THE JOURNAL OF CHEMICAL PHYSICS 144, 014902 (2016)

Coarse-grained model of water diffusion and proton conductivityin hydrated polyelectrolyte membrane

Ming-Tsung Lee, Aleksey Vishnyakov, and Alexander V. Neimarka)

Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey,98 Brett Road, Piscataway, New Jersey 08854-8058, USA

(Received 22 October 2015; accepted 8 December 2015; published online 7 January 2016)

Using dissipative particle dynamics (DPD), we simulate nanoscale segregation, water diffusion, andproton conductivity in hydrated sulfonated polystyrene (sPS). We employ a novel model [Lee et al.J. Chem. Theory Comput. 11(9), 4395-4403 (2015)] that incorporates protonation/deprotonationequilibria into DPD simulations. The polymer and water are modeled by coarse-grained beadsinteracting via short-range soft repulsion and smeared charge electrostatic potentials. The protonis introduced as a separate charged bead that forms dissociable Morse bonds with the base beadsrepresenting water and sulfonate anions. Morse bond formation and breakup artificially mimicsthe Grotthuss mechanism of proton hopping between the bases. The DPD model is parameterizedby matching the proton mobility in bulk water, dissociation constant of benzenesulfonic acid, andliquid-liquid equilibrium of water-ethylbenzene solutions. The DPD simulations semi-quantitativelypredict nanoscale segregation in the hydrated sPS into hydrophobic and hydrophilic subphases, waterself-diffusion, and proton mobility. As the hydration level increases, the hydrophilic subphase exhibitsa percolation transition from isolated water clusters to a 3D network. The analysis of hydrophilicsubphase connectivity and water diffusion demonstrates the importance of the dynamic percolationeffect of formation and breakup of temporary junctions between water clusters. The proposedDPD model qualitatively predicts the ratio of proton to water self-diffusion and its dependenceon the hydration level that is in reasonable agreement with experiments. C 2016 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4938271]

I. INTRODUCTION

The presented paper is the first attempt to model directlyproton diffusion in a hydrated polyelectrolyte using dissipativeparticle dynamics (DPD). Polyelectrolyte membranes (PEMs)are employed in proton exchange membrane fuel cells, anda tremendous experimental and theoretical effort is investedin exploration and prediction of PEM structure and transportproperties. Polyelectrolyte for fuel cells typically consistsof a hydrophobic organic backbone, to which strong acidgroups are attached. Upon hydration (or solvation by otherprotonating compounds that are also used in PEM1), the acidgroups dissociate, releasing protons and making the membraneproton-conductive. Because PEM have both hydrophobicand hydrophilic components, their structure is stronglynon-uniform and may be very complex. Typically, hydratedPEM segregates onto a hydrophilic subphase formed by theacid groups, protons, and polar solvents, and a hydrophobicsubphase formed by the organic backbone. The hydrophilicsubphase forms a network of clusters and channels throughwhich water and protons diffuse. The network morphology isdetermined by the chemical structure of the polyelectrolyteand the hydration level (here denoted as HL and defined as themass of adsorbed water divided by the mass of dry polymer).Thus, the proton transport through PEM is controlled by

a)Author to whom correspondence should be addressed. Electronic mail:aneimark@rutgers.edu. Telephone: +1-848-445-0834.

several phenomena that have very different characteristicspatial and temporal scales: (1) dissociation of the individualacid groups, (2) proton transfer between solvent moleculesand/or acid groups, and (3) overall segregation morphologyof the hydrated PEM.

Due to the wide range of the scales determining water andproton transport in PEM, their modeling typically employshybrid approaches, where different techniques are utilized tomodel the membrane segregation and water/proton diffusionon different levels. The segregation morphology whose typicalscale ranges from several to tens of nanometers is predictedby mesoscale methods, such as thermodynamic modeling,2,3

self-consistent field theory,4–6 MesoDyn,7 coarse grainedmolecular dynamics (CG MD)8–12 or DPD.13–26 Then, thediffusion of water and protons is considered in a static structureobtained from mesoscale modeling.25,27 Alternatively, waterand ion diffusion is modeled in pre-determined “ideal”environments such as cylindrical channels28,29 and then theresults for ideal pores are extrapolated onto the macroscopicsystem using pore network models to predict overall transportproperties30,31 or to obtain insights on pore structure anddiffusion mechanisms from comparison of simulation resultswith experiments.28,32

Because Nafion is the best-known PEM material, mostsimulation studies considered Nafion and other similarperfluorinated polyelectrolytes. The first DPD simulationof Nafion was conducted by Yamamoto and Hyodo.14

The conservative repulsion parameters were estimated from

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the mixing energy calculations conducted with atomisticmodeling. The electrostatic interactions were implicitlymimicked by short-range forces.14 The authors found irregularsegregation morphologies, with reasonable correspondence toexperimental results. They suggested that the proton conduc-tivity might be estimated from water cluster connectivityin the obtained structures. Later, Dorenbos and Suga27 andWu et al.,21,23 employed the same model for studies ofnanostructure and water diffusion in several perfluorinatedionomers that differed by equivalent weight and sidechainlength. Dorenbos and Suga27 estimated water diffusion inthe resulting DPD structures by mapping the segregatedstructure onto a lattice; each lattice site belonged to eithermobile (aqueous) or immobile (organic) subphase. Waterself-diffusion coefficients were estimated using the randomwalks on the lattice digitized replicas. The authors concludedthat this approach cannot be applied to proton conductivity,since the lattice replica of nano-segregated PEM structuresdo not carry any information about the local environmentaround given lattice site and therefore are unable to properlyaccount for the interaction between protons and negativelycharged sulfonate groups. Later, Dorenbos et al.33 used themodels developed for bulk nafion in simulations of the PEMin contact with carbon catalyst support.

