Computer Simulation Study of Model Nafion Membrane in Water ... · consistent field theory (SCFT) [33], we basically confirmed the conclusions of the spheroidal geometry of ionomer

Composite Interfaces 16 (2009) 547–577www.brill.nl/ci

Computer Simulation Study of Model Nafion Membranein Water/Methanol Solvent

Alexander Chertovich a,b,∗, Pavel G. Khalatur b,c and Alexei R. Khokhlov a,b

a Physics Department, Moscow State University, Moscow 119991, Russiab Department of Polymer Science, University of Ulm, Ulm D-89069, Germany

c Institute of Organoelement Compounds, Russian Academy of Science, Moscow 119991, Russia

Received 17 March 2008; accepted 4 September 2008

AbstractUsing NPT and NVT molecular dynamics simulation techniques, we have simulated an atomistic modelof solvated Nafion in the lithium salt form, with the following three main objectives: (i) to obtain detailson the local environment of the lithium cations and to assess the solvent effect on their binding, (ii) toinvestigate the translational and rotational motion of solvent molecules (water and methanol) absorbed inthe polymer matrix, and (iii) to elucidate details of the ionic transport though the hydrophilic regions ofthe membrane and to study the ionic conductivity as a function of solvent (water/methanol) composition.A property which is of central importance for understanding the functional features of Nafion materials,including direct methanol fuel cells, is the ionic conductivity and methanol crossover. We have found thatconductivity parameter is strongly dependent on the solvent composition and determined by the solvationeffects and the spatial distribution of polar sulfonate groups in ion-conductive channels.© Koninklijke Brill NV, Leiden, 2009

KeywordsNafion membrane, molecular dynamic, methanol, diffusion coefficient, ionic conductivity

1. Introduction

Nafion is the well-known example of polytetrafluoroethylene (PTFE)-based PEMmaterials and has, for a long time, been regarded as the state-of-the-art fuel-cellmembrane polymer. After three decades and numerous attempts to create alterna-tive materials, Nafion still holds its position as the proton-conducting polymer ofchoice, by virtue of its superior combination of conductivity and chemical stability.Nevertheless, Nafion is still not without its problems: it is expensive, it has a lowoperating temperature limit, and a relatively high methanol permeability (for directmethanol fuel cells, DMFCs).

* To whom correspondence should be addressed. E-mail: [email protected]

© Koninklijke Brill NV, Leiden, 2009 DOI:10.1163/156855409X450936

548 A. Chertovich et al. / Composite Interfaces 16 (2009) 547–577

Despite extensive experimental and theoretical studies of solvated Nafion mem-branes, up to now there are still many open questions in understanding the inter-connection between their microstructure and dynamical characteristics, includingtransport of ions and absorbed molecules such as water and methanol. This area isaddressed in the present theoretical study, which is based on atomistic moleculardynamics (MD). In this work, we focus on the dynamic properties of physisorbedwater and methanol molecules and on ionic transport phenomena in solvated salt(Li) forms of Nafion. The ionic conductivity is a transport property which, in princi-ple, can be calculated microscopically from the displacement of the mobile ions, ina way similar to the MD calculations of the diffusion coefficient of neutral gaseouspenetrants from their mean-square displacements.

It should be emphasized from the very beginning that the systems modeled usingour MD or similar approach, and most of the coarse-grained and mesoscale mod-els, could be analogous to real perfluorinated sulphonic acid (PFSA) membranematerials to a limited extent, since the size and time scales of the simulations aredramatically smaller than those found in reality.

The unique properties of Nafion and its salts are mainly related to their highconductivity and water permeability. Nafion was originally developed for use as aseparator membrane in chloroalkali cells and has since found widespread use inwater electrolyzers, batteries and methanol membrane fuel cells. It is known thatthe overall performance of the fuel cells is strongly influenced by the conductivityof the Nafion membrane, which itself depends on the solvent content of the mem-brane. It is widely believed that the interesting electrochemical properties of suchsystems derive from the microphase separation of polar (ionic) material from thehydrophobic amorphous matrix. This results in the formation of solvent-containingclusters built up from the ionizable sulfonate groups of Nafion. The clusters aredistributed more or less randomly over the hydrophobic matrix, comprising the flu-orocarbon backbone of the polymer, and already at very low solvent content theycan allow cations to diffuse rapidly through the membrane. Measured ionic conduc-tivities can reach 10−2 S/cm [1] (for comparison, a 0.1 M aqueous solution of NaClslightly above room temperature has a conductivity of around 10−2 S/cm [2]).

Most of the work with Nafion (both theoretical and experimental) was done withwater as a solvent. However, the incorporation of nonaqueous solvents, such as al-cohols, is known to affect the morphology of swollen membranes. Alchohols evenpartly solubilize the polymer [3, 4]. Rivin et al. [5, 6] found that the uptake ofwater–ethanol mixture by Nafion membranes is much higher than the uptake ofpure components. The solubility depends strongly on the composition of the solventmixture and reaches a maximum at approximately 1:1 molar ratio. It was assumedthat ethanol tends to surround the hydrophobic skeleton, while water interacts morestrongly with the ionic side chains and counterions. The elevated uptake of water–methanol mixture by Nafion membranes was observed by Nandan et al. [7]. Pilaret al. [8] performed electron spin resonance studies of Nafion microstructure andfound that the aggregation of Nafion backbone in alcohol is less pronounced than

A. Chertovich et al. / Composite Interfaces 16 (2009) 547–577 549

in water. Pellegrino and Kang [9] found that incorporation of alcohols affects thesolubility and diffusivity of methane and carbon dioxide in Nafion membranesand demonstrated its applicability to gas separation. Preferential removal of alco-hols from aqueous solutions by Nafion membranes was studied experimentally byPasternak and Dorawala [10]. Saito et al. [11] studied PFSA membranes with differ-ent equivalent weight (EW) penetrated water/methanol mixture (molar ratio: 1:1),methanol, ethanol and 2-propanol. They found that the faster proton transport waspromoted more when the membrane was penetrated by smaller alcohol, and the al-cohol permeability is larger for the membranes with smaller EW. Morrone et al.[12] performed ab initio molecular dynamics simulation of the structure and protontransport dynamics of methanol–water solutions. The simulations reveal the exis-tence of separate interpenetrated hydrogen-bonded water and methanol networks,also in agreement with the neutron diffraction data. In addition they predict anom-alously high structural or Grotthuss-type diffusion mechanism of the charge defectwhen the composition of the mixture is 1:1.

To the best of our knowledge, the first endeavor to apply molecular modelingto the study of transport phenomena in Nafion-like systems was undertaken byDyakov and Tovbin [13]. The first all-atom MD simulation of Nafion-like systemwas carried out by Elliott et al. [14]. They investigated sulfonic acid anions similarto Nafion side chains. The results of these simulations confirmed intuitional under-standing of Nafion structure: fluorocarbon and ionic materials spontaneously phaseseparate to form a percolating structure of water-filled channels in a polymer ma-trix. Vishnyakov and Neimark [15] were the first who consider the polymer natureof Nafion structure and calculate minimum energy structures in vacuum. Later theyobserve that the mobility of the fluorocarbon oligomer backbone was enhanced inthe presence of methanol in equimolar quantities with water [16]. But this studydid not consider such properties as diffusion coefficient of different molecules andionic conductivity, and no other proportion except equimolar was under investiga-tion. The same authors studied ‘polymeric’ Nafion (although the molecular weightof modeled Nafion was at least one order of magnitude less that that for the realsystems) at several different water contents [17]. The main conclusion from thisstudy was that ‘polymeric’ simulations revealed a very similar qualitative picture tothat obtained by oligomeric models.

In a more recent study, Urata et al. [18] calculated structure factors from theirmodels to compare with experimental scattering data. They showed that the mole-cular models have a smaller ionic cluster size than real PFSA membranes, whichwas attributed to lower molecular weight of model polymers and, in addition, limi-tations due to finite size and timescale of the simulations. Jang et al. [19, 20] studiedthe effect of monomeric sequence in Nafion-type polymers on their structure andionic transport. They selected two extreme cases of side-chain distribution along thebackbone: random and diblock. The authors concluded that the degree of ‘blocki-ness’ in the real Nafion polymer in something in-between these two extreme cases,but closer to the diblock case. In a recent paper, Brandell et al. [21] compared the


structures of Nafion, Dow and Aciplex polymers at several water contents. Theserepresent the main materials in an industry of PFSA membranes by different com-panies; the main difference between these materials is the minor changes in theside-chain length. It was found that the Nafion side chain has an optimal length (incomparison with others) and the water-channel network connectivity is optimal fora high water-content Nafion system.

