Ind. Eng. Chem. Res. 2001, 40, 4789-4800
4789
Theoretical Studies of Proton Transfer in Water and Model Polymer
Electrolyte Systems
Tao Li, Aaron Wlaschin, and Perla B. Balbuena*
Department of Chemical Engineering, University of South Carolina, Columbia, South Carolina 29208
The interactions of H3O+ with water and model Nafion structures are studied using ab initio,
density functional theory, and molecular dynamics simulations. In the gas phase, H3O+ is solvated
by a well-defined first solvation shell with three water molecules, while an additional water
molecule locates in the second shell. This same structure is preferred in the liquid phase. In the
absence of an electric field, proton transfer between H3O+ and a water molecule proceeds without
barrier if the hydrogen bond length O-H‚‚‚O is smaller than 2.7 Å. When an electric field is
applied in a direction opposite to that of the system dipole, the activation barrier for proton
transport is significantly reduced. The Nafion side chain, ending in a sulfonic group, is folded in
the gas phase, whereas in aqueous media, it becomes stretched, and proton transfer takes place
from the acid group to the water molecules. Results from MD simulations indicate that the
contact ion pair formed between the sulfonic acid anion -SO3- and the hydronium ion is very
stable.
Introduction
Nafion, a perfluorosulfonic acid (PFSA) membrane
developed by DuPont, has been reported as chemically
stable, structurally durable, and proton-conductive.1
One of its well-known applications is in low-temperature-operation polymer membrane electrolyte fuel cell
systems. A typical PFSA has a hydrophobic backbone
that consists of poly(tetrafluoroethylene) (PTFE) and
hydrophilic perfluorovinyl ether pendant side chains
terminated by sulfonate ionic groups.2 The ionic groups
comprise up to 15% of the polymer.3 The molecular
structure of H-Nafion is4
where n can vary between 5 and 13.5 and 100 < m <
1000.1 The polymer equivalent weight, defined as the
weight of dry polymer in grams containing 1 mol of unit
charge, is usually in the range of 1000-2000. The
-SO3H groups tend to dissociate in polar solvents, and
they protonate the solvent. This protonated solvent
species serves as the major charge carrier in the
membrane. Experimental data suggest that the negatively charged -SO3- groups, along with water molecules and positively charged counterions such as H+,
Na+, or K+ depending on the pretreatment, tend to
aggregate and form hydrophilic clusters.5,6 These clusters are dispersed in the continuous hydrophobic fluorocarbon matrix. The cluster network model has been
proposed by Hsu and Gierke to describe the membrane
microstructure.7-9 According to this model, the membrane is formed by clusters and channels, with the ionic
groups located at the inner surface of the 3-5-nmdiameter clusters. As the degree of hydration increases,
* Corresponding author: Perla B. Balbuena. Phone: (803)
777-8022. Fax: (803) 777- 8265. E-mail: [email protected].
the clusters swell and increase in size. Results from
small-angle X-ray measurement of Nafion10 have verified the existence of clusters of about 4-nm diameter,
with channels of 1-nm diameter and length that limit
the transport properties of water and ions. The cluster
model also implies a rather curled polymer side chain
in the membrane. Other researchers have proposed that
the sulfonic acid domains are parallel to each other, that
is, that a lamellar geometry exists in the membrane,
and that the side chain is much more stretched.11
Yeager et al.12 suggested that the hydrated membrane
is formed by two continuous interpenetrating phases
with irregularly shaped clusters. Spectroscopic studies
suggest that the size of the clusters should be smaller
than that proposed by Gierkes’ model.13 The actual
microstructure has not been definitively resolved, and
significant debate still exists about it.14
In fuel cells that utilize hydrogen as a source of fuel,
solvated proton clusters serve as the major charge
carriers. The proton transfer inside the membrane is
usually described by the vehicle mechanism15-17 and by
the hopping mechanism,15,18,19 also known as the Grotthuss model.20 As opposed to proton hopping, the vehicle
mechanism proposes that protons bounded with a
fraction of water molecules travel through the electrolyte via mutual diffusion. To some extent, proton solvation/desolvation can be treated as a special case of
the proton-transfer reaction. Many experiments have
been performed to determine the proton solvation
energy,21 and studies agree that this energy release is
mostly due to the formation of the hydronium ion.22
Three coordination water molecules were found by ab
initio calculations (HF/6-31G* and MP2/6-31G*),23 mass
spectroscopy,24 and ultrafast solvation dynamics experiments,25,26 whereas some earlier diffraction and X-ray
experiments reported the presence of a fourth water
molecule in the first ionic shell of the hydronium ion.27
The hopping mechanism suggests that the proton
propagates along the O-H‚‚‚O hydrogen bond of the
H3O+(H2O) complex. Results from a mixed molecular
dynamics/Monte Carlo (MD/MC) algorithm used to
10.1021/ie010467y CCC: $20.00 © 2001 American Chemical Society
Published on Web 10/09/2001
4790
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
describe the proton transfer by an instantaneous hopping mechanism19 indicate that the proton migration
rate is mainly determined by the lifetime of the solvation complexes. Density functional theory (DFT) studies
of proton transfer from solution to a metal cluster18
reveal that the presence of an external electric field
could effectively accelerate the proton-transfer rate.
The proton transport properties are strongly coupled
with the water content and the microstructure of the
material. The experimentally determined proton diffusion coefficient in a Nafion-117 membrane is 1.4 × 10-9
m2/s,2 whereas in solution at infinite dilution, it is 9.31
× 10-9 m2/s.28 The presence of the hydrophilic pendant
chain -OCF2CF(CF3)OCF2CF2SO3- imposes a dramatic
influence on the solvent structure and the associated
transport properties; thus, structural information regarding the pendant chain is important. Because of the
importance of proton transfer in fuel cell design, studies
of proton solvation and transport processes at the
microscopic level are essential. MD simulations29 have
been used to study the motion of water molecules and
protons in a Nafion membrane,30 modeled as a large
number of identical cylindrical pores. The calculated
electro-osmotic drag coefficients were overestimated
because of the simplified physical model used. Only
when the wall charge density was comparatively low
was the proton diffusion coefficient obtained found to
be of the same order of magnitude as the experimental
value. Applications of a statistical mechanical model31
and a phenomenological model32 were also reported
recently.
In this work, we first examine the minimum-energy
structure and solvation energies of the H+(H2O)n (n )
1-5) complexes using DFT. Next, we investigate the
proton-transfer mechanism by analyzing the potential
energy surface and its corresponding activation barrier,
as well as the effect of an external electric field on proton
transfer between water molecules, as a function of the
oxygen-oxygen distance. The proton-transfer reaction
from the molecular unit CF2dCFOCF2CF(CF3)OCF2CF2SO3H to the water molecule is investigated. The
fully optimized geometry of the ionized molecular unit
CF2dCFOCF2CF(CF3)OCF2CF2SO3- is used in MD
simulations, where we examine the membrane fragment
conformation and proton-transport process in bulk
water and in a model polymer/water system.
