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.

Bibliography

[1] Michael Ball. The Hydrogen Economy: Opportunities and Challenges, chapter

Fossil fuels - supply and future availability. Cambridge University Press, 2009.

[2] Daniel G. Nocera. On the future of global energy. Daedalus, 135(4):112–115, 2006.

[3] Daniel G. Nocera. Personalized Energy: The Home as a Solar Power Station and

Solar Gas Station. ChemSusChem, 2(5):387–390, 2009.

[4] The increase in oil consumption, 2005. [Online; accessed 9-March-2010].

[5] Jiujun Zhang. PEM Fuel Cell Electrocatalysts and Catalyst Layers: Fundamentals

and Applications. Springer, 2008.

[6] Elna Schirrmeister Frank Marscheider-Weidemann and Annette Roser. The Hy-

drogen Economy: Opportunities and Challenges, chapter Key role of fuel cells.

Cambridge University Press, 2009.

[7] P. R. Resnick. A short history of Nafion®. L’Actualite Chimique, 301:144–147,

2006.

[8] A. Ghielmi, P. Vaccarono, C. Troglia, and V. Arcella. Proton exchange membranes

based on the short-side-chain perfluorinated ionomer. Journal of Power Sources,

145(2):108 – 115, 2005.

[9] Stephen J. Paddison. Device and Materials Modeling in PEM Fuel Cells, chap-

ter Proton Conduction in PEMs: Complexity, Cooperativity and Connectivity.

Springer, 2009.

[10] Kenneth A. Mauritz and Robert B. Moore. State of Understanding of Nafion.

Journal of Chemical Information and Computer Sciences, 104:4535–4586, 2004.

130

131

[11] T. D. Gierke, G. E. Munn, and F. C. Wilson. The morphology in nafion perfluori-

nated membrane products, as determined by wide- and small-angle x-ray studies.

Journal of Polymer Science: Polymer Physics Edition, 19(11):1687–1704, 1981.

[12] Howard W. Starkweather. Crystallinity in perfluorosulfonic acid ionomers and

related polymers. Macromolecules, 15(2):320–323, 1982.

[13] Romuald Wodzki, Anna Narebska, and Wojciech Kwas Nioch. Percolation conduc-

tivity in Nafion membranes. Journal of Applied Polymer Science, 30(2):769–780,

1985.

[14] Xueqian Kong. Characterization of Proton Exchange Membrane Materials for Fuel

Cells by Solid State Nuclear Magnetic Resonance. PhD thesis, Iowa State Univer-

sity, 2010.

[15] Laurent Rubatat, Anne Laure Rollet, Gerard Gebel, and Olivier Diat. Evidence

of Elongated Polymeric Aggregates in Nafion. Macromolecules, 35(10):4050–4055,

2002.

[16] K. D. Kreuer. On the development of proton conducting polymer membranes for

hydrogen and methanol fuel cells. Journal of Membrane Science, 185(1):29 – 39,

2001.

[17] Klaus Schmidt-Rohr and Qiang Chen. Parallel cylindrical water nanochannels in

Nafion fuel-cell membranes. Nature Materials, 7(1):75–83, 2008.

[18] Klaus-Dieter Kreuer, Stephen J. Paddison, Eckhard Spohr, and Michael Schuster.

Transport in Proton Conductors for Fuel-Cell Applications: Simulations, Elemen-

tary Reactions, and Phenomenology. Chemical Reviews, 104(10):4637–4678, 2004.

132

[19] Stephen J. Paddison, Reginald Paul, and Jr. Thomas A. Zawodzinski. A Sta-

tistical Mechanical Model of Proton and Water Transport in a Proton Exchange

Membrane. Journal of The Electrochemical Society, 147(2):617–626, 2000.

[20] Stephen J. Paddison, Reginald Paul, and Jr. Thomas A. Zawodzinski. Proton fric-

tion and diffusion coefficients in hydrated polymer electrolyte membranes: Com-

putations with a non-equilibrium statistical mechanical model. The Journal of

Chemical Physics, 115(16):7753–7761, 2001.

[21] S.J. Paddison and R. Paul. The nature of proton transport in fully hydrated

Nafion®. Physical Chemistry Chemical Physics, 4(7):1158–1163, 2002.

