Journal of New Materials for Materials for Electrochemical Systems 4, 197-207 (2001)
c J. New. Mat. Electrochem. Systems
The modeling of molecular structure and ion transport in sulfonic acid based ionomer
membranes
S. J. Paddison∗
Computational Nanoscience Group, Motorola Inc.
Los Alamos Research Park, 4200 W. Jemez Rd, Suite #300, Los Alamos, NM 87544, USA
( Received July 10, 2001 ; receved in revised form September 17, 2001 )
Abstract: Reported here is a molecular modeling investigation into the proton dissociation of the hydrophilic components of hydrated Nafion and
PEEKK membranes and its connection to proton transport within the membrane pores. Minimum energy conformations for trifluoromethane and
para-toluene sulfonic acids with clusters of 1–6 water molecules were obtained using ab initio electronic structure calculations. These calculations
revealed the influence of both the structure and strength of the conjugate base (sulfonate anion) on the dissociation and hydration of the acid.
Although spontaneous dissociation was observed for both sulfonic acids after the addition of three water molecules, the proton (as a hydronium
ion) is less bound for the perfluorinated system than with the aromatic system. This is due to the increased electron stabilization afforded by the
electron withdrawing of the –CF3 group. The molecular membrane specific information was used in combination with membrane morphology
data to compute proton diffusion coefficients for both Nafion 117 and 65% sulfonated PEEKK membranes at hydration levels where the number of
water molecules per sulfonic acid fixed site were: 6, 13, and 22.5; and 15, 23, and 30, respectively. The agreement with pulsed field gradient NMR
diffusion measurements was very good for both membranes across the entire range of membrane hydration, attesting to the substantial predictive
capability of the transport model.
Key words : Nafion , PEEKK, electronic structure, proton transport, mechanism
1. INTRODUCTION
The central component of the PEMFC is the proton exchange
membrane. Nafion , a perfluorinated sulfonic acid ionomer, is
the prototypical membrane because of its favorable chemical,
mechanical, and thermal properties along with high proton conductivity when sufficiently hydrated. There are, however, several draw backs to the practical use of Nafion , which include:
cost, maximum operating temperature (< 90o C), and problems
associated with the transport of water and fuel (e.g. methanol).
This has driven a number of strategies into the design of alternative materials [2].
The polymer electrolyte membrane fuel cell (PEMFC) is deemed
to be a promising alternative energy converter for both portable
and stationary power applications. Much of the progress in the
development of this type of fuel cell is due to the potential use
in vehicles. It is anticipated that the PEMFC will start to replace
the internal combustion engine within the next decade [1]. This
is dependent upon a number of engineering achievements including a reduction in the overall cost of the system. One of the
areas in which progress towards this goal is being made is in research directed towards the development of new materials (catalysts and membranes) that demonstrate improved performance
characteristics accompanied by acceptable manufacturing costs.
∗ email
Many of these novel materials are sulfonated polymers but membranes with sulfonimide functionals are under investigation for
improved thermal stability and proton conductivity [3]. These
imides, however, are faced with similar limitations as the fluorinated sulfonic acid based polymers in terms of cost and water
and methanol crossover.
: [email protected]; Fax: +1 505 663 5150
197
198
S. J. Paddison / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
Distinctly different efforts into the design of advanced and costeffective membranes include: 1) aromatic backbone polymers,
i.e. polyetherketones (PEEKK, PEEK, etc) [4-6]; 2) the inclusion of small inorganic particles like silica [7-9], or zirconium
phosphates and sulfophenylphosphates [9] within the membrane;
3) acid/base blending or covalent cross-linking of polymers
[10,11]; and 4) the complexation of basic polymers (e.g. polybenzimidazole) with oxo-acids [12,13] (e.g. phosphoric acid
[14]). The PEEKK membranes offer definite cost and stability advantages over Nafion membranes, but exhibit substantially
lower conductivity at the lower water contents. The membranes
in 2) and 3) exhibit increased thermal stability (up to 140 ◦ C)
and reduced swelling and methanol and water crossover, but at
a penalty in terms of conductivity and mechanical stability. Finally, the membranes with immobilized acid demonstrate conductivities as high as those seen in the hydrated systems [15],
but with drastically reduced methanol crossover [16].
Although there has been substantial work in the synthesis and
testing of these novel membranes, many performance features
are not well understood. Equally deficient is a fundamental,
molecular-based understanding of the mechanisms of proton
and water transport as a function of membrane morphology and
hydration. Clearly, success in the design of novel membranes
possessing all of characteristics required for use in a commercial fuel cell will require the fundamental physical and mechanistic insight generated from molecular modeling studies.
hydrophilic domains, orientation of the pores in the network,
and evolution of the pore volume with water uptake. These effects all contribute to the performance of the membrane under
fuel cell operating conditions.
The hydrated morphologies and consequent function of the
polyetherketone membranes is somewhat different from the perfluoro polymers. Small angle X-ray scattering (SAXS) experiments [5,19] suggest that there is a less pronounced separation
of the hydrophobic and hydrophilic domains than observed in
Nafion membranes. This along with the greater rigidity of the
aromatic backbone of the polymer result in narrower water filled
pores. In addition, results from pulsed-field gradient NMR measurements indicate that the electroosmotic drag and water permeation is lower in the PEEKK membranes [5,19].
Although the experimental investigations provide a qualitative
understanding of the function of the sulfonic acid based
ionomers (perfluoro and aromatic), the specific details of how
the molecular structure and hydrated morphologies connect with
the transport of protons and water through the membranes is not
at all well known. The development of novel materials meeting
all the requirements for application in fuel cells will require this
fundamental understanding of the function of existing materials.
Thus, we [20-23] have undertaken an extensive modeling effort to understand the hydration and acid dissociation in both of
these classes of ionomers with the aim of connecting these fundamental studies with the function of the hydrated membrane.