2. Computational Details
2.1. Ab Initio Calculations. Ab initio and DFT
calculations were carried out using Gaussian 9433and
Gaussian 98.34 The basis sets used in the calculations
included both 6-31+G (d, p) and 6-311++G (d, P), which
have polarization and diffusion functions on the oxygen
atoms. Full geometry optimizations of the solvation
clusters were carried out using the B3P8635 and
B3PW9136,37 functionals. The calculations of potential
energy curves for proton transfer were carried out only
at the B3P86/6-31+G (d, p) level. A finite external
electric field (EF) was added to the calculation. The field
was generated by a dipole, and the sign of the field was
chosen so that a positive sign was in the direction
parallel to the original dipole direction of the complex.
The EF intensities were in the range of 0.001-0.02 au,
where 1 au corresponds to 5.14 × 10 9 V/cm.
Interactions of the H-terminated side chain [represented by CF2dCFOCF2CF(CF3)OCF2CF2SO3H] with
water were studied at both the HF/6-31+G** and
B3PW91/6-311++G** levels of theory. The ionic side
chain, represented by CF2dCFOCF2CF(CF3)OCF2CF2SO3-, was fully optimized at the HF/6-31+G** level.
2.2. Molecular Dynamics Simulations. MD simulations were carried out using Cerius2.38 Except for the
bulk water simulations, all of the other systems under
consideration contain 500 water molecules, x hydronium
ions, and x SO3--terminated Nafion oligomers, where x
varies between 0 and 2. The Nafion monomer and
oligomer were constructed using the Cerius 2 polymer
builder, and an initial minimization was done using a
combination of the steepest-descent, Newton-Raphson,
quasi-Newton, and truncated-Newton methods.38 We
have chosen n ) 7, which gives an equivalent weight
(EW) of 1165 g of polymer per equivalent of ionic groups
in the H+ form. This structure corresponds to Nafion1200, one of the most extensively studied membranes.
Unless otherwise specified, the simulations were carried
out in the canonical ensemble, i.e., at constant number
of molecules N, volume V, and temperature T. The force
field used has the functional form of the Dreiding force
field39 that has been reported to be effective for polymer
materials. Long-range contributions of the Coulomb
interaction are calculated with the Ewald summation.
The nonbonding potential includes also the short-range
Lennard-Jones terms. Intramolecular terms such as
bond stretching, bending, and torsion are included. The
equilibrium bond distance for the Dreiding force field
is assumed to follow a simple additivity of bond radii,
where the bond radii are based on structural data of
standard reference molecules. The equilibrium bond
angle is assumed to depend only on the central atom.39
Periodic boundary conditions are applied. The simulations are performed at 298.15 and 353.15 K. The
temperature is maintained using a T-damping thermostat.40 The Dreiding force field39 has been also applied
to water, using the SPC/E charges.41 To test the validity
of the Dreiding force field in reproducing the thermodynamic and structural properties of bulk water at
ambient conditions, molecular dynamics simulations of
250 water molecules in the NPT ensemble were carried
out. The temperature is controlled at 298.15 K, and the
pressure is 1 bar. The total length of these simulations
was 100 ps, and the last 50 ps of data were collected
for statistical analysis.
Another set of MD simulations was done to investigate the dynamics of H3O+. To obtain an accurate result
for the proton dynamics, a small time step, 1 fs, was
used in the simulations. The first 50 ps were usually
discarded, and then an additional 200 or 250 ps were
run for the production stage. The hydronium ion diffusion coefficient is calculated on the basis of the mean
square displacement (MSD) as29
〈|r
b(t) - b
r (0)|2〉
tf∞
6t
DMSD ) lim
where r is the position vector and t is the time. As this
equation indicates, the calculated diffusion coefficients
become more accurate with long simulation times. For
short simulation times, the MSD shows a nonlinearity
as a function of time, and this leads to an overestimation
of the diffusion coefficient. If the time evolution of the
MSD is plotted on a logarithmic scale, the slope for
Einstein diffusion should be 1.42 Therefore, for each data
analysis, we plot the change of the MSD with time on a
logarithmic scale, and determine the region where the
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4791
Figure 1. Calculated gas-phase (B3PW91/6-311++G**) proton solvation complexes H+(H2O)n and corresponding atomic Mulliken charges.
(a) n ) 1, (b) n ) 2, (c) n ) 3, (d) n ) 4, (e) n ) 5.
slope is 1. The diffusion coefficient is calculated using
that range of data.
3. Results and Discussions
3.1. Proton Solvation in Water. Figure 1 displays
DFT-optimized structures and representative Mulliken
atomic charges of the H+(H2O)n clusters with n ) 1-5;
the corresponding equilibrium bond lengths are reported
in Table 1. The rO‚‚‚O parameter is defined as the
distance between the H3O+ oxygen and the oxygen of
the hydrogen-bonded water molecule, rO-H is the O-H
bond length in H3O+ or H2O, and rH‚‚‚O is the H‚‚‚O
distance in the OH‚‚‚O hydrogen bond between H3O+
and a water molecule. In general, the optimized structures are in good agreement with other calculated and
experimental results (Table 1). The two different levels
of theory (functional/basis set) tested in this study,
B3P86/6-31+G** and B3PW91/6-311++G**, give comparable geometries, although the O-H bond lengths are
slightly longer at the B3P86 level, whereas the O‚‚‚O
distances are a little bit shorter than those from
B3PW91. For both methods, the calculated OH bond
length in H3O+ is 0.017 Å longer than the OH bond
length in H2O.
The H+(H2O)2 structure (Figure 1b) is highly symmetric, with the proton located exactly between the two
water molecules. This is because the proton is forming
overlaps between the two lone-pair electron orbitals
belonging to two distinct water oxygen atoms separated
by 2.39 Å from each other. The linear H bond formed
between H3O+ and H2O lengthens the O-H bond in
H3O+. Ojamäe and co-workers43,44 reported two different
conformations for H5O2+ based on ab initio MP2 calculations: a structure of C2 symmetry with a symmetric
hydrogen bond, (H2O‚‚‚H‚‚‚OH2)+ and a structure of
Cs symmetry with an asymmetric hydrogen bond,
H3O+‚‚‚H2O. The first conformation is viewed as the
global minimum. Our minimum-energy structure reported in Figure 1b is in agreement with this global
minimum: all bond distances are in close agreement
with those from the MP2 results. Our calculations also
indicate that, at the B3PW91 level, the H3O+‚‚‚H2O
complex of Cs symmetry is not stable.
To locate the minimum-energy structure for the H+(H2O)3 cluster, the two water molecules are initially
located in symmetric positions with respect to H3O+.
After geometry optimization, this symmetry with respect to the hydronium ion remains (Figure 1c). Each
O-H bond in H3O+ that is part of a H bond with a water
molecule is stretched to 1.05 Å, and this length is 0.14
Å smaller than the corresponding bond length in the
H+(H2O)2 cluster (Table 1, B3PW91/6-311++G**).
