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The subject of this thesis is the development of the necessary tools and their application for a better understanding of the morphology and proton transport in the perfluorosulfonic acid short-side-chain membranes.
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UNIVERSITY OF CALGARY
Molecular Modelling of Proton Transport and Structure in the Short-Side-Chain
Perfluorosulfonic Acid Polymer
by
Iordan Hristov Hristov
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF CHEMISTRY
CALGARY, ALBERTA
April, 2010
© Iordan Hristov Hristov 2010
UNIVERSITY OF CALGARY
FACULTY OF GRADUATE STUDIES
The undersigned certify that they have read, and recommend to the Faculty of Graduate
Studies for acceptance, a thesis entitled “Molecular Modelling of Proton Transport and
Structure in the Short-Side-Chain Perfluorosulfonic Acid Polymer” submitted by Iordan
Hristov Hristov in partial fulfillment of the requirements for the degree of DOCTOR OF
PHILOSOPHY.
Neutral Chair, Dr. Jurgen GailerDepartment of Chemistry
Supervisor, Dr. Reginald PaulDepartment of Chemistry
Co-Supervisor, Dr. Stephen J. PaddisonUniversity of Tennessee
Dr. Peter KusalikDepartment of Chemistry
Dr. Arvi RaukDepartment of Chemistry
“Internal” External, Dr. Barry SandersDepartment of Physics and Astronomy
External, Dr. Raymond KapralUniversity of Toronto
Date
“Ere wa rudy fir tha beg shew?”
“Are we ready for the big show? Yes, Mr. Gotchchauk,
I certainly think we definitely have a good chance to be
almost completely ready.”
On the Air, ABC, 1992
Abstract
The subject of this thesis is the development of the necessary tools and their applica-
tion for a better understanding of the morphology and proton transport in the perflu-
orosulfonic acid (PFSA) short-side-chain (SSC) membranes. During recent years these
membranes have been the subject of enhanced interest as potential fuel cell electrolytes
replacing the relatively, better known Nafion membranes. In order to achieve these ends
we developed the mathematical formalism and the necessary algorithm for computing the
force fields that are unique to this material and which reproduce the available data for a
2-unit polymer. Furthermore, this algorithm allows the construction of three dimensional
polymeric structures possessing high molecular weights (MW) and specific morphologies
resulting by the imposition of optional restraints. The polymers thus built were placed
and equilibrated in a periodic simulation cell subject to periodic boundary conditions
(PBC). Special attention was given to the effects of PBC on the virial and the pressure
of the system in view of the fact that the experimentally measured densities of the hy-
drated polymers are not available. We were able to derive an analytic expression capable
of predicting the pore radii in the hydrated membranes as a function of the equivalent
weight (EW) and hydration level.
Having obtained systems with the correct morphologies and under suitable ambient
conditions our next goal was to investigate the transport of the proton through the
aforementioned hydrated pores. In the first instance we assumed the proton to exist
as a hydronium ion and thus enabling the motion of such a relatively heavy particle
to be studied by the application of a classical mechanical molecular dynamics (MD)
technique. The primary object of such a calculation is the diffusion coefficient since these
are experimentally available transport parameters. The results of our calculations showed
excellent agreement with experiment only in the case of the pores with low hydration,
iii
iv
however, at a higher level of hydration the predicted diffusion coefficients are too low
indicating the existence of an alternative more rapid mode of transport.
The obvious alternative mechanism is the well known hopping or Grotthuss mecha-
nism. The most convenient approach is to employ the Empirical Valence Bond (EVB)
method, where the force field is generated on the fly. The generation of the force field in
its most general form is, with the currently available electronic computational facilities,
a prohibitively expensive task since, in principle; it requires a quantum mechanical (QM)
calculation of the electronic energy at a continuum of points on the entire potential en-
ergy surface (PES). In this thesis we have developed a new methodology which we refer
to as the Just-In-Time EVB (JIT-EVB) method that requires such a computation to be
carried out on a much smaller grid of points. Such a computer program, that we have
developed, not only provides an important means for studying the transport of protons in
a highly acidic medium but has also enabled us to garner valuable structural insight. For
example we have found that an ion cage structure composed of sulfonate groups clamps
hydronium ions thereby impeding their diffusion rate.
The numerous advancements in the simulation methodology presented here are ex-
pected to result in significantly improved reliability of the simulations, allowing for accu-
rate structure-property modelling that will ultimately enable the targeted design of new
polymer systems.
Acknowledgements
I thank Dr. Paul for all his support and trust in me during the last six years. A
significant part of this work would have been impossible without the guidance of Dr.
Paddison, for which I am greatly indebted to him. Continuous financial support from
the Alberta Ingenuity Fund and the Natural Science and Engineering Research Council
is gratefully acknowledged.
v
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction to Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Review of Key Experimental Work . . . . . . . . . . . . . . . . . . . . . 81.4 Review of Theoretical Work . . . . . . . . . . . . . . . . . . . . . . . . . 12
I Methodology Development 292 Force Field Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Virial Formulation For Periodic Systems . . . . . . . . . . . . . . . . . . 414 Just-in-Time Empirical Valence Bond Method . . . . . . . . . . . . . . . 55
II Polymer System Studies 685 Constructing The Polymer Systems . . . . . . . . . . . . . . . . . . . . . 696 Designer Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787 Morphology of Hydrated SSC Polymer Systems . . . . . . . . . . . . . . 928 Proton Diffusion in SSC Polymer Systems . . . . . . . . . . . . . . . . . 104
III Conclusions and Future Work 1139 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11410 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
IV Appendixes and Bibliography 116A Short-Side-Chain Force Field . . . . . . . . . . . . . . . . . . . . . . . . . 117B Cross Section Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121C Molecular Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123D JIT-EVB Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . 124Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
vi
List of Tables
1.1 Density of Hydrated Nafion at 300 and 350 K . . . . . . . . . . . . . . . 22
3.1 Virial comparison for a 256 particle Lennard-Jones model . . . . . . . . . 53
5.1 Examples of random polymer sequences generated with different seeds . . 70
6.1 Pore radius as a function of the repeat unit formula (CF2CF2)n−CF2CF−(OCF2CF2SO3H) and the hydration level λ . . . . . . . . . . . . . . . . 91
8.1 Diffusion coefficients obtained from a JIT-EVB simulation of excess protonwith 64 water molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.2 Diffusion coefficients obtained from a JIT-EVB simulation of five triflicacid molecules and 30 water molecules . . . . . . . . . . . . . . . . . . . 110
A.1 SSC force field parameters for the harmonic stretching potential Ebond . . 118A.2 SSC force field parameters for the harmonic bending potential Eangle . . 119A.3 SSC force field parameters for the torsion potential Edih . . . . . . . . . . 119A.4 SSC force field parameters for the non-bonding interactions ECoulomb and
ELJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
D.1 Force field parameters for the anharmonic stretching potential EMorse usedin JIT-EVB simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
D.2 Atom charges used in JIT-EVB simulations . . . . . . . . . . . . . . . . 125
vii
viii
List of Figures
1.1 Different types fuel cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 PFSA based polymers for fuel cell membranes . . . . . . . . . . . . . . . 61.3 Conductivity plot for Nafion and the SSC polymer as a function of the
hydration level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Cluster-network model for the morphology of hydrated Nafion . . . . . . 91.5 Two rows of the hexagonal lattice formed by the polymer chains (shown
as circles) seen end-on . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Morphologies of Nafion at different volume fractions of water . . . . . . . 101.7 Schematic representation of the nano-phase separation in the hydrated
morphology of Nafion and sulfonated polyetherketone derived from exper-iments and modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.8 Parallel water-channel (inverted-micelle cylinder) model of Nafion . . . . 131.9 Idealized membrane pore showing the hydronium ion, water molecules, and
radially symmetric axially periodic distribution of sulfonate SO−3 fixed sites 141.10 Structure of four-unit perfluorosulfate oligomer, optimized in vacuum, wa-
ter and methanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.11 Two monomeric sequences of Nafion 117 with different monomer clustering 181.12 Fully optimized global minimum energy structures of the C6 two sidechain
fragment showing hydration and proton dissociation as additional watermolecules are added . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.13 Fully optimized polymeric fragment and the potential energy profile forrotation about the F2C − CF2 bond along the backbone . . . . . . . . . 21
1.14 Representative configurations of the solvation structures observed in thesimulations using the classical hydronium potential and the EVB potential,which were common for both hydration levels (λ = 7, 15) . . . . . . . . . 24
1.15 Proton mean-square-displacement (MSD) in Nafion and water . . . . . . 261.16 van Hove space-time correlation function for the hydronium-sulfonate ion
pair, given the sulfonate anion as the space-time origin . . . . . . . . . . 271.17 Morphological models of Nafion . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 The seven dihedral angles of the SSC force field illustrated on polymersegments with a protonated sidechains . . . . . . . . . . . . . . . . . . . 34
2.2 Comparison of the ab initio and classic torsion profiles (with contributionsfrom all MM terms) around the C1O-O3 bond . . . . . . . . . . . . . . . 37
2.3 Comparison of the ab initio and classic torsion profiles (with contributionsfrom all MM terms) around the C2S-S6 bond . . . . . . . . . . . . . . . 38
2.4 Energy profile of the dihedral potential for the angle H3–O3H–S6–C2S: inthe SSC force field and a Newman projection of the dihedral angle . . . . 39
ix
3.1 Local region (solid sphere) replicated in a spherical shell. The imagespheres fill up the entire volume of the 3D shell. A special arrangement ofthe images ensures zero forces at the local region boundary . . . . . . . . 42
3.2 If L becomes smaller than 2Rc the interactions in the primary cell, as wellas all its neighbors have to be evaluated explicitly . . . . . . . . . . . . . 43
3.3 Hamiltonian conservation in a short MD trajectory of a Lennard-Jonessystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 The distance between the central particle i and the image particle j′ inthe m-th shell is a function of the angle θ (determining the position of theimage sphere in the shell), the i− j distance r in the primary cell and thedistance between the sphere centers 2mRc . . . . . . . . . . . . . . . . . 50
4.1 Reproduction of the original idea of Grotthuss for proton shuttling betweentwo electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Schematic depiction of the “special pair dance” occurring in the firstsolvation-shell of the hydronium during the long trajectory segments with-out PT events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3 Probability distribution of the proton as a function of the Oa−Ob distanceand the asymmetric stretch coordinate δ = ROaH −RObH . . . . . . . . . 58
4.4 Scatter plot of the proton distribution in a Zundel ion (represented classi-cally as hydronium ion plus water) at 300 K . . . . . . . . . . . . . . . . 59
4.5 A Zundel ion (represented classically as hydronium ion plus water) can ex-ist in two resonance forms, obtained by interconversion between a covalentOH bond (solid line) and a hydrogen bond (dashed line) . . . . . . . . . 61
4.6 PT between two sulfonate groups influenced by a neighbouring Zundel ionin a triflic acid monohydrate solid . . . . . . . . . . . . . . . . . . . . . . 63
4.7 Molecules whose conformations are inside the reactive trigger zone (greenboundary) will be subject to JIT-EVB mixing of the resonance forms toreproduce the ab initio forces along the grid . . . . . . . . . . . . . . . . 66
4.8 At point A a molecule enters the trigger zone. An ab initio calculationhas to be performed to determine the correct mixing of the resonance forms 67
5.1 Flowchart representing the creation of a SSC polymer with random monomersequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Perfluorocyclohexane converging to two distinct local minima dependingon the conformation seed . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Flowchart representing the stepwise process of finding a low energy con-formation for the polymer . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 The interactions needed to be considered in order to optimize the coordi-nates of the new atoms (set A) are those within the set and with the builtatoms (set B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.1 Cross section view of an ideal pore with the shape restraint represented asan outer layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 Perfluoropentane built with a straight backbone along an axis . . . . . . 81
6.3 Single perfluoropentane chain built in a periodic cell with a harmonic bondrestraint between the end carbon atoms . . . . . . . . . . . . . . . . . . . 82
6.4 2D channels formed by two polymer chains (black and green) in a periodiccell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5 Perfluorocyclohexane built in a periodic cell with MIC applied to bothbonding and non-bonding potentials . . . . . . . . . . . . . . . . . . . . . 85
6.6 Two perfluorohexane chains built in an infinite simulation universe withoutany restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.7 Schematic representation of the approach of two polymer strands in theanti conformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.8 Schematic representation of a pore wall as formed by straight polymer chains 886.9 Cross section view of the pore and two polymer chains. . . . . . . . . . . 90
7.1 Polymer morphology snapshots at the end of the production run for thethree levels of hydration . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 Hydrogen bond chain from the snapshot in Fig. 7.1a (λ = 3) using MIC . 957.3 Polymer chain of Fig. 7.1a (λ = 3) without MIC . . . . . . . . . . . . . . 967.4 S-S pair correlation plot (between the ionized sulfonic group sulfur atoms)
for the three hydration levels . . . . . . . . . . . . . . . . . . . . . . . . . 977.5 An ion cage that exhibits very short S-S distances . . . . . . . . . . . . . 987.6 S-O pair correlation plot (between the ionized sulfonic acid sulfur atom
and the hydronium ion oxygen atom) for each of the three hydration levels 1007.7 Fraction of hydronium ions with a given number of sulfonate neighbours
in the SSC polymer for different hydration levels λ . . . . . . . . . . . . 1017.8 Fraction of hydronium ions with a given number of sulfonate neighbours
in the Nafion for different hydration levels λ . . . . . . . . . . . . . . . . 102
8.1 MSD of hydronium ions in the SSC polymer as a function of time andhydration level λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.2 MSD of the CEC in triflic acid solution as a function of time. The hydra-tion level is λ = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
A.1 Atom type labels for the SSC force field . . . . . . . . . . . . . . . . . . 117
B.1 Cross section view of a pore formed by polymer chains . . . . . . . . . . 121
D.1 When a hydronium ion and a water molecule satisfy the trigger conditionsfor hydrogen bond h1 they both become part of a cluster . . . . . . . . . 126
x
Acronyms
ABNR Adopted Basis Newton-Raphson
CEC Center of Excess Charge
CM Center of Mass
DFT Density Functinal Theory
EVB Empirical Valence Bond
EW Equivalent Weight
F3C Flexible Three Center
IPS Isotropic Periodic Sum
JIT-EVB Just-in-Time Empirical Valence Bond
LRC Long Range Correction
MD Molecular Dynamics
MIC Minimum Image Convention
MM Molecular Mechanics
MSD Mean Square Displacement
MW Molecular Weight
PBC Periodic Boundary Conditions
PEM Proton Exchange Membrane
xi
xii
PES Potential Energy Surface
PFSA Perfluorosulfonic Acid
PT Proton Transfer
QM Quantum Mechanics
RDF Radial Distribution Function
RMS Root Mean Square
SASA Solvent-Accessible Surface Area
SD Steepest Descent
SMILES Simplified Molecular Input Line Entry Specification
SSC Short-Side-Chain
1
Chapter 1
Introduction
2
1.1 Introduction to Fuel Cells
Over the last hundred years since the beginning of the industrial revolution the consump-
tion of crude oil has exponentially increased. In the past twenty years alone the world has
consumed more than two thirds of the total amount of oil extracted since the 1880 [1].
According to reliable forecasts there now exists a small window of time (20-50 years) in
which our society will be faced with acute fuel shortages [2]. We are, therefore, at the
present in a transition period in which the global economy must reduce its dependence
on the consumption of non-renewable resources.
While we could envision home and business consumers switching to renewable sources
for their energy needs [3] the situation with the auto industry is far more challenging.
Transportation currently accounts for one third of the energy consumption in the United
States alone and the trends predict it becoming the largest consumer by 2020 [4]. Thus far
no alternative fuel technology has come anywhere near the gasoline/combustion engine
in price or reliability. The challenge is, therefore, not just developing efficient ways of
using alternative fuels in cars but also to have a profitable distribution network for these
fuels along with an enormous industrial capacity to produce them.
The United States is already in the process of laying down the foundations of this
colossal transition by setting forth a plan for a hydrogen-based economy. In the ini-
tial stages the hydrogen can be produced by reforming fossil fuels or by water splitting
employing nuclear energy with the ultimate goal of switching over time to renewable
sources like biomass or water splitting by sunlight, wind energy etc. The attractiveness
of hydrogen starts with its localized production – from nuclear reactors to any of the
alternative energy sources which can be situated in close proximity to the end user. Its
transportation can, in part, utilize the existing infrastructure for distributing natural
gas. When hydrogen makes its way down to our cities and vehicles we will have the
3
cleanest, most environmentally friendly chemical fuel1. This is of paramount importance
as burning hydrogen does not lead to any CO2 emission. Hydrogen is a carbon-neutral
fuel.
Currently most of the hydrogen is used as a mixture with natural gas for combustion
engine vehicles. Hydrogen however has a special place among the other fuels in that its
reaction with O2 can be harvested to produce electricity directly, in an apparatus known
as the fuel cell. The attractiveness of using electric vehicles comes from their highly
efficient engines – they waste half as much energy when compared to a combustion
engine. The list of other fuels that can be converted to electricity in a similar manner is
limited to methanol and ethanol.
The invention of the fuel cell is attributed to Sir William Grove, a Welsh scientist who
in 1839 demonstrated that the electrolysis of water can be reversed, producing electricity
from hydrogen and oxygen [5]. The advancements in other competing technologies like
the dynamo generator which were successfully brought to market in the following years
meant little interest for an electrochemical generator. The interest in converting fuel
into electricity directly was resurrected over a century later with the advent of space
exploration. Commercializing the technology started in the 1980s by Ballard Power
Systems.
Classifying fuel cells can be done either by the type of fuel or, more commonly, by
the type of the electrolyte. A summary of the different types is presented in Fig. 1.1.
Unlike ordinary batteries that have to be recharged by another source of electricity fuel
cells can continuously provide electric current when fed with fuel and oxygen. Proton
exchange membrane (PEM) fuel cells employ a proton-conducting polymer membrane
as the electrolyte between the anode and the cathode. Oxidation of hydrogen on the
1Nowadays hydrogen fuelling stations can be found scattered across parts of Europe and NorthAmerica. In British Columbia a “Hydrogen Highway” was built to link Victoria and Whistler in timefor the 2010 Winter Olympic Games.
4
Figure 1.1: Different types fuel cells. Figure reproduced from Ref. [6].
5
anode releases electrons into the outer circuit and protons into the surrounding medium.
The protons migrate to the cathode side to preserve the electroneutrallity of the system
where they recombine with the electrons and oxygen from the air to produce water. The
membrane separating the two electrodes allows passage of the protons while preventing
the flow of fuel, oxygen or electrons. An alternative to the proton conduction electrolytes
is found in the solid oxide fuel cell. Here the electrolyte system allows passage of oxygen
ions between the electrodes.
The electrolyte is the cornerstone of the fuel cell since the requirements for effective ion
transport dictate the design and operating conditions of the whole cell. The temperature
range that enables ion transport affects the kinetics of the electrochemical reactions
on the electrodes, the choice of catalysts and the acceptable level of impurities in the
fuel. Fuel cells operating under harsh conditions (i.e., corrosive electrolytes, very high
temperatures) would be impractical for mobile applications like powering electric cars or
consumer electronics. On the other hand, low-temperature fuel cells require expensive
catalysts which also makes them unsuitable. The current trend in fuel cell research is
to bridge the gap between these extreme cases in a manner that will provide a safe,
affordable solution for all applications.
The electrolyte systems that we are interested in are polymer based, with a Teflon
backbone and hydrophilic side chains. At the end of the side chains are protogenic groups
including sulfonic acid groups (−SO3H). In order for these membranes to exhibit proton
conductivity they require water. The most well known representatives of this class of
polymers are Nafion (developed by DuPont) and the SSC polymer (developed by Dow)
whose chemical structures are given in Fig. 1.2. Here is an excerpt from the DuPont
announcement from 1969: “A new thermoplastic polymer family - offering features of
both fluorocarbon polymers and ion exchange resins - has been developed by DuPont’s
Plastics Departments. The new composition is currently called XR. It is expected to pro-
6
Nafion3MSSC
Figure 1.2: PFSA based polymers for fuel cell membranes. The fragment in the Nafionstructure shown in red is absent in the SSC polymer, hence the name “short-side-chain”.Typical n values for the SSC polymer are 3 to 6.
vide unique advantages for electrochemical, aerospace, and chemical industries. Based
on advanced fluorocarbon chemistry, the polymer exhibits such features of fluorocarbons
and ionomers as ionic conductivity, permeability, transparency, toughness, chemical in-
ertness, flexibility and adhesion to most substrates . . . The fluorinated polymer contains
pendant sulfonic acid groups, which produce an exceptionally strong acid resin. The
number of sulfonic acid groups can be varied to provide different ion exchange capacity,
electrical and mechanical properties.” [7].
1.2 Motivation
In the mid 1980s Ballard Power Systems demonstrated a significant improvement in fuel
cell performance with the new type of SSC PFSA membrane [8]. The current that the fuel
cell can provide is limited by diffusion processes which supply reactants to the electrodes,
including the trans-membrane shuttling of protons, seen in Fig. 1.1. Polymer systems
that exhibit proton conductivity with limited need for hydration are highly desirable as
this will allow an increase in the operating temperature of the fuel cell above 80 C.
7
In turn, this will allow for cheaper catalysts to be employed on the electrodes. In this
respect, the SSC polymer shows very promising conductivities with only a fraction of the
water required by Nafion (see Fig. 1.3).
material [13]. However, when the two membranes are fully hydrated theproton self-diffusion coefficients are very similar to one another. A furtherunderscoring of the differences that polymer chemistry maymake on protontransport is seen in Figure 12.1(b) where the proton conductivity is plottedfor Nafion (again) and the short side chain (i.e. SSC) PFSA ionomer(originally synthesized by the Dow Chemical Company [14]) and clearlyshows that the membrane with the shorter side chain (i.e. –OCF2CF2SO3H)has much higher conductivity at intermediate water content (i.e. from4–18H2O/SO3H). The reasons for these substantial differences in protonmobilities undoubtedly have to do with the density of the hydrated protonsbut is not fully understood and therefore impetus for molecular-levelstructure/function modeling.
12.1.2. Motivation for Modeling
It is widely appreciated that during operation the PEM in a fuel cell is notuniformly hydrated, with the membrane only partially hydrated on the anodeside and typically fully hydrated (and often ‘flooded’) on the cathode side.This is despite the humidification of the H2 gas stream and removal of waterat the site of reduction. This necessitates that under operation the flux ofprotons occurs across a gradient in the concentration of the water whichimpacts the rate of transport (e.g. see Figure 12.1) and consequently themechanism whereby the proton traverses the electrolyte. Experimentalstudies also indicate that even relatively subtle changes in either the backbone
(a) (b)
10.1
D /
cm2 s
–1
1e-3
1e-4
1e-6
1e-5
1e-7
1e-8
1e-3
1e-4
1e-6
1e-5
1e-7
1e-8
D!
D! sulfonatedDH2O
DH2O polyetherketone
Con
duct
ivity
(S/c
m)
0.15
0.1
0.05
00 5 10 15 20 25 30
NafionDow SSC
Waters per Sulfonatewater volume fraction XV
T = 300 K
NAFION
D!DH2O
pure water
FIGURE 12.1. The dependence of proton mobility on water content. (a) Proton self-diffusion coefficients (D!) of Nafion and sulfonated polyetherketone membranes at300 K plotted as a function of the water volume fraction showing the substantiallygreater proton mobility in the PFSAmembrane as low to intermediate water contents.Taken from Ref. 13. (b) Proton conductivity of Nafion and low EW (!800) Dow SSCmembranes plotted as a function of the water content expressed as the number ofwater molecules per sulfonic acid group showing substantially higher conductivity atintermediate hydration levels. Taken from Ref. [12].
12. Proton Conduction in PEMs 387
Figure 1.3: Conductivity plot for Nafion and the SSC polymer as a function of thehydration level. Figure source: Ref. [9].
Furthermore, this practical observation shows that the membrane structure has a
profound effect on its conductivity and raises the important question whether further
improvements could be achieved through structural modifications. For instance – varying
the structure of the side chain, the type of the protogenic group, the number of CF2
groups in the backbone and so on. However, reliable prediction of structure-property
relationships in these systems is a formidable challenge. Localized proton dissociation and
shuttling requires an accurate QM description, which unfortunately cannot be applied
to the entire membrane channel where these events occur. Long simulation runs and
extensive sampling are required to diminish the effect of fluctuations on the predicted
8
dynamical properties. This is especially true when determining the effect of small, subtle
changes in the polymer structure like an additional CF2 group in the polymer backbone.
The goal of my Ph.D. work was to develop the simulation methodology that would
allow a better understanding of the properties of these systems. In particular, the effect
of structural parameters like the EW and the MW on the morphology of the hydrated
polymers and the proton diffusion rate. The key aspects of these developments are
presented in Part II of the thesis. This is followed in Part III with the practical aspects of
creating the hydrated polymer for computer simulations and the results of our structural
and dynamical property modelling.
Attempts of this nature have been made in the past and a vast amount of literature is
now available. However, for the purpose of this thesis we will review some of the previous
work that is directly related to the present objectives.
1.3 Review of Key Experimental Work
The morphology of the PFSA polymers has been the subject of numerous experimental
studies over the last few decades. A comprehensive review of the current understanding
of the morphology of these systems can be found for example in Ref. [10]. From a his-
torical perspective, the introduction of the inverted micelle model, for the structure of
the PFSA systems by Gierke et al. [11], is no doubt one of great significance. Based on
both wide-angle and small-angle x-ray diffraction data, as well as transmission electron
micrograph imaging and using a process of elimination of other morphologies these au-
thors have arrived at the cluster-network, depicted in Fig. 1.4, as the one best fitting the
experimental data.
Within the context of this model the ionic clustering of the strong sulfonic acid groups,
results in spherical vesicles containing the adsorbed water to produce structures that
9
Figure 1.4: Cluster-network model for the morphology of hydrated Nafion. Figuresource: Ref. [10].
resemble an inverted micelle. Percolation between these clusters is assumed to occur
through narrow channels that interconnect the micelles. However, one important aspect
of the membrane morphology that is not accounted for by the Gierke model is the partial
crystallinity brought about by the polytetrafluoroethylene backbone. Hence, it has been
suggested that a separate crystalline phase exists in the domain where the polymer chains
assemble in a lamellar hexagonal conformation [12], as seen in Fig. 1.5. The side chains
extend perpendicular to the polymer backbone into the ionic cluster domains.
One of the most important the properties of the perfluorosulfonic acid (PFSA) mem-
branes is their hydration level. As the water content increases to a critical fraction of 0.1,
an insulator-to-conductor transition occurs in the membrane, which is attributed to trans-
port through the channel connections between the inverted micelle clusters [13]. Further
swelling of the polymer results in a structural inversion that leads to rod-like polymer
structures suspended in the water phase, as shown in Fig. 1.6. A coherent mechanis-
tic scheme that explains the dilution and swelling of the membranes emerges when the
morphology of the system is described by elongated bundles of polymer chains [15]. The
diameter of these bundles is estimated to be on the order of 40 A with lengths larger
than 1000 A.
10
SO−3
SO−3
SO−3
SO−3
SO−3SO−3
SO−3
SO−3
SO−3
SO−3
Figure 1.5: Two rows of the hexagonal lattice formed by the polymer chains (shown ascircles) seen end-on. Figure reproduced from Ref. [12].
!
!
""!
suggested by Gebel et al. 41, our evaluation of surface-to-volume ratios of the fluoropolymer
(described in Chapter 3) shows that the morphology of Nafion evolves from a matrix with
dispersed water molecules at low hydration level, to the water channel model at medium
hydration level, and to polymer aggregates dispersed in water at high hydration level (Figure
1.6).
F igure 1.6. Morphologies of Nafion at different volume fraction of H2O ( H2O).
1.6 Improved PE M with inorganic nanoparticles
As mentioned in section 1.3, the working temperature of normal PFSA membranes is limited
at around 80 ~ 90 oC, which constrains the tolerance of Pt catalysts to contaminants.
Furthermore, poor proton conductivity at low hydration levels complicates the design of
water management systems. Materials based on e.g. sulfonated poly-arylene ethers 67, poly-
sulfones 68,69, and poly-benzimidazoles 70,71 have been developed as possible alternatives, but
most of them have either lower proton conductivity or insufficient stability.
Another promising approach is to modify PEMs with inorganic oxides and solid acids 30,31,72,73. In general, the oxides, including SiO2, TiO2, ZrO2 and Al2O3, have hygroscopic
properties and are prepared with high surface areas 74,75. Abundant hydroxyl groups on the
oxide surface can strongly retain water molecules and mitigate membrane dehydration at
temperatures above 100 oC. Solid acid nanoparticles, e.g. zirconium phosphate (ZrP) 76,
Figure 1.6: Morphologies of Nafion at different volume fractions of water. Based onsurface-to-volume ratios of the fluoropolymer the morphology of Nafion evolves from amatrix with dispersed water molecules at low hydration level, to the water channel modelat medium hydration level, and, finally, to polymer aggregates dispersed in water at highhydration level. Figure source: Ref. [14].