Sawada et al.34 accounted for possible cross-linkingof the perfluorinated skeleton chains and found that cross-linking leads to much smaller hydrophilic aggregates of only1.8 nm in diameter. Elliott et al.20 combined DPD resultswith experimental SAXS/SANS studies using an originalmodel-independent procedure. The modeling revealed a multi-level membrane organization, with hydrophilic–hydrophobicsegregation on smaller scale and larger scale organizationof the fluorocarbon backbone. This result is consistentwith the NMR studies.35,36 Jorn and Voth37 modeled thenanostructure of segregated polymer with standard short rangeDPD potentials, and then, considered proton transport in thestructures obtained using smoothed particle hydrodynamics(SPH).38 Transport coefficients and coarse-grained forcesfor the polymer backbone, side chain, proton, and waterinteractions were derived from atomistic MD simulations.The proton conductance profiles determined in this simulationat scales of 40 nm are in semi-quantitative agreement withresults of earlier experiments.39,40 The authors also showedthat accounting for the electrostatic interactions is crucial forthe improvement of proton transport modeling with DPD.Albeit not directly related to DPD, discussions on protontransfer regarding to both vehicular and structural diffusionin hydrated PEM of different morphologies should also benoted.37,41–48

DPD models with electrostatics considered implicitlywere applied to other polyelectrolytes, such as sulfonatedpoly(phenylene) sulfone (sPSO2) at different sulfonationlevels (SLs),49 SPEEK,50 and grafted copolymers withvarying type and the attachment of the side chain.17

Explicit treatment of electrostatic interactions in DPD wasenabled by the development of smeared charge models.In this approach, the charge is distributed around thebead center: linear,51 Slater-type exponential,52 Gaussian-type,53 and Bessel-type54 distributions of charge density were

employed in the literature.54 The charge distributions areisotropic and spherically symmetric. The charges interactin an isotropic medium of a uniform dielectric constant ε,although newly introduced polarizable models55 enable moreprecise simulations at extra computational cost. The smearedcharge approach was applied to modeling self-assembly inhydrated metal-substituted Nafion25 that enabled the firstexplicit charge DPD simulation of PEM. The adsorptionisotherms were calculated by Widom trial insertions of waterbeads into the Nafion structures generated with DPD, andthe saturation hydration levels were identified for polymer ofdifferent equivalent weights. Water diffusion was estimated bythe random walk simulation in lattice replicas of the segregatedpolymer. The simulation results were in good agreement withthe experiment on potassium-substituted Nafion.

This succinct review shows that in published mesoscalestudies of polyelectrolytes, dissociation/association of acidgroups was considered indirectly. The degree of dissociationhas to be essentially pre-assumed. For example, in therecent DPD simulation25 of metal-substituted Nafion polymerat low water content, the dissociation degree was fixedand the respective fraction of the alkali metal counterionswas considered as dissociated from their sulfonate groupsand represented by hydrated counter-ion beads; the restof the counterions were kept attached to the sidechains,and such pairs were modeled by neutral beads. Thus, thedissociation degree was determined by the coarse grainingscheme rather than by chemical consideration. Alternatively,each dissociating group may be assigned a fractional chargeaccording to the degree calculated theoretically. This approachwas employed in DPD simulations of α-synuclein thatcontains both carbonic acid and amine groups,56 and inmodeling of ionic diblock surfactants by Posel et al.57 Bothapproaches share the same major drawback: the dissociationof a particular counter-ion is determined by the macroscopicproperties such as hydration or pH. In reality, the dissociationof a proton and its mobility is determined by local environmentaround it. Therefore, it is desirable to embed directlythe dissociation mechanism into the mesoscale simulationforcefield.

Here, we present a direct DPD study of proton diffusion inhydrated polyelectrolytes. The basic approach to incorporationof proton into DPD simulations was suggested recently.61

Proton is modeled as a special P-bead; the protonatingcompounds (e.g., water and acid anions) are modeled asproton-receptive base beads, and they interact with P-beadsby dissociative Morse potential. The model calibration isperformed by matching the simulated proton mobility inbulk water and dissociation degree in dilute solutions ofreference acids to the experimental data. Now, we extend thisapproach onto PEM drawing on the example of sulfonatedpolystyrene (sPS). sPS is a practically important polymerand various sPS based materials (especially block copolymersof sPS and polyolefins) are used in proton exchange fuelcells; their industrial potential is supported by a low cost,as they are generally cheaper than perfluorinated PEMs ofNafion type. In sPS-polyolefin block copolymers, water-swollen sPS forms hydrophilic domains, which are segregatedonto the aqueous subphase formed by the sulfonic acid

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groups surrounded by water and protons and the hydrophobicalkylbenzene subphase. The segregation inside hydratedsPS was observed both experimentally and in atomisticsimulations.58–60 The morphology is irregular and dependson the level of sulfonation, and the scale of segregation isbelieved to be relatively small. Small size of hydrophilicchannels, tangible dissociation constant of the acid group, andability to vary the sulfonation level substantially make sPSan ideal system for examining the advantages and limitationsof our DPD approach. It should be noted that, a DPD modelof benzenesulfonic acid, which is a fragment of sPS, wasdeveloped in our previous work.61

The paper is structured as follows: In Section II, weintroduce the coarse grained model of sPS and describepolymer dissection into beads, coarse-grained forcefield, andits parameterization. We also present the model of proton andthe mechanism of its diffusion with the proposed DPD model.In Section III, we discuss the results of DPD simulations ofhydrated sPS. We describe the morphology of segregation andthe mechanisms of water diffusion at different hydration levels.We also estimate the proton conductivities and compare themwith experimental data. Conclusion is given in Section IV.