The optimized geometry of the H+(H2O)4 complex is
shown in Figure 1d. This structure is close to the global
minimum reported by Ojämae et al.43 Our rH‚‚‚O distance
(1.51 Å) is slightly shorter than the value reported by
Ojämae (1.56 Å). In Figure 1e, the minimum-energy
structure of the H+(H2O)5 complex is shown; however,
4792
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Table 1. Key Structural Parameters for the Proton
Hydrated Clustersa
system
rO-H (Å)
rO‚‚‚O (Å)
rH‚‚‚O (Å)
B3P86/6-31+G** calculations
H2O
0.963
H+(H2O)
0.980
H+(H
0.969 (O1-H3)
2O)2
2.388
1.192 (O1-H7)
1.192 (H7-O4)
H+(H2O)3 0.970 (O1-H2) 2.458 (O1-O8)
1.051 (H7-O8)
0.967 (O8-H10)
1.409 (H7-O1)
H+(H2O)4 0.965 (O1-H2)
0.965 (O4-H5)
1.020 (O8-H9)
[1.017]27
1.508 (H9-O4)
2.527 (O4-O8)
[2.52]27
H+(H2O)5 0.964 (O1-H2) 2.566 (O1-O8)
1.559 (O4-H9)
1.009 (H7-O8) 2.446 (O8-O11) 1.556 (H7-O1)
1.063 (O8-H10) 2.626 (O11-O14) 1.384 (H11-O10)
H2O
B3PW91/6-311++G** calculations
0.960
H+(H2O)
0.978
H+(H2O)2 0.967 (O1-H3)
2.389
1.192 (O1-H7)
1.196 (H7-O4)
H+(H2O)3 0.967 (O1-H2) 2.478 (O1-O8)
1.042 (H7-O8)
0.965 (O8-H10)
1.436 (H7-O1)
H+(H2O)4 0.962 (O1-H2)
0.963 (O4-H5)
1.013 (O8-H9)
[1.017]27
1.538 (H9-O4)
2.547 (O4-O8)
[2.52]27
H+(H2O)5 0.962 (O1-H2) 2.584 (O1-O8)
1.585 (O4-H9)
1.003 (H7-O8) 2.460 (O8-O11) 1.582 (H7-O1)
1.053 (O8-H10) 2.662 (O11-O14) 1.408 (H11-O10)
a Atom numbers in parentheses correspond to Figure 1. Experimental results are in brackets.
it cannot be conclusively identified as the global energy
minimum. The optimized structures of H+(H2O)4 and
H+(H2O)5 suggest that H3O+ is solvated by three water
molecules in the first shell, with a fourth water molecule
located in the second shell. This observation agrees with
an ab initio study23 at a Moller-Plesset level (MP2/
6-31G*)45 that uses a continuum model46 to represent
the hydronium hydration, whereas a Hartree-Fock
(HF) study47 indicated the possibility of the fourth water
being in the first solvation shell. Recent ab initio MD
calculations suggest that the most favorable structures
for hydrated protons in water would be H+(H2O)2 and
H+(H2O)3.48
With the increase of the solvation cluster size, the
hydronium ion rO-H values decrease, whereas the rO‚‚‚O
distance increases. The rH‚‚‚O distances also increase,
except when a water molecule is located in the second
solvation shell. Therefore, the H-bond strength becomes
weaker when the cluster size increases. This change of
bond strength can be related to the amount of charge
transferred between the proton and the solvation water
molecules. In Figure 1b, each solvation water molecule
bears a charge of 0.21 au, whereas the charges are
reduced to 0.07 and 0.05 au for Figure 1c and d,
respectively. For Figure 1e, barely zero charge transfer
occurs between the fourth solvation water and the
H(H2O)3+ ion. Thus, the attractive interactions of the
proton with surrounding water molecules diminish as
more water molecules are added. The atomic charges
in H3O+ also change significantly as the number of
solvation water molecules increases. The charge on the
oxygen atom becomes strongly negative, changing from
-0.20 au in H3O+ to -0.69 au in H(H2O)5+, whereas
the charge on the H atom interacting with water
decreases from 0.57 au in H(H2O)2+ to 0.49 au in
H(H2O)5+. The charge on the noninteracting hydronium
H atom (such as H10 in Figure 1c) also decreases from
0.40 au in H3O+ to 0.32 au in H(H2O)5+.
Table 2 lists the proton solvation energies ∆E, proton
solvation enthalpies ∆H, and proton solvation free
energies ∆G, defined as the corresponding energy differences between the product complexes and the reactants for the reactions
H+ + nH2O S H+(H2O)n
All of the energies and energy differences are slightly
higher at the B3P86 theory level. The solvation reactions are all spontaneous, as indicated by their ∆G
values. Literature-reported values for proton solvation enthalpies are between -254 and -260.5 kcal/
mol,21,22,49,50 which correspond to the calculated values
for n ) 4-5 solvation water molecules (Table 2). It can
also been seen that the initial formation of the hydronium ion contributes most to the total solvation energy
of H+(H2O)n. The other part of the solvation energy is
due to interactions of the hydronium ions with the
solvation water molecules. From our calculations, the
enthalpy for hydronium ion formation is -165.07 kcal/
mol, and Meot-Ner et al.22,50 reported a measured value
of -166.3 kcal/mol. The energy difference between H+(H2O)n and H+(H2O)n-1 becomes smaller with increasing
number of hydration molecules. However, even the
second hydration shell produces a significant increase
(∼13 kcal/mol) of the solvation energy.
Energies of formation for complexes such as H+(H2O)n
are calculated by subtracting the energies of the isolated
monomers from the energy of the complex. In the
calculation of the complex energy, the bonded monomers
are described with the basis set of the whole complex,
while the energies of the isolated monomers are calculated using the monomer basis sets. The energies of
formation will thus be overestimated. This type of error
is called the basis set superposition error (BSSE), which
is known to be relatively important in systems containing hydrogen bonds.51 The standard Boys and Bernardi
counterpoise procedure,52 where the energies of the
isolated monomer are calculated with the basis set of
the complex, has been followed. The BSSE-corrected free
energies of solvation (B3PW91/6-311++G**) are listed
in Table 2. Note that, without the BSSE correction, the
calculated solvation free energies are comparatively
higher than the experimental results,22 with errors
within 5% of the experimental values.
3.2. Proton Transfer between Two Water Molecules: The Hopping Mechanism. According to the
proton hopping mechanism,15 the proton propagates
along the O-H‚‚‚O hydrogen bond direction of the H3O+(H2O)n complexes. This process is shown in Figure 2a,
and the formal reaction is
H3O+ + H2O T H2O + H3O+
The coordinates used to construct the potential energy
curves for the above reaction are the O-O distance R1
and the proton-oxygen distance R2 (Figure 2b). For
each curve, R1 is kept constant while R2 is increased in
increments of 0.01 Å. The calculated potential energy
curves for several values of R1 are displayed in Figure
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4793
Table 2. Summary of Thermodynamic Properties for the Proton Hydration Clustersa
E + ZPE
(hartree)
system
E
(hartree)
ZPE
(hartree)
H2O
H+(H2O)
-76.610 55
-76.887 38
0.021 49
0.034 59
H+(H2O)2
H+(H2O)3
H+(H2O)4
-153.559 73
-230.210 98
-306.854 66
0.057 95
0.083 43
0.108 90
B3P86/6-31+G** calculations
-76.589 07
-76.852 79
-165.49
-165.43
[-166.3]22
-153.501 78
-203.09
-204.25
-230.127 55
-226.12
-227.72
-306.745 76
-244.41
-246.35
H+(H2O)5
-383.489 75
0.133 26
-383.356 49
H2O
H+(H2O)
-76.428 22
-76.704 45
0.021 48
0.034 53
H+(H2O)2
H+(H2O)3
H+(H2O)4
-153.191 21
-229.658 03
-306.117 83
0.057 60
0.083 15
0.108 24
-153.133 61
-229.574 89
-306.009 60
-200.85
-222.50
-240.03
-201.97
-223.95
-241.68
H+(H2O)5
-382.568 48
0.132 74
-382.435 74
-252.19
-254.35
∆E
(kcal/mol)
-258.01
∆H
(kcal/mol)
-260.39
B3PW91/6-311++G** calculations
-76.406 771
-76.669 92
-165.13
-165.07
∆G
(kcal/mol)
BSSE
(kcal/mol)
∆G + BSSE
(kcal/mol)
0.66
-165.41
10.23
13.29
13.31
-183.96
-195.56
-203.91
11.94
-211.62
-166.43
[-158.9]22
-196.4022
-212.12
-222.76
[-205.8]22
-228.94
[-211.4]22
-166.07
[-158.9]22
-194.19
-208.85
-217.23
[-205.8]22
-223.55
[-211.4]22
a Enthalpies and free energies are at 298.15 K. Note that absolute energies (E) and zero-point energies (ZPE) are in hartrees, whereas
relative energies (∆E), enthalpies (∆H), and Gibbs free energies (∆G), as well as basis set superposition error (BSSE) corrections, are in
kilocalories per mole.
