact of the choice of metadynamics parameters can not be completely
denied. However, given the facts that (i) due care was taken in
establishing the metadynamics parameters and that (ii) the present
literature8 suggests the presence of several local minima in the migration
of H atoms on a Pd cluster, we believe that the calculations reported in
this paper represent the correct physical picture of the investigated
phenomenon. We would also like to point out that since this is the first
time that a continuous energy surface of the process is computed, a direct
one-to-one comparison can not be made. As mentioned before, the
metadynamics simulations were started with 2 H atoms, separated by 1.94
a.u. (~1 Å), dissociatively chemisorbed on the Pd4 cluster. This particular
131
Figure 5.3: The three dimensional free energy surface reconstructed from the metadynamics simulation of the system with fixed Pd coordinates. S1, S2, and S3 indicate the key minima in the free energy surface and the images displayed below the plot are the snapshots of the system at corresponding values of the collective variables. The color coding of the atoms is the same as that of Fig.5.1.
S2 S1S3
6 KJ/mol
3.5 KJ/mol 0.5 KJ/mol 5 KJ/mol
S1
S2
S3
132
state of the system corresponds to a local minimum in the free energy
surface showed by the snapshot S1 in Fig.5.1. Analysis of the energy
surface calculated from metadynamics reveals that the system has to cross
an energy barrier of ~6 KJ/mol to come out of this energy well. It is worth
noting that the same energy barrier, when calculated on a system without
any constraints (no fixed Pd coordinates), was found out to be 5.6 KJ/mol.
This suggests that the computed energy surface and barriers are not
significantly altered due to the additional constraint of fixed Pd
coordinates.
After escaping from this local minimum, the system falls into
another energy well S2, which corresponds to the molecular configuration
of H atoms adsorbed on the adjacent edges of the Pd4 tetrahedron. The
configuration of the system with one H atom on the edge of the
tetrahedron and one dangling on the face of the tetrahedron corresponds
to a state with 5 KJ/mol higher energy than that of S2 (just before the
system falls into the energy well S2). As shown in Fig. 5.3, the energy
difference between the configurations at S1 and at S2, i.e. with H atoms
dissociatively chemisorbed on the tip of the Pd4 cluster and with H atoms
attached on the edges of the Pd4 cluster, is very minimal. An unsupported
Pd4 cluster, however, shows a significantly higher energy difference
between the two states.34 Further migration of H atoms from the state S2
requires the system to cross an energy barrier of ~ 3.5 KJ/mol. Though it
could be of interest, it is difficult to comment on the effect of constrained
Pd coordinates on this particular calculated barrier. After crossing the
barrier, H atoms gradually move towards the carbon support, thus taking
the system to an energy state which is 5 KJ/mol lower than that of the H
atoms attached to the edges of the cluster. The state S3 is the lowest energy
level state of the system. It should be noted that unlike an unsupported
Pd4 cluster, the lowest energy state is not with H atoms attached to the
133
edges of the cluster and that as the H atoms move from the tip of the
cluster towards the carbon support, the system keeps moving to lower
energy states. It would have been of great interest to compute the
dynamics and energetics of complete migration of H atom from the Pd
cluster to the carbon support, however, even after running the
metadynamics simulation for a practically feasible timescale, the event
was not observed. The computations were limited to the timescale when
the system reached the state S3 because of the following additional
reasons:
(i) Addition of H atom on a coronene molecule is highly site specific
and no bound state exists on the hollow or bridge site.61 The only possible
binding states are on the central carbon ring and on the edge and outer
edge (where the carbon already has one bound H atom) carbon atoms.61
The Pd4 cluster in our simulations is situated on the central hexagonal
carbon ring of the coronene molecule and hence the spillover of H atom
on the central ring carbon atom is not feasible due to its location right
beneath the cluster. The outer edge carbons are too far for the hydrogen to
reach. Since the only remaining option is the edge carbon atom, which
may not again be in close proximity to the bottom Pd atom from where the
H atom can spillover. Binding of H atom to the edge carbon is also
associated with an energy barrier of 25 KJ/mol (in addition to the barrier
that might be associated with the migration from the Pd cluster) and the
presence of Pd4 cluster may further increase to a level, which may take
impractically long computational time.
(ii) The relatively weaker Pd-C interaction in coronene62 may
significantly increase the energy barrier of the spillover step.33
(iii) To obtain the correct relative depth of the free energy basins in
metadynamics, it is important to stop the simulation after a recrossing
event, i.e., when the system crosses the barrier in the reverse direction48. In
134
our simulation, such a recrossing in the reverse direction and a diffuse
motion of H atoms are observed after the system reached state S3.
Gervasio et al.63 has also demonstrated that overfilling the energy surface,
in an attempt to explore the regions that are too high in energy,
significantly alters the topology of the energy surface, thus giving rise to
false energetic interpretation.
It has been reported in the literature that carbon supported Pd
clusters, upon exposure to hydrogen, form Pd-hydride even before the
hydrogen gets chemisorbed.4, 30 In an attempt to investigate the dynamics
and energetics of the interaction of hydrogen with a Pd-hydride cluster,
we found out that even an icosahedral Pd13 cluster was not big enough to
accommodate hydrogen in an absorbed hydride form and that the
computational cost of performing first-principles molecular dynamics and
metadynamics on a bigger cluster was not affordable (increasing the
cluster size also results in increasing the simulation cell size and the size of
carbon support). However, as can be seen in the snapshots of Fig. 5.4, we
perform the metadynamics simulations on a Pd4 cluster instead, which is
partially saturated with 3 H atoms chemisorbed on the edges of the
cluster. Unlike the bare Pd4 cluster, the Pd4-H3 cluster had a singlet
multiplicity. The starting point of the metadynamics simulation is again
the system with 2 H atoms (coded in red color in Fig. 5.4) dissociatively
chemisorbed on the tip of the Pd4–H3 cluster. The system is equilibrated
before starting the metadynamics simulation. Only the two H atoms
chemisorbed on the tip of the cluster are part of the metadynamics
collective variables and the rest of the H atoms on the edges of the cluster
are allowed to move under the influence of the forces resulting from the
routine Car-Parrinello molecular dynamics. It is observed in this case that
the migration of two H atoms from the tip of the cluster, under the
influence of metadynamics, is also accompanied by their alliance with the
135
Figure 5.4: The three dimensional free energy surface reconstructed from the metadynamics simulation of a system with the Pd4 cluster partially saturated with 3 H atoms. S1 – S6 indicate the key minima in the free energy surface and the images displayed below the plot are the snapshots of the system at corresponding values of the collective variables. The color coding of the atoms is the same as that of Fig.5.1, however, the H atoms included in the collective variables’ definitions are coded in red color.
S3S2S1
S6S5S4
S1S2
S3
S4S5
S6
136
pre-existing H atoms on the edges of the cluster. The separation distance
between them is approximately 1 Å and it suggests that their state is
similar to that of the H atoms that are dissociatively chemisorbed on the
tip of the Pd4 cluster, but still have some non-electron sharing interaction
(cf. S3 in Fig. 5.4). The state S3 with 4 H atoms chemisorbed on two tips of
the Pd4 cluster and 1 H atom chemisorbed on an edge of the cluster is
lower in energy that the state S1 with 2 H atoms chemisorbed on the tip
and 3 H atoms chemisorbed on the edges. This is consistent with our
previous observation from Fig. 5.3 that as the adsorbed H atoms migrate
towards the carbon support, the system falls into lower energy states. The
energy barrier associated with the migration of H atoms from the tip of
the cluster in this case is ~ 3.5 KJ/mol. The migration of H atoms during
the metadynamics simulation reveals several structures that are close in
energy, as shown in Fig. 5.4. Running metadynamics further results in one
of the H atoms moving under the influence of metadynamics to combine
with another H atom, whose movement is not influenced by
metadynamics, to form a desorbed H2 molecule. As expected, the
associative desorption of two H atoms in the form of an H2 molecule
results in a higher energy state of the system. The process is slightly
endothermic. Since the spillover of atomic hydrogen from the Pd cluster is
computationally not feasible in the system investigated due to large
energy barriers, the metadynamics simulation results in the desorption of
the H atom in the form of H2 molecule. The small energy barrier
associated with the associative desorption of H2 molecule from the cluster
is approximately 2.5 KJ/mol and is much less than the barrier for the
desorption of atomic hydrogen chemisorbed on a carbon support.5, 31 This
(small computed energy barrier) is in agreement with and demonstrates
the root cause behind the experimental results4 which show that the
137
desorption of hydrogen chemisorbed on the surface of metal clusters takes
place earlier than the desorption of H atoms from the carbon support.
To investigate the transport of atomic hydrogen on the partially
saturated Pd4 cluster, we repeat the metadynamics simulation, but with
the pre-existing H atoms on the edges of the cluster kept frozen. Figure 5.5
shows the computed energy surface for the simulation. A comparison of
the energy surface in Fig. 5.5 with that in Fig. 5.3 shows significant
similarity. The topologies of the energy surfaces are quite alike, however,
it can be observed that the energy barrier associated with the initial
migration of H atoms from the tip towards the sides of the cluster is
reduced from 6 KJ/mol (as in bare Pd4 cluster) to 2 KJ/mol. After this
initial migration, the system falls into the state S2. The difference between
the energies of the two states S1 and S2 is significantly higher in this case. It
is merely 0.5 KJ/mol in the bare Pd4 cluster, however, it is 6.5 KJ/mol in
this case. After running the metadynamics simulation further, the system
crosses the energy barrier of 3.5 KJ/mol for further migration of the H
atoms. This energy barrier is exactly the same as that in the system with a
bare Pd4 cluster. The lowest energy state observed in this simulation is the
state S5 as shown in Fig. 5.5. Since the starting point S1 state is exactly the
same for both the simulations, with and without frozen pre-existing H
atoms, we can say that the states S2 and S5 which are explored in the case
of frozen H coordinates are energetically more stable than any of the states
that are explored in the case of freely moving pre-existing H atoms (cf. Fig.
5.4). The metadynamics simulation in this case is also run till the point
where the recrossing event takes place in the reverse direction.
Summarizing the above discussion, we perform ab initio molecular
dynamics simulations of dissociative chemisorption of H2 on the tip of the
coronene supported Pd4 cluster and ab initio metadynamics simulations of
the migration of dissociatively chemisorbed H atoms on the tip of (i) a
138
Figure 5.5: The three dimensional free energy surface reconstructed from the metadynamics simulation of a system with the Pd4 cluster partially saturated with 3 H atoms. The coordinates of the 3 H atoms are fixed. S1 – S5 indicate the key minima in the free energy surface and the images displayed below the plot are the snapshots of the system at corresponding values of the collective variables. The color coding of the atoms is the same as that of Fig.5.4.
S1 S2 S3
S4 S5
2 KJ/mol
3.5KJ/mol
6.5 KJ/mol
3.0 KJ/mol
S1
S2S3S4S5
139
pure Pd4 cluster, (ii) a Pd4 cluster partially saturated with 3 H atoms and
(iii) a Pd4 cluster partially saturated with 3 H atoms whose coordinates are
frozen. Though the initial dissociative chemisorption is barrierless, further
migration of H atoms on the cluster is associated with small energy
barriers. The barrier associated with the initial migration of H atoms from
the tip of the cluster is reduced when the cluster has some pre-existing H
atoms chemisorbed on it. We suspect that the presence of absorbed
hydrogen in the form of hydride may also have a similar effect of reducing
the energy barrier in the migration of surface H atoms. Migration of H
atoms from the tip of the cluster towards the support results in the system
moving to lower energy states. In case of an unsupported Pd4 cluster, H
atoms chemisorbed on the edges of the cluster is the energetically most
stable state.8, 34 However, when supported on carbon, the state with H
atoms attached to the tip of the Pd cluster in contact with the carbon
support is energetically the most stable. This shows that the migration of
H atoms on the Pd4 cluster towards the carbon support is an energetically
favourable process. However, the barrier associated with the spillover of
atomic hydrogen from the cluster to the carbon support seemed to be a
large one. This barrier is not only dependant on the metal cluster but also
on the capacity of the carbon support to accept the spillover H atom.
Spillover of hydrogen, if happens on a sub-nano sized cluster, will happen
by migration of H atoms from the metal cluster to the carbon support and
the vacant sites on the cluster will then be replaced by another incoming
hydrogen. Since it is not possible to perform a continuous metadynamics
simulation with changing/dynamic collective variables’ definition, we
partially saturate the cluster with H atoms and perform the metadynamics
simulations. It is however observed that due to the comparatively small
barrier associated with the desorption of hydrogen from the partially
saturated cluster, 2 H atoms recombine to form a H2 molecule and get
140
desorbed from the cluster. To avoid the associative desorption of the H
atoms, the pre-existing H atoms are frozen during the metadynamics
simulations. The presence of pre-existing H atoms on the surface of the
cluster did not significantly alter the topology of the energy surface. The
migration of H atoms from the tip of the cluster that is exposed to
hydrogen to the tip that is in contact with the carbon support is associated
with small barriers and again the significant barrier is the one that is
associated with the transfer of H atom from the Pd atom to the carbon on
the support.
Given the small size of the coronene molecule, physisorption of the
desorbed H2 molecule on coronene is not observed in the simulation. If the
H2 physisorption energy on carbon materials is considered to be
approximately 10 KJ/mol, then it can be said that upon physisorption, the
system would land into a state that is lower in energy than the state S3 (cf.
Fig. 5.4). The dynamics and energetics shown in Fig. 5.4 also suggest that a
small energy barrier is associated with the recombination of H atoms to
form a desorbed H2 molecule. However, as mentioned before, the
spillover of atomic hydrogen from the Pd4 cluster is expected to be
associated with a relatively much larger energy barrier. It could be
possible that in case of sub-nano sized Pd clusters, where the formation of
Pd-hydride does not take place, hydrogen spillover may not take place at
room temperature. When the Pd cluster is exposed to more H2 molecules,
the new molecules may replace the existing molecules and the existing
molecules which are associatively desorbed and are “hot” due to the extra
energy they have may get physisorbed on the support, thus enhancing the
physisorption capacity. The experimental investigations which confirm
the presence of atomic hydrogen chemisorbed on the carbon support4, 30, 35
also report the presence of Pd-hydride and we suspect that the formation
of hydride may play a significant role in the spillover mechanism. One
141
possibility is that it may lower the energy barrier associated with the
migration of surface H atoms from the cluster to the carbon support and
the other is that it may also change the believed mechanism of spillover,
where not the surface H atoms but the H atoms in the hydride phase are
pumped out of the cluster to get adsorbed on the carbon support in an
atomic form.30 The energetics associated with this process will be
completely different than for the migration of surface H atoms from the
cluster.
5.5 Conclusions
The dynamics of the chemisorption and subsequent migration of
hydrogen on a carbon supported Pd4 cluster at room temperature are
simulated using the first-principles molecular dynamics. The initial
dissociative chemisorption is observed to be a barrierless process,
however, the subsequent migration of H atoms is associated with small
energy barriers less than 10 KJ/mol. The dynamics of the migration of H
atoms are accelerated and the energy surface as a function of relevant
coordinates is reconstructed using the first-principles metadynamics
technique. The migration of H atoms from the tip of the cluster towards
the carbon support is an energetically favourable process. Unlike an
unsupported Pd4 cluster, the H atoms attached to the Pd atoms nearest to
the carbon support is the energetically most stable state. Presence of pre-
existing H atoms on the surface of the cluster does not alter the topology
of the energy surface of the migration of H atoms on the cluster
significantly but slightly reduces the associated barriers. The migration of
H atoms from the cluster to the carbon support is associated with a
significantly high energy barrier and the unwillingness of the carbon
support to accept the H atom may further increase this barrier. It is
plausible that the spillover of H atoms from a sub-nano sized Pd cluster
142
may not take place at room temperature and that the formation of Pd-
hydride plays a significant role in the spillover of atomic hydrogen from
the cluster to the support. It is also demonstrated that the associative
desorption of hydrogen in the form of a diatomic H2 molecule is also not a
high energy barrier process and may add to the physisorption capacity of
the carbon support.
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on hydrogen adsorption in single-walled carbon nanotubes through
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destabilization of palladium hydride in the hydrogen uptake of Pd-
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hydrogen spillover in MoO3 and carbon-based graphitic materials. Journal
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150
6 AN INTEGRATED MODEL FOR ADSORPTION-INDUCED STRAIN IN MICROPOROUS SOLIDS
6.1 Summary
Deformation of porous materials during adsorption of gases, driven by
physico- or chemo-mechanical couplings, is an experimentally observed
phenomenon of importance to adsorption science and engineering.
Experiments show that microporous adsorbents exhibit compression and
dilation at different stages of the adsorption process. A new integrated
model based on the thermodynamics of porous continua (assumed to be
linear, isotropic and poroelastic) and statistical thermodynamics is
developed to calculate the adsorption-induced strain in a microporous
adsorbent. A relationship between the strain induced in the adsorbent and
the equilibrium thermodynamic properties of the adsorbed gas is
established. Experimental data of CO2 adsorption-induced strain in
microporous activated carbon adsorbents (Yakovlev et al., Russ. Chem.
Bull. Int. Ed., 54, 2005) is used to fit the model parameters and to validate
the model. Assuming that the initial contraction in a microporous
adsorbent is caused due to an attractive interaction between the adsorbed
gas and the adsorbent, we demonstrate that there also exists a repulsive
interaction amongst the adsorbed gas molecules and that this repulsive
interaction can be correlated to the adsorption-induced strain. The
proposed correlation can be extended to take into account the adsorbate-
151
adsorbent attractive interaction in order to offer a detailed and more
comprehensive explanation of the adsorption-induced strain in
microporous adsorbents.
6.2 Introduction
Adsorption is a phenomenon in which a solid surface with
unbalanced forces, when exposed to a gas, bonds (physically in
physisorption or chemically in chemisorption) to the gas molecules. The
solid substrate is referred to as the adsorbent whereas the gas phase is the
adsorbate. The solid adsorbent is usually a porous medium and the pores
are classified as micropores (< 20 Å), mesopores (20-50 Å), and
macropores (> 50 Å).1 Adsorbents like alumina, silica, activated carbon,
activated carbon fibers, zeolite, to name a few, are extensively used in
separation and purification processes where physisorption takes place
inside the pores and adsorption is analyzed as a volume phenomenon.2 In
the chemical industry, 90% of the chemicals are manufactured using
catalytic processes3 where adsorption is also studied as a 2-D surface
phenomenon. Depending upon the pore size, the type of adsorption, and
the underlying physics either approach is used to describe adsorption
equilibria and kinetics.
Given the environmental, industrial, and chemical importance of
adsorption, extensive experimental and theoretical research has been done
in the field adsorption science and the review article by D browski4
critically summarizes the key developments. Most of these studies
consider that an adsorbent is thermodynamically inert and it only
contributes in inducing an external force field on the adsorbate. This
assumption reduces the adsorption process to a thermodynamic
phenomenon without any mechanics. However, the adsorbent may not be
an inert component during the process of adsorption and this was first
152
shown by Meehan et al.5 when a dimensional change was observed in
charcoal due to CO2 adsorption. Wiig and Juhola6 and Haines and
McIntosh7 also observed dimensional changes in carbonaceous adsorbents
upon adsorption and more recently Levine,8 Yakovlev et al.,9, 10 Day et
al.,11 and Cui et al.12 have also shown that dimensional changes take place
in porous adsorbents during gas physisorption. Deformation and surface
stresses are also caused during chemisorption and those have been
experimentally observed by Men at al.,13 Grossmann et al.,14, 15 and Gsell et
al.,16 to name a few.