II. COARSE-GRAINED MODELS AND SIMULATIONDETAILS

A. Coarse grained model of sPS

The general DPD model adopted in this work operateswith the beads of equal size. To fulfill this requirement, weevaluated the volumes of functional groups from the standardBondi table data used in the UNIFAC group contributionmodel, and we dissected sPS into three types of fragments ofcomparable size represented by soft-core beads as illustratedin Figure 1. sPS monomer contains an ethylbenzene fragmentand, if the monomer is sulfonated, a sulfonic acid group.Ethylbenzene is modeled as a dimer of two hydrophobic beads(B–C), and the deprotonated sulfonated group is modeled asa single S bead. B bead represents four aromatic carbons ofthe benzene ring, and C bead contains two aromatic carbons

and two aliphatic skeleton carbons. Calculations of fragmentvolume and exact mapping of atoms onto the beads are shownin the supplementary material, Section S1.62 S bead containssulfur and three oxygens, making the CG mapping consistentwith four heavy atoms in one DPD bead. To match the volumeof polymer beads, the solvent bead W represents three watermolecules. The corresponding bead size is 0.65 nm, estimatedfrom the bulk density of liquid water.63

We simulate sPS oligomers composed of 20 monomerseach. The monomers are sulfonated in para position accordingthe sulfonation level (defined as the fraction of sulfonatedstyrene monomers and denoted as SL), which is variedfrom 10% to 40%. The sulfonated monomers are uniformlydistributed among the chain, as shown in Figure 1(b). ProtonH+ is modeled separately as an individual bead P, accordingto Ref. 61.

B. DPD forcefield

We employed the conventional DPD scheme64 with thebeads interacting via pairwise conservative soft repulsiveF(C)i j (ri j), bond F(B)

i j (ri j), and electrostatic F(E)i j (ri j) forces,

as well as random F(R)i j (ri j) and velocity dependent drag

F(D)i j (ri j,vi j) forces,

Fi j(ri j) = F(B)i j (ri j) + F(A)

i j (θi j) + F(C)i j (ri j) + F(D)

i j (ri j,vi j)+F(R)

i j (ri j) + F(E)i j (ri j) + F(M)

i j (ri j). (1)

All beads are assigned the equal effective diameter ofRc = 0.65 nm. The soft repulsion force F(C)

i j acts between over-lapping beads: F(C)

i j (ri j) = aI J(1 − ri j/Rc)ri j/ri j at r < Rc,F(C)i j (ri j) = 0 at r ≥ Rc, where aIJ is the repulsion parameter

specific to given bead pair of types I and J. Following thestandard approach to DPD simulations of self-assembly,64 theintra-component repulsion parameters aII between beads ofthe same type are set equal, irrespective to the type. Thebeads are tightly packed at the reduced density of ρRc

3 = 3,common in DPD simulations.64

FIG. 1. (a) Chemical structure of partially sulfonated polystyrene (sPS) and the corresponding DPD model. The beads are connected by 1-2 body bonds (dashedblack lines), and 1-3 body bonds (dashed blue line) to provide chain rigidity. Rigidity on side chain is controlled by harmonic angle (dashed red curves). Detailsof these constraints are discussed in Section II D. (b) sPS at different sulfonation level, denoted as SL vol. %. For example, sPS of SL 10% means there are 2sulfonate groups in a chain of 20 monomers.

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The Langevin thermostat is maintained by random anddrag forces, acting between overlapping beads along thevector ri j connecting the bead centers. Random force F(R)

i j

that accounts for thermal fluctuations is taken proportionalto the conservative force: F(R)

i j (ri j) = σwR(ri j)θi j(t)ri j/ri j,where θi j(t) is a randomly fluctuating in time variablewith Gaussian statistics. Drag force is velocity-dependent:F(D)i j (ri j,vi j) = −γwD(ri j)(ri j ∗ vi j)ri j/r2

i j,2 where vi and v j

are the current velocities of the particles and vi j = v j − vi.We assume the common relationships between the dragand random force weighting functions w(r) and parametersσ and γ that determine the levels of energy fluctuationand dissipation (friction) wD(r) = [wR(r)]2 = (1 − r/Rc)2 atr < Rc and σ2 = 2γkT . This allows one to maintain constanttemperature in the course of simulation via the Langevinthermostat. We assumed γ = 4.5, the value commonly used inDPD simulations of water.64

Beads in the chain are connected by FENE bondsF(B)i j (ri j) = kb(r0 − ri j)/(1 − ((ri j − r0)2/r2

m))ri j/ri j, where Kb

is bond rigidity, r0 is equilibrium bond length, and rm is themaximum bond length. Following our recent papers,56,65–67 inaddition to the nearest neighbor (1-2) bonds, we introducedthe second neighbor (1-3) bonds in order to control the chainrigidity. Because the polymer conformation is an importantfactor affecting the segregated structure of hydrated sPS, whichis relatively complex (compared to linear chains consideredearlier56,65–67) and contains very rigid fragments, we alsointroduced the harmonic angle potentials between certain pairsof nearest neighbor bonds as shown in Figure 1(b): F(A)

i j (θik j)= Kθ

�θ0 − θik j

�. Assuming the bead i and j are separated by

another bead k, θik j refers to the angle between vectors ik andj k, θ0 and Kθ are the equilibrium angle and stiffness.