11 16Jun
200312:17
AR
AR189-12-CO
LOR.tex
AR189-12-CO
LOR.SG
MLaTeX
2e(2002/01/18)P1:G
CE
Figure 1 Schematic representation of the nano-phase separation in the hydrated morphology of Nafion and sulfonated PEEKK derivedfrom experiments and modeling. This scheme illustrates the distinctions in the hydrophilic/hydrophobic separation, connectivity of the waterand ion domains, and separation of the −SO3− groups. Taken from Reference (6) with permission from Elsevier.
Figure 1.7: Schematic representation of the nano-phase separation in the hydrated mor-phology of Nafion and sulfonated polyetherketone derived from experiments and mod-elling. This scheme illustrates the distinctions in the hydrophilic/hydrophobic separation,connectivity of the water and ion domains, and separation of the SO−3 groups. Figuresource: Ref. [16].
The effect of the polymers microstructure and the acidity of the protogenic groups on
the transport properties of the membranes has been recently investigated in the seminal
work of Kreuer [16]. Substitution of the perfluoro-polymer with a less hydrophobic aro-
matic polyetherketone system resulted in narrower, less connected hydrophilic channels
and to larger separations between the sulfonic acid functional groups (see Fig. 1.7). Al-
though the protogenic groups were found to be less acidic, the proton conductivity has
remained high, while the undesirable elecroosmotic drag of water is significantly reduced.
The work also explores polymer blends as a way of improving the characteristics of the
12
membrane, as well as the possibilities of using heterocycles as the proton solvent. Such
heterocyclic compounds (e.g., imidazole) can be immobilized by grafting onto a polymer
backbone, thus providing a non-volatile, proton-conducting medium that operates in the
complete absence of water.
The most solid evidence for the morphology of the hydrated Nafion polymers has
come from the recent work of Schmidt-Rohr et al. [17]. By performing a new analysis on
the existing small-angle scattering data that includes contributions from the crystalline
phase, these authors have been able to show, unequivocally, the occurrence of cylindrical
inverted micelles. The walls of the micelles are formed from straight helical polymer
chains and are lined with the hydrophilic side chains. At a hydration level of 20vol%
the diameter of the channels is about 2.4 nm. Furthermore, not only is the inverted
micelle cylindrical but so is the general shape of the crystalline phase. A schematic
representation of these structures is shown in Fig. 1.8.
1.4 Review of Theoretical Work
In this section we will review some of the modelling work carried out in the last couple
of decades that is most relevant to our own simulations. A much broader-scope review
of the previous computational work can be found in Refs. [10,18].
One of the earliest proton transport studies to include structural level details was
the nonequilibrium statistical mechanical model of Paddison et al. [19]. The idea of
the model is to calculate the hydronium ion diffusion coefficient through the Einstein
relationship D = kT/ζα, where ζα is the Stokes friction coefficient of the hydronium ion.
This friction coefficient can be conveniently determined from a sum of four friction terms,
each calculated as a the ensemble average of a force correlation function:
13
ARTICLES
Ionomer peak
Crystallites
H2O cylinders
q (nm–1)
q (nm–1)
1.00.1 0.3
50 nm
4 nm
10100
CrystalliteH2O channel
I (a.
u.)
0 1 2 3
2
4
6
8
Ionomer peak
H2O
H2O
I (a.
u.)
Matrix knee
10–2
10–1
100
101
a b c d(nm)2 /q!
Figure 2 Parallel water-channel (inverted-micelle cylinder) model of Nafion. a, Two views of an inverted-micelle cylinder, with the polymer backbones on the outside andthe ionic side groups lining the water channel. Shading is used to distinguish chains in front and in the back. b, Schematic diagram of the approximately hexagonal packingof several inverted-micelle cylinders. c, Cross-sections through the cylindrical water channels (white) and the Nafion crystallites (black) in the non-crystalline Nafion matrix(dark grey), as used in the simulation of the small-angle scattering curves in d. d, Small-angle scattering data (circles) of Rubatat et al.17 in a log(I ) versus log(q) plot forNafion at 20 vol% of H2O, and our simulated curve from the model shown in c (solid line). The inset shows the ionomer peak in a linear plot of I(q). Simulated scatteringcurves from the water channels and the crystallites by themselves (in a structureless matrix) are shown dashed and dotted, respectively.
structure was 13% by volume (15% of the dry polymer). Othersimulations with crystallinities between 9 and 15% also gaveacceptable results. The crystallinity from the straight-cylindermodel is probably an overestimate because simulations forundulating channels (see Supplementary Information, Fig. S3)show that correlated undulations of crystallites and waterchannels reduce the scattered intensity at intermediate andhigh q values, while leaving the small-angle upturn due tocrystallites unchanged.
In scattering experiments on non-crystalline Nafion, producedby quenching or solution-casting15,30,31,37, the I(q) curve from thehydrated clusters can be observed selectively. It exhibits a broadregion of I(q) ≈ const. flanked by the ionomer peak and a small-angle upturn that has been proved, by SANS with D2O/H2Ocontrast variation, to be indeed due to the hydrated clusters37;the dashed curve in Fig. 1b shows an example. Our scatteringcurve from the water channels without crystallites (dashed linesin Fig. 2d) reproduces these features within the variation of theexperimental curves; the greater30 or lesser15 steepness of theexperimental small-angle upturn may be attributed to differencesin the tortuosity of the water cylinders in the differently preparedsamples. In particular, it is to be expected that the small-angle upturn from the water channels is more pronounced in asemicrystalline sample, where the crystallites help align the watercylinders, than in a non-crystalline sample, where cylinders maymeander more strongly.
The change in the ionomer peak and other properties of Nafionwith water content is pronounced, see Fig. 3a with data fromGierke et al.1. The shift and intensity increase of the ionomerpeak and small-angle upturn can be reproduced adequately, Fig. 3b,in the water-channel model, with simple swelling of a constantnumber of water channels in a given volume of polymer; forillustration, Fig. 3c shows matching portions of the scatteringdensity for 10 and 28 vol% water. In contrast, Gierke et al. hadto invoke an increase of the number of ionic groups per clusterwith water content1, without specifying the origin of these extragroups. Details regarding the intensity increase in the experimental
and simulated data, which is affected by film-thickness increase andthe excess scattering density of the electron-rich sulphonate groups,are discussed in the Supplementary Information.
SAXS SIMULATIONS FOR OTHER MODELS
For comparison, Figs 4–6 show simulations of small-anglescattering for other models of the Nafion nanostructure,namely Gierke’s model of spherical clusters on a paracrystallinecubic lattice1,6,7, the local-order model4,5, the polymer-bundlemodel16–18,20, hydrated bilayers/slabs10–13 and network models14,15.Models without order9,14,22, which do not produce an ionomerpeak, are discussed in the Supplementary Information.
Gierke’s popular cluster model1,6 with spherical clusters ona paracrystalline cubic lattice, Fig. 4a, would produce scatteringcurves as shown in Fig. 4b. By varying the degree of disorderof the second kind38 in the paracrystalline lattice, the peaksin the scattering curve and in the radial distribution functionP(r) (see insets) can be broadened. These structures can beconsidered as a valid implementation of the ‘local-order model’4,5
of Nafion. In the original version of this model4,5, which wasan attempt to quantify Gierke’s model, an unphysical P(r) witha sharp nearest-neighbour peak separated by a gap down tothe baseline from a long-distance plateau without any othermaxima was used for quantitative calculations. This violates theOrnstein–Zernike equation relating P(r) to the ‘direct’ two-particlecorrelation function39. As predicted, the P(r) in the inset of Fig. 4bshows many sharp peaks when the first peak is sharp, whereasthe radial distribution functions in Fig. 4b,e,h confirm that afeatureless plateau at intermediate distances requires a broadenedfirst maximum that is not separated by a deep gap.
Neither of the simulated scattering curves in Fig. 4b and ematches the features of the experimental data of Fig. 1b. In additionto ionomer peaks that are too sharp, they show a q0, rather thanq−1, power law at small q, which is indeed expected for spheres38.The modulations at ‘large’ q in the log–log plots of Fig. 4 are dueto the form factor for a single particle diameter. Note that a wide
nature materials VOL 7 JANUARY 2008 www.nature.com/naturematerials 77
©!2008!Nature Publishing Group!
Figure 1.8: Parallel water-channel (inverted-micelle cylinder) model of Nafion. a, Twoviews of an inverted-micelle cylinder, with the polymer backbones on the outside andthe ionic side groups lining the water channel. Shading is used to distinguish chains infront and in the back. b, Schematic diagram of the approximately hexagonal packingof several inverted-micelle cylinders. c, Cross-sections through the cylindrical waterchannels (white) and the Nafion crystallites (black) in the non-crystalline Nafion matrix(dark grey), as used in the simulation of the small-angle scattering curves. Figure source:Ref. [17].
14
collection of momentum and position vectors of the N watermolecules each with a mass m, and V(r! ,r) is the total po-tential energy of the system. The latter is assumed to consistof the following four terms:
V"r! ,r#!$i!1
N
V!s" !r!"ri!##V!p"r!#
#$i$ j
N
Vss" !ri"rj!##$i!1
N
Vsp"ri#. "2#
The first term is the interaction potential energy between thehydronium ion and the ith water molecule and is assumed tobe a typical ion–dipole interaction. If the rotational contri-butions are ignored one obtains the simplified expression
V!s" !r!"ri!#%"&2e2
48'2(2kT1
!r!"ri!4, "3#
where ( is the permittivity of the water in the pore, k theBoltzmann constant, and T the temperature.
The second term is the potential energy experienced bythe hydronium ion due to the sulfonate groups. As indicatedearlier, these pendant groups are distributed periodically inthe pore, and if the length of their intrusion within the pore isR") "thus ) is the radial separation distance of the hydro-nium ion from the fixed sites# and axial spacing L/n , and oneassumes that the hydronium ion is transported along the axialcenter of the pore, then this potential energy term is assumedto have the form
V!p"r!#!*0 cos" 2'nz!
L #!en$""e #
'(L K0" 2'n)
L # cos" 2'nz!
L # , "4#
where the sum "in the explicit expression# is over all thefixed groups on each array and z! the axial coordinate ofthe hydronium ion "located at the center of the pore—as specified earlier#. Equation "4# is a simplification of anexact result derived by Grønbech-Jensen, Hummer, and
Beardmore53 using Lekner summations of Coulombic inter-actions in three-dimensional systems having periodicity inone and two dimensions, the former being relevant for ourchosen anionic charge distribution.
The third term in Eq. "2# is the potential energy due towater–water interactions, which are assumed to be dipole–dipole interactions according to
Vss" !ri"rj!#!2&4
3"4'(#2kT1
!r"rj!6, "5#
where, once again, a thermal average has been performedover all rotational angles.
The final term describes the potential energy the watermolecules experience due to the fixed sulfonate groups. Un-der the assumption that the water dipoles are aligned with thefield due to the fixed sites, this term may be approximatedwith the expression
Vsp"ri#%"2'&*0n
eL sin" 2'nziL # . "6#
It should be clear at this point that our system as de-scribed, is an (N#1)-body problem consisting of N watermolecules and a single hydronium ion. In a real membranepore there is one proton for every sulfonate group. Ignoringthe presence of the ‘‘other’’ protons will undoubtedly havecertain ramifications. Perhaps the most significant is that thepresence of the other protons will result in increased shield-ing of the interaction of the anionic groups with the hydro-nium ion and the water molecules. Thus, ignoring these pro-tons will result in overestimating the potential energy+calculated in Eqs. "4# and "6#, and the consequent frictionexperienced by the hydronium ion. However, without spe-cific information concerning the distribution of the protons inthe pore, the effects of the other protons will not be includedin the model at this point. In addition, the effects of proton–proton interactions are not accounted for in our model. Thecontribution of these repulsive interactions to the friction co-efficient will be small. Clearly, at the higher water contents,error"s# introduced by ignoring the other protons become lesssignificant.
The time-dependent distribution of the position and mo-mentum of all the particles of the system,f N#1(p! ,r! ,p,r;t), satisfies the Liouville equation:
i- f N#1
-t !LT f N#1 , "7#
where LT is the Hermitian Liouville operator given by thePoisson bracket
LT!i.HT , /. "8#
The total force on the hydronium ion, F!(r! ,r), may becalculated with the relation
F!"r! ,r#!iLTp!!" $k!1
N-V!s" !r!"rk!#
-r!"
-V!p"r!#
-r!
0 F!s"r! ,r##F!p"r!#, "9#
and the corresponding average force, 1F!2, according to
FIG. 1. Idealized membrane pore showing the hydronium ion, water mol-ecules, and radially symmetric axially periodic distribution of sulfonate"–SO3"# fixed sites.
7755J. Chem. Phys., Vol. 115, No. 16, 22 October 2001 Proton diffusion in polymer electrolyte membranes
Downloaded 22 Sep 2005 to 136.159.235.227. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
Figure 1.9: Idealized membrane pore showing the hydronium ion, water molecules, andradially symmetric axially periodic distribution of sulfonate SO−3 fixed sites. Figuresource: Ref. [20].
ζ
β=
∞∫
0
⟨Fαse
−iLtFαs⟩dt+
∞∫
0
⟨Fαse
−iLtFps⟩dt+
∞∫
0
⟨Fαpe
−iLtFps⟩dt+
∞∫
0
⟨Fαpe
−iLtFas⟩dt
(1.1)
In this equation L is the Liouville operator of the system, and the force subscripts
α, s and p designate the hydronium ion, the solvent water molecules and the pendant
groups (i.e., the sulfonate anions), respectively. For example, the first term on the right-
hand-side of Eq. 1.1 corresponds to the average force experienced by the hydronium ion
due to the solvent molecules, the next term is the average force on the hydronium ion
due to the pendant groups via the solvent medium and so on. For a simple cylindrical
pore model with regular pendant chain distribution (shown schematically in Fig. 1.9) the
last three terms of Eq. 1.1 can be calculated exactly.
The predictive capability of the method was assessed by separate calculations car-
ried out with Nafion membrane pores hydrated with 6 and 13 waters molecules, respec-
15
tively, associated with each fixed anionic site (pendant chain). When combined with
experimentally estimated parameters, the model-predicted proton diffusion coefficients
of 5.05×10−10 and 8.36×10−10 m2/s, are in good agreement with experimental values.
For a Nafion membrane pore with an hydration level of six water molecules per sulfonic
acid group, the model was used to compute friction coefficients for various distributions of
the pendant sites, and for different side chain lengths [20]. The model showed substantial
sensitivity to these parameters and predicted that for pores of fixed volume and a constant
total number of sulfonate groups, the friction on the hydrated proton is the greatest for
distributions with high local sulfonate density. When the radius and length of the pore
were varied, the model demonstrated that the proton diffusion increases with increasing
channel diameter. These calculations, therefore, demonstrate the important predictive
capability of this molecular-based, nonequilibrium statistical mechanical model.
More recently the model was used to predict the diffusion coefficient of protons in a
fully hydrated Nafion membrane with 22.5 water molecules per sulfonic acid group [21].
In this work profiles of the friction and diffusion coefficients were determined across the
radius of the pore, demonstrating that these parameters vary by a full order of magnitude
across the radial cross-section of the pore. The model calculated a diffusion coefficient
for a proton moving along the pore center of 1.92×10−9 m2/s in good agreement with
experimental measurements. In addition, the model also identified that the region within
4 A of the pore center exhibits an intermolecular transfer (Grotthuss mechanism) of the
proton between the water molecules. This is in contrast with the region lying within 8 A
of the wall of the pore, where the transport of the proton is predominantly vehicular in
nature. Although the agreement with the experimental diffusion measurements has been
very good for diffusion along the center of the pore, inclusion of the contributions from a
distribution of protons across the radius of the pore will certainly correspond to a more
realistic modelling of proton transport.
16
In one of the earliest atomistic simulations of the PFSA system the conformations
and hydrophilicity of the side chain in Nafion were examined using Density Functinal
Theory (DFT) [22]. The study has shown that the ether portion of the side chain is
hydrophobic and stiff, while the SO−3 group is strongly hydrophilic and more flexible. In
the absence of explicit solvent molecules the preferred side chain conformation is a folded
(curled up) one.
Similar results were later obtained on for a much larger system using molecular me-
chanics (MM) [23]. In this latter study the force field parameters for the polymer were
chosen to reproduce some of the experimental data, such as the liquid-vapour equilibrium
and the self-diffusion coefficient. A highly folded spiral-like configuration was obtained
for a 10 unit polymer when the randomly bent chain was taken as the initial configura-
tion. MD simulations of shorter oligomers solvated in water and methanol have revealed
a noticeable difference between the backbone conformations in different solvents, as seen
in Fig. 1.10. The skeleton structure in water was observed to be substantially more folded
than in methanol. Additionally, the side chain of the Nafion oligomers was found to be
quite stiff; with only a few conformational transitions being detected. Examining the
hydrogen bond propensity of the side chain, the authors have found that both water and
methanol form stable hydrogen bonds with the oxygens of SO−3 group. All other parts
of the side chain were hydrophobic, including ester oxygens, which showed practically no
tendency for hydrogen bonding. On average, each SO−3 group formed approximately five
hydrogen bonds to the solvent water and four bonds to methanol. The solvent molecules
bonded to the sulfonate group form a pronounced anisotropic first solvation shell.
Following this, the same authors investigated the microphase segregation in hydrated
Nafion membranes at different water contents [24]. As the degree of solvation increased,
the formation of water clusters containing up to about 100 water molecules was observed.
In contrast to the conventional network models, the water clusters did not form a con-
17
(Figure 3a). The folding is caused by intramolecular van derWaals and electrostatic interactions, which turned out to bestrong enough to overcome the gain of the torsional potentialenergy due to folding. None of the optimized structures shownin Figures 2 and 3a are likely to represent the global minimumof the potential energy of the oligomers in a vacuum. However,this analysis shows that even in a vacuum molecular geometryof the oligomers is not entirely dominated by the torsional term.In solution, the intramolecular nonbonded interactions competeto the intermolecular solute-solvent interactions, which favorto stretching of the oligomer. However, the intramolecularentropic contribution to the free energy always favors to foldedstructure of chain molecules; for example, this effect is welldocumented for lipid membranes.50 The three main contributionsto the free energy of conformational transition of a macromol-ecule in solution are (1) intramolecular terms, (2) change inthe enthalpy of solvent-solute interactions, and (3) change inthe enthalpy and entropy of the solvent. The latter is especiallyimportant for hydrophilic molecules in self-associated solventssuch as those considered in the present work. This term cannotbe correctly taken into account in static simulations, like energyminimization. On the one hand, water shows more extensivehydrogen bond network compared to methanol. On the otherhand, methanol is expected to show stronger van der Waalsinteractions with the hydrophobic polytetrafluoroethylene skel-eton of Nafion. Thus, methanol is a better solvent for Nafion;i.e., in a solvated Nafion membrane, the microphase segregationis more pronounced in the case of water.It should be noted, also, that the energy barriers of confor-
mational transitions in condensed phases are much highercompared to those in a vacuum. In the dense membrane matrix
of Nafion conformational transitions in polymer chains aresterically hindered. Therefore, Nafion chains in the membranematrix are not supposed to exhibit the conformations similar tothose observed for the oligomers in a vacuum. Yet, the energyminimization in a vacuum allows one to compare differentintramolecular contributions into the total internal energy of thesystem.In our molecular dynamics simulations of the four-unit
oligomer in water and methanol, the fluorocarbon skeleton
Figure 2. Optimized structures of ten-unit perfluorosulfate oligomerobtained by potential energy minimization in a vacuum (a) stretchedconformation with tortuosity of 2.4, obtained from the regular config-uration with all CCCC dihedrals in trans position, (b) strongly foldedspiral-like configuration obtained from a random structure.
Figure 3. (a) Structure of four-unit perfluorosulfate oligomer, opti-mized in a vacuum (b) snapshot of the molecular configuration of thefour-unit oligomer in water and (c) the same in methanol.
4474 J. Phys. Chem. B, Vol. 104, No. 18, 2000 Vishnyakov and Neimark
Figure 1.10: (a) Structure of four-unit perfluorosulfate oligomer, optimized in vacuum(b) snapshot of the molecular configuration of the four-unit oligomer in water and (c)the same in methanol. Figure source: Ref. [23].
18
Figure 1.11: Two monomeric sequences of Nafion 117 with different monomer clustering:(top) blocky polymer with low degree of randomness (bottom) more dispersed polymerwith high degree of randomness. Figure source: Ref. [25].
tinuous hydrophilic subphase. The cluster size distribution was found to be wide and
evolved in time due to formation and break-up of temporary bridges between the clus-
ters. This dynamic behaviour of the cluster system allowed for the macroscopic transfer
of water and counterions. The calculated diffusion coefficients of water were found to be
of the same order of magnitude as those experimentally measured.
The properties of hydrated Nafion are attributed to its nanophase-segregated struc-
ture in which hydrophilic clusters are embedded in a hydrophobic matrix. However,
prior to the work of Jang et al. (Ref. [25]) there has been little characterization of how
the monomeric sequence of the Nafion chain affects the nanophase-segregation structure
and transport in hydrated Nafion. Using atomistic MD simulations, the authors have
investigated these effects on a hydrated Nafion system with 15 water molecules per sul-
fonate group. Two extreme monomeric sequences were examined, one very blocky and
other very dispersed, as illustrated in Fig. 1.11. Both monomeric sequences produce
a nanophase-segregated structure with hydrophilic and hydrophobic domains. The cal-
culated structure factor shows that the monomer sequence of the polyelectrolyte has a
noticeable effect on the extent of phase-segregation: the blocky sequence has better phase
segregation than the dispersed case. The characteristic dimension of the simulated hy-
drophilic clusters is 50 A for the blocky case and 20-30 A for the dispersed case, which are
in good agreement with the 40-50 A obtained from small-angle scattering experimental
19
observations. This comparison suggests that the real Nafion structure is intermediate
but closer to the blocky case. The interface between the water and polymer phases was
also analyzed to determine how the sulfonate groups are arranged at the interface. It
was found that the latter has a heterogeneous structure, consisting of hydrophobic and
hydrophilic patches. The degree of segregation and size of the patches is larger in the
blocky sequence than the dispersed case. Water transport in these systems was shown
to depend on these structural differences caused by the monomeric sequence. Since the
blocky case leads to larger clusters and channels the observed water diffusion was higher.
This result is consistent with the experimental studies on the difference between Nafion
and sulfonated PEEK. However, no significant difference in the hydronium vehicular dif-
fusion was observed indicating that the proton hopping mechanism must be responsible
for the experimentally observed differences in proton diffusion.
Thereafter, the focus of most studies shifted to include the promising new SSC poly-
mer system under minimal hydration conditions [26]. The number of CF2 groups in the
backbone that separates the side chains affects the connectivity of the terminal sulfonic
acid groups. Specifically, with more than four CF2 groups no hydrogen bonding occurs
between neighbouring sulfonic acid groups on the same backbone in the absence of water.
The number of water molecules required to form a continuous hydrogen-bonded network
between the terminal sulfonic acid groups is also a function of the number of CF2 groups
on the backbone separating the side chains. It has been shown that one, two, and three
water molecules bridged the sulfonic acid groups when five, seven, and nine CF2 units
separated the chains, respectively. The separation along the polymeric backbone of the
side chains affects the minimum amount of water necessary to observe the transfer of
protons to the first hydration shell as demonstrated in Fig. 1.12.
In order to understand the flexibility of both the backbone and side chain in the SSC
PFSA system, a computational first principles based study was conducted on a dry, two
20
C6 Fragment + 4-7 H2Os. The B3LYP/6-311G** fullyoptimized structures for the C6 fragment with nine carbon atomsin the backbone are displayed in Figure 4a-d, with the energiesand selected structural data reported in Tables 1 and 4,respectively. Examination of the calculated binding energiesreveals that the trend in the binding energy calculated per watermolecule is consistent among the three methods. Specifically,the binding energy per water molecule decreased from that withonly a single water molecule (Figure 3a) as the water moleculeswere added, until proton dissociation occurred. With protondissociation, and subsequent separation of the hydronium ionsfrom the sulfonate groups, the magnitude of the binding energyincreased to a value of 14.7 kcal/mol for the fragment with sevenwater molecules when computed on the CP-corrected potentialenergy surface. The lowest energy conformation of the oligo-meric fragment with four added water molecules (Figure 4a)showed no proton dissociation but an increase in the oxygen-hydrogen bond distance of nearly 0.1 Å for both sulfonic acidprotons and some increased separation of the terminal side chaingroups (>0.3 Å). The distance (along the backbone) betweenthe tertiary carbons remained essentially constant in the equi-librium structures as the water molecules were added, indicatinglittle conformational change in the backbone. Of severalminimum energy structures determined after five water mol-ecules were added, the structure in Figure 4b showing dissocia-tion of both protons possessed the lowest energy. In previouswork82,86 that examined the hydration and proton dissociationof single (i.e. isolated) sulfonic acids it was observed that threewater molecules were required to observe the transfer of theproton to the water, but the current result indicates that closeproximity of another sulfonic acid may reduce the number of
water molecules per sulfonic acid required to stabilize theprotonic charge in the first hydration shell. Of additionalsignificance is the result that both dissociated protons appearas ‘Zundel ion’-like (i.e. H5O2+) in the global minimumconformation, with one of the hydrated protons bridging thetwo sulfonate groups. This result is very similar to thatdetermined via AIMD as a stable protonic defect in thetrifluoromethanesulfonic acid monohydrate solid, where in theunit cell (i.e. [CF3SO3H]4) one proton is ‘shuttled’ between twosulfonate groups and another ‘shuttled’ between a pair of watermolecules as a Zundel ion.13 The global minimum energystructures of the same oligomeric fragment with six and sevenadded water molecules are shown Figure 4c and 4d and indicatedthat the additional water has not appreciably changed either thepresence or position of the dissociated protons: they remainZundel-like. The tabulated structural data (Table 4) indicatesthat the additional water molecules have resulted in a greaterseparation of the transferred protons from their conjugate basesand a bringing together of the sulfonate groups.C8 Fragment+ 3-7 H2Os. Binding energies computed from
both uncorrected and ZPE corrected total electronic B3LYP/6-311G** energies, along with those corrected for BSSE withoptimization under the CP method, are reported for the C8fragment in Table 2. Examination of the binding energies asthe number of water molecules is increased again shows asimilar trend as was observed with the smaller C6 fragmentwith the water molecules more tightly bound to the polymericfragment upon dissociation of the protons. This is particularlyevident when comparing the computed BSSE corrected bindingenergy upon dissociation of the proton (i.e. after five H2Os wereadded, Figure 5c) to that immediately prior (i.e. Figure 5b with
Figure 4. Fully optimized (B3LYP/6-311G**) global minimum energy structures of the C6 two side chain fragment showing hydration and protondissociation as additional water molecules are added: (a) no dissociation of either acidic proton with four H2Os, although the O-H bond distancein one of the sulfonic acid groups is lengthened 1.10 Å; (b) both protons dissociation with the hydration of five H2Os; (c and d) the hydratedprotons in Zundel-ion-like configurations that further separate with hydration of six and seven H2Os, respectively.
Modeling of Perfluorosulfonic Acid Membrane J. Phys. Chem. A, Vol. 109, No. 33, 2005 7589
(a) (b)
Figure 1.12: Fully optimized (B3LYP/6-311G**) global minimum energy structures ofthe C6 two sidechain fragment showing hydration and proton dissociation as additionalwater molecules are added: (a) no dissociation of either acidic proton with four H2Os,although the O-H bond distance in one of the sulfonic acid groups is lengthened 1.10 A;(b) both protons dissociation with the hydration of five H2Os. Figure source: Ref. [26].
side chain oligomer [27]. The rotational PES of the various C-C, C-O, C-S, and S-O bonds
were examined with the help of DFT, revealing that the polymer backbone is relatively
stiff, with a barrier of nearly 7.0 kcal/mol (see Fig. 1.13). This barrier corresponds to
the energy difference between the staggered trans and planar cis conformations of the
carbon atoms. Furthermore, the calculations have shown that the stiffest portion of the
side chain is near its attachment to the backbone with the CF −O and O−CF2 barriers
of 9.1 and 8.0 kcal/mol, respectively. The most flexible portion of the side chain occurs
at the point of attachment of the sulfonic acid group where the rotational barrier of the
carbon-sulfur bond was determined to be only 2.1 kcal/mol.