Figure 2. (a) Schematic illustration of Grotthuss proton hopping
mechanism. (b) Coordinates used for potential energy surface scan.
3. Only when R1 is small enough (R1 ) 2.4 Å) does the
the potential energy curve show a single minimum; all
other curves have symmetric double-well shapes. This
observation is in agreement with MP2 and B3LYP
results.43,44 The activation energy (Table 3), defined as
the energy difference between the maximum and minimum points of the curve, decreases significantly as R1
decreases. At fixed R1, the left minimum (Figure 3)
corresponds to the equilibrium of a left water molecule
with a right H3O+, whereas the symmetric minimum
located at the right corresponds to a left H3O+ in
equilibrium with a right water molecule. A transition
state for proton transfer corresponds to the maximum
point in Figure 3, i.e., a proton is located equidistant
between the two water molecules. From Figure 3, we
observe that, with the increase of the R1 distance, the
activation barrier for proton transfer increases sharply,
and at a certain small R1 value, the transfer is barrierfree. As pointed out earlier,19 because the zero-point
energy E0 of a O-H vibration is 5.3 kcal/mol,16 as long
as the O-O distance is smaller than 2.7 Å, the barrier
will be under E0, and the proton will be able to move
Figure 3. Potential energy surface for proton transfer between
two water molecules as a function of the OH distance (R2) and
parametric in the O-O distance (R1), as defined in Figure 2b. The
energy E is the energy difference between the energy of the H+(H2O+)n complex and the energies of the isolated monomers (H3O+
and H2O).
freely between the two solvation water molecules. In
these calculations, we do not consider the dynamics of
the process, which can also influence the proton transfer. Other studies suggest that the rate-determining
step in proton diffusion in water is the reorientation of
the H2O molecules in the first hydration shell of H3O+
to allow proton motion through the water molecules.16
The proton transfer between the H3O+‚‚‚H2O complex
then would take place by quantum mechanical tunneling.16
To investigate the effect of an electric field (EF)
on proton transfer, calculations were repeated at
O-H‚‚‚O distances R1 of 2.7 and 2.8 Å (Figure 4a and
b, respectively). The corresponding activation barriers
are listed in Table 3. From Figure 4a, it can be seen
that even a low-intensity EF destroys the symmetry of
the PES curves reported in Figure 3. In our model, an
electrical field on the x axis is created by a dipole that
4794
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Table 3. Activation Barriers Estimated from a Potential
Energy Surface Scan for Proton Transfer between Two
Water Molecules
R1 (O-O distance)a
(Å)
electric field
(V/cm)
barrier
(kcal/mol)
2.40
2.55
2.60
2.65
2.70
2.80
2.90
3.10
3.50
2.70
2.70
2.80
2.80
2.80
2.80
0
0
0
0
0
0
0
0
0
-5.14 × 106
-25.70 × 106
-5.14 × 107
-10.28 × 107
+5.14 × 107
+10.28 × 107
0
0.25
0.94
1.88
3.21
6.72
10.86
20.75
42.44
2.52
0.55
1.02
0
15.17
-
a The O-O distance is one of the scan coordinates, shown in
Figure 2b.
When a negative EF is applied (Figure 4a), both PES
minima are shifted to larger R2 distances, and the
maximum is shifted to smaller R2 distances. As schematically shown in Figure 4a, a negative field favors
the proton transfer, with the left minimum becoming
less favorable as the field EF increases because of the
proximity of H3O+ to the positive pole. Note that the
corresponding energy increases as the field intensity
increases. Therefore, after proton transfer, the system
stabilizes at a much lower energy minimum, and
consequently, the activation barrier for proton transfer
decreases significantly as the field intensity increases,
thus favoring the hopping transfer mechanism (Table
3). For example, at R1 ) 2.8 Å (Figure 4b), when the
EF intensity is -10.28 × 107 V/cm, the barrier for proton
transfer disappears, and the proton can move freely
from H3O+ to the other solvation water. When the field
direction is reversed (EF > 0), the energy of the system
increases, and the barrier increases sharply (Table 3),
with a very stable minimum corresponding to the H2O‚
‚‚H3O+ complex and a large activation barrier for proton
transfer, which increases with the field intensity. Therefore, an appropriate external electric field lowers the
activation barrier for the proton-transfer reaction and
increases its reaction rate. The reorientation of the
H3O+(H2O) complex as a result of the applied field was
not considered in the calculations, and inclusion of this
effect will possibly change the absolute value of the
barriers. It should also be pointed out that, although
our calculations of the EF effect have been performed
only at the O-H‚‚‚O distances R1 of 2.7 and 2.8 Å, a
monotonic increase of the proton-transfer barrier with
the R1 distance can be expected.
3.3. Structural Properties of the Nafion-Hydronium-Water System. The presence of water causes
dissociation of the -SO3H group of H-Nafion and water
protonation, i.e., protons are transferred from H-Nafion
to water, thus forming the -SO3-‚‚‚H3O+ ion pair
-SO3H + H2O f -SO3-‚‚‚H3O+
Figure 4. (a) Potential energy surface for proton transfer between
two water molecules in the presence of an external electric field
(EF) at R1 ) 2.70 Å. The energies are normalized with respect to
the minima at R1 ) 2.70 Å and EF ) 0. (b) Same at R1 ) 2.80 Å,
with energies normalized with respect to the minima at R1 ) 2.80
Å and EF ) 0.
is aligned with this x axis. We define a positive field as
one that goes from positive to negative along the positive
x-axis direction. Conversely, when the field is negative,
it goes from positive to negative along the negative
x-axis direction, i.e., EF always points in the direction
opposite to the dipole that originates it. In our system,
the positive x axis has the direction O1O4 (Figure 2b).