The electrostatic interactions are modeled using thesmeared charge approach with the Slater-type chargedensity distribution with an exponential decay,52,67 i.e., f (r)=

q

πλ3 exp�− 2r

λ

�, where λ is the effective smearing radius. The

electrostatic force F(E)i j between charged particles i and j in

Eq. (1) is expressed as

F(E)i j (ri j) =

e2qiqj

4πkTε0εrRCr2i j

�1 − exp

�−2RCri j/λri j

� �1 + 2RCri j/λ

�1 + RCri j/λ

��� ri jri j

. (2)

In the long range limit, the electrostatic interaction ofsmeared charges (Eq. (2)) reduces to the Coulomb potential.The standard Ewald summation68 is used to account for long-range electrostatics. The smearing radius is set to λ = 0.25Rcfor all charged beads; this decision was made for technicalreasons (the supplementary material of Ref. 61, Section S2).Similar suggestions can be found in recent DPD studies.54,57,69

The last term F(M)i j (ri j) in Eq. (1) models the forces

between the proton bead P and bases that are water bead Wand sulfonate bead S. The P–base interactions are modelledby the Morse potential, cut and shifted to zero at the cutoffradius rM,

F(M)i j (ri j) = −2αIJKIJ exp

αIJ

(ri j − r0

i j

)×

1 − exp

αIJ

(ri j − r0

i j

)ri j/ri j, at ri j < rM

i j .

(3)

The Morse potential has a minimum at ri j = r0i j and is

characterized by the strength parameter KIJ and effectivesteepness αIJ. F(M)

i j applies to interactions of P beads withbases (S and W beads). That is, a P bead connected to a singleS bead by Morse potential forms a neutral acid. Formationof a new Morse bond between the same P and another base(say, a W bead) leads to a formation of an intermediatecomplex (Figure 2(a)). A P–base pair dissociates when rPS

FIG. 2. (a) Schematics of proton disso-ciating from the sPS side chain to thewater. When protonated sulfonic acid(SO3H) is solvated by water, it forms anS-P-W complex. (b) The overall poten-tial of S-P-W complex (sum of Morseinteraction energy between S and Pbeads, same for W and P beads, andrepulsive interaction energy between Sand W beads) versus proton transfercoordinate (distance between P and Sdivided by distance between S and W).When W bead is just in contact with Sbead (rS-W= 1Rc), the potential has twominima separated by a barrier. As S andW overlap (rS-W < 1Rc), the minimamerge into one, allowing proton bead tomigrate freely from sulfonic acid to thewater bead.

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exceeds rM, leading to a breakup of the complex. The processis associated with a potential barrier that is overcome bythermal fluctuations in the system (Figure 2(b)), which issimilar to actual acts of proton transfer in aqueous solutions.If the Morse pair that dissociates is the original P–S pair, theentire process describes dissociation of the acid in water. InRef. 61, we show how through the sequence of formations andbreakup of Morse bonds: P beads can “hop” between the bases(here sulfonates S and waters W), artificially mimicking theGrotthuss mechanism of proton diffusion.61 The schematics isillustrated in Figure 2. By adjusting the depth and steepnessof the Morse potential, we were able to reproduce both protonmobility in bulk water and the dissociation equilibrium of theacid. Here, we use the same potentials and parameters as insimulations of benzenesulfonic acid.61

C. Non-bonded parameters

Following the most common DPD implementation,63,64

the intra-component repulsion parameters aII are assumedequal for all bead types I and determined from thecompressibility of water, which gives aII = 78.5 kT/Rc forRc = 0.65 nm.61,63 It is worth noting that although moreelaborated approaches can be found in the literature,70

the DPD parameterization63,64 is suitable for this relativelysmall size of coarse-graining. The inter-component repulsionparameters are mapped to the infinite dilution activitycoefficient in solutions of reference compounds for differentbead types.56

We assume that hydrophobic beads B and C have the sameshort-range repulsion parameters with hydrophilic bead types(that is, aBI = aCI, where I = P, W, or S.). In order to estimateaBW, we choose ethylbenzene as a reference compound forhydrocarbon skeleton, and (similarly to Ref. 56) coarse-grainit as a symmetric dimer B–B. The bond length in the B–Bdimer is equal to the length of B–C bond in the coarse grainedsPS model and is found from energy minimized structuresof small sPS fragments (see Table I). We perform standardcanonical MC simulations with Widom trial particle insertionsand determined the calibration dependences γ∞ (aWB) for Wbeads in B–B dimer bath, as well as that for B–B dimerin the W bead bath. Using this approach, we obtain thecalibration correlations between γ∞ and ∆aWB = aWB − aWW.By interpolating the γ∞ values of ethylbenzene and water,which is estimated from the experimental solubility orCOSMO-RS thermodynamic model,71 we obtain aWB values.This calibration procedure is the same as employed for othercompounds,67 and details are given in the supplementarymaterial, Section S2.62

A S bead that models the sulfonate anion is hydrophilic.Because parameterization techniques for ionic species are stillpoorly developed, we set aSW = aWW and imposed strongrepulsion between S and hydrocarbon beads. (It is worthnoting that the value of aSW < aWW is often used whenelectrostatic interactions are implicitly accounted for Ref. 14.)Proton P bead does not experience any short range repulsionfrom the bases or other P beads but is repelled from B andC beads.61 The sPS model parameters are summarized inTable I.

TABLE I. Parameters of DPD model for hydrated sPS.

A. Morse interaction between P bead and bases

Bead pair KPW αPW rPW0 (Rc) rPW

M (Rc)

P–W 16.0 1 0 0.60P–S 18.5 2 0.34 0.66

B. Short-range repulsion parameters, (aIJRc)/kTBead J/bead I W S C and B P

W 78.5 78.5 102.3 0.0S 78.5 78.5 78.5 0.0C and B 102.3 78.5 78.5 102.3P 0.0 0.0 102.3 0.0

C. Bonds

Bond type Forcefield type K (kT /R2c) r0 (Rc) or θ0 rm (Rc)

C–C 1-2 FENE 800.0 0.50 2.0C–(C)–Ca 1-3 FENE 130.0 0.59 3.0B–C 1-2 harmonic 800.0 0.35 ∞S–B 1-2 harmonic 800.0 0.43 ∞

D. Angles

Angle type Forcefield type K (kT /rad2) θ0 (rad)

S–B–C Harmonic 500.0 180◦

B–C–C Harmonic 40.0 90◦

aSecond-neighbor bond between two C beads separated by another C bead.