[22] Stephen J. Paddison and Thomas A. Zawodzinski Jr. Molecular modeling of the

pendant chain in Nafion®. Solid State Ionics, 113-115:333 – 340, 1998.

[23] Aleksey Vishnyakov and Alexander V. Neimark. Molecular simulation study of

Nafion membrane solvation in water and methanol. The Journal of Physical Chem-

istry B, 104(18):4471–4478, 2000.

[24] Aleksey Vishnyakov and Alexander V. Neimark. Molecular Dynamics Simulation

of Microstructure and Molecular Mobilities in Swollen Nafion Membranes. The

Journal of Physical Chemistry B, 105(39):9586–9594, 2001.

[25] Seung Soon Jang, Valeria Molinero, Tahir Cagin, and William A. Goddard.

Nanophase-Segregation and Transport in Nafion 117 from Molecular Dynamics

Simulations: Effect of Monomeric Sequence. The Journal of Physical Chemistry B,

108(10):3149–3157, 2004.

[26] Stephen J. Paddison and James A. Elliott. Molecular Modeling of the Short-Side-

Chain Perfluorosulfonic Acid Membrane. The Journal of Physical Chemistry A,

109:7583–7593, 2005.

133

[27] Stephen J. Paddison and James Elliott. Molecular Flexibility in the Short-Side-

Chain Perfluorosulfonic Acid Membrane. ECS Transactions, 1(6):207–214, 2006.

[28] Stephen J. Paddison and James A. Elliott. The effects of backbone conforma-

tion on hydration and proton transfer in the short-side-chain perfluorosulfonic acid

membrane. Solid State Ionics, 177:2385–2390, 2006.

[29] Stephen J. Paddison and James A. Elliott. On the consequences of side chain

flexibility and backbone conformation on hydration and proton dissociation in per-

fluorosulfonic acid membranes. Physical Chemistry Chemical Physics, 8:2193–2203,

2006.

[30] Stephen J. Paddison and James A. Elliott. Selective hydration of the short-side-

chain perfluorosulfonic acid membrane. An ONIOM study. Solid State Ionics,

178:561–567, 2007.

[31] Arun Venkatnathan, Ram Devanathan, and Michel Dupuis. Atomistic simulations

of hydrated Nafion and temperature effects on hydronium ion mobility. The Jour-

nal of Physical Chemistry B, 111(25):7234–7244, 2007.

[32] R. Devanathan, A. Venkatnathan, and M. Dupuis. Atomistic Simulation of Nafion

Membrane: I. Effect of Hydration on Membrane Nanostructure. The Journal of

Physical Chemistry B, 111(28):8069–8079, 2007.

[33] Ram Devanathan, Arun Venkatnathan, and Michel Dupuis. Atomistic simulation

of nafion membrane. 2. Dynamics of water molecules and hydronium ions. The

Journal of Physical Chemistry B, 111(45):13006–13013, 2007.

[34] Shengting Cui, Junwu Liu, Myvizhi Esai Selvan, Stephen J. Paddison, David J.

Keffer, and Brian J. Edwards. Comparison of the Hydration and Diffusion of Pro-

134

tons in Perfluorosulfonic Acid Membranes with Molecular Dynamics Simulations.

The Journal of Physical Chemistry B, 112(42):13273–13284, 2008.

[35] Matt K. Petersen, Feng Wang, Nick P. Blake, Horia Metiu, and Gregory A. Voth.

Excess Proton Solvation and Delocalization in a Hydrophilic Pocket of the Proton

Conducting Polymer Membrane Nafion. The Journal of Physical Chemistry B,

109:3727–3730, 2005.

[36] Matt K. Petersen and Gregory A. Voth. Characterization of the Solvation and

Transport of the Hydrated Proton in the Perfluorosulfonic Acid Membrane Nafion.

The Journal of Physical Chemistry B, 110:18594–18600, 2006.

[37] Craig K. Knox and Gregory A. Voth. Probing Selected Morphological Models of

Hydrated Nafion Using Large-Scale Molecular Dynamics Simulations. The Journal

of Physical Chemistry B, 114(9):3205–3218, 2010.

[38] Dongsheng Wu, Stephen J. Paddison, and James A. Elliott. Effect of Molecular

Weight on Hydrated Morphologies of the Short-Side-Chain Perfluorosulfonic Acid

Membrane. Macromolecules, 42(9):3358–3367, 2009.