The polymer flexibility studies were later extended with extensive searches for mini-
mum energy structures with 4-7 explicit water molecules [28, 29]. It was shown that the
perfluorocarbon backbone may adopt either an elongated geometry, with all carbons in
a trans configuration, or a folded conformation as a result of the hydrogen bonding of
21
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Figure 1.13: Fully optimized (B3LYP/6-31G**) polymeric fragment and the potentialenergy profile for rotation about the F2C−CF2 bond along the backbone. Figure source:Ref. [29].
the terminal sulfonic acids with the water. These electronic structure calculations show
that the fragments displaying the latter ’kinked’ backbone possessed stronger binding of
the water to the sulfonic acid groups, and also undergo proton dissociation with fewer
water molecules.
Subsequently, the hydration of the SSC PFSA membrane has been explored through
comparing the energetics of a three side chain oligomeric fragment of the polymer [30].
Extensive searches for minimum energy conformations with between 6 and 9 water
molecules revealed that at the lower range of the examined hydration (i.e., 2H2O/SO3H)
the uniform hydration of the sulfonic acid groups results in the lowest energy and there-
fore most favourable state of the system. The calculations have shown that as the degree
of hydration is increased the energetic preference for uniform hydration decreases, disap-
pearing altogether at 3H2O/SO3H. Furthermore, it was found that water distributions
that facilitate a higher degree of dissociation and separation of the protons are important
factors in stabilizing the fragments.
The effects of hydration level and temperature on the nanostructure of an atomistic
22
Table 1.1: Density of Hydrated Nafion at 300 and 350 K. Table source: Ref. [31].
hydration level (λ) density at 300 K, [g/cm3] density at 350 K, [g/cm3]
3.5 1.700 ± 0.038 1.703 ± 0.0346 1.698 ± 0.051 1.677 ± 0.02811 1.686 ± 0.025 1.662 ± 0.02116 1.640 ± 0.023 1.606 ± 0.018
model of a Nafion membrane, as well as the vehicular transport of hydronium ions and
water molecules were examined using classical MD simulations in the paper of Venkat-
nathan et al. [31]. Through the determination and analysis of structural and dynamical
parameters such as radial distribution functions (RDF), coordination numbers, mean
square deviations, and diffusion coefficients, the authors have shown that hydronium
ions play an important role in modifying the hydration structure near the sulfonate
groups. In the regime of low level of hydration, short hydrogen bonded linkages made of
water molecules and sometimes hydronium ions alone give a more constrained structure
among the sulfonate side chains. The work has also examined the density of the Nafion
systems as a function of the hydration level and temperature (see Table 1.1), which is of
paramount importance for determining the correct diffusion coefficients. At 300 and 350
K, the density of the hydrated Nafion system gradually decreases with an increase in the
level of hydration of Nafion. This was attributed to structural relaxation with increasing
hydration that leads to a swelling of the membrane at a given temperature. Temperature
was found to have a significant effect on the diffusion coefficients for both water and
hydronium ions. The diffusion coefficient for water agreed well with experimental data,
while the diffusion coefficient of the hydronium ions was much smaller (6-10 times) which
was attributed to the lack of inclusion of the proton hopping mechanism in the study.
Subsequently, the work was significantly extended in a two part paper examining
the effect of hydration on the membrane nanostructure and the dynamics of water and
23
hydronium ion [32,33]. It was found that at λ less than 7, most of the water molecules and
hydronium ions are bound to the sulfonate groups. The strong binding of hydronium ions
to sulfonate groups prevents vehicular transport of protons. Multiple sulfonate groups
surrounding the hydronium ion offer steric hindrance to hydration of the hydronium ion,
which hinders structural diffusion of protons. Water molecules were mainly found in
the vicinity of the sulfonate groups, while the ether oxygen and backbone were strongly
hydrophobic. In addition to calculating diffusion coefficients as a function of hydration
level, the authors have also determined mean residence time of water and hydronium ions
in the first solvation shell of SO−3 groups. The mean residence time of water decreases
with increasing membrane hydration from 1 ns at a low hydration level to 75 ps at the
highest hydration level studied. The mean residence time of hydronium ions is larger than
the corresponding values for water molecules by a factor of 2.5-4.5. The work provides an
explanation for the experimentally observed characteristic time of slow proton dynamics
in hydrated Nafion in terms of the residence of hydronium ions and water molecules in the
first solvation shell of SO−3 groups. These dynamical changes are related to the changes
in membrane nanostructure.
More recently, a comparative study was carried out to determine the hydrated mor-
phology and proton diffusion coefficients in two different PFSA membranes as functions
of water content [34]. Classical MD simulations were performed on a 1143 EW Nafion
and a SSC PFSA polymer with an EW of 977. The water cluster distributions displayed
distinctive differences at the lower water contents (with 4.4 and 6.4 H2O molecules per
side chain) where hydration of the SSC PFSA membrane tends to produce a more dis-
persed cluster distribution, and thus enhance the connectivity of the clusters by water
channels. On the other hand, the Nafion system characterized by longer and more flexible
side chains, is more amenable to aggregate and form clusters that are more disconnected.
At higher water contents, the cluster differences between the two systems become very
24
trajectories indicates a weak correlation between the dipole ofthe sulfonic group and the dipole of the hydronium ion in thisregion, with a marginal preference for antiparallel configurations.A distance of 3.2 Å is certainly too long to allow direct hydrogenbonding to the hydronium ion and the sulfonic acid oxygen.Further inspection reveals that these configurations are typifiedby “bridging” water molecules (Figure 3, top panel) that arehydrogen bonded to both the classical hydronium and aneighboring oxygen of the same sulfonic acid group, with thesulfonic oxygen participating in no hydrogen bonds. When asolvating water occupies the hydrogen bonding position on thissulfonic oxygen the hydronium is unable to occupy thisintermediate position and instead is found in the solventseparated position. We suggest that these configurations mayrepresent local minima between the CIP and the SSIP. Thesolvent reorganization required by the classical model to allow
for the transition between the CIP and the SSIP is clearly adeficiency, since that model does not allow for rearrangementof the covalent and hydrogen bond configurations. A goal ofthis study is to contrast the classical and MS-EVB potentials inorder to show the need for a model that allows bondingrearrangements (Grotthuss shuttling), so we have chosen not toinvestigate these artificial peaks further.The MS-EVB proton CIP to SSIP transition requires little
solvent reorganization because the excess proton can shuttlethrough the hydrogen bond network. A Zundel-like (H5O2+)cation configuration predominates when the MS-EVB2 protonis adjacent to a sulfonic acid group. One water of the Zundelcation is hydrogen bonded to a sulfonic oxygen, while the otheris located away from SO3- (bottom panel, Figure 3). The protoneasily moves away from the sulfonic ion through a Grotthussshuttle between these two water molecules. Experimentalevidence exists for this kind of depletion of the Eigen-typesolvation structure in favor of the Zundel-type for otherconcentrated strong acids, i.e., concentrated HCl solutions.18,20Simulations recently performed with the new self-consistentiterative multi-state empirical valence bond (SCI-MS-EVB)method,20 which is capable of simulating multiple excessprotons, has replicated this enhancement of the Zundel solvationstructure over the Eigen-type. In the context of these experi-mental results, a classical hydronium potential seems clearlydeficient.The bottom panel of Figure 2 shows the distribution functions
the classical and MS-EVB2 hydronium hydrogen atoms fromthe sulfonic acid oxygen for the low hydration simulation. Thedistinguishing feature of the MS-EVB2 curve is the broadfeature, which replaces the peak around 3 Å in the classicalsimulation. This peak in the classical simulation is composedof the hydronium hydrogens of the CIP hydronium that are notdirectly hydrogen bonded to the sulfonic acid oxygen, as wellas the hydrogens of the hydronium in the intermediate position(top panel, Figure 3) as described above. Since the Zundel cationpredominates in the CIP region for the MS-EVB2 simulation,the distinction between the hydrogens coming from the CIPhydronium and those from the SSIP hydronium is blurred(bottom panel, Figure 3). There is also no appreciable contribu-tion to the MS-EVB2 radial distribution from these intermediatehydronium configurations.Figure 4 illustrates the diffusion coefficients for all proton
containing species as a function of hydration. It has beenpreviously shown15 that the self-diffusion of the water potentialused in the MS-EVB2 model is approximately 30% larger thanthe experimental value and, not unexpectedly, the waterdiffusion for this Nafion simulation is also larger than experi-ment6 for both degrees of hydration. Conversely, the diffusionof the MS-EVB2 excess proton in bulk water was shown inearlier work to be roughly half of the bulk experimental value,and, in turn, the classical hydronium was shown to be abouthalf of the MS-EVB2 excess proton diffusion value.15 This trendalso extends to these simulations and the correspondingexperimental6 data. In the present simulations, one likely reasonfor the apparent under estimation of the diffusion constant ofthe Grotthuss shuttling proton by the MS-EVB2 model (certainlyin the least hydrated simulation) is the effect of “caging” bythe classical hydroniums. The MS-EVB2 proton is artificiallytrapped in a cage formed by classical hydroniums that, in turn,cannot participate in the Grotthuss hopping process importantfor the proton transport. When the level of Nafion hydration isincreased, thereby increasing the distance between the cationsand the extent of the bonding network, the ratio between the
Figure 3. (top panel) Representative configuration of the “intermedi-ate” solvation structures observed in the simulations using the classicalhydronium potential.22,23 Note the water molecule hydrogen bonded(H-bonds in red) to the oxygen adjacent to the inspected oxygen (aster-isk). Extraneous hydrogen bonds, water molecules, and polymer wereexcluded for clarity. (bottom panel) Representative configuration ofthe Zundel structure observed in the simulations using the MS-EVB2potential. Relative to the inspected oxygen (asterisk) one water of theZundel cation is located in the contact ion pair (CIP) position, whilethe other is located in the solvent separated ion pair (SSIP) position.
Letters J. Phys. Chem. B, Vol. 109, No. 9, 2005 3729
trajectories indicates a weak correlation between the dipole ofthe sulfonic group and the dipole of the hydronium ion in thisregion, with a marginal preference for antiparallel configurations.A distance of 3.2 Å is certainly too long to allow direct hydrogenbonding to the hydronium ion and the sulfonic acid oxygen.Further inspection reveals that these configurations are typifiedby “bridging” water molecules (Figure 3, top panel) that arehydrogen bonded to both the classical hydronium and aneighboring oxygen of the same sulfonic acid group, with thesulfonic oxygen participating in no hydrogen bonds. When asolvating water occupies the hydrogen bonding position on thissulfonic oxygen the hydronium is unable to occupy thisintermediate position and instead is found in the solventseparated position. We suggest that these configurations mayrepresent local minima between the CIP and the SSIP. Thesolvent reorganization required by the classical model to allow
for the transition between the CIP and the SSIP is clearly adeficiency, since that model does not allow for rearrangementof the covalent and hydrogen bond configurations. A goal ofthis study is to contrast the classical and MS-EVB potentials inorder to show the need for a model that allows bondingrearrangements (Grotthuss shuttling), so we have chosen not toinvestigate these artificial peaks further.The MS-EVB proton CIP to SSIP transition requires little
solvent reorganization because the excess proton can shuttlethrough the hydrogen bond network. A Zundel-like (H5O2+)cation configuration predominates when the MS-EVB2 protonis adjacent to a sulfonic acid group. One water of the Zundelcation is hydrogen bonded to a sulfonic oxygen, while the otheris located away from SO3- (bottom panel, Figure 3). The protoneasily moves away from the sulfonic ion through a Grotthussshuttle between these two water molecules. Experimentalevidence exists for this kind of depletion of the Eigen-typesolvation structure in favor of the Zundel-type for otherconcentrated strong acids, i.e., concentrated HCl solutions.18,20Simulations recently performed with the new self-consistentiterative multi-state empirical valence bond (SCI-MS-EVB)method,20 which is capable of simulating multiple excessprotons, has replicated this enhancement of the Zundel solvationstructure over the Eigen-type. In the context of these experi-mental results, a classical hydronium potential seems clearlydeficient.The bottom panel of Figure 2 shows the distribution functions
the classical and MS-EVB2 hydronium hydrogen atoms fromthe sulfonic acid oxygen for the low hydration simulation. Thedistinguishing feature of the MS-EVB2 curve is the broadfeature, which replaces the peak around 3 Å in the classicalsimulation. This peak in the classical simulation is composedof the hydronium hydrogens of the CIP hydronium that are notdirectly hydrogen bonded to the sulfonic acid oxygen, as wellas the hydrogens of the hydronium in the intermediate position(top panel, Figure 3) as described above. Since the Zundel cationpredominates in the CIP region for the MS-EVB2 simulation,the distinction between the hydrogens coming from the CIPhydronium and those from the SSIP hydronium is blurred(bottom panel, Figure 3). There is also no appreciable contribu-tion to the MS-EVB2 radial distribution from these intermediatehydronium configurations.Figure 4 illustrates the diffusion coefficients for all proton
containing species as a function of hydration. It has beenpreviously shown15 that the self-diffusion of the water potentialused in the MS-EVB2 model is approximately 30% larger thanthe experimental value and, not unexpectedly, the waterdiffusion for this Nafion simulation is also larger than experi-ment6 for both degrees of hydration. Conversely, the diffusionof the MS-EVB2 excess proton in bulk water was shown inearlier work to be roughly half of the bulk experimental value,and, in turn, the classical hydronium was shown to be abouthalf of the MS-EVB2 excess proton diffusion value.15 This trendalso extends to these simulations and the correspondingexperimental6 data. In the present simulations, one likely reasonfor the apparent under estimation of the diffusion constant ofthe Grotthuss shuttling proton by the MS-EVB2 model (certainlyin the least hydrated simulation) is the effect of “caging” bythe classical hydroniums. The MS-EVB2 proton is artificiallytrapped in a cage formed by classical hydroniums that, in turn,cannot participate in the Grotthuss hopping process importantfor the proton transport. When the level of Nafion hydration isincreased, thereby increasing the distance between the cationsand the extent of the bonding network, the ratio between the
Figure 3. (top panel) Representative configuration of the “intermedi-ate” solvation structures observed in the simulations using the classicalhydronium potential.22,23 Note the water molecule hydrogen bonded(H-bonds in red) to the oxygen adjacent to the inspected oxygen (aster-isk). Extraneous hydrogen bonds, water molecules, and polymer wereexcluded for clarity. (bottom panel) Representative configuration ofthe Zundel structure observed in the simulations using the MS-EVB2potential. Relative to the inspected oxygen (asterisk) one water of theZundel cation is located in the contact ion pair (CIP) position, whilethe other is located in the solvent separated ion pair (SSIP) position.
Letters J. Phys. Chem. B, Vol. 109, No. 9, 2005 3729
Figure 1.14: Representative configurations of the solvation structures observed in thesimulations using the classical hydronium potential (left) and the EVB potential (right),which were common for both hydration levels (λ = 7, 15). Figure source: Ref. [35].
small. The diffusion coefficient of water and hydronium ions are both slightly lower in
the SSC membrane when compared to Nafion, suggesting that structural diffusion by
proton hopping may account for the observed higher conductivities in the SSC PFSA
membrane.
One of the most popular methods to incorporate the Grotthuss mechanism in such
large scale simulations is through the EVB method. One such approach has examined
the solvation properties of the hydrated excess proton in a water cluster of the Nafion
117 membrane [35]. MD simulations were performed with both classical, nondissociable
hydronium cations and with a single excess proton that was treated by the EVB method.
Two levels of hydration were studied (λ = 7, 15), revealing the same marked difference
between the hydronium ion solvation structures, as illustrated in Fig. 1.14. As only a
single excess proton was treated with the EVB formalism, at the low hydration level its
diffusion was artificially reduced by caging from the other classical hydroniums ions in
the simulation cell.
Thereafter, a self-consistent variant of the EVB method was used to allow all excess
25
protons to shuttle via the Grotthuss mechanism [36]. The total proton diffusion was,
then, decomposed into vehicular and Grotthuss components which were found to be of
the same relative magnitude, but with a strong negative correlation, resulting in a smaller
overall diffusion for the Nafion system. By contrast, Grotthuss diffusion accounts for 70%
of the total hydronium diffusion in bulk water, with negligible negative correlation of the
two components as seen in Fig. 1.15. Furthermore, correlated motions between the ion
pair were also examined through the distinct portion of the van Hove correlation function,
as shown in Fig. 1.16. At approximately 425 ps, the function develops a peak that is
two to three times the average hydronium density. So, given that a sulfonate anion
occupied a given position 425 ps earlier, the likelihood of finding a hydronium ion in this
same position is nearly three times greater than that of the uniform hydronium density.
This demonstrates a significant correlation in the local ion pair diffusion. The sulfonate
ions effectively act as proton “traps”, limiting the hydronium diffusion primarily to the
long time correlated ion pair motions. This may in part explain why side chain length
variants of Nafion-like polymers, such as the SSC membrane or Aciplex, exhibit varying
transport rates. A shorter pendant chain may restrain the sulfonate groups from deeply
penetrating the hydrophobic phase and trapping the excess protons in the bulk water
region where transport could be the greatest.
In the largest atomistic simulation carried up to date, Knox et al. investigated six of
the most significant morphological models of hydrated Nafion to compare their structural
properties and behaviour [37]. These models shown schematically in Fig. 1.16 are the
cluster-channel model, the parallel cylinder model, the local order model, the lamellar
model, the rod network model, and a “random” model that does not directly assume any
particular morphology. In order to probe multiple hydrophilic clusters and to accurately
measure the signature scattering these authors used, for the first time in this field, large-
scale systems (∼ 2 million atoms and a box length of ∼ 30 nm). Each system was initially
26
The x component of a representative trajectory (total, discrete,and continuous) is depicted in Figure 6. From the inset of Figure6, the stepwise nature of the discrete portion of the totaldisplacement is more clearly seen. While the continuous portiondevelops with small consistent displacements, the discreteportion proceeds with significant closely spaced multipledisplacements punctuated by intervals of no change, which issymptomatic of relatively long-lived states. Most noteworthyis the near mirroring of the x component of the two displacementvectors, that is to say, the very nearly equal but opposite relativedisplacement of the two contributions. Although the displace-ment vectors of each component need not project onto any givenaxis in this manner, this particular trajectory developed alongthe x-axis in such a manner as to illustrate the interesting andstrong anticorrelation between these two components of the totaldisplacement.The anticorrelation between the discrete and continuous
displacement components is quantified through the MSD plotspresented in Figure 7. Not only is the total diffusion less thanthe sum of its components, the diffusion of either componentis remarkably greater than the total. The strong negative overlap(the last term of eq 5) of these two displacement vectorstherefore results in a total diffusion less than that of eithercomponent. By contrast, discrete diffusion accounts for !70%of the total MS-EVB2 hydronium diffusion (Figure 8) in bulkwater with negligible negative correlation of the vehicular anddiscrete components.3.3. Ion Pair Correlated Diffusion. It has been previously
demonstrated through computer simulation that the diffusionof the protonic defect may be influenced by the motion of thesulfonate anions.28 We have likewise observed here significantcorrelated motion of the ion pair and have quantified the timescale of these correlated motions through the distinct portionof the van Hove correlation function, given by34
Figure 9 depicts this correlation function (normalized by theaverage density) such that the sulfonate anions are chosen asthe space and time origins.At approximately 425 ps, the function develops a peak that
is two to three times the average hydronium density. So, giventhat a sulfonate anion occupied a given position 425 ps earlier,the likelihood of finding a hydronium cation in this sameposition is nearly three times greater than that of the uniformhydronium density. This demonstrates a significant correlationin the local ion pair diffusion with a characteristic period ofapproximately 425 ps. It should be noted that, for long timescomparable to the length of the trajectory and short radialdistances, the data points become relatively sparse. For example,the 3.3 ps-0.5 Å bin about 425 ps and a radial distance of0.75 Å has a value of 2.3 ( 0.2. Longer times and shorterdistances are progressively statistically less reliable.Given the observed strong correlated motion, it is easy to
understand the apparent increase in diffusion seen by Spohr etal.28 upon the transition from a tethered to a flexible model forthe side chain. However, given the long characteristic periodrelative to the total simulation time and the comparatively low
diffusion of both the pendant chain and the associated hydro-nium ion, it seems inappropriate to generalize this increase inlocal diffusion to an increase in macroscopic proton transport.
!! rbCEC1‚! rbCEC1" ) !! rbc‚! rbc" + !! rbd‚! rbd" +
2!! rbc‚! rbd" (5)
GdR!(r, t) )
NR + N!
NRN!
!#i)1
NR
#j)1
N!
"(r - |ri(0) - rj(t)|)" )
FgR!(r, t) (6)
Figure 6. x-Coordinate of a representative trajectory for the CEC; thetotal trajectory (black), the continuous (vehicular, red), and discrete(Grotthuss, blue) components. The inset more clearly displays thestepwise nature of the discrete portion and the continuous nature ofthe vehicular portion.
Figure 7. Total mean-squared displacement (black) and the continuous(red) and discrete (blue) components of the mean-squared displacementin Nafion.
Figure 8. Total mean-squared displacement (black) and the continuous(red) and discrete (blue) components of the mean-squared displacementin bulk water
18598 J. Phys. Chem. B, Vol. 110, No. 37, 2006 Petersen and Voth
The x component of a representative trajectory (total, discrete,and continuous) is depicted in Figure 6. From the inset of Figure6, the stepwise nature of the discrete portion of the totaldisplacement is more clearly seen. While the continuous portiondevelops with small consistent displacements, the discreteportion proceeds with significant closely spaced multipledisplacements punctuated by intervals of no change, which issymptomatic of relatively long-lived states. Most noteworthyis the near mirroring of the x component of the two displacementvectors, that is to say, the very nearly equal but opposite relativedisplacement of the two contributions. Although the displace-ment vectors of each component need not project onto any givenaxis in this manner, this particular trajectory developed alongthe x-axis in such a manner as to illustrate the interesting andstrong anticorrelation between these two components of the totaldisplacement.The anticorrelation between the discrete and continuous
displacement components is quantified through the MSD plotspresented in Figure 7. Not only is the total diffusion less thanthe sum of its components, the diffusion of either componentis remarkably greater than the total. The strong negative overlap(the last term of eq 5) of these two displacement vectorstherefore results in a total diffusion less than that of eithercomponent. By contrast, discrete diffusion accounts for !70%of the total MS-EVB2 hydronium diffusion (Figure 8) in bulkwater with negligible negative correlation of the vehicular anddiscrete components.3.3. Ion Pair Correlated Diffusion. It has been previously
demonstrated through computer simulation that the diffusionof the protonic defect may be influenced by the motion of thesulfonate anions.28 We have likewise observed here significantcorrelated motion of the ion pair and have quantified the timescale of these correlated motions through the distinct portionof the van Hove correlation function, given by34
Figure 9 depicts this correlation function (normalized by theaverage density) such that the sulfonate anions are chosen asthe space and time origins.At approximately 425 ps, the function develops a peak that
is two to three times the average hydronium density. So, giventhat a sulfonate anion occupied a given position 425 ps earlier,the likelihood of finding a hydronium cation in this sameposition is nearly three times greater than that of the uniformhydronium density. This demonstrates a significant correlationin the local ion pair diffusion with a characteristic period ofapproximately 425 ps. It should be noted that, for long timescomparable to the length of the trajectory and short radialdistances, the data points become relatively sparse. For example,the 3.3 ps-0.5 Å bin about 425 ps and a radial distance of0.75 Å has a value of 2.3 ( 0.2. Longer times and shorterdistances are progressively statistically less reliable.Given the observed strong correlated motion, it is easy to
understand the apparent increase in diffusion seen by Spohr etal.28 upon the transition from a tethered to a flexible model forthe side chain. However, given the long characteristic periodrelative to the total simulation time and the comparatively low
diffusion of both the pendant chain and the associated hydro-nium ion, it seems inappropriate to generalize this increase inlocal diffusion to an increase in macroscopic proton transport.
!! rbCEC1‚! rbCEC1" ) !! rbc‚! rbc" + !! rbd‚! rbd" +
2!! rbc‚! rbd" (5)
GdR!(r, t) )
NR + N!
NRN!
!#i)1
NR
#j)1
N!
"(r - |ri(0) - rj(t)|)" )
FgR!(r, t) (6)
Figure 6. x-Coordinate of a representative trajectory for the CEC; thetotal trajectory (black), the continuous (vehicular, red), and discrete(Grotthuss, blue) components. The inset more clearly displays thestepwise nature of the discrete portion and the continuous nature ofthe vehicular portion.
Figure 7. Total mean-squared displacement (black) and the continuous(red) and discrete (blue) components of the mean-squared displacementin Nafion.
Figure 8. Total mean-squared displacement (black) and the continuous(red) and discrete (blue) components of the mean-squared displacementin bulk water
18598 J. Phys. Chem. B, Vol. 110, No. 37, 2006 Petersen and Voth
Figure 1.15: Proton MSD in Nafion (top) and water (bottom). Figure source: Ref. [36].
27
It may very well be that the perceived increase in diffusion issimply an artifact of the correlated motion of the ion pair; thatis, the more labile sulfonate ion of the flexible chain simplydrags the hydronium cation as it diffuses about some meanposition. However, because the pendant chain is ultimatelybound to the comparatively static polymer backbone, the motionof this putative mean position may be inaccessible to theavailable molecular dynamics time scales.3.4. Amphiphilic Association of the Hydronium Cation
and the Hydrophobic Domain. It has recently been demon-strated that the amphiphilic-like character of the hydrated protonobserved near the water liquid-vapor interface35 and waterclusters36-38 extends to other mixed dielectrics such as methanol-water solutions.39 Although the degree of amphiphilic associa-tion may be somewhat potential dependent,40,41 there is com-pelling experimental support 42-45 for the surface enhancementobserved in both empirical force field35,38 and ab initio simula-tions.36,37
Radial distributions were therefore calculated between thehydronium cation and the hydrophobic polymer backbone(including all carbon and fluorine atoms but excluding those ofthe pendant chain) as well as between water and the hydrophobicbackbone. It has been previously demonstrated that the aniso-tropic solvation of the hydronium cation results in a preferentialhydrophobic association in the lone pair region of the ion’ssolvation shells.39 With this in mind, the radial distributionrestricted to a ! steradian solid angle with an apex formed fromthe vector extending from the hydronium hydrogen center-of-mass through the hydronium oxygen (the lone pair region) wasalso calculated. These radial distribution functions are presentedin Figure 10.Although the solvation structures are very similar for the full
water-backbone and hydronium-backbone distributions, thehydronium distribution displays larger populations at shorterdistances. By itself, this is not definitive evidence for thepreferential association of the hydronium lone pair region withthe hydrophobic backbone and is possibly a consequence ofthe hydronium-sulfonate attraction and the proximity of thesulfonate pendant and the polymer backbone. However, therestricted radial distribution shows a significant lone pair regionenhancement of the backbone population over the full distribu-tion. Similar to that which has been previously demonstratedfor the hydrophobic methyl groups of methanol,39 there is a
significant preferential anisotropic association of the hydroniumwith the hydrophobic polymer backbone.
4. Conclusions
The proton transport process about the sulfonate CIP/SSIPregion was found in this work to proceed largely through theGrotthuss shuttling mechanism. A decomposition of the hydro-nium MSD shows that the overall diffusion process is a highlycorrelated exchange between diffusion through vehicular dif-fusion of the transient dominant state and the fluctuating bondtopology, resulting in a relatively small net diffusion. Further-more, the distinct portion of the van Hove correlation functionshows the ion pair diffusion is correlated with a characteristictime scale of several hundred picoseconds.In total, our results indicate that the sulfonate ion significantly
influences the diffusion of the protonic defects in a hydrophilicpocket of Nafion. As the transiently dominant hydronium statediffuses away from the sulfonate ion, the fluctuating bond top-ology “resets” the position of the dominant state back to somemean position relative to the adjacent sulfonate ion. The sul-fonate ions effectively act as proton “traps”, limiting the hydro-nium diffusion primarily to the long time correlated ion pairmotions. This may in part explain why side chain length variantsof Nafion-like polymers, such as the Dow membrane or Aciplex,exhibit varying transport rates. A shorter pendant chain mayrestrain the sulfonate groups from deeply penetrating thehydrophobic phase and trapping the excess protons in the bulkwater region where transport could be the greatest. On the otherhand, perhaps the shorter pendant chains allow the hydratedproton to more closely interact with the hydrophobic portionof the polymer, for which it has a demonstrated affinity, enablingtransport along the hydrophilic/hydrophobic boundary. Thesepossibilities will be more closely explored in future research.
Acknowledgment. This research was supported by theDepartment of Energy Basic Energy Sciences program (grantno. DE-FG02-05ER15724) and the U.S. Army Research Labo-ratory and the U.S. Army Research Office (grant no. DAAD19-03-1-0121). We thank Dr. Kim Wong and Mark Maupinfor their critical reading of the manuscript.
Figure 9. Distinct portion of the van Hove space-time correlationfunction eq 6 for the hydronium-sulfonate ion pair given the sulfonateanion as the space-time origin.
Figure 10. Radial distribution functions for the water oxygen-polymerbackbone (black) and hydronium oxygen-polymer backbone (red). Therestricted radial distribution function for the hydronium oxygen-polymer backbone (red dashed) is restricted to a ! steradian solid anglewith an apex formed from the vector extending from the hydroniumhydrogen center-of-mass through the hydronium oxygen (the lone pairregion).