D. Bonded parameters

Parameterization of bond potentials was performedsimilarly to our recent paper.67 We performed MD simulationof 10-unit oligomer of 100% sulfonated sPS in waterusing the force field parameters from Refs. 59 and 60 andcalculated the distribution of distances between the centersof mass of polymer fragments corresponding to the differentbeads (see Figure 1). The distributions are shown in thesupplementary material, Section S3.62 We fit bond parametersto achieve the best agreement between the MD distributionsof distances between the molecular fragments and the DPDdistributions of distances between the corresponding beads.Unlike previous simulations of linear chain molecules,56,65–67

the conformations of sPS are quite complex, and matching theconformations from atomistic simulations with the precisionobtained for chain molecules67 is not possible. However, theoverall rigidity of the aliphatic skeleton is reproduced well.

III. RESULTS AND DISCUSSION

A. DPD simulations of hydrated sPS

Using the forcefield described above, we perform severalDPD simulations of hydrated sPS at the sulfonation levelsof 10%, 20%, and 40%. From water sorption data inpure sPS72 and sPS–polyolefin block copolymers73 (watersorption in the hydrophobic block was neglected), weestimated the correlation between SL and the saturated HLas HL = 1.44×SL − 3.42. For a given sulfonation level, wemodeled several systems at different hydration, ranging from

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014902-6 Lee, Vishnyakov, and Neimark J. Chem. Phys. 144, 014902 (2016)

half of the saturated hydration to fully saturated sPS. TheDPD simulation is performed with NPT ensembles74 asimplemented in the DL_MESO DPD package,75 with thetarget pressure equal to the pressure of pure coarse-grainedwater (pDPD = 70 at a = 78.5 and ρ = 3). System informationis listed in Table II.

B. Segregation morphology

DPD simulations show that the hydrated sPS is segregatedonto the aqueous and organic subphases. Figure 3 presentsthe morphology of the hydrophilic subphase shown in blueat different sulfonation and hydration levels. Similarly tonafion and other PEM, water forms separate small clustersat low hydration, which make an interconnected networkas the hydration increases. To characterize the segregatedstructure, we employed a geometrical algorithm commonlyapplied to pore structure in solid porous adsorbents, becausethe hydrophilic phase essentially forms a network of pores inthe hydrophobic matrix.25 First, we created a lattice replica ofeach segregated sPS structure.25 The lattice was cubic with agrid of 0.5Rc. Each lattice site was assigned to either mobileaqueous or immobile polymer subphase based on the localcomposition of beads around that particular site. In order toassign a lattice site to either the mobile or immobile subphase,we calculate the site preference as follows:

p(r⃗l) =Ni=1

tiw(r̄i, r̄l),

where

w(r̄i, r̄l) = (1 − |r̄i − r̄l | /Rc)2 at |r̄i − r̄l | < Rc,

w = 0 at |r̄i − r̄l | ≥ Rc. (4)

Here, rl is the radius-vector to the center of lattice site l, ri isthe radius-vector to ith bead, and ti is a mobility coefficientrelated to the bead type: ti = −1 for all mobile W beadsand ti = 1 was assigned to all polymer beads (B, C, S).p(r⃗l) shows whether mobile or immobile beads prevail in theclose vicinity of site l. If p is negative, site l is assigned tomobile (hydrophilic) subphase; otherwise site l is assigned tothe immobile (hydrophobic) subphase. Note that S beads aretreated as a part of immobile subphase. Even though they arehydrophilic, they are rigidly attached to the aliphatic skeleton.The situation differs from that in nafion, where the sidechainsare much more flexible compared to the skeleton and may beconsidered as belonging to either subphase.25,27 The digitizedmorphologies are illustrated in the supplementary material,Section S4.62

The pore size distribution (PSD) for the hydrophilicpore networks was obtained by Poreblazer software version3.0.2.76,77 The distributions for the hydrophilic pore networkswere obtained with algorithms based on the Connellysurfaces78 and applied previously to both irregular79 andregular porous materials.76 Point X in the mobile subphaseis assumed to belong to a pore of size larger than thegiven size d, if there exists a sphere of diameter d thatincludes X but does not include any of the lattice sites

TABLE II. System information and simulation results. Abbreviations: SL, sulfonation level; HL, hydration level;λ, number of water molecule per sulfonate group; NsPS, NW, NP are number of sPS molecules, W beads, and Pbeads; time: simulation time in ns; box length: simulation box size in nm; DP, self-diffusion coefficient of the Pbead (H+); DW self-diffusion coefficient of water in estimated from W bead; αP, degree of dissociation of the Pbead from S bead; σ, proton conductivity calculated from Nernst equation based on DH+ and NP. A/V is theaccessible area per cubic volume. d is the average pore diameter.