[39] Daniel Brandell, Jaanus Karo, Anti Liivat, and John O. Thomas. Molecular dy-

namics studies of the Nafion®, Dow® and Aciplex® fuel-cell polymer membrane

systems. Journal of Molecular Modeling, 13:1039–1046, 2007.

[40] A. M. Lesk. Introduction to Protein Science Architecture, Function and Genomics.

”Oxford University Press”, 2004.

[41] Stephen L. Mayo, Barry D. Olafson, and William A. Goddard. DREIDING: a

generic force field for molecular simulations. The Journal of Physical Chemistry,

94:8897–8909, 1990.

135

[42] Christopher J. Cramer. Essentials of computational chemistry: theories and mod-

els. ”John Wiley & Sons”, 2004.

[43] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numeri-

cal recipes in Fortran 77: the art of scientific computing. ”Cambridge University

Press”, 1992.

[44] Daan Frenkel and Berend Smit. Understanding Molecular Simulation: From Algo-

rithms to Applications. Academic Press, 2002.

[45] Xiongwu Wu and Bernard R. Brooks. Isotropic periodic sum: A method for

the calculation of long-range interactions. The Journal of Chemical Physics,

122(4):044107, 2005.

[46] Xiongwu Wu and Bernard R. Brooks. Using the isotropic periodic sum method to

calculate long-range interactions of heterogeneous systems. The Journal of Chem-

ical Physics, 129(15):154115, 2008.

[47] Kazuaki Takahashi, Kenji Yasuoka, and Tetsu Narumi. Cutoff radius effect of

isotropic periodic sum method for transport coefficients of Lennard-Jones liquid.

The Journal of Chemical Physics, 127(11):114511, 2007.

[48] M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. ”Oxford Univer-

sity Press”, 1990.

[49] Phil Attard. Thermodynamics and Statistical Mechanics: Equilibrium by Entropy

Maximisation. Academic Press, 2002.

[50] Manuel J. Louwerse and Evert Jan Baerends. Calculation of pressure in case of

periodic boundary conditions. Chem. Phys. Lett., 421(1-3):138–141, 2006.

136

[51] H.J.C. Berendsen. Simulating the physical world: hierarchical modeling from quan-

tum mechanics to fluid dynamics, chapter Pressure and virial. Cambridge Univ Pr,

2007.

[52] Glenn J. Martyna, Adam Hughes, and Mark E. Tuckerman. Molecular dynam-

ics algorithms for path integrals at constant pressure. The Journal of Chemical

Physics, 110(7):3275–3290, 1999.

[53] Karsten Meier and Stephan Kabelac. Pressure derivatives in the classical

molecular-dynamics ensemble. The Journal of Chemical Physics, 124(6):064104,

2006.

[54] Jhih-Wei Chu, Bernhardt L. Trout, and Bernard R. Brooks. A super-linear min-

imization scheme for the nudged elastic band method. The Journal of Chemical

Physics, 119(24):12708–12717, 2003.

[55] David J. Keffer, Chunggi Baig, Parag Adhangale, and Brian J. Edwards. A gen-

eralized Hamiltonian-based algorithm for rigorous equilibrium molecular dynamics

simulation in the canonical ensemble. Journal of Non-Newtonian Fluid Mechanics,

152(1-3):129–139, 2008.

[56] H. Flyvbjerg and H. G. Petersen. Error estimates on averages of correlated data.

The Journal of Chemical Physics, 91(1):461–466, 1989.

[57] Theodor Grotthuss. On the decomposition of water and their solution species using

galvanic electricity. Annales de Chimie, 58:54–74, 1806.

[58] I. Ruff and V. J. Friedrich. Transfer diffusion. IV. Numerical test of the correlation

between prototrope mobility and proton exchange rate of H3O+ and OH- ions with

water. The Journal of Physical Chemistry, 76(21):2954–2957, 1972.

137

[59] Wikipedia. Grotthuss mechanism, 2010. [Online; accessed 2-April-2010].

[60] Omer Markovitch, Hanning Chen, Sergei Izvekov, Francesco Paesani, Gregory A.

Voth, and Noam Agmon. Special pair dance and partner selection: Elementary

steps in proton transport in liquid water. The Journal of Physical Chemistry B,

112(31):9456–9466, 2008.