In a series of recent papers [22–24] Devanathan et al. carried out the system-atic simulation study of the Nafion morphology at different degrees of solvation.The hydration levels studied in this work were λ = 1, 3, 5, 7, 9, 11, 13.5 and 20.The microscopic structure was found to be in reasonable agreement with exper-imental observations from neutron scattering [25, 26]. The authors of this workclaim that they can identify free, weakly bound and bound water in hydrated Nafionmembranes. In addition, they studied dynamical properties of water and hydroniummolecules, including diffusion coefficients and mean residence time. These dynam-ical characteristics are in good agreement with the quasi-elastic neutron scattering(QENS) experiment [25] for the system at high hydration levels. Also, they ob-served that temperature had a significant effect on the diffusion coefficients for bothwater and hydronium ions.

In addition to ‘classical’ MD simulations described above, there are alsomesoscale, multiscale and quantum chemistry models in the literature. Not ob-serving all of them, we mention here only the most relevant ones.

In a series papers by our group [27–29], we used Monte Carlo (MC) simula-tions, hybrid self-consistent MC/RISM, and coarse-grained lattice MD (cellular-automaton based) simulations to predict the geometry of microphase separationinside the Nafion hydrated material. It was found that the water and polar sulfonicacid groups are segregated into a three-layer structure with a central water-rich re-gion, an outer layer of side groups strongly associated with water molecules and a‘corona’ of the less polar regions of the side groups that are immersed in the fluoro-carbon matrix. The characteristic length scale of the segregated structure increasedperfectly linearly with water content. This result is in agreement with experimentaldata on Nafion swelling [30–32]. An important feature from these calculations isthat the simulations would seem to be strongly supportive of the view that scatter-ing is a result of the interference between spherical ionic clusters that reorganizeand swell due to the incorporation of water. It is significant that such a result canemerge from a fully polymeric, albeit coarse-grained, molecular model containingno a priori assumptions about the membrane morphology. Recently, using 3-D self-consistent field theory (SCFT) [33], we basically confirmed the conclusions of thespheroidal geometry of ionomer clusters.

Explicit proton and charge delocalization of the excess proton transport, via theGrotthuss hopping mechanism, were treated using the self-consistent multistateempirical valence bond (SCI–MS–EVB) method in Refs [34–36]. The total pro-ton diffusion was decomposed into vehicular and Grotthuss components and werefound to be of the same relative magnitude, but with a strong negative correlation


resulting in a smaller overall diffusion. Using the multiscale quantum mechan-ics/molecular mechanics ONIOM method, combining quantum calculations withina classical MD, Elliott and Paddison [37, 38] observed the formation of Eigen andZundel ions at several hydration levels and concluded that, starting from some hy-dration level, solvent-separated Eigen ions begin to appear, indicating that the mainproton motion will occur in the bulk water, away from sulfonated groups.

The current state of understanding of Nafion morphology can be found in areview by Mauritz and Moore [39]. More detailed discussion related to proton trans-port and local structure of Nafion fragments in the presence of water can be foundin Refs [38, 40].

2. Computational Models and Methods

2.1. Molecular Model of Perfluorinated Ionomer

The pendant chain in Nafion is a perfluorinated double ether chain with the follow-ing structure:

–CFOCF2CF(CF3)OCF2CF2SO3−X+

where X+ is an exchangeable cation. In the presence of polar solvent, the terminalsulfonate groups are easily dissociated to form solvated anion SO3

− and cation X+.The sulfonate groups in bulk Nafion can form ionic clusters (multiplets), embeddedin a hydrophobic fluorocarbon matrix. In the present study, following Elliott et al.[14], we decided to represent the perfluorinated ionomer using only the entire sidechain attached to a trifluoromethyl group. In this case, MD simulation provideshigher molecular mobilities and more significant structural reorganizations of thesystem. On the other hand, it is clear that the use of such a model, rather than afully connected copolymer with comb-like architecture, makes the intramolecularchemical constraints less significant than in the real polymer.

In this study, we consider Nafion in the lithium salt form, that is, X+ = Li+.The structural parameters and partial charges of the 25 atom side-chain anion werecalculated using the semi-empirical quantum chemical method based on the AM1Hamiltonian [41, 42]. Fully optimized molecular geometries were found using thePolak–Ribiere conjugate gradient method. In addition, we performed ab initio self-consistent-field MO calculations both with a small basis set (STO-3G*) and withthe 6-31G** split valence basis set [43]. In this case, the minimum energy structurewas first determined at the semi-empirical AM1 level and then the partial chargeswere refined. The minimum energy structure of the side-chain anion (SHA) togetherwith the partial charges assigned to each atom (giving a net charge of −e) is shownin Fig. 1.

It should be noted here that for the simulations of membrane with a very lowsolvation level, the real charges might differ significantly from formally calculatedcharges. But we claim that the main idea of this research is not to simulate the


Figure 1. Fully optimized geometry of the perfluorinated sulfonate side chain with the partial chargesassigned to each atom.

system which will be maximally similar to experimental data, but to predict andunderstand general behavior at specific conditions.

In our MD simulations, we employed the AMBER-like force field supplementedby some additional parameters corresponding to fluorine atoms (see Refs [14–16]).The total potential energy of the system was represented as a sum of the bondstretching, bond-angle bending and dihedral torsion energies as well as the vander Waals and electrostatic (Coulomb) terms, U = Ub + U� + Uϕ + Uvdw + Ue.Both short-ranged van der Waals and long-ranged electrostatic interactions betweensulfonate fragments, lithium cations and solvent molecules were explicitly includedin the model.


2.2. Lithium Cations

For complicated liquids, such as electrolyte solutions, highly accurate ab initio cal-culations do not necessarily guarantee proper pair potentials needed in computersimulations where the potentials used need to be effective ones, which means thatmany-body interactions (polarization effects) must be incorporated [44]. In view ofthis, we decided not to consider fine details of the ionic interactions, trying to retainthe most essential physical features of the site–site potentials. Although this mayseem less attractive from a fundamental point of view, it is believed that such anapproach gives as good, or even better description than do extensive quantum me-chanical calculations, at least for the complex system under study. In this way, eachcation was, therefore, treated as a positively charged classical particle which wasfree to diffuse between negatively charged perfluorinated sulfonate fragments andneutral solvent molecules. The corresponding Lennard–Jones parameters for Li+were taken from Ref. [43]. In addition to the LJ potential, Coulomb effects weredescribed by the electrostatic potential qiqj /rij with qLi = +1 e.

2.3. Water

There exist several water models which all reproduce structural and thermodynamicproperties observed experimentally with reasonable accuracy. The water modelused in our simulations is the so-called simple point charge (SPC) model [45] ex-tended by Toukan and Rahman [46]. In this model, the oxygen atom carries a charge−0.82 e, and both hydrogen atoms carry charges of +0.41 e. The equilibrium OHbond length is 1 Å, the equilibrium HOH angle is 109◦28′. These values are deter-mined by the combination of a Morse function and harmonic potentials. Parametersfor interactions of lithium ions and water oxygens were calculated from Ref. [43].

2.4. Methanol

Methanol molecules were treated in full atomic detail with the exception of the CH3groups, which were treated as single LJ centers (united atoms). Thus, the moleculeof methanol was described by the three-center potential with the standard OPLSparameters for the CH3 group and the AMBER force field for the O and H atoms.For the OH bond stretching term, the combination of harmonic and Morse potentialswas used [47]. Although the methyl protons were not considered as interaction sites,their masses were properly allowed for in the equations of motion. The molecules,therefore, have the correct masses and moments.

2.5. Composition of Solvent-Containing Ionomer Membrane

The computational unit cell for the MD simulation contained n sulfonate molecularfragments, n lithium cations and ns solvent molecules (water or methanol). Thesolvation level defined by the ratio ns/n was equal to 4. The molar concentrationsof sulfonate fragments, cations and absorbed solvent are 3.2 mol/l, 3.2 mol/l and12.8 mol/l, respectively. This model approximately corresponds to the real systemat EW = 1100 g/equiv, under ambient humidity conditions (the Nafion membrane is


characterized by its equivalent weight, EW, which is defined as the mass of solvent-free polymer containing one mole of terminal sulfonate groups; e.g., for the Nafion117 membrane, EW = 1100 g/equiv). The infinite extent of the cubic simulationcell in three directions was mimicked by the usual periodic boundary conditions.The calculations were carried out for sufficiently large systems containing fromn = 32 to n = 256 sulfonate fragments, leading to the total number of atoms inthe system from 1216 to 9728. Most of the computations were performed with9728 atoms. This system size was satisfactory for sufficiently reliable prediction ofall the dynamic properties considered.