A. System information B. Simulation details

SL (%) HL (%) λ NsPS NW NP Time (ns) Box (nm)

10 5 3 1829 4 182 3658 93.6 18.310 11 6 1739 7 962 3478 52.0 18.420 13 4 1631 9 236 6524 52.0 18.420 19 6 1544 13 064 6176 52.0 18.520 25 8 1465 16 540 5860 52.0 18.540 35 6 1234 21 768 9872 52.0 18.640 45 8 1147 25 944 9176 52.0 18.740 54 10 1077 29 304 8616 52.0 18.7

C. Dynamics properties D. Equilibrium properties

DP (cm2/s) DW (cm2/s) σ (S/cm) αP (%) A/V (nm−1) d (nm)

2.1 × 10−08 1.4 × 10−07 8.0 × 10−05 7.5 0.04 0.89.5 × 10−08 3.4 × 10−07 3.3 × 10−04 19.8 0.11 1.22.3 × 10−07 5.5 × 10−07 1.5 × 10−03 10.4 0.12 1.19.6 × 10−07 1.3 × 10−06 5.9 × 10−03 17.1 0.19 1.42.1 × 10−06 2.3 × 10−06 1.2 × 10−02 22.7 0.23 1.85.9 × 10−06 4.4 × 10−06 5.7 × 10−02 16.9 0.32 1.68.5 × 10−06 5.5 × 10−06 7.5 × 10−02 21.0 0.35 1.91.1 × 10−05 7.0 × 10−06 9.3 × 10−02 24.6 0.39 2.1

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FIG. 3. 3D perspective of hydrated sulfonated polystyrene (sPS) at different sulfonation level (SL) and hydration level (HL). Upper figures are visualized byindividual beads: grey: polymer beads, blue: water beads, yellow: sulfonate beads, and red: proton beads. Lower figures show the 3D structure of the aqueoussubphase (water cluster surfaces are drawn similarly to Refs. 58 and 59).

belonging to the immobile subphase. The total number ofsuch pores provides the integral PSD function, Vm(d), thatis the volume of the space that belongs to the pores ofdiameters larger than d; the respective differential PSD isobtained as the derivative, −dVm/dd. For selected systems,the differential PSDs are shown in Figure 4. Pore sizedistributions for all systems are in the supplementary material,Section S5.62

Figure 5 shows that the segregation scale in hydratedsPS is indeed small compared to nafion, where the equivalentweight is much higher.25 At low hydration levels, the poresizes are comparable with Rc, which means that the typicalwater aggregates only include several beads and may includeonly one. The total volume of the hydrophilic pores is,obviously, proportional to the hydration, and characteristicpore size increases with hydration. Even for 40% sulfonationlevel at saturation conditions, the characteristic pore sizeis limited to 2 nm, and the largest pores observed do notexceed 3 nm. This generally agrees with the MD simulationsof potassium substituted sPS.59 In all systems, the PSD isunimodal; a characteristic pore size can be identified for eachsystem. The distributions are slightly asymmetric with a “tail”

corresponding to larger pores that becomes more prominentas the total hydration increases. Figure 5 shows the pore

FIG. 4. Pore size distributions for sPS of different sulfonation level (SL) atsaturated hydration level (HL). Blue line: SL= 10% and HL= 11%; red line:SL= 20% and HL= 25%; green line: SL= 40% and HL= 55%.

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014902-8 Lee, Vishnyakov, and Neimark J. Chem. Phys. 144, 014902 (2016)

FIG. 5. Accessible surface area per cubic volume (nm−1) of sPS at differentsulfonation (SL) and hydration (HL) levels. Fitting curve and line are forvisualization only.

surface per unit volume area (calculation details describedin the supplementary material, Section S5).62 It is clear thatthe area depends mostly on hydration rather than on thesulfonation level: the results obtained in different systemsfall onto the same curve. Thus, the segregation morphologyfor all systems with the same hydration level is similar andbarely depends on the surface density of the sulfonate groupsat the interface between the hydrophilic and hydrophobicsubphases.

C. Water self-diffusion

We applied two different techniques for evaluation ofwater mobility. First, water self-diffusion can be directlycalculated from the mean square displacement of waterbeads in the course of DPD simulation using the Einsteinrelationship. Since the DPD model cannot reproduce thehydrodynamics of hydrated sPS, we calculate not the diffusioncoefficient per se, but its ratio to that in pure bulk coarse-grained water DW/Dbulk.80,81 The other technique is based onthe random walks simulation25,27 on the lattice replicas ofsegregated sPS structure as described in Section III B. Wemodeled water diffusion within the hydrophilic subphase asa simple random walk of a tracer particle within the mobiledomain of the lattice replicas. Each walk started from arandomly selected lattice site that belonged to the hydrophilicsubphase, and each step was an attempted move to one ofthe six sites that neighbored the current location. The movewas accepted if the attempted site belonged to the mobilesubphase. 104 random walks were performed on each of 400replicas for proper averaging.

The two methods of estimation for water mobility appearto be similar, yet they differ qualitatively. Each randomwalk is conducted in a static lattice replica. That is, anychange in the segregation structure (no matter how minor) isassumed to be infinitely slow compared to the water moleculediffusion. Such consideration excludes dynamic percolationeffects entirely:25,27 if two clusters are not connected, they

will never be connected. Evaluation of DW from DPD MSDsis on the other extreme: the segregation structure is allowedto evolve in the process of simulation, and the soft-core DPDpotentials employed here may significantly overestimate thefluidity of the hydrophobic subphase. The difference betweenthe two methods of water mobility evaluation is demonstratedby Figure 6, which shows the dependence of water self-diffusion on the sulfonation level at saturation conditions.As the sulfonation increases from 10% to 40%, saturationhydration grows from 10% (λ = 3) to 55% (λ = 10), andthe volume of the mobile phase increases, respectively. In40% sulfonates PS, both techniques predict DW/Dbulk ≈ 0.25.Interestingly, this result agrees extremely well with our earlieratomistic MD simulation59 (although the latter was carriedout for metal-substituted PEM that may affect diffusion).In 20% sulfonated PS, the random walk technique shows aslightly lower DW compared to the direct method, and in10% sulfonated PS, the two estimates differ by five ordersof magnitude. The evolution of the segregated structure isthe obvious reason for the difference, since in the course ofDPD simulation, the hydrophilic aggregates are allowed tomerge and split and thus water beads are exchanged and theiroverall mobility increases (these dynamic percolation effectswere initially observed in earlier MD simulations of hydratednafion,82 although insufficient simulation time in MD doesnot allow to make sure such effects are indeed significant83).Experimental data on water mobility in pure sPS are verysparse (most measurements have been performed on blockcopolymers). The gravimetric result from the study by Manojet al.72 is very close to the simulated DW/Dbulk obtaineddirectly from DPD simulations. This means that the dynamicpercolation effects are indeed likely to be very significant inthis system, as they are not accounted for in lattice randomwalk simulations, and the mobility of the W beads in DPDsimulations accurately reproduces the actual mobility of watermolecules in sPS. However, there is also a possibility that the