[61] Dominik Marx. Proton Transfer 200 Years after von Grotthuss: Insights from Ab

Initio Simulations. ChemPhysChem, 7(9):1848–1870, 2006. An addendum to this

article appears in Ref. [100].

[62] Dominik Marx, Mark E. Tuckerman, Jurg Hutter, and Michele Parrinello. The

nature of the hydrated excess proton in water. Nature, 397(6720):601–604, 1999.

[63] Edward F. Valeev and Henry F. Schaefer III. The protonated water dimer: Brueck-

ner methods remove the spurious C1 symmetry minimum. The Journal of Chemical

Physics, 108(17):7197–7201, 1998.

[64] The American Institute of Chemical Engineers. Study of Proton Transport Using

Reactive Molecular Dynamics, Computational Molecular Science and Engineering

Forum, 2009. See also http://utkstair.org/clausius/docs/atoms/RMD/index.html.

[65] Arieh Warshel and Robert M. Weiss. An empirical valence bond approach for

comparing reactions in solutions and in enzymes. Journal of the American Chemical

Society, 102(20):6218–6226, 1998.

[66] Udo W. Schmitt and Gregory A. Voth. Multistate Empirical Valence Bond

Model for Proton Transport in Water. The Journal of Physical Chemistry B,

102(29):5547–5551, 1999.

138

[67] Giuseppe Brancato and Mark E. Tuckerman. A polarizable multistate empirical

valence bond model for proton transport in aqueous solution. The Journal of

Chemical Physics, 122(22):224507, 2005.

[68] Feng Wang and Gregory A. Voth. A linear-scaling self-consistent generalization

of the multistate empirical valence bond method for multiple excess protons in

aqueous systems. The Journal of Chemical Physics, 122(14):144105, 2005.

[69] Diane E. Sagnella and Mark E. Tuckerman. An empirical valence bond model

for proton transfer in water. The Journal of Chemical Physics, 108(5):2073–2083,

1998.

[70] Udo W. Schmitt and Gregory A. Voth. The computer simulation of proton trans-

port in water. The Journal of Chemical Physics, 111(20):9361–9381, 1999.

[71] Yongho Kim, Jose C. Corchado, Jordi Villa, Jianhua Xing, and Donald G. Truhlar.

Multiconfiguration molecular mechanics algorithm for potential energy surfaces of

chemical reactions. The Journal of Chemical Physics, 112(6):2718–2735, 2000.

[72] Eikerling M., Paddison S.J., Pratt L.R., and Zawodzinski T.A. Defect structure

for proton transport in a triflic acid monohydrate solid. Chemical Physics Letters,

368(1):108–114, 2003.

[73] Pathumwadee Intharathep, Anan Tongraar, and Kritsana Sagarik. Ab initio

QM/MM dynamics of H3O+ in water. Journal of Computational Chemistry,

27(14):1723–1732, 2006.

[74] Solvey Solexis. Hyflon® Ion E83 Membranes Data Sheet, 2006. [Online; accessed

9-March-2010].

139

[75] H. L. Morgan. The Generation of a Unique Machine Description for Chemical

Structures-A Technique Developed at Chemical Abstracts Service. Journal of

Chemical Documentation, 5:107–113, 1965.

[76] David Weininger. SMILES, a chemical language and information system. 1. Intro-

duction to methodology and encoding rules. Journal of Chemical Information and

Computer Sciences, 28:31–36, 1988.

[77] CADD Group Chemoinformatics Tools and User Services. Chemical Identifier Re-

solver. [Online; accessed 9-March-2010].

[78] CADD Group Chemoinformatics Tools and User Services. Online SMILES Trans-

lator and Structure File Generator. [Online; accessed 9-March-2010].

[79] N. H. Weiderman and B. M. Rawson. Flowcharting loops without cycles. SIG-

PLAN Not., 10(4):37–46, 1975.

[80] Open Babel: The Open Source Chemistry Toolbox. [Online; accessed 9-March-

2010].

[81] E. Spohr, P. Commer, and A. A. Kornyshev. Enhancing Proton Mobility in Poly-

mer Electrolyte Membranes: Lessons from Molecular Dynamics Simulations. The

Journal of Physical Chemistry B, 106:10560–10569, 2002.