2.6. Initial Configurations and Their Relaxation

In order to obtain a many-molecule system, n identical sulfonate fragments hav-ing fully optimized geometry were initially placed on a regular simple cubic latticespanning a periodic MD box and then were randomly reoriented. To remove theoverlaps between atoms brought about by the omission of the intermolecular in-teractions during the initial generation procedure, a short MD run was performedwith the bending, torsion, and electrostatic interactions switched on, while the LJpotential was introduced very gradually. Then, the cations and solvent moleculeswere distributed in the box and the system was further relaxed. At this initial stage,velocities were rescaled at each time step to compensate for the large amount ofheat produced in the system.

2.7. MD Simulation

The systems were simulated both under NVT and under NPT conditions at T =298 K. The temperature and pressure were maintained by the Nose–Hoover ther-mostat [48, 49] using the modular-invariant method [50]. Most of the computationswere performed for NPT conditions. In the case of NPT simulations, the isotropicpressure was maintained at the required value of 1 bar by loose coupling with a τPof 800 fs and the MD box was kept cubic. The corresponding Nose NVT relax-ation time, τT, was set to 30 fs. The equations of motion were integrated using thereversible double time step algorithm [51] in which all the forces are divided intotwo groups, ‘fast’ and ‘slow’. The first one is due to covalent bonds, bond anglesand dihedral angles; it also includes LJ forces and real-space electrostatic forceswithin a short cutoff distance, Rs, of 5 Å. The second group includes LJ and real-space electrostatic forces on distances between Rs and a ‘long’ cutoff distance Rcas well as reciprocal-space electrostatic interactions. The electrostatic interactionsin the simulations were computed using an Ewald summation technique with sepa-ration parameter, α, of 0.345 Å−1 and reciprocal-space cutoff, κ , of 8.8. With theseparameters for the Ewald sum, the difference between the direct and indirect cal-culations of the reciprocal space contribution to the pressure was less than 0.2 bar.The relative dielectric constant ε was assumed to be 1, since all partial charges aretreated explicitly. The truncation radius for LJ and electrostatic interactions was setto Rc = 10 Å, and long-range corrections to the energy and pressure were made


on the assumption that all the pair correlation functions g(r) = 1 beyond the cut-off [52]. The equations of motion were solved by the Verlet method [53] with ‘fast’time step, tf, of 0.4 fs and ‘slow’ time step ts = 10tf.

Equilibration runs were continued until such properties as potential energy, di-mensions of the MD cell etc. had more or less stabilized. The total length of theequilibration runs was usually 200–300 ps for NPT simulations at P = 1 bar andmore than 400 ps for NVT simulations at a density of 1.8 g/cm3, depending on thewater/methanol composition. Configurations, thermodynamic, and conformationaldata were then stored and analyzed over production runs ranging from 8 to 16 ns.Bulk properties such as the average density in NPT systems and the average pres-sure in NVT systems were found to oscillate about a well-defined mean. In thiswork, we will discuss mainly the results from NPT simulations.

3. Results and Discussion

3.1. Solvent Effect on Binding of Cations

Experimental information about the immediate environment of the cations dis-solved in solvent-containing Nafion membrane is mostly indirect. In principle, thecorresponding data can be obtained via X-ray or neutron scattering measurements,which give pair correlation functions characterizing spatial distribution of atoms.The same functions can be calculated for the model systems under consideration.The pair correlation function (PCF), gAB(r), is defined as the spherically and timeaveraged distribution of interatomic separations r between two species A and B. Wefound the PCFs between the lithium cations and different atoms of the side chainsand solvent molecules. It is clear that the Li+ ions interact most strongly with thenegatively charged oxygen atoms of the sulfonate groups and solvent molecules andwith the positive partial charges on sulfur atoms. These functions, denoted belowas gLi–OS(r), gLi–O(r) and gLi–S(r), respectively, provide important informationabout the state of aggregation of ionic material. They are shown in Figs 2–4 for thewater-containing and methanol-containing systems simulated under NPT and NVTconditions.

A characteristic of all the systems studied is the high degree of short-range (lo-cal) structure that is evident in the Li–OS, Li–O and Li–S correlation functions. Thisfeature is linked to the formation of ionic bonds between oxygens of the sulfonategroups and Li+ and solvation shells around Li+. The tendency is enhanced in pass-ing from the water-containing system to methanol-containing system. The strongerbinding of cations in the latter case is confirmed by the inward shift of the first peakin the Li–OS and Li–S functions and by the increase in intensity of this peak. Inaddition, the formation of second coordination shells is strongly indicated. The po-sition of the first maxima in the Li–O PCFs calculated here for the water-containingsystem under NPT and NVT conditions (r1 = 2.1 Å) is in good agreement withthe room-temperature MD simulations of a 0.55 molal LiI solution in water [54].The main peak in the Li–S correlation functions is shifted toward larger r , relative


Figure 2. Pair correlation functions (a) gLi–O(r) and (b) gLi–OS(r) for the water-containing andmethanol-containing systems simulated under NPT conditions during 16 ns at T = 298 K andP = 1 bar.

to gLi–OS(r). This is due to the repulsive forces between positively charged S andLi+ and to the screening effect of oxygen atoms surrounding sulfur atom so thelithiums cannot get close. Beyond the first and second maxima in PCFs, there arediffuse peaks between 6 and 12 Å. These may be interpreted as ‘free cation peaks’related to the cations that are more loosely associated with the solvent-containingpolar matrix and, thus, can participate in ionic conductivity. In other words, there isno support for the concept of ‘bonded counterions’ in the present simulation.


Figure 3. Pair correlation functions (a) gLi–O® and (b) gLi–OS

® for the water-containing andmethanol-containing systems simulated under NVT conditions during 16 ns at T = 298 K andρ = 1.8 g/cm3.

Table 1 contains time-averaged potential energies, Ue and Uvdw, characterizingelectrostatic and van der Waals interactions between the cations and the sulfonatechains and solvent molecules.

As seen, the electrostatic energy is the dominant term in the total non-bondedenergy. What is relevant here is the fact that the total potential energy, U = Ue +Uvdw, found for the Li/chain and Li/solvent interactions in the water-containingsystems is always a little less than in the methanol-containing counterparts. This


Figure 4. Pair correlation functions gLi–S® obtained for the water-containing and

methanol-containing system simulated under (a) NPT conditions and (b) NVT conditions dur-ing 16 ns at T = 298 K.

Table 1.Time-averaged potential energies, Ue and Uvdw, characterizing electrostatic and van der Waals inter-actions between the cations and the sulfonate chains and solvent molecules

Ensemble Solvent Density

(g/cm3)

Ue (kJ/mol) Uvdw (kJ/mol)

Li/chain Li/solvent Li/chain Li/solvent

NPT H2O 1.611 ± 0.002 −63.5 ± 0.11 −9.08 ± 0.04 0.66 ± 0.01 0.85 ± 0.01CH3OH 1.337 ± 0.002 −54.0 ± 0.03 −7.57 ± 0.05 0.84 ± 0.01 0.53 ± 0.01

NVT H2O 1.8 −70.4 ± 0.15 −9.19 ± 0.06 0.74 ± 0.01 0.80 ± 0.01CH3OH 1.8 −66.4 ± 0.12 −7.22 ± 0.06 0.92 ± 0.02 0.64 ± 0.01


would seem to be at variance with the results presented above for PCFs. However,it should be borne in mind that the PCFs considered here characterize the localenvironment formed around the lithium ions while the value of U is determinedby a sum of interactions between all atoms belonging to given species and, thus,characterizes their overall affinity. The reason for the differences discussed can beunderstood from the fact that the liquid methanol has a lower dielectric permittivitythan bulk water and, as a result, it is a ‘poorer solvent’ for cations. Because of theirpoorer affinity to the methanol (see Table 1), the lithium cations try to associatewith sulfonate groups more closely; this effect is well seen in Figs 2–4. One cansay that the addition of methanol to the system promotes the formation of boundionic states (i.e., ion pairs) but, on the other hand, reduces an overall coupling ofthe polymer and ionic subsystems. Not surprisingly, this leads to the decrease in thesystem density, the result that follows from our NPT simulation (see Table 1). As wewill see below, such features are important for understanding transport phenomena.Of course, in addition to the formation of ion pairs, also larger clusters of ions couldform, including a percolating cluster. It is clear that the mobility of a cluster of ionscan be radically different from the mobility of isolated ions. Some of the relatedproblems will be discussed below.