FIG. 6. Self-diffusion coefficient of water compared to the bulk water mo-bility at different hydration level (SL10% at HL11%, SL20% at HL25%,and SL40% at HL55%). Black line is calculated directly from the MSD ofthe water from the simulation. Red line is calculated by using random walkapproach in 400 static trajectories. The empty square shows the experimentaldata for 11% sPS (H+) at 13 wt. % water adsorption. Filled circle showsatomistic MD result for 40% sPS (Ca2+) at 54 wt. % water uptake.

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014902-9 Lee, Vishnyakov, and Neimark J. Chem. Phys. 144, 014902 (2016)

percolation transition in experiments corresponds to lowerhydration levels compared to simulations, but the absenceof continuous pore network is compensated by the dynamicevolution of the network in DPD.

D. Proton dissociation

The proton self-diffusion and membrane conductivity aredetermined not only by the segregation morphology but alsoby the dissociation of the acid groups, which is modeledexplicitly. The snapshots of the DPD configuration shownin Figure 3 display an expected picture: S beads, someprotonated and some not, are located at the interface betweenthe hydrophobic and hydrophilic subphases. To characterizethe dissociation, we use the same criterion, as in Ref. 61:a P bead is considered as dissociated if no S bead is foundwithin the Morse cutoff radius rM from the P bead. The degreeof dissociation α is the fraction of dissociated P beads. Thedependence of α on the sulfonation and hydration levels isshown in Figure 7.

Naturally, α increases with hydration and is mostlydetermined by λ, the number of water molecules per sulphonicgroup. In all systems, even for the highest λ considered, mostP beads are associated with at least one S bead. It is worthmentioning that the dissociation constant of benzenesulfonicacid is Ka = 0.2, unlike Ka = 1012 for triflic acid84 thatserves as a reference compound for nafion. Assuming thatthe activity coefficients γ of H+ and PhSO−3 ions in the bulkaqueous solution of benzenesulfonic acid are equal to one, onewould obtain α = 0.18 at λ = 10. The γ = H+ = γPhSO−3

= 1assumption is only valid in dilute solutions and is notapplicable in the systems of interest to this work. Yet, itis obvious that one can hardly expect full dissociation ofprotons in concentrated PhSO3H solutions, and therefore, thelow dissociation degree in hydrated sPS is not surprising. Atconstant λ, α decreases with the degree of sulfonation (thatis, with the surface density of the sulfonates at the interfacebetween the hydrophobic and hydrophilic subphases), whichis also expected.

FIG. 7. Degree of proton dissociation α as the function of number of waterper sulfonate group λ. Systems with the same λ are marked based on thesulfonation levels.

In the snapshots shown in Figure 3, one may noticethat in some cases, S and P beads form “clusters” oragglomerates consisting nearly entirely of charged beads withbarely any W beads inside. Morse and electrostatic attractionbetween P and S beads overpower the conservative short-range repulsion between S beads and the entropy that favorscharged bead solvation by W beads. It is also illustratedin the supplementary material, Section S7.62 This effectis probably artificial and shows the shortcomings of thesimplistic DPD potentials, which do not correctly reproducethe actual short-range interactions between the beads. Despitethis shortcoming, the general picture of ion solvation anddissociation in sulfonated sPS is quite reasonable and agreeswith experimental data.

E. Proton mobility

Since the proton is incorporated directly into DPDcalculations, we estimated the ratio of its self-diffusioncoefficient to that of water from the MSDs of the P beads. Inthe water bulk, DP/DW ≈ 3.8: the proton is about four times asmobile as a water molecule thanks to the Grotthuss “hopping”mechanism. In hydrated PEM, the hopping mechanisms arealso in effect: including proton transfer to and from sulfonateions. In the latter case, the proton has to dissociate fromthe sulfonate and then overcome the electrostatic attractionto the sulfonate anion. As a result, the difference betweenwater and protons self-diffusion in poorly hydrated PEM isnot as significant as in the bulk water. Proton may diffuse evenslower than water at low λ.61

Figure 8(a) shows the self-diffusion coefficient of waterand proton in hydrated sPS related to the diffusion coefficientof bulk water. It appears remarkable that DW and DP dataobtained for different sulfonation levels fall onto the samemaster curves and only depend on the hydration (the samefigure in terms of λ is shown in the supplementary material,Section S6).62 Water is indeed more mobile than the proton atlow hydration. Hydration contributes to the proton diffusionmore strongly than to the diffusion of water. At about20% water content, water and proton diffusion coefficientsbecome equal, and at higher hydration, proton diffuses fasterthan water: in sPS with 40% sulfonation DP/DW ≈ 2 at thesaturated hydration of 55%. This ratio is two times lower thanin bulk water. In experiments, a similar qualitative behaviorwas observed for PVDF-grafted-PSSA block copolymer.85

Protons diffuse slower than water at λ = 5, but become fasterthan water at around λ = 10. In sPS, this crossover happensat lower λ = 6-7, because there is no distinct segregatedmorphology such as PVDF-g-PSSA and water diffuses muchslower at low λ compared to that in PVDF-g-PSSA. Interestingenough, proton diffusion coefficient reaches 10−5 cm2/s forsimilar values of λ = 10 in both sPS and PVDF-g-PSSA. Innafion, where dissociation is more facile due to much higherKa, DP is generally higher than in PEMs with benzenesulfonicacid groups86 and DP/DW > 1 even at low λ, but qualitativelyDP(λ) is very similar. Thus, the proposed DPD modelreproduces the mechanism of proton mobility in hydratedPEM at least on a qualitative level.