The solvation of the ion pair to form -SO3-‚‚‚H3O+(H2O)n complexes, the natural next step in aqueous
media, is reported to be irreversible.53,54 The ion-pair
stability was investigated by optimizing (HF/6-31+G**)
the neutral molecular complex structure -OCF2CF(CF3)OCF2CF2SO3H‚‚‚H2O attached to a CF2-CF2 group
representing a small portion of the backbone, as shown
in Figure 5a. In the initial configuration for geometry
optimization, the water molecule is located close to the
-SO3H group, as previous ab initio studies reported
water to be preferentially located at the terminal part
of the side chain.55 It was found that the complex,
formed mainly by H-bond interactions, is more stable
than the -SO3-‚‚‚H3O+ ion pair; therefore, at this level
of theory, there is no evidence of proton transfer from
the side chain to the water molecule. The O-H bond
(O21-H27) of the -SO3H group is stretched by about
0.02 Å because of the H-bond interaction with the water
molecule. The (-SO3)H‚‚‚O(H2) distance is 1.743 Å. The
binding energy of the neutral complex is -10.28 kcal/
mol. The enthalpy of complex formation is -10.59 kcal/
mol, and the free energy change is -2.0 kcal/mol, which
indicates that the formation of the molecular complex
is thermodynamically favorable. The entropic term is
negative and large, T∆S ) -8.59 kcal/mol, indicating
the formation of a highly ordered molecular complex.
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4795
Figure 6. HF/6-31+G**-optimized structure of Nafion pendant
anion chain.
Table 4. Geometric Parameters for Nafion Monomer
Anion Depicted in Figure 6
bond length (Å)
Figure 5. (a) HF/6-31+G**-optimized structure of H-Nafion/
water complex. (b) B3PW91/6-311++G**-optimized structure of
H-Nafion/water complex indicating proton transfer from the acid
group to the water molecule.
It is known that, although the HF geometric parameters
(bond lengths, angles, dihedral angles) are in excellent
agreement with experimental results for many systems,56 the HF method cannot capture some states
where electron correlation effects are important. For this
reason, we re-optimized the same system using DFT.
Figure 5b illustrates the results of a geometry optimization (B3PW91/6-311++G**) of the same system in
Figure 5a, and in its initial configuration, the water
molecule is located close to the -SO3H group. The O-H
bond O21-H27 of the -SO3H group is now stretched
to 1.009 Å because of the H-bond interaction, and
although the H27-O28 distance is still large (1.616 Å),
the proton transfer from the -SO3H group to H2O can
be clearly seen. The charge of H27 increases to 0.51 au,
compared to a value of 0.31 au when no water is present.
This result illustrates the effects of electron correlation
and large basis sets on ion-pair association.
To understand the solvent effect on the chain conformation, we analyzed the anion structure. The fully
optimized (HF/6-31+G**) geometry of the ionized Nafion
side chain CF2CF-OCF2CF(CF3)OCF2CF2SO3- is depicted in Figure 6. Corresponding geometric parameters
are summarized in Table 4. Most of the parameters are
in agreement with the reported calculated data.14 The
distance of the -SO3- group to the backbone (CF2-CF2)
measured between S18 and C22 (Figure 6) is 4.33 Å,
and overall, the side chain is quite folded, with an
estimated diameter of around 8 Å, in agreement with
previous B3LYP/6-31G** calculations.55 On the basis of
this gas-phase folded structure, it has been argued that
S-O
C-S
C-C
C-O
C-F
1.44
1.85
1.54
1.36
1.33
angle (°)
OSO
OSC
SCF
CCS
COC
FCO
CCF
CCC
FCF
115.72
102.65
110.30
116.81
125.23
109.57
110.18
113.87
106.71
108.64
Table 5. Thermodynamic Properties of Bulk Water
Determined From NPT Simulations at 298 K and 1 bar
Using the Dreiding Force Field
density
Evdw
Ecoulomb
EH bond
∆hvap
D
(g/cm3) (kcal/mol) (kcal/mol) (kcal/mol) (kcal/mol) (10-9 m2/s)
0.987
2.616
-12.242
-4.777
14.403
3.350
Gierke’s model,57 a 30-50 Å cluster consisting of 70
sulfonated side chains and 1000 water molecules in a
fully hydrated Nafion membrane, was impossible.55 In
the next section, we report the results of MD simulations to elucidate this point.
3.4. Dynamic Properties of the Nafion-Hydronium-Water System. To verify the accuracy of the
Dreiding force field in representing liquid water, simulations are first performed for ambient water in the NPT
ensemble. The properties obtained are summarized in
Table 5. The average density of the system is 0.987
g/cm3, within 1% of the experimental value.58 The heat
of vaporization59,60 is calculated as
∆hvap ) - Epot(inter)
where Epot(inter) includes all contributions from the
nonbonded intermolecular interactions to the potential
energy. The calculated value of ∆hvap is 14.403 kcal/mol,
which is 3.893 kcal/mol higher than the experimental
value. As seen from Table 5, the inclusion of H-bond
terms in the potential energy accounts for this overestimation. The calculated diffusion coefficient is 3.35
4796
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Table 6. Comparison of the Calculated O-O, O-H, and H-H Radial Distribution Functions of Water at 298 K and 1 bar
Using the Dreiding Force Field with Those of the SPC/E Model and Experimental Values
first peak position
first minimum position
second peak position
second minimum position
model
rOO (Å)
rOH (Å)
rHH (Å)
rOO (Å)
rOH (Å)
rHH (Å)
rOH (Å)
rHH (Å)
rOH (Å)
rHH (Å)
Dreiding
SPC/E 64
expt64
2.89
2.90
2.80
1.95
1.95
1.85
2.67
2.50
2.40
3.41
3.40
3.45
2.39
2.40
2.45
3.13
3.10
3.00
3.31
3.30
3.30
3.92
3.85
3.85
4.30
4.40
4.40
4.70
4.70
4.72
Table 7. Mulliken Atomic Charges (HF/6-31+G**) Used in
MD Simulations
a
molecule
atom
H2O
O
H
-0.84
0.42
q (e)
H3O+
O
H
-0.49
0.49
Nafiona
O19
O20
O21
O1
O11
S18
F8
F15
F17
C2
C5
C6
C12
C13
-0.93
-0.81
-0.83
-0.53
-0.19
2.08
-0.40
-0.39
-0.45
0.94
-0.11
1.70
0.83
0.58
Nafion atom numbers correspond to Figure 6.
× 10-9 m2/s, overestimated in comparison with the 2.3
× 10-9 m2/s experimental result61 at the same temperature.
We have also computed the radial distribution function (RDF), which provides structural information. The
first peak and the first minimum positions (Table 6) of
the distribution functions gOO(r), gOH(r), and gHH(r) agree
well with the experimental RDFs and SPC/E water
model results.62-64 The most important structural discrepancy between the computed and experimental RDF
is that the peak height of the computed gOO(r) is
overestimated with respect to the experimental value.64
Overall, we consider the performance of the Dreiding
model fair for our purposes of testing the relative
mobility of the hydronium in water with and without
the presence of Nafion functional groups.
Thus, to further investigate the proton-transport
process and conformational changes of the hydrophilic
Nafion side chain in the bulk solvent, MD simulations
for both the water/H3O+ and the water/SO3--terminated
Nafion monomer/H3O+ systems were performed. H3O+
is considered rather than H+ assuming that H+ motion
in water takes place either as H3O+ or as H3O+(H2O)n.
Atomic charges for water, H3O+, and the Nafion side
chain extracted from our ab initio results in Figure 6
are included in Table 7.
The calculated diffusion coefficients are collected in
Table 8, along with the MD simulation conditions,
dimensions of the simulation box, and simulation times.