3.2. The Mobilities of Solvent Molecules

The translational and rotational motion of solvent molecules in the polymer matrixcan be characterized by their diffusion constant, D, and reorientational correlationtimes, τ . The self-diffusion coefficient of an individual molecule can be calculatedfrom its center-of-mass mean-square displacement (MSD) using the Einstein rela-tion

D = 1

6t〈|R(t) − R(0)|2〉t→∞, (1)

where the brackets 〈·〉 denote an ensemble average or, equivalently, an average overdifferent time origins t = 0 and R(t) is the center-of-mass position vector of a mole-cule at time t . An alternative route is via the integral of the center-of-mass velocityautocorrelation function (VACF) according to one of the Green–Kubo relations as

D = 1

3

∫ ∞

0〈v(t) · v(0)〉dt. (2)

For the self-diffusion case, the ensemble average is performed over all moleculesas well as over all time origins since R(t) and v(t) are single molecule properties.The corresponding time interval must be sufficiently long to allow for the completedecay of the time correlation functions. The procedure used here for calculatingthe appropriate time averages of the transport coefficients involved the method ofrunning averages [55]. One should note that for some of the runs, the MSDs weretoo noisy to establish unambiguously where the Einstein behavior has been reached.Therefore we preferred to calculate the transport coefficients from the Green–Kuborelations.


From NPT simulations, we found for the diffusion constants: DH2O = 3.41 ×10−6 cm2/s and DCH3OH = 4.08 × 10−6 cm2/s. The value of D calculated forH2O is an order of magnitude smaller than the one in the pure SPC water model(3.6 × 10−5 cm2/s) and a factor of 6 smaller as compared to the experimental valueof 2.4 × 10−5 cm2/s known for bulk water [56]. Hence we conclude that water inthe Nafion membrane diffuses 6–10 times more slowly than bulk water, indicat-ing that the state of water in ionomer environment is significantly different fromthat of ordinary bulk water. In other words, the water motion is affected (sloweddown) by the presence of the polar polymer component. The self-diffusion coeffi-cient of liquid methanol has been measured over a wide range of temperature byradioactive-tracer method [57]. At T = 299.3 K, the experimental value of DCH3OHwas found to be equal to 2.4 × 10−5 cm2/s. From this it is evident that the diffu-sion of methanol in the Nafion membrane is also considerably slower comparedto pure liquid. We can compare the diffusion constant of water in our simulationwith some other experimental results. Porter et al. [58] studied the penetration ofwater vapor through perfluorinated polyether films on concentrated sulfuric acidand found that DH2O = 6 × 10−6 cm2/s. Callaghan et al. [59] demonstrated a dif-fusion constant of H2O in a self-assembling potassium-palmitate/D2O system at338 K of 8.5 × 10−6 cm2/s (at 300 K, this value corresponds to 5.2 × 10−6 cm2/s).These values measured for related systems are in reasonable agreement with ourMD results. In addition, we can compare the translational diffusion coefficientscalculated here with predictions from other MD studies carried out for Nafion-like models. In their NVT simulation of water-containing system Elliot et al. [14]observed surprisingly low molecular mobilities; the resulting diffusion coefficientfor water molecules was found to be 1.14 × 10−8 cm2/s. On the contrary, theNPT simulations of Vishnyakov and Neimark [15] demonstrated that the diffusioncoefficients of water and methanol are only slightly lower than those in the corre-sponding bulk liquids and very close to each other (DH2O = 1.80 × 10−5 cm2/s andDCH3OH = 1.77×10−5 cm2/s). It seems likely that the difference between their andour predictions is due to the fact that the equilibrium system densities consideredin the study [15] were 1.11 g/cm3 in water and 0.842 g/cm3 in methanol, whichactually corresponds to diluted solutions.

A complete description of the reorientational motion of a molecule can be givenin terms of an infinite set of generalized spherical harmonics. For our purposes, itis sufficient to consider only the time autocorrelation functions defined as

Cl(t) = 〈Pl[cos�(t)]〉, (3)

where Pl(cos�) is a Legendre polynomial and �(t) is the angle through which amolecule-fixed vector rotates in a time t . The vectors used were: (i) the dipole vectorμ and (ii) the vector connecting the terminal atoms (H–H for water and H–CH3 formethanol). The autocorrelation functions were calculated for the first and secondLegendre polynomials, P1(cos�) = cos� and P2(cos�) = (3 cos2 � − 1)/2.


Figure 5. The dipole autocorrelation function of water and methanol molecules obtained from an NPTsimulation at T = 298 K and P = 1 bar and averaged over 116 ns.

As an example, Fig. 5 presents the dipole autocorrelation functions (DACF),〈 μ(t) · μ(0)〉, of water and methanol obtained from NPT simulations. In the long-time limit, DACFs display the usual, Debye-like, exponential decay. At very shorttimes, however, there are weak oscillations of a type that is apparently characteristicof strongly associated liquids [60]. It should be noted that the oscillations becomemore pronounced for the second Legendre polynomials and can be related to avibrational motion of the molecule in the potential energy well provided by thehydrogen bond in which the H atoms partake. In our case, the hydrogen bondscan arise both between solvent molecules and between solvent molecules and thesulfonate groups. We found that the mean-square displacement of solvent hydroxylhydrogens relative to molecular center of mass increases initially, showing an anom-alous short-time behavior, and then starts decreasing after ≈0.2 ps, displaying thenearly linear growth with t at t � 1 ps. This can be explained as caused by the oscil-lations of the molecular dipole-moment vectors, which reflects also in the DACFsfor very short times.

To estimate the lifetime of the H bonds, we calculated the angular velocity cor-relation functions (AVCF), 〈ω(t) · ω(0)〉, of solvent molecules and the angularfrequencies corresponding to the oscillations of the dipole vectors. Our analysisshowed that the weak initial glitch in DACF, hardly seen on the time scale of Fig. 5,can be linked to the negative region in the AVCFs which have the following relax-ation times: 0.28 ps and 0.25 ps for the water-containing and methanol-containingsystems, respectively. The data presented above would mean that the lifetime of the


H bonds should be in the range of 0.2–0.3 ps. This estimation can be comparedwith the corresponding value of 0.03–0.07 ps found for pure bulk water (see, e.g.,Ref. [61]). The difference is not surprising, taking into account the possibility ofhydrogen bonding between solvent molecules and the Nafion SO3 groups whichserve as acceptors of a hydrogen bond donated by a solvent molecule.

Having the PCFs calculated, it is easily to estimate the average number of H-bonds formed by the SO3 groups with solvent. Assuming that a hydrogen bond isformed if the distance between the oxygens does not exceed 3.5 Å, we found thatthe oxygens of one sulfonate group can form approximately 10 H-bonds with waterand methanol molecules. This value is consistent with the more rigorous analysis ofVishnyakov and Neimark [16]. However, their hydrogen bond lifetimes consider-ably exceed those predicted in our simulations. Probably, this is due to the fact thatthese authors used another criterion of hydrogen bond lifetime based on residencetimes for oxygen atoms bonded via a hydrogen atom.

The DACFs were fitted to three exponentials as

C1(t) =3∑

i=1

ai exp

(− t

τi

), (4)

with the following condition:∑

i ai = 1. The parameters ai and the relaxation timesτi were calculated from a nonlinear least-squares fit of the simulation data to equa-tion (4). The standard deviation of the fitting was less than 0.01. It is clear that thesmall components in equation (4) die much faster; therefore, the long-time behaviorof C1(t) is controlled by the main component with a maximum characteristic timeτmax. The rotational correlation times, τ (1), defined by

τ (1) =∫ ∞

0C1(t)dt =

3∑i=1

aiτi, (5)

were found to be equal to 10.4 ps for the water-containing system and 15.5 ps forthe methanol-containing system. The rotational correlation times, corresponding tothe C1(t) function defined for the vector connecting two terminal atoms, are 7.74 psand 16.5 ps, respectively. From the comparison of our results with the water simu-lation by Lie and Clementi [62], who found for the dipole relaxation time a valueof 4.1 ps, one can conclude that the rotational motion of water absorbed by theNafion membrane is considerably slowed down compared to pure water. The sameis true for methanol molecules. Indeed, from the MD simulation of liquid methanolat 299.3 K [63], the l = 1 relaxation time for the dipole-moment vector was esti-mated to be 10.1 ps. Thus, solvent molecules, hydrogen-bonded to the sulfonategroups, have roughly 2 times lower rotational mobility.