Since an adsorbent can not be inert during the process of
adsorption, adsorption properties are bound to be affected due to the
dimensional changes occurring in the solid. The degree of deformation
can be as large as 50%17 or as small as 0.1%,10 but even a relatively small
deformation can cause a substantial impact on the experimentally
determined equilibrium thermodynamic characteristics of the adsorption
system.9 For example, Kharitonov et al.18 showed that the differential
molar isosteric heat of adsorption (CO2-activated carbon system) increased
in the initial adsorption region and the authors suggested that it was due
to the contribution of energy from the adsorption deformation
phenomenon. Yakovlev et al.10 further confirmed this fact by
experimentally observing CO2 adsorption-induced strain in a similar
adsorbent and they also estimated the correction to the isosteric heat of
adsorption for the non-inertness of the adsorbent to be 10-15%, when the
strain was not more than 0.1%. In case of CO2 and H2S sequestration, even
though the strain is small it has been reported that large stresses
developed due to adsorption may significantly affect the geomechanical
and permeability characteristics, thereby implicating sorption estimates.11,
12, 19 Also in the case of chemisorption, it has been shown that not only the
properties of the adsorbent but also the adsorbate properties are modified
153
in a strained adsorbent.13, 16 Wu and Metiu20 also showed using atomistic
level modeling of CO adsorption on Pd that the parameters controlling the
thermodynamics of chemisorption, like binding energy and vibrational
frequencies, were altered when the adsorbent was strained.
Recognizing the non-inertness of an adsorbent and realizing the
significance of the effect of adsorption-induced strain and stress on (i) the
adsorption characteristics, (ii) the adsorbent, and (iii) the adsorbate,
theoretical studies have also been performed in this field. Dergunov et al.21
developed a model which took into account the adsorption deformation to
calculate the potential energy of adsorption. They showed that the
maximum contraction in the adsorbent corresponds to the minimum
stress. Recently Pan and Connell22 combined Myers’ solution
thermodynamics approach for adsorption in micropores23 and Scherer’s
strain model24 to calculate gas adsorption-induced swelling in coal.
Serpinskii and Yakubov25 had developed an analytical expression between
the strain and the amount of gas adsorbed using vacancy solution theory
and Hooke’s law, but their expression required the bulk modulus to be a
function of the amount of gas adsorbed, if compression and dilation in the
adsorbent were to be observed. A similar approach was also taken by
Jakubov and Mainwaring,26 where they expressed the strain as a function
of the amount of gas adsorbed and the difference between the chemical
potentials of the gas, adsorbed on a strained and on an unstrained
adsorbent. However, prior information of strain was required to calculate
the chemical potential of the adsorbate that would have been adsorbed if
the solid was prevented from strain. Ravikovitch and Neimark27
developed a non-local density functional theory based model to calculate
the adsorption-induced strain for Kr and Xe adsorption on zeolite. They
were successfully able to reproduce the experimentally observed
contraction and expansion of the adsorbent. When adsorption is treated as
154
a 2-D surface phenomenon, Ibach28 showed that a Maxwell type relation
exists between the dependence of chemical potential of an adsorbate on
surface strain and the dependence of surface stress on coverage.
Weissmüller and Kramer29 studied the metal-electrolyte system, using a
continuum description of a solid adsorbent, where they identified
experimentally measurable state variables in the system and established a
relation between the state variables of the surface and those of an
adsorbate. Lemier and Weissmüller30 also calculated hydrogen
adsorption-induced strain in nanocrystalline Pd using the theory of
thermochemical equilibrium in solids, developed by Larche and Cahn,31
and by expressing the adsorption-induced strain as a function of state
variables, pressure and chemical potential of the adsorbate. Müller and
Saúl32 reviewed some key theoretical contributions in adsorption-induced
stress on a plane surface, including some of the well recognized work of
Ibach.33
In summary, the significant facts in the adsorption-induced stress
and strain literature lead to the following observations: (i) experimental
investigations clearly show the existence and implications of adsorption-
induced strain and stress,5-12, 18, 19 (ii) theoretical studies are needed to
better understand the phenomenon, (iii) some fundamentally significant
research, using thermodynamic and atomistic approaches, has been done
to study the effects of stress and strain on the physics of surface
adsorption,20, 28-30, 33 and (iv) a better understanding of the adsorption-
induced strain, particularly in microporous adsorbents like activated
carbon and zeolites, where an adsorbent first undergoes contraction
followed by an expansion, needs further research in this area. Hence, this
paper focuses on the adsorption-induced strain in microporous
adsorbents. Two previous significant and very valuable contributions in
this problem are: (i) The recent work of Ravikovitch and Neimark27 using
155
non-local density functional theory approach; however, in this work a
simple relation between the adsorption-induced strain and an equilibrium
adsorption property was not established and the strain was assumed to be
entirely due to the deformation of pore space, thereby neglecting the
change in the volume of the solid matrix (A solid adsorbent is composed
of pore space and solid matrix, cf. Fig. 6.2). (ii) Jakubov and Mainwaring26
developed a relation between the adsorption-induced strain and the
difference in the chemical potentials of an adsorbate, when adsorbed on a
strained and on an unstrained adsorbent and they make a similar
assumption of strain in pore space only. However, when they calculate the
difference in the chemical potentials using the difference in adsorption
isotherms, on a strained and on an unstrained adsorbent, the magnitude
of the difference in the isotherms appears to be too large (up to 50%) for
the observed strain ( 0.05%).
In the present paper: (i) we develop a simple relationship between
the adsorption-induced strain and an equilibrium adsorption property by
(ii) taking into account the strain in the solid matrix and in the pore space
and (iii) we show that this relationship can be used to predict the
adsorption-induced strain in microporous adsorbents, and (iv) can
provide a molecular level explanation for the adsorption-induced strain
which to the best of our knowledge has not been previously done. A novel
feature of this model is the integration of adsorbent mechanics with
statistical thermodynamics.
The organization of this paper is as follows. Section 6.3.1 presents
the key equations of the mechanics and thermodynamics of porous
adsorbents34 and the method to calculate the difference between the
chemical potentials of the gas, adsorbed on a strained and on an
unstrained adsorbent. The statistical mechanical model for the chemical
potential difference is described in section 6.3.2 and section 6.3.3 describes
156
a method to predict the adsorption-induced strain by combining the
approaches in section 6.3.1 and 6.3.2. Section 6.4 discusses and validates
the results with previously presented experimental data.10 Section 6.5
presents the conclusions.
Figure 6.1: Schematic of adsorption-induced strain in microporous adsorbents. A typical trend observed in microporous adsorbents where the adsorbent first contracts and then expands.
0
Stra
in in
Ads
orbe
nt
Amt. of Gas Adsorbed
157
Figure 6.2: A Porous adsorbent continuum consisting of; (i) the solid matrix and (ii) the pore space (adapted from Coussy, 2004). It is also referred to as skeleton at some places in the text.
6.3 Model Development
Microporous adsorbents like activated carbon and zeolites exhibit a
typical deformation behaviour where the adsorbent undergoes contraction
in the initial stage of adsorption and later it expands. An illustration of
this behaviour is shown in Fig. 6.1, where the adsorption-induced strain is
plotted as a function of amount of gas adsorbed. . As mentioned above,
the model is based on the integration of the theory of thermoelasticity of
porous continua for the adsorbent and a statistical thermodynamics based
model for the chemical potential of the adsorbate. A necessary and
sufficient description of both the models and their interrelationship is
presented here.
Pore Space(containing adsorbate) Solid Matrix
Skeleton = Empty Pore Space + Solid Matrix
Occluded Pore Space
158
6.3.1 Mechanics of porous adsorbent
The thermodynamics based theory of porous continua, developed
by Maurice Biot and described recently by Coussy,34 is used here to model
the porous adsorbent. The adsorption system is assumed to be a
superimposition of the skeleton continuum (i.e. the solid microporous
adsorbent) and the adsorbate. The skeleton continuum consists of a solid
matrix and a pore space (without the adsorbate) and the adsorbate gas
particles are present in the pore space (in the connected pore space, not in
the occluded pore space) of the skeleton (cf. Fig. 6.2). Let s be the
Helmholtz free energy of the skeleton per unit initial (undeformed)
volume of the skeleton. 0 (cc/gm) is the initial (undeformed) volume of
the skeleton and 0V (cc/gm) is the pore volume (of the connected pore
space of an undeformed skeleton) associated with 0 . With the
assumptions that (i) the porous skeleton continuum is thermoelastic, (ii)
there is no dissipation related to the skeleton and (iii) the deformation in
the skeleton is small, the differential Helmholtz free energy density can be
written as34
dTSPddd sijijs (6.1)
where P is the pressure, is the porosity, T is the temperature, sS is the
entropy, ij are the linearized strain components and ij are Cauchy stress
components. Porosity is defined as the ratio of pore volume (V) of the
connected pore space to the undeformed skeleton volume 0 . Using a
Legendre transformation, the state variables can be changed if equation
(6.1) is expressed in terms of free energy sG as
PG ss (6.2)
Differentiating equation (6.2) and substituting equation (6.1) in it, we get
dTSdPddG sijijs (6.3)
159
The Maxwell relations for equation (6.3) are as follows:
ij
sij
ij
ij STP
, and PS
Ts (6.4)
The strain tensor ij can be split into a shape changing but volume
conserving part (deviatoric) plus a volume changing but shape conserving
part (dilation), giving:
changingvolume
3
1changingshape31
kkkijijij e (6.5)
Then we can introduce a trace E and a deviatoric ije strain:
ii and ijijije31 (6.6)
where ij is the 3D Kronecker delta. A similar decomposition of the stress
tensor gives:
ii31 and ijijijs
31 (6.7)
Introducing equations (6.6) and (6.7), equation (6.3) now reads34:
dTSdPdesddG sijijs (6.8)
The state equations for sij Ss ,,, , with the free energy
TPeGG ijss ,,, formulation, are
T
s
eTP
s
TPij
sij
ijijPGG
eGs
,,,,,
,, ,,,Pe
ss
ijTG
S (6.9)
A material is isotropic when the energy functions only depend on the first
invariant of the strain tensor or the trace of the strain tensor.34 In other
words, the energy functions only depend on the total volume of the
material and are independent of the shape of the material. The elastic
energy part of the total free energy of a linear, isotropic and
thermoporoelastic material is only dependant on the first invariant of the
strain tensor and on the second invariant of its deviatoric part.
160
Differentiating the terms in equation (6.9), with an assumption that the
skeleton is linear and isotropic, gives sij dSddsd ,,, as follows.34
KdTbdPkdd 3 and ijij eds 2 (6.10)
dTNdPbdd 3 (6.11)
TdTCdPKddSs 33 (6.12)
where
2
2sGk (bulk modulus),
TG
K s2
, 2
2
TG
TC s ,P
Gb s2
,
2
21PG
Ns ,
TPGs
2
3 and 2
2
ij
s
eG (shear modulus) (6.13)
Equations (6.10), (6.11) and (6.12) are the constitutive equations of an
adsorbent that is linear, thermoporoelastic and isotropic and when an
adsorbent undergoes any deformation during the process of adsorption.
Since adsorption is an isothermal process we set 0dT in equations (6.10-
6.12).
The interest of the present work is the adsorption-induced strain
and stress. Equation (6.11) can be used to calculate the change in strain
due to change in porosity and pressure and the coefficient b (also referred
as Biot’s coefficient) can be viewed as the relationship between strain
change and porosity change when the pressure and temperature are kept
constantPG
PGb ss
2
. All the previous adsorption-
induced strain modeling efforts have assumed that the strain is completely
due to the change in porosity (b 1) which may not be a completely valid
assumption for all the adsorbents. However, if the occluded pore space is
absent (cf. Fig. 6.2) it can be a valid assumption. Parameter N (also referred
as Biot’s modulus) relates the change in porosity variation to change in
161
pressure variation when strain and temperature are kept constant
PPG
PPG
Nss
2
21 . Equation (6.10) can then be used to
calculate the stress change due to the changes in strain and pressure. From
the Maxwell relations, the coefficient b can also be viewed as the
relationship between stress change and pressure change P
when the strain and temperature are held constant.
In the case of adsorption, the adsorbent skeleton is an open
thermodynamic system which can exchange mass with the surroundings.
The adsorbate gas gets adsorbed in the connected pore space of the solid
adsorbent. If am is the amount of gas adsorbed per unit initial volume of
the adsorbent skeleton ( 0 ) and a is the density of the adsorbed gas in
the porous space then,
aam (6.14)
Differentiating equation (6.14) with respect to pressure (P) and
rearranging, we get
dPdm
dPdm
dPd a
a
aa
a2
1 (6.15)
The term dP
dma and am can be calculated form the adsorption isotherm
and if a model describing the density of the adsorbate in the pores is
available, the change in porosity can be calculated using equation (6.15).
The porosity change can further be used to calculate the total strain and
stress in the adsorbent using equations (6.10) and (6.11) and the
poroelastic properties of the adsorbent. However, it has to be noted that
equation (6.15) calculates the change in porosity based on the difference
between the amount of gas adsorbed in a strained adsorbent
(experimental adsorption isotherm) and the amount of gas that would
162
have been adsorbed if the adsorbent is prevented from strain (using a
model for adsorbate density in pores). Given the extremely small
magnitude of strain in adsorbents, an exceptionally accurate model for the
density of the adsorbate in the pores, a , is needed and any simplistic
analytical model can not be used. Molecular level density computation
techniques may provide the required accuracy in this case. Hence, instead
of correlating the adsorption-induced strain (via porosity) to the amount
of gas adsorbed and to its density in the pore space (as in equation 6.15),
we propose another method where we correlate the adsorption strain (via
porosity) to the chemical potential of the adsorbate. This alternative
approach is based on the following sequence where we calculate the
difference between the chemical potentials of the gas adsorbed on a
deformed and on an undeformed adsorbent using the experimental strain
data:
(I) From equation (6.11) the change of porosity is related to the strain by:
ModulussBiot'
DataStrainalExperimenttCoefficien
sBiot'unknown
1NdP
dbdPd (6.16)
Given the material properties (b, N), using equation (6.16), we calculate
dPd from available experimental strain data.
(II) We calculate P by numerically integrating dPd in step (II) and use
experimental adsorption isotherm Pma to calculate the density of the
adsorbate Pa , using equation (6.14). At this stage we have density as a
function of pressure. ( )a a P .
(III) If the adsorbent would have been prevented from strain and if am is
the amount of gas in an unstrained adsorbent at pressure P then
163
0
000 ,
VPPm aa (6.17)
Figure 6.3: An illustration of adsorption isotherms when the adsorbent undergoes deformation, as shown in Figure 6.1, (full line) and when the adsorbent is prevented from deformation (dashed line).
Using the above mentioned steps (I-III) we can calculate the
adsorption isotherm for the adsorbent if it would have been prevented
from strain. Figure 6.3 shows an illustration of the adsorption isotherms
with strain Pma and without strain Pma , for an adsorbent which
exhibits an adsorption-induced strain as shown in Fig. 6.1. The two
adsorption isotherms can be used to calculate the difference in the
chemical potentials of the gas, adsorbed on a strained and on an
unstrained adsorbent. If sxP is the pressure required get xm amount of gas
adsorbed when the adsorbent undergoes deformation and if wsxP is the
pressure to get the same amount of gas adsorbed (cf. Fig. 6.3 for
illustration) when the adsorbent is prevented from strain then
ln(P
)
Amt. of Gas Adsorbed
xm
wsxP
sxP
164
wsx
sx
Bwsx
sx
diffx P
PTk ln (6.18)
where wsxands
x are the chemical potentials of the adsorbed gas
(Joules/molecule) when mx amount of gas is adsorbed, with strain and
without strain, respectively; Bk is the Boltzmann’s constant. In
formulating equation (6.18) we make use of the fact that the adsorbed
molecules are in equilibrium with the surrounding, unadsorbed gas
molecules adsorbategas . However, it has to be noted that the
usage of equations (6.15-6.17) imply that the constitutive equation (6.11) of
the solid adsorbent (the skeleton in Fig. 6.2) is independent of the type and
characteristics of the adsorbate.
In partial summary, this section: (i) provides a relationship between
the adsorption-induced strain and adsorption-induced porosity change
(equation 6.11) and (ii) illustrates a method to calculate the difference
between the chemical potentials of the adsorbate adsorbed on a strained
and on an unstrained adsorbent ( diffx ), using the relationship in (i) and
the experimental adsorption isotherm. Section 6.3.2 describes a statistical
thermodynamics based model for diffx and section 6.3.3 illustrates a
method to predict the adsorption-induced strain by comparing diffx
calculated using the method presented in this section with that using the
method presented in section 6.3.2. It has to be noted that equations 6.10
and 6.12 are not used in the present work; however, they can be used to
calculate the adsorption-induced stress and entropy once the adsorption-
induced strain is known and this paper presents a method to calculate the
adsorption-induced strain.
165
6.3.2 Chemical potential of the adsorbate
A summary of different approaches used to model the chemical
potential of the adsorbed molecules is given by Kevan.35 Adsorption in
micropores has been previously modeled using Lattice gas models.3, 36-41
According to the lattice gas model, the adsorbed gas is considered as a
layer where the molecules are free to move around but are not allowed to
leave the surface. When the chemical structure of the adsorbent surface
does not change, the adsorbate does not undergo any chemical change
upon adsorption (like molecular dissociation) and there is a complete
absence of adsorbate-adsorbate interaction then the chemical potential of
the adsorbed gas is given as37, 39
int30 ln1
ln qqTkBa (6.19)
where 0 is the binding energy (positive) of an isolated adsorbate
molecule on the surface relative to the molecule far away from the surface
with zero kinetic energy, is the coverage, 3q is the vibrational partition
function of the adsorbed molecule, and intq is the internal partition
function of the molecule. The chemical potential calculated using equation
(6.19) is only applicable for adsorbate molecules that are not interacting
with each other. However, as mentioned before, densely adsorbed
molecules in micropores do exhibit repulsive interactions. The effect of
lateral interactions between the adsorbed molecules can be modeled using
the popular analytical Quasichemical approximation.3, 42 The
Quasichemical approximation has previously been used to model the
lateral interactions between the adsorbed molecules in micro and
mesoporous materials30, 39-41, 43-47 and if the chemical potential is split into
two parts39, 40, i.e. a term due to adsorbate-adsorbent interaction (equation
166
6.19) and a term due to adsorbate-adsorbate interaction, then the later
according to the Quasichemical approximation becomes39, 42, 43
12121ln
21int Tckc Bnna (6.20)
where
TkB
nnexp1141 (6.21)
c is the number of nearest neighbouring adsorbate particles (site
coordination number) of the molecule whose chemical potential is intaa and nn is the strength of interaction between the nearest
neighbor adsorbate molecules. For a repulsive interaction between the
adsorbate molecules, 0nn and for an attractive interaction, 0nn .