The hydrated morphology of the sulfonic acid based ionomers
has a direct bearing on the transport of protons and water in the
membrane. The perfluoro polymers combine extremely high
hydrophobicity of the polytetrafluoroethylene (PTFE) backbone
with the extremely high hydrophilicity of the sulfonic acid functional groups in a single macromolecule. In the presence of water a two-phase system forms consisting of a network of water
containing clusters or pores surrounded by the PTFE medium.
The latter provides the structural and thermal stability of the
membrane and is also responsible for the immobilization of the
dissociated sulfonic acid groups (–SO−
3 ); hence, referred to as
fixed sites. It is within the hydrophilic domains that the transport of water and protons occurs. An interfacial region, therefore, exists consisting of solvated sulfonate-terminated perfluoroether side chains. This interface separates a water region in
the central portion of the pore that is ‘bulk-like’, from the PTFE
backbones. The nature and character of the water in the pores,
however, is by no means well understood.
With either ionomer it is impossible to treat the entire polymer
in an ab initio manner (i.e. a full electron treatment with molecular orbital theory). Treatment of the polymer with empirical
or even semi-empirical methods while computationally feasible, will give conformational results for the polymer interacting
with water that are grossly incorrect due to the incorrect force
fields employed. Therefore for both ionomers the smallest subunit (that includes the hydrophilic sulfonic acid group) of the
polymer that contains the essential membrane specific molecular components is considered. For the Nafion membranes this
sub-unit is trifluoromethane sulfonic acid (triflic acid) and for
the PEEKK membranes it is para-toluene sulfonic acid. Our
modeling work [7-12] has focussed on implementing ‘first principles’ methods in establishing this understanding in sulfonic
acid based ionomer membranes; and here a comparison of the
results obtained for perfluoro membranes with those obtained
for aromatic membranes is reported.
In the modeling work of Eikerling et al [17] the mobility of
the protons occurs via two mechanisms: a surface mechanism
where proton transport proceeds along the array of acid groups
(i.e. via structure diffusion) along the interface , and a bulk
mechanism where the protons are transported with the Grotthuss mechanism. In addition they [18] attempted to model
membrane morphology aspects including connectivity of the
Thus, in the first stage of the modeling of the perfluoro and aromatic ionomers electronic structure calculations of the corresponding sulfonic acid with explicit water molecules are conducted. These computations are used to determine minimum
energy conformations revealing fundamental characteristics of
the acid dissociation and local proton dynamics. In addition
they reveal information concerning the shielding or screening
afforded by the water molecules of the first hydration shell.
The modeling of molecular structure and ion transport . / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
From these fundamental first principles calculations molecular
information is obtained that when combined with experimental studies of the hydrated morphology of the polymer, provide
a set of parameters for implementation in a water and proton
transport model [24-26]. This is the second stage of the modeling effort, and is based on the computation of the proton friction and diffusion coefficients within a PEM pore using a nonequilibrium statistical mechanical framework. An outline of this
model is presented below. Together the molecular and transport
modeling studies provide the means of connecting the molecular scale information of the polymer with the macroscopic (and
experimentally measurable) transport properties of the membrane. Of significant importance is the fact that this bridging
of the different length and time scales is accomplished without
resorting to any ‘fitting’ or adjustable parameters.
2.
2.1
All ab initio self-consistent-field (SCF) molecular orbital calculations were performed using the GAUSSIAN 98 suite of programs [27]. The majority of computations were performed on
a Linux/MPI Beowulf cluster consisting of 216 450 MHz Pentium III processors. Full geometry optimizations, using conjugate gradient methods [28], were undertaken on the CF3 SO3 H
and CH3 C6 H4 SO3 H acids without symmetry constraints using
Hartree-Fock theory with the 6-31G(d,p) split valence basis set
[29]. The HF/6-31G(d,p) minimum energy conformations were
then refined with density functional theory with Becke’s 3 parameter functional (B3LYP) [30] with the same basis set. Water molecules were then systematically added to the B3LYP/631G(d,p) minimum energy structure to obtain successively
larger water clusters of the acid (i.e. –SO3 H + nH2 Os,
1 ≤ n ≤ 6); and the same optimization protocol conducted.
Electrostatic potential derived atom centered partial charges were
obtained for the B3LYP/6-31G(d,p) minimum energy clusters
according to the CHelpG scheme [31].
2.2
The Einstein relation:
Dα =
Proton Transport Model
Complete mathematical details of our model in the form of a
derivation from first principles were presented earlier, [24-26]
and thus, only a brief outline is given here. In our model we
seek to connect the molecular structure of the hydrated polymer,
with the macroscopic (and measurable) quantity of the proton
diffusion coefficient. Factors affecting the coupled transport of
a proton and a water molecule (i.e. a hydronium ion, designated
subscript α in the model equations below) are examined in a
hydrated pore/channel of a PEM ex situ of a fuel cell configuration.
kT
ζα
(1)
establishes the inverse relationship of the diffusion coefficient
with the friction coefficient. While the Stokes relation (ζ =
6πηα) is commonly used to compute friction coefficients for
macroscopic objects moving in viscous media (and with Eq. (1)
forms the Stokes-Einstein formula), in our model we use the
methods of nonequilibrium statistical mechanics to compute the
average force experienced by a hydronium ion moving in the
pore, making use of the fundamental definition of the friction
coefficient:
hFα i = −ζ · vα ,
MODELING AND COMPUTATIONAL METHODS
Electronic Structure Calculations
199
(2)
=
where vα is the velocity (assumed to be constant) of the hydronium ion. Thus, from these two equations it should be clear
that through a computation of the average force, the diffusion
coefficient may be evaluated.