For a diffusive motion, the rotational correlation functions defined by equa-tion (3) decay as exp[−l(l+1)DRt], where DR is the rotational diffusion coefficientwhich is independent of l. To interpret the translational and rotational diffusion co-efficients, D and DR, one may use the Stokes formulae for an effective sphere of


diameter d moving in a medium with viscosity η

D = kBT/3πdη,(6)

DR = kBT/4πd3η.

Since both diffusion coefficients are obtained from the MD simulation, the val-ues of d and η can be calculated independently. We get for the water and methanolmolecules: dH2O = 1.5 Å and ηH2O = 8.7 × 10−3 N · s/m2, dCH3OH = 2.0 Å andηCH3OH = 5.4 × 10−3 N · s/m2. The viscosities are approximately an order of mag-nitude larger than the experimental ones for liquid water (0.8 × 10−3 N · s/m2) andmethanol (0.51 × 10−3 N · s/m2) at 30◦C [64]. At the same time, the effective di-ameters of the molecules are quite reasonable and close to those known for thecorresponding bulk liquids.

Experimentally, information on τ (2) can be obtained from NMR relaxation-timemeasurements. With some assumptions, the spin–lattice relaxation rate for particleswith spin 1/2 is given by [65]

1

T1= 3π2

2 2τ (2)[1 + (ω0τ

(2))2]−1, (7)

where is the quadrupole coupling constant and ω0 is the nuclear Larmor fre-quency. At sufficiently high temperature, i.e., when ω0τ

(2) � 1, the spin–latticerelaxation time T1 is inversely proportional to the correlation time τ (2). Recently,MacMillan et al. [66, 67] have measured proton and deuterium spin–lattice relax-ation times as well as spin–spin relaxation times as a function of hydration andtemperature in hydrated Nafion with a molecular weight of 1100 g/mol. They haveconcluded that water absorbed in Nafion creates an environment of cluster embed-ded in the fluorocarbon matrix. The values of τ (2) measured above some transitiontemperature Tt (at 363 K) were estimated to be 14 ps for 5.9% H2O, 9 ps for 7.7%H2O and 4.5 ps for 15.9% H2O. These hydration levels correspond to 3.8, 5.1and 11.6 water molecules per one sulfonate groups, respectively. For the water-containing system at the hydration level ns/n = 4, our MD simulation predictsτ (2) = 4.45 ps. The correspondence between MD and experimental values is en-couraging. The difference may be caused by the fact that in our NPT simulation thedensity of ionic components is slightly lower compared to the real system as wellas by the fact that model SPC water diffuses faster than real water. Nevertheless,both experimental and theoretical results demonstrate that the rotational motion ofwater molecules is inhibited in the Nafion membrane. Indeed, in the case of bulkwater, the value of τ (2) is characteristically about 1 ps [68]. On the other hand, it isknown that, in bound or surface water, τ (2) ranges from 10 ps to 1 ns [68]. For themethanol-containing system, our calculations give τ (2) = 5.66 ps. This is close tothe NMR data of Gruner and Hertz [69] for liquid methanol.

From a free diffusion model one expects a ration of τ (1) over τ (2) of 3. This re-lationship is violated for solvent molecules absorbed by Nafion, where the ration is


2.3 for H2O and 2.7 for CH3OH. This is another indication of the hindered rotationof absorbed molecules in the Nafion/solvent system.

3.3. Effect of Solvent Composition

At a fixed solvation level, ns/n, where ns is the total number of solvent moleculesin the system, solvent composition can be determined by the mole fraction cm =nm/(nw + nm), nw where nm are the numbers of water and methanol molecules(ns = nw + nm). We performed a series of NPT simulations for different cm whilethe solvation level is fixed at ns/n = 4. In this subsection, we focus on the systemdensity, ρ, and the electrostatic energy of interaction between lithium cations andsulfonate chains, Ue, considering these quantities as a function of cm. They arepresented in Fig. 6.

One sees that the system density decreases as methanol content in the system in-creases. This is accompanied by an increase in the electrostatic energy, which is thedominant term in the total potential energy of ionic interactions. Therefore, we con-clude that, when the fraction of absorbed methanol increases, the system becomesmore ‘swollen’. One could expect that this will result in an increase in molecularmobilities, including diffusion coefficients and ionic conductivity. However, as wewill see below, generally speaking, this is not the case. As a matter of fact, thesituation seems more complicated.

In Fig. 7 we present the translation diffusion coefficients of water and methanolas a function of the solvent composition. As can be seen, the value of DH2O ini-tially increases with cm increasing up to cm ≈ 0.5 and then begins to decrease. Onthe other hand, we observe a monotonous decrease in the diffusion coefficient ofmethanol, DCH3OH, with cm increasing. Note that uncertainties shown in this figure(as well as in the figures given below) were estimated from the corresponding timeautocorrelation functions following the method described by Straatsma et al. [70].

Thus we can conclude that ≈1:1 solvent composition is the most favorable forwater diffusion in the Nafion membrane swollen in the mixed solvent. Going tohigher methanol concentrations, the mobility of methanol molecules drops, despitethe system density decreases (see Fig. 6(a)). A possible reason for this behavior is aninterplay between structural and dynamic correlations among the species involvedin the system. It is clear that a solvent molecule participating in the formation of ahydrogen bond is less involved in the overall circulation of a given solvent. Eachwater molecule can form four H-bonds, both with sulfonate groups and with otherwater molecules or methanol, while a methanol molecule is able only to donate onehydrogen atom and accept one H-bond. At low methanol concentrations, practicallyall water molecules form H-bonds so that their diffusion coefficient is relativelysmall. As the methanol content is increased, some part of the H-bonds formed bywater is replaced by methanol. This leads to an increase in DH2O. However, whenthe total number of water molecules in the system becomes comparable to the num-ber of SO3 groups, practically all of them are bound with these groups. As a result,the fraction of ‘free’ water decreases and DH2O drops. It is well known that water


Figure 6. (a) The system density and (b) the electrostatic energy of interaction between lithium cationsand sulfonate chains as functions of the solvent composition, cm. For each cm, the calculations wereperformed under NPT conditions during 8 ns, at T = 298 K and P = 1 bar.

shows a more extensive hydrogen bond network compared to methanol. Because ofthis, methanol molecules, at their relatively low concentration in the mixed solvent,can be viewed as ‘structural defects’ in the water hydrogen bond network. In ad-dition, they have poorer overall affinity to the sulfonate chains compared to water(see Table 1). Not surprisingly, under these conditions, we observe higher methanolmobility. The mobility reduces as the H2O· · ·SO3 associations are replaced by theCH3OH· · ·SO3 associations with increasing methanol content — behavior that is


Figure 7. The translational diffusion coefficient of (a) water and (b) methanol as a function of thesolvent composition.

well seen in Fig. 7(b). Note that, for any cm, one observes DH2O < DCH3OH, as itshould be taking into account above-mentioned reasoning.

3.4. Diffusion of Cations and Ionic Conductivity

From the velocity autocorrelation functions obtained in our NPT simulations for theindividual Li+ ions one can calculate their self-diffusion coefficients, DLi. Theyare shown in Fig. 8 as a function of the cm parameter characterizing the solventcomposition.

It is seen that, under the conditions considered, the D value goes through a max-imum located near 1:1 solvent composition. This is in line with the observationpresented above for the diffusion coefficient of water. It is easy to understand whyDLi drops when the methanol content increases. As has been mentioned, in thiscase the effective dielectric constant decreases, which promotes the formation of


Figure 8. Self-diffusion coefficient, DLi, as a function of the cm parameter characterizing the solventcomposition.

ion pairs. The binding of the cations by the low-mobile sulfonate chains leads to adecrease in DLi. It is more difficult to explain the behavior of DLi at low methanolcontent. A possible reason is connected with the anomalously high hydration ofthe lithium cations due to their small ionic radius [43]. We do find evidence forthis analyzing the corresponding Li–solvent PCFs (see Figs 2(a) and 3(a)). Both inthe water-containing and methanol-containing system, the lithium ion is found tobe in the cage of about six firmly attached solvent molecules of its first solvationshell. Additional solvent molecules form a second solvation shell around the Li+.The number of water molecules included in this shell is larger than the number ofmethanol molecules by a factor of 1.5. In this situation, one should consider the dif-fusion of Li/water complexes rather than diffusion of individual cations. Becauseof the stronger resulting solvation of the lithium ion in the water environment com-pared to methanol, its dynamic properties are expected to be more influenced bywater, leading to more hindered motion of the ions in this case. Thus we again canclearly see a specific interplay between structural and dynamic correlations in theion-containing Nafion system with absorbed water/methanol solvent.