It has been suggested in the literature that the adsorption-induced
contraction strain in microporous adsorbents is caused due to the strong
attractive forces between the gas molecules and pore walls and as a result
of these attractive forces the gas molecules reach very high densities when
adsorbed in these microporous adsorbents.27, 48-50 But it has also been
shown that due to this dense packing of the gas molecules, repulsive
interactions amongst the gas molecules are developed since the
intermolecular distance is less than the minimum in their potential
curve.27, 48-50 It has to be noted that though there exist repulsive
interactions between the adsorbate molecules, adsorption continues since
the decrease in the free energy of the adsorbed molecule due to the
molecule-adsorbent interaction is larger in magnitude than the increase in
the free energy due to its repulsive interactions with the neighbouring
adsorbate molecules.49
167
Applying the lattice gas model and Quasichemical approximation
(equations 6.19 and 6.20) to calculate chemical potentials sx and ws
x we
get
12121ln
21ln
1ln int30 ws
ws
BwsnnB
wsx TckcqqTk
TkB
wsnnws exp1141 (6.22)
and
12121ln
21ln
1ln int30 s
s
BsnnB
sx TckcqqTk
TkB
snns exp1141 (6.23)
The contractive strain in the adsorbent also results in contraction of the
pore space (equation 6.16) and as a result it may bring the gas molecules
further closer thereby increasing the repulsive interaction amongst them.
However, when the adsorbent undergoes dilation, the repulsive
interactions between the adsorbate molecules are reduced since the
increase in pore space will take the molecules apart. Hence in equations
(6.22) and (6.23), wsnn
snn for a contractive strain and ws
nnsnn for an
expansive strain. Applying equations (6.22) and (6.23) to equation (6.18)
we get,
2121
2121ln
21
ws
ws
s
s
Bwsnn
snn
diffx Tckc (6.24)
It has also been shown by Wu and Meitu20 that the nearest neighbour
interaction nn (CO adsorption on Pd) is larger for a contractive strain and
is smaller for an expansive strain. Based on Wu and Meitu20 and on the
fact that in case of adsorption in micropores, the contraction or expansion
168
of the pore space takes the adsorbed molecules closer or farther,
respectively, we can write
0wsnn
snn (6.25)
Equation (6.24) can then be written as
2121
2121ln
21
0 ws
ws
s
s
Bdiffx Tckc (6.26)
Equation (6.26) expresses the difference between the chemical potentials of
the adsorbate adsorbed on a strained and on an unstrained adsorbent as a
function of the change in the pore space of an adsorbent 0 (and
consequently the adsorption strain via equation 6.16), coverage and
temperature T.
6.3.3 Calculating adsorption-induced strain
The model equation (6.26) correlates diff to the adsorption-
induced porosity change, temperature and coverage. However, diff can
also be calculated for a given porosity ( ) using the experimental
adsorption isotherm data, as shown in section 6.3.1. The difference or
residual,
)18.()26.( eqndiff
eqndiff (6.27)
can thus be minimized to yield the sought after adsorption-induced strain.
Using the model equation (6.26) and experimental adsorption isotherm, an
iterative procedure can be set-up to predict the adsorption-induced
porosity change and hence the adsorption-induced strain. However, the
dependence of diff on is larger than on T and (equation 6.26) and
hence instead of directly comparing diff calculated using equation (6.26)
to that calculated using the method in section 6.3.1, the procedure is
slightly modified so as to cancel out the error introduced in the model
during the fitting procedure.
169
Figure 6.4: Flowchart of the procedure for calculating the adsorption-induced strain. The corresponding explanatory text is given in section 6.6.
(1) Assume a strain function f at
(2) Calculate using eqn. (6.16) and exp. adsorption isotherm at T
(3) Calculate diff using eqn.(6.26)
(4) Calculate wsP using exp. adsorption isotherm and
(5) Calculate sP using wsP and diff , from eqn. (6.18)
(6) Recalculate porosity, ~
(7) Calculate diff~ using ~ in eqn. (6.26)
(8) Calculate the relative error, 2~
diff
diffdiff
Minimized?
End
Y
Update strain function f
N
170
A flowchart of the procedure is shown in Fig. 6.4 and a detailed
explanation is given below in section 6.6.
6.4 Results and discussion
In this section the relationship of adsorption-induced strain in a
linear, isotropic, and poroelastic microporous adsorbent with the chemical
potential of the adsorbate, as developed in section 6.3 is validated using
the experimental strain data of Yakovlev et al.10 The adsorption-induced
strain in microporous activated carbon adsorbent was measured by
Yakovlev et al.10 for CO2 adsorption. Figures 6.5a and 6.5b show the
experimental adsorption isotherms and the experimental adsorption-
induced strain data, adapted from Yakovlev et al.10. The strain data shows
the typical trend observed in microporous adsorbents where the
adsorbent first contracts and then expands. The data at 243 K is used in
the present study for validation and for fitting parameters, where as the
data at 273 K and 293 K is used to validate the calculated results.
171
(a)
(b)Figure 6.5: (a) CO2 Adsorption isotherms and (b) CO2 adsorption-induced strain data (adapted from Yakovlev et al., 2005) at 243 K ( ), 273 K ( ), and 293 K ( ). The adsorbent is microporous activated carbon.
172
Experimental adsorption isotherms and the experimental strain
data are interpolated and a numerical derivative of strain with respect to
pressuredPd is calculated. Material parameters b and N are needed to
calculatedPd (from equation 6.16), however, their values are unknown for
the microporous activated carbon adsorbent. Bouteca and Sarda51 and
Coussy34 have reported orders of magnitude of these properties for
different materials and based on those values, we assume 5.0b and
100N GPa. The porosity of the undeformed activated carbon adsorbent
is reported by Yakovlev et al.10 as 52875.00 . Figure 6.6 shows the
porosity change calculated from the experimental strain data (in figure
6.5b), employing equation (6.16). Using the porosity data, we calculate the
adsorption isotherm if the adsorbent is prevented from deformation
(using equation 6.17). However, since the change in porosity is of the
order of magnitude of 10-3, it is not possible to visually differentiate the
isotherms, with and without strain, hence we do not show the plots of the
calculated isotherms (An exaggerated illustration is shown in Fig. 6.3).
Using the two isotherms we calculate diff (equation 6.18, section 6.3).
Figure 6.7 shows the diff as a function of the amount of gas adsorbed at
243 K, 273 K and 293 K. The chemical potential difference diff data in
Fig. 6.7 and the porosity data in Fig. 6.6, at 243 K, are used to fit the
parameter in equation (6.26). The strength of interaction between the
nearest neighbour adsorbate molecules in an undeformed adsorbent wsnn
is taken equal to TkB (> 0, hence repulsive interaction). is calculated as
the ratio of the amount of gas adsorbed at a pressure to the maximum
adsorption capacity of the adsorbent at that temperature, as determined
from the adsorption isotherm. c, the number of nearest neighbouring
adsorbate particles (site coordination number), is taken as 6. Figure 6.8
173
compares the chemical potential difference diff calculated using the
model equation (6.26) and fitted parameter with the chemical potential
difference calculated using the procedure in section 6.3 (as in Fig. 6.7).
Though a quantitative agreement is obtained, it can be noticed that the
model equation (6.26) underestimates diff at larger strains and it is
attributed to the assumption of linear dependence of the nearest
neighbour interaction on the porosity since the potential curve for the
adsorbate molecules may exhibit a steep variation after a particular
intermolecular distance.
Figure 6.6: Porosity change in the deformed adsorbent calculated using equation (6.16) and experimental strain data10 at 243 K ( ), 273 K ( ), and 293 K ( ).
174
After validating model equation (6.26) (from Fig. 6.8), Fig. 6.9
shows the predicted adsorption-induced strain (using the procedure
described in section 6.3 and Fig. 6.4) in comparison with the experimental
strain data of Yakovlev et al.10 at 243 K, 273 K, and 293 K. The predicted
adsorption-induced strain data matches well with the experimentally
observed strain data. It has to be noticed that the above procedure utilizes
adsorption isotherm as the only experimental signal to predict the strain,
once the model parameter and material parameters b and N are known.
We note that the model is not restricted to adsorption in subcritical
conditions (for CO2, Tcrit = 304.1 K) and is equally applicable to model
adsorption-induced strain in supercritical conditions. It is also
independent of the geometry of pores; cylindrical, spherical or slit shaped.
Figure 6.7: diff (calculated using equation 6.18, section 6.3) as a function of the amount of gas adsorbed; at 243 K ( ), 273 K ( ), and 293 K ( ).
0 2 4 6 8 10 12-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5 x 10-22
0 2 4 6 8 10-4
-3
-2
-1
0
1 x 10-23
175
Exact material parameters b and N were not available for the
present work and though approximate values of these properties have
proven adequate for the “proof of concept”, experimentally determined
values are needed to correlate the skeleton properties with the solid
matrix properties and porosity thereby availing more accurate
understanding of the different stresses developed in an adsorbent and the
structural characteristics of the adsorbent. A correct analysis of the
structural characteristics of the adsorbent is also needed to determine the
effect of adsorption-induced strain on equilibrium sorption properties.
Figure 6.8: diff calculated using equation (6.26) as a function of the amount of gas adsorbed at 273 K ( ) and at 293 K ( ). The filled symbols indicate calculated diff using equation (6.18) and experimental adsorption isotherm (as described in section 6.3).
0 2 4 6 8 10 12-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5 x 10-22
0 2 4 6 8 10-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5 x 10-23
176
A competition between the adsorbate-adsorbent attractive
interaction and the adsorbate-adsorbate repulsive interaction is believed
to govern the type of strain developed in an adsorbent.10, 27, 49 When
adsorption takes place in micropores that are of the size of a few
molecular diameters, the adsorbate molecule attracts the opposite pore
walls due to dispersive interaction and a contractive stress is developed
due to these attractive linkages between the adsorbate and the adsorbent
framework. At lower coverage, as more molecules are adsorbed the total
attractive interaction increases and causes the adsorbent to contract.
Though the contraction in the adsorbent increases the repulsive
interaction (due to the adsorbate molecules coming closer), the
contribution of the dispersive interaction is dominating and thus the
adsorbent continues to contract. However, the adsorbent ceases to contract
when the total repulsive interaction due to the packing of the adsorbate
molecules balances the attractive interactions. Further densification of
adsorbate molecules into the pore space causes the repulsive interaction to
dominate, thereby causing the adsorbent to expand. If the above
mechanism is to be believed, the repulsive interaction is larger in a
contracted adsorbent and smaller in a dilated adsorbent than that of an
undeformed adsorbent when the same amount of gas is adsorbed in both
the cases. Model equation (6.26) calculates diff based on this assumption
and its agreement with the diff calculated using poromechanics and
experimental adsorption isotherm (cf. Fig. 6.8) further validates the
adsorption deformation mechanism. A more detailed analysis including
(i) experimentally determined material parameters and (ii) a more
comprehensive chemical potential model taking into account adsorbate-
adsorbent attractive interaction (equation 6.23), in addition to the
adsorbate-adsorbate repulsion, can further confirm this mechanism and it
177
can also explain why different adsorbate gases induce different strains in
an adsorbent.
Figure 6.9: Predicted CO2 adsorption-induced strain in microporous activated carbon adsorbent at 243 K ( ), 273 K ( ), and 293 K ( ). Filled symbols indicate the experimental data from Yakovlev et al., 2005.
6.5 Conclusions
Adsorption is widely employed in separation, purification and
catalytic applications and adsorption-induced strain is an experimentally
observed phenomenon that affects the adsorbent and the equilibria and
kinetics of adsorption. Microporous adsorbents typically undergo an
initial contraction followed by expansion and though experimental
investigations provide a useful insight, theoretical studies are required to
better understand the phenomenon. The present work assumes the solid
adsorbent to be linear, isotropic and a continuous poroelastic medium and
thermodynamics based constitutive equations for the adsorbent are
coupled with the equilibrium adsorbate chemical potential, modeled using
a lattice gas model and the Quasichemical approximation. The proposed
178
model correlates the difference between the chemical potential of the gas
adsorbed on a deformed adsorbent and that of a gas adsorbed on an
undeformed adsorbent to the adsorption-induced strain. The correlation is
validated using the experimental CO2 adsorption-induced strain data on
activated carbon adsorbent. Based on this correlation, a method which
utilizes the experimental adsorption isotherm data is proposed and is able
to successfully predict the adsorption-induced strain in the activated
carbon adsorbent at 243 K, 273 K and 293 K. The correlation is equally
applicable to subcritical and supercritical gas adsorption, is independent
of the pore geometry and is also in accord with the opinion that the initial
compressive strain in microporous adsorbent is caused due to attractive
interaction between the pore wall and adsorbate and the increased
adsorbate-adsorbate repulsion at higher adsorption causes the adsorbent
to expand. We show that there exists a repulsive interaction between
adsorbed molecules in micropores and we compute the adsorption-
induced strain on the basis of the change in repulsive interaction between
the adsorbate molecules due to the adsorption strain, using approximate
material parameters. The present work can also be extended to a more
comprehensive model that takes into account the adsorbate-adsorbent
attractive interaction and uses measured material parameters.
6.6 Supporting Information
The purpose of this section is to elaborate the procedure used to
predict the adsorption-induced strain, as shown in Figure 6.4. The
adsorption-induced strain is expressed as a polynomial function of the
coverage, and an initial guess of the unknown strain is calculated using an
initial guess for the polynomial coefficients. For the guessed strain
function , the porosity function is calculated using equation
(6.16). It has to be noted that though the strain and porosity are expressed
179
as a function of , they can also be expressed as a function of P using the
experimental adsorption isotherm. Given the porosity function , the
adsorption isotherm of an undeformed adsorbent is calculated wsP , as
explained in section 6.3. diff is computed by using in equation
(6.26) and is then used in equation (6.18) to calculate sP . Thus, having
the two adsorption isotherms, with strain and without strain
lyrespectiveand wss PP , the porosity function is recalculated ~ .
The chemical potential difference diff~ is then calculated by using ~
in equation (6.26). A sum of the relative error 2~
diff
diffdiff
at nine
different values (in the range 0.1-0.99) is then minimized by changing
the polynomial coefficients. This method calculates the adsorption-
induced strain function, , in the entire range 99.01.0 at once. It
is not possible to calculate the strain at a specific value separately
because (i) porosity and strain are related by a differential equation (6.16)
with an initial condition, 0Patand0 0 and (ii) it is not
possible to calculate diffeqn )18.( unless we have porosity as a function of P,
P . It is also verified that, at a specific temperature T, there exists a
unique minimum for the objective function 2~
diff
diffdiff
, in the
strain space. Figure 6.10 shows the objective function plotted against the
distance between the experimental and trial strain curves (at 273 K).
180
Figure 6.10: The objective function 2~
diff
diffdiff
plotted against
the distance between the trial and experimental strain curves at 273 K (as illustrated in the inset).
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19. Viete, D. R.; Ranjith, P. G., The effect of CO2 on the geomechanical
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stress. Surface Science 2004, 556, (2-3), 71-77.
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equilibria: Chemical potentials and adsorption isotherms with correct
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with a modified lattice gas model. Chemical Engineering Communications
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Inc.: New York, 2000; pp 439-480.
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molecules in pores with nonuniform walls. Russian Chemical Bulletin 1999,
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diagrams describing condensation of adsorbate in narrow pores. Russian
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theoretical study of the phase diagrams of simple fluids confined within
narrow pores. Langmuir 1999, 15, (18), 5713-5721.
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behavior of repulsive molecules. Journal of Physical Chemistry B 2005, 109,
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the solid-liquid interface - is the stress due to repulsive interactions
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7 CONCLUSIONS
7.1 General conclusions
7.1.1 Introduction
Bridging the existing knowledge gaps in the interrelationships of
the synthesis, structure and performance of functional carbon based
materials, including those doped with a transition metal, is essential to
design and optimize them for specific applications. The present thesis is
an important step forward in this direction, i.e., to develop/validate and
implement a toolbox that gives a clear and definitive understanding of the
individual components at appropriate and relevant length scales. The key
findings of the multiscale modeling and simulation approach of this thesis
are summarized in the following subsections.
7.1.2 Pore structure computation and analysis (Chapter 2)
Activated carbon fibers, with and without palladium-doping, are
prepared and experimental N2 physisorption isotherms are used to
compute and analyse the pore structure evolution in the fibers and the
effect of palladium doping. A novel statistical mechanical based chi-
theory, the statistical mechanical density functional theory and an
adsorption potential distribution are used to extract the pore structure
information from the experimental adsorption data. These novel methods
187
are contrasted with the traditional adsorption analysis methods and
shortcomings of the conventional methods like the
Brunauer Emmett Teller (BET) method, the Barrett Joyner Halenda
(BJH) method and the t plot are demonstrated. It is found that palladium
doping, using palladium(II) acetylacetonate as the precursor, during the
preparation of the carbon fiber causes (i) the formation of large
macropores (> 5 nm), (ii) a slight increase in the microporosity ( 2 nm) at
lower activation levels and (iii) increase in the mesoporosity (2 5 nm) at
greater activation. The quantitative difference in the pore structure of
activated carbon fibers with and without palladium is attributed to the
tunnelling and agglomeration of nano sized palladium particles and it is
suggested that the chemistry of palladium precursor and carbon precursor
during the fiber preparation process needs to be understood to be able to
control the pore structure of the material.
7.1.3 Crystal structure calculations of palladium(II) acetylacetonate
(Chapter 3)
First principles calculations of the crystal structure of palladium(II)
acetylacetonate are performed using the planewave pseudopotential
implementation of Kohn-Sham electronic density functional theory. The
Goedecker pseudopotential with the local density approximation and the
Troullier Martins pseudopotential with Perdew Burke Ernzerhof
approximation, both reproduced the experimentally observed molecular
and crystal structure reasonably well. The electron localization function
analysis demonstrated that the non planarity of the molecule in the
crystal structure is due to a weak non electron sharing interaction
between the most electronegative carbon atom of the molecule and the
palladium atom of the neighbouring molecule in the lattice.
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7.1.4 Palladium doping of carbon support (Chapter 4)
The underlying chemistry of palladium(II) acetylacetonate carbon
precursor/support interaction is investigated using the Car Parrinello
molecular dynamics. The validated Troullier Martins pseudopotential, in
combination with the Perdew Burke Ernzerhof approximation, is used
for electronic structure calculations. Molecular dynamics simulations and
the electron localization function analysis along the molecular dynamics
trajectory show that palladium(II) acetylacetonate decomposes into two
acetylacetonate ligands in the presence of carbon and that the
acetylacetonate ligand and carbon interaction induces chemical
cross linking in the neighbouring aromatic carbons. These findings not
only validate the experimental premise that, upon mixing, a chemical
interaction takes place between the carbon precursor/support and
palladium precursor but also reveal the molecular mechanism of the
decomposition of the palladium precursor and the chemical changes
taking place in the carbon precursor/support.
7.1.5 Hydrogen interaction with carbon supported palladium cluster
(Chapter 5)
The room temperature dynamics of hydrogen interaction with a
carbon supported palladium cluster are simulated using a model system
of a tetrahedral palladium cluster supported on a coronene molecule.
First principles molecular dynamics simulations are performed using the
Car Parrinello scheme and, when required, the dynamics are accelerated
and energetics are computed using the metadynamics technique. It is
found that after a barrierless dissociative chemisorption, the subsequent
migration of atomic hydrogen from the tip of the cluster towards the
carbon support is energetically favourable and involves energy barriers of
less than 10 KJ/mol. The transfer of surface atomic hydrogen from the
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cluster to the support is however a large energy barrier process and the
support plays a significant role in deciding this barrier. The associative
desorption of hydrogen from the carbon supported palladium cluster is
also observed to be a low energy barrier process.