Perfluorosulfonic Acid Membrane Nafion J. Phys. Chem. B, Vol. 110, No. 37, 2006 18599
Figure 1.16: van Hove space-time correlation function for the hydronium-sulfonate ionpair, given the sulfonate anion as the space-time origin. Figure source: Ref. [36].
six selected morphological model systems of hydrated Nafionwill now be individually described.
A “random” model with randomly placed water moleculesin a box of Nafion polymer chains (basically a model of randomwater shapes and sizes in the polymer)6,29 has been studied. Thismodel has been used before in the literature to try to predict apriori the true structure of Nafion without making any assump-tions.29 Although this model does not directly assume anyparticular cluster shapes or sizes, it does indirectly assume arandom distribution of shapes and sizes, which self-assembleduring the course of MD. This generally leads to a randomdistribution of intercluster spacings. Such uncorrelated or weaklycorrelated behavior lacks a strong scattering peak. Hence, thismodel is not expected to have a well-defined ionomer peak inthe scattering spectra due to the short time scales accessible tosimulation. Much longer time scales, in theory, would changethis model into an accurate depiction of true Nafion, but suchtime scales are not computationally feasible. It should be notedthat Jang et al.29 reported a strong peak in the structure factorof a random model, although this may have mistakenly resultedfrom not spherically averaging the scattering vectors as evi-denced by the incorrectly spaced points in that plot.
The parallel “cylinder” model is the newest morphologicalmodel of Nafion to-date.6 It consists of an array of aligned rigidtubes of varying diameter and spacing filled with water andsurrounded by polymer. No connecting bridge structures havebeen proposed for this model to explain the observed percolationthreshold in Nafion. The mean center-to-center separationdistance between cylinders and the mean cylinder radius wereproposed to be !38 Å and !12 Å, respectively. These samevalues were also used in this study to closely mimic the proposedhypothetical model. In accordance with Schmidt-Rohr’s cylinderdesign, this study used nonoverlapping hard cylinders with athin polymer shell (!7 Å) coating the outside of each one sothat the cylinders were not allowed to touch one another.Cylinders were randomly placed on the basis of these criteriaand using the same replacement and shrinking radius approachof Schmidt-Rohr as well as periodic boundary conditions. Theaxis of each cylinder was placed parallel to the z-direction tomaintain perfect alignment. Cylinder radii were chosen fromthe same “slanted” distribution reported by Schmidt-Rohr. Atotal of 30 cylinders were placed in a 300 Å " 300 Å " 300 Åbox using this approach, which resulted in a total cylindervolume of !17.1% of the total volume of the box, correspondingto the target water content.
The cluster-channel (sphere-rod) model is among the oldestand most well-known morphological models of Nafion. Itconsists of water spheres with connecting water rods between
them, all surrounded by polymer.1,3,6 Typically, the spheres arearranged on a lattice and are approximated with a monodispersesize distribution. However, in the present work, the spheres havebeen randomly placed (nonoverlapping) and given random sizesfrom a broad, slanted distribution, similar to the approach usedfor the cylinder model, to better mimic the intuitive disorder ofamorphous systems and to more closely fit the observedscattering spectra, especially the ionomer peak. Since this modelincludes proposed connecting rods of water between some ofthe spheres, it may be considered a type of network model thatattempts to explain percolation phenomena. The mean center-to-center separation distance between spheres and the meansphere and connecting rod radii were !47, !18, and !5 Å,respectively. We used nonoverlapping hard spheres with a thinpolymer shell (!5.5 Å) coating the outside of each one so thatthe spheres were not allowed to touch one another. Spheres wererandomly placed on the basis of these criteria and using thesame replacement and shrinking radius approach of the cylindermodel as well as periodic boundary conditions. Spheres withinthe average separation distance (!47 Å) from one another werelinked together with rods. Sphere radii were chosen from a“slanted” distribution similar to that from the cylinder model,which basically results from the increasing difficulty of placingadditional spheres as more and more spheres reside in the boxand which is overcome by occasionally shrinking the newlyplaced spheres a little to compensate for this effect. Rod radiiwere chosen from a normal distribution with a 1 Å standarddeviation. A total of 190 spheres with 72 connecting rodsbetween neighboring pairs were placed in a 300 Å " 300 Å "300 Å box using this approach, which resulted in a total spherevolume of !16.6% of the total volume of the box (the total rodvolume was !1%), corresponding to the target water content.
The local order (or hard sphere) model consists of randomlyplaced water spheres (without connecting rods) surrounded bypolymer.1,5,53-55 Geometrically, this model is similar to thesphere-rod model (cluster-channel model), except it does notassume connecting bridges between spheres, and thus, it is nota network model. The mean center-to-center separation distancebetween spheres and the mean sphere radius were the same asthose of the sphere-rod model. The same building approach(randomly placed nonoverlapping hard spheres with a thinpolymer shell and with occasional shrinking sphere radius) wasused as mentioned above, except that no connecting rods wereused for this model. A total of 190 spheres were placed in a300 Å " 300 Å " 300 Å box using this approach, whichresulted in a total sphere volume of !16.9% of the total volumeof the box, which closely matches that of the sphere-rod model.On average, the spheres used in this model were slightly largerthan the spheres of the sphere-rod model to closely match watercontent while maintaining the same number of spheres. Due tothe random nature of the cluster placement algorithm, thelocation and distribution of spheres are both completely uncor-related between the two models.
The lamellar (slab) model consists of alternating parallel slabsor slices, each one filled with water or polymer and sandwichedbetween two slabs of the other.1,6,56 Since it lacks bridgesbetween slabs, it is also not a network model. It is a modelcontaining elongated structures in 2D, whereas the cylindermodel contains 1D elongated structures. The other models donot contain elongated clusters. The mean center-to-centerseparation distance between water slabs and the mean waterslab thickness were !47 Å and !8.5 Å, respectively. Thiscorresponds to a mean polymer slab thickness of !38.5 Å(polymer thickness ) water separation - water thickness).
Figure 1. Morphological models of Nafion shown as schematics. Themodel names used in this work are labeled with their respectiveabbreviations in parentheses. Dark represents water; for clarity, polymeris not shown.
Molecular Dynamics of Nafion J. Phys. Chem. B, Vol. 114, No. 9, 2010 3207
Figure 1.17: Morphological models of Nafion shown as schematics. Dark representswater; for clarity, polymer is not shown. Figure source: Ref. [37].
28
built to closely approximate the proposed hydrophilic cluster structure in a given model.
Formation of connecting bridges between clusters and the resulting percolation was ob-
served at the molecular level in all of the models, with significant involvement of the
sulfonate groups. The solvent-accessible surface area (SASA) was used to measure the
hydrophilic-hydrophobic interfacial area of each model, revealing for the first time large
magnitude SASA values (∼ 1000 m2/g) in the rod model. The authors have pointed out
that such a large interfacial area underscores the strong nanophase segregation behavior
of Nafion. It was suggested that the high interfacial area may also enhance proton trans-
port because of the amphiphilic nature of excess protons, which have been observed to
prefer hydrophilic-hydrophobic interfaces. Most interestingly, the structural and scatter-
ing spectra of the nonrandom models were found to be closely comparable, emphasizing
the insensitivity of the characteristic scattering peak to widely varying geometry and
model differences.
Finally, a dissipative particle dynamics study was carried out recently to elucidate
the role of MW on the hydrated morphology of SSC PFSA membrane [38]. The increase
of MW induces aggregation of the fluorocarbon backbone that minimizes chain bend-
ing forces, while maintaining a phase-separated structure, and results in larger, more
elongated water domains.
Part I
Methodology Development
29
30
Chapter 2
Force Field Development
The quality of any MM simulation is largely determined by the suitability of its force
field. Unfortunately, there is no single, universal force field that can correctly predict all
properties, across the vast array of known compounds. Ideally, any application of a given
force field to a new class of compounds should follow a careful investigation of its validity
within the new context. However, this route was not adopted in the only theoretical study
thus far carried out on a large-scale SSC system [39]. There the force field parameters
were taken directly from the force field employed for Nafion. The inconclusive results
of this study fail to even qualitatively distinguish the proton mobilities in the SSC and
Nafion systems. Hence, for our modelling we have chosen to first modify and tailor the
Nafion force field prior to its application to the SSC polymer.
When examining the structural or dynamic differences between the SSC polymer and
Nafion the question may well be raised whether these differences are indeed present in
the real systems and not just the artefacts of the different force fields. Let us consider
the backbone lengths, LSSC and LNaf , of a SSC polymer and a Nafion polymer strand
with the same number of backbone carbon atoms. This length will be measured at the
minimum energy conformation as found by MM. Since this equilibrium conformation
minimizes the total MM energy of the polymer it is a function of all the terms in the force
field, including the torsion potentials and the non-bonding interactions. The difference
in the lengths can be written as LSSC − LNaf = (L′SSC + δSSC) −(L′Naf + δNaf
)=
(L′SSC − L′Naf
)+(δSSC − δNaf ). Here the primed lengths are the exact lengths (e.g., from
ab initio calculations) and the δ symbols represent the errors of the MM estimates. Let
us now determine how well the exact difference in the lengths L′SSC−L′Naf is reproduced
31
by the force field. From the above formula we can see that a good representation will be
achieved in two cases: if δSSC , δNaf are close in magnitude and of the same sign, or if they
are both close to zero. Therefore, if we have two well parameterized classical force fields,
one for the SSC polymer and one for Nafion (i.e., both δSSC , δNaf are close to zero), we
can get an accurate estimate for the difference in backbone lengths of the two polymers.
On the other hand, one universal force field with the same number of parameters will
necessarily have larger errors. In this case a good estimate can still be achieved, but only
if δSSC , δNaf are, fortuitously, of the same magnitude and sign. Therefore a unique SSC
force field will improve, rather than impede the comparisons with Nafion.
The key feature that our force field aims to reproduce is the conformation of the
polymer molecule. The leading role in determining the conformation is played by the
torsion potential. While the bond lengths and angles of an atom group are restricted
about a single equilibrium value, dihedral potentials are modeled with periodic functions
allowing for a number of low-energy conformations. By specifying the sequence of dihedral
angles one can uniquely determine the morphology of the polymer, an approach widely
used in proteins [40]. Recent ab initio modelling of the SSC system has suggested that
the conformation of the backbone has important effect on chemical properties under low
hydration conditions. Extensive electronic structure calculations of oligomeric fragments
possessing two side chains revealed that the extent of the separation of the sulfonic acid
groups along with the conformation of the backbone significantly affect the propensity
of the acid groups to dissociate [26, 28]. Kinked conformations of the backbone give
rise to closer proximity of the acid groups, stronger binding of the water molecules and
enhanced proton dissociation at lower degrees of hydration. As these properties needed
to be correctly reproduced in our SSC specific force field, we take special care in our
treatment of the torsion potentials (described below).
The force field we present here is based on the generic DREIDING force field devel-
32
oped by Mayo et al. [41] and that has been used and supplemented by several authors
investigating the Nafion system [25,36,35]. Our adaptation of the DREIDING force field
to the SSC polymer is based on extensive ab initio calculations performed by Paddison
and Elliott [29]. Six different torsion profiles were computed in their study of a two-side
chain oligomeric fragment altogether comprising more than 200 geometries. The average
bond lengths and angles were extracted from these structures and their values were used
as the equilibrium parameters r0 and θ0 in the force field potentials. The force constants
of the stretching and bending motion were assumed to be the same as in Nafion (see
Appendix A, Fig. A.1, Tables A.1 and A.2). The atom charges were calculated through
a Mulliken population analysis of the ab initio structures of the SSC fragment, while
the Lennard-Jones parameters ε and σ were assumed to remain unchanged from Nafion
(Appendix A, Table A.4).
The last component of the SSC force field is the torsion potential and it is the corner
stone of our force field since it ties together all the other potentials and ensures agreement
with the ab initio torsion barriers. From Paddison and Elliott’s oligomer geometries one
can further extract any structural information like dihedral angles, including those whose
profiles were not explicitly studied in their work. One such dihedral is the O4:-S6:-C2S:-
C2O (where the colon indicates a wildcard1 ensuring that both the neutral and ionized
side-chains are covered, see Fig. 2.1g). Obtaining a MM representation of the profile of
this dihedral angle and further assuming that it does not change after proton dissociation
makes it possible to go beyond the neutral system modeled in the ab initio work. Another
compelling reason for including this torsion potential in the MM energy balance is the
accompanying exclusion of the 1-4 non-bonding interactions (i.e., between the atoms O4:
and C2O). The Coulomb interaction between these two atoms, especially in the case of an
ionized chain, is quite large which makes fitting the ab initio data very difficult. For that
1In computer (software) technology, a wildcard character can be used to substitute for any othercharacter or characters in a string.
33
reason we have seven torsion potentials as shown in Fig. 2.1) and Appendix A, Table A.3.
It should be noted that our force field does not include all possible dihedral angles that
one can find in the polymer. One such group of dihedral angles that are absent is that
which involves fluorine atoms. The reason we limit the number of dihedral angle types in
the force field is to minimize the computational work during the MD simulation. As our
force field (including the torsion potential) completely replaces the potentials defined in
the generic DREIDING force field [41], any terms from the latter that do not carry over
to the SSC force field should be considered zero. This implies, however, that such terms
are not simply missing but that their effect has been adsorbed into the parameters of
the remaining terms. Our force field should be considered complete in the sense that we
have obtained the best possible correspondence to the ab initio data given the number of
fitting parameters in the torsion potential. In the next few paragraphs we show exactly
how these parameters’ values were obtained.
We define the torsion energy of an oligomeric fragment with atom coordinates Ri as
the difference of the molecule’s ab initio energy and the sum of the energies of the other
bonding and non-bonding potentials:
Etors (Ri) = EQM − Ebonds − Eangles − ECoulomb − ELJ (2.1)
The index i runs over all of the 200 SSC oligomer structures, e.g. R1 represents the
coordinates of the atoms of the SSC structure in the first data point of the first torsion
profile. This assignment bestows an additional role to the torsion term in enabling it to
absorb the error of the model, which arises from the assumption that an ab initio energy
can be split into a sum of MM terms. Alternatively, the torsion energy in Eq. (2.1) can
be viewed as the residual energy after all other classical terms have been accounted. The
torsion energy is modeled as a sum of dihedral potentials according to:
Efit (Ri) =7∑
m=1
Nm∑
l=1
am cos(bmϕim,l − cm) (2.2)
34
C1O
O3
C2O
C2S
S6O4 O4
O3H
H3
F2OF2O
F2S F2S
C1
F1
C1
F1
F1
C1
F1
F1
C1
F1 F1
F1
C1O
O3
C2O
C2S
S6O4 O4
O3H
H3
F2OF2O
F2S F2S
C1O
O3
C2O
C2S
S6O4 O4
O3H
H3
F2OF2O
F2S F2S
C1O
O3
C2O
C2S
S6O4 O4
O3H
H3
F2OF2O
F2S F2S
C1O
O3
C2O
C2S
S6O4 O4
O3H
H3
F2OF2O
F2S F2S
C1
F1
F1
C1O
O3
C2O
C2S
S6O4 O4
O3H
H3
F2OF2O
F2S F2S
(a) (b) (c) (d)
(e)
(f)
(g)
Figure 2.1: The seven dihedral angles of the SSC force field illustrated on polymersegments with a protonated sidechains. Only the first two dihedral angles (a,b) actuallyrequire the protonated form of the sidechain as they include the hydrogen atom or itsadjacent oxygen. A backbone dihedral angle (f) is composed of any four backbone atoms,which can also be terminal or branch carbon atoms. The last dihedral angle (g) is theadditional one defined in this work.
35
where the outer sum is over the seven distinct types of dihedral angles shown in Fig. 2.1
and the inner sum is over all instances of the given type in the oligomer2. We choose to
optimize the set of parameters a,b, c so that the difference between any two oligomer
structures i and j is zero:
Eifit− Ej
fit−(Ei
tors− Ej
tors
)= ∆Ei,j
fit−∆Ei,j
tors≡ ∆Ei,j
fit−tors(2.3)
In the ideal case, where the ab initio torsion profiles match our force field, the ∆Ei,jfit−tors
must vanish. Hence, the absolute value (or the square) of ∆Ei,jfit−tors
is a measure of the
quality of the fit for the oligomer structures i and j. This allows us to define a penalty
function for the entire set as:
P (a,b, c) =Ns∑
i>j
(∆Ei,j
fit−tors
)2
(2.4)
As we do not limit i and j to belong to data points from the same torsion profile our fit
will ensure that the ab initio energy differences are evenly matched across all the profiles.
The form of our dihedral potential, Edih = a cos(bϕ − c) used in calculating Efit,
warrants further description. The constant term that is present in other definitions of
a dihedral potential [42] is redundant in our case owing to the fact that we are only
fitting to energy differences between molecules possessing the same type and number of
dihedral angles (i.e., different conformations of the same oligomer). Even if a constant
term was included in the potential it cannot improve the fit, as it will cancel in Eq. (2.3).
Finally, we note that the absence of such a term has no effect on the forces. When
the penalty function is at a minimum its derivatives (hereafter referred to as “forces”)
with respect to the parameters am, bm, and cm will be zero for all m = 1, 2, ..7. By
deriving analytical expressions for these forces that act on the parameters we can employ
an efficient minimization scheme for P , e.g. a steepest descent (SD) minimization [43].
2For example, there are two instances of the dihedral angle shown in Fig. 2.1g as there are two O4atoms.
36
The first derivative of the potential with respect to each of the unknown parameters is
straight forwardly obtained:
Fam = − ∂P
∂am= −2
Ns∑
i>j
∆Ei,jfit−tors
∂∆Ei,jfit
∂am= −2
Ns∑
i>j
∆Ei,jfit−tors
(∂Ei
fit
∂am−∂Ej
fit
∂am
)
= −2Nm∑
l=1
Ns∑
i>j
∆Ei,jfit−tors
(cos(bmφ
im,l − cm)− cos(bmφ
jm,l − cm)
)
Fbm = − ∂P∂bm
= 2Nm∑
l=1
Ns∑
i>j
∆Ei,jfit−tors
am(sin(bmϕ
im,l − cm)ϕim,l − sin(bmϕ
jm,l − cm)ϕjm,l
)
Fcm = − ∂P∂cm
= −2Nm∑
l=1
Ns∑
i>j
∆Ei,jfit−tors
am(sin(bmϕ
im,l − cm)− sin(bmϕ
jm,l − cm)
)(2.5)
Since an SD algorithm had already been developed in the context of our general MD
code, it was expedient to use this algorithm for the present purposes. The actual im-
plementation involved treating the 7×3 unknowns am, bm, and cm as the coordinates of
seven dummy atoms in 3D space. The forces that act on these atoms were calculated
by Eq. (2.5) with the “energy” of the system being the penalty function P in Eq. (2.4).
The lowest possible value for P along with the corresponding values of the dummy atom
coordinates (i.e., the force field parameters) were obtained at the completion of the min-
imization run and the latter are collected in Appendix A, Table A.3.
The most difficult torsion profile to accurately match was that around the C1O-
O3 bond [29] where the side chain attaches to the polymer backbone (see Fig. 2.1).
This profile has the highest rotational barrier being more than 38 kJ/mol and highly
asymmetric. The latter suggests significant contributions arising from Coulomb and
Lennard-Jones interactions resulting from the crowding of the atoms near the point of
attachment. A comparison of the ab initio results with our MM force field energies is
shown in Fig. 2.2 and indicates very good agreement in most regions of the PES. A
second torsion profile for rotation about the C2S-S6 bond is displayed in Fig. 2.3 and
likewise shows good correspondence between the ab initio data and the MM results.
37
Figure 2.2: Comparison of the ab initio and classic torsion profiles (with contributionsfrom all MM terms) around the C1O-O3 bond.
38
Figure 2.3: Comparison of the ab initio and classic torsion profiles (with contributionsfrom all MM terms) around the C2S-S6 bond.
39
H3
O44O
C2S:
H3O
!10
!5
0
5
10
aco
s(b!!
c)! k
Jm
ol!
1"
!200 !100 0 100 200
! [degrees]
Figure 2.4: Energy profile of the dihedral potential for the angle H3–O3H–S6–C2S: inthe SSC force field (left) and a Newman projection of the dihedral angle (right). Thesmall discontinuity of the potential at ±180 is due to the unconstrained optimization ofthe periodicity parameter in our force field (see Appendix A, Table A.3).
So far we have seen that the simplified dihedral potential a cos(bϕ−c) can be success-
fully parametarized to reproduce the relative energy of SSC conformers. When defining
the torsion energy in Eq. (2.1) we noted the fact that it can also be viewed as the residual
energy of the molecule after all other terms have been accounted for. Therefore, it is not
yet clear if we can assign any physical meaning to the periodicity and phase parameters
b and c, respectively, as we could normally do for torsion potentials [42]. With this in
mind let us now take a closer look at the periodicity and phase parameters shown in
Appendix A, Table A.3. If we plot the dihedral potential for the first angle in the table
that includes OH group of sulfonic acid (shown top left in Fig. 2.1) we get the energy
curve in Fig. 2.4. Two low energy conformations can be seen here at about −150 and
119 separated by a transition state at −16. The location of the minima is close to what
we can predict based solely on the electrostatic interactions. The hydrogen atom is most
strongly attracted to the lone pairs on the double bonded oxygens in the SO3H group.
40
From the Newman projection in Fig. 2.4 we can see that the closest approach to the oxy-
gen atoms takes place when the H3–O3H–S6–C2S: angle is ±120. This coincides well
with what is known from other systems where the energy profile of one fold periodicity
(i.e., for b = 1) is explained in terms of the orientational preference of dipole pairs [42].
Hence, even though our dihedral potentials have exclusively adsorbed the error of the
MM representation they still behave like the regular torsions.
One unexpected result is the value of the periodicity parameter b in the last torsion
profile in Appendix A, Table A.3. For the O4:–S6:–C2S:–C2O dihedral angle (shown
bottom right in Fig. 2.1) we get periodicity of about one half. Such periodicity does
not make sense if we think of the dihedral potential as a truncated Fourier expansion.
It also represents a perplexing case of a dihedral angle which has to be turned 4π be-
fore its energy repeats. The PES of this particular angle was not part of the ab initio
calculations of Paddison and Elliott [29] since for unionized side chains this dihedral is
redundant. Obviously the parameter values obtained here minimize the penalty function
P in Eq. (2.4), which does not include any periodicity considerations. However the one
half periodicity of the O4:–S6:–C2S:–C2O dihedral angle is never actually an issue. Since
ϕ in the potential expression a cos(bϕ− c) is always given in the range [−π, π] the longer
than usual period of this potential function never comes into play. This is not unusual.
The limited range of the angle is what allows us to model a bending angle potential, like
H–O–H in water, with a harmonic potential which is not periodic.
41
Chapter 3
Virial Formulation For Periodic Systems
Later, in this work we present the methodology needed to simulate polymeric structures
of infinite length by ensuring the continuity of the polymer chains across periodic bound-
aries. The rationale for using periodic boundaries in the first place is to minimize the
boundary effects experienced by the atoms near the cell surface. Furthermore, the prop-
erty that we want to obtain from our simulations is the proton diffusion coefficient, or in
other words, the distance that the protons have travelled, and PBC allow seamless move-
ment of all species. Evaluating the interactions in these periodic systems is not trivial
due to the infinite number of interacting pairs. Currently, there is a dichotomy in the
evaluation methods with the Lennard-Jones potentials treated with the cutoff plus Long
Range Correction (LRC) method and the electrostatic potential treated with the Ewald
summation or similar periodic methods [44]. Recently, a distinctively new approach was
introduced in the Isotropic Periodic Sum (IPS) method where a cutoff is combined with
spherical periodic images [45,46]. This method is simple to implement, and most impor-
tantly, valid for all interaction potentials. In this chapter we present the IPS method in
some detail and derive a more accurate expression for its virial1.
Cutoff methods enable the study of large systems by considering only a small, local
region where the interactions are computed exactly. Beyond the cutoff the interactions
are computed based on an approximate model which embodies some of the features of
the system. The LRC method, for example, uses the number density to construct a
structureless, infinite region. The IPS method goes one step further by using the density
along with the exact atom positions of the local region. Thus, in IPS the infinite region
1Here we use Clausius’ meaning of “virial of force” as a function of the forces in the system.
42
i Rc
Figure 3.1: Local region (solid sphere) replicated in a spherical shell. The image spheresfill up the entire volume of the 3D shell. A special arrangement of the images ensureszero forces at the local region boundary (see Ref. [45]).
is represented as local region replicas (referred to as images), arranged uniformly in
spherical shells, see Fig. 3.1. Exact, as well as approximate formulations are available
for general 1/rn and exponential potentials. The formalism developed in Ref. [45] can be
applied to other potentials as well, even those where the summation over the image shells
does not converge (e.g., 1/r). In the simpler case of 3D isotropic systems (i.e., periodic
in all three directions) the more recent paper [46] provides convenient fitting polynomials
that are straightforward to implement in a computer code. The work also extends the
IPS method with a discrete fast Fourier transform for heterogeneous systems.
A comprehensive comparison between the LRC and IPS methods applied to the
Lennard-Jones model can be found in Ref. [47]. The study has shown that the two
methods predict nearly identical results for both thermodynamic and transport proper-
ties when the cutoff is sufficiently long. One interesting exception was the pressure, where
for short cutoffs the results were substantially different and good agreement required a
notably longer cutoff. Conventionally, the choice of the cutoff radius Rc is based on some
43
Rc
L
Figure 3.2: If L becomes smaller than 2Rc the interactions in the primary cell, as wellas all its neighbors have to be evaluated explicitly.
system specific information (e.g., the σ parameter of the Lennard-Jones model) and com-
putational considerations (like the number of steps that can be computed in a given time
interval). The original version of the IPS method assumes that Rc is constant and not
explicitly dependent on the simulation cell dimensions. This assumption has important
repercussions for the virial and pressure regardless of the simulated ensemble. In NPT
simulations with constant Rc the number of particles in the local region will increase
upon compression of the simulation cell, resulting in unnecessarily detailed and long
calculations. If the dimensions of the simulation cell fall below 2Rc we must explicitly
consider the primary cell along with its neighbors, (see Fig. 3.2), a complication rarely
implemented in a computer code. On the other hand, an expansion of the simulation cell
combined with a constant Rc will result in loss of detail, potentially leading to a local re-
gion that is no longer representative of the whole system. However, it is entirely possible
to minimize such side effects resulting from constant cutoff NPT simulations by ensuring
a good initial guess for the density as well as small volume fluctuations. The bigger
44
question, explored in this chapter, is whether, for any simulated ensemble, a cutoff not
explicitly dependent on the simulation cell dimensions would produce the correct virial.
This chapter is organized as follows: We first look into the different contributions to the
potential energy in the IPS method and the modifications necessary for the boundary
term. We then introduce the virial and present two ways of calculating it, and finally
demonstrate the merits of the new virial formulation in two test cases.
The instantaneous potential energy of a system represented by two regions, local
and infinite, can be split into two terms. The energy contribution from the local region
Ulocal is equivalent for any cutoff method since it is calculated exactly. The energy of
the infinite region is labeled here as Uperiodic, to imply that it has been calculated with
the IPS method2. At the boundary between the two regions it is desirable to maintain
continuity of the energy, forces3, and virial in order to avoid pressure artifacts and improve
the numerical stability in MD simulations [44, 48]. Such side effects can be avoided by
shifting the potential at the boundary, which is an approach analogous to the well known
truncated and shifted potentials [44]. The shifting is corrected later by a restoring term.
These two artificial contributions to the energy are collected in Uboundary, giving a total
potential energy:
U = Ulocal + Uperiodic + Uboundary (3.1)
An illustration of the effect of the boundary term on the total energy fluctuations is
presented in Fig. 3.3. Since Uperiodic is calculated from replicas of the local region, it will
suffer from the same boundary truncation inherent in Ulocal. Therefore, the shifting and
restoring of the potentials that happens in Uboundary has to be carried out for both the
2A more explicit notation was used in Ref. [45]. For the potential designated there as εij (rij) we use1/rn. Our Ulocal corresponds to 1
2
∑rj∈Ωi
ε (rij) and Uperiodic to 12
∑rj∈Ωi
φ (rij ,Ωi).
3In the IPS method the forces with respect to atom distances −∂U/∂rij are continuous at theboundary by construction. There are also other forces we have to consider with respect to the dimensionsof the simulation cell and the cutoff radius in order to obtain a correct virial. Therefore the forcecontinuity built in the IPS method does not automatically lead to a virial continuity.
45
0 100 200 300 400 500step
604.8
604.6
604.4
604.2
604.0
603.8
603.6
H [kJ/m
ol]
Uboundary 0
Uboundary=0
Figure 3.3: Hamiltonian conservation in a short MD trajectory of a Lennard-Jonessystem. The curves show the total energy (i.e., kinetic and potential) with (solid line)and without (dashed line) boundary correction. Including the Uboundary term in thepotential energy results in smaller oscillations and a smoother curve.