We can compare the diffusion constants obtained for the Li cations in our studywith those found for the 2.2 molal LiI solution in water from the MD simulationand from experiments at 305 K [71]. These simulation and experimental values ofDLi are 0.7 × 10−5 cm2/s and 1.0 × 10−5 cm2/s, respectively. Inspection of Fig. 8shows that they are considerably larger than the self-diffusion coefficient observedin the water-containing Nafion system. The difference is directly linked with thefact that the Li+ ions tend to form ion pairs with SO3

− groups, competing withsolvent molecules for these sites.


In the standard linear response theory (see, e.g., Ref. [72]), the ionic conductivity,λ, is related to the charge flux autocorrelation function, CFAF, according to thefollowing Green–Kubo relation

λ = 1

3kBT V

∫ ∞

0〈j(t) · j(0)〉dt, (8)

where the flux is given by j(t) = e∑ni

i zivi (t), ni is the number of ions, zi is thecharge of a given ion, e is the elementary charge, and V denotes the system volume.Below, we will discuss the conductivity caused by cations, assuming that only theyare mobile and contribute to the electric current.

It should be emphasized that the ionic conductivity is a collective dynamic prop-erty that reflects mutually correlated motions of particles with long-range electro-static interactions. In this respect it differs from the self-diffusion coefficient of thesame particles even at extremely low concentrations. As can be seen from equa-tion (8), the CFAF contains cross terms (i.e., vivj with i �= j) that account for thecross-correlation between two different particles. Both the MSD of pairs and theVACF of pairs of particles involve cross terms which are responsible for the differ-ence between singlet and pair diffusion. We must, therefore, distinguish betweenthe singlet and collective (pair) self-diffusion treated in terms of the velocity auto-correlation function and the mean-square displacement of particles. The presenceof cross-correlations leads to a reduced diffusion coefficient for pairs in comparisonwith that of single particles, reflected in a steeper decay and a deeper minimum ofthe collective VACF. For the singlet VACF, one can average over all particles ofthe system to improve statistics. On the other hand, in the case of the collectivecorrelation functions, there is no such possibility, since there is only one collectivefunction that manifests itself in greater statistical noise. In our NPT simulations, wewere able to estimate the total error for the CFAF and the collective VACF to beabout 10%.

The value of λ (in siemens per meter S/m = 1/ m) is presented in Fig. 9 asa function of solvent composition, cm. It is seen that, similar to the singlet self-diffusion coefficient of Li+ shown in Fig. 8, the ionic conductivity calculated forthe same ions goes through a maximum. It also occurs in the range of cm whereit is observed for DLi. We believe, again, that this is a direct consequence of thecomplex interplay between electrostatic forces and transport properties, as has beenmentioned above. Unfortunately, we do not have the corresponding experimentaldata at hand to compare with our theoretical predictions. However, it seems that theconductivities calculated in the present work are quite reasonable, at least, in theirorder of magnitude of 10−4 S/m.

In the final part of this subsection, we will briefly consider the mechanism under-lying cation migration and ionic conductivity. The interesting dynamical variable isthe ion-migration trajectory from which the macroscopic transport properties suchas D and λ are calculated. We obtain this trajectory (positions and momenta as afunction of time) for the diffusing particles by numerically integrating the equa-


Figure 9. The ionic conductivity λ (in siemens per meter S/m = 1/ m) as a function of solvent com-position, cm. The value of λ is given by λ = 〈j2(0)〉τ/3kBT V , where τ = τ = ∫ 〈j(t) · j(0)〉/〈j2(0)〉dt

is the characteristic conductivity time.

tions of motion for a given set of initial conditions. In order to shed some lighton the transport mechanism, we analyzed several typical trajectories of 32 individ-ual cations; one of them is presented in Fig. 10 for the water-containing system, atcm = 0. Inspection of these trajectories allows the identification of several patternsfor ion migration in the solvent-containing Nafion matrix.

As can be seen from Fig. 10, where the three-dimensional way and the x-coordinate of the Li+ trajectory are drawn as a function of time for a particularcase, the cation carries out discrete jumps from one binding site (SO3

− group) intoanother. The superposition of all interaction energies of the cation with its envi-ronment creates a potential energy U(x, y, z) for the moving cation as a functionof its migration coordinate. It provides an energetically favorable pathway for thecation through the membrane. The potential energy along this path is characterizedby binding sites, which correspond to the well-defined quasi-equilibrium minimaof U(x, y, z) (potential traps), where the ion shows oscillations for a relatively longtime. Also, there are transition-state sites, that is, the saddle points of U(x, y, z)

which correspond to very rapid crossover from one binding site into a neighboringone.

A few other examples of the 3-D trajectories illustrating typical motion patternsare given in Fig. 11. As seen from Figs 10 and 11, progression of cations from onepotential trap to the next can take different forms. In Fig. 10, we can observe twosuch traps: the ion jumps from one trap into another, stays there for about 50 ps,and then falls back into the first trap. In Fig. 11(a), one observes three transitionsbetween three different traps, while Fig. 11(b) presents an example of a smooth(continuous) motion pattern, when the trapping ion moves together with its binding


Figure 10. The three-dimensional (x, y, z) trajectory and its projection on the x-axis for Li+ iondrawn as a function of time.

site and shows short-time fluctuations, remaining connected to this site. Net cationtransport is the superposition of all these motions along the conductivity grid.

Analyzing the ion-migration trajectories, one can speculate that the long-rangeion motion within the system might be appropriately viewed as a process that isfractal in space or in time with two characteristic space (time) scales. In terms ofsystem morphology, the migration pathways are constrained by hydrophilic con-ductive channels, which exist in the microphase-separated structure of the solvent-containing Nafion. Such channels can be visualized directly by constructing three-dimensional Connolly surfaces. A Connolly surface [73] is the locus of pointsformed by the intersection of the van der Waals surface of a given ensemble of par-ticles with a spherical probe particle of radius R0 which is rolled over the particlesunder consideration. In this way, the global conductive-network morphology canbe studied. An example of the image generated with R0 = 1.4 Å for the subsystemof water molecules and lithium ions is shown in Fig. 12 for the 9728 atom water-containing system. One can observe the formation of the long tortuous channelsrunning through the whole computational box. For the relatively high water contentconsidered, essentially no isolated hydrophilic clusters are presented in the system


Figure 11. The three-dimensional (x, y, z) trajectories of Li+ ion for the water-containing system.

and the majority of the polar component is found on the continuous (effectively in-finite) domain. Such a morphology of the microphase-separated system is favorablefor ionic conductivity. In this case, the sulfonate groups, playing the role of bindingsites for cations, are located close to each other so that the cations can easily jumpfrom one binding site to a neighboring one, demonstrating the site-to-site migrationmechanism. In principle, the cations can travel together with the SO3

− groups as ionpairs, or by the migration of one cation while its contrion SO3

− is released and re-arranges itself into a new environment. Because the polymer itself cannot travel far,the migrating cations are constrained to move along the conductive channels. Con-sequently, we expect an increasing jump rate with decrease in the average distancebetween two adjacent sulfonate groups. The corresponding changes in the spatialdistribution of these groups can be induced by changes in the effective dielectric


Figure 12. Three-dimensional Connolly surface generated for the subsystem of water molecules andlithium ions after NPT simulation for 16 ns for the 9728 atom water-containing system.

permittivity, ε, which, in turn, depends on the solvent composition. It should bestressed that here, however, that there are two competing effects. On the one hand,as ε becomes lower, this promotes the formation of ion pairs and increases theirmutual attraction, resulting in more close contacts between the SO3

− groups andleading, therefore, to more intensive cation migration. On the other hand, it is clearthat very strong binding of cations by the polymer decreases jump rates and a netionic conductivity. Such a behavior is seen in Figs 8 and 9, nicely illustrating therole of these competing effects. Of course, the overall conductivity of the systemnot only depends on the site-to-site transition frequency and the spatial distributionof sulfonate groups in the channels but also on the solvation of the migrating ionsby the low-molecular-weight solvent. In the real system, the ionic conductivity alsoshould depend on the presence of other transported particles in the system (for ex-


ample, Cl− anions). The influence of small mobile anions on the conductivity ofsolvent-containing Nafion membranes is the subject of our future investigation.