7.1.6 Adsorption induced deformation in carbon materials (Chapter 6)
Assuming the carbon material to be a linear, isotropic and
continuous poroelastic medium, a statistical mechanics thermodynamics
based model is developed for calculating the adsorption induced
deformation in their microstructure. A relationship between an
equilibrium adsorption property of the adsorbate and the adsorption
induced deformation is also proposed. The model demonstrates repulsive
interaction between the molecules adsorbed in the micropores and shows
that the competition between the adsorbate adsorbent attractive
interaction and the adsorbate adsorbate repulsive interaction governs the
trend in the adsorption induced deformation.
7.2 Original contributions to knowledge
1. The novel statistical mechanical based chi-theory’s capability to
extract pore structure information from the experimental
physisorption isotherm is quantitatively tested and validated with
real-life, heterogeneous, micro- and mesoporous adsorbents.
2. A quantitative evidence of the shortcomings of the traditional
physisorption based porosity analysis methods, for their application
to heterogeneous, micro- and mesoporous activated carbon fibers, is
provided and is shown that these methods fail to give even a
qualitatively correct picture of the porosity of a structurally
heterogeneous adsorbent (which may further lead to false
interpretations about the porosity controlled functionalities).
190
3. The evolution of the pore structure of activated carbon fibers during
the activation process and the effect of addition of palladium on it are
quantitatively demonstrated. The alteration of the microstructure is
suggested to be controlled by the interactions of palladium and
carbon precursors.
4. Palladium is employed in a large number of catalytic hydrocarbon
reactions and this thesis provides validated and tested palladium
pseudopotentials to be used in the first principles investigations of
these reactions.
5. Acetylacetonate is a common precursor for palladium and unlike in
gas phase, it exhibits a non planar molecular structure in the crystal
lattice. The root cause behind the different molecular structures in the
gas phase and in the crystalline phase is revealed.
6. The long standing experimental hypothesis that, upon mixing,
chemical reaction takes place between palladium(II) acetylacetonate
and carbon precursor/support is confirmed.
7. The molecular details of the interactions between the palladium
precursor and carbon precursor/support, which control the
microstructure, chemical composition and hence the functionality of
palladium-doped carbon materials and are extremely difficult to
comprehend using experimental methods, are revealed.
8. The first principles dynamics of hydrogen interaction with a carbon
supported transition metal cluster, a crucial step in understanding
the hydrogen mediated catalytic reactions and hydrogen storage
mechanisms, are simulated for the first time.
9. Molecular mechanism of and energy landscape and barriers
associated with hydrogen adsorption, transport and desorption on a
carbon supported palladium cluster are reported. These molecular
191
process details can be leveraged to optimally change the composition
and microstructure of metal doped carbon materials.
10. All the first principles calculations performed in this thesis serve as a
benchmark for the parameterization of the very recently developed
reactive force field.
11. A practical, easy to implement and physically sound theoretical
model to predict the adsorption induced microstructural deformation
in carbon materials is developed and validated.
12. An existing premise about the molecular mechanism behind the
unique deformation pattern of microporous carbon adsorbents is
confirmed.
7.3 Recommendations for future work
1. The present thesis quantifies the microstructure of palladium doped
activated carbon fibers and investigates, at molecular level, the
interactions between the palladium precursor and carbon precursor.
However, a direct quantitative correlation between the two could not
be established due to the difference in length scales. It is
recommended that the first principles calculations performed in this
thesis be used to parameterize the recently developed reactive
ReaxFF force field and large scale simulation be performed for the
same system. Force field molecular dynamics can capture large sized
system with hundreds of thousands of molecules and can perform
larger time scale simulations. The formation of palladium clusters
(starting from palladium precursor) and the effect of their migration
on the microstructure and chemical composition evolution can then
be directly evaluated at a molecular level. Similar studies, if
performed using different precursors and metals, can help design
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carbon supported materials with controlled microstructure and
composition.
2. Similarly, the reactive force field calculations are to be performed to
investigate the effect of processing conditions on the microstructure.
Carbon supports are sometimes stabilized by oxidizing them prior to
carbonization and activation. The stabilization step is very difficult to
simulate using first principles molecular dynamics since oxygen at
its ground state exists in a triplet form. First principles calculations
will encounter the multiplicity issue since with increase in the
number of oxygen molecules, not only the multiplicity of the system
may increase but also the number of possible multiplicities increases
factorially. Reactive force field calculations need to be performed to
simulate the stabilization step.
3. Interaction of hydrogen with the carbon supported palladium cluster
is simulated in the present thesis, however, hydride formation and its
effect could not be simulated due to the computational intensity of
the first principles calculations. The reactive force field
investigations (parameterized using the firt principles computations
of the present thesis) can shed light into the effect of formation of
hydride phase on the dynamics and energetics of hydrogen
interaction with a carbon supported palladium cluster. The effect of
pressure can also be taken into account in large scale simulations by
running the molecular dynamics in an NPT ensemble.
4. The magnitude of adsorption induced deformations in carbon
materials is different for different adsorbates. The adsorption
induced deformation model developed in the present thesis can be
implemented to investigate this effect. Molecular simulations can be
used to fix the model parameters. Adsorption induced deformations
in dissociative adsorption will be different (possibly of a larger
193
magnitude) than that in molecular physisorption. The model
developed in this thesis can be extended to investigate the
dissociative adsorption induced deformation.
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A APPENDIX: MOLECULAR MODELING METHODS
A.1 Introduction
Modeling has gained significant importance in basic and applied
sciences research. In the past, the term ‘model’ was thought as a
prototype, made of different shapes of plastic/metal/wooden objects, of
some complex molecule or of a structure. However, in today’s scientific
research, it implies a set of mathematical equations that are capable of
describing a phenomenon/system under investigation. Most of the
models are so complex that an analytical solution is essentially impossible.
Hence, numerical methods are used to obtain the solution and the iterative
nature of these methods makes them convenient to be used on computers.
A computational model can be of a small system like a molecule, a crystal
lattice or a polymer chain, or can be of a macroscopic system like a liquid
solution or a reactor and can be of any phenomenon, may it be a chemical
reaction, a phase transition/separation, adsorption or a mechanical
failure. In any computational modeling effort, a balance between
simplicity and accuracy and that between the system size and the
phenomenon under investigation need to be sought. For example, a model
investigating the reaction pathway between two molecules may need to
and can explicitly take into account the sub-atomic particles of the system,
but a model investigating a hydrodynamic failure may not need to go to
195
that a small length scale and can treat the system (made up of infinite
number of small atoms and molecules) as one continuous medium.
Multiscale computational modeling combines the modeling approaches at
these different length scales in two different ways. (i) Material properties
calculated using atomic level modeling are used as input parameters for
the higher length scales modeling. (ii) An area of interest in the system
that needs more accurate treatment is modelled at an atomic level and the
rest of the system is treated as one continuous medium. Figure A.1 shows
the computational modeling approaches at different length scales. It has to
be noted that with increasing precision and decreasing length scale, the
time scale of the modeling methods also decreases.
The present thesis deals with modeling various aspects of the
structure and performance of carbon based materials for hydrogen storage
and catalytic applications. These carbonaceous catalytic/storage materials,
doped with a transition metal like palladium, are porous materials with a
pore size ranging from a few nanometres to a few microns. It is well
known that molecular hydrogen, when contacted with porous carbon
materials, gets physically adsorbed in the pores.1 However, experimental
evidences suggest that the presence of metal alters the microstructure of
the adsorbent 2, 3 and that it enhances the hydrogen adsorption capacity.4
It is also believed that the presence of palladium leads to the formation of
atomic hydrogen which then gets adsorbed in the supporting carbon
matrix.5, 6 To be able to control the hydrogen adsorption capacity of these
palladium containing porous carbon materials, it is of great importance: (i)
to understand effect of palladium doping on the microstructure of the
carbon support, (ii) to understand the palladium doping mechanism, since
it affects the microstructure of the carbon support and it also controls the
size of palladium particles, (iii) to understand the mechanical changes
taking place in the carbon support during the adsorption which may alter
196
the adsorption performance and (iv) to understand the mechanism of
formation of atomic hydrogen and its migration.
Investigating the above mentioned issues need modeling efforts at
different length and time scales. To model the microstructure/porous
structure of the carbon support and the mechanical deformations taking
place in the support that affect the microstructure, a continuum level or a
mesoscopic model is needed. However, to quantitatively understand the
doping mechanism and the mechanism of formation of atomic hydrogen
and its adsorption, migration and desorption an atomistic level modeling
effort is needed. Given the fact that chemical changes take place during
the process and they modify the chemical nature/bonding of the species,
electronic structure modeling becomes essential.
The necessary and sufficient details of the pore structure modeling
and modeling related to the mechanical deformation of the adsorbent are
provided in the respective chapters, however, since atomic level and
electronic structure modeling (i) is an integral part of this thesis, and (ii) is
rarely practised in chemical engineering, a detailed overview of the
electronic structure and atomistic level modeling methods used in this
thesis is provided in this chapter.
A.2 Molecular Modeling methods
Matter is composed of molecules and molecules can be thought of
as composed of individual atoms or of positively charged nuclei and
negatively charged electrons. Different molecules contain different atoms
(or same atoms in different spatial positions) or they contain different
nuclei and different number of electrons (or same nuclei and same number
of electrons in different spatial positions). These two different ways of
looking at molecules give rise to the two most popular molecular
modeling methods. The former is called as Force Field Method or
197
Molecular Mechanics and the later is called as Electronic Structure
Calculation or First principles or Ab initio Method. In molecular
mechanics, an individual atom is treated as the basic particle and the
potential energy is calculated as a parametric function of the atomic
coordinates. The dynamics of the atoms in molecular mechanics is
modelled by classical Newton’s laws of motion.
Figure A.1: Computational modeling methods at different length and time scales.
However, in electronic structure calculation methods, the positively
charged nuclei and the negatively charged electrons are the fundamental
particles and the interaction between these charged particles give rise to
the potential energy. The following subsections describe the technical
details of molecular mechanics and electronic structure calculations.
10-10 10-9 10-8 10-7 10-6 10-5 10-4
(nm) ( m)Length (m)
10-15
10-12
10-9
10-6
10-3
100
(fs)
(ps)
(ns)
( s)
(ms)
Tim
e(s
ec)
Electronic Structure Model
Atomistic Model
MesoscaleModel
Continuum Level Model
198
A.2.1 Molecular Mechanics
Force field calculations are called as molecular mechanics since
molecules in force fields calculations are described using a ‘ball and
spring’ type of model where the atoms are the “balls” of different sizes
and the bonds are the “springs” of different lengths and stiffness. The
non-bonded interactions like the van der Waals interaction and the
electrostatic interaction are also taken into account in the force field
calculations.
Figure A.2: An illustration of energy terms in molecular mechanics (Adapted from Frank Jensen 7).
The energy in force field calculations is given by a sum of different
terms, where each term contributes for the specific type of deformation in
the species, as given in the following equation.
bonded non bonded
MM stretch bend torsion vdW electrostatic
E E
E E E E E E (A.1)
where stretchE is the energy for stretching a bond between two atoms, bendE
is the energy for bending an angle formed by three bonded atoms, torsionE
is the energy for twisting around a bond and vdWE and electrostaticE are the
energies accounting for van der Waals interaction and electrostatic
interaction between two atoms. Figure A.2 shows a graphical illustration
of the basic terms involved in calculating force field energy. Since the
199
energy is a function of atomic coordinates, the minimum in the energy
corresponding to the most stable configuration can be calculated by
minimizing MME as a function of atomic coordinates.
The stretching energy between two bonded atoms 1 and 2, when
written as a Taylor expansion at the equilibrium bond length, is given as 7
0 0
2 212 12 0 12 00 2
12!stretch
l l
dE d EE E l l l ldl dl
(A.2)
The first derivative at 0l is zero and 0E is usually set zero since it is a zero
point in the energy scale. Hence, equation (A.2) can be written as
0
2 2 212 12 0 12 02
12!stretch stretch
l
d EE l l K l ldl
(A.3)
where stretchK is the force constant. Equation (A.3) is in the form of a
harmonic oscillator. A similar expression for an angle bending is given as 2123 123 0
bend bendE K (A.4)
The harmonic form for stretching and bending, though simple, may not
always be sufficient. In such cases the functional form is extended to
include higher order terms or instead of using a Taylor series expansion, a
Morse potential type function is used which is given below.7
12 012 1 l lMorse dissE E e (A.5)
where dissE is the dissociation energy and is related to the force
constant.
Torsion energy associated with the twisting around bond 2-3, in a
four atom sequence 1-2-3-4 where 1-2, 2-3, and 3-4 are bonded atoms, is
physically different from the bending and stretching energy because (i) the
rotation along the bond can have contributions from bonded and non-
bonded interactions and (ii) the torsion energy has to be periodic, since
after rotating along the bond for 360°, the energy should return to the
200
same value. To take into account the periodicity, the torsion energy is
usually given as 7
1234 1 costorsion torsionE K n (A.6)
where torsionK is the constant, is angle of rotation and n determines the
periodicity.
The van der Waals energy, due to the repulsion and attraction
between the two non-bonded atoms, is usually given in the form of the
popular Lennard-Jones potential as follows 12 60 0
12min 12 124 L J
vdWR RE ER R
(A.7)
where minL JE is the depth of the minimum in the potential and 0R is the
distance at which the potential is zero. The electrostatic energy between
two atoms is usually given by the Coulomb potential as 1 2
1212electrostatic
dielec
Q QER
(A.8)
where 1Q and 2Q are the atomic charges and dielec is the dielectric
constant.
Assigning numerical values to different parameters in the above
described functions is also equally important in force field calculations.
Parameterization of the force field is usually done by reproducing the
structure, relative energies, vibrational spectra obtained from the
electronic structure calculation data and the experimental data. However,
it is also required that the parameters which are fitted in any force field
are transferable amongst different molecules and environments. A
compromise between accuracy and generality needs to be sought.
Different force fields have been developed over the years and some of the
main differences in these force fields are the functional forms of the
energy terms, the number of additional energy terms (other than the basic
201
ones described above) and the information used to fit the parameters in
the force field. Force fields containing simple functional forms, as
described above, are often called as “Harmonic” or “Class I” type Force
fields and those containing more complicated functional forms, additional
terms and sometimes heavily parameterized using electronic structure
calculation methods are called as “Class II” type.7 Depending upon these
factors, there are different force fields for different types of molecules and
Table A.1 lists a few of them.
Table A.1: A list of few common force fields in molecular modeling 8.
Force Field Developers Systems Class
MM2, MM3,
MM4
Prof. Norman
Allinger
Organics/General
Hydrocarbons
II
AMBER Prof. Peter
Kollman
General
Organics/Proteins
I
UFF Prof. William
Goddard
General Between I and II
CHARMM Prof. Martin
Karplus
Proteins I
GROMOS University of
Groningen and
ETH Zurich
Proteins, Nucleic
Acids and
Carbohydrates
I
CFF, TRIPOS Commercial General II
Force field methods are very widely used in computational
modeling community and their ability to provide an understanding of
atomic and molecular motions in different (and large) systems and
phenomena, at a modest computational cost, has contributed greatly to the
scientific research in last two decades. These methods are very popular in
202
investigating systems containing small organic molecules, large
biomolecules like proteins and DNA, polymers etc. They are also several
orders of magnitude faster than the electronic structure calculation
methods. However, there also certain limitations associated with the force
field/molecular mechanics methods.9-17 They are as follows:
1) Out-of-ordinary/Unusual situation 7, 13, 15, 16: Force field methods are
based on various approximate functional forms and their parameters.
Since the parameters are determined using experimental data, these
methods are empirical. Force field methods perform extremely well when
a lot of information about the system under investigation already exists in
the force field. For molecules that are “exotic” or a little “unusual” and for
which there is little information known, the force field methods may
perform poorly. To summarize, the interpolative force field methods may
lead to serious errors when used for extrapolation.
2) Diverse types of molecules 7, 13, 15, 16: Parameterization of a force field
needs a balance between generality and accuracy. The
generality/transferability of a force field can be improved by including
diverse types of molecules in the parameterization process but with a
given functional form of the energy terms, including additional data may
not help. On the other hand, changing the functional form or using
additional terms, may remove the cancellation of the error effect in the
simpler forms. Most of the force fields are restricted for specific types of
molecules.
3) Chemical reactions 13, 18: While performing force field calculations, the
input consists of (i) types of atoms, (ii) interactions between those atoms
(bonded or non-bonded) and (iii) the geometry. The first two factors are
crucial in assigning an appropriate functional form to each interaction in
the system. The force field calculations are appropriate when the type of
every atom and its types of interactions do not change with changes in
203
atomic coordinates. However, during the course of a chemical reaction,
covalent bonds are formed and are broken. Hence, a chemical reaction
leads to different energy functions in reactants and products for the same
atom. The electronic structure of the system also changes significantly
thereby changing the type of an atom (e.g. a carbon that was sp3 before the
reaction may become sp2 or sp after the reaction and vice-versa). These two
factors in a chemical reaction change the fundamental information on
which the force field energy was calculated and thus the energy will not
remain smooth and continuous during the chemical reaction. Hence force
field calculations fail to model a system in which chemical reactions occur.
The harmonic description of the stretching energy would make it
impossible to find parameter values describing the dissociation of a
molecule.
4) Metal systems 7, 9-11, 14, 17, 19-23: Force field methods are believed to be
difficult, if not impossible, to apply to metal compounds and complexes
and especially to transition metal systems. The bonding in metals is much
different than in organic systems. In the case of metal-ligand complexes,
the metal forms a coordinate bond with the complex while in pure
metallic systems, the bonding may vary with the size of the metal cluster.
Since electronic effects can not be taken into account explicitly in force
field calculations, they need to be taken into account implicitly. The key
reasons for less successful implementation of force field methods to model
metal (including transition metals) complexes and compounds are as
follows:
Varied coordination numbers and geometries: In metal-complexes,
(organic or inorganic) coordination number of a metal is the number of
atoms in the ligand to which the metal is bound and in case of metal
clusters, it is defined as the number of nearest neighbour atoms. Transition
metals may exhibit coordination numbers ranging from 1 to 12. There are
204
also more than one ways to organize ligand atoms around the central
metal species, giving rise to isomerism. In case of pure metal clusters,
multiple structures, very close in energy, are present. The geometry may
also differ significantly depending upon the physical state of the system
(solid phase/solution/gas phase). Unlike organic compounds, metal
coordinated compounds possess a much wider structural flexibility and
hence a variety of structural motifs are observed.22-24 This leads to
difficulty in defining the energy functional forms to describe them. Force
field methods are successfully applied to quite a few specific systems 20, 21,
25-27 and a few generalized approaches28-30 have also been developed to
tackle the problem.23, 31 However, whenever force field methods need to be
applied to a new system, very frequently a significant modification is
required to be performed and the predictive power still remains
questionable.9, 14
Varied oxidation states and electronic structures: Another problem
in using force field methods is that transition metals exhibit multiple
oxidation numbers and electronic states (e.g. palladium has oxidation
states of 0, 1, 2 and 4) and separate parameterization needs to be
performed (similar to carbon with sp, sp2, and sp3 hybridization). The
problem is further magnified due to multitude of transition metal
complexes, thus making the parameterization even more difficult.9, 19, 32, 33
Also in pure metallic systems, the nature of bonding and electronic
structure change as the number of atoms in the metal cluster changes (e.g.