46
local and the periodic energy terms. As the system virial includes contributions from
the boundary correction we will first examine the nature of Uboundary in more detail.
The notation and grouping of terms adopted here is slightly different4 from the original
work in order to facilitate the connection with the virial introduced later. However,
the concepts remain the same. The pair-wise sums, Ulocal and Uperiodic, by construction
include the ij-pairs where atom j is inside the local region of i, i.e., rij < Rc (where
rij are calculated after applying the minimum image convention (MIC) with respect to
atom i). For these pairs only, we evaluate the boundary energy after setting atom j at
the boundary of i, i.e., for rij = Rc. Thus, both the restoring and shifting components
of Uboundary can be expressed as:
N∑
i<j
[Ulocal (rij = Rc) + Uperiodic (rij = Rc)] (3.2)
Next, the restoring term will be approximated by removing the rij < Rc requirement.
We introduce U∀local and U∀periodic which represent the same potential functions, but with
extended domains. This implies that all j atoms are considered whether or not they
reside inside the local sphere of i. For the restoring energy we can then write:
Urestoring =4πR3
c
3L3
N∑
i<j
[U∀local (rij = Rc) + U∀periodic (rij = Rc)
](3.3)
The basis of this approximation is that the local region is representative of the whole
system, and the interactions (both local and periodic) of the atoms within the cutoff are
a fraction of the interactions in the system (no limits on rij being imposed). This fraction
is equal to the volume of the sphere divided by the cell volume. Effectively, the distance
dependence has been removed from the restoring term as all rij have been set equal to
Rc, regardless of their actual value. For the shifting term, on the other hand, we can
4In the original formulation the shifting term is grouped with the local and periodic energy terms togive, what the authors call configurational energy. Only the restoring term is referred to as the boundaryenergy Ebound. Thus, the definitions only differ in the grouping of the terms.
47
adopt a more convenient form, with the Heaviside step function used to emphasize the
shifting energy’s depends on rij:
Ushifting =N∑
i<j
[U∀local (rij = Rc) + U∀periodic (rij = Rc)
]Θ (Rc − rij) (3.4)
Finally, the boundary energy is calculated as Uboundary = Urestoring − Ushifting.
An important difference from the original IPS work lies in the atom pairs being con-
sidered in the boundary correction. There the electrostatic and Lennard-Jones restoring
terms (i.e., Ebound) are derived by including all self-pairs, and correspondingly, these pairs
are included in the shifting terms. However, there is no physical basis for considering
the self-pairs in Uboundary. The continuity that the latter brings is only relevant for one
central atom i and a different atom j, moving in and out of the local region of i5. In-
cluding the self-pairs in the boundary correction, however, results in a spurious energy of(
4πR3c
3L3 − 1) [U∀local (rii = Rc) + U∀periodic (rii = Rc)
]per atom, as seen from Eqs. (3.3) and
(3.4). In most practical cases all bonded interactions of atom i (such as bonds, angles and
dihedrals) will be entirely within its local region. Regardless of the type of the bonded
ij-interaction there is never a continuity issue if rij is always less than Rc. Likewise,
all intramolecular interactions in small molecules (like solvents) can be excluded from
Uboundary if these molecules would always fit inside the cutoff sphere. A demonstration
of the effects of an “all-pair” boundary correction will be presented later, following the
calculation of the virial. Since all components of the potential energy are now known we
proceed with the calculation of the virial and pressure.
The thermodynamic pressure is most conveniently derived in the NVT ensemble. A
precise derivation can be found for example in Ref. [49]. Here we follow the simpler
approach presented in Ref. [50] that allows us to focus on periodic systems. Thermody-
namic pressure is the volume derivative of the total energy (sum of potential and kinetic
5However, the self-pairs must be included in the calculation of Uperiodic in Eq. (3.1) in order for theimages to have the same charge and particle density as the local region.
48
energies) at constant temperature: − (dE/dV )T . The kinetic energy contribution to the
pressure is identical to ideal gas pressure resulting in:
PV = NkT − V⟨dU
dV
⟩(3.5)
As usual, for MD simulations, we will assume ergodicity which allows us to consider
instantaneous values for the energy, virial and pressure. The ensemble average of these
quantities can then be calculated from the instantaneous values averaged over the given
trajectory. Following Ref. [50] the negative of the instantaneous virial, V (dU/dV ), can
be written as:
VdU (r, L,Rc)
dV= V
(1
2
∑
i,j
∂U
∂rij
drijdL
+∂U
∂L+∂U
∂Rc
dRc
dL
)dL
dV
= V
(−1
2
∑
i,j
fijdrijdL
+∂U
∂L+∂U
∂Rc
dRc
dL
)1
3L2
=1
3
(−1
2
∑
i,j
fijLdrijdL
+∂U
∂LL+
∂U
∂Rc
LdRc
dL
)(3.6)
The derivation is valid for pairwise additive potentials, with rij calculated after applying
MIC (with respect to atom i). Assuming that the ij-atom distances, rij, scale linearly6
with the dimension of the simulation box L then according to Euler’s theorem we get
L (drij/dL) = rij. Similarly, (for convenience) we choose Rc to be linearly dependent on
L, we get L (dRc/dL) = Rc. This allows us to obtain the virial as a scalar product of the
forces and the arguments of the potential energy:
− V dU (r, L,Rc)
dV=
1
3
(1
2
∑
i,j
fijrij −∂U
∂LL− ∂U
∂Rc
Rc
)(3.7)
As explained in Ref. [50] the standard virial expression∑firi/3 is incomplete for
systems with PBC. An additional −L∂U/∂L term must be included in order to take
into account pressure contributions from changes in the image cell separations. However,
6This choice affects only the instantaneous pressure, but not the ensemble-averaged one. For anin-depth explanation please see Ref. [51].
49
in IPS it is not L that determines the image separations (as in rectangular cells) but
rather the diameter of the local sphere 2Rc. Therefore, a physically correct formulation
of the virial in the IPS method requires the presence of the extra −Rc∂U/∂Rc term. In
order to obtain this term we have to ensure that dRc/dL in Eq. (3.6) is nonzero and
Rc is linearly dependent on L. Accordingly, the cutoff must be an explicit, homogenous
function of the side of the simulation cell of order one, i.e., Rc = Rc (L). It is evident that
the cutoff contribution to the virial will also be found in any constant volume simulation
(i.e., NVE or NVT) since no difficulties are encountered in calculating the derivative of
Rc (L) for a fixed value of the argument. Let us note that the derivation of Eq. (3.7)
does not assume any specific form for U (r, L,Rc), including the periodic and boundary
contribution (if such is present). Consequently it is not just limited to the IPS method.
Examples of this can be found in Refs. [52, 53] where the authors have employed the
cutoff contribution to the virial in LRC, constant volume simulations. Before moving to
the practical illustrations of the theory we will consider whether there is a simpler way
of calculating the virial from Eq. (3.7).
To evaluate Uperiodic in IPS it is necessary to integrate over an angle θ that determines
the location of the image sphere in a given shell and then sum over all shells - from the
one adjacent to the local sphere (with index m = 1) to infinity. Thus, for a 1/rn potential
the IPS energy of one interacting pair in the m-th shell is determined from the integral:
1
2
π∫
0
sin θdθ
(r2 + 4m2R2c + 4mrRc cos θ)n/2
(3.8)
The denominator originates from applying the cosine theorem to the angle π − θ in
Fig. 3.4. The integral depends on both r and Rc, but the integration variable can be
changed to the reduced distances r/Rc obtaining, thereby, a factor of 1/Rnc outside the
integration. From the assumption that r is linearly dependent on L, and the requirement
that this be true for Rc as well, it follows that the IPS energy is homogenous in L of
50
i
2mRc
jr
j’r
m-th shell image
π − θ
θ
Figure 3.4: The distance between the central particle i and the image particle j′ in them-th shell is a function of the angle θ (determining the position of the image sphere inthe shell), the i − j distance r in the primary cell and the distance between the spherecenters 2mRc.
51
order −n. The result is valid for 3D isotropic systems regardless of any approximations
and fittings applied to the integral or the subsequent shell summation over m. The
same homogeneity in L is present in the local energy Ulocal, and can be proven for the
boundary correction as well7. Since all components of the potential energy of Eq. (3.1)
are homogenous in L with an order of −n (for a 1/rn potential) we can calculate the
virial as follows, (see Ref. [49]):
− V dU (r, L,Rc)
dV= −L3dU (r, L,Rc)
dL
dL
dV= −1
3LdU (r, L,Rc)
dL=n
3U (r, L,Rc) (3.9)
Any representation of the infinite region that ensures energy homogeneity equivalent to
that of the local region can make use of Eq. (3.9). Consequently, the above result is valid
for both the IPS and the LRC methods.
The benefit of the above equation over Eq. (3.7) is that only energy fractions are
needed instead of energy derivatives. Thus, a Monte Carlo simulation will not have to
resort to any force evaluations in order to obtain the virial. Moreover, the cutoff force
−∂Uperiodic/∂Rc in the IPS method is as laborious to calculate as the atom pair forces.
In a MD simulation, where the atom pair forces are required to propagate the system,
considerable effort can be saved by avoiding the calculation of the cutoff forces and
employing Eq. (3.9). This remarkably simple result is well known for the local energy,
while its validity for the total energy, including the complicated periodic and boundary
terms, is not obvious. Hence, allowing for a cutoff dependent on system dimensions, and
a more physically accurate picture, simplifies the calculation of the virial.
We now consider a simple analytical example, which is an adaptation of the one
appearing in Ref. [50] for rectangular simulation and image cells. Consider a physical
system containing a spherical region of radius Rc with a single particle located at its
7Since R3c is a homogenous function in L of order three, the volume fraction 4πR3
c/3L3 in Eq. (3.3)
will be zero order in L. The Heaviside step function in Eq. (3.4) has the property Θ (Lx) = Θ (x) forany positive L so it is also zero order in L. Therefore, the boundary correction will have the same orderas the corrected energy.
52
center, along with infinitely many copies distributed in spherical shells of the same density
(as in Fig. 3.1 with i being the only particle). For a 1/rn potential the virial due to the
single particle in the center can be evaluated directly by a summation over the shells.
Instead of integrating over the angle θ (see Fig. 3.4), we use the fact that there are
24m2 + 2 images in the m-th shell (see Ref. [45]), each of them at 2mRc from the center.
Assuming homogeneity of all atom positions in the physical system the Euler theorem
gives the virial as:
Wdirect =n
3U (r, Rc) =
n
3
∞∑
m=1
24m2 + 2
(2mRc)n =
n
3Uperiodic (r = 0, Rc) (3.10)
In the end result Uperiodic (r = 0, Rc) corresponds to the IPS periodic energy calculated
with r set equal to zero. On the other hand, with the original formulation of the virial
we would get:
Woriginal =1
3
N∑
i=1
rifi + Ebound = 0 +4πR3
c
3L3[Ulocal (r = Rc) + Uperiodic (r = Rc, Rc)]
=4πR3
c
3L3
[1
Rnc
+ Uperiodic (r = Rc, Rc)
](3.11)
The only remaining terms above are those coming from the boundary correction. As
we have emphasized before, the boundary correction in the original paper includes the
self-pairs, which when combined with a system dimensions independent cutoff produce
the unexpected result of Eq. (3.11). Let us now see if our modifications can recover the
exact virial. With Eq. (3.9) the IPS virial for the center particle becomes:
Wnew =n
3U (r, L,Rc) =
n
3[Uperiodic (r = 0, Rc) + Uboundary]
=n
3Uperiodic (r = 0, Rc) (3.12)
All boundary terms vanish since our formulation excludes the self-pairs from the bound-
ary correction. We see that this result obtained from an IPS periodic energy calculation
is equivalent to the direct summation result in Eq. (3.10).
53
Table 3.1: Virial comparison for a 256 particle Lennard-Jones model. In the secondcolumn use is made of the virial and boundary terms from the original IPS formulation[45], the third column utilizes the formulation described here, i.e., Eq. (3.9) with self-pairs excluded from the boundary term. Fourth column shows the virials obtained withthe LRC method [53]. All three simulations are constant volume simulations (canonical,canonical and microcanonical, respectively) hence Rc remains constant. The densitycorresponds to the liquid region of a supercritical isotherm with 2.5 times the criticaltemperature. All numerical values are in reduced units.
parameters Woriginala Wnew WLRC
b
ρ = 0.8 T = 3 Rc = 3.4 2108±1c 2079±1 2085
a Simulation details: Our IPS implementation is based on the fitting polynomials appearing inRef. [46]. Starting from random initial positions each system was optimized for one hundred stepswith the SD method [43], followed by one hundred steps of Adopted Basis Newton-Raphson (ABNR)optimization [54]. Random velocities with zero net momentum were assigned corresponding to thetarget temperature of 3. Each system was equilibrated for 2000 steps with the generalized Nose-Hoover method [55] using a thermostat period of 0.0625 and a time step of 0.003. The NVTsimulation was continued in the production run for another 4×105 steps.
b The virial has been calculated as PV − (N − 1) kT using the values from Table IV in Ref. [53].c Lower bounds on the uncertainties were estimated with the method described in Ref. [56].
For our next example we examine a many particle Lennard-Jones model in the state
displayed in Table 3.1. The IPS virial calculated with the new formulation is only about a
quarter of a percent different from the LRC value, whereas with the original formulation
the difference is nearly four times larger (i.e., a 1% difference). This result indicates that
the IPS and LRC virials (and pressures) are actually much closer than had been previously
seen for a state of similar density [47]. Most importantly, the excellent agreement has
been obtained for the relatively short cutoff of 3.4σ. In the LRC method the assumption
is that the pair-correlation function g(r) is unity beyond the cutoff, hence the probability
of finding an atom in the thin spherical layer between R and R + dR is determined by
the number density alone. The IPS method operates on a similar, coarse-grained version
of g (r) = 1, applied to the image spheres, instead of the individual atoms. Thus, in
every spherical shell between (2m− 1)Rc and (2m+ 1)Rc, one finds the exact number
of image spheres (i.e., 24m2 + 2) dictated by the density. This close similarity between
54
the key assumptions of the two methods results in nearly identical virials when they are
calculated in a consistent manner. The IPS method has the noteworthy advantage of
handling electrostatic potentials, allowing a simple, universal way of modelling all types
of interaction in the infinite region.
To summarize, we have seen that in a case where the virial can be calculated exactly
the original formulation of the boundary energy and virial fail. Atom pairs that, due
to their connectivity, remain within the cutoff should not be included in the boundary
correction. In simulations where the system dimensions change the cutoff radius should
be scaled accordingly. Additionally, a −Rc∂U/∂Rc virial term is required to account
for pressure contributions from the image separations. This extra term is necessary
even when the system dimensions are constant. The complete virial can be obtained
conveniently, as a fraction of the potential energy avoiding the calculation of derivatives.
The new virial formulation in the IPS method considerably improves the agreement with
LRC results even for short cutoffs.
55
Chapter 4
Just-in-Time Empirical Valence Bond Method
The computational approach adopted in my work is MM where the motion of the atoms is
treated classically (i.e., using Newton’s laws). Instead of using wave functions and solving
Schrodinger’s equation MM employs fixed force fields where forces are computed from
a mixture of bonding and non-bonding potentials (e.g., bond stretching, angle bend-
ing, dihedral torsion, electrostatic and Lennard-Jones potentials). The success of this
approach relies on good parametrization of these potentials, as discussed in Chapter 2.
Even though the wave function formalism is entirely absent in the classical treatment,
electron density effects such as the polarization can be readily incorporated in the clas-
sical framework. However, force fields are rarely developed to reproduce ab initio data
alone. Physical constants of the substances can also be taken into account. A good
force field for water, for example, would be expected to closely predict the correct dipole
moment, dimer geometry, energy, density, diffusion coefficient etc. In the case of the
hydronium ion mobility, however, MM does not provide the means for describing proton
hopping via Grotthuss-type mechanism, sketched out in Fig. 4.1.
In pure water the mobility of the proton is determined almost entirely by the rate
of this proton hopping [58]. If the estimates are based solely on the vehicular mode of
proton transport even the qualitative predictions of the diffusion constant for different
polymers can be incorrect. In this chapter we examine some of the methods used to
incorporate proton hopping in classical simulations and describe a new approach that we
have developed, referred to as the JIT-EVB method.
The general mechanism of proton hopping was proposed by Grotthuss over 200 years
ago [57]. With the advent of computer simulation and ab initio MD simulation, in par-
56
(which correspond to heterolytically dissociating and reformingindividual H2O molecules) in interchange with the making andbreaking of the associated hydrogen bonds. This process,which involves shifting protons along hydrogen bonds, occursspontaneously at 300 K as a result of the favorable energy scaleinvolved in thermal fluctuations, as qualitatively explained inSection 2.1. The renowned scholar from Leipzig, Theodor Chris-tian Johann Dietrich von Grotthuss (1785–1822) obviously hadsomething like that in mind when he published in 1806: “It isclear that in the whole operation the molecules of water, situ-ated at the extremities of the conductor wires, will alone bedecomposed, whereas all those placed intermediately willchange reciprocally and alternatively their component princi-ples without changing their nature.” and “… all the moleculesof the liquid situated in this circle would be decomposed andinstantly recomposed …” in order to explain electrolysis ofwater in Volta’s Galvanic cell (cited from the English transla-tion[66b] of the original publication in French,[66a] see Figure 5for a facsimile of its front page). In order to illustrate his idea,Grotthuss added two schematic sketches where he drew linearwater wires that connect the cathode with the anode part ofsuch cells (see Figure 6 and its caption for an explanation). This
perception thus led to the alternative designation “Grotthussdiffusion” for the concept of structural diffusion and to the in-troduction of the term “Grotthuss wires/bridges” for such one-dimensional hydrogen-bonded water wires.
Although the paradigm of Grotthuss diffusion has alreadybeen around for 200 years, the Grotthuss mechanism as suchwas unclear until fairly recently. There was, however, thenotion of preferred solvation structures of hydrated protons inthe literature, in particular the complexes proposed by Eigenand collaborators,[67,68] H3O
+ ·ACHTUNGTRENNUNG(H2O)3, on the one hand, and byZundel and co-workers,[69,70] [H2O···H···OH2]
+ , on the other. Inthe latter complex, the proton is shared equally between twowater molecules via an ultrashort, centered hydrogen bond,whereas in the former a hydronium core is solvated by accept-ing three hydrogen-bonded water molecules according toFigure 7. Traditionally, these complexes have been lookedupon as being mutually exclusive, that is, the presence of oneof them rules out the presence of the other, in the proposedexplanation of the nature of the hydrated proton and the Grot-thuss mechanism.[71]
This is in a sense remarkable, since it has been known for along time[26] that small changes in the donor–acceptor distancecan easily induce shifts of the proton along the respective hy-drogen bond, as discussed in relation to Figure 3, and thus in-
Figure 5. Frontispiece of the Grotthuss publication from 1806 in the periodi-cal Annales de Chimie.[66a]
Figure 6. Reproduction of the page containing Figures I and II of the pam-phlet printed 1805 in Rome.[66c] Oxygen, o, and hydrogen, h, are representedby ! and " signs, respectively, and water is represented by neighboring!" pairs, that is, by oh. Note that the distinction between atoms and mole-cules is not clear in the text : “… the molecule of water represented by o,h…” whereas “… the molecules of oxygen situated in …” (cited according tothe English translation).[66b] The water molecules are arranged along a linearchain such as to form a wire (called fil in the French original)[66a] and theelectrolytic decomposition of liquid water manifests itself according to Fig-ure I by giving the rightmost oxygen away to the cathode (marked by a +sign) thus breaking apart a water molecule. The left-over hydrogen, in turn,combines itself with the oxygen of its left water neighbor, the hydrogen ofwhich forms with the oxygen (marked r) of its left neighbor a new watermolecule and so forth. The reverse process happens at the anode (markedby a # sign): hydrogen Q is taken up by the anode, thus breaking apartwater molecule QP, so that oxygen P can form a new water molecule to-gether with hydrogen X from its right neighbor and so forth. Clearly, a con-tinuous wire of water molecules connects in Figure I the cathode with theanode, which allows for a continuous process of breaking and making ofwater molecules along the chain (compare with the modern version inFigure 4). Figure II is an extension of this idea to two coupled Galvanic cells.
ChemPhysChem 2006, 7, 1848 – 1870 ! 2006 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim www.chemphyschem.org 1853
Proton Transfer 200 Years after von Grotthuss
Figure 4.1: Reproduction of the original idea of Grotthuss for proton shuttling betweentwo electrodes [57]. Hydrogen (h) and oxygen (o) are represented by ⊕ and signs,respectively, and water is represented by neighbouring ⊕ pairs, i.e., by oh. The watermolecules are arranged along a linear chain such as to form a wire and the electrolyticdecomposition of liquid water manifests itself by giving the rightmost oxygen away tothe cathode (marked by a + sign), thus breaking apart a water molecule. The leftoverhydrogen, in turn, combines itself with the oxygen of its left water neighbour and soforth.
ticular, we now have a clearer understanding of the events that lead to the translocation
of the excess charge in water [60, 18]. The traditionally postulated solvation species,
namely the Eigen ion (H9O+4 ) and the Zundel ion (H5O
+2 ) are shown to be transient and
mobile due to the interactions with their own solvation shells. Statistical analysis of the
processes leading up to the proton transfer (PT) event have demonstrated a “special pair
dance” in which a distorted Eigen cation constantly changes its closest oxygen atom as
shown in Fig. 4.2. This “dance” represents repeated trials to locate a suitable partner
for the PT act and occurs, on the average, every 40 fs. This resembles the description of
PT events in the early ab inito MD reports, except that partner exchange takes place on
the femtosecond scale, whereas it takes several picoseconds for a PT to take place. The
outcome of the “special pair dance” is equivalent to diffusional rotation of the H3O+,
except that no rotational motion is involved.
Earlier studies of the PES of the translocation process have demonstrated that the
barrier resides in the initial dissociation of a water molecule from the second solvation
shell of the hydronium ion. Once this event occurs the translocation of the proton through
57
forming a SP,20,21 whereas another HB is longer than average.42
This widens the first peak in the RDF but not sufficiently to
split it into a double peak, as expected for the Zundel cation.The latter is a more severe distortion of the SP which occurspredominantly close to a PT event.
The distorted Eigen cation is not static. Rather, its closest(O1x) ligand constantly changes its identity within the firstsolvation-shell. This “SP-dance”, depicted schematically inFigure 13, represents repeated trials to locate a suitablepartner for the PT act. Before the exchange, the probabilityof having an A1 HB to O1y diminishes, whereas after theexchange it increases (Figure 8). This resembles the descrip-tion of PT events in the early AIMD reports,20,21 except thatpartner exchange takes place, on the average, every 40 fs,whereas it takes several picoseconds for a PT to occur.Consequently, we observe in Figure 3 several special-pairswitches between the three first-shell ligands during the longsegments of a “resting” hydronium.
Participation of A1 cleavage in the faster partner exchangeevents (rather than in the slower PT) is in line with previouscalculations showing that A1 HBs are much weaker than typicalwater-water HBs,48 apparently due to the unfavorable interac-tion of a water hydrogen with the positive center. For this reason,the frequency of partner switches depends more weakly ontemperature than PT, so we expect the importance of the SP-dance to diminish at higher temperatures (we have verified thisqualitatively for the qMS-EVB3 potential). Likewise, simula-tions with faster PT (such as the MS-EVB2 and qMS-EVB3)exhibit less of an SP-dance.
The outcome of the SP-dance is equivalent to diffusionalrotation of the H3O+, as suggested by Huckel,12 except thatno rotational motion is involved. This phenomenon rapidlyrandomizes the proton hop direction, so that proton mobilityin water is diffusive rather than coherent (i.e., it does notinvolve correlated hopping over long HB water chains aspreviously anticipated).14 Indeed, time-resolved fluorescencemeasurements from molecules exhibiting excited-state PT to
Figure 13. Schematic depiction of the “special pair dance” occurring in the first solvation-shell of the hydronium in its “resting state” (during thelong trajectory segments of no-PT). Oxygens in red, except for the hydronium oxygen in magenta.
Figure 12. Same as Figure 11 for a quantal MS-EVB3 trajectory. SeeFigures S7-S9 in Supporting Information for the other water models.
9464 J. Phys. Chem. B, Vol. 112, No. 31, 2008 Markovitch et al.
Figure 4.2: Schematic depiction of the “special pair dance” occurring in the firstsolvation-shell of the hydronium during the long trajectory segments without PT events.Oxygens are shown in red, except for the hydronium oxygen in magenta. Figure source:Ref. [60].
58
© 1999 Macmillan Magazines Ltd
2
8.0
5.6
3.2
0.8
–1.6
8.0
5.6
3.2
0.8
–1.6–4.0 –4.0
2.2
2.5
2.8 –1 0 1
2.2
2.5
2.8 –1 0 1
a b
P
Roo (Å)
δ (Å) δ (Å)
2
<
.
h
2 i
Figure 4.3: Probability distribution of the proton as a function of the Oa −Ob distanceand the asymmetric stretch coordinate δ = ROaH −RObH . Both electrons and nuclei aretreated by QM in a (using path integrals), while in b the nuclei are treated classically. Thecolouring scheme in a represents the coordination number (see the figure source Ref. [62]for more details) decreasing from about 4 (yellow) in the Eigen ion regions to about 3.5(blue) in the Zundel ion region. The quantum fluctuations of the nuclei obliterate thedistinction between the peaks seen in b, pointing to a smooth and virtually barrierlessinter-conversion between the Eigen and Zundel ions.
the Zundel ion is fast, with a barrier which is less than the available thermal energy at
room temperature [62]. Quantum fluctuations further stabilize the Zundel ion leading to
a continuous proton distribution along the reaction path as seen in Fig. 4.3.
As a consequence of a practically non-existent barrier, proton shuttling is not a rare
event and should be observed by routine MD simulations (i.e., such simulations that do
not artificially enhance the probability of some events). Whether a PT will be observed or
not depends indeed on how accurately the interactions are calculated along the reaction
path. The issue poses a serious problem for MM simulations as the interaction potentials
are fitted to equilibrium structures (i.e., hydronium ion and water) rather than along
reaction paths. In this regard it is instructive to compare the ab initio proton distribution
functions seen above with that from a MM simulation presented in Fig. 4.4. Even though
59
RO
O
A
δA
Figure 4.4: Scatter plot of the proton distribution in a Zundel ion (represented classicallyas hydronium ion plus water) at 300 K. The distribution is plotted as a function of theOa − Ob distance and the asymmetric stretch coordinate δ = ROaH − RObH . Bondstretching in the hydronium ion is modelled by a Morse potential.
60
the hydronium ion was modelled by a Morse potential which facilitates the OaH bond
dissociation δ values above -0.4 A or oxygen atom separations below 2.5 A were never
observed. Thus, the areas of the plot that would correspond to a symmetrical Zundel ion
structure (i.e., δ = 0 and ROO = 2.38 A) remain empty [63]. The region that the MM
potentials effectively sample corresponds to only the lower left corner of the ab initio
distribution (see the contour plots in Fig. 4.3).
Unfortunately, it is currently not feasible to carry out high quality ab initio calcula-
tions that reliably represent the hydrogen bond on nanoscale structures like the pore in
the hydrated SSC polymer. This is particularly true when long trajectories are required
as is the case when modelling a dynamic process like the diffusion of a proton or water
molecules. In the not too distant past even much smaller systems were beyond the scope
of ab initio calculations thus leading to the development of alternative, classical methods
to model the proton hopping. As will be explained later, the classical approach can offer
another unique advantage that goes beyond the increase in length and time scales.
In order for a proton translocation to occur in classical simulations it will be neces-
sary to manually alter the identity of the species by updating their connectivity matrix,
interaction potentials and parameters. Ideally this switch should take place when δ ≈ 0
to ensure energy and force continuity. However, as we have seen above, the common MM
potentials strongly disfavour δ values near zero. A reactive MD simulation approach cur-
rently under development [64] employs a set of geometrical triggers to identify structures
that are likely to lead to a PT if the molecules move on the correct ab initio energy
surface. When such a structure occurs during a MD run the proton is transferred to the
accepting oxygen atom and the interaction potentials are switched accordingly. Since
this does not occur at δ ≈ 0 the discontinuity in energy and forces requires a short equi-
libration so that the transfer process remains thermoneutral1. The success of such an
1Both the reactant and product of the PT process are Zundel ions, hence the process is thermoneutral.
61
O H
H
H
O
H
H
O H
H
H
O
H
Ha b
O H
H
H
O
H
H
O H
H
H
O
H
H
Figure 4.5: A Zundel ion (represented classically as hydronium ion plus water) canexist in two resonance forms, a and b, obtained by interconversion between a covalentOH bond (solid line) and a hydrogen bond (dashed line). Note the two structures haveidentical geometries.
approach depends on the careful selection of the trigger values so that the reaction rates
can be reproduced.