4. Conclusion

Molecular dynamics simulation of transport phenomena in the condensed phase isa very time-consuming procedure. The numerical effort drastically increases withthe complexity of the model and the number of particles in the system, especially inthe case of charged particles interacting via long-range Coulomb potentials. There-fore, in order to study the dynamic characteristics of solvent-containing Nafionmembranes on a molecular level, we have used a rather simplified model whichnevertheless mimics some of the salient features of these complex microphase-separated solid polyelectrolytes, at least the properties of their hydrophilic regionsknown to be responsible for transport processes and electrochemistry.

We have simulated the model of solvated Nafion in the lithium salt form, with thefollowing three main objectives: (i) to obtain details on the local environment of thelithium cations and to assess the solvent effect on their binding, (ii) to investigatethe translational and rotational motion of solvent molecules (water and methanol)in the polymer matrix, and (iii) to elucidate details of the ionic transport thoughthe hydrophilic regions of the membrane and to study the ionic conductivity as afunction of solvent (water/methanol) composition.

A property which is of central importance for understanding the functional fea-tures of Nafion materials, including methanol membrane fuel cells, is the ionicconductivity. We have found that this parameter is strongly dependent on the solventcomposition and determined both by solvation effects and by the spatial distributionof polar sulfonate groups in ion-conductive channels.

In should be stressed that, although we have only considered the transport ofthe lithium cations and ignored the possibility of charge transfer between ions, theresults obtained are somehow relevant to the problem of protonic conductivity. In-deed, in the water environment, each lone proton associates with a water moleculeand forms a hydroxonium ion H3O+ with a net charge of +1 e. In many respects,the hydroxonium ion behaves like the Li+ ion, especially taking into account thesmall ion radius and mass of the Li+. Hence, it is believed that our main conclusionsconcerning the solvent effect on the ionic conductivity of Nafion in the lithium saltform could be correct for Nafion in its acid form. But we have to remember that ourset of potentials does not include structural diffusion mechanism of proton conduc-tivity (so-called Grotthus jumping). Charge transfer requires a quantum mechanicaltreatment and is, therefore, not within the scope of classical MD simulation tech-niques.

Although the critical Grotthus proton jumping mechanism is missing from ourMD model, it is nevertheless interesting to investigate the classical diffusion insuch a system. We can forecast crossover behavior for different liquid fuels, likemethanol, and usage of Nafion membranes not only as a common use proton-


exchange membrane, but more general ion-exchange membrane for fuel cells andother applications.

References

1. D. D. Morris and X. J. Sun, Water-sorption and transport properties of Nafion 117, Appl. Polym.Sci. 50, 1445–1452 (1993).

2. R. C. Weast and M. J. Astle (Eds), CRC Handbook of Chemistry and Physics, 62nd edn. CRC,Boca Raton, FL, USA (1981).

3. X. Gong, A. Bandis, A. Tao, G. Meresi, Y. Wang, P. T. Inglefield and P. J. James, Self-diffusion ofwater, ethanol and decafluoropentane in perfluorosulfonate ionomer by pulse field gradient NMR,Polymer 42, 6485–6492 (2001).

4. C. R. Martin, T. A. Rhoades and J. A. Ferguson, Dissolution of perfluorinated ion containingpolymers, Anal. Chem. 54, 1639–1641 (1982).

5. D. Rivin and N. S. Schneider, Structure, Dynamics and Charge Transport in Polymeric Materials.Argonne, IL, USA (2000).

6. D. Rivin, C. E. Kendrick, P. W. Gibson and N. S. Schneider, Solubility and transport behavior ofwater and alcohols in Nafion, Polymer 42, 623–635 (2000).

7. D. Nandan, H. Mohan and R. M. Iyer, Methanol and water uptake, densities, equivalental vol-umes and thicknesses of several uni- and divalent ionic perfluorosulphonate exchange membranes(Nafion-117) and their methanol–water fractionation behaviour at 298 K, J. Membr. Sci. 71, 69–80(1992).

8. J. Pilar, J. Labsky and S. Schlick, ESR of soluble Nafion and Nafion membranes spin-labeled viathe ionic bond, J. Phys. Chem. 99, 12947–12951 (1995).

9. J. Pellegrino and Y. S. Kang, CO2/CH4 transport in polyperfluorosulfonate ionomers: effects ofpolar solvents on permeation and solubility, J. Membr. Sci. 99, 163–174 (1995).

10. M. Pasternak and T. G. Dorawala, Preferential removal of alcohols from aqueous solutions byion exchange membranes containing organic counterions, J. Polym. Sci. Part A: Polym. Chem. 29,915–923 (1991).

11. M. Saito, S. Ikesaka, J. Kuwano, J. Qiao, S. Tsuzuki, K. Hayamizu and T. Okada, Mechanismsof proton transport in alcohol-penetrated perfluorosulfonated ionomer membranes for fuel cells,Solid State Ionics 178, 539–545 (2007).

12. J. A. Morrone, K. E. Haslinger and M. E. Tuckerman, Ab initio molecular dynamics simulation ofthe structure and proton transport dynamics of methanol–water solutions, J. Phys. Chem. B 110,3712–3720 (2006).

13. V. A. Dyakov and Yu. K. Tovbin, Molecular dynamics study of the cations, water molecules, andpolymer chains in Nafion type membranes, Russ. Chem. Bull. 44, 1186–1189 (1995).

14. J. A. Elliott, S. Hanna, A. M. S. Elliott and G. E. Cooley, Atomistic simulation and moleculardynamics of model systems for perfluorinated ionomer membranes, Phys. Chem. Chem. Phys. 1,4855–4863 (1999).

15. A. Vishnyakov and A. V. Neimark, Molecular simulation study of Nafion membrane solvation inwater and methanol, J. Phys. Chem. B 104, 4471–4478 (2000).

16. A. Vishnyakov and A. V. Neimark, Molecular dynamics simulation of Nafion oligomer solvationin equimolar methanol–water mixture, J. Phys. Chem. B 105, 7830–7834 (2001).

17. A. Vishnyakov and A. V. Neimark, Molecular dynamics simulation of microstructure and molec-ular mobilities in swollen Nafion membranes, J. Phys. Chem. B 105, 9586–9594 (2001).


18. S. Urata, J. Irisawa, A. Takada, W. Shinoda, S. Tsuzuki and M. Mikami, Molecular dynamicssimulation of swollen membrane of perfluorinated ionomer, J. Phys. Chem. B 109, 4269–4278(2005).

19. S. S. Jang, V. Molinero, T. Cagin and W. A. Goddard, Nanophase-segregation and transport inNafion 117 from molecular dynamics simulations: effect of monomeric sequence, J. Phys. Chem.B 108, 3149–3157 (2004).

20. S. S. Jang, V. Molinero, T. Cagin and W. A. Goddard, Effect of monomeric sequence on nanos-tructure and water dynamics of Nafion 117, Solid State Ionics 175, 805–808 (2004).

21. D. Brandell, J. Karo, A. Liivat and J. O. Thomas, Molecular dynamics studies of the Nafion, Dowand Aciplex fuel-cell polymer membrane systems, J. Mol. Model. 13, 1039–1046 (2007).

22. A. Venkatnathan, R. Devanathan and M. Dupuis, Atomistic simulations of hydrated Nafion andtemperature effects on hydronium ion mobility, J. Phys. Chem. B 111, 7234–7244 (2007).

23. R. Devanathan, A. Venkatnathan and M. Dupuis, Atomistic simulation of Nafion membrane: 1. Ef-fect of hydration on membrane nanostructure, J. Phys. Chem. B 111, 8069–8079 (2007).

24. R. Devanathan, A. Venkatnathan and M. Dupuis, Atomistic simulation of Nafion membrane.2. Dynamics of water molecules and hydronium ions, J. Phys. Chem. B 111, 13006–13013 (2007).

25. M. Pivovar and B. S. Pivovar, Dynamic behavior of water within a polymer electrolyte fuel cellmembrane at low hydration levels, J. Phys. Chem. B 109, 785–793 (2005).

26. D. E. Moilanen, I. R. Piletic and M. D. Fayer, Tracking water’s response to structural changes inNafion membranes, J. Phys. Chem. A 110, 9084–9088 (2006).

27. P. G. Khalatur, A. R. Khokhlov, D. A. Mologin and E. A. Zheligovskaya, Computer simulationstudies of aggregates of associating polymers: influence of low-molecular-weight additives solu-bilizing the aggregates, Macromol. Theor. Simul. 7, 299–316 (1998).