Pd2 has a spin multiplicity of 3, Pd11 has a spin multiplicity of 7 and Pd12
has a spin multiplicity of 5).34
The d-shell electrons 10, 11: In the case of transition metal systems,
the effects due to d-orbital electrons pose further problems in using force
field methods. The structural, spectroscopic and magnetic properties of
transition metal complexes are significantly affected by the d-orbital
205
electrons. Some significant issues are the Jahn-Teller distortion, s-d orbital
mixing etc. Some efforts have been directed towards tackling these
problems in force field methods and the POS (points on a sphere) model
and the LFMM (ligand field augmented molecular mechanics) model have
garnered relatively more attention.11 However, these modifications are
very specific and need significant code writing since they can not be
implemented in standard force field method softwares and to make these
approaches more diversified a lot of parameterization is needed. Another
situation that may hamper the use of these methods is when the system
under investigation contains both, the transition metal complexes and
some routine organic molecules.
A.2.2 Electronic Structure Calculations
To model chemical reactions taking place in a system containing
novel transition metal clusters and complexes and routine organic
compounds, at an atomic level, there is no substitution to the electronic
structure calculation methods. Since it explicitly takes into account the
electronic structure, it also offers an additional advantage of probing and
predicting the bonding and electronic structure changes in the system. The
following sub-sections describe the necessary background material of
electronic structure calculations and give a detailed description of the
methods used in this thesis.
A.2.2.1 Electronic Structure of Atom and Wave-particle duality
An atom consists of electrons, protons and neutrons (Protons and
neutrons are not the most fundamental particles of matter and they are
made up of even smaller particles called quarks. However, these details
are not required and are beyond the scope of this thesis). Electrically
neutral neutrons and positively charged protons are bound together
forming a positively charged nucleus and the negatively charged electrons
206
arrange themselves around the nucleus. There exists electromagnetic
interaction amongst these species. The wave-particle duality is known to
exist for a long time to describe matter and energy in physics and
chemistry and an appropriate mathematical form to describe an object
depends upon its mass (and velocity when relativistic effects need to be
considered). Heavy objects can be treated as “particles only” and hence
can be modelled using classical Newtonian mechanics. However, the
borderline mass for Newtonian mechanics is the mass of a proton (and the
velocity as a fraction of the velocity of light, to neglect relativistic effects).
Electrons are a few orders of magnitude lighter than the neutrons and
protons (and hence the nucleus) and hence they display both wave and
particle like characteristics. The famous double slit-experiment was the first
experimental proof of electrons behaving like a wave. Given the wave like
behaviour of electrons, it is not possible to mathematically treat electrons
using classical Newtonian mechanics and hence they need a special
treatment, i.e. quantum mechanics.
A.2.2.2 Postulates of Quantum Mechanics
Quantum mechanics is nothing but a set of underlying principles
that can describe some of the most fundamental aspects of matter at a sub-
atomic level. The postulates of quantum mechanics are as follows: 7
1. Associated with any particle (like an electron) moving in a force
field (like the electromagnetic forces exerted on an electron due to the
presence of other electrons and nuclei) is a wave function which
determines everything that can be known about the particle.
2. With every physical observable there is an associated operator,
which when operating upon the wavefunction associated with a definite
value of that observable will yield that value times the wavefunction
n n nQ q .
207
3. Any operator associated with a physically measurable property will
be Hermitian **a b a bQ dr Q dr .
4. The set of eigenfunctions of the operator will form a complete set of
linearly independent functions and j j j j jQ q c .
5. For a system described by a given wavefunction, the expectation
value of any property can be found by performing the expectation value
integral with respect to that wavefunction *q Q dr .
A.2.2.3 Schrödinger Equation
The Schrödinger equation,35 which is a second order partial
differential equation, is the most important equation in quantum
mechanics and can describe the spatial and temporal evolution of the
wavefunction of a particle in a given potential. It is given as,
, ,H r t i r tt
(A.9)
where is the reduced Planck’s constant, is the wavefunction and H
is the Hamiltonian operator which is given as follows: 2
2
2H U r
m (A.10)
The first term in the Hamiltonian operator is the kinetic energy operator
and the second term is the potential energy operator. Since equation (A.9)
is a partial differential equation, if the separation of variables method is
used, the wavefunction can be separated into spatial and temporal part as
,r t r f t (A.11)
Inserting equation (A.11) into equation (A.9) gives
1 1 dH r i f tr f t dt
(A.12)
208
The left hand side of equation (A.12) depends only on space while the
right hand side depends only on time. Since these two are completely
independent variables, equation (A.12) can only be true when both the
sides of the equation are constant, i.e.
1 ;H r E H r E rr
(A.13)
E is a constant in equation (A.13). According to postulate number 2 of
quantum mechanics, with every physical observable there is an associated
operator. Since the Hamiltonian is an energy operator, it is intuitive that
the constant E is nothing but the energy of the system. The solution of the
time dependant right hand side part of equation (A.12) can be given as iEtf t e and inserting this solution in equation (A.11) gives,
, iEtr t r e (A.14)
Thus the wavefunction is written as a function with amplitude r and
phase iEte . Inserting equation (A.14) into equation (A.9) gives the time
independent Schrödinger equation as
H r E r (A.15)
and the time dependence can be written as a product of the time
independent function and the phase factor. The phase factor is usually
neglected for time-independent problems.
A nucleus is much heavier than electrons and this large mass
difference also indicates that its velocity is much smaller than that of the
electrons. Hence nuclei exhibit small quantum effects and can be treated
classically. The electrons can adjust instantaneously to any change in the
nuclear coordinates. If we write the time-independent Schrödinger
equation for a system where n denotes nuclei and e denotes electrons and
the nuclear coordinates are denoted as nR and the electronic coordinates
are denoted as er then,
209
, ,sys sys e n sys sys e nH r R E r R (A.16)
where
,sys n e ee e en e n nn n n eH T T V r V r R V R T H (A.17)
T denotes the kinetic energy operator and V denotes the potential energy
operator (columbic interactions between electron-electron, electron-
nucleus and nucleus-nucleus). The total wavefunction of the system
depends on the coordinates and velocities of the electrons and the nuclei.
However, due to the separation of time scales between the electronic and
nuclear motion (nuclei moving much slower than the electrons) it can be
assumed that the nuclei are almost stationary with respect to the electrons.
If the total wavefunction of the system is written as
,sys e e n n nr R R (A.18)
then the Schrödinger equation in a static arrangement of nuclei can be
written as
, ,e e e n e e e nH r R E r R (A.19)
Here the energy eE and the wavefunction e depend only on the nuclear
coordinates and not on nuclear velocities. The total energy of the system
then can be computed from the following equation.
n e n n sys n nT E R E R (A.20)
The energy eE is often called the adiabatic contribution to the energy of
the system and it is shown that the non-adiabatic contributions contribute
very little to the energy. The error in hydrogen molecule is of the order of
10-4 a.u. and as the molecule gets bigger, the nuclei become heavier and
thus the error decreases.7 Thus, with the separation of nuclear and
electronic motion, we can compute the energy eE as a function of different
nuclear coordinates. This way of computing the energy using e provides
a potential energy surface on which the nuclei move. The separation of
210
electronic and nuclear motion in a system, as described above, is called as
the Born-Oppenheimer approximation. Most of the electronic structure
calculations are performed using this approximation (i.e. using equation
A.19 instead of equation A.16) and all the electronic structure calculation
methods described henceforth in this thesis will be using the Born-
Oppenheimer approximation.
A.2.2.4 Solution for Hydrogen atom and Approximate Solution for Helium
The hydrogen atom is the simplest system on which electronic level
calculations can be performed by solving the Schrödinger equation. For
the hydrogen atom, with one electron and a nucleus of charge +1, the time
independent Schrödinger equation can be written as, 2
2
2U r r E r
m (A.21)
where r is the distance of the electron from the nucleus and the potential
energy operator takes into account the columbic interaction between the
electron and the nucleus 1U r r . The analytical solution for the
wavefunction in spherical coordinates is given as,7
32 2 1
1 ,0
1 !2, , . ,2 1 !
l lnlm n l l m
n lr e L Y
na n n (A.22)
where n, l and m are the principal, azimuthal and magnetic quantum
numbers, 0a is the Bohr radius, 02r na , 2 11
ln lL are Laguerre
polynomials and lmY are spherical harmonics.
Once the wavefunction is determined, the square of the
wavefunction at any point gives the probability of finding an electron at
that point. Hence,
1d (A.23)
211
The physical quantity that is associated with the Hamiltonian operator H
is the energy and is given as
H dE H d H
d (A.24)
It is a customary to write the integrals using a “Bra-ket” notation, as
shown in equation (A.24).
The solution of Schrödinger equation for a system containing more
than one electron is more complicated since no analytical solution exists.
An approximate solution needs to be determined even for the helium
atom which contains two electrons and a nucleus. The Schrödinger
equation for this atom can be written as 8
1 2
2 21 2 1 2
1 2 12
1 2 1 2 1 2 1 212
1 1 1 ,2 2
1 , , ,
h h
Z Zr r r
h h H Er
r r
r r r r r r
(A.25)
where Z denotes the charge on the nucleus, subscripts 1 and 2 represent
electrons 1 and 2 and 1 2 and r r represent their positions in space. If we
assume that the two electrons in the atom interact with the nucleus but do
not interact with each other, i.e. 1 2H h h , then equation (A.25) becomes
separable and an exact solution of the individual electron’s wavefunction
can be obtained. The two separate equations are
1 1 1 1 1 1 2 2 2 2 2 2 and h E h Er r r r (A.26)
and the total wavefunction then can be assumed as a product of
individual wave functions 1 2 1 1 2 2,r r r r . The total energy of the
helium atom with non-interacting electrons then can be given as
1 2E E E . To include the correction for repulsion between the two
electrons in helium atom, an additional term can be defined as,
212
1 2 2 2 1 112 12
1 1 and eff effU d U dr r
(A.27)
Since 1 and 2 are known from the non-interacting system, integrals in
equation (A.27) can be evaluated. Using equation (A.27), the effective
Hamiltonian that takes into account the electron-electron repulsion can be
defined as
1 1 1 2 2 2
1 1 1 1 2 2 2 2 1 2 1 2
and
and , , ,
eff eff eff eff
eff eff
H h U H h UH E H E E E
(A.28)
Thus the total energy of the system with interacting electrons, E can be
calculated as
1 2 1 2 1 212
1 2 1 212 12
1 2 1 2 1 2 1 212
True Hamiltonian for He
1 2 1 2 1212
1 2 12
2
1 1
1
1
eff effE E H H h hr
H Hr r
Hr
E d E Jr
E E E J
(A.29)
12J is called as the Coulomb integral. It can be seen that the exact solution
of hydrogen atom plays an important role in calculating an approximate
solution for the helium atom.
Both the electrons in helium are in the 1s orbital and it means that
they have the same principal, azimuthal and magnetic quantum numbers.
However, they differ in the spin since one has a 1 2 spin and the other
has 1 2 spin. In the above discussion, the total wavefunction of the
helium atom is written as the product of individual electron
wavefunctions as
1 2 1 1 2 2, 1 1 1 2s sr r r r (A.30)
213
where the wavefunctions/orbitals 1 and 1s s differ in spin. Pauli’s
exclusion principle states that, for two electrons, the total wavefunction is
antisymmetric 1 2 2 1, ,r r r r . However, if is defined as in
equation (A.30) and we exchange electrons 1 and 2, we get
1 2 2 1
1 2 2 1
, 1 1 1 2 and , 1 2 1 1
, ,
s s s sr r r r
r r r r (A.31)
Equation (A.31) does not obey Pauli’s exclusion principle. Hence the
representation of the total wavefunction needs to be changed. A correct
representation would be
1 21, 1 1 1 2 1 2 1 12
1 1 1 11 =1 2 1 22
s s s s
s ss s
r r
(A.32)
The determinant in equation (A.32) is called as Slater determinant. If the
correct representation of the total wavefunction is put in equation (A.29),
we get
1 2 1 2 1 212 12
12True Hamiltonian for He
12 12
Coulomb Integral
2 1
1 1 1 1 1 2 1 2 1 1 1 1 1 2 1 2 1 12
1 1 1 1 1 1 1 2 1 1 1 2 1 2 1 1 1 2 1 12 2
eff effE E H H h h Hr r
H s s s s s s s sr
E s s s s s s s sr r
12 12
Exchange Integral
12 12
1 2 12 12
1 1 1 1 - 1 1 1 2 1 2 1 1 1 2 1 1 1 1 1 22 2
s s s s s s s sr r
E J KE E E J K
(A.33)
214
A.2.2.5 Linear Combination of Atomic Orbitals (LCAO)
The discussion in the above section is limited to Helium, containing
only two electrons; however, any realistic system will be a polyelectronic
system. Any system consists of molecules and when more than one atoms
form a molecule (by forming electron sharing and non electron sharing
bonds), their electronic structure gets modified. For example, when two H
atoms form a covalent bond to make an H2 molecule, the electronic
structure is different than that of an individual hydrogen atom. According
to the valence bond theory, the individual orbitals of two atoms overlap
and the shared electrons are localized in the overlapped region (the bond)
between the two atoms. However, in electronic structure computations,
the molecular orbital theory is used. According to this theory, the
electrons are not assigned to individual bonds and are considered to
arrange themselves around the molecule, under the influence of the
nuclei. Similar to orbitals in an atom, every molecule is considered to have
a set of molecular orbitals and these molecular orbitals (wavefunction) are
a mathematical construct of the individual atomic orbitals. Molecular
orbitals are expressed as a linear combination of atomic orbitals (LCAO),7
as if each atom were on its own.
1 1 2 2 3 3 ....... n nc c c c (A.34)
where is the molecular orbital, i is an atomic orbital and ic is the
coefficient associated with the atomic orbital i . To make sure that the
orbitals follow the antisymmetric constraint, they are expressed as Slater
determinants (similar to equation A.32). If there are N electrons in the
system with spin orbitals 1 2, , , N then the total wavefunction is given
as,
215
1 2
1 2
1 2
1 2
1 1 12 2 211,2, ,
!
= 1 2
N
N
N
N
NN
N N N
N
(A.35)
Since in a polyelectronic system, it is not possible to obtain analytical
solution, assuming non-interacting electrons (the way it is obtained in the
case of Helium), the atomic orbitals are expressed in the form of basis
functions. A basis function can be of any type; exponential, Gaussian,
polynomial, cube function, planewaves, to name a few. Though these
basis functions need not be an analytical solution to an atomic Schrödinger
equation, they should properly describe the physics of the problem and
these functions go to zero when the distance between the nuclei and the
electron becomes too large. The two types of basis functions commonly
used to construct atomic orbitals are the Slater type orbital and the
Gaussian type orbital. The functional form of the Slater type orbitals is as
follows8
1, , , ,, , , n rn l m l mr NY r e (A.36)
where N is a normalization constant, is a constant related to the effective
charge of the nucleus and ,l mY are spherical harmonic functions. The
Gaussian type orbitals are given as follows8
22 2, , , ,, , , n l rn l m l mr NY r e (A.37)
The Slater type basis functions are superior to the Gaussian type basis
function since less number of Slater orbitals are required to get a given
accuracy and the physics of the system is better described using the Slater
type orbitals.7
The minimum number of basis functions that are needed to
describe any system is that which can just accommodate the number of
216
electrons present in the system. For ex. two sets of s functions (1s and 2s)
and a set of p functions (2px, 2py and 2pz) are required to describe the first
row elements in the periodic table. Increased accuracy can be obtained
using a larger number of basis functions. A detailed overview on different
types of basis functions is beyond the scope of this thesis and hence not
discussed here. The basis functions used in all the computations in this
thesis are planewaves and the pertaining details will be discussed in
section A.2.4.
A.2.2.6 Hartree-Fock Calculations
While performing electronic structure calculations in a
polyelectronic system, we are aiming to calculate the molecular orbitals
(and the energy). Once the type and numbers of basis function are
decided, the molecular orbitals are formed as a linear combination of
atomic orbitals, as in equation (A.34), and the wavefunction is expressed
in the form of Slater determinant, as in equation (A.35). Since there is no
“exact” wavefunction, we need to determine the coefficients of equation
(A.34) which will give the best possible solution. The variational principle
is used to calculate these coefficients and ultimately the wavefunction. It
states that the energy calculated from an approximate wavefunction will
always be bigger than the “true” energy of the system.7 This implies that
the closer the approximate wavefunction to the actual solution, lower will
be the energy of the system. Hence to obtain the best possible
wavefunction, we need to determine the set of coefficients that will result
in the minimum energy of the system i.e. the electronic energy of the
system is minimized with respect to the coefficients. The numerical
procedure is as follows.
If there is an N electron system for which the Hamiltonian is given
as
217
21 12
N N N N
i i i ji i j
i ij
ZHr r r r
h g
(A.38)
and the wave function is expressed as a Slater determinant, as in equation
(A.35) , then the energy is given as,
1 2 1 2
1 2 1 2
2
= 1 2 1 2
1 2 1 2
=2
N N N
i iji i j
N
N i Ni
N N
N ij Ni j
N N N Ni
i i i i ij iji i i ji
E h g
N h N
N g N
Z J Kr
(A.39)
where
12 1 2 1 2 12 1 2 2 112 12
1 11 2 1 2 and 1 2 1 2J Kr r
(A.40)
If every orbital is doubly occupied then,
,
2 2n n
ii ij iji i j
H h J K (A.41)
Varying the electronic energy as a function of orbitals and equating it to
zero we get,
, ,2 2 2
0 2
n n n
ii ij ij ij i j iji i j i j
n n
i j j i ij jj ji
h J K
h J K (A.42)
where ij are the Lagrange multipliers that are introduced because the
minimization of the energy needs to be performed under the constraint
that the molecular orbitals remain orthogonal and normalized.
218
Since the wavefunction is a determinant, the matrix ij can be
brought into diagonal form as 0 and ij ii i ,
2n
i j j i i i i ij
h J K F (A.43)
Expressing i
M
ic we get,
M M
i i i ic cF (A.44)
where ijc are the coefficients and M is the number of basis functions.
Multiplying equation (A.44) by a specific basis function and integrating
gives the following equation
and |F S
Fc Sc
F (A.45)
where F is called as the Fock Matrix and S is called as the overlap matrix.
Equation (A.45) is an eigenvalue problem and the Fock matrix needs to be
diagonalized to get the coefficients of the molecular orbitals. But it has to
be noted that the Fock matrix can only be calculated if the coefficients are
known. Hence, the Hartree-Fock procedure to perform electronic level
calculations starts with an initial guess for the coefficients, then calculating
the Fock matrix and then recalculating the new set of coefficients by
diagonalizing the Fock matrix. The procedure is repeated till the
coefficients used to form the Fock matrix are the same to those emanating
from the diagonalization of the Fock matrix. The convergence on the
coefficients implies that the system is at the minimum energy.
A.2.2.7 Semi-empirical methods
The electronic structure calculation methods, like the Hartree-Fock
method described above, are computationally very expensive and a
219
significant amount of computational effort is needed to calculate and
manipulate the integrals that are calculated in the process. Hence, to
reduce the computational cost of electronic structure calculations, some
approximations are made to these integrals, particularly to the two-
electron, multicentre integrals. These integrals are either neglected or are
parameterized using empirical data. Such methods are called as semi-
empirical methods.36 Some common semi-empirical methods are Zero-
differential overlap (ZDO), Complete neglect of differential overlap
(CNDO), Intermediate neglect of differential overlap (INDO), Neglect of
diatomic differential overlap (NDDO), Modified neglect of diatomic
overlap (MNDO), Austin model-1 (AM1), Parametric model-3 (PM3) etc.36
A detailed discussion of all these methods is not included in this thesis.