The interconversion between the hydronium ion and water need not be as abrupt, as
in the case of the reactive MD method. Instead of occurring at some particular value of
δ we can imagine a gradual transition between the identities of the two species. At any
instance in time the interactions in the Zundel ion can be represented by a superposition
of two resonance forms, displayed in Fig. 4.5. The mixing between the a and b characters
of the classical Zundel ion can be calibrated such that its overall properties are similar
to that of an ab initio Zundel ion. The original EVB formulation developed thirty years
ago [65] determines the mixing coefficients from the ground state solution of the secular
equation: ∣∣∣∣∣∣∣
Ha − E Hab
Hab Hb − E
∣∣∣∣∣∣∣= 0 (4.1)
The potential energy of resonance forms a and b, Ha and Hb respectively, are calculated
classically using valence bond theory (i.e., as sum of stretching and bending potentials
plus non-bonding Lennard-Jones and Coulombic potentials). When the ground state
energy E is known one can solve for the empirical off-diagonal term Hab and vice versa.
Thus, the ab initio PES of the Zundel ion needs to be obtained first in order to determine
62
the appropriate value for Hab. Once this is done the properties (e.g., the location of
the positive charge) of any Zundel ion occurring in our simulation can be obtained by
summing the coefficients of the resonance forms.
Quite early in the development it was realized that a single, global value for the
empirical parameter Hab does not lead to a good agreement between the ab initio and
EVB PES. It is possible to overcome this problem if the Hab term is determined on the
basis of the current reaction coordinates ROO and δ. As is the case with any classical
approach, it is often desirable to reproduce experimentally determined properties, such
as the proton diffusion coefficient. This can be achieved by adjusting the parameters of
the intermolecular interaction potentials (such as the charge and the Lennard-Jones pa-
rameters ε and σ) which affect the diagonal terms Ha and Hb in the secular determinant.
The EVB method has currently evolved to include features such as more resonance forms
that include the solvating water molecules [66], solvent polarization [67], self consistency
in multiproton systems [68], nuclear quantum effects [69, 70] and so on. Another closely
related method that has gained considerable popularity is the Multi-Configuration MM
method [71].
Despite these advancements modelling the structural diffusion of proton in the SSC
polymer with the EVB method remains challenging. There is nothing, inherently limiting
the applicability of EVB to PT between a hydronium ion and water only. The side
chains of the polymer carry the strong sulfonic acid group −SO3H which are rendered
deprotonated in the presence of water. Typically, the ab initio PES can be determined
from a 10x10 grid in the ROO, δ plane. However, in the highly concentrated, proton rich
medium present in the hydrated polymer there are not only different pairs of donors and
acceptors but also coupled PTs as shown in Fig. 4.6. The increase in the number of
resonance forms means a higher order EVB matrix and more off-diagonal terms to fit.
Determination of the ab initio PES must now be carried out in a grid of 10x10x10x10
63
a b
c d
O
OS
OS
O
O
O
H
O
H
HH
OH H
O
OS
OS
O
O
O
H
O
H
HH
OH H
O
OS
OS
O
O
O
H
O
H
HH
OH H
O
OS
OS
O
O
O
H
O
H
HH
OH H
Figure 4.6: PT between two sulfonate groups influenced by a neighbouring Zundel ion(see Ref. [72] for more details) in a triflic acid monohydrate solid. Here we see the fourpossible resonance forms of the process obtained by interchanging covalent bonds (solidlines) with hydrogen bonds (dashed lines).
64
points. Coupled three-PTs would be impossible to parametrize since a grid with 1×106
points is now required. Hence, computing the EVB matrices of a SSC system would
require more calculations than a direct ab initio MD run. In the next few paragraphs
we shall introduce the JIT-EVB method demonstrating how a significant reduction in
the number of grid points can be achieved. Firstly, we investigate the possibility of
developing a way of simplifying the manner in which EVB mixes the resonance forms.
The solution of the secular equation in the EVB method yields both the ground state
energy and the corresponding mixing coefficients for the resonance forms. Alternatively,
we could directly fit these coefficients such that the desired ab initio energy (or forces)
are reproduced. For instance, in the case of two resonance forms a and b the classical
energy could be calculated as caEa + cbEb, this time ca and cb being the unknown empir-
ical parameters. Just like the off-diagonal terms in EVB are dependent on the reaction
coordinates ROO, δ so are the mixing coefficients ca and cb. The convenience such sim-
plification brings becomes apparent for higher order EVB matrices where the number
of unknown off-diagonal terms starts to exceed the number of the resonance forms. We
should not, however, be concerned about such an approach causing a deterioration of the
final result. Since, even though the EVB mixing might be more sophisticated it is not
more accurate.
The success of the EVB method in reproducing the ab initio PES is entirely due
to the reasonable interaction potentials and the extensive parametrization, both in the
diagonal and off-diagonal terms. Since the diffusion coefficient is a dynamic property
obtainable by MD, our primary concern is reproducing the ab initio forces, rather than
the energy. For this reason it is preferable to fit the parameters ca and cb such that
the total force caFa + cbFb matches the ab initio force2. It is important to address a
2In our method each resonance form coefficient is calculated as a product of two polynomial functions
of the reaction coordinates ROO and δ, e.g., c′i =4∑j=0
sjRjOO
4∑j=0
tj (1− δ/ROO)j . The primed coefficients
65
commonly held misconception about the forces in the EVB method regarding the assumed
“orthonormality” of the resonance states. Such an assumption allows the forces to be
calculated through the Hellmann-Feynman theorem. However, as explained above, the
false consequences of even the most serious assumption (i.e., the correspondence between
the classical potentials in the resonance forms and the QM states) can be mitigated by
extensive parametrization. The problem is that the parametrization that reproduces the
ab initio energies along the grid is not the same one that will reproduce the forces.
Borrowing the idea of geometrical triggers from the reactive MD method, in JIT-
EVB we use these triggers to switch from the unique, classical potentials of the reacting
species to a linear combination of such potentials in the resonance forms. In other
words, we do not automatically transfer the proton but merely allow for this to occur
gradually. Since the conformations outside the trigger zone are considered unreactive, no
grid points are required there, see Fig. 4.7. However, the fact that a certain conformation
(or a grid point) belongs to the reactive zone does not automatically mean that such a
conformation will ever be encountered during simulation. We can reduce the number
of grid points even further by excluding such areas of the reactive zone that have low
probability of occurring. To see how this can be achieved let us examine Fig. 4.8. With
our EVB method the ab initio PES and the fitting are performed on the fly, just as soon
as a molecule requires it. Hence the name of the method “Just-in-Time” EVB. Once
a grid point has been evaluated by an ab initio calculation and the fitting obtained the
information is saved to a database. Subsequently, whenever a system finds itself in the
vicinity of the same grid point the information is looked up from the database without
the need to perform another ab initio calculation. With the JIT-EVB method a system
where coupled PT events can occur can be studied immediately, without the prerequisite
are then squared and normalized such that ci are always positive and correspond to the probability offinding resonance form i. Hence, the fitting parameters are actually the polynomial coefficients s and t.For more details, please refer to Appendix D.
66
Figure 4.7: Molecules whose conformations are inside the reactive trigger zone (greenboundary) will be subject to JIT-EVB mixing of the resonance forms to reproduce theab initio forces along the grid. Outside of the reactive zone a single resonance form issufficient, hence, JIT-EVB is not invoked.
of tens of thousands of grid point evaluations.
Finally for this chapter we will address an issue that is of paramount importance in
classical proton hopping simulations. Pure ab inito methods can handle PBC through
periodic wave functions. Here it does not matter if the proton donor and the proton
acceptor belong to the same periodic cell as the MIC always applies. In a classical
simulation, however, this is not the case. Let us consider Fig. 4.5. If the acceptor oxygen
is in a different periodic cell then resonance form b (which depicts a hydronium ion on
the left) will be misleading – the new covalent OH bond will actually span across periodic
cells. Thus, resonance form b will have a very high potential energy. In this situation the
standard EVB method which uses the ground state solution of Eq. (4.1) will invariably
lean toward resonance form a. It is not surprising then that all EVB results reported to
date underpredict the proton diffusion coefficient. This flaw of the classical simulation
is not limited to EVB. Even in a simple QM/MM approach once a molecule leaves
the (periodic) QM zone there will be an immediate jump in the potential energy and
67
A
B
A
Figure 4.8: At point A a molecule enters the trigger zone (top). An ab initio calculationhas to be performed to determine the correct mixing of the resonance forms. As withthe regular EVB method, the fitting obtained at this point is considered valid withinsome neighbourhood determined by the grid spacing. The system is propagated until themolecule either exits the trigger zone or drifts outside the validity radius. When the latterhappens (say at point B) a new ab initio calculation and parametrization are carried out(bottom) and so on. The molecule explores the PES using the fitting parameters of itsclosest grid point - A, B etc. Beyond the trigger zone boundary the molecule returns toa unique classical representation in the resonance form with the highest contribution.
forces, unless all its hydrogen atoms are in the same periodic cell as their oxygen anchor.
For this reason QM/MM studies of proton transport in bulk water have been unable to
incorporate PT events [73].
The problem is not unique to the OH stretching potentials. Both bending and dihe-
dral potentials that involve mobile hydrogen atoms can also yield spurious results. The
solution is quite simple by introducing what we call MIC bonding potentials which we
describe in Chapter 6. Application of the JIT-EVB method to water and triflic acid is
discussed in Chapter 8.
Part II
Polymer System Studies
68
69
Chapter 5
Constructing The Polymer Systems
It is evident that an experimental chemist wishing to study the properties of a new com-
pound first needs to obtain a sample of it. If computational chemistry does, indeed,
provide the tools and methods to explore the properties of compounds then a given
compound of interest must be a well defined chemical entity that can be modelled. Con-
sequently, one would expect to know the formula of the compound, its structure and
conformation in order to uniquely predict its properties. For the hydrated polymer sys-
tems of my study the exact formula, structure and conformation are unknown. The goal
of this chapter is to present the methodology we have developed to create a “sample” of
the SSC polymer that will be later subjected to computational testing. In the next few
paragraphs we proceed toward this goal starting from determining the polymer formula,
then building its chemical structure and finally generating plausible 3D conformations.
The SSC polymer is a copolymer of tetrafluoroethylene and a sulfonyl fluoride vinyl
ether with the general formula CF3 [(CF2CF2)n − CF2CF (OCF2CF2SO3H)]N CF3. The
synthesis of the polymer does not allow perfect control of the monomer ratio in the
polymer chain, resulting in a lack of a well defined value for n. However, a typical MW
of the polymer unit (i.e., its EW) is 830± 20g/mol [74], corresponding to the presence of
between five and six tetrafluoroethylene groups in the backbone for every sulfonic acid
group. Hence, a reliable model of the SSC polymer needs to incorporate some degree of
randomness in generating the molecular formula and structure.
With the advent of computer databases new chemical notations have emerged that
make it possible to uniquely represent and digitize chemical structures. In the mid 60s
H. L. Morgan introduced the Chemical Abstract’s registration system that assigns an al-
70
Table 5.1: Examples of random polymer sequences generated with different seeds.
seed na N Polymerb
1 5 2 TTSTSSTTTTTT2 5 2 TSTTTTTTTTTT3 6 3 TTTTTTSTSTSTTTTTSTTTT
a The general formula of the polymer is CF3 [(CF2CF2)n − CF2CF (OCF2CF2SO3H)]NCF3.
b The shorthands T and S are used in place of tetrafluoroethylene and sulfonyl fluoride vinyl ether,respectively
phanumeric sequence to chemical structures [75]. Another notations system that emerged
nearly twenty years later is the SMILES system [76]. It has gained considerable popular-
ity due to its simple notation rules. For example, tetrafluoroethylene is represented by
the string sequence FC(F)=C(F)F, with brackets used to indicate side chains attached on
the left. If we know the name of the compound or its CAS registry number the SMILES
representation can be conveniently obtained using a simple on-line converter [77]. An-
other useful on-line tool incorporates a graphical structure editor that directly generates
the SMILES sequence [78]. With the help of the SMILES system one can reduce the
problem of randomness in the polymer formula to randomness of a string. The algorithm
we use to determine the SMILES sequence of a random polymer is presented in the flow
chart in Fig. 5.1. Even though randomness is a favourable feature in the monomer se-
quence the final result should be reproducible. This is accomplished by specifying a seed
for the random number generator. When the program is executed with the same seed it
will produce the same random sequence of monomers regardless of the hardware or the
operating system it runs on. Some examples are shown in Table 5.1.
The algorithm we have presented here for the generation of random comb copolymers
can be extended to higher levels of branching by manipulating the SMILES representation
directly. For example, introducing a branch in the polymer chain can be carried out
by a simple replacement of one atom by the SMILES formula of the branch. Such
71
iterate over the N polymer units
input the chemical
formula as n, N
number > 1/(n+1)?Yes No
iterate over the n+1 monomer
units
generate a random number between 0
and 1
SMILES = 'C(F)(F)F-'
'-C(F)(F)-C(F)(F)' appended to SMILES
'-C(F)(F)-C(F)(O-C(F)(F)-C(F)(F)-S(=O)(=O)O)' appended to SMILES
'-C(F)(F)F' appended to SMILES
input the random
seed
Figure 5.1: Flowchart representing the creation of a SSC polymer with random monomersequence and an average formula of CF3 [(CF2CF2)nCF2CF (OCF2CF2SO3H)]N CF3.The flowchart is traversed starting from the top (always traversing the right sub-flowchartbefore proceeding downward). The style used is a variation of the one described inRef. [79].
72
manipulations are easily scriptable in any high-level language like Python, or can be
carried out in a text editor.
Having determined the polymer formula and represented its structure in the SMILES
notation it is now necessary to translate this information into a set of bonds, angles and
dihedrals which collectively constitute what is known as the connectivity matrix. This
information is used by the interaction evaluators in our code to calculate the energy,
forces and virial of the system. Generating the connectivity matrix is done from the
SMILES formula with the help of the OpenBabel [80] chemical expert system.
The hardest part, however, lies in the last step of the building process i.e., obtaining
the 3D conformation of the polymer. The nature of the hydrated polymer systems poses
an immense challenge for the prediction of their conformation, perhaps even eclipsing
those encountered with proteins. Determining the conformation from X-ray measure-
ments is not possible as only a small part of the polymer is crystalline. In fact, unlike
proteins in their native state, the hydrated polymers are characterized as non-equilibrium,
“living systems” [81]. It is useful to estimate the size of the conformational phase space
for our polymer systems. The MW of Nafion has been estimated to be in the range
of 105 − 106g/mol [10]. Assuming a similar range for the SSC polymer and a typical
EW of 850 g/mol the smallest number of polymer units N is found to be 118. In each
polymer unit we have n tetrafluoroethylene monomers and one side chain monomer with
a total of 2(n + 1) backbone dihedral angles. A simple estimate of the number of con-
formations of the entire polymer can be done if we assume that each dihedral angle has
a limited number of preferred conformations (e.g. trans, gauche plus and minus). Thus,
we get the astronomical number of about 10788 conformers1 for even the shortest SSC
structure. This enormous conformational phase space cannot be traversed by any com-
1If each of the 2(n+ 1)N dihedral angles in the polymer backbone has c possible conformations thenthe total number of conformations for the polymer would be c2(n+1)N . Taking c = 3, n = 6 and N =118 we get approximately 10788.
73
Figure 5.2: Perfluorocyclohexane converging to two distinct local minima dependingon the conformation seed. For seed=1 a twisted chair structure is obtained (left). Forseed=5 an ideal chair structure emerges whose energy is about 12.7 kJ/mol lower (right).
putational means in the time that the universe has existed! More importantly, even the
real polymer would never be able to sample the entire phase space. Consequently, the
polymer will explore only a small part of the entire phase space volume, limited to some
neighbourhood of the initial configuration. This implies that for any sufficiently long
polymer the 3D conformation will depend on the initial conditions. Here, again, it is
desirable to have reproducibility of the conformations obtained by the polymer builder,
which can be achieved by introducing another random seed. In fact, even short systems
like perfluorinated cyclohexane already settle into different local minima, displayed in
Fig. 5.2. In summary, the random monomer sequence issue is a separate one from the
issue of the random 3D conformation of the polymer. For a given monomer sequence we
can generate a large number of conformations by specifying unique initial conditions for
the 3D conformation search. Thus, an appropriate estimation of the properties of the
polymer system should necessarily include a statistical average over both the monomer
sequences (in random polymers) and the initial conditions.
As demonstrated above, the enormity of the size of the phase space severely limits the
74
usefulness of statistical averaging over randomly picked conformations. For this reason
the 3D conformation search algorithm we have developed aims at building and folding
the growing polymer chain as it may occur during the polymer synthesis. It is based on
the following rules:
• A newly added group of atoms to the growing end of the polymer (e.g., CF2, SO3
etc.) will have time to find its energy minimum before the next addition takes
place.
• The part of the polymer already built is much larger than the newly added group,
so the interactions with the latter will not lead to conformational changes in the
rest of the polymer.
• In the case of steric clashes the entire polymer will relax its structure but only to
the point of avoiding the particular clash allowing a new addition to take place.
The algorithm we use to determine the 3D conformation of the polymer according to the
above rules is presented in Fig. 5.3. The general idea is to split the optimization of the
whole polymer into a sequence of short optimizations performed only on few of atoms at
a time. These optimizations are fast, not because the rest of polymer is kept frozen, but
due to the reduced number of interactions being considered, see Fig. 5.4. Our MD code
was specifically developed to accommodate the interaction set switching that takes place
in the polymer building process.
A distinct advantage of our polymer builder is that the desired polymer can be built
directly into the periodic simulation box. Even though a polymer chain can span many
simulation cells it always experiences the interactions of the periodic images and folds
accordingly at every addition step. This is in stark contrast to the technique that is
currently employed where a polymer chain is built in an infinite simulation box first and
then put into a periodic box. The latter technique complicates the folding of the polymer
75
energy below threshold?
No
iterate over all dihedral bundles
bundle all dihedrals with the same central bond
find the new atoms in the bundle (with yet unknown
coordinates)
set the interaction calculators to consider only (new atoms, new atoms +
built atoms) interaction pairs
run a short SD optimization to find the new atom
coordinates, built atoms are frozen
set the interaction calculators to consider only
(built atoms, built atoms) interaction pairs
add the new atoms to the list of built atoms
run a short SD optimization to find the built atom
coordinates
run a final SD/ABNR optimization of the whole
polymer
input the conformation random seed
Figure 5.3: Flowchart representing the stepwise process of finding a low energy con-formation for the polymer. The bundling of atoms by dihedral angles ensures that ateach addition step the correct local conformation is obtained. The optimization of thecoordinates of the new atoms starts from randomly chosen values controlled by the con-formation seed. At the end of the stepwise process the polymer structure is furtheroptimized until its gradient has been sufficiently reduced allowing an MD simulation tocommence.
76
F
C
F
O
F
F
F
F
A
B
Figure 5.4: The interactions needed to be considered in order to optimize the coordinatesof the new atoms (set A) are those within the set and with the built atoms (set B). Onlyin the case of steric clashes it is necessary to relax the entire built polymer, in which caseall interactions are considered (i.e., including those within set B).
enormously due to the sheer size of the conformational phase space. One solution that
alleviates the problem is to use multiple copies of a single, much shorter polymer chain,
which can be equilibrated more rapidly. However, the resulting increase in polymer chain
mobility in this segmentation approach may affect the transport properties of both water
and hydronium ions.
Finally, the polymer system needs to be hydrated in order for any proton transport
to take place. We know, a priori, the amount of water needed from the λ parameter
(i.e., the number of water molecules per sulfonic acid group). The way we achieve the
hydration is by superimposing the built polymer onto a simulation box filled with water
molecules. The size of the water box coincides with the size of the box in which the
polymer was built and is determined by the density of the desired system. The potential
energy of each water molecule, in the box, is calculated by taking into account both the
water-water and water-polymer interactions. The water molecules that overlap with the
77
polymer will possess very high potential energy due to the repulsive term in the Lennard-
Jones potential. To reduce the number of the water molecules to the desired hydration
level the excess water molecules are removed, starting with those with the highest energy.
Once the correct composition of the hydrated polymer has been achieved we carry out
two cycles of optimization. In the first one the coordinates of the water molecules are
optimized, while the polymer structure remains frozen. This is done in order to avoid
rupturing the polymer. In the second optimization cycle the entire system is allowed to
move.
So far we have seen the approach adopted for the building of 3D conformations that
correspond to a specific local minimum determined by the seed of the conformation search
algorithm. In the next chapter we show how we can simulate polymer systems with a
specific morphology.
78
Chapter 6
Designer Structures
The methods described in the previous chapter were aimed at building polymer systems
whose morphology was uniquely determined by the monomer sequences and the initial
conditions. The most interesting morphologies are seen in a class of polymers known
as block copolymers where each chain contains long, covalently bonded homopolymer
units. Such polymers are able to spontaneously form ordered phases with nanometer
scale structures [82]. The large variety of shapes accessible to the copolymer opens
the door to designing membranes with unique morphologies and potentially desirable
properties. Theoretical studies have already addressed the effect of blockiness of Nafion
on the observed morphology and properties [25]. Blocky structures with high degree of
clustering of the same monomers were found to exhibit larger size hydrophilic domains
and better proton conductivity.
As useful as it is, the bottom up approach that relates monomer sequences to mor-
phology and properties has a flip side. For instance, it would be a daunting task to
isolate the effect of a single feature, like the depth of the side chain protrusion into the
pore on the proton diffusion rate. Numerous polymer systems will have to be tested,
with their side chain protrusion depths averaged over the production runs in the hope
that statistically significant differences can be observed. In order to avoid a complicated
cross-correlation analysis, the selected systems must only differ in the side chain depths
and maintain equivalence amongst all other features, i.e. they should posses the same
pore radius, side chain density and so on. In this chapter we will introduce a different
approach that allows us to work directly at the monomer morphology level.
Previously, a kinetic model was developed based on a non-equilibrium statistical me-
79
chanical approach [19, 20, 83, 21] for direct determination of the effects of geometrical
parameters of the membrane pores on the diffusion of hydronium ions. This top down
approach relating membrane morphology to measurable properties may be applied to
any hydrated polymer electrolyte system. The only system-specific parameters used in
the model are the radius of the pore, the protrusion of the anionic side chains into the
pore and their distribution within the pore. These studies revealed that a larger pore
radius, shorter side chains and more uniform side chain distribution will lead to higher
proton diffusion coefficients [20].
An alternative approach was presented by Spohr et al. [81,84,85] in which the polymer
system is divided into structureless pore walls, represented by a simple volume exclusion
potential and the explicit side-chain fragments, treated with an all atom potential. Ex-
cluding the motion of a large part of the polymer has allowed simulation times of tens of
nanoseconds revealing slow conformational changes in the side-chain dihedral angles.
By the very nature of these models the picture for the proton transport will be
severely altered by changes of the interaction potentials. Therefore, our goal here will be
to preserve all the interactions in the real system, without simplifying any of the bonding
or non-bonding potentials. Thus, the effect of the polymer on the proton transport
inside the pores will be brought about by the actual backbone and side chains. The
only modification we will introduce will be an external potential whose function is to
restrain the polymer backbone to the desired shape of the pore. A comparison with the
traditional restraining scheme is depicted in Fig. 6.1. As we will find out later in the
chapter, once the polymer system has been built the shape restraining potential can be
turned off, in some special cases.
We begin by looking at a simple example of a polymer with a straight backbone. As
this is the preferred conformation arising from the low energy anti conformation of the
backbone dihedral angles, such a restraint may at first appear to be redundant. However,
80
water phase
polymer wall
shaperestraint
water phase
shaperestraint
Figure 6.1: Cross section view of an ideal pore with the shape restraint represented asan outer layer. Traditionally the pore shape is enforced by restraints acting directly onthe water phase, where the proton transport occurs (left). In our scheme the restrainingpotential acts only on the polymer backbone leaving the side chains and the water phaseshielded from the external field (right).
any sufficiently long strand of the polymer will collapse onto itself as a consequence of
the action of the Lennard-Jones and Coulombic forces. To prevent such a collapse and,
furthermore, to orient the backbone along a specific axis we introduce a restraining
potential proportional to r2, where r is the distance from the axis in the normal plane.
Because of the zig-zag conformation of the tetrahedral carbon atoms the restraint is
applied only to every alternate carbon atom, see Fig. 6.2. Despite the presence of the
straight backbone restraint which prevents each strand from twisting and intertwining
with itself, we can, by placing other strands in a parallel fashion ensure that the favourable
inter-strand interactions still persist. Therefore, we build bundles of straight polymer
chains, analogous to the ones constituting the membrane pores [17].
In principle, it is possible to apply the restraining potential of Fig. 6.2 individually
to each strand in the bundle. However, this particular form of restraint gives rise to
some computational artefacts. Due to the distance r being defined with respect to the x
coordinate axis, the restraining potential acts as an external field. In the presence of such
fields (be it restraining, electric, or gravitational) the system is no longer invariant with
81
Figure 6.2: Perfluoropentane built with a straight backbone along an axis. A harmonicrestraining potential of the form kr2 (where r is the distance from the x-axis in theyz-plane) is applied to the bottom three carbon atoms (shown in green).
respect to translations and rotations. Therefore, neither the linear momentum nor the
angular momentum will be conserved [86]. These complications become apparent when
the system requires temperature or pressure control1. Therefore, it is highly desirable
to use restraints that conserve the total linear momentum of the system. One way of
achieving this objective, for the straight backbone polymer, is to restrain the end-to-end
distance in the polymer chain. Such a restraint corresponds to a new bonding potential
being defined between the end atoms. However, as with any bonding potential, the
orientation of the participating atoms is undefined. Hence the end-to-end restraint will
produce a straight polymer with an arbitrary orientation in space, see Fig. 6.3. The
problem is that in order to create the bundles all polymer strands need to have one
consistent orientation, regardless of what that orientation may be. With the end-to-end
restraint we cannot easily enforce the consistency in orientation between the different
1Only the relative momenta of the individual atoms with respect to the center of mass (CM) shouldbe thermostated [55]. Similarly, the ideal gas component of pressure which is proportional to the kineticenergy of the atoms should exclude the CM contribution [87]. Such methods are well known but notwidely implemented in computer code due to the non-symplectic coordinate transformations that needto take place. Furthermore, calculating transport properties also becomes more complicated in suchsystem as explained in Ref. [88].
82
Figure 6.3: Single perfluoropentane chain built in a periodic cell with a harmonic bondrestraint between the end carbon atoms (shown in green). Also shown is one periodicimage in the +x direction to emphasize the lack of continuity across periodic boundaries(see the text). The conformation deviates slightly from the ideally straight one due tothe non-bonding interactions with the periodic images.
83
Figure 6.4: 2D channels formed by two polymer chains (black and green) in a periodiccell. Routine simulations will exhibit features whose length is bound by the dimensionsof the cell (left). However, when the polymer chains are continuous across the periodicboundaries the pores they form will be infinite (right). In the latter case, the absence ofend effects may permit smaller dimensions for the periodic cell.
strands without resorting back to using external potentials.
The size of the simulated systems should be as large as the methods and the time
constraints allow. As we are interested in the proton transport occurring in the membrane
pores we can achieve long, realistic size pores in two cases. The first and obvious one is
when the simulation cell itself is large. However, the more appealing one is when there is
continuity at the boundary, such that an infinite pore can be obtained. These two case
are shown in Fig. 6.4. Unfortunately, the end-to-end restraint does not automatically
lead to boundary continuity. To conclude, the end-to-end restraint is a step in the right
direction as it removes the external field and conserves the linear momentum. But we
still need to go even further in order to control the orientation of the polymer strands.
When calculating the non-bonding interactions in a periodic system we know that
we first have to apply MIC, then calculate the distances r, and finally, calculate the
corresponding 1/rn potential. On the other hand, for bonding potentials multiples of the
84
periodic cell dimensions (say Lx, Ly and Lz) are not subtracted before obtaining r. The
only reason it is done this way is because of the assumption that the bond lengths are
always much smaller than the cell’s dimensions, hence MIC is automatically satisfied.
Nevertheless, let us see what happens when we apply the convention to the familiar
example of Fig. 5.2 with the molecule (perfluorocyclohexane) placed in a periodic cell.
The result is shown in Fig. 6.5. Amazingly, the two types of structures depicted have
the exact same connectivity matrix. Applying MIC for the bonding potentials allows for
bonds, angles and dihedrals2 to be calculated with the image atoms, across the periodic
boundary. We have already seen the need for such potentials when proton hopping is
involved. Here it is possible to obtain an infinite, straight polymer chain (bottom) that is
significantly more stable than the cyclic structure (top). The infinite polymers that can
be generated in this manner will have a straight conformation only when the dimension
of the periodic cell is commensurate with the straight chain length. Looking more closely
at the bottom conformation we note that:
• Since the straight polymer conformation is enforced by the dimensions of the pe-
riodic cell simulations where the dimension can change (e.g., NPT ensemble) can
disrupt any structure modelled with the MIC bonding potentials.