The pore of the PEM is assumed to possess a cylindrical geometry with length L and cross sectional radius R, filled with
N water molecules each possessing a dipole moment µ. The
dissociated sulfonic acid functional groups (–SO−
3 ) in the pore
are modeled as n radially symmetric axially periodic arrays of
fixed ions (i.e. point charges) each possessing a charge of -e.
Now the average force experienced by a hydronium in such a
pore maybe calculated from the standard relation (from statistical mechanics):
Z
hFα i(rα ) =
dr dp Fα (rα , r)ρ(rα , p, r),
(3)
where rα denotes the position of the hydronium ion; and the
average (integration) is over the position, r, and conjugate momentum, p, of all the N water molecules, of the net force on the
hydronium weighted with a phase space distribution function
ρ(rα , p, r). This distribution function is obtained from the more
general time-dependent distribution function, a solution of the
time evolution or Liouville equation:
i
∂ρ(rα , p, r, t)
= Lo ρ(rα , p, r, t),
∂t
(4)
where Lo is the Liouville operator for our pore system with a
coordinate reference system moving with constant velocity vα .
The Liouville operator is defined by the Poisson bracket:
Lo = i{Ho (rα , p, r), },
(5)
200
S. J. Paddison / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
where Ho (rα , p, r) is the Hamiltonian for the pore. The total
energy of the pore will consist of the kinetic energy of all the
water molecules and the net potential energy, V (rα , r), due to
two-body interactions of the water molecules, hydronium ion,
and fixed sites according to:
Ho (rα , p, r) =
N
X
m(vi + vα )2
2
i=1
+ V (rα , r)
(6)
where m is the mass and vi the velocity of the ith water molecule.
The latter term in Eq. (6) consists of the following four terms:
V (rα , r) = −
N
X
i=1
+
N
X
i<j
−
µ2 e2
1
+ Ψo cos
48π 2 ε2 kT |rα − ri |4
2πnzα
L
2µ4
1
2
3(4π) kT |ri − rj |6
N
X
2πµΨo n
i=1
eL
sin
2πnzi
L
(7)
where ε is the permittivity of the water in the pore, k the Boltzmann constant, T the temperature, and Ψo the amplitude of the
potential energy due to interaction of the hydronium ion with
the –SO−
3 groups (see Ref. [26] for details). These respective
contributions to the potential energy of the system are due to:
1) interactions of the hydronium ion with the water molecules;
2) interaction of the hydronium ion with the arrays of the fixed
sites; 3) water-water interactions; and 4) interactions of the water molecules with the fixed sites. A formal solution of Eq. (4)
is:
−iLo t
ρ(rα , p, r, t) = e
ρ(rα , p, r, 0)
= e−iLo t ρeq (rα , p, r),
(8)
where ρeq (rα , p, r) is the distribution function under equilibrium conditions (see Ref. [25] for details of its exact form). A
non-equilibrium stationary state (moving with the ion) and described by the distribution function in Eq. (3), is obtained in the
limit of t → ∞ in Eq. (8) (see once again Ref. [25] for the
exact form). The total force required in Eq. (3) is determined
by taking the action of the Liouville operator on the momentum
of the hydronium ion. Combining these results, one obtains an
expression for the scalar friction coefficient of the hydronium
that consists of four force-force correlation functions:
ζα =
β
3
Z
∞
dt (hFαs e−iLo t Fαs i0 + hFαs e−iLo t Fps i0
0
+ hFαp e−iLo t Fps i0 + hFαp e−iLo t Fαs i0 ) (9)
where β = 1/kT , and the forces Fαs , Fps , Fαp are between:
the hydronium ion and the water molecules; the fixed sites and
the water molecules; and the hydronium ion and the fixed sites,
respectively. We explicitly evaluate only the latter three terms in
Eq. (9), taking their sum to be a correction, ζ (c) , to the friction
coefficient of the proton in bulk water, i.e.:
ζ (c) = ζ2 + ζ3 + ζ4
(10)
Examination of the first force-force correlation function, ζ1 , indicates it involves only the force the water exerts on the hydronium ion (Fαs ) and so is taken to be either the friction coefficient of a hydronium ion in bulk water calculated with the
Stokes relation, or the friction coefficient of a proton in bulk
water derived from experimental diffusion measurements. The
choice of the numerical value of ζ1 is dependent on the characteristics of water (in the pore) through which the proton moves
and is discussed below.
3.
RESULTS AND DISCUSSION
3.1
Explicit water cluster calculations
As indicated earlier, from the B3LYP/6-31G** minimum energy structure of the acid a single water molecule was brought
in proximity to the sulfonic acid portion of the molecule, and
optimizations performed: first at the HF/6-31G** level and then
at the B3LYP/6-31G**. Successive water molecules were then
added one at a time without any bias or constraints as to the
starting geometry prior to the optimization.
3.1.1
Trifluoromethane sulfonic acid
The minimum energy conformations (at the B3LYP/6-31G**
level) of triflic acid with the successive addition of six water
molecules are presented in Figures 1 and 2. Structural parameters consisting of the –O H distance (i.e. oxygen of the sulfonic acid/sulfonate group to the acidic hydrogen) and the –
O· · · H· · · OH2 distance (again, the distance from the oxygen
on the sulfonic acid/sulfonate group to the oxygen of the water molecule/hydronium ion) are collected in Table 1. The atom
types are designated in the first figure, i.e. Figure 1(a). It is
interesting to note that despite the fact that CF3 SO3 H is a ‘superacid’ no dissociation of the proton occurs with either the addition of the first water molecule or even after a second water molecule is added; the former result was reported earlier
[20]. The CF3 SO3 H + H2 O conformation shows that the water molecule forms a somewhat shorter (than typical, ∼2.8 Å)
The modeling of molecular structure and ion transport . / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
201
Table 1: Structural parameters for water clusters of CF3 SO3 H + nH2 Os and CH3 C6 H4 SO3 H + nH2 Os (n=1–6)
Number of Water
Molecules, n
0
1
2
3
4
5
6
CF3 SO3 H
r(-SO2 O-H· · · OH2 ) r(-SO2 O· · · H· · · OH2 )
Å
Å
0.973
–
1.020
2.595
1.059
2.496
1.562
2.556
1.721
2.658
1.739
2.693
3.679
4.243
CH3 C6 H4 SO3 H
r(-SO2 O-H· · · OH2 ) r(-SO2 O· · · H· · · OH2 )
Å
Å
0.972
–
1.007
2.650
1.033
2.564
1.437
2.488
1.455
2.500
1.433
2.487
3.196
3.645
hydrogen bond with the acidic proton and adopts an overall ‘sixmembered ring’ with the –SO3 H group. Table 1 reveals that the
–SO3 —H bond distance has increased by 0.086 Å after the second water molecule has been added over that observed in the
minimum energy conformation of CF3 SO3 H (0.973 Å). However, no dissociation of the proton is observed even after a second water molecule is added.