However, these semi-empirical methods share a limitation of force field
methods, i.e., their performance on unknown species. The extent of this
limitation is not as severe as in the case of force field calculations though.
Most of these methods also use the minimal basis function only.
A.2.2.8 Post Hartree-Fock
The electronic structure calculations are performed in the Hartree-
Fock procedure by selecting the type and number of basis functions,
forming a Slater determinant and then obtaining the coefficients in an
iterative fashion, as explained in the previous section. The accuracy of
these calculations can be improved (or the energy of the system can be
further lowered) by increasing the number of basis functions used. The
Hartree-Fock wavefunction can provide 99% accuracy in calculating the
“true” energy of the system and hence suffices the purpose in a lot of
cases. However, in certain situations, this 1% difference becomes very
crucial to describe correctly the physics of the system. In order to further
improve the accuracy of electronic structure calculation methods beyond
220
the Hartree-Fock procedure, it is required to identify the root cause of this
slight inaccuracy. Assuming that a sufficiently large basis set is used, the
other possible source of error is due to electron-electron interaction
treatment in the Hartree-Fock procedure. Though the electron-electron
interaction is taken into account, as can be seen in equation (A.39), it has to
be noted that each electron in the Hartree-Fock procedure sees an average
field of all other electrons and the motion of each and every electron is not
correlated. In other words, an electron does not see each and every other
electron as individual point charge so as to avoid it as much as possible. It
also does not allow electrons to cross each other. Hence, any method that
can improve the electron correlation this way will definitely lead to a
lower energy of the system than that calculated using the Hartree Fock
procedure. These methods are called as electron correlation methods and
the electron correlation energy is the difference between the “true” energy
of the system and the energy calculated using the Hartree Fock procedure.
One of the ways to further improve the electron correlation effects
in electronic structure calculations is to include additional (Slater)
determinants. Addition of determinants in the mathematical formulation
can also be seen physically as addition of some unoccupied/virtual
orbitals to the system, something very similar to addition of excited state
orbitals to the ground state orbitals of the system.0 1 1 2 2HF
corrrelation c c c (A.46)
where HF is the single determinant wavefunction obtained using the
Hartree-Fock procedure and i are the additional determinants. There are
3 main methods that are used to take into account electron correlation
effects, viz., Configuration Interaction (CI)37, Coupled Cluster (CC)38 and
Møller-Plesset (MP)39 and they differ in ways to calculate the coefficients
in equation (A.46). Further details of these methods are beyond the scope
of this thesis and hence not provided here.
221
A.2.3 Density Functional Theory (DFT)
The previous section described the electronic structure calculation
methods (Hartree-Fock, Semi-empirical and Post-Hartree-Fock) based on
the multielectron wavefunction of the system. However, there exists one
more theoretical approach to perform electronic structure calculations and
it is called as density functional theory. As the name suggests, the energy
of the system containing N electrons that repel each other, get attracted to
the nuclei by Columbic interaction and follow Pauli’s exclusion principle
is calculated using the density of electrons instead of the wavefunction.
The following sections describe the mathematical formulation of the
density functional theory and the necessary details.
A.2.3.1 Origin and formulation of DFT
The Schrödinger wave equation for an N electron system is given
as
21 12
N N N N
i i j iij i
Z Er r
r
(A.47)
and, as mentioned before, the ground state electronic density of the
system can be calculated as
1 1 1 1 1 12 , , 2, , , , 2, ,r N dx d dN r x N r x N (A.48)
Equations (A.47) and (A.48) show that for a given potential r , it is
possible to compute the ground state density r , via the electronic
wavefunction . However, Hohenberg and Kohn40 showed that there
exists an inverse mapping with which it is possible to obtain an external
potential, if the ground state density is provided. In their seminal paper,40
they also showed that this inverse mapping can be used to calculate the
222
ground state energy of a multielectron system using the variational
principle, without having to resort to calculate the wavefunction. If the
wave function is expressed as
(A.49)
then since , the wavefunction can be written as
(A.50)
Using the variational principle, the energy of the system can be calculated
as,
min min
min ee
E H H
T V (A.51)
where2 1= , and
2 eeee
ZT Vr r
.
Equation (A.51) can be separated as,
3
E min
min
eeT V
F
d r r r F
(A.52)
The energy is thus minimized over the density and not over the
wavefunction. However, in equation (A.52), the functional form of F
is unknown.
The inverse mapping of density on potential is also valid for a
system of non-interaction electrons and Thomas-Fermi,41, 42 even before
the Hohenberg-Kohn theorem,40 provided a way to calculate the energy of
such a non-interacting system by replacing the wavefunction with electron
density as follows. For a non-interacting system,
223
23 3
1 1
23 3
1 1
2
2
N N
i i i ii i
N N
i ii i
E d r r r d r r r r
d r r r d r r r (A.53)
As shown in equation (A.53), Thomas-Fermi just replaced the
wavefunction with the electron density. They also showed that the kinetic
energy part of a homogeneous electron gas is given as,
2 32 3 5 33 310sT d r r (A.54)
Hence according to Thomas-Fermi approach the energy of the non-
interacting system can be given in terms of the electron density as,
2 32 3 5 3 3
1
3 310
N
iE d r r d r r r (A.55)
This Thomas-Fermi approach of calculating the kinetic energy part of the
non-interacting system of electrons is used to develop a scheme to
evaluate the functional F (for a system with interacting electrons) and
the details are provided in the following section.
A.2.3.2 Kohn-Sham formulation
To calculate the unknown functional F consisting of the
electronic kinetic energy part and the electron-electron interaction part,
Kohn and Sham43 came up with a scheme which calculates the total
energy of the system using a combination of the density functional theory
and orbital/wavefunction approach. A detailed description is as follows.
Kohn-Sham introduced a non-interacting system for which the
Hamiltonian can be given as
212ks KSh r (A.56)
If an orbital corresponding to the above Hamiltonian (usually referred as
Kohn-Sham orbitals) is given in the form of a determinant as
224
KS 1 2 3 1 2 3 ... N N (A.57)
then the Schrödinger equation will be
KS KS KS KSh E (A.58)
In the Kohn-Sham formulation, the potential KS r is defined in such a
way that the electron density calculated using the Kohn-Sham orbitals KS
is equivalent to the exact density of the electron-interacting system with
the actual potential r . Using the Kohn-Sham orbitals, the kinetic
energy of the non-interacting system can be given as
S KS KST T (A.59)
From equations (A.56-A.59), with the actual wavefunction of the system as
, the ground state electronic energy of the system can be given as,
3
3 3 312
ee ee
K
S
K s ee
XC
E T V T d r r r V
Tr r
T d r r r d rd rr r
U
T T V U
E
(A.60)
where sT is the kinetic energy which can be calculated using the Kohn-
Sham orbitals. XCE is the only unknown function in the above
equation that needs to be approximated. It accounts for less than 10% of
the total energy.8
It is possible to calculate the total ground state energy of the
interacting system using the following steps:
(i) Define the Kohn-Sham non-interacting system
(ii) Calculate the density and kinetic energy of the system and
(iii) Calculate the ground state energy
225
However, it is only possible if the Kohn-Sham potential KS r is known.
The methodology to determine the potential is as follows.
From equations (A.56) and (A.58) we get
3minKS S KSE T d r r r (A.61)
where 0 0KS SKS
E T (A.62)
From equation (A.60), the actual ground state energy of the system is
given as
3min S XCE T d r r r U E (A.63)
where 30 0S XCrT EE r d rr r
(A.64)
Combining equations (A.62) and (A.64), the Kohn-Sham potential can be
obtained as
3 XCKS
r Er r d rr r
(A.65)
If an approximate functional for XCE is known, the Kohn-Sham non-
interacting system can be solved to obtain the Kohn-Sham orbitals KS
and the density . Using this information the total energy of the system
can be calculated using equation (A.63). However, since the Kohn-Sham
potential depends on density the above equations need to be solved
iteratively.
This thesis uses the Kohn-Sham formulation of the density
functional theory. To use the Kohn-Sham Density Functional theory, it is
needed to have an appropriate functional form for XCE and the
following section describes the approaches to obtain it.
226
A.2.3.3 Local density approximation and Generalized Gradient Approximation
From equation (A.60) it can be seen that the XCE term accounts for
(i) the difference between the actual/exact electronic kinetic energy of the
system and the Kohn-Sham non-interacting electronic kinetic energy
K sT T and (ii) the difference between the exact electron-electron
interaction energy and U (analogous to the Coulomb integral term J
in the Hartree-Fock method, as shown in equation A.39). In other words, it
accounts for the exchange-correlation energy XC X CE E E . The two
common approximations for the functional XCE are the local density
approximation (LDA) and the generalized gradient approximation
(GGA).44
As described in section A.2.3.1, the energy of non-interacting,
homogeneous electron gas is given using equation (A.55) as
2 32 3 5 3 33 310
d r r d r r r . However, it only takes into account
the kinetic energy part and the electron-nuclear interactions and neglects
the electron-electron interaction energy. A next step in this procedure is to
take into account the Coulomb interaction energy given by the term U ,
thus extending the Thomas-Fermi energy to
2 32 3 5 3 3 3 33 1310 2TF
r rE d r r d r r r d rd r
r r (A.66)
Equation (A.66) is a step ahead from the completely non-interacting
system but it still does not take into account the exchange and correlation
energy. Dirac 45 developed an exchange energy formula for the uniform
electron gas as
227
1 34 3
1 31 3
3 34
3 34
X
X
E r dr
E r
(A.67)
Since,
XC CXXC X C
E EEE E E (A.68)
Ceperley and Alder 46 determined the exact values of CE using numerical
simulations and Vosko et al.47 interpolated those values to obtain an
analytical function for the same. This analytical expression is given in the
spin-dependant form (if the electrons have spins and , then the total
electron density is given as ) as
2 4 42
2
, ,
,0 1 ,1 ,00
Cs c s
c s a s c s c s
E r r
fr r r r f
f
(A.69)
where
4 3 4 3
2 1 3
2 21
2 2
/ 2 20 0 10
2 2 20 0
1 1 1 22
2 1
2 4ln tan24
2 2 4ln tan24
c a
f
x b c bx bx c x bc b
x Ax x b xbx c b
x bx c x bx c x bc b
(A.70)
sx r and 0, , ,A x b c are fitting constants. sr is the effective volume
containing an electron and the spin polarization is given as
. Thus, for a uniform electron gas, the total energy of
the system can be given as,
228
2 32 3 5 3 3
1 33 3 4 3 3
3
3 3101 3 32 4
+
TF
C
E d r r d r r r
r rd rd r r d r
r r
Er d r
(A.71)
In the Kohn-Sham density functional theory approach (as discussed in
section A.2.3.2) , when the exchange correlation term XCE in equation
(A.60) is assumed to be equal to that of the uniform electron gas 1 3
4 3 3 33 3 +4
CXC
EE r d r r d r as shown in equation
(A.69), it is referred to as Local Density Approximation or LDA.
The Generalized gradient approximation or GGA is an
improvement over the LDA since it implements the gradient correction as 3 g ,GGA
XCE d r (A.72)
For the GGA to be practically useful, it is again important that it has an
analytical form like LDA. Different popular formats of the GGA48-54 are
used in the electronic structure calculations, but only the Perdew-Burke-
Ernzerhof approximation52 will be discussed here since it is used in this
thesis.
Similar to the LDA, the XCE term is again separated into the
exchange and correlation parts in GGA and in the Perdew-Burke-
Ernzerhof approximation,52 the correlation term is given as 3, , , GGA
C C s sE r H r t d r (A.73)
where t is the dimensionless density gradient given as 2 st k ,
is the scaling factor given as 2 3 2 31 1 2 and
229
1 32 2 24 3sk me . The functional form of H in equation (A.73)
is given as
22 2 2 3 2
2 2 4
0.066725 10.031091 ln 10.031091 1
AtH e me tAt A t
(A.75)
where1
3 2 2 20.066725 exp 0.031091 10.031091 CA e me
The exchange energy term for this GGA approximation is defined as 3GGA
X x XE F s d r (A.76)
where s is another type of reduced density gradient given as 0.52 2 1.2277sr me t and the functional form of XF is as follows.
20.8041 0.804
0.2195110.804
XF ss
(A.77)
This thesis uses the local density approximation (LDA) and the Perdew-
Burke-Ernzerhof (PBE) generalized gradient approximation in the Kohn-
Sham density functional theory, as described above.
A.2.4 Planewave-pseudopotential Methods
The above sections describe the multielectron system’s electronic
structure calculations using the Kohn-Sham density functional theory
within the framework of Born-Oppenheimer approximation. Even though
the density functional theory takes into account the electron correlation at
the computational cost of Hartree-Fock theory, it is computationally still a
difficult task to apply this method to an extended system like crystals or
bulk soft matter. The solution to this problem is to define a tractable size
of system, which when repeated periodically in all spatial directions will
represent the bulk i.e. to apply periodic boundary conditions to the cell
containing a set of atoms/molecules. If the cell is defined by vectors 1a , 2a
230
and 3a , then its volume is given as 1 2 3a a a . General lattice vectors
are integer multiples of these vectors as
1 1 2 2 3 3L N N Na a a (A.78)
A particular atomic arrangement which is repeated periodically, to mimic
the actual system of interest, can reduce the computational cost of the
calculations since the computations are restricted to the lattice. However,
in such an arrangement the effective potential also has a periodicity as
eff effr L r (A.79)
The resultant electron density will also be periodic. Given the periodic
nature, the potential can be expanded as a Fourier series as 55
31 where iGr iGreff eff eff eff
G
r G e G r e d r (A.80)
where the vector G forms the lattice in the reciprocal space which is
generated by the primitive vectors 1b , 2b and 3b in such a way that
2i j ija b , ij being the Kronecker delta. Thus, the volume of the
primitive cell in the reciprocal space is given as 31 2 3 2b b b .
Bloch’s theorem56 states that if r is a periodic potential
r L r , then the wavefunction of a one electron Hamiltonian of
the form 212
r is given as planewave times a function with the
same periodicity as the potential. Mathematically it can be written as
.ik rk kr e u r where k ku r u r L (A.81)
Alternatively the Bloch’s theorem can be written as55
ik rk kr L e r (A.82)
231
It also has to be noted that planewaves are the exact orbitals for
homogeneous electron gas.44 Since the function ku r is periodic, it can be
expanded as a set of planewaves as55
1 k iG rGk
G
u r c e (A.83)
Combining equations (A.81) and (A.83) we get,
1 i G k rkGk
G
r c e (A.84)
where kGc are complex numbers. While performing electronic structure
calculations of a system with periodic boundary conditions (periodic
potential) using the Kohn-Sham implementation of density functional
theory and planewaves as basis functions, k r can be seen as the Kohn-
Sham orbitals. In that case the Kohn-Sham density functional theory
equations can be rewritten as
212 eff k k kr r r (A.85)
where eff r accounts for nucleus-electron and electron-electron
interactions and the electron density is given as
2 332
2Fk kr r E d k (A.86)
Factor 2 in the above equation is to take into account both the electron
spins and is the step function which 1 for positive and 0 for negative
function arguments. The integration in equation (A.86) is over the
Brillouin zone (primitive cell in the reciprocal space).
As explained above, using the Bloch’s theorem, a problem of an
extended system with an extended number of electrons is transformed
into a problem within a small periodic cell. This improvement may not be
very satisfactory if the integration in equation (A.86) needs to be
232
performed at each and every point in the reciprocal space or the k-space
since there are infinite numbers of k-points. However since the electronic
wavefunction at k-points close to each other will be very similar, it is
possible to replace the above integral as a discrete sum over limited
number of k-points.
2 33
int
1 2
F kk kk po s
r E d k fN
(A.87)
The error due to this approximation can be minimized by using large
number of k-points.57 Usually it is practised to converge the energy of the
system with respect to the number of k-points in the calculation. However,
as the size of the simulation cell in real space gets larger the size of the
reciprocal space cell becomes smaller. This implies that as the simulation
cell in real space becomes bigger, the k-space and hence the number of k-
points becomes smaller. If the simulation cell is large enough then a single
k-point (often referred as -point) is also good enough for the calculation
purpose. Hence this thesis uses only the -point calculations.
A finite number of planewaves are required to perform the
computations. Since the accuracy of the Kohn-Sham potential is
dependant on the accuracy of basis set used for Kohn-Sham orbitals, it is
apparent that a larger basis set would result in a more accurate Kohn-
Sham potential. If plane waves are the basis set then the Kohn-Sham
potential needs to be converged with respect to the number of
planewaves. From equation (A.84), it can be seen that a higher modulus of
G would result in higher number of planewaves. Hence, a limit is placed
in the calculations where the G vectors with a kinetic energy lower than
the specified cut-off are only considered for a calculation. 2 2
2 cutG Em
(A.88)
233
The precision of the planewave implementation of Kohn-Sham density
functional theory approach is thus dependant on the parameter cutE , as
per equation (A.88). Some of the advantages and disadvantages of using
planewaves as basis functions are
Planewaves are orthonormal and energy independent.
Planewaves are not biased to any particular atom. Hence the entire
space is treated on an equal footing and hence does not cause the basis
set superposition error (due to overlap of individual atom’s basis set)
The conversion between real and reciprocal space representations can
be efficiently performed using Fast-Fourier transform algorithms and
hence the computational cost can be decreased by performing the
calculations in the reciprocal space.
However, it can not take advantage of the vacuum space in the
simulation cell by avoiding having a basis set in that region.
Since the planewaves are independent of the positions of atoms,
Hellman-Feynman theorem can be used to compute the forces15,
thereby reducing the computational cost. In other words, if a basis set
is dependant on the nuclear coordinates, then while calculating the
forces the derivatives of the coefficients (with respect to nuclear
coordinates) associated with the basis set also need to be computed.
However if the basis set is independent, the variationally minimized
coefficients can be used, as it they are, to compute the forces.
i i j ji j
i j i ji jHellman Feynmann n n
c H cE Hc c
R R R (A.89)
The valence wavefunctions are nodal in the core region of the atom
(Pauli’s exclusion principle) and hence a large number of planewaves
are needed to represent these large oscillations.
234
For a practical application of planewaves approach, a solution to
the nodal structure of valence wavefunctions problem needs to be identified.
The solution is to use the frozen core approximation i.e. to treat the core
electrons and the nucleus as a pseudocore. The consequence is that the
electron-nuclear potential will also have to replaced by a pseudopotential.
Since this pseudopotential eradicates the core electrons from the system, it
is very important that the pseudopotential takes into account the electron-
nucleus interaction (as if shielded by the core electrons) and the electron-
electron interaction (the classical Columbic and exchange-correlation
interaction between the valence and core electrons). Hence the
pseudopotential is angular momentum dependant as well. Due to this
pseudopotential, the all electron wave function also gets replaced by the
pseudo-wavefunction. Outside a certain cut-off radius the pseudopotential
matches the true potential of the system and the pseudo-wavefunction
matches the true wavefunction of the system (cf. Fig. A.3).
Additionally, it is worth noting that the contribution of core
electrons to chemical bonding is negligible and only the valence electrons
play a significant part in it. The core electrons play an important part in
the calculation of the total energy though and this implies that the
removal of core electrons will also result in lower energy differences
between different configurations, thereby reducing the efforts in achieving
the required accuracy. As mentioned before, less number of planewaves is
required than with the all-electron electron approach.