• Once the initial bias to connect with the image atoms has been removed, the
simulation can proceed as a normal, unbiased and unrestrained one that conserves
the total linear momentum.
• The orientation of the straight polymer is uniquely specified by setting one dimen-
sion (e.g., Lx) to be commensurate with the straight chain length. Thus, consistent
orientation will be found in bundles of the same chain.
2Since all angles are calculated from bond distances using the cosine theorem, the MIC bending anddihedral potentials are the same potentials but with all distances involved calculated after applying MIC.
85
Figure 6.5: Perfluorocyclohexane built in a periodic cell with MIC applied to bothbonding and non-bonding potentials. Closest periodic images are also shown. The samerandom seed and optimization procedure was used in the top and bottom cases, i.e.brute force SD optimization followed by an ABNR minimization. In the bottom case wehave picked two linked carbon atoms (shown in green), set their coordinates to equal theend-to-end distance in a straight C6 chain and kept these two atoms frozen during theSD stage. All atoms were allowed to move during the ABNR stage. Not only does thestraight conformation persist, it is also about 114 kJ/mol lower in energy than the cyclicconformation shown at the top.
86
a
b
c
F
F
a
C
C
F
F
C
F
b
F
F
F
C
Figure 6.6: Two perfluorohexane chains built in an infinite simulation universe withoutany restraints. Their closest segments are highlighted in green. The two segments areslightly offset vertically, with the carbon atom a being near the midpoint of the b − cbond. The distances are 4.91 A between the ab atoms and 5.13 A between the ac atoms.Shown on top are the approximate Newman projections revealing the presence of thegauche conformation in both segments.
• Pores formed by the infinite polymers (whether straight or not) will exhibit bound-
ary continuity (as in Fig. 6.4, right), allowing unimpeded proton transport across
periodic boundaries.
Simulating structures formed by polymer bundles requires knowledge of the effective
width of the polymer chains. In this paragraph we will try to determine this width.
Additionally this will allow us to establish the link between the polymer formula, the pore
size and the hydration level. Let us examine the conformation of two perfluorohexane
chains as they approach one another, as depicted in the bottom of Fig. 6.6. As we
87
Figure 6.7: Schematic representation of the approach of two polymer strands in the anticonformation. When these are straight perfluorocyclohexane chains, in a periodic cell(built with MIC potentials as in Fig. 6.5), the separation between the chains is about6.24 A.
can see from the Newman projections the preferred conformation is, sadly, not the anti
conformation. The two chains are interlocked with the less bulky fluorine side, while
the carbon atoms are on the opposite side. The gauche conformation that these strands
adopt enable the close approach of one fluorine atom from one strand to the opposing
carbon-carbon bond (e.g., the fluorine atom at a to the b− c bond). On the other
hand, all carbon-carbon bonds in a purely anti conformation strand will recede behind
the fluorines, as illustrated in Fig. 6.7. The crowding that is present here makes the
interlocking more difficult. Indeed, two perfectly straight chains can be expected to be
significantly farther from each other.
The motivation for this chapter was to develop the methodology that will link specific
geometrical factors (e.g., pore radius) to phenomena that occur inside the pore, like
proton diffusion. Unfortunately, we do not know how to maintain a perfect cylindrical
pore (with an unambiguous pore radius) using realistic, twisted polymer chains. Since the
pore walls only provide the boundary conditions for water and hydronium ion movement
in the pore, we could neglect the details of the intra-wall interactions. Thus, we will
assume that whether the pore walls are composed of twisted or straight polymer chains,
88
L
a
b
Figure 6.8: Schematic representation of a pore wall as formed by straight polymer chains(shown as columns). In the vertical direction the MIC potentials can be used to enablepore continuity across the periodic boundaries. To control the lateral movements of thechains an additional end-to-end restraints can be applied, i.e. between carbon atoms aand b.
their effect on the water phase will be the same. However, this requires us to hold the
straight chains closer together so to achieve the same inter-chain separation as it would
be found with twisted chains. This can be easily accomplished by employing end-to-end
type restraints (similar to the ones in Fig. 6.3) across the straight chains that form the
pore walls, Fig. 6.8. The pore walls we construct in this manner are not impermeable to
small molecules like water and hydronium ion. Placing additional bare polymer chains
(i.e., without sulfonic acid groups) in the four corners of the simulation cell will provide a
realistic model for the pore surroundings where any small, polar species will be naturally
excluded.
Finally, in this chapter, we establish the connection between the repeat unit of the
polymer, the pore radius and the hydration level. The volume of the pore can be found
from the volume of the water phase and the volume of the polymer forming the pore
89
walls:
Vpore = Vwaterphase
+ Vpolymer (6.1)
Let us label the pore radius R and choose the pore length L such that it contains
only one repeat unit (i.e., one sulfonic acid group) with the formula (CF2CF2)n −
CF2CF (OCF2CF2SO3H). The value of L can be calculated as L = 2(n + 1)lc where lc
is the chain length per backbone carbon atom. Thus, for a cylindrical pore we obtain:
Vpore = πR22(n+ 1)lc (6.2)
If the pore walls are comprised of m polymer chains distributed along the circumference
of the pore and pointing inward then the total number of sulfonic acid groups in the pore
will also be m. With the hydration level λ defined as the number of water molecules per
sulfonic acid group we arrive at mλ as the total number of water molecules. Accordingly,
using VH2O as the volume of a single water molecule we can write:
Vwaterphase
= mλVH2O (6.3)
The volume of the polymer can be found from the number of chains m, the volume of
a single polymer chain Vchain and a factor f (R,w) which represents the cross section of
the chain that is inside the pore:
Vpolymer = mVchainf (R,w) +mV sidechain
= mπ (w/2)2 Lf (R,w) +mV sidechain
= mπ (w/2)2 2(n+ 1)lcf (R,w) +mV sidechain
(6.4)
In estimating Vchain in the above equation we have assumed a cylindrical shape of radius
w/2 and length L. In our model w was the effective width of a bare (i.e., without side
chains) polytetrafluoroethylene molecule. Accordingly, the volume of the side chain has
to be explicitly included in Eq. (6.4). As the side chains are expected to be entirely
inside the pore, their volume is not subject to the f factoring. The form of f (R,w)
90
O
A
B
2α
R w
2
w
2
|OA| = |OB| = R
|AB| = w
AOB = 2α = 2π/m
sin α =w
2R
∴ m = π/ arcsin w
2R
Figure 6.9: Cross section view of the pore and two polymer chains.
can be obtained from the intersection of the two circles defining the pore and polymer
chain cylinders. The details are given in Appendix B. For brevity here we will keep the
shorthand notation f (R,w). When the last three equations are substituted into Eq. (6.1)
we get the following relation:
πR22(n+ 1)lc = mλVH2O +mπ (w/2)2 2(n+ 1)lcf (R,w) +mV sidechain
(6.5)
The number of polymer chains m along the circumference of the pore can be linked to R
and w as shown in Fig. 6.9. After substitution of the expression for m in Eq. (6.5)
and some rearranging we finally arrive at the following transcendental equation for
R(n, λ, lc, w, VH2O, V side
chain
):
R2 arcsin( w
2R
)− π (w/2)2 f (R,w) =
λVH2O + V sidechain
2(n+ 1)lc(6.6)
The only free variables in this expression are n and λ representing the polymer formula
and the hydration level. Numerical solutions for a number of cases are listed in Table 6.1.
Not surprisingly, increasing the hydration level swells the pores leading to larger pore
91
Table 6.1: Pore radius as a function of the repeat unit formula (CF2CF2)n − CF2CF −(OCF2CF2SO3H) and the hydration level λ, calculated by numerically solving Eq. (6.6).
n EW [g/mol] λ R [nm] a
3 578 3 1.326 1.6813 2.52
5 778 3 1.006 1.2413 1.80
7 978 3 0.836 1.0213 1.44
a The following parameter values were used: lc = 0.131 nm, w = 0.5 nm, VH2O = 0.0312 nm3,V sidechain
= 0.156 nm3. The chain length per backbone atom lc is calculated as the projection of a
carbon-carbon bond along the cylinder axis, i.e. lc = r0 sin (θ0/2), with the equilibrium values forthe bond length and angle taken from Tables A.1 and A.2 in Appendix A. The estimate for theeffective width of the polymer chain cylinder w is based on the values we have seen in Fig. 6.6. AMonte Carlo method was used to calculate the volumes of a water molecule VH2O and the side chainV sidechain
as explained in Appendix C.
radii. On the other hand, increasing the EW results in longer pores with extra volume
to fill along their axes. As the water molecules fill up this space the pores get thinner.
The numerical solutions we have obtained here for a SSC polymer with n = 7 are slightly
below the pore diameters found in a comparable Nafion system [17]. This is indeed
expected as Nafion has a longer side chain that requires more volume. With Eq. (6.6)
it is now possible to predict the effect of different factors on the pore radius in any
applicable polymer system. In particular, it allows us to use the appropriate values for
the dimensions of the pore and simulation cell when modelling the diffusion of species.
92
Chapter 7
Morphology of Hydrated SSC Polymer Systems
Previously we introduced the idea of designer systems where the morphology of the poly-
mer was controlled by some external restraint. The employed restraints were geometrical
in nature and they acted on the polymer backbone. However, the currently available syn-
thetic methods allow only variations in the EW and hydration level of the SSC polymer.
Thus, it is of great practical interest to establish the effect of these parameters on the
morphology and transport properties of the polymer. In this chapter we examine the local
morphology of a hydrated SSC polymer as a function of the hydration level λ. Given that
our modelling is based on a force field developed specifically for the SSC polymer, the
morphology discussed below is expected to reflect that of the real polymer system. For the
system studied here the EW was set at 578 g/mol corresponding to three tetrafluoroethy-
lene units and a general formula CF3 [(CF2CF2)3 − CF2CF (OCF2CF2SO3H)]40CF3.
The complete simulation details are given below.
Simulation details: All atom simulations of the above polymer strand were carried out at three
hydration levels of 3, 6 and 13 water molecules per sulfonic acid group. The simulation procedure begins
with building the polymer with a random conformation in a 3D periodic box, as described in Chapter 5.
The size of the simulation box is determined by the desired density of 1.67 g/cm3 for the hydrated
polymer systems, a value that was taken from Nafion [25]. The non-bonding interactions are calculated
with the IPS method [45,46] with the cut off of the Coulomb and Lennard-Jones interactions at 1.5 nm.
All optimizations performed in this chapter (unless otherwise stated) consist of first carrying out a SD
minimization [43] followed by an ABNR minimization [54]. After the polymer structure was optimized
it was introduced into a simulation box filled with flexible three center (F3C) water molecules [89]. The
interaction energy between each water molecule and the polymer was then calculated and all but the
93
required 40λ water molecules removed, beginning with the H2O molecules possessing the highest energy
(i.e., those that overlap with the polymer). This procedure results in three hydrated polymer systems
with 120, 240 and 520 water molecules and a total of 1768, 2128 and 2968 atoms, respectively. The
water phase is first optimized (SD only) keeping the polymer frozen, followed by optimization of the
whole system (polymer and water). All sulfonic acid protons are then transferred to their second closest
neighbour water molecule producing solvent separated hydronium ions and the sulfonate terminated
side chains. Following proton dissociation, the atoms in the side chain are renamed according to our
scheme (see Fig. A.1), and the corresponding new potentials are used henceforth. The system is again
optimized in stages first the hydronium ions (SD only, keeping water and polymer frozen), then water
and hydronium ions (SD only, keeping the polymer frozen) and finally, the whole system. The optimized
hydrated polymer systems are then equilibrated at 300 K for up to 450 ps using an NVT simulation.
The NVT run is stopped when the predicted change in the potential energy is less than 1 % per ns. All
data is collected from a 2 ns NVE production run with data points saved every 200 steps. The time
step in all MD simulations is 1 fs.
Snapshots of typical morphologies at the three different water contents (λ = 3, 6, 13)
are shown in Fig. 7.1. Cursory examination of Fig. 7.1 shows the increasing distinctions
in the distribution of the water and hydronium ions through the polymer as the water
content is increased; an observation which is consistent with studies of Devanathan et
al. in Nafion membranes over a similar hydration range [32]. The water appears to only
form isolated clusters or domains with the sulfonate groups and hydronium ions at the
lowest hydration level (λ = 3). However, connectivity between these domains begins
to emerge at λ = 6 and might be considered as an appearance of channels. There is
also the emergence of a separation of the hydronium ions from the sulfonate groups by
several water molecules. At the highest hydration level (λ = 13) these channels appear to
permeate the morphology of the system. The sulfonate groups and hydronium ions are
found at much greater average separations from each other. However, contact ion pairs
94
Figure 7.1: Polymer morphology snapshots at the end of the production run (2 ns)for the three levels of hydration: a) λ = 3, b) λ = 6, c) λ = 13. Each system shownrepresents a 3 A deep cross section of a 3 × 3 supercell so that the continuity can beobserved in the x and y directions. The three supercells are drawn to scale. Hydroniumion oxygen atoms (blue) and sulfur atoms (yellow) are emphasized. The other atomsseen are oxygen (red), hydrogen (white), carbon (light blue) and fluorine (green).
95
Figure 7.2: Hydrogen bond chain from the snapshot in Fig. 7.1a (λ = 3) using MIC.The sulfonic acid group oxygens have been left out. The numbers on the sulfur atomsrepresent the index of the side chain (1 through 40). Colouring scheme as in Fig. 7.1.
between one sulfonate group and two hydronium ions are still visible in this snapshot
(Fig. 7.1c).
Careful examination of the central section of Fig. 7.1a reveals the presence of a hydro-
gen bond chain that spans about 1/3 of the supercell. This particular chain is composed
of the following species: (going clockwise) H2O − H3O − SO3 − H3O − H2O − SO3 −
H3O − H2O − SO3 − H3O − SO3 − H3O − H2O, which is shown in Fig. 7.2. The side
chain index numbers (on the sulfur atoms) indicate that in this particular snapshot the
sulfonate groups are not from neighbouring side chains but are 2 side chains apart in
the case of the 1-4 hydronium/water bridge, and 5 side chains apart in the case of the
31-37 hydronium ion bridge. One should note that the distances between the sulfur
atoms shown in Fig. 7.2 are measured after applying MIC. These distances are also used
in the pair-correlation plots presented later. However, since the polymer spans several
simulation cells the separation of the same sulfonate groups in the actual polymer is
quite different. The single polymeric fragment of Fig. 7.1a is shown in Fig. 7.3 without
96
Figure 7.3: Polymer chain of Fig. 7.1a (λ = 3) without MIC. Water and hydronium ionsare left out for clarity. The sulfur atoms from the hydrogen bond bridges in Fig. 7.2 areemphasized.
applying MIC and with both the water molecules and hydronium ions left out for clarity.
This rendition of the macromolecule reveals much greater separation between the sulfur
atoms (e.g. 22.7 A for 1-4 and 25.6 A for 31-37). Furthermore it is evident that the poly-
mer backbone consists of a few almost perfectly straight segments. This preference in
the conformation of the backbone with the CF2 groups forming a helical pitch is similar
to that determined in the ab initio calculations of Paddison and Elliott for a two unit
oligomer [26] and a three unit oligomer [30] and this result in our classical simulations is
similar despite the polymer being placed in a periodic simulation box. The periodicity
of the model probably allows for the formation of hydrogen bond bridges while avoiding
any significant bending of the backbone. Similarly, the real polymer may organize itself
into straight bands with different threads held together by intermolecular hydrogen bond
bridges.
97
Figure 7.4: S-S pair correlation plot (between the ionized sulfonic group sulfur atoms)for the three hydration levels.
Further quantitative information concerning molecular interactions in the hydrated
SSC polymer systems is obtained in examining the RDF, denoted as g(R) . The in-
teractions of the sulfonate groups are quantified through the sulfur-sulfur g(R) and are
plotted in Fig. 7.4 for all three hydration levels. These pair correlation plots are derived
from trajectory snapshots taken every 200 steps during the entire production run. One
striking feature in this data is the intense peak for λ = 3 at approximately 4.5 A. This
separation between the sulfur atoms is significantly smaller than the one found in the
case of only a single bridging hydronium ion (see Fig. 7.2), and is actually very close
to the separation between a sulfur atom and a hydronium ion in a contact ion pair (see
Fig. 7.6 and the discussion below). It is also somewhat smaller than observed in the
simulations of Devanathan et al. [32] with 1134 EW Nafion where at a similar degree
of hydration they observed a broader peak with a shoulder at 4.6 A and a maximum at
98
Figure 7.5: An ion cage that exhibits very short S-S distances. The rest of the side chainatoms in the polymer have been omitted for clarity. The two lower hydronium ions arealso fully coordinated with three sulfonic acid groups (only two are shown).
about 5.2 A. Scanning the trajectories in this part of the plot revealed a close packing
structure that is shown in a wire-frame rendition of the atoms in Fig. 7.5 (similar colour
convention as before).
The hydronium ions in these structures are found in ion cages with all three hydrogen
atoms hydrogen-bonded to the oxygen atoms in three different sulfonate groups. There
is no water to mitigate the strong ionic interactions in such clusters and this results in
very small separations of the sulfur atoms. Since the hydronium ions in these cages are
immobilized to a large extent it is insightful to know what fraction of the hydronium
ions exist in such a state and its dependence on the degree of hydration. This analysis
is presented later on in the chapter. The intensity of the 4.5 A peak greatly diminishes
for λ = 6 broadening into two peaks at around 5.0 and 7.5 A with a probability of
about one. This may be viewed as being entirely determined by the number density of
the sulfur atoms and suggests there is little interaction between the sulfonates at this
99
water content. The peak completely disappears at the highest hydration level with the
majority of the sulfonate groups seemingly very well separated by over 10 A, the average
separation of the tertiary carbons (i.e., CFO) when the backbone carbon atoms are in an
anti conformation [26]. Another type of contact ion pair between the sulfonate groups
and the hydronium ions are those where the ionic cages are more labile - either due to
the incorporation of water or to missing hydronium ions. This is the case of the hydrogen
bond bridges in Fig. 7.2. The observed S-S separation in this type of bridges gives rise to
the broad peak near 7.5 A that is visible in the pair-correlation plot for both λ = 3, 6. Not
surprisingly, these peaks of the labile ionic cages also disappear in the highest hydration
case.
As we have seen from the S-S pair correlation plot with an increase in hydration the
ion cages release some of the trapped hydronium ions. Another effect is the increase
in the spatial separation between the hydronium ion and the sulfonate groups once the
cages break down. This can be seen from the S-O pair correlation plot in Fig. 7.6. The
plot for λ = 3 exhibits a sharp peak for the contact ion pair just below 4 A. A shoulder
on the left side of the peak is also visible that can be attributed to the fully “sulfonated”
hydronium ions in the ion cages. The typical S-O separation there is between 3.58 and
4.0 A. Another broad peak for the more hydrated systems is visible at about 6 A that
corresponds to the solvent separated species sulfonic acid group/water/hydronium ion.
As the ion cages disappear with the increase in λ, the intensity of the first peak goes
down and it becomes more Gaussian. At the same time it becomes more likely to find
the hydronium ions away from the sulfonate groups.
Now we turn our attention to a way of quantifying these observations. What we are
interested in is the mobility of the hydronium ions which is directly linked to the resilience
of the ion cages. As we have seen above any defects in the ion cage composition lead
to much greater separations between the hydronium ion and the sulfonic acid group
100
Figure 7.6: S-O pair correlation plot (between the ionized sulfonic acid sulfur atom andthe hydronium ion oxygen atom) for each of the three hydration levels.
101
Figure 7.7: Fraction of hydronium ions with a given number of sulfonate neighbours inthe SSC polymer for different hydration levels λ. The numbers have been averaged fromthe production run trajectory (2 ns).
making the hydronium ion more effective as a charge carrier in the polymer membrane.
In Fig. 7.7 we show a histogram of the number of sulfonate groups present in the first
solvation shell (4.3 A) of a hydronium ion. When the S-O separation between the two
species is within this cut off the sulfonic acid is considered a neighbour of the hydronium
ion. The histogram is drawn in a way so that a direct comparison can be made with the
corresponding plot for Nafion [32], as seen in Fig. 7.8. For the SSC polymer at λ = 3 about
1/4 of the hydronium ions are trapped in ion cages with three sulfonic acid neighbours.
Roughly, equal numbers of hydronium ions are found with one sulfonic acid neighbour,
while slightly more have two neighbours. The trend in the histogram indicates that as
the hydration level increases there is a dramatic drop in the number of fully “sulfonated”
hydronium ions (with three sulfonic group neighbours) and a significant increase in the
102
Å at ! ) 3, to 6.7 Å at ! ) 13.5 and 20. The position of thefirst peak in our gS-S(r) differs from that in united-atomsimulations.11,14 Cui et al.14 observed a sharp first peak at !4Å at all hydration levels and nonmonotonic trends in gS-S(r)with increasing membrane hydration. In contrast, Urata et al.11observed the first peak shifting from !5 Å at ! ) 2.8 to !6 Åat ! ) 35.4 and the peak becoming shorter and broader withincreasing hydration. This is consistent with our finding thatwith increasing !, the average S-S coordination number at agiven distance decreases. The average distance from a sulfuratom within which another sulfur atom can be found (coordina-tion number nS-S(r) ) 1) is 5.3, 5.6, 6.0, 6.5, 6.6, 7.0, 7.1, and7.9 Å, respectively, at ! ) 1, 3, 5, 7, 9, 11, 13.5, and 20. Urataet al.11 found this distance increasing from 4.6 Å at ! ) 2.8-7.7 Å at ! ) 35.4 in their united-atom simulation. Thus, oursimulations reveal that the sulfonate groups move away fromeach other with increasing membrane hydration in qualitativeagreement with the findings of previous simulations.9,11,15 Thecalculation of this sulfur-sulfur distance for different levels ofmembrane hydration in the present work establishes a classicalbenchmark for sulfonate-sulfonate separation in hydratedNafion.Figure 4b shows that the changes in gOh-Oh(r) with increasing
! are similar to the changes in gS-S(r), especially at lowhydration levels. This suggests that hydronium ions may bestrongly bound to sulfonate groups at low ! consistent with thedistribution of hydronium ions seen in Figure 3a. The area undergOh-Oh(r) decreases with increasing ! indicating that thecoordination number of hydronium oxygen atoms around eachother, nOh-Oh(r), decreases. The average distance from ahydronium ion within which another hydronium ion can befound (nOh-Oh(r) ) 1) is 5.6, 5.9, 6.1, 6.4, 6.6, 6.8, 7.1, and 7.6Å, respectively, at ! ) 1, 3, 5, 7, 9, 11, 13.5, and 20. Withincreasing membrane hydration, the hydronium ions move awayfrom each other just as the sulfonate groups move apart.Panels c and d of Figure 4 represent the sulfonate-hydronium
interaction in the form of gS-Oh(r) and gOs-Oh(r), respectively.For all values of !, gS-Oh(r) shows a dominant peak between3.85 and 3.92 Å. In contrast, the united-atom simulations ofCui et al.14 give the first peak at 4 Å for the four ! values theyexamined. In our work, there is an additional peak at 3.2 Åonly for ! ) 1, which represents the contact ion pair (closestapproach of H3O+ and SO3-). When water molecules areincluded, this peak disappears as water molecules pull the H3O+
away from the SO3-. When ! increases from 1 to 7, the areaunder the curve up to 4.3 Å (first hydration shell) decreasesdrastically, while the area under the next peak increases. Thisshows that as ! increases, H3O+ moves away from the firsthydration shell of the sulfonate group and in to the second shellextending from 4.3 to 6.8 Å. This is confirmed by the plot ofgOs-Oh(r) in Figure 4d. A sharp first peak occurs at a sulfonateoxygen-hydronium oxygen separation of !2.5 Å (contact ionpair) and a broader second peak is at !4.8 Å. The area under
these first two peaks decreases with increasing !, while the areaunder the curve for the third neighbor shell located between!5.2 and 7.2 Å increases with increasing !. This indicatesincreasing separation with increasing !. In agreement with Cuiet al.,14 we do not observe an “apparently artificial peak” at 3.2Å that Petersen et al.17 observed in their gOs-Oh(r) from classicalsimulations, which they attributed to the limitations of theclassical hydronium ion. While the classical hydronium ionmodel does not incorporate proton shuttling (see ref 21), itslimitations may have been overstated in the literature. Oursimulations show that the hydronium ion can move away fromthe sulfonate group without the use of potentials that allowproton transfer.The second peak in gOs-Oh(r) has been attributed by Cui et
al.14 to the sulfonate group and hydronium ion being separatedby a layer of water molecules. This is true for large values of! as we will show later in our discussion. However, oursimulations show that the second peak occurs even when thereare only hydronium ions and no water molecules present in thesystem (! ) 1). Therefore, at small values of !, this peak canbe attributed to multiple hydronium ions being present near eachsulfonate group as a result of the sulfonate groups being closeto each other. Evidence of this can be found by visualexamination of the bottom right corner of Figure 3a. Furtherevidence in the form of the average number of hydroniumoxygens within 4.3 Å of sulfur atoms (first hydration shell cutoffchosen based on gS-Oh(r)) from 10 000 configurations is listed
TABLE 3: The Average Coordination Numbers ofHydronium Ions around Sulfur (nsh) and Water Moleculesaround Sulfur (nsw) in Nafion as a Function of !
! nsh nsw1 2.46 0.003 2.05 2.235 1.59 3.657 1.14 4.269 0.97 4.8111 0.77 5.3413.5 0.76 5.3320 0.49 5.79
Figure 5. Distribution of sulfonate neighbors of hydronium ions inNafion for various hydration levels, !, indicated by the legend.
Figure 6. Percentages of hydronium ions that have at least 1 sulfonateneighbor (square) and multiple sulfonate neighbors (triangle) as a func-tion of hydration level, !, The percentage of nondiffusing hydrogen atomsobtained from neutron scattering experiment25 is represented by circles.
8074 J. Phys. Chem. B, Vol. 111, No. 28, 2007 Devanathan et al.
Figure 7.8: Fraction of hydronium ions with a given number of sulfonate neighbours inNafion for different hydration levels λ. Figure source: Ref. [32].
number of free hydronium ions (with zero neighbours). It is intriguing that the SSC
curves for λ = 3, 6, 13 resemble the Nafion plots of Devanathan et al. but for a higher
hydration level - namely λ = 5, 9, 20 water molecules. This allows the SSC system to
have the same amount of free hydronium ions as Nafion but at a lower water content.
Another feature that emerges is that the correspondence between the two membranes
changes with the hydration level - while at the lowest water content the improvement
is ∆λ = 2 (i.e. 3 → 5) at the highest content the improvement is already ∆λ = 7 (i.e.
13 → 20). These simulation results can be used to explain the differences found in the
experimental conductivity plots for the two polymer systems [18]. In particular, it was
found that a low EW the SSC polymer system has a superior conductivity to Nafion at
the same hydration level, and with increasing the water content this superiority becomes
even more pronounced.
In conclusion, by carefully developing a SSC specific force field we have obtained new
insights into the morphology of the SSC polymer systems as a function of the hydration
103
level. The polymer backbone is able to assume straight configurations while at the same
time forming strong hydrogen bond bridges. Ion cages with fully “sulfonated” hydronium
ions are typical of the low hydration level and explain the previously unresolved peaks
(in Nafion) in the S-S and S-O pair correlation plots. The trends seen in a structural
parameter representing the number of sulfonic groups around a hydronium ion allows us
to explain the differences between the proton mobility of Nafion and the SSC polymer,
which is discussed in the next chapter.
104
Chapter 8
Proton Diffusion in SSC Polymer Systems
In this chapter we present the results of the work we have carried out on the simulation
of vehicular proton diffusion in the SSC polymer. In addition, two other systems were
examined – one involving a single excess proton in water and a second one consisting of a
multiproton triflic acid solution. These latter systems were considered, solely, as tests for
our JIT-EVB method. The lessons we have learned from these experimental runs, and
in particular from triflic acid, give us the confidence needed to apply the new method to
polymer system.
First, we present a brief description of the manner whereby the diffusion process was
followed in the simulations. In the polymer systems all side-chains were considered ion-
ized, hence the simulations were performed on a mixture comprised of tethered sulfonate
groups, hydronium ions and water molecules. In the absence of proton hopping events
the proton diffusion is naturally determined from the diffusion of the hydronium ions,
more specifically, from the trajectory of the oxygen atom in H3O+. Similarly, in the
case of the excess proton in water studies, carried out with the JIT-EVB method, proton
diffusion was examined through the oxygen atom associated with the excess charge. This
representation of the proton diffusion is commonly referred to as the Center of Excess
Charge (CEC) representation. Finally, in the simulation of triflic acid solution (which,
like the polymer systems was considered to be fully ionized) an oxygen atom from a
sulfonate group could potentially be a CEC as well, if a hydronium ion donates a proton
back to SO−3 .