Figure 2: Fully optimized (B3LYP/6-31G**) conformations
of water clusters of Triflic acid: a) CF3 SO3 H + 4 H2 O; b)
CF3 SO3 H + 5 H2 O; b) CF3 SO3 H + 6 H2 O.
Figure 1: Fully optimized (B3LYP/6-31G**) conformations of
water clusters of Triflic acid: a) CF3 SO3 H + H2 O; b) CF3 SO3 H
+ 2 H2 O; b) CF3 SO3 H + 3 H2 O.
After a third water molecule is added, see Figure 1(c), a spontaneous dissociation of the acidic proton is observed during the
B3LYP/6-31G** optimization. The formation of a hydronium
ion is favored through the formation of hydrogen bonds with the
two water molecules and one of the oxygens of the now formed
triflate anion. The dissociated state is adopted as a result of the
excess positive charge being stabilized in the hydrogen bonding
network, and the excess electron density (due to the breaking
of the –SO3 —H bond) sufficiently delocalized by the electron
withdrawing –CF3 group. It is the combination of these two
effects that result in a minimum energy conformation for the
cluster showing a dissociated proton. The separation as measured by the distance of the oxygen on the hydronium ion to the
sulfonate oxygen from which the proton left, is about the mean
of that observed in Eigen (H9 O4 + , 2.60 Å) and Zundel (H5 O2 + ,
2.50 Å) ions in bulk water [32].
The clusters formed with four and five water molecules, see Figure 2 (a) and (b), are similar to that observed with three water
molecules in that the hydronium ion forms a contact ion pair
with the triflate anion. However, the hydronium ion adopts a
position further away from the anion as the number of water
molecules is increased from 3 to 5 (see Table 1).
Finally, when the sixth water molecule is added we witness a
complete separation of the excess proton (as a hydronium ion)
202
S. J. Paddison / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
from the anion (Figure 2 (c)). This result was consistently observed with optimizations began from several different starting
geometries. Here the hydronium ion forms a true Eigen ion as
it is hydrogen bonded to three water molecules with an average O—O distance of 2.56 Å. Of additional significance is the
observation that the hydronium ion is nearly twice the distance
away from the anion (4.243 Å) as was observed in the contact
ion pair minimum energy conformations (2.556 Å – 2.693 Å).
This suggests that with sufficient water (i.e. with 6 H2 O’s) the
proton is shielded from direct electrostatic interaction with the
sulfonate anion. Clearly, in the context of the hydrophilic terminations of a Nafion polymer, this observation will have direct
consequences on the conductivity of the membrane. This result
also suggests that the ‘first’ hydration shell of the triflate anion
is made up of five water molecules.
in the structural parameters of the various water clusters.
Figure 4: Fully optimized (B3LYP/6-31G**) conformations of
water clusters of p-Toluene sulfonic acid: a) CH3 C6 H4 SO3 H +
4 H2 O; b) CH3 C6 H4 SO3 H + 5 H2 O; b) CH3 C6 H4 SO3 H + 6
H2 O.
Figure 3: Fully optimized (B3LYP/6-31G**) conformations of
water clusters of p-Toluene sulfonic acid: a) CH3 C6 H4 SO3 H
+ H2 O; b) CH3 C6 H4 SO3 H + 2 H2 O; b) CH3 C6 H4 SO3 H + 3
H2 O.
3.1.2
Para-toluene sulfonic acid
The fully optimized structures (at the B3LYP/6-31G** level) of
para-toluene sulfonic acid with the successive addition of one
through six water molecules are displayed in Figures 3 and 4.
Once again, the atom types have been identified in the first figure, Figure 3(a). In comparing these minimum energy conformations with those obtained with triflic acid one notices a number of qualitative similarities: 1) the conformation with a single water molecule adopts the same ‘six-membered ring’ structure with the –SO3 H group; 2) upon the addition of the third
water molecule, the proton spontaneously dissociates from the
acid forming a hydronium ion, hydrogen bonded to two water molecules; and 3) separation of the hydronium ion from the
sulfonate anion does not occur until six water molecules are
added. There are, however, important quantitative differences
A similar set of structural parameters as was tabulated for triflic acid are collected for para-toluene sulfonic acid in Table
1. The oxygen–hydrogen bond distance in the minimum energy conformation of both acids without the addition of any water molecules is essentially the same (0.97 Å). However, after
the addition of the first and second water molecules this bond
stretches to a greater extent in triflic acid (compare columns 2
and 4 in Table 1) than with para-toluene sulfonic acid. In addition, the water molecule, hydrogen bonded to the acidic proton,
adopts a closer position to the sulfonic acid (compare columns
3 and 5 in Table 1) for the former system. After dissociation
of the proton occurs (at n=3) the opposite is observed. Here,
the hydronium ion in the water clusters of the aromatic sulfonic
acid adopts a position that is closer to the sulfonate anion than is
observed in the perfluorinated sulfonic acid. This trend is continued with the further addition of water molecules; and after
separation of the ions occurs (n=6), the difference in the separation distances of the two ions is substantially greater with triflic
acid. All of these conformational differences may be rationalized in terms of the differences in the strength of the acids and
conjugate bases.