28. P. G. Khalatur, S. K. Talitskikh and A. R. Khokhlov, Structural organization of water-containingNafion: the integral equation theory, Macromol. Theor. Simul. 11, 566–586 (2002).

29. D. A. Mologin, P. G. Khalatur and A. R. Khokhlov, Structural organization of water-containingNafion: a cellular-automat-based simulation, Macromol. Theor. Simul. 11, 587–607 (2002).

30. M. Fujimura, T. Hashimoto and H. Kawai, Small-angle X-ray scattering study of perfluorinatedionomer membranes. 1. Origin of two scattering maxima, Macromolecules 14, 1309–1315 (1981).

31. T. D. Gierke, G. E. Munn and F. C. Wilson, The morphology in Nafion perfluorinated membraneproducts, as determined by wide- and small-angle X-ray studies, J. Polym. Sci. 19, 1687–1704(1981).

32. J. A. Elliott, S. Hanna, A. M. S. Elliott and G. E. Cooley, Interpretation of the small-angle X-rayscattering from swollen and oriented perfluorinated ionomer membranes, Macromolecules 33,4161–4171 (2000).

33. D. Y. Galperin and A. R. Khokhlov, Mesoscopic morphology of proton-conducting polyelectrolytemembranes of Nafion type: a self-consistent mean field simulation, Macromol. Theor. Simul. 15,137–146 (2006).

34. E. Spohr, P. Commer and A. A. Kornyshev, Enhancing proton mobility in polymer electrolytemembranes: lessons from molecular dynamics simulations, J. Phys. Chem. B 106, 10560–10569(2002).

35. M. K. Petersen, F. Wang, N. P. Blake, H. Metiu and G. A. Voth, Excess proton solvation and de-localization in a hydrophilic pocket of the proton conducting polymer membrane Nafion, J. Phys.Chem. B 109, 3727–3730 (2005).

36. M. K. Petersen and G. A. Voth, Characterization of the solvation and transport of the hydrated pro-ton in the perfluorosulfonic acid membrane Nafion, J. Phys. Chem. B 110, 18594–18600 (2006).


37. S. J. Paddison and J. A. Elliott, Selective hydration of the ‘short-side-chain’ perfluorosulfonic acidmembrane. An ONIOM study, Solid State Ionics 178, 561–567 (2007).

38. J. A. Elliott and S. J. Paddison, Modelling of morphology and proton transport in PFSA mem-branes, Phys. Chem. Chem. Phys. 9, 2602–2618 (2007).

39. K. A. Mauritz and R. B. Moore, State of understanding of Nafion, Chem. Rev. 104, 4535–4586(2004).

40. K.-D. Kreuer, S. J. Paddison, E. Spohr and M. Schuster, Transport in proton conductors forfuel-cell applications: simulations, elementary reactions and phenomenology, Chem. Rev. 104,4637–4678 (2004).

41. J. J. P. Stewart, MOPAC: a semiempirical molecular orbital program, J. Comput. — Aided Mol.Des. 4, 1–105 (1990).

42. X. Li and Z.-Z. Yang, Study of lithium cation in water clusters: based on atom-bond electroneg-ativity equalization method fused into molecular mechanics, J. Phys. Chem. A 109, 4102–4111(2005).

43. P. C. Hariharan and J. A. Pople, The influence of polarization functions on molecular orbitalhydrogenation energies, Theo. Chem. Acta 28, 213–222 (1973).

44. K. Heinzinger, Computer simulations of aqueous electrolyte solutions, Physica B 131, 196–216(1985).

45. H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren and J. Hermans, Interaction models forwater in relation to protein hydration, in: Intermolecular Forces, B. Pullman (Ed.), pp. 331–342.Reidel, Dordrecht, Germany (1981).

46. K. Toukan and A. Rahman, Molecular-dynamics study of atomic motions in water, Phys. Rev. B31, 2643–2648 (1985).

47. P. A. Giguere, The infra-red spectrum of hydrogen peroxide, J. Chem. Phys. 18, 88–92 (1950).48. S. Nose, A molecular dynamics method for simulations in the canonical ensemble, Mol. Phys. 57,

187–190 (1986).49. W. G. Hoover, Canonical dynamics: equilibrium phase-space distributions, Phys. Rev. A 31, 1695–

1697 (1985).50. G. J. Martyna, D. J. Tobias and M. L. Klein, Constant pressure molecular dynamics algorithms,

J. Chem. Phys. 101, 4177–4189 (1994).51. M. Tuckerman, B. J. Berne and G. J. Martyna, Reversible multiple time scale molecular dynamics,

J. Chem. Phys. 97, 1990–2001 (1992).52. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids. Clarendon, Oxford, UK (1987).53. L. Verlet, Computer ‘experiments’ on classical fluids. I. Thermodynamical properties of Lennard–

Jones molecules, Phys. Rev. 159, 98–103 (1967).54. Gy. I. Szasz and K. Heinzinger, Hydration shell structures in a LiI solution at elevated temperature

and pressure — a molecular dynamics study, Earth Planet. Sci. Lett. 64, 163–167 (1983).55. B. J. Alder, D. M. Gass and T. E. Wainwright, Studies in molecular dynamics. VIII. The transport

coefficients for a hard-sphere fluid, J. Chem. Phys. 53, 3813–3826 (1970).56. K. Krynicki, C. D. Green and D. W. Sawyer, Pressure and temperature dependence of self-

diffusion in water, Faraday Discuss. Chem. Soc. 66, 199–208 (1978).57. R. L. Hurle and I. A. Woolf, The effect of isotopic substitution on self-diffusion in methanol under

pressure, Aust. J. Chem. 33, 1947–1952 (1980).58. J. J. Porter, J. K. Klassen and G. M. Nathanson, Penetration of water vapor through perfluorinated

polyether films on concentrated sulfuric acid, Langmuir 12, 5448–5450 (1996).59. P. T. Callaghan, M. A. Le Gros and D. N. Pinder, The measurement of diffusion using deuterium

pulsed field gradient nuclear magnetic resonance, J. Chem. Phys. 79, 6372–6380 (1983).


60. R. W. Impey, P. A. Madden and I. R. McDonald, Spectroscopic and transport properties of water.Model calculations and the interpretation of experimental results, Mol. Phys. 46, 513–539 (1982).

61. D. C. Rapaport, Hydrogen bonds in water network organization and lifetimes, Mol. Phys. 50,1151–1162 (1983).

62. G. C. Lie and E. Clementi, Molecular-dynamics simulation of liquid water with an ab initio flexi-ble water–water interaction potential, Phys. Rev. A 33, 2679–2693 (1986).

63. M. Haughney, M. Ferrario and I. R. McDonald, Molecular-dynamics simulation of liquidmethanol, J. Phys. Chem. 91, 4934–4940 (1987).

64. R. C. Weast, S. S. Selby and C. D. Hodgman, Handbook of Chemistry and Physics, 50th edn,p. F-36. Chemical Rubber Company, Cleveland, OH, USA (1968).

65. A. Abragam, Principles of Nuclear Magnetism. Clarendon Press, Oxford, UK (1948).66. B. MacMillan, A. R. Sharp and R. L. Armstrong, An NMR investigation of water absorbed in

Nafion, Polymer 40, 2471–2480 (1999).67. B. MacMillan, A. R. Sharp and R. L. Armstrong, An NMR relaxation in Nafion — the low tem-

perature regime, Polymer 40, 2481–2485 (1999).68. J. T. Edsall and H. A. McKenzie, Water and proteins. II. The location and dynamics of water in

protein systems and its relation to their stability and properties, Adv. Biophys. 16, 53–183 (1983).69. M. Gruner and H. G. Hertz, The structure of liquid mixtures as studied by nuclear magnetic

relaxation times, Adv. Mol. Relax. Processes 3, 75–80 (1972).70. T. P. Straatsma, H. J. C. Berendsen and A. J. Stam, Estimation of statistical errors in molecular

simulation calculations, Mol. Phys. 57, 89–95 (1986).71. G. I. Szasz, K. Heinzinger and W. O. Riede, Self-diusion and reorientational motion in an aqueous

LiI solution. A molecular dynamics study, Ber. Bunsenges. Phys. Chem. 85, 1056–1059 (1981).72. D. A. McQuarrie, Statistical Mechanics. Harper and Row, New York, USA (1976).73. M. L. Connolly, Solvent-accessible surfaces of proteins and nucleic acids, Science 221, 709–713

(1983).

Documents

Computer Simulation Study of Model Nafion Membrane in Water ... · consistent field theory (SCFT) [33], we basically confirmed the conclusions of the spheroidal geometry of ionomer