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FIG. 8. (a) Diffusion coefficient of water and proton estimated from the MSD of W beads and P bead, at different sulfonation (SL) and hydration levels (HL). (b)Proton conductivity calculated from the diffusion coefficient and proton concentration. Black lines are derived from the experimental data at different sulfonationlevel. Conductivity of sPS (sulfonation level from 10% to 20% at fully hydrated state) at 22 ◦C and 60 ◦C is interpolated to obtain the conductivity at 30 ◦C.

To compare calculated mobilities with experimental data,we estimated the conductivities of hydrated sPS from protonself-diffusion coefficients using the Nernst equation,

σH+ =F2

RTDH+CH+, (5)

where F is the Faraday constant and CH+ is the overall protonconcentration. Figure 8(b) presents the proton conductivityin sPS fully saturated with water. The experimental data arederived from Ref. 87. The experimental conductivities areonly available for sulfonation levels between 10% and 20%;the water content is not reported in Ref. 87, but we calculatedit from sulfonation using the experimental sorption data. Thecalculated conductivities are of the same order of magnitudeas the experimental ones but show more gradual dependenceon the hydration level. In particular, we underestimate theconductivity at high hydration. The possible reason are theartificial S–H “clusters” noted above, since the protons thatbelong to such cluster are less mobile in simulations comparedto the experiment.

IV. CONCLUSIONS

We applied the DPD proton transfer model developedearlier61 for the systems with protonation equilibria to model-ing self-assembly and transport in sulfonated polystyrene,which is of special practical interest for the hydrophiliccomponent in several prospective proton-exchange block-copolymer membranes for fuel cells. The proposed approachoffers several important advances: (1) the smeared chargemodel allows for explicit consideration of electrostatic inter-actions, (2) the dissociating Morse bonds between the proton(which is introduced as a separate bead) and the base beads(water and the anion) establish the dissociation equilibriumbetween protonated and deprotonated forms of the acid, and(3) the proton transfer model artificially mimics Grotthussmechanism of proton diffusion. The Morse potentials areparameterized from the experimental properties independentof the system under consideration: parameterization is based

on the known proton mobility in pure water and the benzene-sulfonic acid dissociation constant and does not involve anydata on the hydrated polymer. Additionally, the short-rangeconservative repulsion parameters are parameterized from theinfinite dilution activity coefficients of reference compounds,and the bond potentials are parameterized from atomistic MDsimulations of sPS fragments.

DPD simulation provides a physical insight into thehydration-induced nanoscale segregation of sPS into irregularhydrophilic and hydrophobic subphases. The segregationmorphology, its evolution upon hydration, and characteristicscales are correctly reproduced. Sulfonate groups are locatedat the interface between the hydrophilic and hydrophobicsubphases. Acid is partially dissociated; the dissociationdegree increases approximately linearly with the hydrationlevel and depends on the sulfonation level. With the increaseof hydration, the connectivity of the hydrophilic subphaseincreases with the clear percolation transition from the systemof water clusters to the 3D network of transport channels.The hydrophilic subphase can be considered as a networkof pores in the hydrophobic matrix. For the first time, wehave performed a quantitative analysis of the area of thehydrophilic-hydrophobic interface and the size distribution oftransport pores, which provides additional information on thestructure of self-assembled polyelectrolytes.

The most interesting results are found by modelingwater and proton transport in nano-segregated sPS. Waterself-diffusion was estimated directly from the mean squaredisplacement of water beads in the process of DPDsimulation and showed a remarkable agreement withexperimental results (at low hydration) and atomistic MDresults (at high hydration). We found a confirmation forthe dynamic percolation effects predicted earlier in atomisticMC simulation of Nafion.25 We aslo performed the randomwalk simulation on the digitized replicas of segregatedstructure snapshots, which showed much slower diffusionat low hydration levels, but the same diffusion at higherhydration levels. We attribute this difference to the dynamicpercolation effects:82 water aggregates merge and break up

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014902-11 Lee, Vishnyakov, and Neimark J. Chem. Phys. 144, 014902 (2016)

due to thermal fluctuations mimicked in the course of DPDsimulation, and this dynamic effect is ignored in static digitizedreplicas of structure snapshots. Proton mobilities obtainedfrom DPD are in reasonable agreement with experimentaldata: they are lower than the mobility of water molecule atlow hydration levels and higher at high hydration levels. Theconductivities estimated from the self-diffusion coefficientsusing the Nernst equation are also in qualitative agreementwith the experiments.

Overall, the modeling approach presented here ispromising for mesoscale simulations of proton-exchangemembranes. A lack of reliable techniques for parameterizationof the short-range interactions between the charged speciesas well as of charge smearing parameters remains a seriousproblem; we suspect that the unphysical clustering betweensulfonate and proton beads found in some systems originatesfrom underestimation of the short-range repulsion. Alterna-tively, the activity coefficient and/or the radial distributionfunctions in reference electrolyte solutions can be usedas target properties for parameterization of short-rangeinteractions and charge distributions.

ACKNOWLEDGMENTS

We thank Dr. Lev Sarkisov for recommendations oncalculations of pore size distributions and surface areas. Thiswork is supported by NSF Grant No. DMR-1207239 andDTRA Grant No. HDTRA1-14-1-0015. The authors declareno competing financial interest.

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