Triflic acid is a superacid and substantially stronger (in a Lewis
acid sense) than para-toluene sulfonic acid. Upon dissociation
of the proton, electron density on the sulfonate anion is delocalized in both systems. However, in the aromatic system it is
delocalized in the π-ring; and in the perfluorinated system it
is withdrawn and stabilized by the –CF3 group. The conjugate
Lewis base (i.e. the sulfonate anion) that is formed is stronger in
the case of the para-toluene sulfonate anion than for the triflate
anion. The strength of the conjugate base has a direct bearing
The modeling of molecular structure and ion transport . / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
on the position of the dissociated proton: the proton will interact more strongly in the case of the stronger conjugate base, i.e.
para-toluene sulfonate. This will be quantified as a function of
the number of water molecules in the acid/anion cluster through
examination of the computed partial atomic charges.
3.1.3
Partial charge distributions in the acid-water clusters
The atom centered partial charges and dipole moments (as calculated with the CHelpG routine – see above) of the water clusters of CF3 SO3 H and CH3 C6 H4 SO3 H are presented in Tables
2 and 3. Examination of Table 2 shows the substantial positive
charge residing on the sulfur atom and the negative charge on
the oxygen atoms. The charge on the former decreases significantly after dissociation, while the negative charge on the latter
increases upon dissociation of the proton. It is also interesting
to note that there remains slightly more negative charge on the
oxygen from which the proton resided through out the addition
of the water molecules than on the other two sulfonate oxygens.
203
one would possibly expect to see greater differences in the electron density residing on these atoms in light of the structural
differences of the minimum energy conformations. However,
examination of the total negative charge residing on the sulfonic/sulfonate oxygen atoms (see the sixth column in Tables
2 and 3) shows the difference between the two systems: there
is consistently greater negative charge on the oxygens in the
aromatic clusters. This reveals (in a quantitative manner) the
increased basicity of the CH3 C6 H4 SO3 ¯ anion. The strength of
the conjugate Lewis base for either anion, therefore, is manifest
in the position adopted by hydronium ion: a stronger conjugate
basis will result in a closer (through electrostatic interaction)
equilibrium position for the hydronium ion.
Table 2: Atom centered partial charges and dipole moment for
CF3 SO3 H + nH2 Os (n=1–6)
Number
Atom,
of Water
–SO3 H
Molecules
n
S
O†
O
O
0
0.8436
-0.4866
-0.4308
-0.3794
1
0.7950
-0.5085
-0.4652
-0.3889
2
0.8460
-0.5178
-0.5074
-0.4106
3
0.6484
-0.5373
-0.4887
-0.4879
4
0.6185
-0.5226
-0.5134
-0.4881
5
0.5812
-0.5163
-0.4887
-0.4533
6
0.6376
-0.4978
-0.4703
-0.4703
†Oxygen atom to which acidic proton is/was bonded.
Total charge
on oxygen
atoms
Dipole
Moment
-1.2968
-1.3626
-1.4358
-1.5139
-1.5241
-1.4583
-1.4384
2.7219
3.3996
4.2398
4.0042
1.8784
2.6558
4.8486
Table 3: Atom centered partial charges and dipole moment for
CH3 C6 H4 SO3 H + nH2 Os (n=1–6)
Number
Atom,
of Water
–SO3 H
Molecules
n
S
O†
O
O
0
0.9136
-0.5482
-0.4483
-0.4861
1
0.7884
-0.5636
-0.4520
-0.5041
2
0.7354
-0.5803
-0.4884
-0.5048
3
0.7459
-0.5947
-0.5512
-0.5511
4
0.8359
-0.6048
-0.5995
-0.5538
5
0.8121
-0.6108
-0.5882
-0.5573
6
0.6556
-0.5493
-0.5452
-0.5073
†Oxygen atom to which acidic proton is/was bonded.
Total charge
on oxygen
atoms
Dipole
Moment
-1.4826
-1.5197
-1.5735
-1.6970
-1.7581
-1.7563
-1.6018
4.8514
2.6626
3.4458
2.0463
1.0546
3.1181
5.6153
Figure 5: Total atomic charge as computed with CHelpG routine
in G98 [27] for the acid or anion as a function of the number
water molecules in the cluster.
3.1.4
The total charge on the sulfonic acid prior to dissociation
of the proton and total charge on the sulfonate anion following
dissociation, is plotted as a function of the number
of water molecules in the cluster in Figure 5. It is interesting to note the similarity in the negative excess charge lying
on the atom centers of the molecule throughout the range of associated water molecules: there is only slightly more negative
charge
on
CF3 SO3 H/CF3 SO−
than
residing
on
3
CH3 C6 H4 SO3 H/CH3 C6 H4 SO3 ¯. This is a little surprising as
Binding energies in the acid-water clusters
Incremental water binding energies were calculated for the addition of the six water molecules to both sulfonic acids according
to the relation:
∆Eb = E [acid(H2 O)n ] − E [H2 O]
− E [acid(H2 O)n−1 ] (11)
where all energies (total electronic) are those computed at the
B3LYP/6-31G** level. The numerical values are presented in
Table 4. In comparing the two sulfonic acids, these binding energies indicate that generally the water molecules in the clusters
with triflic acid are more loosely bound to the acid than with
para-toluene sulfonic acid. One also notices that with the aromatic acid there is very little difference in the binding energies
204
S. J. Paddison / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
for the addition of the first 5 water molecules (∼-16 kcal/mol);
the sixth water molecule is much more loosely bound having a
substantially (> 5 kcal/mol) lower binding energy. This latter
observation would suggest that the first hydration shell for the
p-toluene sulfonic acid probably consists of 5 water molecules.