235
Figure A.3: The wavefunction of the system under the nuclear potential and under the pseudopotential and AE pseudo pseudoZ r .
Hamann, Schluter and Chiang58 laid down a set of conditions for a
good pseudopotential. Those are
1. The all electron and pseudo valence eigenvalues agree for a particular
atomic configuration.
2. The all electron and pseudo-wavefunction agree beyond a chosen core
radius cutr .
rcutr
pseudo
AE
pseudo
Zr
236
3. The logarithmic derivative of both the wavefunctions agree at cutr i.e.
ln lncut cut cut cut
pseudoAEAE pseudo
r r AE pseudor r
d ddr dr
.
4. Though inside cutr the pseudo and all electron wavefunctions and the
respective potentials differ, the integrated charge densities for both
agree i.e.22 3 3
0 0
cut cutr r
AE pseudod r d r .
5. The first energy derivatives of the logarithmic derivatives of both the
wavefunctions agree at cutr .
All the pseudopotentials that satisfy condition 4 are called as norm-
conserving pseudopotentials since the “norm” is conserved. The
pseudopotentials used in this thesis are norm-conserving
pseudopotentials. When a pseudopotential is developed, above conditions
(equivalency of energies and the first derivatives) are satisfied for the
reference energy, however, with a change in the chemical environment of
the atoms, the eigenstates will be at a different energy. Hence, for practical
application of the pseudopotential, it has to have the capability of
reproducing the above equalities with the all electron wavefunction in
different chemical environments and in a wider range of energies. It was
shown that the norm-conserving condition enhances this transferability58.
The two key aspects associated with any pseudopotential are “softness”
and “transferability”. A soft pseudopotential means that fewer
planewaves are needed, more electrons are frozen in the pseudocore and a
large cutr is employed. However, to make the pseudopotential more
transferable, fewer electrons should be frozen in the pseudocore (more
electrons treated explicitly), small cutr is employed and hence more
planewaves are required. A balance need to be sought between softness
and transferability, while developing the pseudopotential.
237
In addition to the conditions listed above, Hamann, Schluter and
Chiang58 also provided with a methodology to generate the norm-
conserving pseudopotentials. Generation of pseudopotential begins with
the all electron calculations. The atomic potential is multiplied by a cut-off
function so as to eliminate the strong attractive part. The parameters of the
cut-off function are adjusted to give eigenvalues equal to the all electron
calculations and the pseudowavefunctions which will agree with the all
electronic wavefunction after the cut-off radius. The total potential is then
calculated by inverting the Schrödinger equation. The total
pseudopotential acting on the valence electrons is then screened by
subtracting the classical Columbic potential and the exchange-correlation
potential, to obtain the ionic pseudopotential. Kerker59 and Troullier-
Martins60 simplified the above procedure of constructing the
pseudopotential by modifying the valence wavefunction instead of
modifying the potential, as suggested by Hamann-Schluter and Chiang.58
The Troullier-Martins wavefunction is of the following form 1 2 4 6 8 10 12
0 2 4 6 8 10 12expll r r c c r c r c r c r c r c r (A.90)
where the coefficients are determined using the Hamann-Schluter-Chiang
conditions.
Since the norm-conserving pseudopotential that can be developed
using the above procedure, while satisfying the Hamann, Schluter and
Chiang criteria, is angular momentum dependant (spherical symmetry
due to the potential of the nucleus and angular momentum dependency
due to the core electrons), it can be written in a generalized form as61
0
ll lm
pseudol m l
r P (A.91)
where l is the azimuthal quantum number , , ,s p d f , m is the magnetic
quantum number and lmP is a projector on angular momentum
238
functions. An approximate way is to treat one specific angular momentum
(typically the highest value of l for which the pseudopotential is
generated) as the local part and the non-local part then consists of the
difference between this local part and the actual angular momentum
dependant pseudopotential. The pseudopotential is then written as
,
l lmpseudo local
l m
semilocal
r r P (A.92)
Thus, the local potential describes the interaction outside the pseudocore
and the non-local part describes the interaction with core electrons. This
type of pseudopotential is also called as semi-local pseudopotential since
(i) the local part has one specific angular momentum and (ii) the projection
operators act only on the angular variables of the position vector. The
energy resulting from the local part of the pseudopotential can be
calculated conveniently as 3locald r r r , however, the non-local part
is slightly more complicated due to the fact that the operator in the
planewave basis set does not have a simple form in both, the real and the
k-space. The two methods used to calculate the contribution of the non-
local part of the pseudopotential to the energy are as follows:
1. Gauss-Hermite Integration62:
The energy associated with the semi-local potential semilocal is given
in the reciprocal space as
ˆ
1 ˆ = ,
semilocal i semilocal i i semilocal ii i G G
i i semilocali G G
E G G G G
c G c G G G (A.93)
If the angular momentum projector operator is given in the form of
lmlmP G Y G , where lmY are spherical harmonics, and a spherical
239
wave expansion for G is used then a simplified form (analogous to
semilocal in equation A.92) for ˆsemilocal is given as,
2
,
2
16ˆsemilocal lm lml m G G
ll l
Y G S G Y G S G
r dr r J Gr J G r (A.94)
where lJ is the Bessel function of the first kind with integer order l . The
first part of equation (A.94) can be calculated analytically, however, the
second part is numerically integrated using the Gaussian quadrature
formula such that 2i i
ir f r dr w f r .
2. Kleinman-Bylander method63:
The scheme proposed by Kleinman and Bylander involves the
following potential operator which substitutes the potential operator
ˆsemilocal .
, ,
, ,
l ll m l m
KB ll m l m
G G (A.95)
where ,l m is the atomic pseudowavefunction and lpseudo local . The
energy associated is then given as
, , ,,
=
KB i KB ii
l li i l m l m l m
i l mG G
E
c G c G C G G (A.96)
If a unit operator is inserted between the bra and ket of the element
,l
l mG , the wavefunction ,l m is written in the form of radial
wavefunction and spherical harmonics as ,l m l lmY and a spherical
wave expansion for G is used then
2, 4l l l
l m lm l lG i S G Y G r drJ Gr (A.97)
240
Substituting equation (A.97) into equation (A.96), we get the final
expression for the energy as 2
,,
2 2
16KB l m i i
i l mG G
l llm lm l l l l
E C c G c G S G S G
Y G Y G r drJ Gr r dr J Gr (A.98)
It has to be noted that the Kleinman-Bylander scheme is computationally
more efficient than the Gauss-Hermite numerical integration scheme to
calculate the energy contribution of the semi-local part of the
pseudopotential, however, constructing an accurate and transferable
pseudopotential using the Kleinman-Bylander scheme can be challenging
due to its complex form.64
It is also possible to generate the pseudopotential directly in an
analytical form in such a way that it fulfils the Hamann-Schluter-Chiang
conditions. The pseudopotential is separated in such a way that the local
part is completely independent of the angular momentum and the non-
local part accounts for the angular momentum. One of the most popular
pseudopotential with this type of construction is the Goedecker
pseudopotential65, 66 where the local part is given as2
2 4 6
1 2 3 4
1exp22
ionlocal
locallocal
local local local
Z r rr erfr rr
r r rc c c cr r r
(A.99)
where ionZ is the charge on the pseudocore and localr is the range of the
ionic charge distribution. The non-local part of the pseudopotential is
given as
241
22 2
1
4 1 2, , 1
22 2
4 1 2
1exp2
, 24 1 2
1exp2
24 1 2
l i
l l lnon local lm ijl i
l m i j l
l j
l
l jl
rrr
r r Y hr l i
rrr
r l jlmY
(A.100)
where lmY r are spherical harmonics and is the gamma function. The
benefit of the above pseudopotential is that it has an analytical expression
in the Fourier space (or k-space) as well. The parameters of the above
pseudopotential are computed by minimizing the objective function
which could be a sum of differences of properties calculated using the
pseudopotential and the all electron calculations.
This thesis uses the planewave-pseudopotential (Troullier-Martins
pseudopotential with the Gauss-Hermite integration scheme and the
Goedecker pseudopotential) implementation of the Kohn-Sham density
functional theory (with LDA and Perdew-Burke-Ernzerhof GGA
approximation), as described above.
A.2.5 Optimization techniques
The electronic structure calculation methods described in sections
A.2.2 – A.2.4 are used to calculate the ground state energy of the system at
a particular nuclear (atomic) configuration. The energy calculated using
the electronic structure calculation methods is a function of the nuclear
configurations only. However, an arbitrarily chosen nuclear
configuration/geometry may not be the most stable (lowest energy) one
and hence it is needed to minimize the system energy with respect to the
nuclear configuration. Finding the stationary point of the system where
242
the first derivative of energy with respect to the nuclear configuration is
zero 0E R is an important aspect of molecular modeling. Finding
the nearest stationary point of the domain in which the system is lying can
be done with some of the most common minimization/optimization
methods like the Newton-Raphson, Steepest-Descent etc. For example, if a
Newton-Raphson method is used then the steps involved in the
optimization process will be as follows:
1. Calculate initialE R
2. Calculate numerically initialR
E R and initialR
H R where H is the
Hessian matrix.
3. Continue the iteration scheme 11
initial initialinitial R R
R R H R E R till the
convergence is reached 0E R .
This type of energy minimization leads to a local minimum and the
minimized configuration may not be the “true” configuration which lies at
the global minimum. Two of the most widely used methods in molecular
simulations to find the global minimum more reliably are Molecular
Dynamics and Monte-Carlo. Monte-Carlo method is based on making
random changes to the system configuration. A specific criterion is
defined to accept or reject these random changes, thus helping the system
to move towards lower energy configuration states. Molecular dynamics
on the other hand is a more physical method since every atom (nucleus) is
assigned a finite velocity at the initial system configuration. Consecutive
configurations are then generated by solving the Newton’s equations of
motion for all the atoms. The velocity of the atoms is dependant on the
kinetic energy of the system, which in turn governs the experimentally
measurable quantity of the system, i.e. the temperature
243
21 12 2kinetic BE mV k T . Thus, molecular dynamics also accounts for
the finite temperature effect on the system. In molecular dynamics, the
system explores the potential energy surface part with the energies lower
than the kinetic energy of the system. Molecular dynamics is also a better
representation of the physical state of the system since most of the systems
are at some finite temperature and the temperature dependant dynamics
observed in this method are the real dynamics of the system. The forces
acting on every atom of the system are calculated from the kinetic energy
and the potential energy (the energy calculated using the electronic
structure calculations or force field methods) of the system at each
configuration during the molecular dynamics run. Details of the molecular
dynamics method are described in the next section.
Figure A.4: An illustration of a 1-dimensional potential energy surface of a system.
Ener
gy
Trial Configuration
Global Minimum Local Minimum (Newton-Raphson, Steepest Descent)
(Molecular Dynamics, Monte-Carlo)
244
A.2.5.1 Molecular Dynamics algorithm
As mentioned before, nuclei are orders of magnitude heavier than
electrons and hence they can be treated using the classical Newtonian
mechanics i.e. their motion can be calculated using Newton’s equations of
motion. If the system contains n atoms with coordinates 1 2, , , nR R R R
and the potential energy calculated from electronic structure calculations
at a fixed configuration R is E R , then the Newton’s second law of
motion Force mass acceleration can be written in the differential form
as2
2
dE d RmdtdR
(A.101)
It can be shown that equation (A.101) conserves the total
(kinetic+potential) energy of the system. If RT is the kinetic energy then, 2
2
2
2 2
2 2
12
from equation (2.101)
0
R R
R
d E T dE dT dE dR d dRmdt dt dt dt dt dtdR
dE d RmdtdR
d E T d R dR dR d Rm mdt dt dt dt dt
(A.102)
To calculate the evolution of atomic positions with time, equation (A.101)
needs to be solved. The procedure is described below and is shown in Fig.
A.5. The advancement of nuclear coordinates in a small timestep t can
be given by the Taylor series expansion as,2 3
2 31 2 3
1 12! 3!
i i ii i
dR d R d RR R t t tdt dt dt
(A.103)
If we go one timestep back, then a similar expression can be written as 2 3
2 31 2 3
1 12! 3!
i i ii i
dR d R d RR R t t tdt dt dt
(A.104)
Adding equations (A.103) and (A.104) gives,
245
22
1 1 22 ii i i
d RR R R tdt
(A.105)
It can be seen that equation (A.105) calculates the configuration at a
particular timestep using the configurations of two previous timesteps
and the error is of the order 4t . To start the molecular dynamics
calculations this way, the configuration at a timestep before the starting
configuration has to be known and it can be approximated as
0 1 0 0R R dR dt t .
Figure A.5: The Velocity Verlet Molecular dynamics algorithm.
The numerical method described above is also referred as Verlet
algorithm.67 However, some of the disadvantages of this method are that it
requires storage of two sets of positions and velocities do not appear
explicitly (which is needed when a simulation needs to be run at a
246
constant temperature). It may also give rise to a numerical instabilities
since a small number 2t containing term is added to the difference
between two large numbers 12 i iR R . Hence the Velocity Verlet
algorithm,68 as described below, is a more popular scheme to solve
molecular dynamics equations.2
21 2
2 21 1
2 2
12!
12
i ii i
i i i i
dR d RR R t tdt dt
dR dR d R d R tdt dt dt dt
(A.106)
A more general formulation of the above described equations of
motion can be done in the form of Lagrangian. Lagrangian, in classical
mechanics, is defined as the difference between the kinetic energy and the
potential energy of the system. The benefit of using Lagrangian
formulation is that it is not restricted to a particular type of coordinate
system. If q are the coordinates and q are their time derivatives then the
Lagrangian can be written as7
£ , RT Eq q q q (A.107)
The equations of motion in the Lagrangian formulation can be obtained
from the following Euler-Lagrange equation.
£ £ 0ddt q q
(A.108)
where q is treated as a variable. In Cartesian coordinates Rq and
dR dtq and in that case equation (A.108) reduces to the Newton’s
equation (A.101) as follows.
247
2
2
2
£ , £ ,£ £ 0
12R
dR dRR Rd d dt dtdt dt RdR
dt
dRd mdtd dT dE d dE
dt dtdR dRdR dRd ddt dt
d dE d R dEdRm mdtdt dtdR dR
q q
(A.109)
A.2.5.2 Nose-Hoover thermostat
As shown above, the molecular dynamics algorithm described in
the previous section conserves the total energy. Hence the system falls
naturally under the microcanonical ensemble where the total number of
atoms, the system’s volume and its total energy are conserved. Since the
kinetic energy is the difference between the total energy (constant) and the
potential energy (changing with atomic positions) of the system, it may
vary significantly during the course of a molecular dynamics run, thereby
causing temperature variations. When the objective of any simulation
study is to support/validate/predict the experimental data, it is more
appropriate to be able to perform the simulations at constant temperature
condition (There is also an ensemble where the pressure of the system
needs to be maintained constant but its discussion is beyond the scope of
this thesis). One of the most popular ways to perform molecular dynamics
simulations at constant temperature is to integrate the system with a heat
bath. Heat transfer occurs between the system and the heat bath so as to
keep the system at a constant temperature. The mathematical formulation
for such a heat bath was provided by Nose and Hoover 69, 70 and hence is
commonly referred as Nose-Hoover thermostat. In this formulation, to
248
the physical system of n particles with coordinates R , potential energy E
and velocities dR dt R , an artificial dynamical variable s representing
the heat bath is added. This additional variable has mass M (actual unit
of M is 2energy time ) and velocity s . The interaction between the actual
system and the heat bath is obtained through this parameter s . It acts as a
time scaling parameter as 1t s t where t is the time interval in the
real system and t is the time interval of the extended system containing
the real system and the heat bath. As a consequence, the atomic
coordinates remain similar in both the cases but the velocities are
modified as71
1 and R R R Rs (A.110)
The definition of parameter s and its interaction with the real system as a
heat bath is more intuitive in equation (A.110) where it can be seen as a
velocity scaling parameter 2Temperature velocity . With such an
integrated heat bath, the extended Lagrangian of the system, analogous to
equation (A.107), becomes
2 2 2
1
1 1£ , , , ln2 2
n
i Bi
R R s s ms R E R s k T sM (A.111)
where is equal to the degrees of freedom or 1real realDOF DOF ,
the first two terms are the kinetic and potential energies of the real system
and the last two terms are the kinetic and potential energies of the heat
bath. It was shown that this form of the potential energy of the heat bath
results in canonical ensemble of the real system.70 It has to be noted that
the sign of s determines the direction of heat flow. If 0s then heat flows
into the real system and if 0s then heat flows out of the real system. The
equations of motion derived from the Lagrangian of equation (A.111) are
249
1 2 1
1 1 2 2
1
2i i ii
n
i i Bi
ER m s s sRR
s s m s R k TM
(A.112)
where the first term represents the equation of motion of the real variables
and the second term represents the equation of the s variable. Equation
(A.112), representing the extended system, can be numerically integrated
based on the timestep t . However, the atomic coordinates and velocities
of the real system will evolve at a timestep of 1t s t . This implies that,
if the molecular dynamics algorithm is implemented using the thermostat
of equation (A.112), then the real system will evolve at uneven time
intervals. Hence the equations were reformulated69 from extended system
to real system as follows. Since 1t s t , d dsdt dt .
2 2
2
, , ,
, and
s s s ss s s s ss R RdE dER sR R s R ssRdR dR
(A.113)
The Lagrangian equations of motion (A.112) according to above
transformation then become
1
1 2
21
1
1
i i ii
n
i i B ni
i ii
ER m RR
Tm R km R
M
(A.114)
where 1s s . It can be seen that the magnitude of parameter M
determines the coupling between the heat bath and the real system. A
very large value of M will result in poor temperature control (or a
microcanonical or NVE ensemble) whereas a very small value of M will
result in high frequency oscillations.
250
The Nose-Hoover thermostat, as described above, is one of the
most widely used thermostat method in molecular dynamics simulations.
However, it has been reported that this type of thermostat suffers from the
problem of non-ergodicity72, 73 for systems with certain types of
Hamiltonians. A very similar thermostat method, called as Nose-Hoover
chain thermostat72 cures this problem. In this method, the thermostat
applied to the real system is thermostatted by another similar thermostat
and so on. The mathematical formulation for the Nose-Hoover chain
thermostat is given as follows.
11
1 21 1 1 2
21
1
1 22 2 1 1 2 3
1
i i ii
n
i i B ni
i ii
B
ER m RR
Tm R km R
k T
M
M M
(A.115)
A.2.6 Car-Parrinello Molecular Dynamics
Molecular dynamics in atomic level modeling can be classified into
following two main categories:
1. Classical molecular dynamics using force field: The potential energy
E R and forces dE dR are calculated using force-field methods or
molecular mechanics, as described in section A.2.1 and the equations of
motion are solved as described in section A.2.5.
2. Ab Initio molecular dynamics: The potential energy and forces are
calculated using electronic structure calculations described in sections
A.2.2 – A.2.4 and the equations of motion are solved as described in
section A.2.5.