In all of our studies we have chosen to evaluate the diffusion constant from the Einstein
105
relation using the MSD of the CEC:
D = limt→∞
⟨|R(t)−R(0)|2
⟩
6t(8.1)
We have found that the alternative Green-Kubo relation [86] which employs the velocity
autocorrelation functions is inconvenient to use owing to the poor convergence in the
t→∞ limit.
The proton diffusion data for the SSC polymer system is based on the same simulation
runs1 discussed in Chapter 7. The average of the MSD of the hydronium ions for the
three hydration levels is shown in Fig. 8.1. One should note the crucial importance of the
initial configuration averaging in order to obtain accurate results. The predicted values
for the diffusion coefficient are 2.84×10−7, 1.36×10−6, and 3.47×10−6 cm2/s for water
contents of 3, 6, and 13, water molecules per sulfonic acid group. The agreement with
the experimental results of Kreuer et al. [90] is nearly quantitative at the lowest water
content. This is a strong indication that the dominating transport mechanism under
minimal hydration may be vehicular. Still, even under these conditions proton hopping
does occur, as demonstrated recently by accurate ab initio MD simulations [91]. The
proton hopping is only localized and does not lead to significant displacement of the
protons. On the other hand, the diffusion coefficients for λ = 6 and λ = 13 are much
lower than those obtained from the experimental measurements suggesting an increased
contribution from structural diffusion in these systems.
Since the force field used in the present calculation is based on the one used in Ref. [25]
to study the hydronium ion diffusion in Nafion, it is, indeed, remarkable to see how well
the two polymer systems are differentiated in the simulations. The only hydration level
investigated for the Nafion system was λ = 15 where one finds an average hydronium
diffusion coefficient about 2.5 times smaller than the one we found here for λ = 13. This1Additional simulation details: The H3O
+ diffusion coefficients were calculated from an average overfive initial configurations R(0), taken from the same trajectory about 25 ps apart, using the Einsteinrelation, Eq. (8.1). A second averaging is done over the diffusion coefficients of the 40 hydronium ions.
106
Figure 8.1: MSD of hydronium ions in the SSC polymer as a function of time andhydration level λ. The slope of the curves gives directly 6D in units cm2/s. The averagetemperature from the 2 ns NVE production runs was 315 K.
107
is in qualitative agreement with the conductivity difference between a SSC polymer of
EW 800 and Nafion [18,90]. As the conductivity of the polymer samples is proportional
to the total diffusion coefficient (and not just the vehicular component) it is important to
know if the vehicular contribution changes significantly between the two polymer systems.
In the case of Nafion the computed vehicular diffusion coefficient corresponds to 19.5%
of the total measured diffusion, whereas in the case of the SSC polymer the computed
value is about 20.4%. Since the hydration level of the Nafion system is a bit higher (i.e.,
λ = 15) it is reasonable to expect a higher contribution from the structural diffusion
and a concomitant reduction of the vehicular component. We can conclude that the
two polymer systems exhibit practically the same partition between their vehicular and
structural modes of proton transport at least in the case of λ between 13 and 15 water
molecules per sulfonic acid group.
The failure of the classical MM approach to account for the major part of the diffusion
coefficient was the motivation that obliged us to develop the the JIT-EVB method. While
reproducing experimental results is the ultimate goal we have chosen not to resort to
any system-specific fitting parameters, including specially developed force fields for the
species involved. This is in stark contrast to the alternative EVB implementations which
rely heavily on extensive, system-specific parametrization. The only kinds of parameters
used in the JIT-EVB method are those which determine the quality of the ab initio
PES fit. These, for example, are the validity radius around a grid point (i.e., the grid
point spacing), the order of the polynomials used in the fitting etc. Consequently, the
agreement between the JIT-EVB simulations and experiment is controlled by the ab
initio method used.
The first system that we tested the JIT-EVB method on was an excess proton with
64 water molecules. This is a very small simulation system and the goal here is not to
extract any physical constants, but rather to determine the simulation parameters (e.g.,
108
Table 8.1: Diffusion coefficients obtained from a JIT-EVB simulation of excess protonwith 64 water molecules in a cubic periodic box. Five runs were started with differentinitial configurations. The diffusion constant including both the vehicular and Grotthusscomponents appears in the second column. The third column shows the number of gridpoints in the reactive trigger zone (i.e., the number of ab initio calculations and fittingsperformed). Last column shows the root mean square (RMS) difference between the QMand classical forces in the superposition of resonance forms.
Run # D [cm2/s] grid points Frms [kJ mol−1 nm−1]
1 9.3×10−5 6 8432 1.9×10−5 5 8193 3.7×10−6 3 8534 8.9×10−8 6 7335 1.7×10−5 4 883
a Simulation details: the forcefield parameters used for all JIT-EVB simulations are given in Ap-pendix D. Proton hopping was not allowed during the equilibration. Accordingly, the MIC bondingpotentials were not invoked in this stage. Non-bonding interactions were treated with the IPSmethod [46] using the maximum cutoff (i.e., half a box length). Excess proton and 64 F3C watermolecules [89] were placed randomly in a periodic box of 1.2429 nm length on all sides. The systemswere optimized first with the SD method [43]. Random velocities with zero net momentum were thenassigned corresponding to the target temperature of 298 K. Each system was equilibrated for 5×103
steps with the generalized Nose-Hoover method [55] in the canonical ensemble using a thermostatperiod of 72.0 fs and a time step of 1 fs. After the systems were equilibrated proton hoppingwas turned on and MIC applied to the bonding potentials. The details of the JIT-EVB simulationare given in Appendix D. Diffusion data was collected during a 50 ps production run with a timestep of 0.2 fs. In each of the five trajectories an averaging of the CEC MSD was done over fourinitial configurations, taken 250 steps apart. One Berendsen thermostat [93] was used for the atomsoutside any reactive zones and separate thermostats were used for each of the reactive zones. Allthermostats had the same target temperature of 298 K, while their periods were 1 ps for the formerand 0.1 ps for the latter kind.
time step, thermostat frequency, grid spacing etc.) that lead to stable MD trajectories.
Still, a comparison can be made between our results (shown in Table 8.1) and diffusion
results from a Car-Parrinello ab inito MD [92] study of the same system.
Not surprisingly, due to the small system size we observe a high variance in the
predicted diffusion constant – nearly three orders of magnitude difference in some cases
which underlines the need for extensive sampling (or larger systems) before sound results
can be obtained. Never the less, in some cases we see significant enhancement of the
diffusion constant compared to the experimental vehicular diffusion in bulk water (about
109
2.3×10−5 cm2/s). A more sensible comparison, however, is with the results of ab initio
MD simulations as our JIT-EVB method is designed to match those forces. Depending on
the functional and the initial conditions Car-Parrinello MD results estimate the proton
diffusion constant between 0.5×10−5 and 2.1×10−5 cm2/s [92]. For comparison, the
average of the data in Table 8.1 is 2.6×10−5 cm2/s. It can be expected that with an
increase of the simulation size our results will further increase [94] bringing diffusion
constant closer to the experimental value of 9.307×10−5 cm2/s [58]. Despite the high
statistical noise in the diffusion constant these preliminary tests of the JIT-EVB method
reveal a highly consistent number of grid points and fit quality (last two columns). In
some cases as few as three grid points are all that is needed to represent the active region
of the PES.
In our EVB method the interacting species are represented as a superposition of
resonance forms, with the coefficients selected such that the desired (i.e., ab initio) forces
are reproduced. The last column in the table shows the difference with the QM forces
remaining after the fit. In order to put the values in perspective, we note that the largest
difference seen in the first entry (883 kJ mol−1 nm−1) has the magnitude of the restoring
force in a water OH bond stretched to 1.04 A. The limitations of DFT in accurately
describing hydrogen bonds are well known and they stem from non-local correlations
in the van der Waals forces [95]. As the PBE functional that was used here does not
incorporate any van der Waals correction terms we can conclude that at least some of the
difference between the DFT and the classical forces is due to these terms. Accordingly,
by representing the DFT forces as a superposition of classical forces in the resonance
forms, the JIT-EVB method can improve the agreement with experiment, particularly
in systems with hydrogen bonds where the van der Waals interactions are important.
Lastly we show the diffusion results for triflic acid at the intermediate hydration level
of λ = 6, see Table 8.2. The MSD of the CEC as a function of time is shown in Fig. 8.2.
110
Table 8.2: Diffusion coefficients obtained from a JIT-EVB simulation of five triflic acidmolecules and 30 water molecules. The final density obtained for the system after relax-ation appears in the first column. The diffusion constant including both the vehicularand Grotthuss components appears in the second column. The third column shows thenumber of grid points in the reactive trigger zone (i.e., the number of ab initio calcula-tions and fittings performed). Last column shows the RMS difference between the QMand classical forces in the superposition of resonance forms.
ρ [g/cm3] D [cm2/s] grid points Frms [kJ mol−1 nm−1]
1.556 1.1×10−5 101 881
a Simulation details: the forcefield parameters used for all JIT-EVB simulations are given in Ap-pendix D. Proton hopping was not allowed during the equilibration. Accordingly, the MIC bondingpotentials were not invoked in this stage. Non-bonding interactions were treated with the IPSmethod [46] using the maximum cutoff (i.e., half a box length). Five triflate ions F3CSO
−3 , five
hydronium ions H3O+ and thirty F3C water molecules [89] were placed randomly in a periodic box
of 1.2893 nm length on all sides, corresponding to a density of 1 g/cm3. The system was optimizedfirst with the SD method [43] for 1500 steps. Random velocities with zero net momentum were thenassigned corresponding to the target temperature of 298 K. Each system was equilibrated for 5×103
steps with the generalized Nose-Hoover method [55] in the canonical ensemble using a thermostatperiod of 81.6 fs and a time step of 1 fs. This was followed by 3×104 equilibration steps in theNPT ensemble performed with the generalized Nose-Hoover method [87] using a barostat period of1 ps. The final density for the system was 1.556 g/cm3. After the system was equilibrated, protonhopping was turned on and MIC applied to the bonding potentials. The details of the JIT-EVBsimulation are given in Appendix D. Diffusion data was collected during a 60.6 ps production runwith a time step of 0.2 fs. In each of the five trajectories an averaging of the CEC MSD was doneover four initial configurations, taken 250 steps apart. One Berendsen thermostat [93] was used forthe atoms outside any reactive zones and separate thermostats were used for each of the reactivezones. All thermostats had the same target temperature of 298 K, while their periods were 1 ps forthe former and 0.1 ps for the latter kind.
111
[R(t
)−
R(0
)]2
[nm
2]
t [ps]
Figure 8.2: MSD of the CEC in triflic acid solution as a function of time. The hydrationlevel is λ = 6.
The diffusion constant obtained here is nearly one order higher compared to the vehicular
diffusion constant calculated for the SSC polymer. The JIT-EVB predicted diffusion is
also higher than the experimental one of the SSC polymer at this hydration level. As our
simulation lacks the polymer component that tethers the end acid groups and obstructs
the free movement of hydronium ion such an increased mobility is expected. Of note is
the significantly higher number of grid points required in this system. This is due to the
wide variety of hydrogen bond complexes that occur in this multiproton, concentrated
acid solution. The acid dissociation is a favourable, though not barrierless process. For
this reason the acid was introduced as triflate and hydronium ions. In this manner the
diffusion can be sampled immediately (after some equilibration) without requiring that
112
we first observe the rare proton dissociation events.
In these preliminary examples of the JIT-EVB method we have obtained qualitatively
reasonable results. Application of the method to larger systems, including the SSC
polymer is currently being carried out.
Part III
Conclusions and Future Work
113
114
Chapter 9
Conclusions
The subject of this thesis was the development of the necessary tools and their application
for exploring the morphology and proton transport in the PFSA SSC membranes. A
major part of the work related to the forces in the modelled systems since they are of
paramount importance, affecting the dynamics, kinetic energy, temperature, pressure in
the system, and so on. As the simulation methodology was MM, all forces are in principle
derived from the force field. For our studies we have used a special force field that works
well for the SSC polymer. Furthermore, the traditional view that a force field needs to
be known beforehand has been replaced by the method we have developed for creating
the force field parametrization on the fly, during a MD simulation. With this approach
we were able to simulate bond breaking and making events by switching between force
field parameters, depending on the current value of the reaction coordinates. It is hoped
that these developments will broaden the concept of classical MM simulations.
The second pivotal point of the work was overcoming the problem of high dimen-
sionality. For example, a conformational search algorithm (like the brute force geometry
optimization) that works well for a small number of degrees of freedom is absolutely
useless for a system like a polymer. Another example is the parametrization of the PES
of a reaction with multiple degrees of freedom. An important conclusion here is that
even the real systems, that exist on a much larger timescale, do not necessarily sample
the entire available phase space. Accordingly, we have developed the methodology that
only operates on small, tractable regions of the phase space. This approach has allowed
us to build systems of thousands of atoms in conformations that may resemble the real
systems and parametarize a PES surface only in the most frequently visited regions.
115
Chapter 10
Future Work
The wide scope of the methodology developed and presented in this thesis has, unfor-
tunately, presented serious time limitations on the number cases we were able to study
in the time allowed. A natural extension of the work, in particular the proton hopping
studies, would include bigger systems where the polymer backbone is also included in
the JIT-EVB simulation. Establishing a structure–properties relationship in the SSC
polymer can then be undertaken by varying the monomer ratios, EW distribution, the
total MW of the polymer etc. As we have seen in the last chapter, a relatively high
number of grid points may be required in the multi-proton, concentrated acid systems.
Simulations that are run in parallel on equivalent systems will be beneficial if the code
allows pooling of the grid points between the computers. Thus the PES will be required
to be parameterized only once. Once the grid points are known for a particular system
it will be possible to smoothly interpolate the forces in the reactive regions of the PES.
This will allow for energy conservation in the JIT-EVB method, and possibly, an increase
of the time step. Since the JIT-EVB method is quite general it may find applications in
other areas, particularly biological systems.
Part IV
Appendixes and Bibliography
116
117
Appendix A
Short-Side-Chain Force Field
C1O
O3
C2O
C2S
S6O4 O4
O3HH3
C1
F1
F1
C1T
F1T
F1T
F1T
F2OF2O
F2S F2S
F1O
C2SI
S6IO4I O4I
O4I
O2
H2H2
H2
O1
H1H1
ionized side-chain
terminal groupwater
hydronium ion
Figure A.1: Atom type labels for the SSC force field. Water and hydronium ion areshown for completeness only, their force field parameters are as described in Ref. [25]
118
Table A.1: SSC force field parameters for the harmonic stretching potential Ebond.
Atom types r0 [nm] k [kJ mol−1 nm−2]
H1 O1 0.1000 209200H2 O2 0.0982 454364H3 O3H 0.0974 292880O3H S6 0.1628 292880O4: S6: 0.1451 585760S6: C2S: 0.1886 292880C2S: C2O 0.1555 292880C: F: 0.1348 253240C:O O3 0.1392 292880C1:O C1: 0.1566 292880C1: C1: 0.1564 179627
a Ebond = 0.5k (r − r0)2b Force constants k are taken from Ref. [25].c A colon (:) in the atom names represents a wildcard (e.g. C: covers C1, C1T, C1O, C2O, C2S, C2SI,
etc.).
119
Table A.2: SSC force field parameters for the harmonic bending potential Eangle.
Atom types θ0 k [kJ mol−1 rad−2]
H1 O1 H1 109.47 502.08H2 O2 H2 113.40 330.64C: C: C: 113.64 471.45F1O C1O C1: 106.43 472.04O3 C1O C1: 112.23 470.70O3 C1O F1O: 105.60 470.70C: C: F: 108.62 472.04F: C: F: 109.02 470.70F2O C2O O3 111.82 470.70C2S: C2O O3 106.96 470.70C2O C2S: S6: 114.52 490.91F2S C2S: S6: 106.76 478.15O4: S6: O4: 123.71 509.36O4 S6 O3H 108.07 509.36C2S: S6 O3H 100.36 456.31C2S: S6: O4: 107.09 456.31C1O O3 C2O 126.40 470.70
a Eangle = 0.5k (θ − θ0)2b Force constants k are adapted from Refs. [25] and 23.c A colon (:) in the atom names represents a wildcard (e.g. C: covers C1, C1T, C1O, C2O, C2S, C2SI,
etc.).
Table A.3: SSC force field parameters for the torsion potential Edih.
Atom types a [kJ mol−1] b c [deg]
H3 O3H S6 C2S: 7.4185 1.3349 -21.037O3H S6 C2S: C2O -9.5835 1.1453 17.673S6: C2S: C2O O3 -34.0433 1.0306 -0.931C2S: C2O O3 C1O 8.4275 1.1671 14.264C2O O3 C1O C1: 38.2339 0.9381 32.494C1: C1: C1: C1: 5.6356 1.0258 -49.553O4: S6: C2S: C2O 12.0085 0.4842 -6.642
a Edih = a cos(bϕ− c)b A colon (:) in the atom names represents a wildcard (e.g. C: covers C1, C1T, C1O, C2O, C2S, C2SI,
etc.).
120
Table A.4: SSC force field parameters for the non-bonding interactions ECoulomb and ELJ .
Atom type q ε [kJ mol−1] σ [nm]
H1 0.4100 0.0418400 0.08018O1 -0.8200 0.7732032 0.31655H2 0.4606 0.0418400 0.08018O2 -0.3818 0.7732032 0.31655F1 -0.2709 0.2075264 0.30249C1O 0.4462 0.3978984 0.34730F1O -0.2741 0.2075264 0.30249C2S 0.3715 0.3978984 0.34730C2SI 0.3234 0.3978984 0.34730H3 0.3882 0.0004184 0.28464F2O -0.2569 0.2075264 0.30249F2S -0.2414 0.2075264 0.30249O3H -0.5140 0.4004088 0.30332S6 1.2412 1.4392956 0.35903S6I 1.0237 1.4392956 0.35903C1 0.5497 0.3531296 0.34599O4 -0.4512 0.4004088 0.30332O4I -0.5876 0.4004088 0.30332O3 -0.5392 0.4004088 0.30332
a ECoulomb = k q1q2r , with k = 138.93547 kJ mol−1 nm
b ELJ = 4√ε1ε2
[(σ1+σ2
2r
)12 −(σ1+σ2
2r
)6]
c The Lennard-Jones parameters and are obtained from Ref. [25].
121
Appendix B
Cross Section Factor
The pore model presented in Fig. B.1 emphasizes that each polymer chain (shown as
cylinder) is considered only partially inside the pore. When two circles of radii R and
Figure B.1: Cross section view of a pore formed by polymer chains. In our model eachpolymer chain (represented by a cylinder) is bisected by the radius of the pore. Thepart of the polymer chains on the inside is shown in yellow, the part outside the pore inbrown. The remaining pore volume is filled with side chains and water molecules (notshown).
r separated by a distance d intersect each other the common area can be found as in
Eq.(14) of Ref. [96]:
A (R, r, d) = r2ArcCosd2 + r2 −R2
2dr+R2ArcCos
d2 − r2 +R2
2dR
− 1
2
√(d+ r −R) (d− r +R) (−d+ r +R) (d+ r +R) (B.1)
122
Since the centers of the polymer chains lie on the pore surface we can make the substi-
tution d = R. For the radius r we will use half the effective width of the polymer chain,
i.e. w/2. This leads to a cross section area of:
A (R,w) =1
4w2ArcSec
4R
w+R2ArcCos
(1− w2
8R2
)− w
8
√16R2 − w2 (B.2)
We define the cross section factor f (R,w) as the ratio of the cross section area A (R,w)
and the polymer chain area π (w/2)2 resulting in:
f (R,w) =A (R,w)
π (w/2)2 =1
πArcSec
4R
w+
4R2
πw2ArcCos
(1− w2
8R2
)− 1
2π
√16R2
w2− 1 (B.3)
123
Appendix C
Molecular Volume
Here we explain the method used to determine the molecular volume of a water molecule
and the polymer side chain. A trial point will be considered lying within the molecule if
it is within σ/2 from any atom, σ being the Lennard-Jones parameter. For every atom
type we use the corresponding value of σ from Table A.4 in Appendix A. Trial points are
picked randomly within the smallest rectangular box that the molecule will fit in. For
a trial point within the molecule the counter Nhit is increased by one. The molecular
volume is then calculated as Vmol = (Nhit/Ntotal)Vbox.
The molecule conformation used for the volume estimate was obtained from a geom-
etry optimization in an infinite simulation universe, with no other species present. The
total number of trial points Ntotal was 1×105 for water and 2×106 for the side chain1,
resulting in molecular volumes of VH2O = 0.0312 nm3 and V sidechain
= 0.156 nm3. To
check the validity of this approach one can compare the known density of water, about
1.0 g/cm3, to the value of 0.96 g/cm3 that follows from the above molecular volume.
1For the volume estimate of the side chain we consider only the CF2CF2SO3H fragment, i.e. withoutthe branching oxygen atom, as the latter falls within the volume of the backbone cylinder.
124
Appendix D
JIT-EVB Simulation Details
In our previous calculations the harmonic stretching potentials were used for all molec-
ular bonds. However, In order to facilitate the proton hopping mechanism in EVB a
more suitable potential is the anharmonic Morse potential. This new potential has been
employed for modelling the OH bonds in hydronium ion and the sulfonic acid group.
Furthermore, the atomic charges in H3O+ are also different from those that were used in
our vehicular diffusion studies, and have been updated in accordance with the hydronium
ion model of Ref. [67]. All other force field parameters not listed here remain unchanged
from those found in Appendix A. Of particular interest may be the potential of water,
for which we employ the standard F3C model [89] without any modifications. Even
when proton hopping is not allowed (e.g., during the initial equilibration stage) the force
field parameters listed below are still used in the conventional manner (i.e. without such
complications as mixing of resonance forms or applying MIC to the bonding potentials).
Table D.1: Force field parameters for the anharmonic stretching potential EMorse usedin JIT-EVB simulations. Atom labels appear in Fig. A.1.
Atom types D [kJ mol−1] a [nm−1] r0 [nm]
H2 O2 603.0 11.85 0.098H3 O3H 603.0 11.85 0.098
a EMorse = D(1− ea(r0−r)
)2b Force field parameters for hydronium ion are taken from Ref. [67]. The same parameters are assumed
valid for the OH bond in the sulfonic acid group.
In Chapter 4 the idea of geometrical triggers were introduced. These triggers, that
define the extent of the reactive zone of the PES, are delineated by the following criteria:
• distance between the hopping proton Hd and the accepting oxygen Oa below 2.2 A.
125
• distance between the donor and acceptor oxygens Od and Oa below 2.85 A.
• angle HdOdOa below 30.
• angle OdOaGa between 110 and 180 (where Ga is along the bisector of the angle
in the accepting water, or the sulfur atom in SO−3 ).
At every step during the simulation the molecular configurations are checked against
these triggers. If no structures satisfy the trigger conditions the properties of the whole
system are calculated conventionally, without invoking the JIT-EVB method. If, on the
other hand, a structure does satisfy the trigger conditions it is denoted as a cluster,
illustrated in Fig. D.1.
Once we have determined the clusters that represent the reactive zones the next step
is to determine the resonance forms for each of those clusters. Combinatorics gives 2n
resonance forms for a cluster with n reactive hydrogen bonds. For example, in Fig. D.1a
we would have two resonance forms, four resonance forms in the case of sulfonate (b)
and eight in the case of water wire (c). However, resonance forms that correspond to
rare events like the ionization of water will, in general, have a very small contribution.
Thus, we choose to exclude all combinations of PTs that will lead to over-protonation of
the accepting species or under-protonation of the donors. For each cluster we go through
each of the acceptable resonance forms, change the atom types according to the new
Table D.2: Atom charges used in JIT-EVB simulations. Atom labels appear in Fig. A.1.
Atom type q
H2 0.33O2 0.01F2S -0.2107
a ECoulomb = k q1q2r , with k = 138.93547 kJ mol−1 nm
b Charges on the hydronium ion are taken from Ref. [67]. The charge on the fluorine atoms is adjustedsuch that the total charge of the triflate ion F3CSO
−3 is minus one.
126
OH
H
H
OS
O
O
OH
H
H
O
HH
O
HH
h1 h1
h2 OH
H
H
OS
O
O
OH
H
H
O
HH
O
HH
O H
H
H
O
H
H O H
H
O
H
Hh1 h2 h3
a b
c
Figure D.1: When a hydronium ion and a water molecule satisfy the trigger conditionsfor hydrogen bond h1 they both become part of a cluster (a). If the same hydroniumion is hydrogen-bonded to a sulfonate group and the bond h2 also satisfies the triggerconditions, the cluster will include both the sulfonate group and the water (b). A thirdexample with three hydrogen bonds is the water wire (c). Hence a cluster contains allspecies connected by reactive hydrogen bonds. For simplicity the charges on the specieshave been omitted. Hydrogen bonds are depicted with dashed lines.
127
identity of the atoms (e.g. the hydronium ion oxygen O2 becomes a water oxygen O1
etc.), update all force field parameters and calculate the properties of the resonance form
(i.e., energy, gradient and virial). The resonance form properties are calculated from the
bonding terms entirely within the cluster, and the non-bonding interactions within the
cluster plus its periodic images.
Having determined the properties of the resonance forms we now have a basis in which
to expand the ab initio forces. At this point an ab initio calculation is carried out for
each cluster, supplemented with capping atoms if needed1. There is no restriction on the
QM method used to obtain the forces, other than that it must be done with the same
PBC. Here we have used the ASE library [97] combined with the GPAW calculator [98]
that performs a DFT calculation using plane-waves. The PBE functional was used with
the default atomic setups. This ab initio calculation gives us the first grid point in the
reactive zone of the PES.
Now we fit the QM forces to a superposition of resonance form forces, i.e. F =∑i
ciFi.
The fit will be most accurate for the current conformation of the cluster. As the geometry
of the cluster changes, the fit coefficients ci must be updated accordingly. To overcome
this problem each coefficient ci (where i designates the resonance form) are considered
product of two polynomials of the reaction coordinates: c′i =4∑j=0
sjRjOO
4∑j=0
tjwj, where
w = 1 − δ/ROO, δ = ROdHd− ROaHd
. The primed coefficient c′i have to be squared and
normalized in order to correspond to a probability, which is achieved through the relation
ci = (c′i)2
/∑i
(c′i)2. When resonance form i requires PTs across multiple hydrogen bonds
(e.g., the resonance form corresponding to a hydronium ion on the right in the water wire
of Fig. D.1c) the reaction coordinates ROO and w are averaged over all participating
hydrogen bonds. Accordingly, in this example ROO = (ROO,h1 +ROO,h2 +ROO,h3) /3
and w = (wh1 + wh2 + wh3) /3. Thus the unknown fitting coefficients are the sj and tj in
1In the case of sulfonic groups the carbon atom is part of the cluster, while the ab initio calculationincludes three extra capping fluorines.
128
the expansion polynomials of each resonance form coefficient c′i. The fitting is performed
using the basic particle swarm optimization method [99]. Twenty walkers were employed
in the method for 50 optimization cycles.
The cluster properties used for the force fitting have been calculated with the as-
sumption that clusters are isolated from each other and the rest of the atoms. This
however is not true, so in order to obtain the properties of the whole system additional
interactions have to be taken into account. Once the resonance form coefficients ci are
known each cluster is put in the resonance form with the highest contribution. This is
done to ensure that upon exit from the reactive zone the clusters will be in their correct
resonance form. The charge and Lennard-Jones parameters of the clusters are updated as
a superposition of the resonance form parameters, e.g. the charge on a particular cluster
atom is calculated as q =∑i
ciqi. The non-bonding interactions are now recalculated to
include all atoms, whether in a cluster or not, employing the exclusion lists of the highest
contribution resonance states. The bonding interactions are calculated as sum of two
classes. In the first class are terms that are entirely within some cluster. Such terms are
subject to the resonance form mixing. The second class of bonding terms are those that,
at least partially, are outside any clusters. Such terms are unaffected by the resonance
forms and are evaluated and summed directly, without any weighting.
As explained in Chapter 4 the parametrization obtained for the first grid point is
assumed valid in some neighbourhood around it, determined by the validity radius. Here
we use a value of 0.387 A for this radius. One should note that the described fitting
procedure is not performed for equivalent clusters. Thus, a system with many hydronium
ions may need only a couple of clusters to be parametrized, for instance, one for Zundel
ion type clusters and another one for Eigen ion clusters. All parametrization is kept on
file. As the simulation progresses more grid points are accumulated. Accordingly, there
is a build-up stage of the simulation where most of the ab initio work is done, while in
129
the latter stages hardly any QM calculations are necessary. As the molecules explore the
reactive zone PES they employ the parameters of their closest grid point.
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