Table 4: Incremental binding energies (∆BE) and standard free
energies (∆Go ) (kcal mol−1 ) for the addition of the nth water
molecule to CF3 SO3 H and CH3 C6 H4 SO3 H
Number of Water
Molecules, n
CF3 SO3 H
CH3 C6 H4 SO3 H
∆BE ∆Go ∆BE
∆Go
1
-17.4 -5.6 -16.1
-3.7
2
-15.6 -5.3 -16.7
-7.0
3
-21.0 -6.1 -16.4
-1.5
4
-16.6 -4.4 -16.0
-5.2
5
-20.3 -6.6 -16.2
-2.0
6
-19.2 -5.6 -21.3
-7.2
The change in the standard Gibbs free energy for the reactions
described by Eq. (11) are also tabulated in Table 4. These were
calculated from Go ’s computed with G98 [27] under the harmonic oscillator approximation. It is interesting to note that
only small fluctuations are observed in the ∆Go ’s for triflic acid
while the ∆Go ’s computed for the toluene sulfonic acid show
substantial variance. Comparing these free energy changes reveals that the dissociation of the proton (@ n = 3 for both acids)
is much more favorable in the perfluorinated acid (- 6.1 vs.
-1.5 kcal/mol).
3.2 Computation of the proton diffusion coefficients
Friction and diffusion coefficients were computed for both
Nafion 117 and 65% sulfonated PEEKK membranes at ambient temperature (298.15 K), each at three distinct water contents. The input parameters needed in these computations were
taken from small-angle X-ray scattering (SAXS) measurements
[5,19,33]. This information is collected together in Table 5
and includes specifically: the radius of the pore (R), the length
of the pore (L), the total number of water molecules in the
pore (N), the total number of fixed sites in the pore (fs ), the
number of axially positioned radially symmetric arrays of sulfonate groups (n), the average separation distance of the sulfonate groups (dSO− ), the average radial distance the hydro3
nium ion is from the sulfonate groups (τ ), and the amplitude of
the periodic potential (Ψo ). The numerical value of the last parameter is based on the assumption that the dominant contribution to the proton mobility is due to the hydronium ion moving
along the center of the pore.
Examination of the parameters in Table 5, clearly shows that at
similar degrees of hydration, the Nafion membrane pores are
of much larger diameter. It is also apparent that with increas-
ing membrane hydration, the size of the water-ion domains (i.e.
pores) increases beyond that due solely to the increase in the
number of water molecules associated with each fixed site. This
trend is witnessed for the perfluorinated and aromatic membranes and is tracked by observing the increase in the value of
the total number of fixed sites (fs ) and the total number of water
molecules (N). With the aromatic membranes, the average separation distance of the sulfonate groups remains constant (9 Å)
over the range of hydration. This is not observed in the Nafion
membranes, where with increasing hydration there is observed
an increase in the separation of the fixed sites. Finally, it is
worth noting that the amplitude (and thus strength) of the electrostatic field due to the presence of the anionic groups along the
walls of the pore ranges over nearly two orders of magnitude.
Clearly, the increase in the radius of the pore has a substantial
impact on the electrostatic field experienced by a hydronium ion
moving along the center of the pore.
The friction coefficient correction terms (ζ2 , ζ3 , ζ4 - see Eq.
(10)) were computed at equally spaced intervals of 1 Å along the
length of each pore. Average values of these correction terms,
along with the selected ‘base value’ of the friction coefficient,
ζ1 , were then used to calculate the proton diffusion coefficient
according to Eq. (1). The choice of the value of ζ1 is explained
and justified below.
3.2.1 Hydrated Nafion
The choice of the ‘base value’ of the friction coefficient (ζ1 ) is
based on a consideration of the character as revealed in the permittivity of the water in the pore. The numerical value of ζ1 was
taken to be either that computed with the Stokes relation for a
hydronium ion in bulk water (2.69×10−12 kg s−1 ), or that derived from experimental diffusion measurements of a proton in
bulk water (4.42×10−13 kg s−1 ) [34]. For membrane hydration
levels where the water in the pores is relatively bound through
hydrophilic/electrostatic interactions with the –SO−
3 groups, the
proton is transported as a H3 O+ suggesting the appropriateness
of the former value of ζ1 . For membranes at higher water contents, where the water in the pores is more like bulk water, transport of the proton will occur (at least to some extent) through
transfer from water molecule to water molecule (i.e. the Grotthuss mechanism); and therefore, the latter value for ζ1 is the
applicable value for inclusion for this transport mechanism
To quantify the character of the water, we recently constructed
a model using an equilibrium statistical mechanical formulation
to compute the permittivity of water in PEM pores as a function
of: 1) pore size; 2) distribution of the fixed sites; and 3) radial
separation distance of the proton from the anionic groups [35].
The permittivity of water in the pores of Nafion membranes at
water contents of λ = 6, 13, 22.5 computed with this model suggest that much of the water at the two lower water contents is
bound through interaction with the –SO−
3 fixed sites. This is
The modeling of molecular structure and ion transport . / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
205
Table 5: Transport model input parameters for Nafion and 65% sulfonated PEEKK membrane pores.