251
Classical molecular dynamics is computationally much less expensive
than ab initio molecular dynamics since ab-initio molecular dynamics
require electronic structure calculations. This also means that the time and
length scales that can be accessed by classical molecular dynamics are
much larger than those by ab initio molecular dynamics. However, as
mentioned before, classical molecular dynamics is inadequate to model
chemically complex systems (where electronic structure and bonding
patterns change due to reactions, system contains transition metal
compounds and simulations give rise to many novel molecular species)
and hence ab initio molecular dynamics remains the only option. In ab
initio molecular dynamics, when the electronic structure calculations
(optimizing the wavefunction for a fixed nuclear configuration) are
performed after every molecular dynamics time step, it is called as Born-
Oppenheimer molecular dynamics and the Lagrangian associated with the
Born-Oppenheimer molecular dynamics (without the thermostat) is
21£2BO i i
im R H (A.116)
The above Lagrangian is appended to include the thermostat when ab
initio molecular dynamics is performed in a canonical ensemble. The
electronic structure calculations in the Born-Oppenheimer molecular
dynamics can be performed using Hartree-Fock, semi-empirical or Kohn-
Sham DFT methods and can be performed explicitly for all the electrons or
using the planewave-pseudopotential approach. Born-Oppenheimer
molecular dynamics has the capability to significantly leverage the field of
molecular dynamics by extending it to incorporate complex, diverse and
less known systems in material science and chemistry. However, due to
the associated computational expense, it did not gain the popularity that
would justify its usefulness. A breakthrough in the field of ab initio
molecular dynamics was brought by Car-Parrinello74 with the
252
introduction of a modified method that would reduce the computational
cost of ab initio molecular dynamics. Unlike Born-Oppenheimer molecular
dynamics, the Car-Parrinello scheme does not require the optimization of
the wavefunction (or density) to be performed after every molecular
dynamics timestep and it ensures that the electronic wavefunction stays
close to the optimized value throughout the course of the molecular
dynamics simulation. The details of Car-Parrinello method are given in
the following section.
A.2.6.1 Car-Parrinello Scheme
Born-Oppenheimer molecular dynamics is a combined modeling
approach where the electronic motion is treated purely quantum
mechanically and the nuclear motion is treated classically and hence
electronic structure calculations (or wavefunction optimization) are
required after every molecular dynamics step. If this quantum-classical
two component system is mapped on to a completely classical formulation
such that the movement of the electronic structure (optimized before the
molecular dynamics) also can be followed classically, then it might be
possible to avoid the electronic structure calculation after every time step
and is needed only once. In other words, instead of optimizing the
wavefunction for the modified Hamiltonian H of the system after every
molecular dynamics step, the wavefunction is directly propagated using a
classical formalism. This is the fundamental idea behind the Car-Parrinello
scheme.74
The (potential) energy of the system, calculated using electronic
structure calculations, is a function of nuclear coordinates but it can also
be considered as a functional of the wavefunction which in turn consists of
individual atom wavefunctions (as a Slater determinant). As discussed
before, the forces on the nuclei are calculated by differentiating the
253
Lagrangian with respect to nuclear coordinates. Similarly, if a Lagrangian
is defined such that it encompasses the motion of nuclei and the
propagation of the electronic structure, then the forces on the wavefunction
(so as to propagate it along time) can be calculated by taking a functional
derivative of the Lagrangian with respect to the wavefunction. In this way
the quantum mechanically calculated electronic wavefunction can be
propagated classically. Car and Parrinello formulated this Lagrangian as 74
2
Potential EnergyKinetic Energy of Kinetic Energy of nuclei wavefunction
1£ constraints2CP i i i j j
i jm R H (A.117)
where j is the orbital/wavefunction of the thj electron in the system,
(unit of is 2energy time ) is the fictitious mass associated with the
wavefunction and the constraints in equation (A.117) can be some external
constraints on the system or internal constraints like orthonormality. It has
to be noted that the kinetic energy term 12 i j j
j has no relation
with the physical quantum kinetic energy and is completely fictitious. The
Euler-Lagrange equations will then be
£ £ £ £ and CP CP CP CP
i i i i
d ddt R R dt
(A.118)
From equation (A.118), the equations of motion become
constraints
constraints
i ii i
i ii i
m R HR R
H (A.119)
Equation (A.119), in conjunction with the Nose-Hoover thermostat as
described in section A.2.5.2, can then be numerically solved using the
velocity verlet algorithm, as described in section A.2.5.1. The kinetic
energy of the nuclei (and the system) is 2i i
im R and thus the temperature
254
of the system 2i i
iT m R . Similarly the fictitious temperature associated
with the wavefunction is 12 i j j
j. A very small value of is
chosen so as to keep the fictitious temperature of the wavefunction very
low. The reason for this choice will become clear later. The main concerns
about the Car-Parrinello scheme would be as follows: (i) If the electronic
wavefunction follows the Born-Oppenheimer surface through out the
molecular dynamics simulation, (ii) How the forces calculated using Car-
Parrinello Lagrangian are equal to Born-Oppenheimer molecular
dynamics forces and (iii) How the total energy of the system is conserved
(microcanonical ensemble).15, 75
At the beginning of the Car-Parrinello molecular dynamics scheme,
the electronic wavefunction is optimized for the initial nuclear
configuration. When the nuclei start moving, their motion changes the
electronic structure of the system thereby changing the wavefunction
representing the minimum in the energy at an instantaneous nuclear
configuration. According to the Car-Parrinello Lagrangian, the
wavefunction is also propagated classically according to equation (A.119).
As mentioned before, the Car-Parrinello molecular dynamics will follow
the Born-Oppenheimer molecular dynamics when the wave function
propagated using the Car-Parrinello scheme will result in the same
electronic energy as that from the wavefunction which is optimized for the
modified Hamiltonian. This is only possible when no additional energy
from an external system (i.e. from the classical nuclear system) is
transferred to the classically propagated quantum electronic system;
because if the energy transfer occurs then the quantum system which is
propagated classically need to be treated quantum mechanically so as to
bring the electronic system to its minimum energy level. Also the “extra”
energy transferred to the wavefunction will result in larger forces on the
255
wavefunction (in addition to i
H ), thereby deviating from the
Born-Oppenheimer surface. In other words, if the energy from the nuclei
is not transferred to the electronic system, the Car-Parrinello scheme
should follow the Born-Oppenheimer molecular dynamics. This
adiabaticity is achieved due to the virtue of the timescale difference
between the very fast electronic motion and the slow nuclear motion. It is
shown that when a small perturbation in the minimum energy state of a
system results in some force on the wavefunction, then the minimum
frequency related to the dynamics of the orbital is75
12
mingape E
(A.120)
where gapE is the energy difference between the highest occupied and the
lowest unoccupied orbital. The parameter gapE in equation (A.120) is
determined by the physics of the system, however, the parameter is
completely fictitious and hence can be fixed to an arbitrary value. The
frequency range of the orbital dynamics thus can be switched towards the
higher side by choosing a very small value of . If a sufficiently small
value is chosen then the power spectra emerging from the wavefunction
dynamics (fast motion, high frequency) can be completely separated from
that emerging from the nuclear dynamics (slow motion, low frequency). If
these two power spectra do not have any overlap in the frequency domain
then there will not be any energy transfer from the nuclei to the
wavefunction. Thus the fictitious kinetic energy of the wavefunction
12 i j j
j will remain constant. This fictitious kinetic energy is also a
measure of the correctness of the Car-Parrinello implementation. In this
way if the wavefunction or electronic structure during the Car-Parrinello
256
molecular dynamics follows the Born-Oppenheimer surface then the
forces acting on the nuclei i
HR
will be the same as Born-
Oppenheimer molecular dynamics. The physical energy of the system is 2
i ii
m R H , whereas the Car-Parrinello energy is
2 12i i i j j
i jm R H . Since
212 i j j i i
j im R H (A.121)
and the fictitious kinetic energy remains constant, from statistical
mechanical point of view, it can be said that the system is under
microcanonical ensemble where the total energy of the system is
conserved. Figure A.6 shows various energies of a model system of bulk
crystalline silicon during the car-Parrinello run.
Figure A.6: Variation of different energies during the Car-Parrinello molecular dynamics run of bulk silicon. Adapted from Pastore and Smargiass75
A minute inspection of Fig. A.6 shows that the oscillations in the
fictitious kinetic energy are a mirror image of the oscillations in the
2 12i i i j j
i jm R H
2i i
im R H
H
12 i j j
j
257
potential energy of the system (a few order of magnitude difference
though). These can be attributed to the pull applied by the classically
moving nuclei on the classically propagating wavefunction.
This thesis uses the Car-Parrinello molecular dynamics scheme as
described above and the necessary details of the accuracy of the molecular
dynamics run as described in chapter 4 of the thesis.
A.2.7 Metadynamics
The Car-Parrinello molecular dynamics scheme, even though
significantly reduces the computational cost of ab initio molecular
dynamics, can not access the time scales more than a few picoseconds
when hundreds of atoms are present in the simulation system (even with
the state of the art computational servers). With the technological
evolution in computer hardware and reduction in costs, researchers are
hoping to increase the accessible length and timescales of ab-initio
molecular dynamics simulations. However, to run a simulation equivalent
to hundreds of nanoseconds (which is possible using classical force field
molecular dynamics) and to be able to simulate realistic phenomena
which take place at much larger timescales than that can be accessed by ab
initio molecular dynamics, in addition to the development of
computational hardware, it also becomes necessary to implement methods
that can accelerate the events somehow to make them happen earlier. Ab-
initio molecular dynamics is usually employed when chemical reactions
are taking place in the system and if the reaction of interest is associated
with a large energy barrier then the timescale for the reaction is large and
thus becomes difficult to be accessed using ab initio molecular dynamics.
Hence, a system may be stuck in a local minimum in the energy surface
and may take too long computational time to cross the energy barrier to
reach the global minimum (cf. Figure A.6).
258
The molecular dynamics simulation can be run at a higher
temperature so as to accelerate the event which is expected to take place at
a longer timescale in real system, however, this may cause some
undesirable and unrealistic events to happen in the system. There are
several methods that have been implemented in the literature in the past
to overcome this difficulty including umbrella sampling,76 nudged elastic
band,77 finite temperature string method,78 transition path sampling,79
milestoning,80 multiple timescale accelerated molecular dynamics,81 to
name a few. The most recent of all these methods is called metadynamics82
and it has the following advantages:
1. It encompasses several benefits of all the above mentioned methods
2. It can accelerate the rare events so as to be able to see them in a
realistic computational simulation time.
3. It can also be used to reconstruct the energy surface so as to get
quantitative information about the energy landscapes and barriers.
4. It is coupled with the Car-Parrinello molecular dynamics by
Iannuzzi et al. very recently.83
The following two sub-sections describe the concept and mathematics
behind the metadynamics technique.
259
Figure A.7: An illustration showing a large energy barrier for the system to go from a state A to the more stable state B.
A.2.7.1 Concept
The metadynamics technique, as described by Laio and Gervasio,82
is based on the principle of filling up the energy surface with potentials.
As shown in Figure A.8, if the energy surface is plotted as a one
dimensional function of a particular reaction coordinate and the system is
residing in the potential well ‘A’ from which it is taking too long to escape
due to the energy barriers then (cf. Figure A.7),
1. The potential well ‘A’ is filled up with small potentials so that
system slowly comes to a higher energy position.
2. As soon as the middle potential well is filled up, the system escapes
to the potential well ‘B’ on the left through the saddle point S1.
3. The potential well ‘B’ is then gradually filled till the saddle point S2
is reached then the system escapes to the potential well ‘C’.
4. Gradually, well ‘C’ is filled.
260
5. The system is forced to cross the energy barriers to reach the global
energy minimum by filling up the potentials and if the potentials
and the positions of their deposition are tracked, then the energy
well can be reconstructed.84, 85
Figure A.8: The system initially placed in well A goes to the global minimum in well C after filling up the energy surface. Adapted from Laio and Gervasio82
A.2.7.2 Extended Car-Parrinello Lagrangian for metadynamics
As described above, the metadynamics technique is based on filling
up the energy surface by dropping potentials at small time intervals in the
coordinate space of interest. Though this method can be implemented in
any type of molecular dynamics techniques (classical and ab initio), the
mathematics relevant to the implementation of this method in the Car-
Parrinello scheme is described in this thesis, as originally given by Ianuzzi
AB
CS1S2
E
Coordinate
—— Well ‘A’ is filled
—— Well ‘B’ is filled
—— Well ‘C’ is filled
261
et al.83 and further extended by Laio et al.82, 86-88 If is a vector of the
collective variables (coordinates of interest) that form the energy well to be
filled or that form the energy surface of interest (for ex. it can be the bond
distance between two hydrogen atoms if the dynamics and energy surface
of hydrogen dissociation is studied or it can be some coordination number
if a more complex phenomenon like protein conformation is studied), the
metadynamics approach treat them as additional variables in the system
and the car-Parrinello Lagrangian is then extended as 83
21£ £ ,2MTD CP cv cv cv cv cv cv cv
cv cvm k R t (A.122)
where £CP is the Lagrangian defined in equation A.117, the first term
indicates the kinetic energy of the collective variables, the second term is
the harmonic restraining potential and the last term is the potential that is
dropped to fill the energy well in the collective variable space at
different time intervals. Defining the kinetic and potential energy of the
additional collective variables allows controlling their dynamics in the
canonical ensemble using a suitable thermostat. The dynamics of ionic and
electronic (fictitious) motion are separated in the Car-Parrinello molecular
dynamics by choosing an appropriate value for the mass associated with
the fictitious kinetic energy of the wavefunction (as described in section
A.2.6.1). Analogous to the original Car-Parrinello scheme, the dynamics of
the collective variables are separated from the ionic and fictitious
electronic motion by choosing an appropriate value for the fictitious mass
cvm of the collective variables. If the fictitious mass cvm is large then the
dynamics of the collective variables will be slow and thus can be
separated from the ionic dynamics. The forces acting on the collective
variables are due to the potential energy cv cv cv cvk R , and due to the
potential drops ,cv t . Hence the dynamics of the collective variables are
262
also dependant on cvk . It has been shown that the extra term in
metadynamics introduces an additional frequency for the motion of
collective variables as cv cvk m 86. The force constant cvk is chosen such
that the collective variables are close to the actual coordinates of the
system. If a small cvk is used, it will result in a large variation in the
collective coordinate even with a small potential drop. However, a very
large value may result in very small timestep and excessive long
computational time. Since cvk is selected by the above constraints, it sets
limitation on the value of cvm so as to maintain the adiabaticity.
The potential used to fill the energy well at a time t is given as 82
22 1
2 4, exp exp2 2i
i i ii
cv MTDt t i
t Hw w
(A.123)
where parameter MTDH is the height of the Gaussian. The first term in the
above functional form of the potential is a typical Gaussian which is then
multiplied with another Gaussian of width 1i iiw . This
mathematical manipulation narrows the width of the potential in the
direction of the trajectory, thus depositing the potentials close to each
other in the direction of the trajectory. The values of the height and width
of Gaussian depends upon the topology of the energy landscape of the
system under investigation.86 The procedure to choose optimum values of
the parameters in the potential is provided in chapter 5 of the thesis along
with further details about implementing the metadynamics method to a
particular system.
Metadynamics using an extended Car-Parrinello Lagrangian has
recently been implemented to study glycine and pyrite surface
interactions,89 isomerisation of alanine dipeptide,90 azulene to naphthalene
rearrangement,91 proton diffusion in molecular crystals,92 to name a few.
263
A.2.8 Electronic Structure analysis methods
Ab initio calculation methods described in sections A.2.2—A.2.4 aim
at getting an accurate electronic structure around the fixed nuclei and
methods described in sections A.2.5—A.2.7 describe how the movement of
the nuclei at a finite temperature can be coupled with the electronic
structure calculation methods. In all these methods, the system as
composed of heavy nuclei immersed in the sea of light electrons which are
moving at high velocities and are treated quantum mechanically (their
position and momentum can not be determined simultaneously above a
specific accuracy). However, a more visually appealing picture of the
chemistry of the system is in terms of atoms that are held together by
covalent or ionic or electrostatic or van de Waals bonds (as in molecular
mechanics). Though the outcome of the electronic structure calculations is
the electronic energy and the wavefunction, it would of great advantage if
this information can be interpreted in the bonding and non-bonding type
formulation. Though the electronic structure calculations do not provide
this information exclusively, it has to be noted that this information is
hidden in the wavefunction (or electron density) calculated using these
methods. An extensive literature is available on the mathematical
treatment of the wavefunction (or electron density) to compute these
properties of interest. These mathematical treatment methods are also
called as wavefunction analysis methods and some of the most commonly
used methods are Mulliken population analysis, 93 Bader’s theory of atoms
in molecules,94 Hirshfeld charges,95 electrostatic potential, Becke and
Edgecombe’s electron localization function,96 to name a few. The details of
all these methods will not be discussed in this thesis, except for the
electron localization function, since this method is used extensively here.
The details of electron localization function analysis are described in the
following sub-section.
264
A.2.8.1 Electron Localization Function (ELF)
Electron localization function was proposed by Becke and
Edgecombe96 as a measure of localization of electrons in the real space of
the system in the following functional format.12
1 ELFELF
HOM
D rr
D r (A.124)
where ELFD r is the probability of finding second like spin electron near
the reference point and is given as 221 1( )
2 8
n
ELF ii
r
D r (A.125)
The summation in equation (A.125) is over n spin orbitals (either 12 or
12 spin) and is the Kohn-Sham orbital. is the density of electrons
with one particular spin. The term HOMD r in equation (A.124) is the
corresponding term for the uniform electron gas with density equal to the
local value of r in the system and is given as
5 52 3 33 310HOMD r (A.126)
It can be noted that ( )ELFD r can itself be a measure of localization of
electrons, however, it will vanish in the regions dominated by same spin
orbitals and will have a very small value in the region where actual
localization of electrons exists, as in the formation of covalent bonds. It
also does not have an upper or a lower bound. Hence, Becke and
Edgecombe proposed the normalization method, as can be seen in
equation (A.124), so that the values of the localization function remain
bounded between 0 and 1. Higher value of the electron localization
function indicates higher localization and a value of 0.5 indicates
localization equivalent to homogeneous electron gas.
265
Using the benefit of the fact that, similar to electron density, the
electron localization function is also a continuous and differentiable scalar
field Silvi, Savin and coworkers97-99 further showed that its topological
analysis can be used to classify the nature of interaction between species.
The gradient of electron localization function will determine the local
maxima, local minima and saddle points in the electron localization
function field. The local maximum, also called as an attractor, plays a key
role in the classification of chemical bonds. If the domain of ELF, a spatial
region bounded by closed ELF isosurface, contains only one attractor, it is
called as irreducible domain, whereas, if a domain contains more than one
attractor, it is called as reducible domain. As the ELF value is increased, a
reducible domain separates into two or more irreducible domains. The
reduction of these localization domains gives rise to distinguishable
valence basins. The synaptic order of a valence basin, i.e. the number of
atomic core basins in contact with the valence basin, is used to
characterize the chemical interaction as electron-sharing or non electron-
sharing. An electron-sharing interaction will always have a disynaptic
valence basin. The spatial arrangement of these irreducible disynaptic
basins containing an attractor was first used by Silvi and Savin to classify
the covalent electron sharing interactions between the species, as shown in
Fig. A.9. Similarly attractors are also present in the non-electron sharing
type of interaction, however, they are monosynaptic in nature (lone pairs
of electrons). Details of the analysis of the electron localization function
based on the valence basin separation and the attractors are discussed in
chapter 3 of the thesis.
266
Figure A.9: A 2-D illustration showing the difference in the topology of the electron localization function isosurfaces for ethane, ethylene and ethyne.
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