[82] Yushu Matsushita. Precise Molecular Design of Complex Polymers and Morphology

Control of Their Hierarchical Multiphase Structures. Polymer Journal, 40(3):177–

183, 2008.

[83] Reginald Paul and Stephen J. Paddison. Effects of dielectric saturation and ionic

screening on the proton self-diffusion coefficients in perfluorosulfonic acid mem-

branes. The Journal of Chemical Physics, 123(22):224704, 2005.

140

[84] E. Spohr. Molecular Dynamics Simulations of Proton Transfer in a Model Nafion

Pore. Molecular Simulation, 30(2/3):107–115, 2004.

[85] P. Commer, C. Hartnig, D. Seeliger, and E. Spohr. Modeling of Proton Transfer in

Polymer Electrolyte Membranes on Different Time and Length Scales. Molecular

Simulation, 30(11/12):755–763, 2004.

[86] J. M. Haile. Molecular Dynamics Simulation: Elementary Methods. ”John Wiley

& Sons”, 1997.

[87] David J. Keffer, Chunggi Baig, Parag Adhangale, and Brian J. Edwards. A gen-

eralized Hamiltonian-based algorithm for rigorous equilibrium molecular dynamics

simulation in the isobaric-isothermal ensemble. Molecular Simulation, 32(5):345–

356, 2006.

[88] David J. Keffer, Carrie Y. Gao, and Brian J. Edwards. On the Relationship be-

tween Fickian Diffusivities at the Continuum and Molecular Levels. The Journal

of Physical Chemistry B, 109(11):5279–5288, 2005.

[89] Michael Levitt, Miriam Hirshberg, Ruth Sharon, Keith E. Laidig, and Valerie

Daggett. Calibration and Testing of a Water Model for Simulation of the Molecu-

lar Dynamics of Proteins and Nucleic Acids in Solution. The Journal of Physical

Chemistry B, 101(25):5051–5061, 1997.

[90] K.D. Kreuer, M. Schuster, B. Obliers, O. Diat, U. Traub, A. Fuchs, U. Klock, S.J.

Paddison, and J. Maier. Short-side-chain proton conducting perfluorosulfonic acid

ionomers: Why they perform better in PEM fuel cells. Journal of Power Sources,

178(2):499 – 509, 2008.

[91] Robin L. Hayes, Stephen J. Paddison, and Mark E. Tuckerman. Proton Trans-

port in Triflic Acid Hydrates Studied via Path Integral Car-Parrinello Molecular

141

Dynamics. The Journal of Physical Chemistry B, 113(52):16574 – 16589, 2009.

[92] Sergei Izvekov and Gregory A. Voth. Ab initio molecular-dynamics simulation of

aqueous proton solvation and transport revisited. The Journal of Chemical Physics,

123(4):044505, 2005.

[93] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, and J. R.

Haak. Molecular dynamics with coupling to an external bath. The Journal of

Chemical Physics, 81(8):3684–3690, 1984.

[94] In-Chul Yeh and Gerhard Hummer. System-Size Dependence of Diffusion Coeffi-

cients and Viscosities from Molecular Dynamics Simulations with Periodic Bound-

ary Conditions. The Journal of Physical Chemistry B, 108(40):15873–15879, 2004.

[95] Biswajit Santra, Angelos Michaelides, and Matthias Scheffler. On the accuracy

of density-functional theory exchange-correlation functionals for H bonds in small

water clusters: Benchmarks approaching the complete basis set limit. The Journal

of Chemical Physics, 127(18):184104, 2007.

[96] Eric W. Weisstein. “Circle-Circle Intersection.” From MathWorld – A Wolfram

Web Resource. [Online; accessed 9-March-2010].

[97] S. R. Bahn and K. W. Jacobsen. An object-oriented scripting interface to a

legacy electronic structure code. COMPUTING IN SCIENCE & ENGINEERING,

4(3):56–66, 2002.

[98] J. J. Mortensen, L. B. Hansen, and K. W. Jacobsen. Real-space grid imple-

mentation of the projector augmented wave method. PHYSICAL REVIEW B,

71(3):035109, 2005.

[99] Wikipedia. Particle swarm optimization, 2010. [Online; accessed 9-March-2010].

142

[100] Dominik Marx. Proton Transfer 200 Years after von Grotthuss: Insights from Ab

Initio Simulations. ChemPhysChem, 8(2):209–210, 2007.