Parameter
Nafion 117
65% sulfonated PEEKK
Water content, λ
6
13
22.5
15
23
30
R, Å
8
14
16
7
9.5
12
L, Å
30
56
64
40
48
56
N
216
1001
1800
375
828
1470
fs
36
77
80
25
36
49
n
6
8
8
5
6
7
dSO− , Å
6
7
8
9
9
9
3
τ, Å
4
10
12
6.4
9
11
−Ψo , J
5.17×10−22 2.10×10−23 4.67×10−24 3.63×10−22 3.69×10−23 7.48×10−24
in contrast to the results computed for fully hydrated Nafion
where a significant fraction (in the center of the pore) of the water is similar to bulk water. With these results, the value for ζ1 at
the two lower hydration states was taken to be that derived with
the Stokes relation, and for that at the highest water content, the
value derived from experimental proton diffusion measurements
in bulk water.
Computed proton diffusion coefficients, Dα , for Nafion membranes are compared with experimentally measured values, Dexp ,
[5,19,36] in a plot as a function of water content in Figure 6.
This comparison reveals that in all cases the calculated values
are slightly lower (specifically, from 8% to 15% lower) than the
experimental values. The agreement with the experiments is
nevertheless quite good, probably within the error of the measurements. We had previously suggested that this may in part
be due to the treatment of the transport of the proton in a classical manner (i.e. via the vehicular mechanism) and the consequent failure to account for transport via the Grotthuss mechanism [25]. While no contribution from intermolecular (waterwater) proton transfer was included in the calculation of proton
diffusion coefficients for the Nafion membrane pores (at λ=6
and 13), this was included for the fully hydrated membrane (i.e.
λ=22.5) in the choice of ζ1 . Despite this, the computed values
of Dα are still lower than suggested by experiment. Possible
reasons for this will be discussed in the next section.
3.2.2
Hydrated 65% sulfonated PEEKK
A similar criteria was used to select a value for ζ1 for the PEEKK
membranes at each of the water contents; i.e. calculation of the
permittivity of the water in each pore with our recently derived
equilibrium statistical mechanical model [35]. This theoretical
investigation indicated that the relative permittivity of the water
in the center of a pore at the lowest water content (i.e. λ=15)
is about 67; approximately 16% less than that of bulk water;
while at the two higher water contents, the permittivity of the
water is 80 even for distances of up to 1.5 Å from the center
of the pore. Based of these results ζ1 was taken to be that computed with the Stokes relation for a hydronium ion in bulk water
Figure 6: Computed and experimental proton diffusion coefficients in Nafion 117 and 65% sulfonated PEEKK membranes
as a function of water content.
(2.69×10−12 kg s−1 ) for λ=15, and that derived from experimental diffusion measurements of a proton in bulk water
(4.42×10−13 kg s−1 ) for λ=23 and 30.
The computed proton diffusion coefficients for the 65% sulfonated PEEKK membranes are plotted in Figure 6 along with
the corresponding experimentally measured values [5,19]. Once
again the agreement with the pulsed field gradient NMR measurements is good, and the calculated values lower than the experimental values.
This consistent result of the calculated values being smaller than
the experimental values suggests that the model over estimates
the effect of the –SO−
3 groups in retarding the mobility of the
proton. The electrostatic field generated by the anionic fixed
sites is probably too high (in the model) due to the neglect of
the presence of the additional protons. Clearly, these protons
206
S. J. Paddison / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
will increase the shielding, over that due only from the water molecules, of the hydronium ion from interaction with –
SO−
3 groups and reduce the effects of the latter on the water
molecules in the pore. Thus, inclusion of additional protons
in our model will result in a decrease in the magnitudes of the
computed friction coefficient correction terms and a consequent
increase in the calculated proton diffusion coefficient. The main
reason the presence of additional protons were not included in
the original formulation of the transport model was due to the
fact that the distribution of these protons is not known to any
degree of certainty. While others have assumed a Boltzmann
distribution for the protons within the pore [17,18] this neglects
proton dissociation effects due to differences in conjugate anionic bases as suggested by the explicit water calculations presented earlier. Furthermore, it has been recently argued that
such a continuum distribution overestimates shielding effects in
pores with radii of less than two Debye lengths [37]. Nevertheless, the inclusion of additional protons in the model is an
important aspect that needs to be addressed in future work.
4.
CONCLUSION
Proton dissociation, stabilization, and separation were examined through a series of explicit water electronic structure calculations for hydration of trifluoromethane sulfonic acid and
para-toluene sulfonic acid. These first principles computations
have provided molecular scale information of acidity and local proton dynamics occurring in hydrated Nafion and PEEKK
membranes. The minimum energy conformations of the water
clusters with the acids indicate that due to the greater electron
stabilization afforded via the electron withdrawing –CF3 group
(over the aromatic π-ring), the proton (as an hydronium ion) of
the perfluorinated acid is stabilized through hydrogen bonding
with neighboring water molecules within the cluster at a greater
distance from the conjugate base than observed in the aromatic
acid. In addition, these first principles results indicate that once
the first hydration shell is constructed around the sulfonic acid,
more effective shielding of the proton from the anion by the water molecules is witnessed with the perfluorinated system. This
suggests that at relatively low hydration levels proton mobility
will be greater in Nafion than in PEEK membranes. Experiments support this theoretical finding.
These results were then used in conjunction with hydrated polymer morphologies derived from SAXS measurements as molecular/polymer specific input in a water and proton transport model
employing a non-equilibrium statistical mechanical framework.
This model was used to compute proton diffusion coefficients
for Nafion 117 and 65% sulfonated PEEKK membranes at hydration levels where the number of water molecules per sulfonic
acid fixed site were: 6, 13, and 22.5; and 15, 23, and 30, respectively. The agreement with pulsed field gradient NMR diffusion measurements was very good for both membranes across
the entire range of membrane hydration. This investigation has
demonstrated the substantial predictive capability of the transport model as there are no ‘fitting’ or adjustable parameters in
the model, and no attempt was made to match the experimental
values.
REFERENCES
[1] D. P. Wilkinson, Interface 10, 22 (2001).
[2] O. Savadogo, J. New Mater. Electrochem. Syst. 1, 47
(1998).