Because of the limited size of the simulation box, the
diffusion coefficients reported hereafter are local, i.e.,
they reflect the influence of the neighboring functional
groups. The calculated H3O+ diffusion coefficient in
water (2 × 10-9 m2/s) is slightly lower than that for the
self-diffusion of pure water, 3.35 × 10-9 m2/s (Table 5).
Note that we report the diffusion coefficient of H3O+,
instead of that of H+. For that reason, the value is
comparable to those of a larger ion such as Rb+,65
Figure 7. Snapshot of simulation 1 (Table 8) illustrating a H3O+
solvation configuration in the liquid phase, where the background
water molecules have been removed for clarity.
whereas the experimental value15 reported for H+
diffusion in water is 9.31 × 10-9. Figure 7, a snapshot
of simulation 1 (Table 8), illustrates the dynamic
behavior of the first shell of water molecules surrounding the hydronium ion. Compared to the results shown
in Figure 1e, the distances from the H atoms in H3O+
to the water oxygen atoms have been stretched by more
than 0.3 Å. On the other hand, the three-water structure
of Figure 1d and e is distorted in Figure 7, indicating
an exchange of the water molecules located at the left
of the figure, probably because of a thermal hydrogenbond breaking in the second solvation shell, as found
recently by ab initio MD simulations.48
According to our model, at 298.15 K, H3O+ diffuses
4-4.5 times faster in pure water than in the presence
in a Nafion/water environment (Table 8). Although the
diffusion coefficients in simulations 1-3 (Table 8) were
calculated for hydronium ions not attached to the anion
group, the electrostatic interaction between the membrane and the hydronium ion, and the possible structure
change of the solvation waters because of the presence
of the ionic chain, might account for the smaller diffusion coefficient in the Nafion membrane. The proton
diffusion values in simulations 1 and 3 differ very
slightly because of the negligible difference in hydration
conditions for the Nafion chains between the two cases.
Simulation 2 was conducted at a higher temperature,
353.15 K, an upper limit for fuel cell operation. At this
temperature, the model predicts a 4-fold increase in the
H3O+ diffusion coefficient with respect to that at room
temperature (Table 8). Experimentally, Kreuer4 has
reported a 2-fold increase in the proton diffusion coefficients in highly hydrated membranes (Table 8). Thus,
our simulation results agree qualitatively with the
proton-transport behavior observed experimentally.
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4797
Table 8. Calculated and Experimental H3O+ Diffusion Coefficients in Bulk Water and Nafion/Water Systems
system
simulation
no.
unit cell
(Å)
temperature
(K)
total simulation
time (ps)
diffusion coefficient
(10-9 m2/s)
exptl value
(10-9 m2/s)
500 water molecules
0
24.64
298.15
100
2.00
Nafion (1 monomer)/
500 water molecules
1
2
24.77
24.77
298.15
353.15
250
300
0.45
1.78
1.40 at 298 K2
0.65 at 298 K4
Nafion (2 monomers)/
500 water molecules
3
25.74
298.15
250
0.50
1.2 at 353 K4
Changes of the side-chain conformation in the hydrated state of Nafion membrane with respect to the
gas-phase structure were investigated by analyzing the
distance between the first backbone carbon atom (C22
in Figure 5) and the sulfur atom (S18) in the pendant
chain. The variation of this distance during the simulation is displayed in Figure 8. Two measurements were
evaluated under the conditions of simulations 1 and 3
indicated in Table 8. The calculated gas-phase C22S18 distance (Figure 6) is 4.33 Å, whereas in simulation
1, the average distance is 7.50 Å. In simulation 2, the
distance increases to 8.15 Å. Thus, the side chain
stretches dramatically because of the interactions of the
-SO3- group with solvation water molecules and H3O+.
It can be concluded, in agreement with previous reports,
that the unfolding of side chain is favored in the aqueous
environment.55 For the two-unit Nafion oligomer, the
distance between the two neighboring -SO3- groups
during the simulation is around 13 Å. Consequently, the
side chains can be considered independent of each other.
Aldebert et al.66 reported an estimated experimental
distance of 18 Å between two -SO3- groups in a Nafion
membrane with EW ) 1100. Thus, it can be inferred
by comparison with our simulation results that the
polymeric backbone between the -SO3- groups is not
very rigid. Figure 9a and b illustrates the radial
distribution functions gSF(r) calculated at 298.15 and
353.15 K, respectively. These functions provide the
position of the F atoms surrounding the sulfonic group,
therefore indicating the structural changes of the side
chain as a function of temperature. Comparing Figure
9a and b, we observe that the positions of the first two
peaks, corresponding to the first two sets of F atoms in
Figure 6, remain unchanged from 298 to 353.15 K. The
next two peaks show only slight changes, but the last
peak is shifted to higher separations, because of the
high-temperature-induced chain elongation illustrated
in Figure 8.
At the beginning of simulation 3 (two-unit oligomer
in water at 298.15 K), one H3O+ was deliberately located
close to one of the -SO3- groups, to study the coordination of H3O+ with the sulfonic acid anion. Figure 10
illustrates that H3O+ remained coordinated to the
-SO3- group during the 250-ps simulation. On average,
H3O+ is 3.75 Å away from the -SO3- group, and this
distance shows only very slight fluctuations. Figure 11
illustrates the structure of the solvated hydronium ion
in contact with the sulfonic group, where all of the
background water molecules and part of the polymer
Figure 8. Length of the side chain measured as the distance
between C22 and S18 in Figure 6 during MD simulation; simulation states specified in Table 8.
Figure 9. (a) Radial distribution function gSF(r) illustrating the
F atoms surrounding a central S atom in the Nafion chain. (a) At
298.15 K, (b) at 353.15 K.
4798
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
Figure 10. Time evolution of the distance between the -SO3group and H3O+ from simulation 3 (Table 8).
and the energy required for additional water molecules
attached by linear hydrogen bonds reduced as n increases.
In the gas phase, when the O-O distance is less than
2.7 Å, the proton-transfer barrier is lower than the zeropoint energy for vibration of the OH bond, and the
transfer can take place readily. In the presence of an
external electric field, the symmetry of the double-wellshaped PES curve is destroyed. When the applied field
is opposite to the dipole direction of the complex, the
activation barrier diminishes dramatically.
Calculated H3O+ diffusion coefficients in bulk water
are found to be 4-4.5 times faster than those in model
Nafion/water systems. At 353.15 K, the diffusion coefficient in model Nafion/water system increases 4 times
with respect to that at ambient temperature conditions.
The Nafion sulfonic acid-terminated side chain is folded
in the gas phase, whereas in aqueous environment, MD
results reveal that much more stretched side chains are
favored and that the stretching increases when the
temperature increases from 298.15 to 353.15 K. DFT
calculations indicate that proton transfer from the acid
group is favored and that an ion complex -SO3-‚‚‚H3O+
is formed, whereas results from MD simulations show
that this contact ion pair formed is very stable and that
the proton-transfer reaction from the side chain to water
is irreversible.
Acknowledgment
This work is supported by the National Science
Foundation Career Award Grant CTS-9876065 and by
the Army Research Office Grant DAAD19-00-1-0087.
A.W. was an undergraduate participant, supported
under Grant NSF/REU EEC-9732345. The use of computational facilities at the National Center for Supercomputing Applications (NCSA) at the University of
Illinois, Urbana-Champaign, and at NERSC is gratefully acknowledged.
Literature Cited
Figure 11. Snapshot from simulation 3 illustrating the structure
and solvation of the ion pair SO3-H3O+.
structure have been removed for clarity. Although the
relevant distances between the sulfonic group and the
hydronium ion are elongated with respect to those in
Figure 5b, the ion pair remains in close contact. It can
be concluded that the contact ion pair formed between
the cation and the acid anion is very stable and that
the protonation of the solvent by H-Nafion is preferred
and irreversible, in agreement with experimental studies.54
4. Conclusions
Stable gas-phase proton hydration complexes H+(H2O)n with n ) 1-5 are formed by exothermic reactions. The major contribution to the enthalpy and free
energy of formation comes from the formation of H3O+,
(1) Heitner-Wirguin, C. Recent Advances in Perfluorinated
Ionomer Membranes: Structure, Properties and Applications. J.
Membr. Sci. 1996, 120, 1.
(2) Verbrugge, M. W.; Hill, R. F. Analysis of Promising Perfluorosulfonic Acid Membranes for Fuel-cell Electrolytes. J. Electrochem. Soc. 1990, 137, 3770.
(3) Schlick, S. Ionomers: Characterization, Theory and Applications; CRC Press: Boca Raton, FL, 1996.
(4) Kreuer, K. D. On the Development of Proton Conducting
Materials for Technological Applications. Solid State Ionics 1997,
97, 1.
(5) James, P. J.; Elliott, J. A.; McMaster, T. J.; Newton, J. M.;
Elliott, A. M. S.; Hanna, S.; Miles, M. J. Hydration of Nafion
Studied by AFM and X-ray Scattering. J. Mater. Sci. 2000, 35,
5111.
(6) Porat, Z.; Fryer, J. R.; Huxham, M.; Rubinstein, I. Electron
Microscopy Investigation of the Microstructure of Nafion Films.
J. Phys. Chem. 1995, 99, 4667.
(7) Gierke, T. D.; Hsu, W. J. The Cluster-Network Model of Ion
Clustering in Perfluorosulfonated Membranes; ACS Symposium
Series No. 180; American Chemical Society: Washington, D.C.,
1982.
(8) Gierke, T. D.; Mumn, G. E.; Wilson, F. C. Morphology of
Perfluorosulfonated Membrane Products. J. Polym. Sci. 1981, 19,
1687.
(9) Hsu, W. J.; Barkley, J. R.; Meakin, P. Ion Percolation and
Insulator-to-Conductor Transition in Nafion Perfluorosulfonic Acid
Membranes. Macromolecules 1980, 13, 198.
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4799
(10) Yeo, S. C.; Eisenberg, A. Physical Properties and Supermolecular Structure of Perfluorinated Ion-Containing (Nafion)
Polymers. J. Appl. Polym. Sci. 1977, 21, 875.
(11) Rebrov, A. V.; Ozerin, A. N.; Svergun, D. I.; Bobrva, L. P.;
Bakeev, N. F. Study of Aggregation of Macromolecules of Perfluorosulfinated Ionomer in Solution by the Small-Angle X-rayScattering Methodology. Polym. Sci. U.S.S.R. 1990, 32, 1593.
(12) Yeager, H. L.; Steck, A. Cation and Water Diffusion in
Nafion Ion Exchange Membranes: Influence of Polymer Structure.
J. Electrochem. Soc. 1981, 128, 1880.
(13) Falk, M. An Infrared Study of Water in Perfluorosulfonate
(Nafion) Membranes. Can. J. Chem. 1980, 58, 1495.
(14) Vishnyakov, A.; Neimark, A. Molecular Simulation Study
of Nafion Solvation in Water and Methanol. J. Phys. Chem. B
2000, 104, 4471.
(15) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, U.K., 1994.
(16) Conway, B. E.; Bockris, J. O. M.; Linton, H. Proton
Conductance and the Existence of the H3O‚ ion. J. Phys. Chem.
1956, 24, 834.
(17) Eikerling, M.; Kornyshev, A. A.; Stimming, U. Electrophysical Properties of Polymer Electrolyte Membranes: A Random
Network Model. J. Phys. Chem. B 1997, 101, 10807.
(18) Ohwaki, T.; Yamashita, K. Electric Field Effects on Proton
Transfer Reactions at Metal Electrodes: A DFT Study on H+(H2O)2/Pt(111) and Ag(111). Electrochemistry 1999, 67, 1214.
(19) Schmidt, R. G.; Brickmann, J. Molecular Dynamics Simulations Study of a Hydronium Ion in Liquid Water with Implementation of the Proton Transfer by Means of a Hopping Mechanism. Solid Ionics 1995, 77, 3.
(20) Grotthuss, C. J. T. d. Sur la decomposition de l’eau et des
corps qu’elle tient en dissolution a’l’aide de l’electricite galvanique.
Ann. Chim. 1806, LVIII, 54.
(21) Lim, C.; Bashford, D.; Karplus, M. Absolute pKa Calculations with Continuum Dilelectric Methods. J. Phys. Chem. 1991,
95, 5610.
(22) Meot-Ner, M.; Speller, C. V. Filling of Solvent Shells About
Ions. 1. Thermochemical Criteria and the Effects of Isometric
Clusters. J. Phys. Chem. 1986, 90, 6616.
(23) Tunon, I.; Silla, E.; Bertran, J. Proton Solvation in Liquid
Water. An Ab Initio Study Using Continuum Model. J. Phys.
Chem. 1993, 97, 5547.
(24) Lau, Y. K.; Ikuta, S.; Kebarle, P. Thermodynamics and
Kinetics of the Gas-phase Reactions: H3O+(H2O)n-1 + H2O )
H3O+(H2O)n. J. Am. Chem. Soc. 1982, 104, 1462.
(25) Lee, J.; Robinson, G. W.; Webb, S. P.; Philips, L. A.; Clark,
J. H. Hydration Dynamics of Protons from Photon Initiated Acids.
J. Am. Chem. Soc. 1986, 108, 6538.
(26) Robinson, G. W. Proton Charge Transfer Involving the
Water Solvent. J. Phys. Chem. 1991, 95, 10386.
(27) Triolo, R.; Narten, A. H. Diffraction Pattern and Structure
of Aqueous Hydrochloric Acid Solutions at 20 °C. J. Chem. Phys.
1975, 63, 3624.
(28) Newman, J. S. Electrochemical Systems; Prentice Hall:
Englewood Cliffs, NJ, 1973.
(29) Allen, M. P.; Tildesley, D. J. Computer Simulation of
Liquids; Oxford University Press: Oxford, U.K., 1990.
(30) Din, X.-D.; Michaelides, E. E. Transport Processes of Water
and Protons Through Micropores. AIChE J. 1998, 44, 35.
(31) Paddison, S. J.; Paul, R.; Zawodzinski, T. A. A Statistical
Mechanical Model of Proton and Water Transport in a Proton
Exchange Membrane. J. Electrochem. Soc. 2000, 147, 617.
(32) Eikerling, M.; Kornyshev, A. A.; Kuznetsov, A. M.; Ulstrup,
J.; Walbran, S. Mechanisms of Proton Conductance in Polymer
Electrolyte Membranes. J. Phys. Chem. B 2001, 105, 3646.
(33) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M.
W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.;
Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham,
M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Peng, C.
Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle,
E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.;
Defrees, D. J.; Baker, J.; Stewart, J. P.; Head-Gordon, M.;
Gonzalez, C.; Pople, J. A. Gaussian 94, revision E.1; Gaussian
Inc.: Pittsburgh, PA, 1997.
(34) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G.
E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery,
J. A.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J.
M.; Daniels, A. D.; Kudin, K. N.; Strain, O. F. M. C.; Tomasi, J.;
Barone, B.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.;
Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P.
Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ciolovski, J.; Ortiz, J. V.; Stefanov,
V. V.; Liu, G.; Liashensko, A.; Piskorz, P.; Komaroni, I.; Gomperts,
R.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C.
Y.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P. M.
W.; Johnson, B.; Chen, W.; Wong, M. W.; Andres, J. L.; HeadGordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98, revision 0.1;
Gaussian Inc.: Pittsburgh, PA, 1998.
(35) Perdew, J. P.; Wang, Y. Accurate and Simple Density
Functional for the Electronic Exchange Energy Generalized Gradient Approximation. Phys. Rev. B 1986, 33, 8800.
(36) Becke, A. D. Density functional thermochemsitry II. The
effect of the Perdew-Wang generalized gradient correlation
correction. J. Chem. Phys. 1992, 97, 9173.
(37) Becke, A. D. Density-functional thermochemistry. III. The
role of exact exchange. J. Chem. Phys. 1993, 98, 5648.
(38) Cerius2 User’s Manual; Molecular Simulations Inc.: San
Diego, CA, 1997.
(39) Mayo, S. L.; Olafson, B. D.; Goddard, W. A. Dreiding: A
generic force field for molecular simulations. J. Phys. Chem. 1990,
94, 8897.
(40) Berendsen, H. J. C.; Postma, J. P. M.; Gunsteren, W. F.
v.; DiNola, A.; Haak, J. R. Molecular Dynamics with Coupling to
an External Bath. J. Chem. Phys. 1984, 81, 3684.
(41) Berendsen, H. J. C.; Postma, J. P. M.; Gunsteren, W. F.
v.; Hermans, J. Intermolecular Forces; Reidel: Dordrecht, The
Netherlands, 1981.
(42) Ennari, J.; Elomaa, M.; Sundholm, F. Modeling a Polyelectrolyte System in Water to Estimate the Ion Conductivity.
Polymer 1999, 40, 5035.
(43) Ojämae, L.; Shavitt, I.; Singer, S. J. Potential Energy
Surfaces and Vibrational Spectra of H5O2+ and Larger Hydrated
Proton Complexes. Int. J. Quantum Chem: Quantum Chem. Symp.
1995, 29, 657.
(44) Ojämae, L.; Shavitt, I.; Singer, S. J. Potential Models for
Simulations of the Solvated Proton in Water. J. Chem. Phys. 1998,
109, 5547.
(45) Moller, C.; Plesset, M. S. Note on an Approximation
Treatment for Many-Electron Systems. Phys. Rev. 1934, 46, 618.
(46) Tomasi, J.; Persico, M. Molecular Interactions in Solution: An Overview of Methods Based on Continuous Distributions
of the Solvent. Chem. Rev. 1994, 94, 2027.
(47) Kalström, G. Proton Transport in Water Modeled by a
Quantum Chemical Dielectric Cavity Model. J. Phys. Chem. 1988,
92, 1318.
(48) Marx, D.; Tuckerman, M. E.; Hutter, J.; Parrinello, M. The
Nature of the Hydrated Excess Proton in Water. Nature 1999, 397,
601.
(49) Noyes, R. M. Thermodynamics of Ion Hydration as a
Measure of Effective Dielectric Properties of Water. J. Am. Chem.
Soc. 1962, 84, 513.
(50) Meot-Ner, M. Comparative Stabilities of Cationic and
Anionic Hydrogen-Bonded Networks. Mixed Clusters of WaterMethanol. J. Am. Chem. Soc. 1986, 108, 6189.
(51) Jensen, F. Introduction to Computational Chemistry;
Wiley: New York, 1999.
(52) Boys, S. F.; Bernardi, F. The Calculation of Small Molecular Interaction by the Difference of Separate Total Energies.
Some Procedures with Reduced Error. Mol. Phys. 1970, 19, 553.
(53) Laporta, M.; Pegoraro, M.; Zanderighi, L. Perfluorosulfonated Membrane (Nafion): FT-IR Study of the State of Water
with Increasing Humidity. Phys. Chem. Chem. Phys. 1999, 1, 4619.
(54) Buzzoni, R.; Bordiga, S.; Ricchiardi, G.; Spoto, G.; Zecchina,
A. Interaction of H2O, CH3OH, (CH3)2O, CH3CN, and Pyridine
with the Superacid Perofluorosulfonic Membrane Nafion: An IR
and Raman Study. J. Phys. Chem. 1995, 99, 11937.
(55) Paddison, S. J.; Zawodzinski, T. A. Molecular Modeling of
the Pendant Chain in Nafion. Solid State Ionics 1998, 113-115,
333.
(56) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A.
Ab Initio Molecular Orbital Theory; John Wiley & Sons: New York,
1986.
(57) Hsu, W. Y.; Gierke, T. D. Ion Transport and Clustering in
Nafion Perfluorinated Membranes. J. Membr. Sci. 1983, 13, 307.
(58) Handbook of Chemistry and Physics, 77th. ed.; Lide, D.
R., Ed.; CRC Press: Boca Raton, FL, 1997.
4800
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001
(59) Fuller, T. F.; Newman, J. Experimental Determination of
the Transport Number of Water in Nafion 117 Membrane. J.
Electrochem. Soc. 1992, 139, 1332.
(60) Müller-Plathe, F. An All-Atom Force Field for Liquid
EthanolsProperties of Ethanol-Water Mixtures. Mol. Simul.
1996, 19, 133.
(61) Colonomos, P.; Wolynes, P. G. Molecular Theory of Solvated Ion Dynamics. II. Fluid Structure and Ionic Mobilities. J.
Chem. Phys. 1979, 71, 2644.
(62) Koneshan, S.; Rasaiah, J. C.; Dang, L. X. Computer
Simulation Studies of Aqueous Solutions at Ambient and Supercritical Conditions Using Effective Pair Potential and Polarizable
Potential Models for Water. J. Chem. Phys. 2001, 114, 7544.
(63) Dang, L. X. Importance of Polarization Effects in Modeling
the Hydrogen Bond in Water Using Classical Molecular Dynamics
Techniques. J. Phys. Chem. B 1998, 102, 620.
(64) Chialvo, A. A.; Cummings, P. T. Microstructure of Ambient
and Supercritical Water. Direct Comparison between Simulation
and Neutron Scattering Experiments. J. Phys. Chem. 1996, 100,
1309.
(65) Oelkers, E. H.; Helgeson, H. C. Calculation of the Transport Properties of Aqueous Species at Pressures to 5 kB and
Temperatures to 1000 °C. J. Solution Chem. 1989, 18, 601.
(66) Aldebert, P.; Dreyfus, B.; Pineri, M. Small-Angle Neutron
Scattering of Perfluorosulfonated Ionomers in Solution. Macromolecules 1986, 19, 9, 2651.
Received for review May 29, 2001
Revised manuscript received August 24, 2001
Accepted August 28, 2001
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