[3] J. J. Sumner, S. E. Creager, J. J. Ma, and D. D. DesMarteau, J. Electrochem. Soc. 145, 107 (1998).
[4] K. D. Kreuer, Solid State Ionics 97, 1 (1997).
[5] K. D. Kreuer, J. Membr. Sci. 185, 29 (2001).
[6] G. Alberti, M. Casciola, L. Massinelli, and B. Bauer, J.
Membr. Sci. 185, 73 (2001).
[7] R. A. Zoppi, I. V. P. Yoshida, and S. P. Nunes, Polymer 39,
1309 (1997).
[8] A. S. Aricò, P. Cretí, P. C. Antonucci, V. Antonucci, Electrochem. Solid State Lett. 1, 66 (1998).
[9] B. Bonnet, D. J. Jones, J. Rozière, L. Tchicaya, G. Alberti, M. Casciola, L. Massinelli, B. Bauer, A. Peraio, E.
Ramunni, J. New Mater. Electrochem. Syst. 3, 87 (2000).
[10] J. Kerres, A. Ullrich, F. Meier, T. Häring, Solid State Ionics 125, 243 (1999).
[11] W. Zhang, G. Dai, J. Kerres, Acta Polym. Sin. 5, 608
(1998).
[12] J. C. Lassègues, in Proton Conductors: Solids, Membranes and Gels – Materials and Devices, Ed. P. Colomban, Cambridge University Press, Cambridge, MA, 1992,
pp. 311-328.
[13] J. S. Wainwright, J.-T. Wang, R. F. Savinell, M. Litt, H.
Moaddel, C. Rogers, J. Electrochem. Soc. 94, 255 (1994).
[14] T. Dippel, K. D. Kreuer, J. C. Lassègues, D. Rodriguez,
Solid State Ionics 61, 41 (1993).
[15] A. Bozkurt, M. Ise, K. D. Kreuer, W. H. Meyer, G. Wegner, Solid State Ionics 125, 225 (1999).
[16] J. T. Wang, S. Wasmus, R. F. Savinell, J. Electrochem. Soc.
143, 1225 (1996).
[17] M. Eikerling, A. A. Kornyshev, A. M. Kuznetsov, J. Ulstrup, and S. Walbran, J. Phys. Chem B 105, 3646 (2001).
[18] M. Eikerling, A. A. Kornyshev, and U. Stimming, J. Phys.
Chem. 101, 10807 (1997).
The modeling of molecular structure and ion transport . / J. New Mat. Electrochem. Systems 4, 197-207 (2001)
[19] M. Ise, Ph.D.-Thesis, University Stuttgart, 2000.
[20] S. J. Paddison, L. R. Pratt, T. A. Zawodzinski, and D. W.
Reagor, Fluid Phase Equilibria 150, 235 (1998).
[21] S. J. Paddison and T. A. Zawodzinski Jr., Solid State Ionics
115, 333 (1998).
[22] S. J. Paddison, L. R. Pratt, and T. A. Zawodzinski Jr., J.
New Mater. Electrochem. Sys. 2, 183 (1999).
[23] S. J. Paddison, L. R. Pratt, and T. A. Zawodzinski Jr.,
in “Proton Conducting Membrane Fuel Cells II”, Eds. S.
Gottesfeld and T. F. Fuller, The Electrochemical Society
Proceedings Series, Pennington, NJ, 98-27, 1999, p. 99.
[24] S. J. Paddison, R. Paul, and T. A. Zawodzinski Jr., in “Proton Conducting Membrane Fuel Cells II”, Eds. S. Gottesfeld and T. F. Fuller, The Electrochemical Society Proceedings Series, Pennington, NJ, 98-27, 1999, p. 106.
[25] S. J. Paddison, R. Paul, and T. A. Zawodzinski Jr., J. Electrochem. Soc. 147, 617 (2000).
[26] S. J. Paddison, R. Paul, and T. A. Zawodzinski Jr., J.
Chem. Phys., 115, 7753 (2001).
[27] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski,
J. A. Montgomery Jr., R. E. Stratmann, J. C. Burant, S.
Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M.
C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R.
Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford,
J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, D. K. Malick, A. D. Rabuck, K. Raghavachari,
J. B. Foresman, J. Cioslowski, J. V. Ortiz, A. G. Baboul,
B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R. L. Martin, D. J. Fox, T. Keith,
M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong,
J. L. Andres, C. Gonzalez, M. Head-Gordon, E. S. Replogle, and J. A. Pople, Gaussian 98 (Revision A.9), Gaussian Inc., Pittsburgh, PA, 1998.
[28] H. B. Schlegel, J. Comp. Chem., 3, 214 (1982).
[29] P. C. Hariharan, J. A. Pople, Theo. Chim. Acta., 28, 213
(1973).
[30] A. D. Becke, J. Chem. Phys., 98, 5648 (1993).
[31] C. M. Breneman, K. B. Wiberg, J. Comp. Chem., 11, 361
(1990).
[32] M. E. Tuckerman, D. Marx, M. L. Klein, and M. Parrinello, Science 275, 817 (1997).
[33] T. D. Gierke, G. E. Munn, and F. C. Wilson, J. Polym. Sci.
Polym. Phys. 19, 1687 (1981).
207
[34] David R. Lide, Editor-in-Chief, Handbook of Chemistry
and Physics, 80th Ed., p. 5-94, CRC Press, Boca Raton
(1999).
[35] R. Paul and S. J. Paddison, J. Chem. Phys., 115, 7762
(2001).
[36] T. A. Zawodzinski, J. Davey, J. Valerio, and S. Gottesfeld,
Electrochim. Acta 40, 297 (1995).
[37] B. Corry, S. Kuyucak, and S. H. Chung, Chem. Phys. Lett.
320, 35 (2000).